APPLIED AERODYNAMICS
APPLIED AERODYNAMICS
By L. Baifstow, F.R.S., C.B.E., Associate of the
Royal College of Science in Mechanics ; Whitworth
Scholar ; FeHow and Member of Council of the Royal
Aeronautical Society, etc.
IViiA Illustrations and Diagrams. %vo.
AEROPLANE STRUCTURES
By A. J. Sutton Pippard, M.B.E., M.Sc, Assoc.M.
Inst.C.E., Fellow of the Royal Aeronautical Society,
and Capt. J. Laurence PRiTCHAkD, late R.A.F.,
Associate Fellow of the Royal Aeronautical Society.
With an Introduction by L. Bairstow, F.R.S.
With Illustrations attd Diagrams. Svo.
THE AERO ENGINE
By Major A. T. Evans and Captain W. Grylls
Adams, M.A.
fVttk Illustrations and Diagrams. Svo.
THE DESIGN OF SCREW
PROPELLERS
With Special Reference to their Adaptation for
Aircraft. By Henry C. Watts, M.B.E., B.Sc,
F.R.Ae.S., late Air Ministry, London.
Wit/i Diagrams. SVo.
LONGMANS, GREEN AND CO.
LONDON, NEW YORK, BOMBAY, CALCUTTA, AND MADRAS
APPLIED AERODYNAMICS
\
BY
LEONARD BAIRSTOW, F.R.S., C.B.E.,
EXPERT ADVISBR ON AERODYNAMICS TO THE AIR MINISTRY : MEMltER OF THE ADVISORY COMMITTEE
FOR AERONAUTICS, AIR INVENTIONS COMMITTEE, ACCIDENTS INVESTIGATION COMMITTER,
AND ADVISORY COMMITTEE ON CIVIL AVIATION; LATE SUPERINTENDENT OF THE
AERODYNAMICS DEPARTMENT OF THE NATIONAL PHYSICAL LABORATORY
WITH ILLUSTRATI0N5i AND DIAGRAMS
<;^
4
LONGMANS, GREEN AND CO
39 PATERNOSTER ROW, LONDON
FOURTH AVENUE & 30th STREET, NEW YORK
BOMBAY, CALCUTTA, AND MADRAS
1920
AH rights reserved
61 0
PREFACE
^HE work aims at the extraction of principles of flight from, and the
Uustration' of the use of, detailed information on aeronautics now
livailable from many sources, notably the publications of the Advisory
"iJommittee for Aeronautics. The main outlines of the theory of flight
are simple, but the stage of application now reached necessitates careful
examination of secondary features. This book is cast with this distinction
in view and starts with a description of the various classes of aircraft,
both heavier and lighter than air, and then proceeds to develop the
laws of steady flight on elementary principles. Later chapters complete
the detail as known at the present time and cover predictions and
analyses of performance, aeroplane acrobatics, and the general problems
of control and stability. The subject of aerodynamics is almost wholly
based on experiment, and methods are described of obtaining basic
information from tests on aircraft in flight or from tests in a wind
channel on models of aircraft and aircraft parts.
The author is anxious to acknowledge his particular indebtedness to
the Advisory Committee for Aeronautics for permission to make use of
reports issued under its authority. Extensive reference is made to those
reports which, prior to the war, were Jssued annually ; it is understood
that all reports approved for issue before the beginning of 1919 are now
ready for publication. To this material the author has had access, but
it will be understood by all intimately acquainted with the reports that
the contents cannot be fully represented by extracts. The present
volume is not an attempt at collection of the results of research, but a
contribution to their application to industry.
For the last year of the war the author was responsible to the
Department of Aircraft Production for the conduct of aerodynamic
research on aeroplanes in flight, and his thanks are due for permission
to make use of information acquired. For permission to reproduce
photographs acknowledgment is made to the Admiralty Airship Depart-
ment, Messrs. Handley Page and Co., the British and Colonial Aeroplane
Co., the Phcenix Dynamo Co., Messrs. D. Napier and Co., and H.M.
Stationery Office.
L. BAIRSTOW.
Hampton Wick,
October 6th, 1919.
CONTENTS
CHAPTER I
GENERAL DE8CEIPTI0N OF STANDARD FORMS OF AIRCRAFT
PAGE
Introduction — Particular aircraft — The largest aeroplane— Biplane— Monoplane-
Plying boat— Pilot's cockpit — Air-cooled rotary engine— Vee-type air-cooled
engine— Water-cooled engine — Rigid airship— Non-rigid airship— Kite balloons . 1
CHAPTER II
THE PRINCIPLES OF PLIGHT -^
(I) TJie aeroplane. Wings and wing lift — Resistance or drag — Wing drag — Body drag <
— Propulsion, airscrew and engine — Climbing — Diving — Gliding — Soaring — Extra
weight — Flight at altitudes — Variation of engine power with height— Longitudinal /
balance — Centre of pressure — Down wash — Tail-plane size — Elevators— EfEor^/
necessary to move elevators — Water forces on flying boat hull 18
{II) Lighter-than-air craft. Lift on small gas container — Convective equilibrium —
Pressure, density and temperature for atmosphere in convective equilibrium —
Lift on large gas container— Pitching moment due to inclination — Aerodynamic
forces, drag and power — ^Longitudinal balance — Equilibrium of kite balloons —
Three fins — Position relative to lower end of kite wire — Insufficient fin area . . 58
CHAPTER III
GENERAL DESCRIPTION OP METHODS OF MEASUREMENT IN AERODYNAMICS .
AND THE PRINCIPLES UNDERLYING THE USE OP INSTRUMENTS AND
SPECIAL APPARATUS
Measurement of air speed — Initial determination of constant of Pitot — Static tube —
Effect of inclination of tube anemometer — Use on aeroplane — Aeroplane pressure
gauge or airspeed indicator — Aneroid barometer — Revolution indicators and
counters — Accelerometer — Levels— Aerodynamic turn indicator — Gravity con-
trolled airspeed indicator — Photomanometer — Cinema camera — Camera for re-
cording aeroplane oscillations — Special experimental modifications of aeroplane —
Laboratory apparatus — Wind channel — Aerodynamic balance — Standard balance
for three forces and one couple for body having plane of symmetry — Example of
use on aerofoil-lift and drag — Centre of pressure — Use on model kite balloon —
Drag of airship envelope — Drag, lift and pitching moment of complete model
aeroplane — Stability coefficients — Airscrews and aeroplane bodies behind airscrews
— Measurement of wind speed and local pressure — Water resistance of flying boat
hull — Forces duo to accelerated fluid motion — Model test for tautness of airship
envelope 73
viii CONTENTS
CHAPTER IV
DESIGN DATA FROM THE AERODYNAMICS LABORATORIES
PAdR
(I) Straight flying. Wing forms — Geometry of wings — Definitions — Aerodynamics of
wings, definitions — Lift coefficient and angle of incidence — Drag coefficient and
angle of incidence— Centre of pressure coefficient and angle of incidence — Moment
coefficient and angle of incidence — Lift/drag and angle of incidence — Lift/drag
and lift coefficient — Drag coefficient and lift coefficient — Effect of change of wing
section — Wing characteristics for angles outside ordinary flying range — Wing
characteristics as dependent on upper surface camber — Effect of changes of lower
surface camber of aerofoil — Changes of section arising from sag of fabric — Aspect
ratio, lift and drag — Changes of wing form which have little effect on aerodynamic
properties — Effect of speed on lift and drag of aerofoil— Comparison between
monoplane, biplane and triplane — Change of biplane gap — Change of biplane
stagger — Change of angle between chords — Wing flaps as means of varying wing
section — Criterion for aerodynamic advantages of variable camber wing— Changes
of triplane gap — Changes of triplane stagger — Partition of forces between planes
of a combination, biplane, triplane — Pressure distribution on wings of biplane —
Lift and drag from pressure observations — Comparison of forces estimated from
pressure distribution with those measured directly — Resistance of struts — stream-
line wires — Smooth circular wires and cables — Body resistance — Body resistance
as affected by airscrew — Resistance of undercarriage and wheels — Radiators and
engine-cooling losses — Resistance of complete aeroplane model and analysis into
parts — Relation between model and full scale — Downwash behind wings —
Elevators and effect of varying position of hinge — Airship envelopes— Complete
model non-rigid airship^Drag, lift and pitching moment on rigid airship —
Pressure distribution round airship envelope 116
(II) Body axes and non-rectilinear flight. Standard axes — Angles relative to wind —
Forces along axes — Moments about axes— Angular velocities about axes —
Equivalent methods of representing given set of observations — Body axes applied
to wing section — Longitudinal force, lateral force, pitching moment and yawing
moment on model flying boat hull — Forces and moments due to yaw of aeroplane
body fitted with fin and rudder — Effect of presence of body and tail-plane and of
shape of fin and rudder on effectiveness of latter — Airship rudders — Ailerons and
wing flaps — Balancing of wing flaps — Forces and moments on complete model
aeroplane — Forces and moments due to dihedral angle — Change of axes and
resolution of forces and moments — Change of direction without change of origin
— Change of origin without change of direction — Formulae for special use with
the equations of motion 214
CHAPTER V
AERTAIi MANOEUVRES AND THE EQUATION OF MOTION
Looping— Speed and loading records in loop — Spinning — Speed and loading records in
a spin — Roll — Equations cf motion — Choice of co-ordinate axes — Calculation of
looping of aeroplane — Failure to complete loop — Steady motions including turn-
ing and spiral glide — Turning in horizontal circle without side slipping — Spiral
descent — Approximate method of deducing aerodynamic forces and couples on
aeroplane during complex manoeuvres — Experiment which can be compared with
calculation — More accurate development of mathematics of aerofoil element
theory — Forces and moments related to standard axes — Autorotation — Effect of
dihedral angle during side slipping — Calculation of rotary derivatives .... 242
CONTENTS ix
CHAPTER VI
AIRSCREWS
PAOB
General theory — Measurements of velocity and direction of air flow near airscrew —
Mathematical theory— Application to blade element — Integration to obtain thrust
and torque for airscrew — Example of detailed calculation of thrust and torque —
Effect of variation of pitch — diameter ratio — Tandem airscrews — Botational
velocity in slip stream — Approximate formulae relating to airscrew design —
Forces on airscrew moving non-axially — Calculation of forces on element —
Integration to whole airscrew — Experimental determination of lateral force on
inclined airscrew — Stresses in airscrew blades — Bending moments due to air
forces — Centrifugal stresses — Bending moments due to eccentricity of blade
sections and centrifugal force — Formulae for airscrews suggested by considera-
tions of dynamical similarity 281
CHAPTER VII
FLUID MOTION
Experimental illustrations of fluid motion — Remarks on mathematical theories of
aerodynamics and hydrodynamics — Steady motion — Unsteady motion — Stream
lines — Paths of particles — Filapent lines — Wing forms — Elementary mathe-
matical theory of fluid motion — Frictionless incompressible fluid — Stream
function — Flow of inviscid fluid round cylinder — ^Equations of motion of
inviscid fluid— Forces in direction of motion — Forces normal to direction of
motion — Comparison of pressures in source and sink system with those on model
in air — Cyclic motion of inviscid fluid — Discontinuous fluid motion — Motion in
viscous fluids — Definition of viscosity — Experimental determination of the
coefficient of viscosity ^ 343
CHAPTER VIII
DYNAMICAL SIMILARITY AND SCALE EFFECTS
Geometrical similarity — Similar motions — Laws of corresponding speeds — Principle
of dimensions applied to similar motions — Compressibility — Gravitational attrac-
tion— Combined effects of viscosity, compressibility and gravity — Aeronautical
applications of dynamical similarity — Aeroplane wings — Variation of maximum
lift coefficient in model range of vl — Resistance of struts — Wheels — Aeroplane
glider as a whole — Airscrews 372
CHAPTER IX
THE PREDICTION AND ANALYSIS OF AEROPLANE PERFORMANCE
Performance — Tables for standard atmosphere — Rapid prediction -Maximum speed
- Maximum rate of climb - Ceiling — Structure weight - Engine weight— Weight
of petrol and oil — More accurate method of prediction— General theory— Data
required - Airscrew revolutions and flight speed -Level flights — Maximum rate
of climb — Theory of reduction from actual to standard atmosphere —Level flights
— Climbing— Engine power — Aneroid height— Maximum rate of climb— Aero-
CONTENTS
PAOE
dynamic merit^Ghange of engine without change of airscrew — Change of weight
carried — Separation of aeroplane and airscrew efficiencies — Determination of
airscrew pitch — Variation of engine power with height — Determination of aero-
plane drag and thrust coefficient — Evidence as to twisting of airscrew blades
in use 395
CHAPTER X
THE STABILITY OP THE MOTIONS OP AIRCRAFT
(I) Criteria for stability. Definition of stability —Record of oscillation of stable
aeroplane — Records taken on unstable aeroplane — Model showing complete
stability — Distinguishing features on which stability depends — Degree of stability
— Centre of pressure changes equivalent to longitudinal dihedral angle — Lateral
stability of flying models — Instabilities of flying models, longitudinal and lateral
— Mathematical theory of longitudinal stability — Equations of disturbed longi-
tudinal motion — Longitudinal resistance derivatives — Effect of flight speed on
longitudinal stability — Variation of longitudinal stability with height and loading
— Approxirqate formulae for longitudinal stability — Mathematical theory of
lateral stability — Equations of disturbed lateral motion — Lateral resistance
derivatives— Effect of flight speed on lateral stability — Variation of lateral
stability with height and loading — Stability in circling flight — Equations of
disturbed circling motion — Criterion for stability of circling flight— Examples of
general theory — Gyroscopic couples and their effect 'on straight flying — Stability
of airships and kite ballooiis — Theory of stability of rectilinear motion — Remarks
on resistance derivatives for lighter-than-air craft — Critical velocities — Approxi-
mate criterion for longitudinal stability of airship — Approximate criterion for
lateral stability of airship — Effect of kite wire or mooring cable 447
(II) The details of the disturbed motion of an aeroplane. Longitudinal disturbances —
Formulae for calculation of details of disturbances — Effect of gusts — Effect of
movement of elevator or engine throttle — Lateral disturbances — Formulae for
calculation of details of lateral disturbances — Effect of gusts — Effect of move-
ment of rudder or ailerons — Continuous succession of gusts — Uncontrolled flight
in natural wind — Continuous use of elevator — Elimination of vertical velocity —
Controlled flight in natural wind — Analysis of effect on flight speed of elimination
of vertical velocity 517
APPENDIX
The solution of algebraic equations with numerical coefficients in the case where
several pairs of complex roots exist 551
INDEX 561
LIST OF PLATES
PACING PAGE
Fourteen tons of matter in flight . . 1
Fighting Biplane Scout iO
High-speed Monoplane . . 11
Large Flying Boat 12
Cockpit of an Aeroplane 13
Rotary Engine — Air-cooled Stationary Engine . . 14
Water-cooled Engine 14
Nearly completed Rigid Airship ......... 15
Rigid Airship ... . ...'.... 15
Kite Balloons . 17
Non-rigid Airship 16
Experinaental arrangement of Tube Anemometer on an Aeroplane 81
Wind Channel 95
Model Aeroplane arranged to show Autorotation .^ 266
Viscous Plow round Disc and Strut 344
Eddies behind Cylinder (N.P.L.) 345
Eddying Motion behind Struts 349
Viscous Flow round Flat Plate and Wing Section 350
Plow of Water past an Inclined Plate. Low and High Speeds 378
Flow of Air past an Inclined Plate. Low and High Speeds' 378
Very Stable Model — Slightly Stable Model 452
Stable Model with two Real Fins — Model which develops an Unstable Phugoid
Oscillation— Model which illustrates Lateral Instabilities 456
FOURTEEN TONS OF MATTER IN FLIGHT
CHAPTER I
GENERAL DESCRIPTION OF STANDARD FORMS OF AIRCRAFT
Introduction
In the opening references to aircraft as represented by photographs of
modern types, both heavier-than-air and Hghter-than-air, attention will
be more especially directed to those points which specifically relate to the
subject-matter of this book, i.e. to applied aerodynamics. Strictly in-
terpreted, the word " aerodynamics " is used only for the study of the forces
on bodies due to their motion through the air, but for many reasons it is
not convenient to adhere too closely to this definition. In the case of
heavier-than-air craft one of the aerodynamic forces is required to counter-
balance the weight of the aircraft, and is therefore directly related to a
non- dynamic force. In lighter- than- air craft, size depends directly on
the weight to be carried, but the weight itself is balanced by the buoyancy
of a mass of entrapped hydrogen which again has no dynamic origin. As
the size of aircraft increases, the resistance to motion at any predetermined
speed increases, and the aerodynamic forces for lighter-than-air craft
depend upon and are conditioned by non-dynamic forces.
The inter-relation indicated above between aerodynamic and static
forces has extensions which affect the external form taken by aircraft.
One of the most important items in aircraft design is the economical
distribution of material so as to produce a sufficient margin of strength
for the least weight of material. Accepting the statement that additional
resistance is a consequence of increased weight, it will be appreciated that
the problem of external form cannot be determined solely from aerodynamic
considerations. As an example of a simple type of compromise may be
instanced the problem of wing form. The greatest lift for a given resistance
is obtained by the use of single long and narrow planes, the advantage being
less and less marked as the ratio of length to breadth increases, but remaining
appreciable when the ratio is ten. Most aeroplanes have this " aspect
ratio " more nearly equal to six then ten, and instead of the single plane
a double arrangement is preferred, the effect of the doubhng being an
appreciable loss of aerodynamic efficiency. The reasons which have led
to this result are partly accounted for by a special convenience in fighting
which accompanies the use of short planes, but a factor of greater im-
portance is that arising from the strength desiderata. The weight of
wings of large aspect ratio is greater for a given lifting capacity than that
of short wings, and the external support necessary in all types of aeroplane
is more difficult to achieve with aerodynamic economy for a single than
for a double plane. Aerodynamically, a limit is fixed to the weight
1 B
2 APPLIED AEEODYNAMICS
carried by a wing at a chosen speed, and for safe alighting the tendency
has been to fix this speed at a httle over forty miles an hour. This gives a
lower limit to the wing area of an aeroplane which has to carry a specified
weight. The general experience of designers has been that this limit is
a serious restriction in the design of a monoplane, but offers very little
difl&culty in a biplane. In a few cases, three planes have been superposed,
but the type has not received any general degree of acceptance. For
small aeroplanes, the further loss of aerodynamic efficiency in a triplane
has been accepted for the sake of the greater rapidity of manoeuvre which
can be made to accompany reduced span and chord, whilst in very large
aeroplanes the chief advantage of the triplane is a reduction of the overall
dimensions. Up to the present time it appears that an advantage remains
with the biplane type of construction, although very good monoplanes and
triplanes have been built.
The illustration shows that aircraft have entered the stage of " engineer-
ing " as distinct from " aerodynamical science " in that the final product
is determined by a number of considerations which are mutually restrictive
and in which the practical knowledge of usage is a very important factor
in the attainment of the best result.
Although air is the fluid indicated by the term " aerodynamics," it
has been found that many of the phenomena of fluid motion are independent
of the particular fluid moved. Advantage has been taken of this fact in
arranging experimental work, and in a later chapter a striking optical
illustration of the truth of the above observation is given. The distinction
between aerodynamics and the dynamics of fluid motion tends to disappear
in any comprehensive treatment of the subject.
In the consideration of aerial manoeuvres and stability the aero-
dynamics of the motion must be related to the dynamics of the moving
masses. It is usual to assume that aircraft are rigid bodies for the purposes
indicated, and in general the assumption is justifiable. In a few cases, as
in certain fins of airships which deflect under load, greater refinement may
be necessary as the science of aeronautics develops.
It will readily be understood that aerodynamics in its strict inter-
pretation has little direct connection with the internal construction of
aircraft, the important items being the external form ahd the changes of
it which give the pilot control over the motion. As the subject is in itself
extensive, and as the internal structure is being dealt with by other writers,
the present book aims only at supplying the information by means of
which the forces on aircraft in motion may be calculated.
The science of aerodynamics is still very young, and it is thirteen years
only since the first long hop on an aeroplane was made in public by Santos
Dumont. The circuit of the Eiffel Tower in a dirigible balloon preceded
this feat hj only a short period of time. Aeronautics attracted the
attention of numerous thinkers during past centuries, and many historical
accounts are extant dealing with the results of their labours. For many
reasons early attempts at flight all fell short of practical success, although
they advanced the theory of the subject in various degrees. The present
epoch of aviation may be said to have begun with the publication of the
STANDAED FOEMS OF AIECEAFT 3
experiments made by Langley in America in the period 1890 to 1900.
The apparatus used was a whirling arm fitted with various contrivances
for the measurement of the forces on flat plates moved through the air at
the end of the arm.
One line of experiment may perhaps be described briefly. A number
of plates of equal area were made and arranged to have the same total
weight, after which they were constrained to remain horizontal and to
fall down vertical guides at the end of the whirhng arm. The time of fall
of the plates through a given distance was measured and found to depend,
not only on the speed of the plate through the air, but also on its shape.
At the same speed it was found that the plates with the greatest dimension
across the wind fell more slowly than those of smaller aspect ratio. For
small velocities of fall the time of fall increased markedly with the speed
of the plate through the air. By a change of experiment in which the
plates were held on the whirling arm at an inclination to the horizontal
and by running the arm at increasing speeds the value of the latter when
the plate just lifted itself was found. Eepetition of this experiment
showed that a particular inclination gave less resistance than any other
for the condition that the plate should just be airborne.
From Langley's experiments it was deduced that a plate weighing two
pounds per square foot could be supported at 35 m.p.h. if the inchnation
was made eight degrees. The resistance was then one-sixth of the weight,
and making allowance for other parts of an aeroplane it was concluded
that a total weight of 750 lbs. could be carried for the expenditure of 25
horsepower. Early experimenters set themselves the task of building a
complete structure within these limitations, and succeeded in producing
aircraft which hfted themselves.
Langley put his experimental results to the test of a flight from the
top of a houseboat on the Potomac river. Owing to accident the aero-
plane dived into the river and brought the experiment to a very early end.
In England, Maxim attempted the design of a large aeroplane and
engine, and achieved a notable result when he built an engine, exclusive
of boilers and water, which weighed 180 lbs. and developed 360 horse-
power. To avoid the difficulties of deahng with stabihty in flight, the
aeroplane was made captive by fixing wheels between upper and lower
rails. The experiments carried out were very few in number, but a lift
of 10,000 lbs. was obtained before one of the wheels carried away after
contact with the upper rail.
For some ten years after these experiments, aviation took a new
direction, and attempts to gain knowledge of control by the use of aero-
plane ghders were made by Pilcher, Lihenthal and Chanute. From a hill
built for the purpose Lihenthal made numerous glides before being caught
in a powerful gust which he was unable to negotiate and which cost him
his hf e. In the course of his experiments he discovered the great superiority
of a curved wing over the planes on which Langley conducted his tests.
By a suitable choice of curved wing it is possible to reduce the resistance
to less than half the value estimated for flat plates of the same carrying
capacity. The only control attempted in these early gliding experiments
4 APPLIED AEEODYNAMIOS
was that which could be produced by moving the body of the aeronaut
in a direction to counteract the effects of the wind forces.
In the same period very rapid progress was made in the development
of the light petrol motor for automobile road transport, and between liiOC
and 1908 it became clear that the prospects of mechanical flight had
materially improved. The first achievements of power-driven aeroplanes
to call for general attention throughout the world were those of tw(f
Frenchmen, Henri Farman and Bleriot, who made numerous short flights
which were limited by lack of adequate control. These two pioneers took
opposite views as to the possibilities of the biplane and monoplane, but
in the end the first produced an aeroplane which became very popular
as a training aeroplane for new pilots, whilst the second had the honour
of the first crossing of the English Channel from France to Dover.
The lack of control referred to, existed chiefly in the lateral balance of
the aeroplanes, it being difficult to keep the wings horizontal by means
of the rudder alone. The revolutionary step came from the Brothers Wright
in America as the result of a patient study of the problems of gliding. A
lateral control was developed which depended on the twisting or warping
of the aeroplane wings so that the lift on the depressed wing could be
increased in order to raise it, with a corresponding decrease of lift
on the other wing. As the changes of lift due to warping were accompanied
by changes of drag which tended to turn the aeroplane, the Brothers
Wright connected the warp and rudder controls so as to keep the aeroplane
on a straight course during the warping. The principle of increasing the
lift on the lower wing by a special control is now universally apphed, but
the rudder is not connected to the wing flap control which has taken the
place of wing warping. From the time of the Wrights' first public flights
in Europe in 1908 the aviators of the world began to increase the duration
of their flights from minutes to hours. Progress became very rapid, and
the speed of flight has risen from the 35 m.p.h. of the Henri Farman to
nearly 140 m.p.h. in a modem fighting scout. The range has been
increased to over 2000 miles in the bombing class of aeroplane, and the
Atlantic Ocean has recently been crossed from Newfoundland to Ireland by
the Vickers' *i Vimy " bomber.
As soon as the problems of sustaining the weight of an aeroplane and
of controlUng the motion through the air had been solved, many investiga-
tions were attempted of stabihty so as to elucidate the requirements in
an aeroplane which would render it able to control itself. Partial attempts
were made in France for the aeroplane by Ferber, See and others, but the
most satisfactory treatment is due to Bryan. Starting in 1903 in collabora-
tion with WilUams, Bryan apphed the standard mathematical equations
of motion of a rigid body to the disturbed motions of an aeroplane, and the
culmination of this work appeared in 1911. The mathematical theory
remains fundamentally in the form proposed by Bryan, but changes have
been made in the method of application as the result of the development
of experimental research under the Advisory Committee for Aeronautics.
The mathematical theory is founded on a set of numbers obtained from
experiment, and it is chiefly in the determination of these numbers that
STANDAED FOEMS OF AIECEAFT 5
development has taken place in recent years. Some extensions of the
mathematical theory have been made to cover flight in a natural wind
and in spiral paths.
Experimental work on stability on the model scale at the National
Physical Laboratory was co-ordinated with flying experiments at the
Royal Aircraft Factory, and the results of the mathematical theory of
stability were apphed by Busk in the production of the B.E. 2c. aeroplane,
which, with control on the rudder only, was flown for distances of 60 or 70
miles on several occasions. By this time, 1914, the main foundations of
aviation as we now know it had been laid. The later history is largely
that of detailed development under stress of the Great War.
The history of airships has followed a different course. The problem
of support never arose in the same way as for aeroplanes and seaplanes,
as balloons had been known for many years before the advent of the air-
ship. The first change from the free balloon was little more than the
attachment of an engine in order to give it independent motion through
the air, and the power available was very small. The spherical balloon
has a high resistance, its course is not easily directed, and the dirigible
balloon became elongated at its earhest stages. The long cigar-shaped
forms adopted brought their own special difficulties, as they too are difficult
to steer and are inclined to buckle and collapse unless sufficient precautions
are taken. Steering and management has been attained in all cases by
the fitting of fins, both horizontal and vertical, to the rear of the airship
envelope, and the problem of affixing fins of sufficient area to the flexible
envelope of an airship has imposed engineering limitations which prevent
a simple application of aerodynamic knowledge.
The problem of maintenance of form of an airship envelope has led to
several solutions of very different natures. In the non-rigid airship the
envelope is kept inflated by the provision of sufficient internal pressure,
either by automatic valves which hmit the maximum pressure or by the
pilot who hmits the minimum. The interior of the envelope is divided
by gastight fabric into two or three compartments, the largest of which
is filled with hydrogen, and the smaller ones are fully or partially inflated
with air either from the slip stream of an airscrew or by a special
fan. As the airship ascends into air at lower pressure the valves to the
air chambers open and allow air to escape as the hydrogen expands, and
so long as this is possible loss of lift is avoided. The greatest height to
which a non-rigid airship can go without loss of hydrogen is that for which
the air chambers or balloonets are empty, and hence the size of the
ballodnets is proportioned by the ceiHng of the airship.
If the car of an airship is suspended near its centre, the envelope at
rest has gas forces acting on it which tend to raise the tail and head. The
underside of the envelope is then in tension on account of the gas Uft,
whilst the upper side is in compression. As fabric cannot withstand
compression, sufficient internal pressure is applied to counteract the effect
of the lift in producing compression.
The car of the non-rigid airship is attached by cables to the underside
of the envelope, and as these are inchned, an inward pull is exerted which
6 APPLIED AEEODYNAMICS
tends to neutralise the tension in the fabric. For some particular internal
pressure the fabric will tend to pucker, and special experiments are made
to determine this pressure and to distribute the pull in the cables so as
to make the pressure as small as possible before puckering occurs. The
experiment is made on a model airship which is inverted and filled with
water. The loads in the cables, their positions and the pressure are all
under control, and the necessary measurements are easily made. The
theory of the experiment is dealt with in a later chapter.
In flight the exterior of the envelope is subjected to aerodynamic
pressures which are intense near the nose, but which fall off very
rapidly at points behind the nose. From a tendency of the nose to blow
in under positive pressure, a change occurs to a tendency to suck out at
a distance of less than half the diameter of the airship behind the nose,
and this suction, in varying degrees, persists over the greater part of the
envelope. At high speeds the tendency of the nose to blow in is very
great as compared with the internal pressure necessary to retain the form
of the rest of the envelope, and a reduction in the weight of fabric used is
obtained if the nose is reinforced locally instead of maintaining its shape
by internal pressure alone. In one of the photographs of this chapter the
reinforcement of the nose is very clearly shown.
The problem of the maintenance of form of a non-rigid airship is
appreciably simplified if the weight to be carried is not all concentrated
in one car.
In the semi-rigid airship the envelope is still of fabric maintained to
form by internal pressure, but between the envelope and car is interposed
a long girder which distributes the concentrated load of the car over the
whole surface of the envelope. This type of airship has been used in
France, but has received most development in Italy ; it- is not used in this
country.
Eigid airships depend upon a metal framework for the maintenance
of their form, and in Germany were developed to a very high degree of
efficiency by Count Zeppelin. The largest airships are of rigid construction
and have a gross lift of nearly seventy tons. The framework is usually
of a light aluminium alloy, occasionally of wood, and in the future steel may
possibly be used. The structure is a light latticework system of girders
running along and around the envelope and braced by wires into a stiff
frame. In modern types a keel girder is provided inside the envelope at
the bottom, which serves to distribute the load from the cars and also
furnishes a communication way. The number of cars may be four or more,
and the bending under the lift of the hydrogen is kept small by a careful
choice of their positions. Some of the transverse girders are braced inside
the envelope by a number of radial wires, the centres of which are joined
by a wire running the whole length of the airship along its axis. In the
compartments so produced the gas-containers are floated, and the lift is
transferred to the rigid frame by the pressure on a'netting of small cord.
The latticework is covered by fabric in order to produce a smooth
unbroken surface and so keep down the resistance. Speeds of 76 m.p.h.
have been reached in the latest British types of rigid airship, and the return
STANDAED FORMS OF AIRCRAFT 7
journey of many thousands of miles across the Atlantic has been made
by the R 34 airship.
The duties for which the aeroplane and seaplane, non-rigid and rigid
airships are suitable probably differ very widely. The heavier-than-air
craft have a distinct superiority in speed and an equally distinct inferiority
in range. The heavier-than-air craft must have an appreciable speed at
first contact with the ground or sea, whilst airships are very difficult to
handle in a strong wind. It is to be expected that each will find its position
in the world's commerce, but the hurried growth of the aeronautical
industry under the stimulus of war conditions has led to a state without
precedent in the history of locomotion in that the means of production
have developed far more rapidly than the civil demands.
In Britain, in particular, the progress of aeronautics has been assisted
by the publications and work of the Advisory Committee for Aeronautics,
and the country has now a very extensive literature on the subject. The
Advisory Committee for Aeronautics was formed on April 30, 1909, by
the Prime Minister " For the superintendence of the investigations at the
National Physical Laboratory, and for general advice on the scientific
problems arising in connection with the work of the Admiralty and War
Office in aerial construction and navigation." The committee has worked
in close co-operation with Service Departments, which have submitted for
discussion and subsequent publication the results of research on flying
craft. The Royal Aircraft Factory has conducted systematic research on
the aerodynamics of aeroplanes, and the Admiralty Airship department has
taken charge of all lighter-than-air craft. Standard tests on aircraft
have also been carried out at Martlesham Heath and the Isle of Grain
by the Air Ministry. The collected results were pubHshed annually
until the outbreak of war in 1914, and are now being prepared for
publication up to the present date. These publications form by far the
greatest volume of aeronautical data in any country of the world, and from
them a large part of this book is prepared.
In January, 1910, M. Eiffel described a wind channel which he had
erected in Paris for the determination of the forces on plates and aero-
plane wings, the first results being published later in the same year. The
volumes containing Eiffel's results formed the first important contribution
to the technical equipment of an aeronautical drawing office, and are
well known throughout Britain. The aerodynamic laboratory was a
private venture, and experiments for designers were carried out without
charge, but with the rights of pubhcation of the results.
For the Italian Government, Captain Crocco was at work on the
aerodynamics of airships, and published papers on the subject of the
stability of airships in April, 1907. He has since been intimately connected
with the development of Italian airships. The chief aerodynamics
laboratory, prior to 1914, in Germany was the property of the Parseval
Airship Company, but was housed in the Gottingen University under the
control of Professor Prandtl. Some particularly good work on balloon
models was carried out and the results pubHshed in 1911, but in 1914
the German Government started a National laboratory in Berhn under
8 APPLIED AERODYNAMICS
the direction of Prandtl, of which no results have been obtained in this
country. Some of the German writers on stabihty were following closely
along parallel lines to those of Bryan in Britain, and had, prior to 1914,
arrived at the idea of maximum lateral stability.
The other European laboratory of note was at Koutchino near Moscow,
with D. Eiabouchinsky as director. This laboratory appears to have been
a private estabhshment, and played a very useful part in the development
of some of the fundamental theories of fluid motion. The practical demand
on the time of the experimenters appears to have been less severe than in
the more Western countries.
A National Advisory Committee for Aeronautics was formed at
Washington on April 2, 1915, by the President of the United States.
Reports of work have appeared from time to time which largely follow
the lines of the older British Committee and add to the growing stock of
valuable aeronautical data.
Before dealing with specific cases of aircraft it may be useful to compare
and contrast man's efforts with the most nearly corresponding products
of nature. Between the birds and the man-carrying aeroplane there are
points of similarity and difference which strike an observer immediately.
Both have wings, those in the bird being movable so as to allow of flapping,
whilst those in the aeroplane are fixed to the body. Both the bird and
the aeroplane have bodies which carry the motive power, in one case
muscular and in the other mechanical. Both have the intelligence factor
in the body, the aeroplane as a pilot. The aeroplane body is fitted with
an airscrew, an organ wholly unrepresented in bird and animal life, the
propulsion of the bird through the air as well as its support being achieved
by the flapping of its wings. In both cases the bodies terminate in thin
surfaces, or tails, which are used for control, but whilst the aeroplane has
a vertical fin the bird has no such organ. The wings of a bird are so mobile
at will that manoeuvres of great complexity can be made by altering their
position and shape, manoeuvres which are not possible with the rigid wings
of an aeroplane. In addition to the difference between airscrew and flap-
ping wings, aeroplanes and birds differ greatly in the arrangements for
alighting, the skids and wheels of the aeroplane being totally dissimilar
to the legs of the bird.
The study of bird flight as a basis for aviation has clearly had a marked
influence on the particular form which modern aeroplanes have taken,
and no method of aerodynamic support is known which has the same
value as that obtained from wings similar to those of birds. The fact that
flapping motion has not been adopted, at least for extensive trial, appears
to be due entirely to mechanical difficulties. In this respect natural
development indicates some limitation to the size of bird which can fly.
The smaller birds fly with ease and with a very rapid flapping of the wings ;
larger birds spend long periods on the wing, but general information
indicates that they are soaring birds taking advantage of up currents
behind cliffs or a large steamer. With the still larger birds, the emu and
ostrich, flight is not possible. The history of bird-life is in strict accordance
with the mechanical principle that structures of a similar nature get
STANBAKD FORMS OF AIRCRAFT 9
relatively weaker as they get larger. Man, although he has steel and a
large selection of other materials at his disposal, has not found anything
so much better than the muscle of the bird as to make the problem of
supporting large weights by flapping flight any more promising than the
results for the largest birds. In looking for an alternative to flapping
the screw propeller as developed for steamships has been modified for aerial
use, and at present is the universal instrument of propulsion.
The adoption of rigid wings in large flying machines in order to obtain
sufficient strength also brought new methods of control. Mechanical
principles relating to the effect of size on the capacity for manoeuvre show
that recovery from a disturbance is slower for the larger construction.
The gusts encountered are much the same for birds and aeroplane, and
the slowness of recovery of the aeroplane makes it improbable that the
beautiful evolutions of a bird in countering the effects of a gust will ever
be imitated by a man-carrying aeroplane. In one respect the aeroplane
has a distinct advantage : its speed through the air is greater than that
of the birds, and speed is itself one of the most effective means of combating
the effect of gusts.
Further reference to bird flight is foreign to the purpose of this book,
which relates to information obtained without special attention to the
study of bird flight.
The airship envelope and the submarine have more resemblance to
the fishes than to any other living creatures. Generally speaking, the form
of the larger fishes provides a very good basis for the form of airships.
It is curious that the fins of the fish are usually vertical as distinct from
the horizontal tail feathers of the bird, and the fins over and under the
central body have no counterpart in the' airship. Both the artificial and
hving craft obtain support by displacement of the medium in which they
are submerged, and rising and falling can be produced by moderate changes
of volume. The resemblance between the fishes and airships is far less
close than that between the birds and aeroplanes.
General Description of Particular Aircraft
A number of photographs of modern aircraft and aero engines are
reproduced as typical of the subject of aeronautics. They will be used to
define those parts which are important in each type. The details of the
motion of aircraft are the subject of later chapters in which the conditions
of steady motion and stability are developed and discussed.
The Aeroplane. — The frontispiece shows a large aeroplane in flight.
Built by Messrs. Handley Page & Co., the aeroplane is the heaviest yet
flown and weighs about 30,000 lbs. when fully loaded. Its engines develop
1500 horsepower and propel the aeroplane at a speed of about 100 miles
an hour. It is of normal biplane construction for its wings, the special
characteristics being in the box tail and in the arrangement of its four
engines. Each engine has its own airscrew, the power units being divided
into two by the body of the aeroplane, each half consisting of a pair of
engines arranged back to back. One airscrew of each pair is working in
10 APPLIED AEEODYNAMICS
the draught of the forward screw, and this tandem arrangement is as yet
somewhat novel.
Biplane (Fig. 1). — Fig. 1 shows a single-seater fighting scout, the StE. 6,
much used in the later stages of the war. Its four wings are of equal length,
and form the two planes which give the name to the type. The lower wings
are attached to the underside of the body behind the airscrew and engine
cowl, whilst the upper wings are joined to a short centre section supported
from the body on a framework of struts and wires. Away from the body the
upper and lower planes are supported by wing struts and wire bracing,
and the whole forms a stiff girder. In flight the load from the wings is
transmitted to the body through the wing struts and the wires from their
upper ends to the underside of the body. These wires are frequently
referred to as lift wires. The downward load on the wings which accom-
panies the running of the aeroplane over rough ground is taken by " anti-
lift " wires, which run from the lower end^ of the wing struts to the centre
section of the upper plane.
In the direction of motion of the aeroplane in flight are a number of
bracing wires from the bottom of the various struts to the top of the
neighbouring member. These wires stiffen the wings in a way which
maintains the correct angle to the body of the aeroplane, and are known
as incidence wires. The bracing system is redundant, i.e. one or more
• members may break without causing the collapse of the structure.
The wings of each plane will be seen from the photograph to be bent
upwards in what is known as a dihedral angle, the object of which is to
assist in obtaining lateral stability. For the lateral control, wing flaps
are provided, the extent of which can be seen on the wings on the left of
the figure. On the lower flap the lever for attachment of the operating
cable is visible, the latter being led into the wing at the front spar, and
hence by pulleys to the pilot's cockpit. The positions of the front and rear
spars are indicated by the ends of the wing struts in the fore and aft
direction, and run along the wings parallel to the leading edges.
The body rests on the spars of the bottom plane, and carries the engine
and airscrew in the forward end. The engine is water-cooled, and the
necessary radiators are mounted in the nose immediately behind the air-
screw. Blinds, shown closed, are required in aeroplanes which climb to
great heights, since the temperature is then well below the freezing-point
of water, and unrestricted flow of air through the radiator during a glide
would lead to the freezing of the water and to loss of control of the engine.
The bUnds can be adjusted to give intermediate degrees of coohng to
correspond with engine powers intermediate between ghding and the
maximum possible.
Alongside the body and stretching back behind the pilot's seat is one
of the exhaust pipes which carry the hot gases well to the rear of the aero-
plane. The' pilot's seat is just behind the trailing edge of the upper wing.
Above the exhaust pipe and near the front of the body is a cover over the
cylinders on one side of the engine, the cover being used to reduce the air
resistance.
The airscrew is in the extreme forward position on the aeroplane, and
/**'
^''
STANBAED FOEMS OF AIECEAFT 11
has four blades. The diameter is fixed in this case by the high speed of
the airscrew shaft, and not, as in many cases, by the ground clearance
required for safety when running over the ground.
Below the body and under the wings of the lower plane is the landing
chassis. The frame consists of a pair of vee-shaped struts based on the
body and joined at the bottom ends by a cross tube. The structure is
supported by a diagonal cross-bracing of wires. The wheels and axle are
held to the undercarriage by bindings of rubber cord so as to provide
liexibihty. The shocks of landing are taken partly by this rubber cord
and partly by the pneumatic tyres on the wheels. With the aeroplane
body nearly horizontal the wheel axle is ahead of the centre of gravity
of the aeroplane, so that the effect of the first contact with the ground is
to throw up the nose, increasing the angle of incidence and drag. If the
speed of alighting is too great the lift may increase sufficiently to raise
the aeroplane off the ground. The art of making a correct landing is one of
the most difficult parts to be learnt by a pilot.
The tail of the aeroplane is not clearly shown in this figure, and
description is deferred.
With an engine developing 210 horsepower and a load bringing the gross
weight of the aeroplane to 2000 lbs., the aeroplane illustrated is capable
of a speed of over 130 m.p.h. and can climb to a height of 20,000 feet.
The limit to the height to which aircraft can climb is usually called the
" ceiling."
Monoplane (Fig. 2). — The most striking difference from Fig. 1 is the
change from two planes to a single one, and in order to support the wings
against landing shocks, a pyramid of struts or " cabane " has been built
over the body. From the apex of the pyramid bracing wires are carried
to points on the upper sides of the front and rear spars. The lower bracing
wires go from the spars to the underside of the body, and each is duplicated.
On the right wing near the tip is a tube anemometer used as part of
the equipment for measuring the speed of the aeroplane. In biplanes the
anemometer is usually fixed to one of the wing struts, as the effect of the
presence of the wing on the reading is less marked -than in the case now
illustrated.
In this type of aeroplane, made by the British & Colonial Aeroplane
Coy., the engine rotates, and the airscrew has a somewhat unusual feature
in the " spinner " which is attached to it. The airscrew has two blades
only, and this type of construction has been more common than the four-
bladed type for reasons of economy of timber. The differences of
efficiency are not marked, and either type can be made to give good
service, the choice being determined in some cases by the speed of rotation
of the airscrew shaft of an available engine.
The undercarriage is very similar to that shown in Fig. 1. On one
of the front struts is a small windmill which drives a pump for the petrol
feed. Windmills are now frequently used for auxiliary services, such as
the electrical heating of clothing and the generation of current for the
wireless transmission of messages.
The tail is clearly visible, and underneath the extreme end of the body
12 APPLIED AEEODYNAMICS
is the tail-skid. This skid is hinged to the body, and is secured by rubber
cord at its inner end, so as to decrease the shock of contact with the ground.
The horizontal plane at the tail is seen to be divided, the front part or
tailplane being fixed, whilst the rear part or elevator is movable at the
pilot's wish. The control cables go inside the fuselage at the root of the
tail plane. Underneath, the tail plane is seen to be braced to the body ;
above, the bracing wires are attached to the fin, which, like the tail plane,
is fixed to the body. The rudder is hidden behind the fin, but the rudder
lever for attachment of the control cable can be seen about halfway up
the fin.
The pilot sits under the " cabane," and his downward view is helped
by holes through the wings. Immediately in front of him is a wind screen,
and also in this instance a machine-gun, which fires through the airscrew.
Flying-boat (Fig. 3). — The difference of shape from the land types is
marked in several directions, as will be seen from the illustration relating to
the Phoenix " Cork " flying-boat P. 5. The particular feature which gives
its name to the type is the boat structure under the lower wing, and this
replaces the wheel undercarriage of the aeroplane in order to render possible
ahghting on water. The flying boat is shown mounted on a trolley during
transit from the sheds to the water. On the underside of the boat, just
behind the nationality circles, is a step which plays an important part in
the preHminary run on the water. A second step occurs under the wings
at the place of last contact with the sea during a flight, but is hidden by
the deep shadow of the lower wing.
Underneath the lower wing at the outer struts is a wing float which
keeps the wing out of the water in any slight roll. The wing structure is
much larger than those of Figs. 1 and 2, and there are six pairs of inter-
plane struts. The upper plane is appreciably longer than the lower, the
extensions being braced from the feet of the outer struts. The levers on
the wing flaps or ailerons are now very clearly shown ; owing to the
proximity of waves to the lower wing, ailerons are not fitted to them.
The tail is raised high above the boat and is in the sHp streams from
the two airscrews. As the centre line of the airscrews is far above the
centre of gravity, switching on the engine would tend to make the flying-
boat dive, were it not so arranged that the slip-stream effect on the tail
is arranged to give an opposite tendency. The fin and rudder are clearly
shown, as are also the levers on the rudder and elevators. Besides having
a dihedral angle on the wings, small fins have been fitted above the top
wings as part of the lateral balance of the flying-boat.
The engines are built on struts between the v/ings, and each engine
drives a tractor airscrew. The engines are run in the same direction,
although at an early stage of development of flying-boats the effects of
gyroscopic action of the rotatory airscrews were eliminated by arranging
for rotation in opposite directions. This was found to be unnecessary.
The tail of the flying-boat has been especially arranged to come into the
slip stream of the airscrews, but in aeroplanes this occurs without
special provision or desire. Not only does the airscrew increase the air-
speed over the tail, but it alters the angle of incidence and blows the tail
Fig. 4. — Cockpit of an aeroplane.
STANDARD FORMS OF AIRCRAFT IB
up or down depending on its setting. There is also a twist in the slip stream
which is frequently unsymmetrically placed with respect to the fin and rudder
and tends to produce turning. The effects of switching the engine on and
olf may be very complex.
In order to ease the pilot's efforts many aeroplanes are fitted with an
adjustable tail plane, and if they are stable the adjustment can be made
so as to give any chosen flying speed without the application of force to
the control stick.
Pilot's Cockpit (Fig. 4). — The photograph of the "Panther" was taken
from above the aeroplane looking down and forward. At the bottom of the
figure is the edge of the seat which rests on the top of the petrol tank. Along
the centre of the figure is the control column hinged at the bottom to a rock-
ing shaft so that the pilot is able to move it in any direction. By suitable
cable connections it is arranged that fore-and-aft movement depresses or
raises the elevators, whilst movement to right or left raises or lowers the
right ailerons. Some of the connections can be seen ; behind the control
column is a lever attached to the rocking shaft and having at its ends the
cables for the ailerons. The cables can be seen passing in inchned directions
in front of the petrol tank. On the near side of the control column but
partly hidden by the seat is the link which operates the elevators.
In the case of each control the motion of the colunm required is that
which would be made were it fixed to the aeroplane and the pilot held
independently and he attempted to pull the aeroplane into any desired
position. In other words, if the pilot puUs the stick towards him the nose
of the aeroplane comes up, whilst moving the column to the right brings
the left wing up.
On the top of the control column is a small switch which is used by the
pilot to cut out the engine temporarily, an operation which is frequently
required with a rotary engine just before landing.
Across the photograph and a little below the engine control switches
is the rudder bar, the hinge of which is vertical and behind the control
column. The two cables to the rudder are seen to come straight back
under the pilot's seat. In the rudder control the pilot pushes the rudder
bar to the right in order to turn to the right.
Several instruments are shown in the photograph. In the top left
corner is the aneroid barometer, which gives the pilot an approximate
idea of his height. In the centre is the compass, an instrument specially
designed for aircraft where the conditions of use are not very favourable
to good results. Immediately below the compass and partly hidden by
it is the airspeed indicator, which is usually connected to a tube anemometer
such as was shown in Fig. 2 on the edge of the wing. Still lower on the
instrument board and behind the control column is the cross-level which
indicates to a pilot whether he is side-slipping or not. To the right of
the cross-level are the starting switches for the engine, two magnetos being
used as a precautionary measure. Below and to the right of the rudder
bar is the engine revolution-indicator.
14 APPLIED AEEODYNAMICS
Engines
Air-cooled Rotary Engine (Fig. 5a). — In this type of engine, the B.Ej 2,
the airscrew is bolted to the crank case and cyHnders, and the whole then
rotates about a fixed crankshaft. The cylinders, nine in number, develop a
net brake horsepower of about 230 at a speed of 1100 to 1300 revolutions per
minute. The cylinders are provided with gills, which greatly assist the cool-
ing of the cylinder due to their motion through the air. Without any forward
motion of the aeroplane, cooling is provided by the rotation of the cylinders,
and an appreciable part of the horsepower developed is absorbed in turning
the engine against its air resistance. Air and petrol are admitted through
pipes shown at the side of each cylinder, and both the inlet and exhaust
valves are mechanically operated by the rods from the head of the cylinder
to the crank case. The cam mechanism for operating the rods is inside
the crank case. The hub for the attachment of the airscrew is shown in
the centre.
A type of engine of generally similar appearance has stationary
cylinders and is known as " radial." It is probable that the cooUng losses
in a radial engine are less than those in a rotary engine of the same net-
power, but no direct comparison appears to have been made. The
effectiveness of an engine cannot be dissociated from the means taken to
cool its cylinders. The resistance of cylinders in a radial engine and
radiators in a water-cooled engine should be estimated and allowed for
before comparison can be made with a rotary engine, the losses of which
have already been deducted in the engine test-bed figures. For engines
with stationary cylinders test-bed figures usually give brake horsepower
without allowance for aerodynamic cooling losses.
Vee-type Air-cooled Engine (Fig. 6b). — The engine shown has twelve
cylinders, develops about 240 horsepower and is known as the
E.A.F. 4d. The cylinders are arranged above the crank case in
two rows of six, with an angle between them, hence the name given
to the type. In order to cool the cylinders a cowl has been provided,
so that the forward motion of the aeroplane forces air between the
cylinders and over the cylinder heads. At the extreme left of the photo-
graph is the airscrew hub, and in this particular engine the airscrew is
geared so as to turn at half the speed of the crankshaft, the latter making
1800 to 2000 r.p.m. To the right of and below the airscrew hub is one of
the magnetos with its distributing wires for the correct timing of the
explosions in the several cylinders. At the bottom of the photograph are
the inlet pipes, carburettors, petrol pipes and throttle valves.
Water-cooled Engine (Fig. 6). — Water-cooled engines have been used
more than any other type in both aeroplanes and airships. The two
photographs of the Napier 450 h.p. engine show what an intricate
mechanism the aero engine may be. The cylinders are arranged in three
rows of four, each one being surrounded by a water jacket. The feed-
pipes of the water-circulating system can be seen in Fig. 6& going from
the water pump at the bottom of the picture to the lower ends of the
cylinder jackets, whilst above them are the pipes which connect the
Fig. 5 (a). — Rotary engine.
Fig. 5 (&). — Air-cooled stationary engine.
iji
"3)
Pm
\
STANDAED FOEMS OF AIECEAFT 15
outlets for the hot water and transmit the latter to the radiator.
The camshafts which operate the inlet and exhaust valves run along
the tops of the cylinders, and are carefully protected by covers ; the
inclined shafts, ending in gear cases at the top> connect the camshafts
with the crankshaft of the engine.
The inlet pipes for the air and petrol mixture are shown in Fig. 6a ;
they are three in number, each feeding four cylinders and having its own
carburetter. The magnetos are shown in Fig. 66, on either side of the
engine, with the distributing leads taken to supporting tubes along the
engine. The same illustration also shows the location of the sparking
plugs and the other end of the magneto connecting wires.
The airscrew is geared 0-66 to 1, and runs at about 1300 r.p.m. ; the hub
to which it is attached is clearly shown in Fig. 6a.
The engine is well known as the " Napier Lion," and was especially
designed for work at altitudes of 10,000 feet and over. It represents
the furthest advance yet made in the design of the aero-motor.
Airships
The Rigid Airship (Figs. 7 and 8). — Eigid airships have been made with
a total lift of nearly 70 tons, a length of 650 feet, and a diameter of envelope
of about 80 feet. They are capable of extended flight, being afloat for
days at a time whilst travelling many thousands of miles. The speeds
reached with a horsepower of 2000 are a little in excess of 75 miles an hour.
A photograph of a recent rigid airship is shown in Fig. 7. The sections
of the envelope are polygonal, and the central part of the ship cylindrical.
The head and tail are short and give the whole a form of low resistance.
Still later designs have a much reduced cyUndrical middle body and con-
sequent longer head and tail, with an appreciably lower resistance.
To the rear of the airship are the fins which give stabiUty and control,
and in the instance illustrated the four fins are of equal size. The control
surfaces, elevators and rudders^ are attached at the rear edges of the
fixed fins.
The airship has three cars ; each contains an engine for the driving
of a pair of airscrews. For the central car the airscrews are very clearly
«hown, but for the front and rear cars they have been turned into a hori-
zontal position to assist the landing, and are seen in projection on the side
of the cars, so that detection in the figure is much more difficult than for
those of the central car. Below each of the end cars is a bumping bag to
take landing shocks, whilst rope ladders connect the cars with a communica-
tion way in the lower part of the envelope.
Valves are shown at internals along the ships, one for each of the gas
containers, and serve to prevent an excess of internal pressure due to the
expansion of the hydrogen. As arranged for flight, rigid airships can reach
a height of 20,000 feet before the valves begin to operate. Fig. 8, E 34,
shows the gas containers hanging loosely to the metal frame, which is just
being fitted with its outer coverings. In the centre of the figure the
skeleton is clearly visible, and consists of triangular girders running along
16 APPLIED AERODYNAMICS
the ship and rings running round it. Two types of ring are visible, one
of which is wholly composed of simple girders, whilst the second has king-
posts as stiffeners on the inside. From the corners of this second frame
radial wires pass to the centre of the envelope and form one of the divisions
of the airship. The centres of the various radial divisions are connected
by an axial wire, which takes the end pressure of the gas bags in the case
of deflation of one of them or of inclination of the airship. The cord netting
against which the gas bags rest can be seen very clearly. The airship is
one built for the Admiralty by Messrs Beardmore.
The Non-rigid Airship (Fig. 9). — The non-rigid type of construction
is essentially different from that described above, the shape of the envelope
being maintained wholly by the internal gas pressure. The N.S. type of
airship illustrated in Fig. 9 has a gross weight of 11 tons, and with 500 h. p.
travels at a little more than 55 m.p.h. The length is 262 feet, and the
maximum width of the envelope 57 feet. Fig. 96 gives the best idea of
the cross-section of this type of airship, and shows three lobes meeting in
well-defined corners. The type was originated in Spain by Torres Quevedo
and developed in Paris by the Astra Company. It contains an internal
triangular stiffening of ropes and fabric between the corners. The
satisfactory distribution of loads on the fabric due to the weight of the
car and engines is possible with this construction without necessitating
suspension far below the lower surface of the envelope. Fig. 9c, taken
from below the airship, shows the wires from the car to the junction of
the lobes at the bottom of the envelope, and these take the whole load
under level- keel conditions. To brace the car against rolling, wires are
carried out on either side and fixed to the lobes at some distance from the
plane of symmetry of the airship. The principle of relief of stress by
distribution of load has been utilised in this ship, the car and engine
nacelles being supported as separate units. Communication is permitted
across a gangway which adds nothing of value to the distribution of
load.
The engines are two in number, situated behind the observation car,
and each is provided with its own airscrew. Beneath the engines and also
below the car are bumping bags for use on ahghting.
As the shape of the airship is dependent on the internal gas pressure,
special arrangements are made to control this quantity, and the fabric pipes
shown in Fig. 9c show how air is admitted for this purpose to enclosed
portions of the envelope. The envelope is divided inside by gastight
fabric, so that in the lower lobes both of the fore and rear parts of the
airship, small chambers, or balloonets, are formed into^ which air can be
pumped or from which it can be released. The position of these balloonets
can be seen in Fig. 9c, at the ends of the pair of long horizontal feed
pipes ; they are cross connected by fabric tubes which are also clearly
visible. The high-pressure air is obtained from scoops lowered into the
slip streams from the airscrews, the scoops being visible in all the figures,
but are folded against the envelope in Fig. 9a. Valves are provided in
the feed pipes for use by the pilot, who inflates or deflates the balloonets
as required to allow for changes in volume of the hydrogen due to variations
f^
r
I
o
6
STANDAED FOEMS OF AIECBAFT 17
of height. Automatic valves are arranged to release air if the pressure
rises above a chosen amount.
The weight of fabric necessary to withstand the pressure of the gas
is greatly reduced by reinforcing the nose of the airship as shown in
Fig, 9b. The maximum external air force due to motion occurs at the
nose of the airship, and at high speeds becomes greater than the internal
pressure usually provided. The region of high pressure is extremely local,
and by the addition of stiffening ribs the excess of pressure over the
internal pressure is transmitted back to a part of the envelope where it is
easily supported by a small internal pressure. Occasionally the nose of
an airship is blown in at high speed, but with the arrangements adopted
the consequences are unimportant, and the correct shape is recovered by
an increase of balloonet pressure.
The inflation of one balloonet and the deflation of the other is a control
by means of which the nose of the airship can be raised or lowered, and so
effect a change of trim, but the usual control is by elevators and rudders.
In the N.S. type of airship the rudder is confined to the lower surface, and
the upper tin is of reduced size. This, the largest of the non-rigid airships,
is the product of the Admiralty Airship Department from their station
at Kingsnorth, and has seen much service as a sea-scout.
Kite Balloons (Fig. 10). — The early kite balloon was probably a German
type, with a string of parachutes attached to the tail in order to keep
the balloon pointing into the wind. The lift on a kite balloon is partly
due to buoyancy and partly due to dynamic lift, the latter being largely
predominant in winds of 40 or 50 m.p.h. The balloon is captive, and may
either be sent aloft in a natural wind or be towed from a ship. Two types
of modern kite balloon are shown in Fig. 10, (a) and (6) showing the latest
and most successful development. To the tail of the balloon are fixed
three fins, which are kept inflated in a wind by the pressure of air in a
scoop attached to the lower fin. With this arrangement the balloon
swings slowly backwards and forwards about a vertical axis, and travels
sideways as an accompanying movement.
The kite wire is shown in Fig. 10& as coming to a motor boat. The
second rope which dips into the sea is an automatic device for maintaining
the height of the balloon. The general steadiness of the balloon depends
on the point of attachment of the kite wire, and the important difference
illustrated by the types Fig. 10 (a) and (c) is that the latter becomes
longitudinally unstable at high-wind speeds and tends to break away,
whilst the former does not become unstable. The general disposition
of the rigging is shown most clearly in Fig. 10a, where u rigging band
is shown for the attachment of the car and kite line.
CHAPTEE II
THE PRINCIPLES OF FLIGHT
(i) The Aeroplane
In developing the matter under the above heading, an endeavour will be
made to avoid the finer details both of calculation and of experiment. In
the later stages of any engineering development the amount of time devoted
to the details in order to produce the best results is apt to dull the sense
of those important factors which are fundamental and common to all
discussions of the subject. It usually falls to a few pioneers to establish
the main principles, and aviation follows the rule. The relations between
lift, resistance and horsepower became the subject of general discussion
amongst enthusiasts in the period 1896-1900 mainly owing to the researches
of Langley. Maxim made an aeroplane embodying his views, and we can
now see that on the subjects of weight and horsepower these early in-
vestigations established the fundamental truths. Methods of obtaining
data and of making calculations have improved and have been extended
to cover points not arising in the early days of flight, and one extension
is the consideration of flight at altitudes of many thousands of feet.
The main framework of the present chapter is the relating of experi-
mental data to the conditions of flight, and the experimental data will be
taken for granted. Later chapters in the book take up the examination
of the experimental data and the finer details of the analysis and prediction
of aeroplane performance.
, Wings. — The most prominent important parts of an aeroplane are the
wings, and their function is the supporting of the aeroplane against gravita-
tional attraction. The force on the wings arises from motion through the
air, and is accompanied by a downward motion of the air over which the
wings have passed. The principle of dynamic support in a fluid has been
called the " sacrificial " principle (by Lord Bayleigh, I beheve), and stated
broadly expresses the fact that if you do not wish to fall yourself you must
make something else fall, in this case air.
If AB, Fig. 1 1; be taken to represent a wing moving in the direction of the
arrow, it will meet air at rest at C and will leave it at EE endued with a
downward motion. Now, from Newton's laws of motion it is known that
the rate at which downward momentum is given to the fluid is equal to
the supporting force on the wings, and if we knew the exact motion of
the air round the wing the upward force could be calculated. The problem
is, however, too difficult for the present state of mathematical knowledge,
and our information is almost entirely based on the results of tests on
models of wings in an artificial air current.
18
THE PKINCIPLES OF PLIGHT
19
The direct measurement of the sustaining force in this way does not
involve any necessity for knowledge of the details of the flow. It is usual
to divide the resultant force R into two components, L the lift, and D
the drag, but the essential measurements in the air current are the magni-
tude of R and its direction y, the latter being reckoned from the normal
to the direction of motion. The resolution into lift and drag is not the
only useful form, and it will be found later that in some calculations it is
convenient to use a Hne fixed relative to the wing as a basis for resolution
rather than the direction of motion.
No matter by what means the results are obtained, it is found that the
supporting force or lift of an aeroplane wing can be represented by curves
such as those of Fig. 12. The lifting force depends on the angle a (Fig. 11)
which the aerofoil makes with the relative wind, and it is interesting to
Fig. 11.
find that the lifting force may be positive when a is negative, i.e. when the
relative wind is apparently blowing on the upper surface. The chord, i.e.
the straight line touching the wing on the under surface, is inclined down-
wards at 3° or more before a wing of usual form ceases to Uft.
The lift on the wing depends not only on the angle of incidence and
of course the area, but also on the velocity relative to the air, and for
full-scale aeroplanes the Hft is proportional to the square of the speed, at
the same angle of incidence. Of course in any given flying machine the
weight of the machine is fixed, and therefore the lift is fixed, and it follows
from the above statement that only one speed of flight can correspond
with a given angle of incidence, and that the speed and angle of incidence
must change together in such a way that the lift is constant. This relation
can easily be seen by reference to Fig. 12. The curve ABODE is obtained
by experiment as follows : A wing (in practice a model of it is used and
20
APPLIED AEEODYNAMICS
multiplying factors applied) is moved through the air at a speed of 40 m.p.h.
In one experiment the angle of incidence is made zero, and the measured
lift is 340 lbs. This gives the point P of Fig. 12. When the angle of
incidence is 5° the lift is 900 lbs., and so on. In the course of such an
experiment, there is reached an angle of incidence at which the hft is a
maximum, and this is shown at D in Fig. 12 for an angle of incidence of
17° or 18°. For angles of incidence greater than this it is not possible to
carry so much load at 40 m.p.h. Without any further experiments it is
now possible to draw the remainder of the curves of Fig. 12. At B the lift
for 40 m.p.h. has been found to be 610 lbs. At Bi it will be 610 X (|-g)2 lbs.,
INCLINATION OF CHORD (DEGREES)
Fig 12. — Wing lift and speed.
at B2 610 X (f ~|})^ lbs., and so on, the lift for a given angle being proportional
to the square of the speed.
Now suppose that the wings for which Fig. 12 was prepared are to be
used on an aeroplane weighing 2000 lbs. At 35 m.p.h. the wings cannot be
made to carry more than 1530 lbs., and consequently the aeroplane will
need to get up a speed of more than 35 m.p.h. before it can leave the ground.
At 40 m.p.h., as we see at D, the weight can just be lifted, and this con-
stitutes the slowest possible flying speed of that aeroplane. The angle of
incidence is then 17 to 18 degrees. If the speed is increased to 50 m.p.h.
the required lift is obtained at an angle of incidence rather less than 9°,
and so on, until if the engine is powerful enough to drive the aeroplane at
100 m.p.h. the angle of incidence has a small negative value.
THE PEINCIPLES OF FLIGHT 21
It will be noticed that in this calculation no knowledge is needed of
the resistance of the aeroplane or the horsepower of its engine. The
angle of incidence for any speed is fixed entirely from the lift curves.
A common size of aeroplane in flying order weighs roughly 2000 lbs.
The area of the four wings adopted in order to alight at 40 m.p.h. comes
to be approximately 360 sq. feet. Flying at the lower speeds is almost
entirely confined to the last few seconds before alighting.
Resistance or Drag. — All the parts of an aeroplane contribute to the
resistance, whereas practically the whole of the hft is taken by the wings.
The resistance is usually divided into two parts, one due to the wings and
the other due to the remainder of the machine. The reason for this is
that the resistance of the wings is not even approximately proportional
to the square of the flying speed, because of the change of angle of incidence
of the wings already shown to occur ; on the other hand, the resistance of
each of the other parts is very nearly proportional to the square of the speed.
At low flight speeds the resistance of the wings is by far the greater
of the two parts, whilst at higher speeds the body resistance may be
appreciably greater than that of the wings.
Drag of the Wings, — The curves for the drag of the wings correspond-
ing with those of Fig. 12 for the hft are given in Fig. 13. The curve marked
ABODE in Fig. 13 is obtained experimentally, usually at the same time as
the similarly marked curve of Fig. 12. It shows the drag of the wings
when travelhng at 40 m.p.h. at various angles of incidence. At 0° the
drag is httle more than 30 lbs., whilst at 16^ it is 300 lbs. Bi is got from
B by increasing the drag at the same angle of incidence in proportion to
the square of the speed.
It has already been shown that there can only be one angle of incidence
of the main planes for any one speed, and from Fig. 12 the relation between
angle and speed for an aeroplane weighing 2000 lbs. was obtained. At a
speed of 40 m.p.h. an angle of 17-5° was found, and point E of Fig. 13 shows
that the resistance would then be 560 lbs. The points Ei, E2, E3 and E4
similarly show the drag at 50, 60, 70 and 100 m.p.h. If the aeroplane
is supposed to be flying slowly, i.e. at 40 m.p.h., and the speed be gradually
increased, it will be seen that the drag due to the wings diminishes very
rapidly at first from 560 lbs. at 40 m.p.h. to 130 lbs. at 50 m.p.h., and
reaches a minimum of 99 lbs. at about 60 miles an hoar, after which a
marked increase occurs. Contrary to almost every other kind of loco-
motion, a very considerable reduction of resistance may result from
increasing the speed of the aeroplane. It will be seen later that the
r(!duction is so great that less horsepower is required at the higher speed.
Drag 0! the Body, Struts, Undercarriage, etc.— The drag of the aero-
plane other than the* wings is usually obtained by the addition of the
measured resistances of many parts. The actual carrying out of the opera-
tion is one of some detail and is referred to later in the book (Chapter IV.). •
For present purposes it is sufficient to know that as the result of experi-
ment, these additional resistances amount to about 50 lbs. at 40 m.p.h.,
and vary as the square of the speed, so that at 100 m.p.h. the additional
resistances have increased to 312 lbs.
22
APPLIED AEE0DYNAMIC8
It is now possible to make Table 1 showing the resistance of the aero-
plane at various speeds, and to estimate the net horsepower required to
propel an aeroplane weighing 2000 lbs. The losses in the organs of pro-
pulsion will not be considered at this point, but will be dealt with almost
immediately when determining the horsepower available.
A rough idea of the brake horsepower of the engine required for
500
400
300
200
ANGLE OF INCIDENCE DEGREES
Fio. 13. — Wing drag and speed.
horizontal flight can be obtained by assuming a propeller efficiency of
60 per cent, in all cases. It will then be seen thai the aeroplane would
just be able to fly with an engine of 45 horsepower at a speed of
approximately 50 m.p.h. At 70 m.p.h. the brake horsepower of the
engine would need to be nearly 80, whilst to fly at 100 m.p.h. would
need no less than 225 horsepower. By various modifications of wing area
the horsepower for a given speed can be varied considerably, but the
example given illustrates fairly accurately the limits of speed of an
w
T» aerc
THE PEINCIPLES OF FLIGHT
23
aeroplane of the weight assumed ; e.g. an engine developing 100 horse-
power may be expected to give a flight-speed range of from 40 m.p.h.
to 80 m.p.h. to an aeroplane weighing 2000 lbs.
TABLE 1.-
-Abboplani! Dbao
AND SfBED.
Speed of flight
Besistance of wings
Resistance of rest of
Total resistance
Net horsepower
(m.p.h.).
alone (lbs.).
aeroplane (lbs.).
(lbs.).
required.*
40
660
60
610
65
60
130
78
208
28
60
97
113
210
34
70
100
153
253
47
100
195
312
507
134
The Propulsive Mechanism. — Up to the present the calculations have
referred to the behaviour of the aeroplane, without detailed reference to
the means by which motion through the air is produced. It is now
proposed to show how the necessary horsepower is estimated in order that
the aeroplane may fly. This estimate involves the consideration of the
airscrew.
An airscrew acts on the air in a manner somewhat similar to that of
a wing, and throws air backwards in a continuous stream in order to
produce a forward thrust. The thrust is obtaine(i for the least ex-
penditure of power only when the revolutions of the engine are in a very
special relation to the forward speed.
Increase of the speed of revolution without alteration of the forward
speed of the aeroplane leads to increased thrust, but the law of increase is
complex. Increasing the speed of the aeroplane usually has the effect of
decreasing the thrust, again in a manner which it is not easy to express
simply. Calculations can be made to show what the airscrew will do
under any circumstances, but the discussion will be left to a special chapter.
One simple law can, however, be deduced from the behaviour of air-
screws, and is of much the same nature as that already pointed out for the
supporting surfaces. It was stated that, if the angle of incidence is kept
constant, the lift and drag of a wing increase in proportion to the square
of the speed. Now in the airscrew, it will be found that the angle of
incidence of each blade section is kept constant if the revolutions are
increased in the same proportion as the forward speed, and that under
such conditions the thrust and torque both vary as the square of the
speed. If from a forward speed of 40 m.p.h. and a rotational speed of
600 r.p.m. the forward speed be increased to 80 m.p.h. and the
rotational speed to 1200 r.p.m., the thrust will be increased four times.
Given a table of figures, such as Table 2, which shows the thrust of
an airscrew at several speeds of rotation when travelhng at 40 m.p.h.
through the air, results can be deduced for the thrust at other values of
the forward speed in the manner described below.
* By net horsepower is here meant the power necessary to drive the aeroplane if a
perfectly efficient means of propulsion existed. The conditions are very nearly satisfied by
an aeroplane when gliding.
24
APPLIED AEBODYNAMICS
The figures in Table 2 would be obtained either by calculation or by
an experiment. Tests on airscrews are frequently made at the end of a
long arm which can be rotated, so giving the airscrew its forward motion.
Actual airscrews may be tested on a large whirling arm, or a model air-
screw may be used in a wind channel and multiplying factors employed
to allow for the change of scale.
TABLE 2. — AiRSCEEW Thrust and Speed.
Forward speed 40 m.p.h.
Eevs
per minute.
Thrust (lbs.).
500
0
800
162
1100
374
1400
620
It will be noticed from Table 2 that the airscrew gives no thrust until
rotating faster than 500 r.p.m. At lower speeds than this the airscrew
would oppose a resistance to the forward motion, and would tend to be
turning as a windmiU. When the subject is entered into in more detail
it will be found that the number of revolutions necessary before a thrust
is produced is determined by the " pitch " of the airscrew. The term
" pitch " is obtained from an analogy between an airscrew and a screw,
the advance of the latter along its axis for one complete revolution being
known as the " pitch." Whilst there are obvious mechanical differences
between a solid screw turning in its nut and an airscrew moving in a
mobile fluid, the expression has many advantages in the latter case and
will be referred to frequently. For the present it is not necessary to know
how pitch is defined.
The numbers given in Table 2 correspond with the curve marked
ABC in Fig. 14. To deduce those for any other speed, say 60 m.p.h., the
first column is multipUed by |£'and the second by (|o)^ giving the
following table : —
TABLE 3. — ArasoBEW Thbttst and Speed.
Forward speed 60 m.p.h.
Eevs. per minute.
Thrast (lbs).
750
1200
1650
2100
0
365
842
1400
It will be noticed that the airscrew must now be rotating much more
rapidly than before in order to produce a thrust. The remaining curves
of Fig. 14 were produced in a similar way, and relate to speeds of the
THE PKINCIPLES OF FLIGHT
25
aeroplane which were considered in the supporting of an aeroplane weigh-
ing 2000 lbs. The thrust necessary to support the aeroplane in the air at
speeds of 40, 50, 60, 70 and 100 m.p.h. has been obtained in Table 1, and
using Fig. 14 it is now possible to obtain the propeller revolutions which
are necessary to produce this required thrust. The points are marked
C, C], C2, C3 and C4. To produce a thrust of 610 lbs. at 40 m.p.h. the
propeller must be turning at about 1380 r.p.m., as shown at the point C.
As the speed rises to 50 m.p.h. the engine may be shut down very appre-
ciably, the revolutions being only 930. For higher velocities of flight the
600
200
O 1,000 T.pm 2,000
AIRSCREW REVOLUTIONS,
Fig. 14. — Thrust and speed.
necessary revolutions increase steadily, until at 100 m.-p.h. the rate of rota-
tion is over 1600 r.p.m. The engine may, however, not be powerful enough
to drive the propeller at these rates, and it is now necessary to estimate,
in a manner similar to that for thrust, how much horsepower is required.
The initial data given in Table 4 are again assumed to have been
TABLE 4. — AiBSOEEw Horsepowee and Speed.
Forward speed 40 m.p.h.
Revs, per minute.
Horsepower.
500
800
1100
1400
3 0
27
70
167
26
APPLIED AEEODYNAMICS
obtained experimentally, and the figures from this table are plotted in
Fig. 15 in the curve ABC. To obtain the curve for 60 m.p.h. the first
column of Table 4 is multiplied by f {] and the second by (40)^, obtaining
the numbers given in Table 5.
TABLE 5. — Airscrew Horsepower and Speed.
Forward speed 60 in.p.h.
Bevs. per minute.
Horsepower.
760
1200
1660
2100
101
91
266
710
200
ISO
4 L
A
HORSEPOWER
/
f
4.
#
'3 /
ii
\OOm.p.h
i.ooo T.p.m
AIRSCREW REVOLUTIONS.
Fig. 15. — ^Horsepower and speed.
2,ooo
The curves so obtained for various flight speeds indicate zero horse-
power before the airscrew has stopped. The speeds are lower than those
for which the thrust has become zero, and indicate the points at which
the airscrew becomes a windmill. In an aeroplane, however, the resistance
to turning of the engine would greatly reduce the speed at which the wind-
mill becomes effective below that indicated for no-horsepower, and stoppage
of the petrol supply to the engine would often result in the stoppage of the
airscrew.
THE PRINCIPLES OF FLIGHT 27
From Figs. 14 and 15 it is now easy to find the brake horsepower of
the engine which would be necessary to drive the aeroplane through the
air at speeds from 40 to 100 m.p.h. From Fig. 14 it is found that the
aeroplane when travelhng at 50 m.p.h through the air needs an airscrew
speed of 930 r.p.m. To drive the airscrew at this speed is seen from Fig. 15,
point Ci, to need 39 horsepower. For other speeds the horsepower is
indicated by the points C, C2, C3 and C4, and the collected results are
given in Table 6.
TABLE 6. — ^Aeboplabte Hobsefoweb and Speed.
Speed of aeroplane
(m.p.h.)
Horsepower of engine
necessary for fli^t.
40
60
60
70
100
156
39
48
66
188
On Fig. 15 a line OP has been drawn which represents the work which
a particular engine could do at the various speeds of rotation ; this again
is an experimental curve. The engine is supposed to be giving 120 h.p.
at 1200 r.p.m. It will be seen, from Fig. 15, that the engine is not powerful
enough to drive the aeroplane at either the lowest or the highest speeds for
which the calculations have been made. For many purposes the information
given in Fig. 15 is more conveniently expressed in the form shown in Fig. 16,
where the abscissa is the flight speed of the aeroplane. The curve ABODE
of the latter figure is plotted from the points C, Ci, C2, C3 and C4 of Fig. 15,
while the line FGH corresponds with the points B, Bi, B2,'^B3 and B4.
The first curve shows the horsepower required for flight, and the second
the horsepower available. From the diagram in this form it is easily seen
that the point F represents the slowest speed at which the aeroplane can
fly, in this case 40*3 m.p.h., and that H shows the possibility of reaching a
speed of nearly 93 m.p.h.
Fig. 16 shows more than this, for it gives the reserve horsepower at any
speed of flight. This reserve horsepower is roughly proportional to the
speed at which the aeroplane can chmb, and the curve shows that the best
climbing speed is much nearer to the lower limit of speed than to the
upper limit.
General Remarks on Figs. 12-16. — Calculations relating to the flight speed
of an aeroplane are illustrated fairly exactly by the curves in Fig. 12-16.
As the subject is entered into in detail many secondary considerations will
be seen to come in. The difl&culties will be found to consist very largely
n the determination of the standard curves marked ABCDE in the figures,
cind the analysis of results to obtain these data constitutes one of the more
laborious parts of the process. The compHcation is very largely one of
detail, and should not be allowed to obscure the common basis of flight
conditions for all aeroplanes as typified by the curves of Figs. 12-16,
28
APPLIED AEEODYNAMICS
Climbing Flight. — In the more general theory of the aeroplane it is
of interest to show how the previous calculations may be modified to
include flights other than those in a horizontal plane. The rate at which
an aeroplane can climb has already been referred to incidentally in con-
nection with Fig. 16.
It is clear from the outset that the air forces acting on the aeroplane
depend on its speed and angle of incidence, and are not dependent
on the attitude (or inclination) of the aeroplane relative to the direction
of gravity. If the aeroplane is flying steadily, the force of gravity
acting on it will always be vertical, whilst the inclination of the wind
forces will vary with the attitude of the aeroplane. If the aeroplane
is cUmbing the airscrew thrust will need to be greater than for horizontal
flight, whilst if descending the thrust is reduced and may become zero
or negative. There is a minimum angle of descent for any aeroplane when
2O0
/a
/
i
ISO
B
/
G
^A-
—
HORSEPOWER
F
ENGINE
/
POWER AVA
ILABUe
/
\
( — ACTUAL HO
REQUIRED F
ISEPOWER
)R FLIGHT.
SO
\
r""^
c
o
SPEEOOF flight (MILES PER HOUR THROUGH THE AIR)
Fia. 16. — Horsepower and speed for level flight.
the airscrew is giving no thrust, and this angle is often referred to as the
" angle of glide for the aeroplane." More correctly it should be referred
to as the " least angle of glide."
The method of calculation of gliding and climbing flight is illustrated
in Fig. 17, which is a diagram of the forces acting on an aeroplane in free
flight but with its flight path inclined to the horizontal.
In horizontal flying it will be assumed that the direction of the thrust is
horizontal, in which case it directly balances the resistance of the remainder
of the aeroplane to motion through the air. In the above diagram this
statement means that T = D. Similarly the weight of the aeroplane is
exactly counterbalanced by the lift on the wings, i.e. L = W. The angle of
incidence of the wings may be varied by adjustment of the elevator, in
which case the thrust would not strictly lie along the wind. If necessary
a slight complication of formula could be introduced to meet this case, but
the effect of this variation is small, and, in accordance with the idea on
THE PEINCIPLES OF FLIGHT
^9
which this chapter is built, is omitted in order to render the main effects
more obvious.
Now suppose that the angle of incidence of the aeroplane is kept
constant, by moving the elevator if necessary, and that the thrust is altered
by opening the throttle of the engine until the aeroplane climbs at an angle
6 as shown. Because the angle of incidence has been kept constant the
relative wind will still blow along the same line in the aeroplane, now in
position oX', but the thrust will not now exactly balance the resistance
)r the hft the weight of the aeroplane.
The relations between weight, speed and thrust may be expressed in
lifferent ways, but the following is the most instructive. If the force W
[be resolved along the new axes of D and L into its components Wi and W2,
it will be seen immediately that Li must exactly counterbalance W2 as
Z 2'
X HORIZONTAL
LINE
Fio. 17.
for horizontal flight. Since the angle of incidence has not been altered for
where u is the velocity of the aeroplane
the cHmb, it follows that— „= -\
U^ Ui'
through the air in horizontal flight, and Mj the velocity when cHmbing.
Since L^ = W2 and W2 is less than W, it will be seen that the velocity of
cHmbing flight is less than that for horizontal flight if the angle of incidence
is unaltered. The relation is easily seen to be
,2
cos^ (1)
Ui'
u*
From the balance of forces along the axis of Di it is clear that Ti = Wi
+ Di, or the thrust is greater than the drag by a fraction of the weight of
the aeroplane. If chmbing at 1 in 6 this fraction is Jth. Since at the
same attitude drag varies as the square of speed, the relation between
thrust, weight and resistance can be put into the form
Ti = Wsin^ + Dcos^ (2)
where D is the aeroplane resistance in horizontal flight.
30
APPLIED AERODYNAMICS
Equation (2) can now be used to show how diagrams 12 and 13 may be
altered to allow for inclined flight. In the first place the ordinates of
Fig. 13, which, after addition of the drag of the body, show the value
of D for many angles of incidence, need to be decreased by multiplying
by cos 6 to give D cos 6. The effect of this multiplication is verv small
as a rule. At 10° the factor is 0*985, and at 20°, 0-940. For a very
steep spiral glide at say 45°, the difference between cos 6 and unity becomes
important, cos 6 being then 0-707.
To the value of D cos 6 is to be added a term W sin 9 in order to obtain
the thrust of the airscrew when climbing at an angle 6. We may then
make a table as below, using figures from Table 1 to obtain the second
column.
TABLE 7. — Theust when climbing.
Speed of flight
(m.p.h.).
Drag in horizontal
flight X cos 0 (lbs.).
W sin fl. fl - 5"
(lbs.).
Airscrew thrust when
climbing at 6° (lbs.).
40
50
60
70
100
608
208
210
253
506
174
174
174
174
174
782
382
384
427
680
The angle of cHmb was chosen arbitrarily at 5°, and to complete the
investigation of the possibiHties of climb Table 7 would be repeated for
other angles. Using Pigs. 14 and 15 for the airscrew as for horizontal
flight, we may now calculate the horsepower required for flight when
climbing, and so obtain the figures of Table 8.
TABLE 8. — HoESEPOWEE when climbing.
Speed of flight
(m.p.h.).
(lbs.).
Thrust. From Table 7.
a.p.m. from Fig. 14
and previous column.
Horse-power. From
Fig. 15 aud previous
column.
40
50
60
70
100
782
382
384
427
680
1600
1170
1220
1340
1760
85
96
116
At the lowest and highest speeds of the table the horsepower required
is far greater than that available, and the figures are not within the range
of Fig. 16.
We may now proceed to plot the horsepower of Table 8 against speed
to obtain a diagram corresponding with Fig. 16. The new curve marked
AiBiCiDi in Fig. 18 compared with ABCDE as reproduced from Fig. 16
shows an increase of nearly 60 h.p. at all speeds due to the chmb at 5°.
The highest speed of flight is shown by the intersection of AiBjCiDi with
FGH at H5. FGH is the horsepower available, and is the same as the
similarly marked curve of Fig. 16. The highest speed is 78-4 m.p.h., and
THE PEINCIPLES OF PLIGHT
81
since the angle 6 is constant along AiBiCiDi the rate of chmb will be
greatest at this point for the conditions assumed. Kate of climb, Vg, is
commonly estimated in feet per minute, and we then have
Max. Vcfor^ = 6*' = 88 X Vmph X sin 0
= 88 X 78-4 X 0-0875
= 604 ft. per min.
The calculations shown in Tables 7 and 8 have been repeated for other
angles of climb and one angle of descent to obtain corresponding curves
zoo
A
V
e
POWER AVAII
ABLE
/
f
4
ISO
ORSt
lOO
'^ / ^
>OWER
Hg^a^
Ho/
>
1
\ x^
^c,
y/^
\
B|
^^^y.^
/EL FLIGMj/
50
w
■"^
^^
C
-"'"^
O
( — — "'"^
30 4-0 SO 60 70 eo 90 lOO
SPEED OF FLISMT (MILES PER HOUR;
Fia. 18. — ^Horsepower and speed for climbing flight.
in Fig. 18. The intersections H_5, Hq, etc., then provide data for Table 9
below.
TABLE 9. — Rate of Climb and Speed.
Angle of climb.
Maximum flight speed
(m.p.h.).
Maximum rate of climb
for given angle.
-5°
101-6
— 783 ft.-min.
0
93
0 „
5°
78-4
+ 604 „
7°
67-5
+ 725 „
8°
59-5
+ 727 „
8-5°
56-2
+ 730 „
^ 8-9°
51-6
+ 702 „
10°
Flight not possible.
Table 9 shows that the rate of cHmb varies rapidly with the flight speed
in the neighbourhood 100 m.p.h. to 80 m.p.h., but that from 65 m.p.h. to
56 m.p.h. the value of rate of climb varies only from 725 to 730. This
illu:?trates the well-known fact that the best rate of cHmb of an aeroplane
- not much affected by small inaccuracies in the flight speed.
The table shows another interesting detail ; the maximum angle of
32
APPLIED AEBODYNAMICS
climb is S^'O, but the greatest rate of climb occurs at a smaller angle.
For reasons connected with the control of the aeroplane an angle of 8° or
thereabouts would probably be chosen by a pilot instead of the 8°-b
shown to be the best.
Diving. — By " diving " is meant descent with the engine on, as
distinguished from a ghde in which the engine is cut off. If the engine be
kept fully on it is found that the speed of rotation of the airscrew rises
higher and higher as the angle of descent increases. There is, however,
an upper limit to the speed at which an aeroplane engine may be run with
safety, and in our illustration an appropriate limit would be 1600 r.p.m.
The speed of rotation corresponding with H_5 was 1550 r.p.m., and it will
be seen that the new restriction will come into operation for steeper
descent. Fig. 14, if extended, would now enable us to determine the thrust
of the airscrew at any speed without reference to the horsepower, but it
will be evident that the Hmits of usefulness of each of the previous figures
have been reached, and an extension of experimental data is necessary to
cover the higher speeds.
The fact that under certain circumstances forces vary as the square
of forward speed of the aeroplane suggests a more comprehensive form of
presentation than that of Figs. 12, 13, 14 and 15, and the new curves of
Figs. 19 and 20 show an extension of the old information to cover the
new points occurring in the consideration of diving. The values of the
extended portion are so small that on any appreciable scale it is only
possible to show the range corresponding with small angles of incidence
and for small values of thrust and horsepower.
TABLE 10. — Airscrew Thrust when diving.
Wr.p.in.
Speed (m.p.h.). V^p^
n = iedc.
Thrust
V^m.p.h.
Prom Fig. 20.
100
120
140
160
160
13-3
11-4
100
00406
00088
-0-0095
-00180
thrust
The curve connecting :^f^ — and speed is shown in Fig. 21.
m.p.h.
Instead of equation (2) will be used the equation
Ti Wsin6> , Di
^ m.p.h. '
m.p.h.
m.p.h.
(3)
The use of Di instead of D cos 6 is convenient now since the drag in
level flight at high speeds is not determined in any other calculation. In
compiling Table 11 some angle of path such as —10° is chosen, and various
speeds of flight are assumed. From these speeds the third column is
THE PEINCIPLES OF PLIGHT
83
calculated and gives one of the quantities of Fig. 19, The value ;^ = O'l 97
(Table 11) occurs at an angle of — 0°']G (Fig. 19), and from the same figure
LIFT
ANGLE OF INCIDENCE
Fia. 19. — Lift and drag of aeroplane at very high speeds.
0.05
DRAG
V^,
mpK
0.0
//
0 0002
SEPOWER
THRUST
/
/ HOP
" mp/i
0.02
O
\J3
mph
O OOOI
0
v.
fim
iph
0
5 1
y / 'p ^
MORS
EPOWER
J THRUST
/ ^mp/l
-o.oa
-^
^^/
Fio. 20.-
— Thius
b and horsepowe
r of airscrew at
very high speeds
.
the corresponding value of ^ is read off as 0*0506. Column 5 of
34
APPLIED AEEODYNAMICS
Table 11 follows from the known weight of the aeroplane and columns
1 and 2, and the last column of ^ is the sum of the preceding
T
columns in accordance with equation (3). The values of ;^ from
Table 11 are plotted in Fig. 21 and marked with the appropriate value
TABLE . — Akgle of Descent and Speed when divino.
D,
T
0
Speed
Li W cos e
V2
Wsln«
V
Angle of path.
(m.p.h.).
V2" \ni
From col. 8 and
V2
by adding cols.
Fig. 19.
4 and 6.
-10°
100
0-197
0-0506
-00348
00168
110
0163
0-0620
-0-0287
00233
-20°
110
0156
0-0525
-0-0670
-00045
120
0131
0-0540
-0-0475
+0-0065
-30»
120
0120
0-0550
-0-0695
-0-0146
130
0103
0-0562
-0-0592
-00030
140
0-088
0-0575
-0-0511
+0-0064
-60®
130
0-059
0-0600
-0-1025
-0-0425
140
0-051
00612
-0-0884
-00272
150
0045
0-0620
-00770
-0-0150
-80°
150
0-015
0-0660
-0-0877
-0-0217
160
0-014
0-0660
-0-0770
-0-0110
-90°
150
0
0-0680
-0-0890
-0-0210
160
0
0-0680
-0-0784
-0-0104
of 0. The intersection at A of the curve 6 = —10° and the curve
thrust
from Table 10 shows the speed at which the aeroplane must be flying in
order that the airscrew shall be giving the thrust required by equation (3).
The results shown in Fig. 21 can be collected in a form which shows
how the resistance of an aeroplane is divided between the aeroplane and
T
airscrew. At A the speed is 110 m.p.h. and the value of tt- is 0*0240,
and hence T=290 lbs. Equation (3) then shows that Di=290 lbs.— W sin 6
=290 lbs. +348 lbs. = 638 lbs. Eepetition of the process leads to Table 12.
TABLE 12.— Speed and Deag
WHEN DIVING.
Angle of descent.
Flight speed
(m.p.h.).
Aeroplane drag
fibs.).
Airscrew drag
(lbs.).
Wing drag (lbs.).
0
10°
20°
30°
60°
80°
90°
93
110
121
130-6
150-5
156
154-6
437
638
789
958
1406
1618
-437
-290
-105
+ 42
+326
+382
167
261
333
427
700
874
Examination of the table shows that a moderate angle of descent is
sufficient to produce a considerable increase of speed. The maximum
THE PEINCIPLEB OF FLIGHT
35
flight speed is reached before the path becomes vertical, but the value is
little greater than that for vertical descent. The terminal speed of our
I ypical aeroplane is 155 m.p.h. With the hmitation placed on the airscrew
liat its revolutions should not exceed 1600 p.m. it will be noticed from
column (4) of Table 12 that the thrust ceases at about 125 m.p.h., and
that at higher speeds the airscrew offers a resistance which is an appreciable
fraction of the total. At the terminal velocity the total resistance is
divided between the airscrew, wings and body in the proportions 19*1 per
cent., 43'7 per cent, and 37*2 per cent, respectively.
If the curve of horsepower of Fig. 20 be examined at the terminal
T . n
velocity it will be found that the value of ^ (—0*016) gives to :^ a value
of 10*4, and the horsepower is then negative. This means that the air-
screw is tending to run as a windmill, and the horsepower tending to drive
Fio. 21. — Angle of descent and speed in diving.
it is about 150. A speed much less than 155 m.p.h. would provide
sufficient power to restart a stopped engine, since 30 h.p. would probably
suffice to carry over the first compression stroke. This means of restarting
an engine in the air is frequently used in experimental work.
Gliding. — In ordinary flying language " gliding " is distinguished from
" diving " by the fact that in the former the engine is switched ofif. If the
revolutions of the airscrew be observed the angle of ghde can be calculated
as before. There is, however, one special case which has considerable
interest, and this occurs when the engine revolutions are just such as to
give no thrust from the airscrew. Fig. 20 shows, for our illustration,
that the revolutions per minute of the airscrew must then be 12*5 times
the speed of the aeroplane in miles per hour. If the revolutions be limited
to 1600 p.m. as before, the highest speed permissible is 128 m.p.h. Fig. 20
shows that the engine would then need to develop about 85 horsepower,
and would be throttled down but not switched off.
The special interest of gUdes with the airscrew giving no thrust will
36
APPLIED AEEODYNAMICS
be seen from equation (2) by putting Ti = 0 when the rest of the equation
gives
•n
(4)
xTr = —tan 9
W
where D is the drag in horizontal flight at the same angle of incidence as
during the glide, and consequently ^^ is the well-known ratio of lift to
ro
/
— N
.
/b
\
•^ tje. RATIO (
>R LIFT TO ORA
3
^
0
/
\
/
so 5 lO 15 20
ANGLE OF INCIDENCE OF WINGS. (DEGftEES)
Fig. 22. — Aeroplane efficiency and gliding angle.
drag. This depends only on angle of incidence, and (4) may be generalised
to
^1=^^=-'-" («)
The negative sign implies downward flight, and we see that the gliding
1 *fii
angle ^ is a direct measure of the -^ — of an aeroplane, i.e. of its
drag
aerodynamic efficiency as distinct from that of the airscrew. In practice
it is not possible to ensure the condition of no thrust with sufficient accuracy
THE PEINCIPLES OF PLIGHT 37
for the resulting value of ^^ *o be good enough for design purposes. It is
better as an experimental method to test with the airscrew stopped and
to make allowance for its resistance. The -prr of the aeroplane and
uft ^
airscrew is then the quantity measured by —tan 0. The least angle of glide
is readily calculated from a curve which shows the ratio of hft to drag for
the aeroplane. The curve given in Fig. 22 is obtained from the value of
the body drag and the numbers used in plotting Figs. 12 and 13.
The value of drag for the aeroplane is least when the ordinate of the
curve in Fig. 22 is greatest, and will be seen to be only ^75 of the hft.
If then an aeroplane is one mile high when the engine is throttled down
to give no thrust, it will be possible to travel horizontally for 9-3 miles
before it is necessary to aHght. Should the pilot wish to come down more
steeply he could do so either by increasing or decreasing the angle of
incidence of his aeroplane. For the least angle of ghde Fig. 22 shows the
angle of incidence to be about 7 degrees, and by reference to Fig. 12 it
will be seen that the flying speed is between 50 and 60 m.p.h., probably
about 54 m.p.h. To come down in. a straight line to a point 6 miles away
from the point vertically under him from a point a mile up, the pilot could
choose either the angle of |° and a speed of about 90 m.p.h. or an angle of
15° and a speed of about 42 m.p.h. From Fig. 12 it will be seen that 15°
is hear to the greatest angle at which the aeroplane can fly, and it will be
shown later that the control then becomes difficult, and for this reason
large angles of incidence are avoided. If a pilot wishes to descend to some
point almost directly beneath him, he finds it necessary to descend in a
spiral with a considerable " bank " or lateral incUnation of the wings of
the aeroplane. It is not proposed to analyse the balance of forces on a
banked turn at the present stage, but it may here be stated that for the
same angle of incidence of the wings an aeroplane descends more rapidly
when turning than when flying straight. For an angle of bank of 45°
the fall for a given horizontal travel is increased in the ratio of ^2 : 1.
Soaring. — In considering the motion of an aeroplane it has so far been
assumed that the air itself is either still or moving uniformly in a horizontal
direction, so that chmbing or descending relative to the air is equal to
chmbing or descending relative to the earth. The condition corresponds
with that of the motion of a train on a straight track which runs up and
down hill at various points. If the air be moving the analogy in the case
of the train would lead us to consider the motion of the train over the ground
when the rails themselves may be moved in any direction without any
eontrol being possible by the engine driver. If the rails were to run
backwards just as quickly as the train moved forward over them, obviously
the train would remain permanently in the same position relative to the
,^iound. If the rails move more quickly backwards than the train moves
lorward, the train might actually move backwards in spite of the engine
driver's efforts. Of course we know that such things do not happen to
38
APPLIED AEEOBYNAMICS
trains, but occasionally an aeroplane flying against the wind is blown
backwards relative to an observer on the ground. Mying with the wind
the pilot may travel at speeds very much greater than those indicated in
our earlier calculations. The motion of the aeroplane may be very
irregular, just as would be the motion of the train if the rails moved side-
ways and up and down as well as backwards and forwards, with the
difference that the connection between the air and aeroplane is not so rigid
as that between a train and its rails. The motion of an aeroplane in a
gusty wind is somewhat complicated, but methods of making the necessary
calculations have aheady been developed, and will be referred to at a more
advanced stage.
If the rails in the train analogy had been moving steadily upwards
with the train stationary on the rails, the train might have been described
as soaring. The train would be Hfted by the source of energy lifting the
rails. Similarly if up-currents occur in the air, an aeroplane may continue
to fly whilst getting higher and higher above the ground, without using
any power from the aeroplane engine. This case is easily subjected to
numerical calculation. Lord Rayleigh and Prof. Langley have shown that
soaring may be possible without up-currents, if the wind is gusty or if it
has different speeds at different heights. Such conditions occur frequently
in nature, and birds may sometimes soar under such conditions. Continued
flight without flapping of the wings usually occurs on account of rising
currents. These may be due to hot ground, or round the coasts more
frequently to the deflection of sea breezes by the chffs near the shore. Gulls
may frequently be seen travelling along above the edges of cliffs, the path
following somewhat closely the outline of the coast. Other types of soaring
are scarcely known in England.
To calculate the upward velocity of the air necessary for soaring in the
case of the aeroplane already considered, it is only necessary to refer back
to the gliding angles and speeds of flight. Values obtained from Figs.
12 and 22 are collected in Table 13 for a weight of 2000 lbs.
TABLE 13.— SoAEiNG.
Angle of incidence,
from Fig. 12-
Speed of flight
(m.p.h.), fromFig. 12.
Gliding angle, from
Fig. 22.
Vertical velocity of
fall with engine cut
off (m.p.h.).
17°-5
5°0
3*0
-0«'-2
40
50
60
70
100
1 in 3-3
1 in 9-2
1 in 9-0
1 in 7-9
1 in 3-95
121
5-4
6-7
8-8
25-3
The figures in the last column of Table 13 are readily obtained from
those in columns 2 and 3. At 60 m.p.h. and a gliding angle of 1 in 9 the
falling speed is %^- m.p.h., i.e. 0*7 m.p.h. as in column 4. The least velocity
of rising wind is required at a speed just below that of least resistance,
and in this case amounts to about 5*4 m.p.h. or nearly 8 feet per second.
. Winds having large upward component velocities are known to exist.
i
THE PEINCIPLES OF FLIGHT 39
In winds having a horizontal component of 20 m.p.h. an upward velocity
of G or 7 m.p.h. has been recorded on several occasions.* In stronger
winds the up -currents may be greater, but in all cases they appear to be
local. One well-authenticated test on the climbing speed of an aeroplane
shows that a rising current of about 7 miles an hour existed over a
distance of more than a mile. The chmbing speed of the aeroplane had
been calculated by methods similar to those described in the earlier pages
of this book and found to be somewhat less than 400 feet per min. ; the
ireneral correctness of this figure was guaranteed by the average perform-
ance of the aeroplane. On one occasion, however, the recording baro-
graph indicated an increase of 1000 feet in a minute, and it would appear
that 600 feet per minute of this was due to the fact that the aeroplane
was carried bodily upwards by the air in addition to its natural climbing
rate. At 60 miles an hour the column traversed per minute is a mile, as
already indicated.
The possibility of soaring on up-currents for long distances does not
seem to be very great. It will be noticed, from the method of calculation
given for Table 13, that the speed of the up-current required for supporting
a flying machine at a given gliding angle is proportional to the flying
speed. Hence birds having much lower speeds can soar in less
strong up-cuirents than an aeroplane. The local character of the
up-currents is evidenced by the tendency for birds when soaring to keep
over the same part of the earth.
The Extra Weight a given Aeroplane can carry, and the Height to
which an Aeroplane can climb. — So far the calculations have been
made for a fixed weight of aeroplane and for an atmosphere
as dense as that in the lower reaches of the air. It will often
happen that additional weight is to be carried in the form of extra
passengers or goods. Also during warfare, in order to escape from hostile
aircraft guns, it may be necessary to climb many thousands of feet above
the earth's surface. The problem now to be attacked is the method of
estimating the effects on the performance of an aeroplane of extra weight
and of reduced density. The greatest height yet reached by an aeroplane
is about 0 miles, and at such height the barometer stands at less than 10 ins.
of mercury ; it is clear from the outset that the conditions of flight are
then very different from those near the ground. In order to climb to such
heights the weight of the aeroplane is kept to a minimum and the reserve
horsepower made as great as possible. The problem is easily divisible
into two distinct parts, one of which relates to the power required to
support the aeroplane in the air of lower density and the other of which
deals with the reduction of horsepower of the engine from the same cause.
The latter of the two causes is of the greater importance in limiting the
height of climb.
It has already been pointed out in connection with Fig. 1 2 that the
lifting force on any aeroplane varies as the square of the speed so long
is the angle of incidence is kept constant. Now suppose that the weight
* Report of the Advisory Committee for Aeronautics, 1911-12, p. 315.
40
APPLIED AERODYNAMICS
of the aeroplane is increased in the ratio M^ : 1 by the addition of load
inside the body, i.e. where it does not add to the resistance directly. In
order that the aeroplane may Uft without altering its angle of incidence,
it is necessary to increase the speed in the proportion of M : 1. This in-
crease will apply with equal exactness to the revolutions of the airscrew,
and the simple rule is reached that if an aeroplane has its weight increased
in the ratio M^ • 1 and its speeds in the ratio M : 1, flying will be possible
at the same angle of incidence for both loadings.
From the previous analysis it wiU be reaUzed that the increase of
speed necessary to give the greater lift involves an increase in the resistance
proportional to M^ and to balance this an increase of propeller thrust also
proportional to M^, The method of finding horsepower shows that the
increased horsepower is in the ratio of M^ : 1 to the old horsepower. Leav-
ing the variation of density alone for the moment, new calculations for
other loads could be made as before. Since Fig. 15 exists for the old
loading a simpler method may be followed.
The curves OP and C1C2C3C4 of Fig. 15 are reproduced in Fig. 23 below,
with an increase of scale for the airscrew revolutions. The two further
curves of Fig. 23 marked 3000 lbs. and 4000 lbs. are produced as shown
in Table 14 in accordance with the laws just enunciated.
TABLE 14. — Inobbasbd Loading.
Weight -
2000 lbs.
Weight =
3000 lbs.
Weight =
4000 lbs.
R.p.m.
from curve
OC,C,C^C,.
Horsepower
from curve
cc^c,c^c„
R.p.m.
from col. (1),
by multiplying
by -1/8OOO
^ 2000
i.e. by 1-226.
Horsepower,
from col. (2),
by multiplying
, /3000\3/2
^ V2OOO/
i.e. by 1-84.
E.p.m.
from col. (1),
by multiplying
by a/4000
^ 2000 .
i.e. by 1-414.
Horsepower
from col. (2),
by multiplying
by (4000\3/2
"^ V2OOO/
i.e. by 2-88.
930
960
1000
1060
1100
1200
40
f41\
/46\
154/
(S)
/57\
175/
71
1140
1166
1225
1286
1346
1470
73
/75)
183/
/83\
199/
/ 94\
1118/
fl06j
1138/
131
1316
1346
1416
1490
113
|116\
\127f
fl27l
\163/
144
Fig. 23 shows that the aeroplane would still fly with a total load of
4000 lbs. At top speed the airscrew speed has fallen from 1525 to
1470 r.p.m. owing to the extra loading. It is easy to calculate the maximum
load which might be carried, since Fig. 23 shows that the airscrew would
in the limiting case be making about 1400 r.p.m. and delivering 135 horse-
power. If, then, we find 4/^^, i.e. 2-28, and multiply 2000 lbs. by this
number, 4560 lbs. will be obtained as the limiting load which this aeroplane
can carry. It will be seen that each 1000 lbs. of load carried now requires
about 30 horsepower.
THE PEINCIPLES OF FLIGHT
41
Corresponding calculations based on Fig. 23 in an exactly analogous
way to those of Table 14 on Fig. 16 have been made. The details are not
I 50
800 IPOO 1,200 l,-400 Ij500 ' l,800
AIRSCREW REVOLUTION'S.
Fia. 23. — Effect of additional weight on horsepower and airscrew revolutions.
given, but the results are shown in Fig. 24, and it can be seen how the speed
of flight is affected by the increased loading.
150
B
f="
]
^
*^
:^
MOR
E POWER
lOO
\
^.OOO LBS
^"/
/^
__^
y^
S.OOOLBS
^^
D
SO
2.0C
OLBs\
^,,,„.-^^'^
c"
—^
O
30 -VO 50 60 70 80 90
SPEED OF FLIGHT M.P.H.
Fig. 24.— Effect of additional weight on the speed of flight.
The curves FGH and BCDE are reproduced from Fig. 16, whilst those
marked 3000 lbs. and 4000 lbs. are the results of the new calculations.
The first very noticeable feature of Fig. 24 is the small difference of
42 APPLIED AERODYNAMICS
top speed due to doubling the load, the fall being from 93 m.p.h. to 86
m.p.h. The effect on the slowest speed of flight is very much greater, for
the least possible speed of steady horizontal flight is 64 m.p.h. with a load
of 4000 lbs., instead of 40 m.p.h. with a load of 2000 lbs. The difficulties
of landing are much increased by this increase of minimum flying speed.
Fig. 24 can be used to illustrate a point in the economics of flight. The
subject will not be pursued deeply here, since more comprehensive methods
will be developed later. If it be decided that a speed of 90 m.p.h. is
desirable for a given service, it is seen that 2000 lbs. can be carried for an
expenditure of 129 horsepower, 3000 lbs. for 138 horsepower, and 4000 lbs.
for 152 horsepower. If these numbers are expressed as " horsepower per
thousand pounds carried," they become 65, 46 and 38, showing a progressive
change in favour of the heavy loading. The difference is very great, and
obviously of commercial interest. Variation of loading is not the
only factor leading to economy, but the impression given above from a
particular instance may be accepted as typical of the aeroplane as we now
know it.
It should be remembered that the present calculations refer to
increased load in an existing aeroplane. Any new design for an original
weight of 4000 lbs. would differ from the prototype probably both in size
and in the power of its engine.
|Flight at Altitudes ot 10,000 £eet and 20,000 feet.— At a height of
10,000 feet the density of the air is relatively only 0'74 of that near the
ground, and we now inquire as to the effect of the change. The experi-
mental law is a simple one, and states that at the same attitude and speed
of flight the air force is proportional to the air density.
The new performance at 10,000 ft. may be calculated from that near
ground-level by a process somewhat analogous to the one followed for
variation of weight. At the same angle of incidence it is possible to produce
the same lift in air of different densities by changing the speed, and the law
is that 0-V2 * is constant during the change.
The power required is not the same since the speed has increased as
\/-, and hence the horsepower has also increased as \/-. We then
get the following simple rule for the aeroplane and airscrew, that flight
at reduced density is possible at the same angle of incidence if the speed
of flight and the speed of rotation of the airscrew are increased in proportion
to /v/ ; the horsepower required for flight is also increased in the pro-
(T
portion \/-.
(T
Table 15 shows how the calculations are made.
Prom columns 3 and 4 of Table 15 the curve AiBiCi of Fig. 25 is drawn
to represent the horsepower necessary for flight at 10,000 feet. The
original curve for unit density is shown as ABC.
* o- is the relative density, while p is used for the mass per unit volume of the fluid or
absolute density.
THE PBINCIPLES OF FLIGHT
43
TABLE 15. — Flying at Gbeat Heights.
Flight near the ground
Flight at 10,000 ft.
Flight at 20,000 ft.
where the relative density
where the relative density
where the relative density
is unity.
is 0-74.
is 0-5S5.
B.p.m.
Horsepower,
R.p.m.
Horsepower
column (1)
column (2)
column (1)
column (2)
from Table 14, from Table 14,
multiplied
1
multiplied
1
multiplied
1
multiplied
column (1). column (1).
'"Ti
"V,
""7-^
by-i-
i.e. by 116.
i.e. by 116.
ie. by 1-37.
1.6. by 1 37.
930
40
1080
46
1260
54-5
950
(41\
(45/
1100
/47\
\52}
1295
/56\
161/
1000 : {^)
1160
/52>
\63/
1360
(61\
\73|
lom 1 {^}
1220
/59\
174/
—
—
1100
m
1280
f66\
— —
1200
71
1390
82
— —
800
1,800
I.OOO l,200 l,-4-00 1,600
AIRSCREW REVOLUTIONS.
Fig. 25. — Effect of variation of height on horsepower and airscrew revolutions.
Variation of Engine Power with Height. — The horsepower of an
engine in an average atmosphere falls off more rapidly than the density
and curves of variation have been derived experimentally. For a height
of 10,000 ft, the horsepower at any given speed of rotation is found to be
0-09 of that where the density is unity. The curve OiPi of Fig. 25 is
obtained from OP by multiplying the ordinates by 0-69. The pair of curves
44
APPLIED AEEODYNAMICS
OiPi, AiCiBi now refers to flight at 10,000 ft. and the revolutions of the
engine at top speed, i.e. at B, will be seen to be a little less than those at
the ground. The reserve horsepower for climbing will be seen to be much
reduced, and is little more than half that at the low level.
There must come some point in the ascent of an aeroplane at which
a new curve for OP will just touch the new curve for ABC, and the density
for which this occurs will determine the greatest height to which the aero-
plane can climb. This point is technically known as the " ceiling." A
repetition of the calculation for a height of 20,000 ft. shows this height
as being very near to the ceiHng. The drop in airscrew revolutions at top
speed (Bii) is now well marked.
The corresponding curves for flight speed and horsepower have been
calculated and are shown in Fig. 26. The curves for " horsepower required"
AVAILABLE
•4-0 SO 60 70 80
SPEED OF FLIGHT M.P.H.
Fig. 26. — Effect of height on the speed of flight.
and speed are obtained from those at ground-level (Fig. 16) by multiplying
both abscissae and ordinates by —~ . The horsepowers at maximum and
Vff
minimum speeds are given by the points Aj, Bj, An and B^ of Fig. 25
and fix two points on each curve of horsepower available, and hence
fix-; the maximum and minimum speeds. The speeds at ground-level,
10,000 ft. and 20,000 ft. are found to be 93 m.p.h., 89 m.p.h. and 79 m.p.h.,
showing a marked fall with increased height.
The increase of the lowest speed of level steady flight is of little im-
portance since landing does not now need to be considered.
Another item in the economics of flight is illustrated by Fig. 26. The
load carried is 2000 lbs. at all heights, but at a speed of 90 m.p.h. the
horsepowers required are 129 near the ground, 99 at 10,000 ft. and 82 at
20,000 ft., i.e. 64, 49 and 41 horsepower per 1000 lbs. of load carried. The
intense cold at great heights such as 20,000 ft. must be offset against the
THE PEINCIPLES OP FLIGHT
45
obvious advantages of high flying in reduced size of engine and in petrol
consumption.
This completes the general exposition of those properties of an aero-
plane which are generally grouped under the heading " Performance."
Before passing to the more mathematical treatment of the subject a short
account will be given of the longitudinal " balance "of an aeroplane in
flight.
Longitudinal Balance. — The fmiction of the tail of an aeroplane is
to produce longitudinal balance at all speeds of steady flight. In the search
for efficient wings it has been found that the best are associated with a
property which does not lend itself to balance of the wings alone. In the
earher part of the chapter we have considered the forces acting on a wing
and on an aeroplane without any reference to the couples produced, and
the motion of the centre of gravity of the aeroplane is correctly estimated
in this way provided the motion can be maintained steady. We now
proceed to discuss the couples called into play and the method of dealing
with them.
}Centre o! Pressuie. — ^Pig. 27 shows a drawing of a wing with the position
of the resultant force marked on it at various speeds of steady flight. The
Pio. 27- — Resultant wing force and centre of pressure.
lengths of the lines show the magnitude, and a standard experiment fixes
both the magnitude and position. The intersection of the line of the re-
sultant and the chord of the section is called the centre of pressure, and
at 100 m.p.h. the intersection, CP of Fig. 27, occurs at 0*58 of the chord
from the leading edge. The most forward position of the centre of pressure
occurs at about 50 m.p.h., and is situated at 0*32 of the chord from the
leading edge.
46
APPLIED AERODYNAMICS
One of the conditions for steady flight requires that the resultant force
on the whole aeroplane shall pass through the centre of gravity of the
SPEED M.P.H
30 40 50 60 70 80 90 100
4-0 SO 60 70 80 90 lOO
SPEED M.P.H.
[j^Uitt^^J FiG« 28. — Longitudinal balance.
aeroplane, and it is impossible to find any point near the wing for
which the condition is satisfied at all speeds. It will be supposed that
THE PRINCIPLES OP PLIGHT
47
the centre of gravity is successively at the points A, B and C of Fig. 27,
and it will be shown how to produce the desired effect by means of a tail
plane with adjustable angle of incidence. Table 16 shows the values of
resultant force and the leverages about the point A in terms of the chord
of the aerofoil c, and finally the couple in terms of the previous quantities
tabulated.
TABLE 16.— Wing Moments.
Flight speed
(m.p.h.).
Angle of inci-
dence of wings.
Resultant
force (lbs.).
Distance from A.
Ma
W5
40
50
60
70
100
17°-5
4"»-9
3°0
2100
2020
2020
2020
2060
-0-135 0.
-0 125 c.
-0-146 0.
-0-180 c.
-0-342 0.
-0-177
-0-101
-6-082
-0-074
-0-070
The moments for the points B and C are obtained by a repetition of
the process followed for A. The resulting figures have been used to draw
the curves of Fig. 28, which are marked A, B, C.
These couples are to be balanced by the tail plane, and the first point to
be considered is the effect of the down current of air from the wings on the
air forces acting on the tail plane. The angle through which the air is
deflected is called the " angle of downwash," and is denoted by " e ."
Downwash. — In the consideration of wing lift it was seen that the down-
ward velocity of the air is directly related to the lift on the wing. Ex-
15°
/
ANGLE OF WIND
TOWING CHORD
BUTBEHINDTHE
/
/
<
5
MVj
/
€
A
O
X
n
1
>
^c
O 10° 20°
/nc//n<Bt/on of Chord
of l/V/n^ OL .
Fio. 29. — Downwash from wings
perimentally it is found to be very nearly proportional to the lift for
various angles of incidence, and a typical diagram showing " downwash "
ia given in Fig. 29.
48
APPLIED AEKODYNAMICS
The upper straight Une AB of Fig. 29 shows the angle of the chord
of the wings relative to the air in front of the wings, whilst CD shows the
angle at the tail. The chord of the tail plane will not usually be parallel
to the chord of the wings, -and its setting is denoted by a<. Pig. 30 will
make the various quantities clear.
D CCt TAILSETTING
ANGLE.
FiQ. 30.
For an angle of incidence a at the wings we have at the tail an angle of
wind relative to AP of a — e, and the tail plane being set at an angle c^ to
AP, for the angle of incidence of the tail plane is given by the relation
a' = a — € + Oj (6)
Tail planes are usually symmetrical in form, and the chord is taken as
0.08
0.07
O. 06
O. 05
LIPT
0.04-
O.03
0.0 2
O.Ol
L'
npjt.
/
TAIL PLANE
CHORD
)F 55 SQ.FT
4- FT.
/
/
/
/
0'
m.pJt^
/
__...,.--^
O
0,004
0.003
DRAG
tn.ph
O.0O2
O.OOI
IO«
20"
cC' INCLINATION OF TAIL PLANE TO WIND
Fig. 31. — Lift and drag of tail plane.
the
median line of the section. Fig. 31 shows curves of :^
tail Kft
and
'in.p.h.
-— ^ for a typical tail plane suitable for an aeroplane weighing 2000 lbs.,
V2ni.pji. . . . -
of 55 sq. feet area and a chord of 4 feet. As nothing is lost in the prmciple
of balance by the omission of terms depending on the change of centre of
p
THE PEINOIPLES OF FLIGHT
49
^ issure of the tail plane, such terms will be ignored, and the force on the
ail plane mil always be assumed to pass through the point P.
If the distance from A to P be denoted by ?a the equation for moment
if the tail about A is
moment =— ^^{L' cos (a — e) + D' sin (a— c)}
>i' more conveniently
moment f L' , , , D' . , J
•"A * m.p.h. ( V m.p.h. V ^m.p.h. )
The calculation proceeds as in Table 17.
TABLE 17.— Tail Moments.
* «/ = 0,
h " 2'7c.
* m.p.h.
a
a — €
a'
L'
D'
Y'
cV-*
1^ 40
17° -5
8° -6
8° -6
0 0504
0 0006
-0134
60
8°-7
3°0
3°-0
00181
0-0002
-0-049
60
4°-9
1°0
l°-0
0-0055
0-0002
-0-015
70
S'O
0°0
0°0
0-0000
0-0002
-0-000
^^100
-0*2
— 1°-6
-l*>-6
-0-0091
0-0002
+0-025
KWhen the aeroplane is in equilibrium the couple given in the last column
ust be equal to, but of opposite sign to, that on the wings. Couples due
to the tail are therefore plotted in Fig. 28 with their sign reversed. The
intersections of the various curves then show the speeds of steady flight
for various tail settings.
The differences between Figs. 28a, 28&, 28c, correspond with the
differences in the position of the centre of gravity, i.e. with A, B and C.
They are considerable and important.
Fig. 28a shows that equilibrium is not possible within the flight range
40 m.p.h. to 100 m.p.h. until the tail setting is less than —3°, the speed
being then 100 m.p.h. For 0^=— 5° the speed for equilibrium is 65 m.p.h.,
and for a< =— 10°, 49 miles per hour.
Fig. 28b shows that the aeroplane is almost in equilibrium at all speeds
' »r the same setting, a^ = 0, the statement being most nearly correct at
speeds of 70 m.p.h. to 100 m.p.h. To change from 50 m.p.h. to 41 m.p.h.
the tail-plane setting needs to be altered from +1° to —2°.
Fig. 28c is to a large extent a reversal of Fig. 28a. The angle of tail setting
must exceed +2° to bring the equihbrium position within the flight range
40 m.p.h. to 100 m.p.h At a,=-f2° the speed is 102 m.p.h., for af=+5°
it is 81 m.p.h., and for 04= +10° it is 60 m.p.h. To reduce the speed
further would need still greater angles, and the tail plane passes its critical
iiiglo. It might not be possible in this case to fly steadily at 45 m.p.h.
i he same might be true for a position of the centre of gravity of the aero-
plane further forward than A.
50
APPLIED AERODYNAMICS
If we regard the variation of tail setting as a control, we see that both
A and C are positions of the centre of gravity which lead to insensitiveness,
whilst position B leads to great sensitivity. An example is then reached of
a general conclusion that greatest sensitiveness is obtained for a particular
position of the centre of gravity, and that for ordinary wings this point is
about 0*4 of the chord from the leading edge. We shall see that this
conclusion is not greatly modified if the tail plane be reduced in area.
Consider, now, the aeroplane with its centre of gravity at A, flying at
an angle of incidence of 3°'0 and a speed of 70 m.p.h., but with a tail setting
of —10°. The wings are then giving a couple — O-OScV^, which tends to
SPEED M.RM
-4-0 50 60 70 80 90 lOO
Fig. 32. — Longitudinal balance with small tail plane.
decrease the angle of incidence and to put the aeroplane in a condition!
suitable for higher speed, whereas the equihbrium position for this taill
setting is at a lower speed. The tail is, however, exerting a couple of!
-\-0'14cV^, and this tends in the opposite direction and overcomes the
couple due to the wings. It is almost certain that the aeroplane would be
stable and settle down to its speed of 49 m.p.h. if left to itself with the
tail plane fixed at —10°.
Fig. 28c shows the reverse case ; the wing moment being greater than the
tail moment, the aeroplane would be unstable. It is not proposed to
discuss stabihty in detail here, but it should be noted that the simple
criteria now employed are only approximate, although roughly correct.
It can now be seen that greatest sensitivity to control occurs when the
k
THE PRINCIPLES OF FLIGHT 51
stability is neutral ; putting the centre of gravity forward reduces the
sensitivity and introduces stability, whilst putting the centre of gravity
back reduces the sensitivity and makes the aeroplane unstable.
Tail Plane of DiflEerent Size. — For positions A and B of the centre of
gravity of the aeroplane calculations have been made for a tail area of
85 sq. feet instead of 55. The effect is a reduction of the moment due to
the tail n the proportion of 35 to 55 for the same tail setting and aeroplane
speed. The results are shown in Fig. 32. For neither positions A nor B
is the character of the diagram greatly altered, the chief changes being the
smaller righting couple for a given displacement, as shown by the smaller
angles of crossing as compared with Fig. 28. A tail-setting angle of
—10° with position A now only reduces the speed to 58 m.p.h., and it is
probable that the tail plane would reach its critical angle at lower speeds
of flight.
For position B the diagram shows a smaller restoring couple at low
speeds and a somewhat greater disturbing couple at high speeds.
Small tail planes tend towards instability, but the effect of size is not
so marked as the effect of the centre of gravity changes represented by
A, B and C. The control may not be sufficient to stall the aeroplane when
its centre of gravity is at A. This tends to safety in flight.
Elevators. — ^Many aeroplanes are fitted with tail planes which can be
set in the air. The motions provided for this purpose are slow, and the
control is normally taken by the elevators. The effect of the motion of
the elevators is equivalent to a smaller motion of the whole tail plane,
and Fig. 33 shows a typical diagram for variation of Hft with variation of
angle of elevators, the lift being the only quantity considered of sufficient
importance for reproduction.
The ordinate of Fig. 38 is the value of ^ for a tail plane and elevators
of 55 sq. feet area, of which total the elevators form 40, per cent. The
abscissae are the angles of incidence of the tail plane, and each curve corre-
sponds with a given setting of the ^levators. The angle of the elevators
is measured from the centre Hne of the tail plane, and is positive when
the elevator is down, i.e. making an angle of incidence greater than the
tail plane. For elevator angles between —15° and +15° the curves
are roughly equally spaced on angle, but after that the increase of lift
with further ncrease of elevator angle s much reduced.
The diagram may be used for negative settings by changing the signs
of both angles and of the lift. This foUows because the tail plane has a
symmetrical section.
From the diagram at A, it wiU be seen that an elevator setting of 5°
produces an^of 0*015, and this would also be produced by a movement of
the whole tail plane and elevators through 2°-6 (B, Fig. 33). For this par-
ticular proportion of elevator to total tail surface the angle moved through
by the elevator is then about twice as great for a given lift as the movement
of the whole tail surface. Variations of tail-plane settings of 10° were seen
to be required (Fig. 28) if the centre of gravity of the aeroplane was far
52
APPLIED AEEODYNAMICS
forward, and this would mean excessive elevator angles, an angle of over
20° being indicated at C for +10°. These elevators are large, and it will
be seen that an aeroplane may be so stable that the controls are not suffi-
cient to ensure flight over the full range otherwise possible. For the centre
o£ gravity at position B, Eig. 28, the elevator control is ample for all
purposes.
o.io
20<=
•O.04.
-0.06
-0.08
Fig. 33. — Lift of tail plane and elevator for different settings.
Effort necessary to move the Elevators.— The muscular effort required of
the pilot is determined by the moment about the hinge of the forces on the
elevator, and it is to reduce this effort that adjustable tail planes are used.
If it be desired to fly for long periods at a speed of 70 m.p.h. the tail plane
is so set that the moment on the hinge is very small. For large aeroplanes
balancing of controls is resorted to, but there is a limit to the approach
to complete balance, which will ultimately lead to relay control by some
mechanical device. The mmediate scope of this section will be limited
to unbalanced elevators in which the size is fixed at 40 per cent, of the total
tail plane and elevator area.
It has been seen that the lift on the tail is the important factor in longi-
tudinal balance, and so we may usefully plot hinge moments on the basis
of lift produced. In the calculation a total area of 55 sq. feet will be
assumed so as to compare directly with the previous calculations on tail
setting.
T /
The curves of Fig. 34 may be used for negative values of ^- if M^ and
V2
the tail incidence are used with the reversed sign.
THE PEINCIPLES OF FLIGHT
53
As there are now two angles at disposal another condition besides that
of zero total moment must be introduced before the problem is definite.
The extra condition will be taken to be that which puts the aeroplane
" in trim " at 70 m.p.h., this expression corresponding with flight with no
force on the control stick. The force on the control stick being due to the
moment of the forces on the elevators about its hinge, the condition of
" trim " is equivalent to zero h nge moment.
O.OI
0
o
I^
O.
OS
L'
V2
v^-^
:::::
■■^v^
^~
.^
-0,01
^
^
^,
^^*^-N
s.
N,
^
V2
\
N
fAILINC
,^-100
idW;e
1\
\
^
\
0.02
s
5°
\°
Xs
>
]\o°
EL
TOTA
EVATOI
L ARE
? ARE,
\ 55 S
\ 22 S
q.FT.
Q.FT.
O.IO
Fig. 34. — Hinge moment of elevators.
For position A of the centre of gravity of the aeroplane the forces on
the control stick are worked out in Table 18.
TABLE 18. — Forces on Control Stick.
Speed
M,
1 e\'^
L'
M»
Ma
Force on pilot's
(in.p.h.).
Table 16.
y.
V=
hand.
40
-0-177
-0 065
-0°-9
0-0155
25
12-5 lbs. pull
50
-0101
-0037
-6"'-5
0-0030
8
4
60
-0-082
-0 030
-8''-5
0-0010
4
2
70
-0074
-0-027
-9° -5
0-0000
0
0
100
-0070
-0026
-ir-1
-00010
-10
5 lbs. push
The value of -^ is taken from Table 16, and from it ..— for the tail
is calculated by dividing by Z^j (2*7 c). From Fig. 34 we then find that for
T »
^Vg =—0-027 (70 m.p.h.), the hinge moment is zero if the tail incidence is
— 9°-5. Equation (6) and the figures m column (3) of Table 17 then show
the tail setting to be — 9°'5, and the angles of incidence at other speeds to be
those given in column 4 of Table 18. From columns 3 and 4 of Table 18..
54
APPLIED AEKODYNAMICS
Ma
the values of :^ can be determined by use of Fig. 34 (see column (5),
V2
M,
Table 18). M;^ is easily calculated from :p^, and the force on the pilot's
hand is then calculated by assuming that his hand is 2 feet from the pivot
of his control stick. A positive moment at the elevator hinge means a
pull on the stick.
Before commenting on the control forces the results of similar calcula-
tions for positions B and C of the centre of gravity of the aeroplane are
given in Table 19 in comparison with those for A.
TABLE 19. — FoEOES on Control Stick for Different
Positions of Centre of Gravity.
Speed (m.pJi.).
A.
B.
C.
40
60
60
70
100
12-5 lbs. pull
4 M
2
0
6 lbs. push
0
3 lbs. push
2 „
0 „
5 lbs. pull
16 lbs. push
5
0 „
20 lbs. pull
Consider position C first ; at 100 miles per hour the pilot is pulling
hard on his control stick. It has already been seen that the aeroplane is
unstable with the centre of gravity at C, and one result of this is a tendency
to dive without conscious act of the pilot. The result of a dive is an
increase of speed, and Table 19 shows that an increase of pull may be ex-
pected. At a moderate angle of dive the pull may become so great that
the pilot is not strong enough to control his aeroplane, which may then
get into a vertical dive or possibly on its back. A skilful pilot can recover
his correct flying attitude, but the aeroplane in the condition represented
by C is dangerous.
Position A shows the reverse picture ; the aeroplane is stable and does
not tend to dive without conscious effort by the pilot. It needs to be
pushed into a dive, and if the force gets very great owing to increase of
speed it automatically stops the process.
The aeroplane which is lightest on its controls is still that with the
centre of gravity at B, but it is further clear from Table 19 that an im-
provement would be obtained by a choice of centre of gravity somewhere
between A and B.
(ii) Forces on the Float op a Flying Boat
A diagram illustrating the form of a very large flying boat hull is shown
in Fig. 85, the weight of the flying machine being 32,000 lbs. The
design of a flying boat hull has to provide for taxying on the water prior
to flight and for alighting. When once in the air the problem of the motion
of a flying boat differs little from that of an aeroplane, the chief difference
II
THE PEINCIPLES OF FLIGHT
56
ing that the airscrews are raised high above the centre of gravity in
order to provide good clearance of the airscrews from waves and any
green water which might be thrown up. The present section of this
chapter is directed chiefly to an illustration of the forces and couples on
a flying boat in the period of motion through the water.
Experiments on flying boat hulls have usually been made on models
at the William Froude National Tank at Teddington, but in one instance
a flying boat was towed by a torpedo-boat destroyer, and measurements
of resistance and inclination made for comparison with the models. The
comparison was not complete, but the general agreement between model
and full scale was satisfactory.. Such phenomena as the depression of the
bow due to switching on the engine and " porpoising " are reproduced in
the model with sufficient accuracy for the phenomena to be kept under
control in the design stages of a flying boat.
In making tests of floats in water, Froude's law of corresponding speeds
^^is used, since the greater part of the force acting on the float arises from
THRUST -^
AIRSCREW AXIS
DATUM LINE
Fio. 35. — Flying boat hull.
the waves produced, and if the law be followed it is known on theoretical
grounds that the waves in the model will be similar to those on the full
scale. The law states that a scale model should be towed at a speed equal
to the speed of the full scale float multiplied by the square root of the scale.
A one-sixteenth scale model of a flying boat hull which taxies at 40 m.p.h.
will give the same shape of waves at 10 m.p.h. The forces on the full scale
ate then deduced from those on the model by multiplying by the square of
the scale and the square of the corresponding speeds, i.e. by the cube of
the scale. Similarly, moments vary as the fourth power of the linear
dimensions for tests at corresponding speeds.
As the float is running on the surface of the water, the forces on it
depend on the weight supported by the water as well as on the speed and
inclination of the float, and this complexity renders a complete set of
experiments very exceptional. The full scheme of float experiments
which would eliminate the necessity for any reference to the aerody-
namics of the superstructure would give the lift, drag and pitching moment
of a float for a range, of speeds and for a range of weight supported. From
56
APPLIED AEEODYNAMICS
such observations and the known aerodynamic forces and moments on the
superstructure for various positions of the elevator, the complete conditions
of equilibrium could be worked out in any particular case.
A less complete series of experiments usually suffices. At low air speeds
the lift from the wings is not very great, and at the speed of greatest float
resistance not so much as one quarter of the total displacement at rest.
At higher speeds, but still before the elevators are very effective, the attitude
of the wings is fixed by the couples on the float and does not vary greatly.
A satisfactory compromise, therefore, is to take the angle of incidence
of the wings when the constant value has been reached, and to calculate
from it and the known properties of the wings the speed at which the whole
load will be air-borne. At lower speeds the air-borne load is taken as pro-
portional to the square of the air speed. After a little experience this part
6000
5000
4000
RESISTANCE (LBS)
& LIFT -i-IO
3000
2000
1000
20 30 40 50
SPEED OVER WATER (MP H.)
Fig. 36. — Water resistance of a flying boat hull.
of the calculation presents no serious difficulty, and the curve of " lift
on float " shown in Fig. 36 is the result for the float under consideration.
At rest on the water the displacement was 32,000 lbs. ; at 20 m.p.h., 29,000
lbs. ; at 40 m.p.h., 19,000 lbs., and had become very small at 60 m.p.h.
For the loads shown by the lift curve, the float took up a definite angle
of inclination to the water, which is shown in the same figure. The re-
sistance is also shown in one of the curves of Fig. 36. The angle of incidence
depends generally on the aerodynamic couple of the superstructure, and
the part of this due to airscrew thrust was represented in the tests. By
movement of the elevator this couple is variable to a very slight extent at
low speeds, but to an appreciable extent at high speeds.
The first noticeable feature of the water resistance of the float is the
rapid growth at low speeds from zero to 5400 lbs. at 27 m.p.h., where it is
17 per cent, of the total weight of the flying boat. At higher speeds the
THE PKINCIPLES OF FLIGHT
57
P
^^fcesistance falls appreciably and will of course become zero when the lift
^^Bn the float is zero. If the aerodynamic efl&ciency of the flying boat is
^Hb at the moment of getting off, the air resistance is 4000 lbs,, and with
^ negligible error the air resistance at other speeds may be taken as pro-
tportional to the square of the air speed, since the attitude is seen to be
pearly constant at the higher and more important speeds. By addition
of the drags for water and air a curve of total resistance is obtained which
caches a value of a Httle over 6000 lbs. at a speed of 30 m.p.h., rises
lowly to 6600 lbs. at 50 m.p.h., and then falls rapidly to less than 5000 lbs.
fter the flying boat has become completely air-borne the resistance again
creases with increase of speed.
The additional information required to estimate the drag of a seaplane
efore it leaves the water is thus obtained, and the method of calculation
roceeds as for the aeroplane. The drag of the wings is estimated, and to
8000
eooo
4000
RESISTANCE
LBS.
2000
— \ — \ — \ — I — r-
CONSTANT SPEED 55 MPH
OF FORCES ON FLOAT'
ABOUT C G
DISPLACEMENT AT REST 32,000 LBS
DISPLACEMENT AT AN6LE OF 8°8 & A
SPEED OF 55 MPH 7,500 LBS
INCLINATION OF FLOAT (degrees)
Fio. 37. — Pitching moment on a flying boat hull.
10
+ 100,000
+ 50,000
MOMENT
LBS. FT
O
- 50,000
- 100,000
it is added the drag of the float, including its air resistance. To the sum
is further added the resistance of the remaining parts of the aircraft.
The calculation of the speed and horsepower of the airscrew follows the
same fundamental lines as for the aeroplane, and differs from it only in
the extension of the airscrew curves to lower forward speeds. The same
extension would be needed for a consideration of the taxying of an aero-
plane over an aerodrome. The extension of airscrew characteristics is
easily obtained experimentally, or may be calculated as shown in a later
chapter.
The evidence on longitudinal balance is not wholly satisfactory, but an
example of a test is given in Fig. 37, which shows a series of observations at
a constant speed, the resistance and the pitching moment being measured
for various angles of incidence. In the experiment the height of the model
from still water was hmited by a stop, and it is improbable that under
these circumstances the load on the float would correctly supplement the
load on the wings. Treating the diagram, however, as though equilibrium
58 APPLIED AEEODYNAMICS
of vertical load had been attained, it will be noticed that the pitching
moment was zero at 8° "8, and that at smaller angles the moment was
positive, and thus tended to bring the float, if disturbed, back to S^'S.
For greater angles of incidence the moment changed very rapidly, but for
smaller angles the change was very much more gradual, and it is interest-
ing to compare the magnitude with that applicable by suitable elevators
on the superstructure. For the present rough illustration the aerodynamic
pitching moment due to a full use of the elevators may be taken as
20V2ni.p.h. lbs. -feet, and if balanced so that the pilot can use the full
angle a couple of 60,000 lbs. -feet at 55 m.p.h. is obtained. A couple of
this magnitude is sufficient to change the angle of the float from 9 degrees
to 4 degrees, and the pilot has appreciable control over the longitudinal
attitude some time before leaving the water.
(iii) LiGHTER-THAN-AiR CrAFT
All Ughter-than-air craft obtain support for their weight by the utihsa-
tion of the differences of the properties of two gases, usually air and
hydrogen. In the early days of ballooning the difference in the densities
of hot and cold air was used to obtain the lift of the fire balloon, whilst
later the enclosed gas was obtained from coal. Very recently, hehum has
been considered as a possibility, but none of the combinations produce so
much lift for a given volume as hydrogen and air, since the former is the
lightest gas known. The external gas is not at the choice of the aeronaut.
At the same pressure and temperature air is 14*4 times as heavy as pure
hydrogen, and the lift on a weightless vessel filled with hydrogen and
immersed in air would be — -— of the weight of the air displaced.
14*4
HeHum is twice as dense as hydrogen, whilst coal gas is seven times as
dense, and is never used for dirigible aircraft.
Some of the problems relating to the airship bear a great resemblance
to problems in meteorology. As in the case of the aeroplane, the stratum
of air passed through by the airship is very thick, the limit being about
20,000 feet, where the density has fallen to nearly half that at the surface of
the earth. As the lift of an airship depends on the weight of displaced
air, it will be seen that the lift must decrease with height unless the volume
of displaced air can be increased. It is the limit to which adjustment
of volume can take place which fixes the greatest height to which an airship
can go. The gas containers inside a rigid airship are only partially inflated
at the ground, and under reduced pressure they expand so as to maintain,
at least approximately, a lift which is independent of height. The process
of adjustment, which is almost automatic in a rigid airship, is achieved by
automatic and manual control in the non-rigid type, air from the balloonets
being released as the hydrogen expands. In both types, therefore, the
apparent definiteness of shape does not apply to internal form.
The first problem in aerostatics which will be considered is the effect,
on the volume of a mass of gas enclosed in a flexible bag, of movement from
one part of the atmosphere to another. The well-known theorems relating
p
THE PRINCIPLES OF FLIGHT
59
V(
I
the properties of gases will be assumed, and only the applications de-
veloped. The gas is supposed to be imprisoned in a partially inflated
liexible bag of small size, the later condition being introduced so as to
eliminate secondary effects of changes of density from the first example.
The gas inside the bag exerts a pressure normal to the surface, whilst other
pressures are applied externally by the surrounding air. At B, Fig. 38,
the internal pressure will be greater than that at A by the amount necessary
to support the column of gas above it. If w be the weight of gas per unit
volume, the difference of internal pressure at B and A is wh. Similarly if
i' be the weight of air per unit volume, the difference of external pressures
w'h, and the vertical component of the internal and external pressures at
and B is {w' — w)h. Now for the same gases (w'— w) is constant, and the
ilement of lift is proportional to h and to the horizontal cross-section of
the column which stands on B. Adding up all the elements shows that
e total lift is equal to the pro-
luct of the volume of the bag and
le difference of the weights of unit
rolumes of air and the enclosed
itas. At ordinary ground pressure
id temperature, 2116 lbs. per sq.
)ot and 15° C, the value of w' for
air is 0-0763 lb. per cubic foot,
whilst w for hydrogen would be
0*0053 ; w' — w for air and pure
hydrogen would therefore be 0*0710
lb. per cubic foot. In practice
pure hydrogen is not obtainable,
and under any circumstances be-
comes contaminated with air after
a little use. Instead of the figure
0*071 values ranging from 0*064 to
0*068 are used, depending on the
purity of the enclosed gas.
If a suitable weight be hung to the bottom of the flexible gas-bag the
whole may be made to remain suspended at any particular place in the
atmosphere. What will then happen if the whole be raised some thousands
of feet and released ? Will the apparatus rise or fall ?
The effect of an increase of height is complex. In the first place, the
density of the air falls but with a simultaneous fall of pressure, and the
hydrogen expands so long as full inflation has not occurred. For certain
conditions not greatly different from those of an ordinary atmosphere the
increased volume exactly counterbalances the effect of reduced density,
and equiHbrium is undisturbed by change of height. The problem involves
the use of certain equations for the properties of gases. If f be the pressure,
w the weight of unit volume, and t the absolute temperature of a gas, then
(8)
Fig. 38.
For air, R
f = Ez^t
> 95*7, and for hydrogen, R = 1375, p being in lbs. per sq. foot.
60 APPLIED AEEODYNAMICS
w in lbs. per cubic foot, and t in Centigrade degrees on the absolute scale of
temperature.
When a gas is expanded both its temperature and pressure are changed,
and unless heated or cooled by external agency during the process the
additional gas relation is
where y is a physical constant for the gas and equal to 1 '41 for both air and
hydrogen, pq, Wq and Iq, are the values of p, w and t, which existed at the
beginning of the expansion.
Inside the flexible bag gas weighing W lbs. has been enclosed at a
pressure po ^^^ ^ density Wq. The volume displaced at any other pressure
W ...
is — , and as was seen earlier, the lift on the bag when immersed in air is
w ^
the volume displaced multiplied by the difference of the weights of unit
volumes of air and hydrogen. The equation is therefore
W
Lift =:■ — (w' — w)
w ^
==w('^-l) (10)
If the bag be so small that p has sensibly the same value inside and out,
equation (8) shows that the weights of unit volumes of the two gases vary
inversely as their absolute temperatures, and equation (10) shows that
the lift is independent of position in the atmosphere if the temperatures
of the two are the same. If the bag be held in any one place equality of
temperatures will ultimately be reached, but for rapid changes in position,
equation (9) shows the changes of temperature to be determined by the
changes of pressure. It is now proposed to investigate the law of variation
of pressure with height which will give equilibrium at all heights for rapid
changes of position.
CoNVBCTiVE Equilibrium
If for the external atmosphere equation (9) is satisfied, the gas inside the
bag expands so as to keep the lift constant. Eeplace the hydrogen by air,
and in jiew surroundings at the reduced pressure reconsider the problem of
equilibrium. It will be found that the pressures inside and outside the bag
are equal at all points, and the fabric may then be removed without
affecting the condition of the air. The conditions are, however, those for
equilibrium, and the air would not tend to return to its old position. It is
obvious that no tendency to convection currents exists, although the air
is colder at greater heights. The quantity which determines the
possibility or otherwise of convection is clearly not one of the three used
in equations (8) and (9). A quantity called " potential temperature " is
employed in this connection, and is the temperature taken by a portion of
gas which is compressed adiabatically from its actual state to one in which
THE PEINCIPLES OF FLIGHT
61
its pressure has a standard value. In an atmosphere in convective equiU-
brium the potential temperature is constant. If the potential temperature
rises with height equilibrium is stable, whilst in the converse case up and
down currents will be produced.
Applying the conclusions to the motion of an airship with free expansion
to the hydrogen containers, it will be seen that in a stable atmosphere the
lift decreases with height for rapid changes of position, and hence the airship
is stable for height. In an unstable atmosphere the tendency is to fall
continuously unless manual control is exerted. Calculations for an atmo-
sphere in convective equilibrium are given below, and are compared with
the observations of an average atmosphere.
Law of Variation of Pressure, Density and Temperature in an Atmo-
sphere which is in Convective Equilibrium. — Since the increase of pressure
at the base of an elementary column of air is equal to the product of the
I^Bie negative sign indicating decrease of pressure with increase of height*
Using equation (9) to substitute for w converts equation (11) into
dp
-^ = — w
dh
(11)
dp _
dh~
id the solution of this is
h =
<0
y — l Wot \po/ )
(12)
(13)
which clearly gives /i = 0 when jp = j^q. For the usual conditions at the foot
'f a standard atmosphere, Pq = 2116 and Wq = 0*0783, and for these values
equation (13) has been used to calculate values of p for given values of h.
Values of relative density and temperature follow from equation (9).
The corresponding quantities for a standard atmosphere are taken from a
table in the chapter on the prediction and analysis of aeroplane performance.
TABLE 20.
Atmosphere
in convective equilibrium.
Standard atmosphere.
Height
(ft.).
Potential
Relative
Relative
Temperature
Relative
Relative
Temperature
temperature
pressure.
1
density.
Centigrade.
pressure.
density.
Centigrade.
for standard
atmosphere.
0
1000
1025
+ 9
1000
1-026
+ 9
9
5,000
0-827
0-895
- 6
0-829
0-870
+ 1-6
15
10,000
0-676
0-776
-21
0-684
0-740
- 6
25
16,000
0-546
0-668
-37
0-560
0-630
-16
31
20,000
0-435
0-568
-52
0-456
0-535
-26
37
25,000
0-340
0-476
-67
0-369
0-448
-35
45
30,000
02()2
0-305
-82
0-296
0-374
-44
53
62 APPLIED AEEODYNAMICS
It will be seen from Table 20 that the fall of temperature for convective
equilibrium is very nearly three degrees Centigrade for each 1000 feet of
height. In the standard atmosphere the fall is less than two degrees
for each 1000 ft. of height, i.e. the potential temperature rises as the height
increases and indicates a considerable degree of stability.
Lift on a Gas Container of Considerable Dimensions. — In the first
example the container was kept small, so that the gas density was sensibly
the same at all parts. In a large container the quantity — which occurs
in equation (10) is not constant, since for the hydrogen in the container
and for the air immediately outside the density varies with the height of
the point at which it is measured. To develop the subject further, con-
vective equihbrium inside and outside the gas-bag will be assumed, and
equation (13) used to define the relation between pressure and height.
The equation in new form is
Po ^ y Po ^
and for values of h less than 5000 feet the second term in the bracket is
small in comparison with unity. The expression may then be expanded
by the binomial theorem and a limited number of terms retained. The
expansion leads to
'^^LO^' ('*)
Vo Po 2yVpo'
where Wq and ^q are the values of p and w at some chosen point in the gas,
say its centre of volume, and h is measured. above and below this point.
For a difference between ground-level and h = 5000 feet the terms of (14)
are 1, — 0*185 and 0*012, and the terms are seen to converge rapidly. On
the difference of pressure between the two places the accuracy of (14) as
given is about 1 per cent. For any airship yet considered the accuracy
of (14) would be much greater than that shown in the illustration, and may
therefore be used as a relation between pressure and height in estimating
the lift of an airship.
If ^2 be the pressure at B, Pig. 38, due to internal pressure, and X2 ^^e
angle between the normal to the envelope at B and the vertical, the
contribution to the lift is —p2 cos X2 X element of area at B. If a column
be drawn above B, the horizontal cross-section is equal to cos X2 X element
of area at B, and the value of the latter quantity is equal to an
increment of volume, 8 (vol.), divided by h, or, what is the same thing,
by hi — ^2. The total lift is then given by the equation
gross lift = /'fi^-2S(vol.)-/*^^^=|^8 (vol.) . . (15)
J hi — /^2 J III — Al2
where the pressures for the air are indicated by dashes.
From equation (14) the necessary values for use in equation (15) can
be deduced, since
I^H* THE PBINCIPEES OF FLIGHT 63
^aor by drogeu inside with a similar expression for air outside. Equation (1 6)
becomes
gross lift = (Wo' - wo)/|l -^y--^-(''i + ''2)|s (vol.)
= « - w„) vol. - 1 . (<)!^i%! I'tl+hB (vol.) . (17)
Tbe term {wq — Wq) vol. is that which would be obtained by considering
the hydrogen and air of uniform density Wq and Wq respectively. The
econd term depends on the mean height of the points A and B above the
entre of volume, and in a symmetrical airship on an even keel the quantity
—= — - is zero for all pairs of points and the second integral vanishes.
i the axis of the airship is incHned the integral of (17) must be examined
further. For a fully inflated form which has a vertical plane of symmetry
he average value of -i-^r — - for any section is equal to x sin 0, x being the
distance from the centre of volume along the axis, and the section being
normal to the axis. The element of volume is then equal to the area of
cross-section multiplied by dx, and
lh+hs{Yo\.) = smeJAxdx (18)
his integral is easily evaluated graphically for any form of envelope, but
for the purposes of illustration a cylinder of length 21 and diameter d will
be used. The first point is easily deduced, and shows that the gross lift
of an inclined cyHnder is the same as that on an even keel. GeneraHsing
from this, it may be said that for an airship the gross Uft is not appreciably
affected by the inclination of the axis, and the hft may be calculated from
the displacement and the difference of densities at the height of the centre
of volume.
Pitching Moment due to Inclination of the Axis. — ^Moments will be
taken about the centre of volume of the airship. To do this it is only
necessary to multiply the lift of an element by —x before the integration
in (17) is performed. The first term will be zero, whilst the second has
a value equal, for the cyUnder, to
Pitching moment = - . ^-^ ^ sin 6 i Ax^dx
y Po J-i
=:|.i.(<)!:ZiV.Asin^.Z3 . . (19)
To appreciate the significance of (19) consider a numerical case. A
height of 15,000 feet in a convective atmosphere has been chosen as corre-
sponding with fully expanded hydrogen containers. The pressure is here
1150 lbs. per square foot, and Wq is 0-0433. The value of Wq is of no
importance. An airship 70 feet in diameter and of length 650 feet shows
for an inclination of 15° a couple of more than 25,000 Ibs.-ft., and to.
64 APPLIED AEEODYNAMICS
counteract this a force of 90 lbs. on the horizontal fin and elevators would
be needed. The couple may, however, occur when the airship has no
motion relative to the air, in which case it is balanced by a moment due
to the weight of the airship, which in the illustration would be 100,000
lbs. A movement of 3 ins. would suffice, whilst the movement caused by
a pitch of 15° would be about 8 feet. The effect is then equivalent to a
reduction of metacentric height of 3 per cent.
Equation (19) shows that the pitching moment increases rapidly with
the length of the ship, but in these cases the type of construction adopted
reduces the moment to a small amount. The length of the airship is divided
into compartments separated by bulkheads which can support a consider-
able pressure. In each compartment is a separate hydrogen container,
and the arrangement is therefore such that the gas cannot flow freely
from end to end of the airship. This greatly reduces the changes of
density due to inclination of the axis, and so reduces the pitching moment.
The arrangement also effectively intervenes to prevent surging of the
hydrogen, which might increase the pitching moments as a result of the
effects of inertia of the hydrogen.
It may therefore be concluded that the result of displacing air by
hydrogen is a force acting upwards at the centre of the volume of the
displaced air, and with suitable precautions in large airships no other
consequences are of primary importance.
Forces on an Airship due to its Motion through the Air
The aerodynamics of the airship is fundamentally much simpler than
that of the aeroplane. This follows when once it is appreciated that the
attitude relative to the wind does not depend on the speed of the airship.
The most important forces are the drag, which varies as the square of
the speed, and the airscrew thrust, which also varies as the square of the
speed since it counterbalances the drag. A secondary consequence of the
variation of thrust as the square of the speed is that at all speeds the
airscrew may be working in the condition of maximum efficiency, a state
which was not possible in the aeroplane for an airscrew of fixed shape.
It is true that dynamic lift may be obtained* from an airship envelope,
but this has not the same significance as in the case of the aeroplane, since
height can be gained apart from the power of the engine. The number
of experiments from which observations of drag for airships can be deduced
with accuracy is very small, and the figures now quoted are based on full
scale observations and speed attained, together with a certain amount
of analysis based on models of airships both fully rigged and partially
rigged.
The two illustrations chosen correspond with the non-rigid and rigid
airships shown in Figs 7-9, Chapter I. The N.S. type of non-rigid
airship has a length of 262 feet and a maximum width of 57 feet.
The gross lift is 24,000 lbs., and the result of the analysis of flight tests
shows that the drag in pounds is approximately O-TTV^mph. The drag is
made up in this instance in the proportions of 40 per cent, for the envelope,
THE PEINCIPLES OF FLIGHT 65
'65 per cent, for the car and rigging cables, and 25 per cent, for the vertical
and horizontal fins, rudder and elevators. The horsepower necessary to
propel the airship depends on the efficiency of the airscrew, tj, the relation
being
0-77V3„.p.„. = 375.7?.B.H.P (20)
It has already been mentioned that the airscrew if correctly designed
would always be working at its maximum efficiency at all speeds and a
reasonable value for the efficiency is 0"75. At maximum power the two
engines of the N.S. type of airship develop 520 B.H.P., and from equation
(20) it is then readily found that the maximum speed of the airship is
57*5 m.p.h. The drag at this speed is 2500 lbs.
For a large rigid airship, 693 feet in length and with an envelope 79 feet
in diameter the drag in lbs. was l*25V2mpii, and the gross lift 150,000 lbs.
The drag of the envelope was about 60 per cent, of the total, with cars and
rigging accounting for 30 per cent, and fins and control surfaces for 10 per
cent. It will be noticed that the envelope of the rigid airship has a greater
proportionate resistance than that of the non-rigid, and this is largely
accounted for by the smaller relative size of the cars and rigging in the
former case.
The relation between horsepower and speed has a similar form to (20),
and is
l-25V3^.pj,. =3 37577 B.H.P (21)
With engines developing 1800 B.H.P. and an airscrew efficiency of 0'75 equa-
tion (21) shows a maximum speed of 74 m.p.h. The drag is then 6800 lbs.
A convenient formula which is frequently used to express the resistance
of airships is
Kesistance in lbs. = C . p . V2 (vol.)* .... (22)
where C is a constant defiuaing the quality of the airship for drag. The
advantage of the formula is that C does not depend on the size of the
airship or its velocity or on the density of the air, but is directly affected
by changes of external form. In the formula p is the weight in pounds
of a cubic foot of air divided by g in feet per sec. per sec, V is the
velocity of the airship in feet per sec, and " vol." is the volume in cubic
feet of the air displaced by the envelope. For the non-rigid airship above,
the value of C is 0*03, and for the rigid airship C = 0'016.
Longitudinal Balance of an Airship. — For an airship not in motion,
balance is obtained by suitable adjustment of the positions of the weights
carried. A certain amount of alteration of " trim " can be obtained by
transferring air from one of the balloonets of a non-rigid airship to another.
Fig. 9, Chapter I., shows the pipes to the two balloonets which are about
120 feet apart. One pound of air moved from the front to the rear produces
a couple of 120 Ibs.-ft. If the centre of buoyancy of the hydrogen be taken
as 10 feet above the centre of gravity and the weight of the airship is
24,000 lbs., the couple necessary to displace the airship through one degree
is 4200 Ibs.-feet, and would require a movement of 35 lbs. of air from one
balloonet to the other. By this means sufficient adjustment is available
G6 APPLIED AEEODYNAMICS
for the trim of the airship when not in motion. In the rigid airship a
similar control can be obtained by the movement of water-ballast from
place to place.
When in motion the aerodynamic forces introduce a new condition of
balance which is maintained by movement of the elevators. The couples
due to movements of the elevators are very much greater than those
arising from adjustment of the air between the balloonets, a rough figure
for the elevators of the N.S. type of airship being SV^j^pu. Ibs.-feet per
degree of movement of the elevator. At a speed of 40 m.p.h.the couple due
to one degree change of elevator position is 8000 Ibs.-feet, and so would
tilt the airship through an angle of about 2°. For a sufficiently large
movement of the elevators considerable inclination of the axis of an air-
ship could be maintained at high speeds, and the airship then has an
appreciable dynamic lift. For the N.S. type of airship about 200 lbs. of
dynamic lift or about 1 per cent, of the gross lift is obtained at 40 m.p.h.
for an inchnation of the axis of one degree.
The various items briefly touched on in connection with longitudinal
balance are more naturally developed in considering the stability of
airships, since it is the variation from normal conditions which constitutes
the basis of stability, and apart from a tendency to pitch and yaw the control
of an airship presents no fundamental difficulties.
Equilibrium of Kite Balloons
The conditions for the equilibrium of a kite balloon are more complex
than those for the airship. The kite balloon has its own buoyancy, which
is all important at low wind speeds but unimportant in high winds. The
aerodynamic forces of lift and drag and of pitching moment are all of
importance, and in addition there is the constraint of a kite wire. It is
now proposed to consider in detail the equilibrium of the two types of kite
balloon shown in Fig. 10, Chapter I., and to explain why one of them is
satisfactory in high winds and the other unsatisfactory.
A diagram of a kite balloon is shown in Fig. 39, on which are marked
the quantities used in calculation. Axes of reference are taken to be
horizontal and vertical, with the origin at the centre of gravity. If
towed, the kite balloon would be moving along the positive direction
of the axis of X, whilst in the stationary balloon the wind is
blowing along the negative direction of the axis. The axis of Z is
vertically downward, and the pitching moment M is positive when it tends
to raise the nose of the balloon. The kiting effect results from an in-
clination, a, of the axis of the kite balloon to the relative wind. The
buoyancy due to hydrogen has a resultant F which acts upwards at the
centre of volume of the enclosed gas, a point known as the centre of
buoyancy (CB of Fig. 39). The kite wire comes to a puUey at D, which
runs freely in a bridle attached to the balloon at the points E and H.
The point D moves in an ellipse of which E and H are the foci, and for a
considerable range of inclination the point of virtual attachment is at A,
the centre of curvature of the path of D.
THE PEINCIPLES OF FLIGHT
67
By arranging the rigging differently the point of attachment could be
transferred to B. To effect this the pulley at D is removed, the points
E and H moved nearer the axis of the balloon, the wires from them meeting
the kite wire at B. The details of the calculations follow the same routine
for all points of attachment, and the effects illustrated will be those of
changing from type Fig. 10a to type Fig. 10c with a fixed attachment
and those due to changing the point of attachment of type Fig. lie from
A to B of Fig. 39. The co-ordinates of the point of attachment (or virtual
point of attachment) of the kite wire are denoted by a and c respectively
for distances along the axis of X and Z. The length of the kite balloons
considered in these pages was about 80 feet, and the maximum diameter
27 feet.
Fig. 39. — Equilibrium of a kite balloon.
Kite Balloon with three Fins (Figs. 10a and 10&). — For a particular
xample of this type the weight of the balloon structure was 1500 lbs.,
>ind at a height of 2000 feet the buoyancy force F was 2085 lbs. For
various angles of inclination of the balloon the values of the lengths a, c
I lid f were calculated from the known geometry of the balloon. The results
if the calculations are given in Table 21 below.
A model of the kite balloon was made and tested in a wind channel,
-0 that for various angles of inclination, a, the values of the lift, drag and
1 ero dynamic pitching moment about the centre of gravity were measured.
I ho observations were converted to the full size by multiplying by the
n[uare of the scale for the forces and by the cube of the scale for moments,
i Extensions of observations to speeds higher than those of the wind channel
vere made by increasing the forces and moment in proportion to the square
of the wind speed.
68
APPLIED AEEODYNAMICS
Erom Eig. 89 it will be seen that the components of the tension of the
kite wire are very simply related to the lift and drag of the kite balloon.
The relations are
T2 = lift + E-Wf ^^^
The total pitching moment is obtained by taking moments of the forces
about CG and adding to them the couple from aerodynamic causes
other than lift and drag. The resultant moment must be zero for any
position of equilibrium, and hence
M + TiC - Tgtt + E/ = 0
(24)
TABLE 21.
Inclination of the axis
of the balloon to
Co-ordinates of the position of the point of
attachment of the kite wire.
Horizontal distance
between centre of
gravity and centre of
horizontal,
a
a
(ft.).
c
(ft.).
buoyancy,
(ft.).
0
26-8
36-2
13-5
5
29-9
33-8
120
10
32-7
310
10-5
15
35-3
281
8-8
20
37-6
24-8
7-1
25
39-6
21-6
5-3
Since E — W is constant and equal to 585 lbs., T2 differs from the lift
by a constant amount, and in tabulating the results of experiment Tj and
T2 have been used directly instead of drag and lift. The value of the
aerodynamic moment about the centre of gravity, i.e. M of equation (24),
is given in the second column of Table 22 for various wind speeds, whilst
the value of the whole of the left-hand side of (24) for various angles of
incidence and for a range of speeds is shown in the sixth column of the
table. From an examination of the figures in columns (3) and (4) it will
be seen that for the same angle of incidence the aerodynamic pitching
moment and the drag vary as the square of the wind speed. A similar
result will be found for T2 — 585.
EquiHbrium occurs when the figures in the last column of Table 22
change sign, and an inspection shows a progressive change of angle of
incidence from about 12° "5 for no wind to a little more than 15° at a wind
speed of 80 m.p.h. A positive moment tends to put the nose of the balloon
up and so increase the angle of incidence, the effect being a tendency
towards the position of equilibrium.
The figures for no wind give a measure of the importance of the couples
due to reserve buoyancy, and by comparison with those due to a combination
of buoyancy and aerodynamic couples and forces at 80 m.p.h. it will be
realised that the equihbrium of a kite balloon in a high wind depends
almost wholly on the aerodynamic forces and couple. This is an illustra-
tion of a law which appears on many occasions, that effects of buoyancy
THE PRINCIPLES OF FLIGHT
69
ire only important in determining the attitude of floating bodies at very
low relative velocities. The theorem applies to the motion of flying boats
)ver water, and explains a critical speed in the motion of airships.
I
TABLE 22.
Wind
speed
(m.p.h.).
a
(degrees).
Aerodynamic
pitching moment,
M
(Ibs.-ft.)
Drag = Tj
(lbs.).
LUt+585=T2
(lbs.).
Total pitching
moment about C.G.
(Ibs.-ft.).
0
0
_
_
585
12,480
5
—
—
585
7,520
10
—
—
585
2,790
15
—
—
585
- 2,300
20
—
—
585
- 7,200
25
—
—
585
- 12,100
20
0
2,030
126
607
18,500
5
4,650
144
763
11,740
10
7,340
172
889
5,630
15
9,470
225
1,027
- 2,170
20
9,870
309
1,210 •
- 13,170
25
10,250
424
1,400
- 25,000
40
0
8,290
506
675
36,700
5
18,600
578
1,298
24,390
10
29,400
690
1,801
13,750
15
37,900
900
2,353
- 1,460
20
39,450
1,236
3,085
- 31,000
25
41,000
1,696
3,845
- 63,500
60
0
18,700
1,136
787
66,800
5
41,900
1,299
2,185
45,500
10
66,100
1,550
3,320
27,700
15
85,200
2,025
4,560
- 620
20
88,800
2,786
6,210
- 60,330
t-
25
92,300
3,816
7,920
-128,200
80
0
33,200
2,024
945
109,300
5
74,400
2,312
3,440
75,000
10
117,500
2,760
5,450
46,900
15
151,500
3,600
7,660
470 '
20
157,800
4.944
10,580
-102,800
25
164,000
6,784
13,630
-218,500
The tension in the kite wire for each of the positions of equihbrium is
obtained from Table 22, since it is equal to the square root of the sum of
the squares of Tj and T2. The values are given in Table 23 below.
TABLE
23.
Wind speed
Tension in Idte wire
(m.p.h.).
(Iba.).
0
585
20
990
40
2460
60
4960
80
8480
70 APPLIED AEEOBYNAMICS
At 80 m.p.h. the tension in the kite cable has been increased to more
than 14 times its value for no wind. Had the rigging been so arranged
that the angle of incidence for equihbrium was 25°, Table 22 shows that
the force would have been 80 per cent, greater than at 15°, and conversely
a reduction of tension would have been produced by rigging the kite-
balloon so as to be in equilibrium as a smaller angle of incidence. The
effect of change of position of the point of attachment of the kite wire will
now be discussed.
The aerodynamic pitching moment on the kite balloon is seen from
column 3 of Table 22 to tend to raise the nose of the balloon at all angles of
incidence. The couple due to buoyancy depends on the point of attach-
ment of the kite wire, and the nose will tend to come down as this point
is moved nearer the nose. At high speeds it has been seen that the
buoyancy couples are unimportant in their effects on equilibrium, and that
the only variations of importance are those which affect the couples due to
the tension in the kite wire.
Since TiC — T2a is greater than M, as may be seen from Table 22, it
follows that to obtain equilibrium at a lower angle of incidence the former
quantity must be increased. TiC — T.2a is the moment of the kite wire
a;bout the centre of gravity, and can be increased by moving the point of
attachment forward. Changing the vertical position is much less effective,
since the kite wire is more nearly vertical than horizontal.
Before the calculation of equilibrium can be said to be complete, an
examination of the resultant figure taken by the rigging will need to be
made to ensure that all cords are in tension. In reference to Fig. 39 it will
be observed that ED and HD will be in tension if the line of the kite wire
produced falls between them. A running block ensures this condition,
but a joint at D might produce different results. The virtual point of
attachment would move to E or H if HD or ED became slack.
Position o£ a Kite Balloon relative to the Lower End of the Kite Wire. —
When equilibrium has been attained the position in space of the kite balloon
is determined by the length of kite wire and its weight and by the forces
on the balloon. The equilibrium of the balloon has been dealt with, and
its connection to the kite wire is fully determined by the tensions Ti and T2.
The wind forces on the wire being negligible the curve taken by the wire is
a catenary, and the horizontal component of the tension in the wire is
constant at all points. Define the co-ordinates of the upper end of the wire
relative to the lower end by ^ and ^, and the weight of the wire rope per
unit length by w. The equation of the catenary is then
? = ^|cosh|^(^ + A)-cosh^A} . . .(25)
where A is a constant so chosen that ^ = 0 when ^ = 0, i.e. the distances
are measured from the lower end of the kite wire : the equations for a
catenary can be found in text-books on elementary calculus. The length
of the kite wire to any point is given by
^=^['^"^li^^+^)-'^^^li4 • • • ^^^^
THE PEINCIPLES OF FLIGHT
and the vertical component of the tension in the wire is
T2 = Tisinh,^($ + A). .
71
(27)
As an example take the equilibrium position at 40 m.p.h. : —
Ti = 880 lbs., Ta = 2300 lbs., S = 2000 ft., w = 0*15 lb. per ft. run.
From equation (27) and a table of hyperbolic sines the value of { -f A is deduced
as 9920 feet. Using both equations (26) and (27) the value of A is found as 9160 feet,
and hence | = 760 feet.
Using the values of | + A and A in equation (25) shows that ^ = 1850 feet.
The kite balloon is then 1850 feet up and 760 feet back from the foot of the cable,
T
Had the cable been quite straight its inclination to the vertical would have been tan~"^ ~>
^2
and the height of the balloon would be 2000
and its distance back
For this assumption the height would be 1870 feet and the dis.
2000 y—-
tance back from the base 715 feet.
From the above example it may be concluded that the wire cable is
nearly straight and that a very simple calculation suffices for a moderate
wind. Since Table 22 shows that the ratio of Ti to T2 does not change
much at high speeds, it follows that the kite balloon will- be blown back to
a definite position as the result of light winds, but will then maintain its
position as the wind velocity increases.
Kite Balloon with Large Veitical Fin and Small Horizontal Fins (Fig. 10c).
— As the calculations follow the lines already indicated the results will
be given with very little explanation. The object of the calculations is to
draw a comparison between the two forms of kite balloon and to show the
difference due to form of fins and point of attachment of the kite wire.
In the new illustration the balloon will be taken to have the weight,
1500 lbs., and buoyancy, 2085 lbs., used for the calculations on the kite
balloon with three fins. In one case the point of attachment will be taken
as A and will correspond with the running attachment at D, whilst in a
second case an actual attachment at B will be used. The points A and B
are marked on Fig. 39, and corresponding with them is the table of dimen-
sions below.
TABLE 24.
a
A. Running attachment of
kite wire.
B. Fixed attachment of
kite wire.
A and B.
Angle of
(ft.).
inclination
(degrees).
(ft.).
(ft.).
(ft.).
(ft.).
0
190
- 4-6
26-6
+ 3-3
12-6
' 10
17-9
- 7-8
25-8
- 11
100
20
16-2
-10-8
25-2
- 5-7
7-2
30
14-2
-135
23-8
- 9-9
4.4
40
11-6
-15-7
21-7
-140
11
72
APPLIED AEEODYNAMICS
Only the values of pitching moment and tensions in the wire for a
speed of 40 m.p.h. will be given, as they suffice for the present purpose of
illustrating the limitation of the type.
TABLE 25.
a
Angle of
inclination
(degrees).
Aerodynamic
pitching moment.
M
(Ibs.-ft.).
(lbs.).
T2
(lbs.).
Total pitching moment.
A. Running
attachment.
B. Fixed
attachment.
0
10
20
30
40
5,150
29,000
51,100
66,000
70,900
500
596
885
1,435
2,490
685
1,196
1,855
2,460
3,375
18,000
23,700
26,500
20,800
-4,800
18,100
18,400
14,400
2,400
-34,900
An examination of the last two columns of Table 25 will show that with
the running attachment of kite wire the angle of equilibrium is 39°, and for
the fixed attachment a = 31°. Both angles are much greater than those
sliown in Table 22 for the same wind speed, and at higher speeds the results
would be still less favourable to the type. The point of attachment will
be seen from Table 24 to have been moved forward more than 6 feet
Ijetween positions A and B, and is already inconveniently placed without
having introduced sufficient correction. It may therefore be concluded
that the horizontal fins shown in Fig. 10c are wholly inadequate for the
control of a kite balloon in a high wind.
CHAPTEE III
WNERAL DESCRIPTION OF METHODS OF MEASUREMENT IN AERO-
DYNAMICS, AND THE PRINCIPLES UNDERLYING THE USE OF
INSTRUMENTS AND SPECIAL APPARATUS
lERODYNAMics as we now know it is almost wholly an experimental
science. It is probably no exaggeration to say that not a single case of
iuid motion round an aircraft or part is within the reach of computation.
Phe effect of forces acting on rigid bodies forms the subject of dynamics,
md is a highly developed mathematical science with which aeronautics
intimately concerned. Such mathematical assistance can, however,
lonly lead to the best results if the forces acting are accurately known, and
pt is the determination of these forces which provides the basic data on
[which aeronautical knowledge r,ests. Two main methods of attack are
common use, one of which deals with measurements on aircraft in flight,
md the other with models of aircraft in an artificial wind under laboratory
[conditions. The two hues of investigation are required since the possi-
^bilities of experiment in the air are limited to flying craft, and are unsuited
to the analysis of the total resistance into the parts due to wings, body,
undercarriage, etc. On the model side the control over the conditions of
experiment is very great and the accuracy attainable of a high order.
There is, however, an uncertainty arising from the small scale, which
makes the order of accuracy of application to the full scalo less than that
of the measurement on the model. The theory of the use of models is
of sufficient importance to warrant a separate chapter, and the general
result there reached is that with reasonable care in making the experiments,
observations on the model scale may be applied to aircraft by increasing
the forces measured in proportion to the square of the speed and the square
of the scale.
The full development of the means of measurement would need many
chapters of a book and will not be attempted. This chapter aims only at
explaining the general use of instruments and apparatus and the precautions
which must be observed in applying quite ordinary instruments to experi-
mental work in aircraft. As an example of the need for care it will be shown
that the common level used on the ground ceases to behave as a level in
the air, although it has a sufficient value as an indicator of sideshpping
for it to be fitted to all aeroplanes.
In very few of the cases dealt with are the instruments shown in
mechanical detail, but an attempt has been made to give sufficient descrip-
tion to enable the theory to be understood and the records of the instruments
appreciated. The particular methods and apparatus described are mostly
British as produced for the service of the Air Ministry, but with minor
73
74 APPLIED AEEODYNAMICS
variations may be taken as representative of the methods and apparatus
of the world's aerodynamic laboratories.
The Measurement o£ Air Velocity. — A knowledge of the speed at which
an aircraft moves through the air is perhaps of greater importance in
understanding what is occurring than any other single quantity. Its
measurement has therefore received much attention and reached a high
degree of accuracy. For complete aircraft the instruments used can be
calibrated by flight over measured distances, corrections for wind being
found from flights to and fro in rapid succession over the same ground.
The reading of the instruments is found to depend on the position of certain
parts relative to the aircraft, and in order to avoid the complication thus
introduced experiments will first be described under laboratory conditions.
All instruments which are used on aircraft for measuring wind velocity,
i.e. anemometers, depend on the measurement of a dynamic pressure
difference produced in tubes held in the wind. The small windmill type
of anemometer used for many other purposes has properties which render
it unsuitable for aerodynamic experiments either in flight or in the labora-
tory. One form of tube anemometer is shown in Fig. 40 so far as its essential
working parts are involved. It
]=3S
consists of an inner tube, open
at one end and facing the air
current ; the other end is con-
^^^^^ ! ! I . nected to one side of a pressure
*^^^^^ : ; ! ' \ g^^g©- -^n outer tube is fixed
^'""^"'" I I — I""* """^ concentrically over the inner, or
io flinch Pitot, tube and the annulus is
Fig. 40,— Tube anemometer. Open to the air at a number of
small holes ; the annulus is con-
nected to the other end of the pressure gauge, and the reading of the
gauge is then a measure of the speed.
For the tube shown the relation between pressure and speed may be
given in the form
t?ft..3.=66-2\//i (1)
where v is the velocity of air in feet per sec, and h is the head of water in
inches which is required to balance the dynamic pressure. The relation
shown in (1) applies at a pressure of 760 mm, of water and a temperature
of 15°"6 C, this having been chosen as a standard condition for experiments
in aerodynamic laboratories. For other pressures and temperatures
equation (1) is replaced by
%.-B.-66-2>/^^ (2)
where a is the density of the air relative to the standard condition.
Except for a very small correction, which will be referred to shortly,
the formula given by (2) apphes to values of v up to 300 ft.-s.
The tube anemometer illustrated in Fig. 40 has been made the
subject of the most accurate determination of the constant of equations
METHODS OF MEASUKEMENT 75
(1) and (2), but the exact shape does not appear to be of very great
importance.
As a result of many experiments it may be stated that the pressure in
the inner tube is independent of the shape of the opening if the tube has
a length of 20 or 30 diameters. The actual size may be varied from the
smallest which can be made, say one or two hundredths of an inch in
diameter, up to several inches.
The external tube needs greater attention ; the tapered nose shown in
Fig. 40 may be omitted or various shapes of small curvature substituted.
The rings of small holes should come well on the parallel part of the tube
and some five or six diameters behind the Pitot tube opening. The diameter
of the holes themselves should not exceed three hundredths of an inch in
a tube of 0*3 inch diameter, and the number of them is not very im-
portant. When dealing with measurements of fluctuating velocities it is
occasionally desirable to proportion the number of holes to the size of the
opening of the Pitot tube in order that changes of pressure may be trans-
mitted to opposite sides of the gauge with equal rapidity. This can be
achieved by covering the whole of the tubes by a flexible bag to which
rapid changes of shape are given by the tips of the fingers. By adjustment
of the number of holes the effect of these changes on the pressure gauge
can be reduced to a very small amount.
The outside tube should have a smooth surface with clearly cut edges
to the small holes, but with ordinary skilled workshop labour the tubes can
be repeated so accurately that calibration is unnecessary. The instrument
is therefore very well adapted for a primary standard.
Initial Determination of the Constant of the Pitot-Static Pressure Head.
— The most complete absolute determination yet made is that of Bram-
well, Eelf and Fage, and is described in detail in Eeports and Memoranda,
No. 72, 1912, of the Advisory Committee for Aeronautics. The anemometer
was mounted on a whirling arm of 30-feet radius rotating inside a building.
The speed of the tube over the ground was measured from the radius of
the tube from the axis of rotation and the speed of the rotation of the
arm. The latter could be maintained constant for long periods, so that
timing by stop-watch gave very high percentage accuracy. The air in
the building was however appreciably disturbed by the rotation of the
whirling arm, and when steady conditions had been reached the velocity
of the anemometer through the air was only about 93 per cent, of that
over the ground. A special windmill anemometer was made for the
evaluation of the movement of air in the room. It consisted of four large
vanes set at 30 degrees to the direction of motion, and the rotation of
these vanes about a fixed axis was obtained by counting the signals in
a telephone receiver due to contact with mercury cups at each rotation,
"^ome such device was essential to success, as the forces on the vanes were
^o small that ordinary methods of mechanical gearing introduced enough
friction to stop the vanes. A velocity of. one foot per second could be
measured with accuracy. To caUbrate this vane anemometer it was
mounted on the whirling arm and moved round the building at very low
speeds ; any error due to motion of air in the room is present in such
76
APPLIED AEEODYNAMICS
calibration, but as it is a 7 per cent, correction on a 7 per cent, difference
between air speed and ground speed the residual error if neglected would
not exceed 0*5 per cent. As, however, the 7 per cent, is known to exist
the actual accuracy is very great if the speed through the air is taken as
93 per cent, of that over the floor of the building. The order of accuracy
arrived at was 2 or 3 parts in 1000 on all parts of the measurement.
To determine the air motion in the building due to the rotation of the
whirling arm, the tube anemometer was removed, the vane anemometer
placed successively at seven points on its path, and the speed measured.
For the main experiment the tube anemometer was replaced at the
end of the arm, and the tubes to the pressure gauge led along the arm to
its centre and thence through a rotating seal in which leakage was prevented
by mercury. As a check on the connecting pipes the experiment was
repeated with the tube connections from the gauge to the anemometer
reversed at the whirling arm end. The pressure difference was measured
on a Chattock tilting gauge described later.
The results of the tests are shown in Table 1 below.
TABLE 1.
Speed over the floor of
the building
(feet per sec).
Speed of the air over the
floor of the building
(feet per sec).
Speed of tube anemo-
meter through the air
(feet per sec).
21-8
1-5
20-3
1-006
301
2-2
27-9
1-001
33-6
2-4
31-2
1-017?
39-8
2-9
36-9
0-994
431
31
40-0
1-005
48-8
3-6
45-2
1-004
511
3-8
47-3
1001
Connecting tubes reversed.
21-2
1-6
19-7
0-991
31-2
2-2
29-0
0-991
37-6
2-7
34-9
0-993
45-5
3-4
421
1000
48-8
3-6
45-2
1-000
511
3-8
47-3
1-002
63-4
3-9
49-5
1-001
Mean value of ~/~^=i =
or neglecting the doubtful reading
1-0005
0-9997
The pressure readings on the gauge were converted into " head of air,"
%, and the value of \/ ^- is a direct calculation from the observations of
pressure and velocity. Its value is seen to be unity within the accuracy
of the experiments, the average value being less than ^V per cent, different
from unity.
METHODS OF MEASUEEMENT 77
For this form of tube anemometer the relation is
%.-s.=V2i;ifeet of fluid (3)
In this equation the relation given is independent of the fluid and would
ipply equally to water. Most aerodynamic pressure gauges, however, use
[water as the heavy liquid, and the conversion of (3) to use h in inches of
rater for an air speed v leads to equation (1).
The determination so far has given the difference of pressure in the two
tubes of the anemometer. The pressure in each was compared with the
•pressure in a sheltered corner of the building, and it was found that in the
annular space the effect of motion was negligible. The method of ex-
periment now involves a consideration of the centrifugal effect on the air
in the tube along the whirling arm, since there is no longer compensation
by a second connecting pipe.
If p be the pressure at any point in the tube on the whirling arm at
an angular velocity a>, the equation of equihbrium is
dj> == poihdr (4)
and as the air in the tube is stationary the temperature will be constant,
so that
V='^-P (5)
Pi
where pi and p^ are the pressure and density at the inner end of the tube.
Equations (4) and (5) are readily combined, and the integration leads to
Po _^QiPii>o'IPi (6)
Vi
where jj^ is the pressure at the outer end of the whirling arm tube and v^
the velocity there. The difference of pressure ^o-pj can be calculated
from an expansion of (6) to give
Vo-Pi = 8p=.y,vXl+i^\^o'+ - - ■) . . • (7)
or in terms of the velocity of sound, Oj,
%' = W.t+|.0%...| . . . .(8)
At 300 ft.-s. the second term in the brackets is about 2 per cent, of
the first ; in the experiments described above it is imimportant.
The expression Ipv^ occui-s so frequently in aerodynamics that its relation to (3)
will be developed in detail. Squaringjhoth sides of^(3) gives
v^ = 2gh (9)
Multiply both sides by ^p to get
lpv^ = pgh (10)
= hp (11)
The weight of unit volume of a fluid is pg, and the value of pgh is the difference
of pressure per imit area between the top and bottom of the column of fluid of height h.
If p be in slugs per cubic foot and v in feet per sec. , the pressure hp is in lbs. per square
foot. The equation is, however, applicable in any consistent set of dynamical imits.
78 APPLIED AEEODYNAMICS
A comparison of equations (8) and (11) brings out the interesting result
that the difference of pressure between the two ends of the tube of the
whirling arm is of the same form as to dependence on velocity at flying
speeds as the pressure difference in a tube anemometer of the type
shown in Fig. 40. The velocity in (11) is relative to the air, whilst in (8)
the velocity is related to the floor of the building. Had the air in the
building been still so that the two velocities had been equal, the differences
of pressures in the anemometer and between the ends of a tube of the
whirling arm would have been equal to a high degree of approximation.
One end of the pressure gauge being connected to the air in a sheltered
part of the building, equation (8) can be used to estimate the pressure in
either of the tubes of the anemometer. The important observation was
then made that the air inside the annular space of the tube anemometer
at the end of the arm was at the same pressure as the air in a sheltered
position in the building. This is a justification for the name " static
pressure tube," since the pressure is that of the stationary air through which
the tube is moving. The whole pressure difference due to velocity through
the air is then due to dynamic pressure in the Pitot tube, which brings the
entering air to rest. A mathematical analysis of the pressure in a stream
brought to rest is given in the chapter on dynamical similarity, where it
is shown that the increment of pressure as calculated is
8p=^lpv^[l+iQ\ . .| (13)
where a is the velocity of sound in the undisturbed medium, and the second
term of (13) is the small correction to equation (2) which was there referred
to. At 300 ft.-s, the second term is 1-5 per cent, of the first, and (13) is
therefore appHcable with great accuracy.
The principles of dynamical similarity (see Chapter Vlll.) indicate for
the pressure a theoretical relationship of the form
8p = iP"Xa' v) ^^^^
which contains the kinematic viscosity, v, not hitherto dealt with, and I,
which defies the size of the tubes and is constant for any one anemometer.
V .
The function may in general have any form, but its dependence on - m
this instance has been shown in equation (13). The experiments on the
whirling arm have shown that the dependence of the function on viscosity
over the range of speeds possible was negUgibly small. The limit of range
over which (13) has been experimentally justified in air is limited to 50 ft.-a.
It is not however the speed which is of greatest importance in the theory
vl
•of the instrument, but the quantity -. If this can be extended by any
means the validity of (13) can be checked to a higher stage, and the ex-
tension can be achieved by moving the tube anemometer through still
water which has a kinematic viscosity 12 or 13 times less than that of air.
A velocity of 20 ft.-s. through water gives as much information as a velocity
METHODS OF MEASUEEMENT
79
of 250 ft.-s. through aur, and the experiment was made at the William Eroude
National Tank at Teddington. The anemometer was not of exactly the
rfame pattern as that shown in Fig. 40, but differed from it in minor particu-
lars and has a slightly different constant.
The results of the experiments are shown in Table 2.
TABLE 2.
Speed (ft. per sec).
Equivalent speed in air
(ft. per sec).
V¥ »' Vife
^20
1-00
30
—
0-99
Air
40
—
0-99
50
—
0'99
60
—
0-98
/ 2-88
37
0-98
3-92
61
1-00
5-07
65
0-99
6-78
75
0-99
6-80
88
0-99
7-80
101
0-99
9-69
125
0-97
9-85
128
0-98
10-82
140
0-99
1104
142
100
Water
1115
144
0-98
11-98
154
0-96
1310
169
0-97
14-24
184
0-97
14-52
187
0-99
14-76
190
0-99
16-06
207
0-98 -
16-92
218
0-97
L
17-59
227
0-99
i
18-49
238
0-99
1
19-86
256
0-99
I \2010
259
0-98
The values
of V'
'2gh
shown in the last column vary a little above
id below 0-99, and the table may be taken as justification for the use of
luation (13) up to 300 ti.-s. The difference between 0-99 and 1-00 may
rly be attributed to changes of form of the tube anemometer from that
lown in Fig. 40. In the case of water the velocity of sound is nearly
)0 ft. per sec, and the second term of (13) is completely neghgible.
rom Table 2 it may thus be deduced that the constant of equation (1) is
idependent of v up to the highest speeds attained by aircraft.
Effect o! Inclination of a Tube Anemometer on its Readings.— It would
ive been anticipated from the accuracy of calibration attained that the
ressure difference between the inner and outer tubes is not extremely
ansitive to the setting of the tubes along the wind. At inclinations of
^, 10° and 15° the errors of the tube anemometer illustrated in Fig. 40 are
per cent., 2-5 per cent, and 4-5 per cent, of the velocity, and tend to
►ver-estimation if not allowed for.
80
APPLIED AERODYNAMICS
Use of Tube Anemometer on an Aeroplane. — ^Anemometers of the
general type described in the preceding pages are used on aeroplanes
and airships. In the aeroplane the tubes are fixed on an interplane
strut about two-thirds of the way up, and with the opening of the Pitot
one foot in front of the strut. The position so chosen is convenient, since
it avoids damage during movements of an aeroplane in its shed, but is
not sufficiently far ahead of the aeroplane as to be free from the disturbance
of the wings. Although the anemometer correctly indicates the velocity
of air in its neighbourhood it does not register the motion of the aeroplane
relative to undisturbed air. The effect of disturbance is estimated for
each aeroplane by flights over a marked ground course, and Fig. 41 illus-
trates a typical result. The air immediately in front of the aeroplane is
pushed forward with a speed varying from 2 per cent, of the aeroplane
speed at 100 m.p.h. to 7 per cent, at 40 m.p.h.
How is this correction to be applied ? Does it depend on true speed
or on the indicator reading ? In order to answer these questions it is
110
105
SPEED
FACTOR
X
^■^
■ —
100 1 1 1 1 \ 1 1
40 50 60 70 80 90 100
60 70 80 90
INDICATED AIRSPEED. M.P.H.
Fig. 41.
necessary to anticipate the result of the analysis in later chapters. The
pressure gauge inside the aeroplane cockpit indicates a quantity which
may be very different from the true speed, the quantity actually measured
being of the type shown in equation (11). Allowing for the interference of
the aeroplane, it is found that the reading depends on the density of the air,
the speed of the aircraft and its inclination. The incUnation of the aero-
w
plane is fixed when — ^ is known, w bemg the loading of the wings in lbs.
per square foot, «t the relative density of the air, and v the true velocity.
The quantity ah occurs often and is called " indicated air speed " or some-
times " air speed." For aeroplanes designed for a long journey during
which the consumption of petrol and oil is an appreciable proportion of
the total weight the correction should be apphed to (—\ v. For an
aeroplane which flies with its total weight sensibly constant it will be
seen that w is constant, and that the inclination of the aeroplane is de-
termined by <jv^, and it is to this quantity therefore that the cahbration
corrections for position should apply.
Fig. 42. — Experimental arrangement of tube anemometer on an aeroplane.
METHODS OF MEASUEEMENT 81
For accurate experimental work it is very desirable that the correction
for position should be as small as possible, and at the Eoyal Aircraft Estab-
lishment it has been found that projection of the tube anemometer some
6 feet ahead of the wings reduces the correction almost to vanishing point.
The arrangement is shown in Fig. 42. To the front strut is attached a
wood support projecting forward and braced by wires to the upper and
lower wings. The two tubes of the anemometer are separated in the in-
strument used, the Pitot tube being a short distance below the static pressure
tube ; the combination is hinged to the forward end of the wooden support,
and is provided with small vanes which set it into the direction of
the relative wind. The two tubes to the pressure gauge pass along the
wooden support, down the strut and along the leading edge of the wing to
the cockpit. The thermometer used in experimental work is shown on
the rear strut.
On tjie aeroplane illustrated the residual error did not exceed 0*5 per
cent, at any speed, and there was no sign of variation with inclination of
the aeroplane.
Aeroplane "Pressure Gauge" or "Air-speed Indicator." — At 100
m.p.h. the difference of pressure between the two tubes of a tube
anemometer is nearly 5 ins. of water, and readings are required to about
one-tenth of this amount. The instruments normally used depend on
the deflection of an elastic diaphragm, to the two sides of which the tubes
from the anemometer are connected. The various masses are balanced so
as to be unaffected by inclinations or accelerations of the aeroplane. The
instruments are frequently calibrated on the ground against a water-gauge,
and have reached a stage at which trouble rarely arises from errors in the
instrument.
The scale inscribed on the dial reads true speed only for exceptional
conditions. Were the tube anemometer outside the field of influence of
the aeroplane the scale would give true speeds when the density was
equal to the standard adopted in the aerodynamic laboratories. For the
average British atmosphere this standard density occurs at a height of
about 800 feet, above which the " indicated air speed " is less than the true
speed in proportion to the square root of the relative density. Apart from
calibration corrections due to position of the tube anemometer on the
aeroplane the indicator reading at 10,000 ft. needs to be multiplied by
1-16 to give true speed. At 20,000 feet and 30,000 feet the corresponding
factors are 1-37 and 1*64 respectively, these figures being the reciprocals
of the square roots of the relative densities at those heights.
Used in conjunction with a thermometer and an aneroid barometer
the speed indicator readings can always be converted into true speeds
through the air.
Aneroid Barometer. — The aneroid barometer is a gauge which gives
the pressure of the atmosphere in which it is immersed. Its essential
part consists of a closed box of which the base and cover are elastic
diaphragms, usually with corrugations to admit of greater flexibility.
The interior of the box is exhausted of air, and the diaphragms are
connected to links and springs for the registration and control of the
82
APPLIED AERODYNAMICS
motion which takes place owing to changes in the atmospheric pressure.
At a height of 30,000 feet the pressure is about one-third of that at the
earth's surface, and the aneroid barometer for use on aircraft is required
to have a range of 5 lbs. per sq. inch to 15 lbs. per sq. inch. The forces
called into operation on small diaphragms are seen to be great, and the
supports must be robust. All but the best diaphragms show a lag in
following a rapid change of pressure, and the instrument cannot be relied
on to give distance from the ground when landing chiefly for this reason.
The aneroid barometer is used in accurate aerodynamic work solely
as a pressure gauge. It is divided however into what is nominally a scale
of height, in order to give a pilot an indication as to his position above the
earth. There is no real connection between pressure at a point and height
above the earth's surface, and the scale is therefore an approximation
25,000
20,000
HEIGHT
Feet
15,000
10,000
5,000
•
^
\
^r
\
/
/
\
/
V
20
80
40 60
TIME- Minutes
Fig 43. — Barogram taken during a flight.
100
120
only and was rather arbitrarily chosen. If 7i be the height in feet which
is marked on the aneroid barometer, and p is the relative pressure, a
standard atmosphere being at a pressure of 2116 lbs. per sq. foot, the
relation between In, and j? is
/i = - 62,700 logiop (15)
The relation is shown in tabular form in the chapter on the prediction and
analysis of aeroplane performance.
The aneroid barometers used for the more accurate aerodynamic
experiments are indicators only, and the readings are taken by the pilot
or observer. In some cases recording barometers are used, and Fig. 43
illustrates an example of the type of record obtained during a climb to the
ceiling and the subsequent descent. The rapid fall in the rate of climb is
clearly shown, for the aeroplane reached a height of 10,000 feet in 10 mins.,
but to climb an additional 10,000 feet, 27 mins. were required. The return
METHODS OF MEASUREMENT
88
to earth from this altitude of 24,000 feet occupied three-quarters of an
hour. The lag of the barometer is shown at the end of the descent, and
corresponds with an error in height of 200 or 300 feet, or about 1 per cent,
of the maximum height to which the aeroplane had climbed.
Revolution Indicators and Counters. — Motor-car practice has led to
the introduction of revolution indicators, and these have been adopted
in the aeroplane. Many instruments depend for their operation on the
tendency of a body to fly out under the influence of a centrifugal accelera-
tion, the rotating body being a ring hinged to a shaft so as to have relative
motion round a diameter of the ring. The ring is constrained to the shaft
by a spring, the amount of distortion of which is a measure of speed of
rotation of the shaft. Various methods of calibration of such indicators
are in use, and the readings are usually very satisfactory. For the most
accurate experimental work the indicator is used to keep the speed of rota-
tion constant, w^hilst the actual speed is obtained from a revolution counter
and a stop-watch.
The air-speed indicator, the aneroid barometer and the revolution
indicator are the most important instruments carried in an aeroplane,
both from the point of view of general utility and of accurate record of
performance. Many other instruments are used for special purposes, and
those of importance in aerodynamics will be described.
Accelerometer. — The most satisfactory accelerometer for use on aero-
planes is very simple in its main idea, and is due to Dr. Searle, F.R.S.,
working at the Royal Aircraft Establishment during the war. The essential
part of the instrilment is illustrated in Fig. 44, and consists of a quartz fibre
bent to a 'Semicircle and rigidly attached to a base block at A and B. If
the block be given an acceleration normal to the plane of the quartz fibre
the force on the latter causes a deflection of the point C relative to A and
Diam? fooo '"
(b)
Fig. 44. — Accelerometer.
B, and the deflection is a measure of the magnitude of the acceleration.
By the provision of suitable illumination and lenses an image of the point
C is thrown on to a photographic film and the instrument becomes re-
cording. The calibration of the instrument is simple : the completed
instrument is held with the plane of the fibre vertical, and the vertex then
lies at C as shown in Fig. 44 (b). With the plane horizontal the film record
shows Ci for one position and C2 for the inverted position, the differences
CCi and CC2 being due to the weight of the fibre, and therefore equal to
the deflections due to an acceleration of g, i.e. 32*2 feet per sec. per sec.
84 APPLIED AERODYNAMICS
The stiffness of the fibre is so great in comparison with its mass
that the period of vibration is extremely short, and the air damping is
sufficient to make the motion dead beat. As compared with the motions
of an aeroplane which are to be registered, the motion of the fibre is
so rapid that the instrumental errors due to lag may be ignored. Pig. 45
shows some of the results recorded, the accelerometer having been strapped
to the knee of the pilot or passenger during aerial manoeuvres in an
aeroplane.
In the records reproduced the unit has been taken as g, i.e. 32*2 feet
per sec. per sec, and in the mock flight between two aeroplanes it may be
noticed that four units or nearly 130 ft.-s.^ was reached. The interpre-
tation of the records follows readily when once the general principle is
appreciated that accelerations are those due to the air forces on the aero-
plane. To see this law, consider the fibre as illustrated in Fig. 44 (a) when
Held in an aeroplane in steady flight, the plane of the fibre being horizontal.
A line normal to this plane is known as the accelerometer axis, and in the
example is vertical. 8ince the aeroplane has no acceleration at all, the
fibre will bend under the action of its weight only and register g ; in
the absence of lift the aeroplane would fall with acceleration g, and the
record may then be regarded as a measure of the upward acceleration
which would be produced by the lift if weight did not exist. If the motion
of the aeroplane be changed to that of vertical descent at its terminal
velocity, the acceleration is again zero and the weight of the fibre does not
produce any deflection. Again it is seen that the acceleration recorded
is that due to the air force along the accelerometer axis, and this theorem
can be generalised for any motion whate^ver. The record then gives the
ratio of the air force along the accelerometer axis to the mass of the
aeroplane.
Coribider the pilot as an accelerometer by reason of a spring attachment
to the seat. His accelerations are those of the aeroplane, and his apparent
weight as estimated from the compression of the spring of the seat will be
shown by the record of an accelerometer. ^^^len the accelerometer
indicates g his apparent weight is equal to his real weight. At
four times g his apparent weight is four times his real weight, whilst
at zero reading of the accelerometer the apparent weight is nothing.
Negative accelerations indicate that the pilot is then held in his seat by
his belt.
Examining the records with the above remarks in mind shows that
oscillations of the elevator may be made which reduce the pilot's apparent
weight to zero, and an error of judgment in a dive might throw a pilot
from his seat unless securely strapped in. In a loop the tendency during
the greater part of the manoeuvre is towards firmer seating. Generally,
the first effect occurs in getting into a dive, and the second when getting
out. It will be noticed that in three minutes of mock fighting the great
preponderance of acceleration tended to firm seating, and on only one
occasion did the apparent weight fall to zero.
Levels. — The action of a level as used on the ground depends on the
property of fluids to get as low as possible under the action of gravity.
METHODS OP MEASUEEMENT
85
START
^ri'^kr
.^.^_>«v ^
1.1
M ^tnm I
' r - 1
TAKING OFF
-BUMPS WHILE FLYING LOW DOWN
.
aAa^
^
^vw
O6CILLATI0N OF ELEVATOR '.
:^^'^^
nj^^wApW
-^
QUICK DIVE AND FLATTEN OUT
ROLL
^3
SPIN
•w»>i4
^ i.<*j>.rfj^j
^_
IF"
LANDING TAXYJNG IN
Vertical divisions every 15 seconds. Horironi-al lines at- rnuii-iplesoF'
9-
l\
k A
A
. ^
na
1
1
iU
A
J
^i
V
i..^.
Iv
yv,
jf^
\A
Ai
'r\.
5^**
/(v
]^
V »- ■•
V
V
MOCK FIGHT
Fio. 45. — Accelerometer records.
86
APPLIED ABEODYNAMICS
In a spirit-level the trapped bubble of gas rises to the top of the curved
glass and stays where its motion is horizontal. In this way it is essentially
dependent on the direction of gravity and not its magnitude. The prin-
ciples involved are most easily appreciated from the analogy to a pendulum
which hangs vertically when the support is at rest. In an aeroplane the
support may be moving, and unless the velocity is steady the inclination
is affected. Eef erring to Fig. 46 (a) a pendulum is supposed to be suspended
about an axis along the direction of motion of an aeroplane, and P is the
projection of this axis. In steady motion the centre of gravity of the bob
B will be vertically below P ; if P be given a vertical acceleration a
and a horizontal acceleration /, the effect on the inclination can be
found by adding a vertical force ma and a horizontal force mj to the bob.
CWF
W=/^y
Tn(g^a)
Mertical.
Fig. 46. — The action of a cross -level.
The pendulum wiU now set itself so that the resultant force passes through
P, the inclination Q will be given by the relation
tan B =
g + a
(16)
and the pendulum will behave in all ways as though the direction of
gravity had been changed through an angle 6 and had a magnitude equal
to V(3 + a)^+p.
The accelerations of P are determined by the resultant force on
the aeroplane, i.e. as shown by Mg. 46 (6), by the lift, cross-wind force and
wfeight of the aeroplane. The equations of motion for fixed axes are
ma == L . cos ^ + C.W.F. sin (f) — mg . . . (17)
and w/ = L . sin 0 - C.W.F. cos .^ (18)
From equations (17) and (18) it is easy to deduce the further equation
C.W.F.
— / cos <l> + {g + a) Bin cf> =
m
(19)
METHODS OF MEASUEEMBNT
87
pwhere <f) is the inclination of the plane of symmetry of the aeroplane to
[the vertical.
From (19) follows a well-known property of the cross-level of an aero-
fplane, for if the aeroplane is banked so as not to be sideslipping the cross-
[wind force is zero, and
tan0 = -i- (20)
ii.e. the angle of bank of the aeroplane is equal to 6, the inclination of a
pendulum to the vertical. To an observer in the aeroplane the final
[^position of the pendulum during a correctly banked turn is the same as
[if it had originally been fixed to its axis instead of being free to rotate.
The deviation of a cross-level from its zero position is then an indication
[of sideslipping and not of inclination of the wings of the aeroplane to the
[ horizontal.
There is no instrument in regular use which enables a pilot to maintain
(an even keel. In clear weather the horizon is used, but special training
[is necessary in order to fly through thick banks of fog. By a combination
of instruments this can be achieved as follows : an aeroplane can only fly
straight with its wings level if the cross-level reads zero, and vice versa.
: The compass is not a very satisfactory instrument when used alone, as it
[is not sensitive to certain changes of direction and may momentarily give
i an erroneous indication. It is therefore supplemented by a turn indicator,
[which may either be a gyroscopic top or any instrument which measures
[the difference of velocity of the wings through the air. This instrument
[laakes it possible to eliminate serious turning errors and so produce a
condition in which the compass is reliable. Straight flying and a cross-
I level reading zero then ensures an even keel.
Aerodynamic Turn Indicator. — ^An instrument designed and made
[by Sir Horace Darwin depends on the measurement of the difference of
[velocity between the tips of the wings of an aeroplane as the result of
[.turning. The theory is easily developed by an extension of equation (8),
[where it was shown that the difference of pressure due to centrifugal force
[on the column of air in a horizontal rotating tube was
8p = ipVo^ (21)
[where p was the air density, v the velocity of the outer end of the pipe of
iwhich the inner end was at the centre of rotation. The difference of
[pressure between points at different radii is then seen to be
8p=^lp{v,^-Vi^) (22)
/here Vi is the velocity of the inner end of the tube. If an aeroplane
[has a tube of length I stretched from wing tip to wing tip, the difference
[of the velocities of the inner and outer wings is col cos (f> due to an angular
[velocity co, and equation (22) becomes
8p = pvcol cos ^ (23)
f "where v is the velocity of the aeroplane and (f> is the angle of bank. For
[slow turning cos 'f> is nearly unity, and the pressure difference between
88
APPLIED AEEODYNAMICS
the wing tips is proportional to the rate of turning of the aeroplane. To
this difference of pressure would be added the component of the weight
of the air in the tube due to banking were this latter not eliminated by the
arrangement of the apparatus. The tube is open at its ends to the atmo-
sphere through static pressure tubes on swivelling heads, and the pressure
due to banking is then counteracted by the difference of pressures outside
the ends of the tube. Turning of the aeroplane would produce a flow of
air from the inner to the outer wing, and the prevention of this flow by a
delicate pressure gauge gives the movement which indicates turning.
Gravity Controlled Air-speed Indicator. — The great changes of apparent
weight which may occur in an aeroplane make it necessary to examine
very carefully the action of instruments which depend for their normal
properties on the attraction of gravity. In the case of the accelero-
meter and cross-level the result has been to find very direct and simple
uses in an aeroplane, although these were not obviously connected with
(a)
Direction
Sf motion
Fig. 47. — The action of a gravity controlled air-speed indicator.
terrestrial uses. A special use can be found for a gravity controlled air-speed
indicator, but the ordinary instrument is spring controlled to avoid the
special feature now referred to. The complete instrument now under
discussion consists of an anemometer of the Pitot and static pressure tubes
type with connecting pipes to a U-tube in the pilot's cockpit. The U-tube
is shown diagrammatically in Fig. 47, the limbs of the gauge being marked
for static pressure and Pitot connections. When the aeroplane is in
motion the difference of pressure arising aerodynamically is balanced by
a head of fluid, the magnitude of this head la being determined for a given
aerodynamic pressure by the apparent weight of the fluid. The two tubes
of the gauge may be made concentric so as to avoid errors due to tilt or
sideways acceleration, and the calculations now proposed wiU take advantage
of the additional simplicity of principle resulting from the use of concentric
tubes.
The relation between the aerodynamic pressure and the head h can
be written as
hf>v^ = h.p,,{g cos di+ J) (24)
where h is the constant of the Pitot and static pressure combinations as
METHODS OF MEASUEEMENT 89
affected by inclination of the aeroplane, p is the air density, and v the velocity
of the aeroplane. On the other side of the equation, h is the head of
fluid, /3j^ the weight of unit volume of the fluid as ordinarily obtained, &i
the inclination of the gauge to the vertical, and / the upward acceleration
of the gauge glass along its own axis. In steady flight/ is zero and cos 9^
so nearly equal to unity that its variations may be ignored. Equation (24)
then shows that h is proportional to the square of the indicated air speed
which would be registered by a spring controlled indicator.
The special property of the gravity controlled air-speed indicator is seen
by considering unsteady motion. Fig. 47 (&) shows the necessary diagram
from which to estimate the value of /. The hquid gauge is fixed to the
aeroplane with its axis along the Une AG, and its inclination to the vertical
will depend on the angle of climb 6, the angle of incidence a, and the
angle of setting of the instrument relative to the chord of the wings oq.
The relation may be
^j = ^ 4- a — ao (25)
The forces on the aeroplane are its weight, mg, and the aerodynamic
resultant E acting at an angle y + 90° to the direction of motion. It
then follows that
mf = E cos (y — a 4- ao) —wgf cos ^1 . . . . (26)
or g cos di +/ = — cos (y — a + oq) . . . . (27)
and combining equations (24) and (27) gives the fundamental equation
for h.
h = -^ fcmp.^ (28)
p^E cos (y — a + ao)
As the result of experiments on aeroplanes it is known that the lift
L = Ecosy = fcii)^;2S (29)
where kj^ is known as a lift coefficient and depends only on the angle of
incidence of a given wing and not on its area S or speed v. Equation (28)
can then be expressed as
^ ^ m fc cos y ,g^.
p^S fei cos (y — a + tto)
The first factor of this expression is constant, whilst the second is a
function only of the angle of incidence if the engine and airscrew are stopped.
If the engine be running the statement is approximately true, a small
error in lift being then due to variation of airscrew thrust unless the air-
screw speed be kept in a definite relation to the forward speed.
The result of the analysis is to show that in imsteady flight as well as
in stf ady flight the reading of the gravity controlled air-speed indicator
depends on the angle of incidence of the aeroplane and not on the speed.
For all wings the quantity fci, has a greatest value ; cos y and
cos (y — a + tto) are nearly unity for a considerable range of angles, and the
ratio required by (30) is exactly unity when oq === a. The value of h then
90
APPLIED AEEODYNAMICS
has a minimum value for an aeroplane in flight, and this minimum gives the
lowest speed at which steady level flight can be maintained. The instru-
ment is therefore particularly suited to the measurement of " stalling speed."
Although not now used in ordinary flying, the advantages of an instrument
which will read angle of incidence on a banked turn or during a loop are
obvious for special circumstances. The advantage as an angle of incidence
meter is a disadvantage as a speed indicator, for there is no power to
indicate speeds after stalling. Given sufficient forward speed the control
of attitude is rapid, but the regaining of speed is an operation essentially
involving time, and the spring controlled air-speed indicator gives the pilot
earliest warning of the need for caution.
■tpnm
8
iO
12
13
14
15
16
Fig. 48. — Photo-manometer record.
Photo-manometer. — ^From the discussion just given of the air-speed
indicator it wiU be realised that a U-tube containing fluid may be used
to measure pressures if the aeroplane is in steady flight, and a convenient
apparatus for photographing the height of the fluid has been made and
METHODS OF MEASUEEMENT 91
used at the Eoyal Aircraft Establishment. A considerable number of
tubes is used, each of which communicates with a common reservoir at one
end and is connected at the other to the point at which pressure is to be
measured. In the latest instrument the tubes are arranged round a half-
cylinder and are thirty in number, and the whole is enclosed in a hght-
tight box. Behind the tubes bromide paper is wound by hand and rests
against the pressure gauge tubes ; exposure is made by switching on a
small lamp on the axis of the cylinder.
A diagram prepared from one of the records taken in flight is shown in
Fig. 48, which shows nineteen tubes in use. The outside tubes are connected
to the static pressure tube of the air-speed indicator, and the line joining
the tops of the columns of fluid furnishes a datum from which other pres-
sures are measured. The central tube marked P was commonly connected
to the Pitot tube of the air-speed indicator, whilst the tubes numbered 1-16
were connected to holes in one of the wing ribs of an aeroplane.
The method of experiment is simple : the bromide paper having been
brought into position behind the tubes, the aeroplane is brought to a steady
state and maintained there for an appreciable time, during which time the
lamp in the camera is switched on and the exposure made. The proportions
of the apparatus are sufficient to produce damping, and the records are
clear and easily read to the nearest one-hundredth of an inch.
Considerable use has been made of the instrument in determining the
pressures on aeroplane wings, on tail planes and in the shp streams of
airscrews.
Cinema Camera. — A method of recording movements of aircraft has
been developed at the Eoyal Aircraft Establishment by G. T. E. Hill, by
the adaptation of a cinema camera. The camera is carried in the rear seat
of an aeroplane, and the film is driven from a small auxiliary windmill.
This aeroplane is flown level and straight, and the camera is directed by
the operator towards the aeroplane which is carrying out aerial manoeuvres.
The possible motions of the camera are restricted to a rotation about a
vertical and a horizontal axis, and the position relative to the aeroplane is
recorded on the film. From the succession of pictures so obtained it is
possible to deduce the angular position in space of the pursuing aeroplane.
Analytically the process is laborious, but by the use of a globe divided into
angles the spherical geometry has been greatly simplified, and the camera
is a valuable instrument for aeronautical research.
Camera for the Recording of Aeroplane Oscillations. — ^A pinhole
camera fixed to an aeroplane and pointed to the sun provides a trace
of pitching or rolling according to whether the aeroplane is flying to or
from the sun or with the sun to one side. A more perfect optical camera
for the same purpose has been made and used at Martlesham Heath, the
pinhole being replaced by a cyhndrical lens and a narrow slit normal to
the fine image of the sun produced by the lens. The record is taken on
a rotating film, and a good sample photograph is reproduced in Fig. 49.
The oscillation was that of pitching, the camera being in the rear seat of
an aeroplane and the pilot flying away from the sun. At a time called
1 minute on the figure the pilot pushed forward the control column until
92
APPLIED AERODYNAMICS
the aeroplane was diving at an angle of nearly 20 degrees to the horizontal,
and then left the control column free. The aeroplane, being stable, began
to dive less steeply, and presently overshot the horizontal and put its nose
up to about 11 degrees. The oscillation persisted for three complete
periods before being appreciably distorted
by the gustiness of the air. The period
was about 25 seconds, and such a record
is a guarantee of longitudinal stability.
Fig. 50 is a succession of records of
the pitching of an aeroplane, the first of
which shows the angular movements of
the aeroplane when the pilot was keeping
the flight as steady as he was able. The
extreme deviations from the mean are
about a degree. The second record fol-
lowed with the aeroplane left to control
itself, and the fluctuations are not of
greatly different amphtude to that for
pilot's control. The periodicity is however
more clearly marked in the second record,
and the period is that natural to the aero-
plane. The third record shows the natural
period ; as the result of putting the nose of the aeroplane up the record
shows a well-damped oscillation, which is repeated by the reverse process
of putting the nose up.
Photographs of lateral oscillations have been taken, but for various
reasons the records are difficult to interpret, and much more is necessary
Fig. 49. — Stability record.
-lO
-5
o
-2
o
O
t
10.
CONTROLLED
^^^VHM^^g^iiH^
-10
I 2 3
Minutes.
-5
o
-2
o
O
o
2
10
UNCONTROLLED
^VV^A^vvA
o
I 2 3
Minutes.
Fig. 60. — Control record.
-10
-5
o
-2
o
0
i
IQ
C. D
O I 2 3
Minutes.
before the full advantages of the instrument are developed as a means of
estimating lateral stability.
Special Modifications of an Aeroplane for Experimental Purposes. —
Fig. 51 shows one of the most striking modifications ever carried
out on an aeroplane, and is due to the Eoyal Aircraft Establishment.
The body of a BE2 type aeroplane was cut just behind the rear cockpit,
METHODS OF MEASUBEMENT
93
I
k
94 APPLIED AEEODYNAMICS
and the tail portion was then hinged to the front along the underside of
the body. At the top of the body a certain amount of freedom of rotation
about the hinge was permitted, the conditions " tail up " and " tail down "
being indicated by lamps in the cockpit operated by electric contacts at
the limits of freedom.
To the rear portion of the body were fixed tubes passing well above
the cockpit and braced back to the tail plane by cables. From the top of
this tube structure wires passed through the body round pulleys in the
front cockpit to a spring balance. The pull in these wires was variable
at the wish of an observer in the front seat, and was varied during a flight
until contact was made, first tail up and then tail down as indicated by the
lamps. The reading of the balance then gave a measure of the moments
of the forces on the tail about the hinge. In order to leave the pilot free
control over the elevators without affecting the spring balance reading the
control cables were arranged to pass through the hinge axis.
The aeroplane has been flown on numerous occasions, and the apparatus
is satisfactory in use.
Several attempts have been made to produce a reliable thrust-meter
for aerodynamic experiments, but so far no substantial success has been
achieved. The direct measurement of thrust would give fundamental
information as to the drag of aeroplanes, and the importance of the subject
has led to temporary measures of a different kind. It has been found that
the airscrews of many aeroplanes can be stopped by stalling the aeroplane,
and at the Eoyal Aircraft Establishment advantage has been taken of this
fact to interpose a locking device which prevents restarting during a glide.
The airscrew when stopped offers a resistance to motion, but the airflow
is such that the conditions can be reproduced in a wind channel for an
overall comparison between an aeroplane and a complete model of it. It
has already been shown that the angle of glide of an aeroplane is simply
related to the ratio of lift to drag, and this furnishes the necessary key to
the comparison.
Laboratory Apparatus. The Wind Channel. — The wind channel is one
of the most important pieces of apparatus for aerodynamical research,
and much of our existing knowledge of' the details of the forces on aircraft
has been obtained from the tests of models in wind channels. The types
used vary between different countries, but all aim at the production of a
high-speed current of air of as large a cross-section as possible. The
usefulness depends primarily on the product of the speed and diameter of
the channel and not on either factor separately, and in this respect the
various designs do not differ greatly from country to country. Measuring
speeds in feet per sec. and diameters in feet, it appears that the product vD
reaches about 1000. The theory of the comparison will be appreciated by
a reading of the chapter on dynamical similarity, and except for special
purposes the most economical wind channels are of large diameter and
moderate speed, the latter being 100 ft.-s. and between the lowest and
highest flying speeds of a modern aeroplane.
Fig. 52 shows a photograph of an English type of wind channel as
built at the National Physical Laboratory. It is of square section and
I
METHODS OF MEASUBEMENT
95
Itands in the middle of a large room, being raised from the floor on a light
letal framework. The airflow is produced by a four-bladed airscrew
dven by electro-motor, and the airscrew is situated in a cone in the centre
the channel, the cone giving a gradual transition from the square forward
action to the circular section at the airscrew. The motor is fixed to the
ir wall of the building and connects by a line of shafting to the airscrew.
]he airscrew is designed so that air is drawn in to the trumpet mouth
shown at the extreme left of Fig. 52, passes through a cell of thin plates
to break up small vortices, and thence to the working section near the
open door. Just before the end of the square trunk is a second honeycomb
to eliminate any small tendency for the twist of the air near the airscrew
to spread to the working section. After passing through the airscrew the
air is deUvered into a distributor, which is a box with sides so perforated
that the air is passed into the room at a uniform low velocity. This part
10 20 30 Secs. 40 50 60
Ordinate- Perccnrage change of Velocity. Wind Channel wirhoul- distribui-or.
'^-^\^^'^\l^^i^/f\^^ ft
10 20 Secs. 30 40 50
Ordinate- Percentage change oF Velocity. WiND Channel with distributor.
Fig. 53. — The steadiness of the airflow in wind channels.
of the wind channel has an important bearing on the steadiness of the
airflow.
The speed of the motor is controlled from a position under the working
section, where the apparatus for measuring forces and the wind velocity
is also installed.
Over the greater part of the cross-section of the channel the airflow
is straight and its velocity uniform within the limits- of ± 1 per cent. The
rapidity of use depends to a large extent on the magnitude of the fluctua-
tions of speed with time, and Figs. 53 (a) and 53 (b) show the amount of these
in aparticular case when the channel was tested without a distributor
and with a good distributor. Without the distributor the velocity changed
by ± 5 per cent, of its mean value at very frequent intervals, and as this
would mean changes of force of ± 10% on any model held in the stream, it
would follow that the balance reading would be sufficiently unsteady to
be unsatisfactory. With the distributor the fluctuations of velocity rarely
96 APPLIED AEEODYNAMICS
exceeded ±0-5 per cent., or one-tenth of the amount in the previous
illustration.
A great amount of experimental work has been carried out on the design
of wind channels, and the reports of the Advisory Committee for Aero-
nautics contain the results of these investigations. Although the results
of wind-channel experiments form basic material for a book on aero-
dynamics the details of the apparatus itself are of secondary importance,
and the interested reader is referred for further details to the reports
mentioned above.
Aerodynamic Balances. — The requirements for a laboratory balance are
so varied and numerous that no single piece of apparatus is sufficient to
meet them, and special contrivances are continually required to cope with
new problems. Some of the arrangements of greatest use will be illustrated
diagrammatically, and again for details readers will be referred to the
reports of the Advisory Committee for Aeronautics, Eiffel and others.
The first observations of forces and moments which are required are
those for steady motion through the air, and in many of the problems,
symmetry introduces simplification of the system of forces to be measured.
For an airship the important force is the drag, whilst for the aeroplane,
lift, drag and pitching moment are measured. For the later problems of
control and stability, lateral force, yawing and rolling couples are required
when the aircraft is not symmetrically situated in respect to its direction
of motion through the air.
At a still later stage the forces and couples due to angular velocities
become important, and for lighter- than-air aircraft it is necessary to measure
the changes of force due to acceleration and the consequent unsteady
nature of the airflow. The problems thus presented can only be dealt with
satisfactorily after much experience in the use of laboratory apparatus,
but the main lines of attack will now be outlined.
Standard Balance for the Measurement of Three Forces and One Couple
for a Body having a Plane of Symmetry. — The diagram in Fig. 54 will
illustrate the arrangement. AB, AE and AF are three arms mutually at
right angles forming a rigid construction free to rotate in any direction
about a point support at A. The arm AB projects upwards through the
floor of the wind channel, and at its upper end carries the model the air
forces on which are to be measured. Downwards the arm AB is extended
to C, and this hmb carries a weight Q, which is adjustable so as to balance
the weight of any model and give the required degree of sensitivity to
the whole by variation of the distance of the centre of gravity below the
point of support at A. The arm AB is divided so that the upper part
carrying the model can be rotated in the wind and its angle of attack
varied ; this rotation takes place outside the channel.
The arms AE and AF are provided with scale pans at the end, and by
the variation of the weights in the scale pans the arm AB can be kept
vertical for any air forces acting. The system is therefore a " null .
method, since the measurements are made without any disturbance of the
position of the model.
Moment about the vertical axis AB is measured by a bell-crank lever
METHODS OP MEASUEEMENT
97
GHI, which rests against an extension of the arm AF and is constrained
by a knife-edge at H. The moment is balanced by weights in a scale pan
hanging from I. It is usually
found convenient to make
this measurement by itself,
and a further constraint is
introduced by a support J,
which can be raised into con-
tact with the end C of the
vertical extension AC. It is
not then necessary to have
the weights hung from E and
F in correct adjustment.
The force along the axis
AB can be measured by two
steelyards which weigh the
whole balance. These are
shown as KPN and CMO, the
points P and M being knife-
edges fixed to a general sup-
port from the ground. At C
and K the support to the
balance is through steel points,
and the weight of the balance
is taken by counterweights
hung from 0 and N. Varia-
tions of vertical force due to
wind on the model are mea-
sured by changes of weight in
the scale pan of the upper
steelyard.
Suitable damping arrangements are provided for each of the motions,
and the part of the arm AB which is in the wind is shielded by a guard
fixed to the floor of the channel.
Example of Use on an Aerofoil : Determination of Lift and Drag. — ^For
this purpose the arms KPN and CMO are removed and the arm IH is locked
so as to prevent rotation of the balance about a vertical axis. The aerofoil
is arranged with its length vertical, and is attached to the arm AB by a
spindle screwed into one end. A straight-edge is clamped to the underside
of the aerofoil, and by sighting, is made to He parallel to a fixed line on the
floor of the wind channel, this Une being along the direction of the wind.
The zero indicator on the rotating part of the arm AB is then set, and the
weights at Q, E and F are adjusted until balance is obtained with the
'requisite degree of sensitivity.
In order that this balance position shall not be upset by rotation of
the model about the arm AB it is necessary that the centre of gravity of
the rotating part shall be in the axis of rotation, and by means of special
counterweights this is readily achieved.
Fig. 54
98 APPLIED AEEODYNAMICS
The values of the weights in the scale pans at E and F then constitute
zero readings of drag and Hft. The arms AE and AF are initially set to be
along and at right angles to the wind direction within one-twentieth degree,
whilst the axis AB is vertical to one part in 6000. The wind is now pro-
duced, and at a definite velocity the weights in the scale pans at E and E
which are needed for balance are recorded ; the difference from the zero
values gives the lift and drag at the given angle of incidence. The model
is then rotated and the weights at E and F again changed, and so on for
a sufficient range of angle of incidence, say —6° to +24°.
Centre of Pressure. — ^For this measurement the lock to the arm IH is
removed and the vertical axis constrained by bringing the cup J into
contact with C. The weights on the scale pans at E and F are then in-
operative, and the weights in the scale pan at I become active. For the
angles of incidence used for lift and drag a new series of observations is
made of weights in the scale pan at I. From the three readings at each
angle of incidence the position of the resultant force relative to the axis
AB is calculated. The model being fixed to the arm AB, the axis of rota-
tion relative to the model is found by observing two points which do not
move as the model is rotated. This is achieved to the nearest hundredth
of an inch, and finally the intersection of the resultant force and the chord
of the aerofoil, i.e. the centre of pressure, is found by calculation from the
observations.
The proportions adopted for the supporting spindle are determined
partly by a desire to keep its air resistance very low and partly by an effort
to approach rigidity. The form adopted at the National Physical Labora-
tory is sufficiently flexible for correction to be necessary as a result of the
deflection of the aerofoil under air load. Almsot the whole deflection
occurs as a result of the bending of the spindle, and as this is round, the
plane of deflection contains the resultant force. A little consideration will
then show that the moment reading (scale pan at 1) h unaffected by
deflection, and that the lift and drag are equally affected. The corrections
to lift and drag are small and very easily applied, whereas corrections for
the aerodynamic effects of a spindle, although small, are very difficult to
apply. As a general rule it may be stated that corrections for methods
•of holding are so difficult to apply satisfactorily when they arise from
aerodynamic interference, that the lay-out of an experiment is frequently
determined by the method of support which produces least disturbance
of the air current. The experience on this point is considerable and is
growing, and only in prehminary investigations is it considered sufficient
to make the rough obvious corrections for the resistance of the holding
spindle.
Example of Use on a Kite Balloon. — ^For the symmetrical position of
a kite balloon the procedure for the determination of hft, drag and moment
is exactly as for the aerofoil, the model kite balloon being placed on its
side in order to get a plane of symmetry parallel to the plane EAF. Any
observer of the kite balloon in the open wiU have noticed that the craft
swings sideways in a wind, slowly and with a regular period. Not only
has it an angle of incidence or pitch, but an angle of yaw, and the condition
METHODS OF MEASUREMENT
99
can be represented in the wind channel by mounting the kite balloon
model in its ordinary position and then rotating the arm AB. There is
not now a plane of symmetry parallel to EAF, and the procedure is some-
what modified. The model is treated as for the aerofoil so far as the taking
of readings on the scale pans E, F and I is concerned, after which the arm
IH is locked and the two steelyards brought into operation for the measure-
ment of upward force.
The readings are now repeated with the model upside down in order to
allow for the lack of symmetry, and the new weights in the scale pans
E and F are observed. With the aid of Fig. 55 the reason for this can
be made clear. A' will be taken as a point in the model and also on the
axis of AB, and from A' are drawn Unes parallel to AE and AF. The
complete system of forces and moments on the model can be expressed
by a drag along E'A', a cross-wind force
along F'A', a hft along A'B', a rolling
couple L' about A'E' tending to turn
A'F' towards A'B', a pitching couple M'
tending to turn A'E' towards A'B', and
a yawing couple N' tending to turn A'F'
towards A'E'. Now consider the mea-
surements made on the balance. The
force A'B' was measured directly on the
two steelyards, whilst the couple N' was
determined by the weighing at I.
Denoting the weighings at E and F
by El and R2 with distinguishing dashes,
it will be seen that
Ei'=M'-fLdrag . . . (31)
md R2' =L' + Z . cross-wind force (32)
where I is the length AA'. Neither
reading leads to a direct measure of drag
or cross-wind force. Invert the model
about the drag axis A'E' so that A'F' becomes A'F" and A'B' becomes
A'B". As the rotation has taken place about the wind direction the
forces and couples relative to the model have not been changed in any
way, and it will follow that the drag and rolling moment are unchanged.
The lift, cross-wind force, pitching moment and yawing moment have the
same magnitude as before, but their direction is reversed relative to the
balance. Instead of equations (31) and (32) there are then two new
equations :
Bi" = - M' + L drag (33)
K2" = L' — i . cross-wind force .... (84)
It will then be seen from a combination of the two sets of readings that
Bi^ + Ei"
Fig. 55.
drag
21
(35)
100 APPLIED AEEODYNAMICS
cross-wind force = ^ , — — (36)
^,^-Rl;VR£ ^3^^
j^,^Bi--Bi" ^gg^
The result of the experiment is a complete determination of the forces and
couples on a model of unsymmetrical attitude, and the generalisation to
any model follows at once.
Although the principle of complete determination is correct the method
as described is not satisfactory as an experimental method of finding L' and
M', although it is completely satisfactory for drag and cross-wind force.
The reason for this is that the moment ixdrag is great compared with
M', and a small percentage error in it makes a large percentage error in
M'. If however I be made zero, equations (31) and (32) show that both
L' and M' can be measured directly, and various arrangements have been
made to effect this. No universally satisfactory method has been evolved,
and the more complex problems are dealt with by specialised methods
suitable for each case.
The balance illustrated diagrammatically in Fig. 54 is often used in
combination with other devices, such as a roof balance, and various special
arrangements will now be described.
Drag of an Airship Envelope. — ^For a given volume the airship envelope
is designed to have a minimum resistance, and for a given cross-section of
model the resistance is appreciably less than 2 per cent, of that of a flat
plate of the same area put normal to the wind. For sufficient permanence
of form and ease of construction models are made soHd and of wood, and
the resistance of a spindle of great enough strength and stiffness is a very
large proportion of the resistance of the model. Further than this, it is
found that such a spindle affects the flow over the model envelope to a
serious extent and introduces a spurious resistance up to 25 per cent, of
that of the envelope. As a consequence of the difficulties experienced at
the National Physical Laboratory a method of roof suspension was devised,
and is illustrated in Fig. 56. The model is held from the roof of the wind
channel by a single wire, the disturbance from which is very small, and the
drag is transferred to the balance by a thin rod projecting from the tail
and attached by a flexible joint to the vertical arm. The force is measured
by weights in the scale pan as in the previous case.
The weight of the model produces a great restoring force in its pendulum
action, but this is counteracted by making the balance unstable, so that
sufficient sensitivity is obtained. The correction for deflection of the
spindle is easily determined and applied. Further, the resistance of the
supporting wire can be estimated from standard curves, as its value is a
small proportion of the resistance to be measured.
The method has now been in use for a considerable period, and has
displaced all others as an ultimate means of estimating the drag of bodies
of low resistance.
METHODS OF MEASUEEMENT
101
■
^^^Hffhe model tends to become laterally unstable at high wind speeds,
s^ana in that case the single supporting wire to the roof is replaced by two
« wires meeting at the model and coming to points across the roof of the
hannel which are some considerable distance apart. The necessary
recautions to ensure the safety of a model are easily within the reach of
careful experimenter.
CAanne/ Roof
WIND
)«» >-
Supporting l/Vire
yyyyyyy//yy/y//yyy/y/yy^y///^//////////y/y'^y''y/y/yy^y/y'y''''yyy^'^'^'^yyy^y>'y^^
&
y Top of Balance
Fig. L6. — Measurement of the drag of an airship envelope model.
Drag, Lift and Pitching Moment of a Complete Model Aeroplane. —
The method described for the aerofoil alone becomes unsuitable for the
testing of a large complete model aeroplane. The ends of the wings are
usually so shaped that the insertion of a spindle along their length is difficult
for small models and the size inadequate for large models. Kecourse is
then made to a suspension on wires, the arrangement being indicated by
Fig. 57. The model is inverted for convenience, and from two points, one
on each wing, wires are carried to a steelyard outside the wind channel
and on the roof. These wires are approximately vertical and take the weight
of the model, and any downward load due to the wind. The pull in them
is measured by the load in the scale pan hung from the end of the steelyard
at U. The knife-edge about which the steelyard turns is supported on
stiff beams across the channel.
Another point of support is chosen near the end of the body, and in
the illustration is shown at B as situated on the fin. B is attached by a
flexible connection to the top of the standard balance, which is arranged
as indicated in the diagram to measure the direction and magnitude of
the force at B.
The angle of incidence of the wings is altered by a change of length
of the supporting wires RS, and although this wire is very long as compared
102
APPLIED AEKODYNAMICS
with the horizontal movements of E, it is necessary to take account of the
inclination of the supporting wires. The weight added at E measures the
drag, except for a small correction for the incUnation of the wires ES ;
the weight added at N measures the couple about E, and this point can be
chosen reasonably near to the desired place without disturbing the lay-out
of the experiment. The weights added at U and N measure the lift, with
an error which is usually negligible. The corrections for deflection of
^^^^^^^^^^^^^^r^^^y^y^y^^^^^y^yyyc^^^^^yyy^y^^^^^
WIND
^
'-<^^^^^^-
Channel Roof
^/'//////////////////
1
C M
' IT
^
Fig. 57. — Measurements of forces and couple on a complete model aeroplane.
apparatus and inchnation of wire involve somewhat lengthy formulae as
compared with the aerofoil method described earlier, but present no
fundamental difficulties. As an experimental method the procedure
presents enormous advantages over any other, and is being more exten-
sively used as the science of aerodynamics progresses.
Stability Coefficients. — It will be appreciated, once attention is \
drawn to the fact, that the forces on an oscillating aircraft are different ,
from those on a stationary aeroplane, and that the forces and moment on '
METHODS OF MEASUEEMENT
103
aeroplane during a loop depend appreciably on the angular velocity.
The experiment to be described applies more particularly to an aeroplane
for a reason given later.
By means of wires or any alternative method, an axis in the wind channel
is fixed about which the aeroplane model can rotate, and a rigid arm GFD
(Fig. 58) connected to the model is brought through the floor of the
channel and ends in a mirror at D. The angular position of the model at
any instant is then shown by the position of the image of the lamp H on
the scale K, the ray having been reflected from the mirror D. The arm
GFD is held to the channel by springs EF and FG, and in the absence of
wind in the channel will bring the model and the image on the scale to a
definite position. The model if disturbed will oscillate about this position
as a mean, and by adjustment of the. moment of inertia of the oscillating
Mirror
Pio. 58. — ^The measurement of resistance derivatives as required for the theory of stability.
system and the stiffness of the spring the period can be made so long that
the extremes of successive oscillations can be observed directly on the scale.
The mechanical arrangements are such that the damping of the oscil-
lation in the absence of wind is as small as possible, and considerable
success in the elimination of mechanical friction has been attained. When
reduced as much as possible the residual damping is measured and used
as a correction. In the description to follow the instrument damping will
be ignored.
The diagram inset in Fig. 58 wiU show why the forces and moments
on the model depend on the oscillation. A narrow flat plate is presumed
to be rotating about a point 0, from which it is distant by a distance I.
If the angular velocity be g then the velocity of the plate normal to the
current will be Zg,and the relative wind will be equal to ig and in the opposite
direction. Compounding this normal velocity with the wind speed V
104 APPLIED AERODYNAMICS
shows a wind at an inclination a such that tan o,=M, and this will produce
both a force and a couple opposing the angular elocity. If the angle is
small the force on the plate wiU be proportional to the angle, and also to
the square of the speed, and hence proportional to the product of the
angular velocity and the forward speed.
The equation of motion of the model in a wind may then be expressed as
Be+fiYe-{-he = o (39)
where B is the moment of inertia, /x a constant depending on the lengths I
and areas of the parts of the model, particularly the tail, h a constant
depending on the stiffness of the spring, and 6 the angular deflection of
the model. The q used earlier is equal to 6.
The solution of equation (39) can be found in any treatise on differential
equations, and is
^0 sin e
where 6q is the value of 6 at zero time, and t is a constant giving the phase
at zero time. For the present it is sufficient to note that equation (40)
represents a damped oscillation of the kind illustrated in Fig. 58. At
zero time the value of 6 is shown by the point A and is a maximum. The
other end of the swing is at B, and the oscillation continues with decreasing
amplitude as the time increases. The curve has two well-known charac-
teristics ; the time from one maximum to the next is always the same as
is the ratio of the amplitudes of successive oscillations. The changes of
the logarithms of the maximum ordinates are proportional to the differences
of the times at which they occur, and the constant of proportionality is
known as the " logarithmic decrement."
In the experiment the measurement of the logarithmic decrement is
facilitated by the use of a logarithmic scale at K. The ends of successive
swings are observed on this scale, and the observations are plotted against
number of swings. The slope of the line so obtained divided by the time
of a swing is the logarithmic decrement required, and from equation (40)
is equal to ^^ . This expression shows that the damping is proportional
to the wind speed, and the experimental results fully bear out the property
indicated.
Before the value of /x can be deduced it is necessary to determine the
moment of inertia B, and this is facilitated by the fact that in any practicable
apparatus the value of k does not depend appreciably on the wind forces,
k
and that the ratio is very much greater than the square of the logarith-
mic decrement. With these simplifications equation (40) shows that
/B
periodic time =3 277 W^ (41)
METHODS OF MEASUEEMENT 105
k is determined by applying a known force at F and measuring the angular
deflections. B is then calculated from the observed periodic time and
equation (41). Even were the air forces appreciable the determination
of B would present little additional calculation.
The observations have now been reduced to give the value of /x, and
consequently the couple fjuYd or [xYq which is due to oscillation of a model
in a wind. Corrections for scale are then applied in accordance with the
laws of similar motions.
Some of the quantities which have been determined in this way are
very important in their effects on aeroplane motion. The one just de-
scribed is the chief factor in the damping of the pitching of an aeroplane.
Others are factors in the damping of rolling and yawing.
An allied series of measurements to those in a wind channel can be
made by tests on a whirling arm. One of the effects is easily appreciated.
If an aeroplane model be moved in a circle with its wings in a radial direction
the outer wing will move through the air faster than the inner, and if the
wings are at constant angle of incidence this will give a greater Hft on the
outer wing than on the inner. The result is a rolling moment due to turn-
ing. In straight flying an airman may roll his aeroplane over by producing
a big lift on one side, but this is accompanied by an increased drag and a
tendency to yaw. Hence rolling may produce a yawing moment, and in a
wind channel the amount may be found by rotating a model about the
wind direction and measuring the tendency of one wing to take a position
behind the other. The apparatus for the last two factors has not yet been
standardised, and few results are available. Further reference to the factors
is given in another chapter ; they are generally referred to in aeronautical
work as " resistance derivatives."
Airscrews and Aeroplane Bodies behind Airscrews. — The method
to be described is applicable particularly simply when the model is of such
size that an electromotor for driving the airscrew is small enough to be
completely enclosed in the model body. In other cases the power is
transmitted by belting or gear, and although the principle used is the same
the transmission arrangements introduce troublesome correction in many
cases owing to their size and the presence of guards. The diagrammatic
arrangement is shown in Fig. 59. The motor is supported by wires, a pair
from points on the roof coming to each of the point supports at C and D.
This arrangement permits of a parallel motion in the direction CD, together
with a rotation about an axis through the points C and D. Movement
under the action of thrust and torque is prevented by attaching the rod
DM to the aerodynamic balance by a flexible connection. The thrust is
measured by weights in the scale pan at E, and the torque by the weights
below F.
The body has a similar but independent suspension from the roof, and
as shown, rotation about EP and movement along EP is prevented by
the wires from L to the floor of the wind channel. By such means the body
is fixed in position in the channel irrespective of any forces due to thrust
or torque.
The speed of the airscrew is measured by revolution counter
106
APPLIED AEBODYNAMICS
and stopwatch, the counter being arranged to transmit signals to a
convenient point outside the channel. In order to keep the speed steady-
it is usual to employ some form of electric indicator under the
control of the operator of the electromotor regulating switches. Torque
and thrust are rarely measured simultaneously, one or other of the beams
AF or AE being locked as required. To make a measurement of thrust
the scale pan at E is loaded by an arbitrary amount, and the wind in the
channel turned on and set at its required value. The airscrew motor is
G J H K
Fig. 59. — The measurement of airscrew thrust and torque.
then started, and its revolutions increased until the thrust balances the
weight in the scale pan ; the revolutions are kept constant for a suffi-
cient time to enable readings to be taken on a stopwatch. The readings
are repeated for the same wind speed but other loads in the scale pan,
and finally the scale-pan reading for no wind and no airscrew rotation is
recorded.
After a sufficient number of observations at one wind speed the range
may be extended by tests at other wind speeds, including zero, before the
beam AE is locked and the torque measured on AF. Torque readings are
obtained in an analogous manner to those of thrust.
METHODS OF MEA8UEEMENT 107
It will be noticed that in this experiment the influence of the body on
thrust and torque is correctly represented. In one instance wings and
undercarriage were held in place in the same way as the body.
The resistance of the body in the airscrew slip stream is measured by
releasing the tie wires SL and TL and connecting L to the top of the balance.
M is disconnected from the balance and tied to the floor of the channel so
as to fix the motor. For a given wind speed and a number of speeds of
rotation of the airscrew the body resistance is measured by weights in the
scale pan at E. It is found that the increase in the body resistance is
proportional to the thrust on the airscrew and may be very considerable.
The effect of the body on the thrust and torque of the airscrew is relatively
small ; both effects are dealt with more fully in later chapters.
The apparatus is convenient and accurate in use, and when it can be
used has superseded other types in the experiments of the National
Physical Laboratory. For smaller models finahty has not been reached,
and aU methods so far proposed offer appreciable difficulties. In this
connection the provision of a large wind channel opens up a new field of
accurate experiment on complete models in that the airscrew, hitherto
omitted, can be represented in its correct running condition.
Measurement of Wind Velocity and Local Pressure. — The pressure
tube illustrated in Fig. 40 is used as a primary standard anemometer,
and during calibration of a secondary anemometer is placed in the wind
channel in the place normally occupied by a model. This secondary
anemometer consists of a hole in the side of the channel, and the difference
between the pressure at this hole and the general pressure in the wind
channel building is proportional to the square of the speed. The special
advantage of this secondary standard is that it allows for the determina-
tion of the wind speed without obstructing the flow in the channel, and
only a personal contact with the subject can impress a full realisation of
the effect of the wind shadows from such a piece of apparatus as an
anemometer tube. A very marked wind shadow can be observed 100
diameters of the tube away.
For laboratory purposes the pressure differences produced by both the
primary and secondary anemometers are measured on a sensitive gauge
of the special type illustrated in Fig. 60. Designed by Professor Chattock
and Mr. Fry of Bristol the details have been improved at the National
Physical Laboratory until the gauge is not only accurate but also con-
venient in use. The usual arrangement is capable of responding to a differ-
ence of pressure of one ten-thousandth of an inch of water, and has a total
range of about an inch. For larger ranges of pressure a gauge of different
proportions is used, or the water of the normal gauge is replaced by mercury.
The instrument does not need calibration, its indications of pressure being
calculable from the dimensions of the parts.
In principle the gauge consists of a U-tube held in a frame which may
be tilted, and the tilt is so arranged as to prevent any movement of the
fluid in the U-tube under the influence of pressure apphed at the open
ends. The base frame is provided with three levelling screws which support
it from the observation table. The frame has, projecting upwards, two
108
APPLIED AEKODYNAMICS
spindles ending in steel points and a third point which is adjustable in
height by a screw and wheel, and the three points form a support for the
upper frame. A steel spring at one end and a guide at the other are
sufficient with the weight of the frame to completely fix the tilting part
in position. Eigidly attached to this upper frame is the glasswork which
essentially forms a U-tube ; to facihtate observation the usual horizontal
limb is divided, one part ending inside a concentric vessel which is connected
to the other part of the horizontal limb. Above the central vessel is a
further attachment for the filling of the gauge. Were the central vessel
completely filled with water, flow from one end of the gauge to the other
would be possible without visible effect in the observing microscope shown
as attached to the tilting frame. Incipient flow is made apparent by the
introduction of castor oil in the central vessel for a distance sufficient to
cover the otherwise open end of
the inner tube. The surface of
separation of the water and
castor oil is very sharply defined
and any tendency to distortion
is shown by a departure from
the cross wire of the microscope,
and is corrected by a tilting of the
frame. In this way the effects of
viscosity and the wetting of the
surfaces of the glass vessels are
reduced to a minimum. The film
is locked by the closing of a tap
in the horizontal limb, and the
gauge then becomes portable.
A point of practical conveni-
ence is the use of a salt-water
solution of relative density 1-07
instead of distilled water, as the
castor oil in the central vessel
Fio. 60. — Tilting pressure gauge.
then remains clear for long periods. A gauge of this construction
carefully filled will last for twelve months without cleaning or refilHng. A
fracture of the castor oil water surface is followed by a temporarily dis-
turbed zero, but full accuracy is rapidly recovered. The zero can be reset
by the levelling screws after such break, and ultimately by transference
of salt water from one limb of the U-tube to the other.
As used in the wind channels of the National Physical Laboratory
a reading of about 600 divisions is obtained at a wind speed of 40 ft.-s.,
and the accuracy of reading is one or two divisions determined wholly by
the fluctuations of pressure. Speeds from 20 ft.-s. to 60 ft.-s. are read
with all desirable accuracy on the same gauge ; lower speeds are rarely
used, and gauges of the same type but larger range are used up to the
highest channel speeds reached.
Chattock tilting gauges have also been used extensively for the measure-
ment of local pressures on models of aircraft and parts of aircraft. If
METHODS OF MEASUREMENT 109
the wing section be metal., holes are drilled into it at suitable points, each
of which is then cross- connected to a common conduit tube. The whole
system is arranged to have an unbroken surface in the neighbourhood of
the surface holes, and the conduit pipe is led to some relatively distant
point before a gauge connection is provided. Before beginning an experi-
ment all the surface openings are closed with soft wax or *' plasticene,"
and the whole system of tubing tested for airtightness. Until this has
been attained no observations are taken, and in the case of a complex
system it is often difficult to secure the desired freedom from leakage.
Once satisfactory, the surface holes are opened one at a time and the
pressure at this point measured for variations of the various quantities,
such as wind speed, angle of incidence, angle of yaw, etc.
The connection made as above determines the pressure on one limb
of the tilting gauge, but it is clear that the readings of the gauge will
also depend on the pressure appUed at the other limb. This pressure,
usually through a secondary standard, is almost invariably taken as the
pressure in the static pressure tube of the standard anemometer when in
the position of the model. This static pressure differs little from the
pressure at the hole in the side of the wind channel, which is the point
usually connected to the other Hmb of the tilting gauge. A standard table
of corrections brings the pressure to that of the static-pressure tube
of the standard anemometer.
For large wood models the tube system used in pressure distribu-
tion is made by inserting a soft lead composition tube below the
surface and making good by wax and varnish. Holes at desired points
are made with a needle and closed with soft wax when not in use. This
method is applied to airship models in most cases, but a variant of value
is the use of a hollow metal model, the inside of which is connected
to the tilting gauge, and through the shell of which holes can be drilled
as required.
The determination of local pressures in this way is one of the simplest
precise measurements possible in a wind channel. If the number of
observations is large the work may become lengthy, but errors of import-
ance are not easily overlooked. Any errors arise from accidental leakage,
and general experience provides a check on this since the greatest positive
pressure on a body is calculable, and the position at which it occurs is known
with some precision. Measurements have been made over the whole
surface of a model wing for a number of angles of incidence, over an air-
ship envelope for angles of yaw, over a cyhnder and over a model tail
plane. The latter experiment covered the variations of pressure due to
inclination of the elevators. An example will be given later showing the
accuracy with which the method of pressure distribution can be used to
measure the lift and drag of an aerofoil. It wiU be understood that skin
friction is ignored by the method, and that the pressure measured is that
normal to the surface. A series of experiments by Fuhrmann at Gottingen
University showed that for small holes the reading of pressure was inde-
pendent of the size of the hole, and the conclusion is supported by experi-
ments at the National Physical Laboratory.
no APPLIED AEEODYNAMICS
The Water Resistance o! Flying-Boat Hulls. — Experiments on the
resistance of surface craft are made by towing a model over still water.
The general arrangement of the tank consists of a trough some 500 to 600
feet long, 30 feet wide and 12 feet deep. Along the sides are carefully laid
rails which support and guide a travelUng carriage, the speed of which is
regulated by the supply to the electromotors mounted above the wheels.
The first 100 to 150 feet of the run are required to accelerate to the final
speed, and a rather larger amount for stopping the carriage at the end of
the run. Speeds up to 20 feet per sec. can be reached, and the time avail-
able for observation isthen limited to fifteen seconds, so that all the measure-
ments are most conveniently taken automatically. At lower speeds the
time is longer, and direct observation of some quantities comes easily within
the limits of possibility.
The water resistance of a flying-boat hull is associated intimately with
the production of waves, and the law followed in the tests is known as
Froude's law, and states that the speed of towing a model should be less
than that of the full-size craft in the proportion of the square root of the
relative linear dimensions. This rule is dealt with in greater detail in the
chapter on dynamical similarity, where it is shown that once the law is
satisfied the forces on the full scale are deduced from those on the model
by multiplying by the cube of the relative linear dimensions.
The flying boat at rest is supported wholly by the reaction of the water,
and the displacement is then equal to the weight of the boat. As the air
speed increases, part of the weight is taken by the wings until ultimately
the whole weight comes on to the wings and the flying boat takes to the
air. The testing arrangements are shown diagrammatically in Fig. 61.
Points of attachment of the apparatus to the tank carriage are indicated
by shaded areas. The model of the flying- boat hull is constrained to move
only in a vertical plane, but is otherwise free to take up any angle of
incidence and change of height under the action of the forces due to motion.
The measuring apparatus is attached at A by free joints, the resistance
being balanced by a pull in the rod AB, and the air lift from the wings being
represented by an upward pull in the rod AD. The trim of the boat can
be changed by the addition of weight at P, and the angle for each trim is
read on the graduated bar N, which moves with the float.
The upper end of the rod AD moves in a vertical guide, and a wire cord
passing over pulleys to a weight 0 gives the freedom of vertical adjustment
mentioned, together with the means of representing the air lift. The pull
in the rod AB is transmitted to a vertical steelyard EFG and is balanced
in part by a weight hung from G, and for the remainder by the pull in the
spring HJ. From J there is a rod JK operating a pen on a rotating drum,
whilst other pens at L and M record time and distance moved through the
water. The record taken automatically is sufficient for the determination
of speed and resistance.
Since the model is free to rotate about an axis through A, the observa-
tions of pull in AB and of lift in AB are sufficient, in addition to the obser-
vation of inchnation, to completely define the forces of the model at any
speed. The conditions of experiment can be varied by changes in the weights
METHODS OF MEASUREMENT
111
o
wwb. §!R^
QO
112
APPLIED AEEODYNAMICS
at 0 and P, and the whole of the possibilities of motion for the particular
float can be investigated.
The observations include a general record of the shape of the waves
formed, the tendency to throw up spray or green water, or to submerge
the bow. Occasionally more elaborate measurements of wave form have
been made. Flying boats of certain types bounce on the water from point
to point in a motion known as " porpoising," and by means of suitable
arrangements this motion can be reproduced in a model.
Forces due to Accelerated Fluid Motion. — In aviation it is usual
to assume that the forces on parts of aeroplanes depend only on the veloci-
ties of the aeroplane, linear and angular, and are not affected appreciably
by any accelerations which may occur. A little thought will show that
this assumption can only be
justified as an approxima-
tion, for acceleration of the
aircraft means acceleration
of fluid in its neighbourhood,
with a consequent change
of pressure distribution and
total force on the model. In
recent years the examination
of the effects of acceleration
on aerodynamic forces has
become prominent in the
consideration of the stability
of airships. To estimate its
importance recourse is had
to experiments on the oscil-
lations of a body about a
state of steady motion, and
the principle may be illus-
trated for a sphere. Fig. 62
shows an arrangement which
can be used to differentiate between effects due to steady and to unsteady
motion. The sphere is mounted on a pendulum swinging about the point
A, the sphere itself being in some liquid such as water. On an extension
of the pendulum at D is a counterweight which brings the centre of mass
of the pendulum to A, so that the whole restoring couple is due to the
springs at EP and EG and the eccentric counterweight 0.
The moment of inertia about A will be denoted by I, and the oscilla-
tions will be such that 6 is always a small angle and within the limits
sin 6 ='6 and cos ^ = 1 . The equation of motion may be written as
Fig. 62. — Forces due to acceleration of fluid motion.
Id = &Wi - hd -j{v + w, e)
(42)
where 6Wi is the couple due to the counterbalance weight at C, kd is the
restoring couple arising from the springs at EE and EG, and/(v -\-ld, 6) is the
hydrodynamic couple. The linear velocity of the centre of the sphere is
METHODS OF MEASUREMENT 113
V + W, whilst the hnear acceleration is proportional to 6. A somewhat
similar equation to (42) could be written down in which 6 was not restricted
to be small, but the general solution is unknown until / is completely
specified. With the special assumption / can be expanded in powers of
6 and 6 and powers higher than the first neglected, leading to
j{^ + ie/d)^j{v,o)-\-ld^^4-d% . . . .(43)
where /(y, o) is the hydrodynamic couple when the motion is steady. The
counterbalancing couple 6Wi will be taken equal to/(y, o) as a condition
of the experiment, and equation (42) becomes
The resulting motion indicated is a damped oscillation of the type
already dealt with in equation (89). The logarithmic decrement and the
periodic time are
5/
r-^
dV 1 rrt .^ / / ^
log dec. = -7- , and T = ^n / -j + (bg dec.)2 (45)
2(l + ^) / ./ 1+^
dO' I 'V d$
and from the observation of the logarithmic decrement and the periodic
if
time the value of — can be deduced from (45). I may be determined
de
by an experiment in air (or vacuo if greater refinement is attempted),
whilst k is measured as explained in connection with equation (39). It
will be noticed that the acceleration coefficient -4. occurs as an addition
de
to the moment of inertia, and might be described as a " virtual moment
of inertia." In translational motion it would appear as a " virtual mass."
The idea of virtual mass is only possible in those cases for which / can be
expanded as a linear function of acceleration. The case of small oscilla-
tions is one important instance of the possibility of this type of expansion.
In the case of the sphere the virtual mass appears to be about 80 per
cent, of the displaced fluid ; for an airship moving along its axis the
proportion is about 25 per cent., and for motion at right angles over 100
per cent. The accelerations of an airship along and at right angles to its
axis are therefore reduced to three-quarters and half their values as esti-
mated by a calculation which ignores virtual mass. On the other hand
no appreciable correction for heavier-than-air craft is suspected, and a
few experiments on flat plates show that the efifect of accelerations of the
fluid motion on the aerodynamic forces is not greater than the accidental
error of observation.
Model Tests on the Rigging of an Airship Envelope.—Calculations
relating to the rigging of the car of a non-rigid airship to the envelope
become very complex when they are intended to cover flight both on an
I
114
APPLIED AEEODYNAMICS
even keel and when inclined as the result of pitching. Advantage is taken
of a theorem first propounded in 1911 by Harris Booth in England and by
Crocco in Italy. A model of the envelope is made with rigging wires
attached, and is held in an inverted position by the wires, which pass over
pulleys and carry weights at their free ends. The model is filled with water,
and a sufficient pressure apphed to the mterior of the envelope by con-
nection to a head of water.
The arrangement is shown diagrammatically in Fig. 63, the number
of wires having been chosen only for illustration and not as representing
any real rigging. A beam NO carries a number of pulleys F, E, D, which
can be adjusted in position along the beam so as to vary the inclinations
Fia. G3. — Experiment to determine the necessary gas pressure in a non-rigid airship.
of the rigging wires AF, BE and CD. The tensions in these rigging wire
are produced by weights K, H and G. The model being inflated with water!
the pressure can be varied by a movement of the reservoir L, and can b^
measured on the scale M. The points F, E and D will be on the car of ai
airship, and the geometry of the rigging and the loads in the wires will be*
known approximately from calculation or general experience. Once this
point has been reached an experiment consists of the gradual lowering of
the reservoir L until puckering of the fabric takes place at some point or
other. By carefully adjusting the positions of the rigging wires and the
loads to be taken by them it may be possible to reduce the head of water
before puckering again takes place, and by a process of trial and error the
best disposition of rigging is obtained.
METHODS OF MEASUEEMENT 115
The relation of the experiment to the full scale is found by the principles
of similarity. The shape of the envelope is fixed by the difference between
the pressures due to hydrogen and those due to air. The internal pressure
can be represented by the effect of the head in a tube below the envelope,
the length of the hydrogen column produced being an exactly analogous
quantity to the length of the column of water in the model experiment.
In the model the shape of the envelope depends on the difference between
water and air, and the pressures for a given head are 900 times as great
as that for hydrogen and air at ground-level, or 1050 times as great as at
10,000 feet. The law of comparison states that the stresses in the fabric
of the model envelope will be equal to those in the airship if the scale is
VOOO, i.e. 30, for ground-level, or Vi050, i.e. 32-4, for 10,000 ft. The
necessary internal pressure to prevent puckering of the airship envelope
fabric is calculated from the head of hydrogen obtained by scaling up the
head of water.
The method neglects the weight of the fabric, but the errors on this
account do not appear to be important.
CHAPTEE IV
DESIGN DATA FROM THE AERODYNAMICS LABORATORIES
PAET I. — Straight Flying
The mass of data relating to design, particularly that collected under the
auspices of the Advisory Committee for Aeronautics, is very considerable
and will be the ultimate resort when new information is required. The
reports and memoranda have been collected over a period of ten years,
part of which was occupied by the Great War. To this valuable material
it is now becoming essential to have a summary and guide, which in itself
would be a serious compilation not to be compressed into even a large
chapter of a general treatise. Some general line of procedure was neces-
sary therefore in preparing this chapter in order to bring it within reason-
able compass, and in making extracts it was thought desirable in the
first place to give detailed descriptive matter covering the whole subject
in outline. In scarcely any instance has a report been used to its full
extent, and readers will find that extension in specific cases can be obtained
by reference to original reports. Although detailed reference is not given,
the identity of the original work will almost always be readily found in
the published records of the Advisory Committee for Aeronautics.
A second main aim of the chapter has been the provision of enough
data to cover all the various problems which ordinarily arise in the aero-
dynamic design of aircraft, so that as a text-book for students the volume
as a whole is as complete as possible in itself.
The chapter is divided into two parts, which correspond with a natural
physical division. In the first, " Straight Flying," the measurements
involved are drag, lift and pitching moment, and have only passing refer-
ence to axes of inertia. " Non-rectilinear flight " is, however, most suit-
ably approached from the point of view of forces and moments relative
to the moving body, and the second part of the chapter opens with a
definition of body axes and the nomenclature used in relation to motion
about them. The first part of the chapter is not repeated in new form in
the second, as the transformations are particularly simple and it is only in
the case of complete models that they are required. In its second part
this chapter, in addition to dealing with the data of circling flight, gives
some of the fundamental data to which the mathematical theory of
stability is applied.
Wing Forms. — The wings of an aeroplane are designed to support its
weight, and their quality is measured chiefly by the smallness of the re-
sistance which accompanies the lift. The best wings have a resistance
which is little more than 4 per cent, of the supporting force. Almost the
116
DESIGN DATA FEOM AEKODYNAMICS LABOEATOEIES 117
'^Bwhole of our knowledge of the properties of wing forms as dependent on
^^shape and the combinations of more than one pair of wings is derived from
tests on models and is very extensive. The most that can ever be expected
from flight tests is the determination of wing characteristics in a limited
number of instances, and it is fortunate for the development of aeronautics
that the use of models leads to results applicable to the full scale with httle
uncertainty. The theory of model tests and a comparison with full scale
is given in the chapter on Dynamical Similarity, and in the present chapter
typical examples are selected to show how form affects the characteristics
ot aeroplane wings without special reference to the changes from model
to full scale.
Wing forms, owing to their importance, are described by a number of
terms which have been standardised by the Eoyal Aeronautical Society,
Some of these are reproduced below, and are accompanied by ex-
planatory sketches in Figs, 64
and 65. Wing forms may be
so complex that simple defini-
tion is impossible, but in all
cases the geometry can be
fixed by sufficiently detailed
drawings. The complex defini-
tions are less important than,
and follow so naturally from,
the simple ones that they will
be ignored in the definitions
now put forward, and readers
are referred to the Glossary of
the Eoyal Aeronautical Society
for them.
Geometry of Wings : Defini-
tions.— The simplest form of
wing is that illustrated in Fig.
64 (a) by the full lines. In plan
the projection is a rectangle
of width G and length ^
(U)
ANGLE OF SWEEP BACK
(b)
<C)
-DIHEDRAL ANGLE
Two
(d)
Fio. 64.
ANGLE OFJ
FORWARD
STAGGER
wings together make a plane
of " span " s and " chord " c.
In the standard model s is
made equal to six times c, and the ratio is known as the " aspect ratio."
A section of the wings parallel to the short edges is made the same as
every other and is called the " wing section." The area of the projection,
i.e. sxc, is the " area of the plane " and has the S5'mbol S.
Departures from this simple standard occur in all aeroplanes, the
commonest change being the rounding of the wing tips. A convenient
way of accurately recording the shape is illustrated in Fig. 80, where
contours have been drawn. The leading edges of the wings may be
inclined in the pair which go to form a plane, and the inclinations are
118 APPLIED AERODYNAMICS
called the angle of sweepback if in plan, Fig. 64 (&), and dihedral angle
if in elevation, Fig. 61 (c).
When two planes of equal chord are combined the perpendicular
distance between the chords is called the- " gap," whilst the distance of
the upper wing ahead of the lower is defined by the " angle of stagger,"
Fig. 64 (d). Similar definitions apply to a triplane.
For tail planes, struts, etc., the chord is taken as the median line of a
section, and in general the chord of an aerofoil is the longest line in a section,
and the area its maximum projected area.
With these definitions it is possible to proceed with the description of
the forces on a wing in motion through the air, and an account of the
tables and diagrams in which the results of observation are presented.
Aerodynamics of Wings : Definitions (Fig. 65).— In the standard model
wing the attitude relative to the wind is fixed by the inchnation of
the chord of a section to the direction of the relative wind. The angle a is
known as the " angle of incidence." The forces on the wing in the standard
atmosphere of a wind
channel are fixed by the
angle a, the wind speed V,
and the area of the model.
No matter what the rela-
tion between the angle,
^.^^^^ r7>v^ velocity and forces, the
^"""■"T^IT'''^'^---^^ latter can always be com-
WIND DIRECTION O 4 r^--..^^^*^^ i , i i. j u
pletely represented by a
force of magnitude E, Fig.
65, in a definite position
AB. Various alternative
methods of expressing this possibility have current use. The resultant R
may be resolved into a lift component L normal to the wind direction and
a drag component D along the wind. If y be the angle between AB and
the normal to the wind direction, it will be seen that the relation between
L and D and R and y is
L = Rcosy, D=Rsiny (1)
The position of AB is often determined by the location of the point C,
which shows the intersection with the chord of the section. It is equally well
defined by a couple M about a point P at the nose of the wing, M being due
to the resultant force R acting at a leverage p. The sign is chosen for
convenience in later work. The point P may be chosen arbitrarily ; in
single planes it is usually the extreme forward end of the chord, in biplanes
the point midway between the forward ends of the chords, and in triplanes
the forward end of the chord of the middle plane.
The next step in representation arises from the result of experiments.
It is found that for all sizes of model and for all wind speeds, the angle y
is nearly constant so long as a is not changed, and that the ratio CP to PQ
is also little affected. On the other hand, the magnitude of R is nearly
proportional to the plane area and to the square of the speed. On theoretical
I
DESIGN DATA FEOM AERODYNAMICS LABORATORIES 119
grounds it is found that the magnitude is also proportional to the density
of the air. Putting those quantities into mathematical form shows that
CP R
PQ,y,and^-g-^-, (2)
are all nearly independent of the size of the model or the wind speed
during the test. The quantities are therefore peculiarly well suited for
a comparison of wing forms and the variation of their characteristics
with angle of incidence. The first quantity is clearly the same whether C P
and PQ are measured in feet or in metres, and is therefore international.
Similarly, the radian as a measure of angle and the degree are in use all
over the civilised world. The third quantity can be made international
by the use of a consistent dynamical system of units.*
Quantities which have no dimensions in mass, length and time are
denoted by the common letter fe, are particularised by suffixes and referred
to as coefficients. The following are important particular cases as applied
to wings, and are derived from the three already mentioned (2), by the
ordinary process of resolution of forces and moments : —
CP \
Centre of pressure coefficient ^ /ccp = p7=. of Fig. 65
Lift coefficient
Drag coefficient
Moment coefficient
. (3)
pV2Sc ;
* The choice of units inside the limits of djmamical consistency leads to difficulties between
the pure scientist and the engineer. Whilst both agree to the fundamental character of mass as
differentiated from weight, usage of the word " pound " as a unit for both mass and weight or
force is common. To the author it appears that any system in which such confusion can occur
is defective, and in England part of the defect lies in the absence of a legal definition of force
which has any simple relation to the workaday problems of engineering. Thus, in aeronau-
tics, the English-speaking races invariably speak of the thrust of an airscrew in pounds and
of pressures in pounds per square inch or per square foot. The whole of the difficulty does not
lie here, for the metric system has separate names for force and mass, and yet the French
aeronautical engineer expresses air pressure in kilogrammes per square metre instead of the
roughly equal quantity megadynes per square metre, which is consistent with his system of
units. It would appear that the conception of weight as a unit of force is so much simpler
than that of mass acceleration that only students wiU systematically use the latter. If we were
now to make the weight of the present standard of mass into a standard of force by specifying
g at the place of measurement as some number near to 322 and introduce a new unit of mass
.32 2 times as great as our present unit, it appears to the author that the divergencfe of language
between science and engineering would disappear. In this belief, the standard indicated above
has been adopted throughout this book from amongst those in current use at teaching insti-
tutions, as being the best of three alternatives. The rather ugly name of " slug " was given
to this unit of mass by some one unknown. The standard density of air in aeronautical
experiments is 0-00237 slug per cubic foot, and not 0-0765 lb. per cubic foot. To meet
objections as far as possible full use has been made of non-dimensional coefficients, so that in
many cases readers may use their own pet system without difficulty in applying the tables
of standard results.
120
APPLIED AEKODYNAMICS
All results obtained in aerodynamic laboratories apply also to a
non-standard atmosphere if the expressions (3) are used, but the speed of
test usually quoted applies only to air at 760 mm. Hg and a temperature of
15°-6C.
Fig. QG shows how the various quantities of (3) are arranged in presenting
results. The independent variable of greatest occurrence is " angle of
-
fa)
LIFTCOEFFIC
ENT y^
- -*L
/
y^
10 (
ANG
> lODEGREeS 20
LE OF INCIDENCE = a
DRAG COEFFICIENT
0 07
006
-
O05
_
f^ J
004
DRAG COEFFICIENT
/
003
-
/
O02
y
1
OOI
=— ^^'^''^
J
O
1 1 1
1 1
0-2 OS O* , O-S O 6
LIFT COEFFICIENT = ftt
CENTRE OF
-PRESSURE
COEFFICIENT
ANGLE OF INCIDENCE = a
o \
0
0 1
oz
"^>v^^
fd)
MOMENT
COEFFICIENT
^-\
1
0 C
10 DEGREES ZO
ANGLE OF INCIDENCE 3 O.
lOOECREES 20
ANGLE OF INCIDENCE = d
Fig. 66. — Methods of illustrating wing characteristics.
incidence," but for many purposes the lift coefficient /c^ is used as an
independent variable. The reasons for this will appear after a study of
the chapter on the Prediction and Analysis of Aeroplane Performance.
The useful range of angle of incidence in the flight of an aeroplane is
from— 1° to +15°, and model experiments usually exceed this range at
both ends. An example is given a little later in which observations were
taken for all possible angles of incidence, but this case is exceptional.
DESIGN DATA FROM AERODYNAMICS LABORATORIES 121
Fig. 66 (a). Lift Coefficient and Angle of Incidence. — For angles of in-
cidonce which give rise to positive hft the curve of Uft coefficient against
angle of incidence has an initial straight part, the slope of which varies
little from one wing to another. At some angle, usually between 10 and
20 degrees, the lift coefficient reaches a maximum value, and this varies
appreciably ; the fall of the curve after the maximum may be small or
great, and the condition appears to correspond with an instability of the
fluid motion over the wing. The maximum lift coefficient is very im-
portant in its effect on the size of an aeroplane, since it fixes the area for
a given weight and landing speed. By rearrangement of (3) it will be
seen that
'=m. <*'
and in level flight L is equal to the weight of the aeroplane. Near the
ground, the air density p does not vary greatly, and for a chosen landing
speed the area required is inversely proportional to the lift coefficient fe^.
The ratio of total weight to total area is often spoken of as loading and is
denoted by w, and equation (4) shows that the permissible loading is
proportional to the lift coefficient.
TABLE 1.
Lift Coefficient, Loading and Landing Speed.
Landing speed.
Loading (lbs. per sq. ft.).
ft.-s.
m.p.h.
k^ = 0-4'
kj, = 0-6
k^ = 0-8
20
13-6
0-38
0-57
0-76
40
27-3
1-52
2-28
303
• 60
40-9
3-4
51
6-8
80
54-5
61
91
12-2
100
68-2
9-6
14-2
19-0
Table 1 shows, for a possible range of lift coefficients, the values of wing
loading which may be used for chosen landing speeds. It will be noticed
that the size of an aeroplane is primarily fixed by the weight and landing
speed, and only to a secondary extent by possible changes of lift coefficient.
For an aeroplane weighing 2000 lbs., using wings having a maximum lift
coefficient of 0-6, the areas required are 3,500, 390 and 141 sq. ft. for landing
speeds of 20, 60 and 100 ft. per sec. In normal practice the area varies
from 250 to 350 sq. feet for an aeroplane weighing 2000 lbs., but in earlier
designs an area of 700 or 800 sq. feet would have been considered appro-
})riate. The difficulties of landing are much increased by heavy wing
loading, and at speeds of 50 m.p.h. and upwards prepared grounds with a
smooth surface are required for safety.
It is important to bear in mind the above restriction on the choice of
wing area, for efficiency calls for loadings which are prohibited on this
score.
122 APPLIED AEEODYNAMICS
Pig. 66 (&). Drag CoeflScient and Angle of Incidence. — The curve is shown
to the same scale as lift coefficient, but is rarely used in this form although
the numbers are given in tables for all wing forms tested under standard
conditions. The smallness of the ordinates over the flying range for any
reasonable scale of drag at the critical angle of lift is the chief reason for
a limited use of this type of diagram.
Fig. 66 (c). Centre of Pressure Coefficient and Angle of Incidence. — Con-
siderable variation in curves of centre of pressure occur in wing forms,
but that illustrated is typical of the present day high-speed wing. The
curve has two infinite branches occurring near to the angle of zero lift,
and the changes in this region are great. For larger angles of incidence
the changes are smaller in amount, and the curve has an average position
about one-third of the chord behind the leading edge of the wings. The
exact position of infinite centre of pressure coefficient is defined by the
angle at which the resultant force (E of Fig. 65) becomes parallel to the chord,
and therefore depends to some extent on the definition of the chord. If
the centre of pressure moves forward with increase of angle of incidence,
the tendency of the wing is to further increase the angle and is therefore
towards instability. Turning up the trailing edge of a wing may reverse
the tendency, as will appear in one of the illustrations to be given.
Fig. 66 (d). Moment Coefficient and Angle of Incidence. — The infinite
value of centre of pressure coefficient near zero lift has no special significance
in flight, and it is often more convenient to use a moment coefficient.
The curve has no marked peculiarities over the flying range, but may be
very variable at the critical angle of lift.
Fig. 66 (e). Lift/Drag and Angle of Incidence. — The ratio of lift to drag
is one of the most important items connected with the behaviour of aero-
plane wings, and in level steady flight is the ratio of the weight of an
aeroplane to the resistance of its wings. The curve starts from zero when
the lift coefficient is zero, and rapidly reaches a maximum which may be
as great as 20 to 25, and then falls more slowly to less than half that value
at maximum lift coefficient. It is obvious that every effort is made to use
a wing at its best, i.e. where :_^ is a maximum, but the limitation of
landing speed can be seen to affect the choice as below. Denoting the
speed of flight by V and the landing speed by V;, it will be seen that the
condition of constant loading requires that
PzV,2(fci,U,.=/,V2fci. (5)
. Equation (5) can be arranged in a more convenient form as
a*V = v/^^^-.V, (6)
where o- is the relative density of the atmosphere at the place of flight, and
<T*V will be recognised as indicated airspeed. The whole of the right-hand
side of (6) is fixed by the landing speed and the wing form if kj^ be chosen
as the lift coefficient for maximum lift /drag, and hence the indicated air speed
for greatest efficiency is fixed.
r
DESIGN DATA FKOM AEKODYNAMICS LABOEATOEIES 123
Referring to Figs. 66 (a) and 66 (e) it will be found that kj^ has a maximum
value of 0-54 and a value of 0-21 for maximum ^. This shows an indicated
air speed of 1 -6 times the landing speed. As applied to an aeroplane the
theorem would use the lift/drag of the complete structure and not of the
wings alone, and the number 1-6 is much reduced. Near the ground the
speed of most efficient flight is well below that of possible flight, but
the difference becomes less at great heights. For high-speed fighting
scouts the ratio of lift to drag for the wings may be only 10 instead of the
best value of 20, and it becomes important to produce a wing which has a
high value of lift to drag at low lift coefficients. This is the distinguishing
characteristic of a good high-speed, wing, and appears to be unattainable
at the same time as a high lift coefficient.
Fig. 66 (J) . Lift/Drag and Lift Coefficient.— The remarks on Fig. 66 (e) have
indicated the importance of the present curve, and particular attention
has been paid to the development of wing forms having a high speed value
of j^ at a lift coefficient of O'l and as high a value as possible at a lift co-
efficient 0*9 times as great as the maximum, the latter being important in
the climbing of an aeroplane. It will thus be seen that in modern practice
the maximum lift/drag of a wing is not the most important property of its
form as an intrinsic merit, but only as it is associated with other properties.
Equation (6) suggests that the quantity under the root sign is important
as an independent variable, and this is recognised in certain reports on
wing form.
Fig. 66 {g). Drag Coefficient and Lift Coefficient. — The diagram is con-
venient in its relation to a complete aeroplane, for the change from the curve
for wings alone is almost solely one of position of the zero ordinate. A
tangent from the new origin shows the value of the maximum lift /drag of the
aeroplane and the lift coefficient at which it occurs. The diagram shows more
clearly than any other that the useful range of flying positions lies within
the limits 0*01 and 0*05 for drag coefficient, and that small changes of lift
coefficient and therefore of indicated air speed produce large changes of
drag near the critical angle. The indicated air speed at the critical angle
of lift is known as the " stalling speed," and has been used in these notes
as identical with " landing speed." The latter is, however, always greater
than the former for reasons of control over the motion of the aeroplane at
the moment of alighting.
Particular Cases of Wing Form
Effect of Change of Section (Fig. 67 and Tables 2-5).— The shape of the
section of the standard model aerofoil is conveniently given by a table of
the co-ordinates of points in it, the chord being taken as a standard from
which to measure and the front end as origin. For two wings, R.A.F. 15
for high speed and R.A.F. 19 for high maximum lift coefficient, the co-
ordinates which define their shapes are given in Table 2 below. The length
of the chord is taken as unity, and all other linear measurements are given
124
APPLIED AERODYNAMICS
in terms of it. It will be seen that R.A.P. 15 has a maximum height above
the chord of 0'068, and this number is often called the upper surface camber
o
0-2
/
y
r
CENTRE 0(=^ 4
PRESSURE 1
--
—
-^
==^
_ __
^ .^ ^
JI —
.^
-—.
0'4
0-6
0-8
lO
COEFFICIE
NT ...
'
p*"^
T
• " "
/
L
L
1 1 1
ANGLE OF INCIDENCES
L_^ \ \ 1
'-a
1
O 10 DEGREES
Fig. 67. — Effect of change of wing section.
20
for the wing section. The other wing of the table, R.A.P. 19 has an upper
surface camber of 0-152, or more than twice that of R.A.P. 15, and this
DESIGN DATA FEOM AEEODYNAMICB LABORATORIES 125
difference is characteristic of the difference between high speed and high
hft wings.
TABLE 2.
Shapes of Wing Sections.
R.A.F. 15.
B.A.F. 19.
Dist&uc© from
leading edge.
Height of upper
Heiglit of lower
Height of upper
Height of lower
surface.
surface.
surface.
surface.
0
0013
0013
0012
0-012
001
0027
0 008
0034
0 003
002
0035
0-005
0051
0001
003
0041
0-003
0065
0-000
004
0045 -.
0-002
0-076
0-000
0-05
^0047^ '
0-001
0-085
0-001
006
0052
0-001
0093
0003
008
0057
0-000
0-107
0-008
012
0 063
0001
0-127
0-021
016
0-066
0003
0140
0-034
0-22
0-068
0006
0150
0054
0-30
0-067
0-008
0152
. 0-060
0-40
0-065
0-008
0-147
0-075
0-50
0062
0-006
0-134
0-072
0-60
0-056
0-002
0-117
0062
0-70
0-048
0-000
0095
0050 • "
0-80
0040
0001
0-071
0034
0-90
0-029
0003
0-043
0017
0-95
0-023
0 005
0-026
0-008
0-98
0-017
0-006
0015
0-002
0-99
0-015 ^
0-007
0-01 1
0-000
100
0-010
0010
0-006
0-006
The aerodynamic properties of these wings are compared with those of
a plane which is defined by a rectangular section of which the width is
one-fiftieth of the length. A less precise but more obvious definition of
the shapes of the sections is given in Fig. 67 above the diagrams showing
their aerodynamic properties. Tables 3-5 show data for the sections in
the form in which they appear in the reports of test from an aerodynamic
laboratory.
Table 3 is compiled from results given by Eiffel in his book " La Re-
sistance de I'air et I'Aviation," and is sufficient to show variations of hft,
drag and centre of pressure at all angles of incidence. The lift coefficient
has a first maximum of 0-400 at 15° and a second of the same magnitude
at about 30°. The drag steadily increases from a minimum at 0° angle of
incidence to a maximum of 0*590 when perpendicular to the wind. It is
interesting to notice that the lift at 15° is two-thirds of the maximum
possible force on the plate. The lift/drag ratio of 7*0 is very small and
occurs at a lift coefficient of 0-18, where it is not of the greatest use. One
feature of the table is of interest as showing that the centre of pressure
moves back as the angle of incidence increases, and it is this property which
makes it possible to fly small mica plates. With the centre of gravity
adjusted to he at one-third of the chord by attaching lead shot to the
126
APPLIED AEEODYNAMICS
leading edge, thin mica sheets can be made to Hy steadily across a
room.
TABLE 3.
Forces and Moments on a Fi^at Plate.
Angle of
incidence
(degrees).
Lift coefficient.
Drag coefficient.
Lift
Drag
Centre of
pressure
coefficient.
Moment coefficient
about the leading
edge.
0
0-000
0-019
00
0-26
000
5
0-177
0-025
7-0
0-27
-0-05
10
0-340
0-067
61
0-33
-Oil
15
0-400
0-114
3-5
0-33
-014
20
0-388
0-144
2-7
0-39
-0-16
30
0-400
0-235
1-7
0-41
-0-19
40
0-380
0-320
1-2
0-43
-0-21
60
0-335
0-400
0-85
0-45
-0-23
60
0-275
0-475
0-68
0-47
-0-26
70
0-190
0-540
0-35
0-48
-0-28
80
0-100
0-580
017
0-49
- 0-29
90
0 000
0-590
0-00
0-50
-0-30
Tables 4 and 6 are representative tables of wing characteristics in their
best form, The intervals in angle of incidence are usually 2°, with inter-
polated values at small angles of incidence where the ratio of lift to drag
is varying most rapidly. All the terms which occur have been defined,
and the characteristics of the wings are most easily seen from the curves
of Fig. 67, which was produced from the numbers in Tables 2-5.
TABLE 4.
R.A.F. 16 Aebofoil.
Size of plane, 3" x 18". Wind speed, 40 ft.-s.
Angle of
Lift
Centre of
Moment coefficient
incidence
Lift coefficient.
Drag coefficient.
pressure
about leading
(degrees).
Drag
coefficient.
edge.
-6
-0170
0-0310
-5-50
0-167
+0-028
-4
-0-087
0-0156
-5-61
0-034
+0-003
-2
-00163
0-0099
-1-66
-0-725
-0011
-1
-fO-0173
0-0085
2-03
0-822
-0-014
0
0-057
0-0082
6-96
0-404
-0023
1
0-107
0 0084
12-7
0-362
—0038
2
0-164
0-0104
16-0
0-350
-0-057
3
0-203
0-0123
166
0.327
-0067
4
0-242
0-0148
16-4
0-307
-0-075
6
0-312
0-0205
15-2
0-278
-0-087
8
0-387
0-0277
14-0
0-268
-0-104
10
0-454
0-0363
12-6
0-274
-0-124
12
0-519
0-0460
11-2
0-280
-0-145
14
0-538
0-0630
8-9
0-280
-0-160
16
0-530
0-100
5-3
0-341
-0-183
18
0-476
0148
3-2
0-394
-0-197
■I the
DESIGN DATA PROM AERODYNAMICS LABORATORIES 127
The first noticeable feature of the lift coefficient curves is, that whilst
the plate only begins to lift at a positive angle of incidence, the high speed
wing R.A.F. 15 lifts at angles above —1-5° and the high lift wing at —8°.
This feature is common to aU similar changes of upper surface camber.
The surprising fact is well established that an aeroplane wing may Uft
with the wind directed towards the upper surface.
TABLE 5.
R.A.P. 19 Aerofoil.
Sire of plane, 3' X 18". Wind speed, 40 ffc.-s.
Angle of
Lift
Centre of
Moment coefflcient
incidence
Lift coefflcient.
Drig coefflciebt.
pressure
about leading
(degrees).
Drag
coeflEicient.
edge.
-12
-0063
0-0750
-0-83
+0-218
+0017
-10
-0-038
0-0648
-0-59
+0-130
+0-006
- 8
+0-006
0-0541
+0-11
-21-04
-0-021
— 6
0-050
0-0550
1-11
+0-758
-0-034
- 4
0103
0-0390
2-64
0-588
-0-059
- 2
0189
0-0351
5-4
0-612
-0-093
- 1
0-246
00359
6-8
0-487
-0-120
0
0-302
0-0371
8-1
0-472
-0-142
+ 1
0-358
0-0381
9-4
0-449
-0-161
2
0-413
00396
10-4
0-434
-0-180
4
0-516
0-0438
11-8
0-412
-0-214
6
0-591
0-0506
11-7
0-387
-0-230
8
0-662
0-0617
10-7
0-369
-0-245
10
0-737
0-0740
10-0
0-356
-0-262
12
0-797
0-0865
9-2
0-348
-0-278
14
0-845
0-1012
8-3
0-341
-0-288
15
0-531
0-1420
3-74
0-339
-0184
16
0-531
01515
3-52
0-339
-0187
18
0-529
0-1716
3-08
0-343
-0-191
20
0-531
0-189
2-81
0-344
-0-194
All the lift coefficient curves show a maximum at 14**, but the values
are very different, being 0-40 for the plate, 0*54 for R.A.F. 15 and 0-84 for
R.A.F. 19. This is partly due to a progressive increase in the average
slope of the curves, the values being 0-035, 0*040 and 0*045, but much more
to the increase of range of angle between zero lift and maximum lift co-
efficient. The very high lift coefficient of 0*84 given by R.A.F. 19 appears
to be highly critical, and the maximum is followed by a rapid fall, so that at
an angle of incidence of 20 degrees the difference between the wings is
greatly reduced. At still greater angles the effects of differences of wing
form tend to disappear.
The curves giving the ratio of lift to drag show a different order to
the curves for lift coefficient, for the plate gives a maximum of 7, R.A.F. 15
of 16*6, and R.A.F. 19 of 12*0, It is therefore clear that there is some
section which has a maximum lift to drag ratio. R.A.F. 15 is the outcome
of many experiments on variation of wing section, none of which has given
a higher ratio under standard conditions. As is usual in the case of
variations near a maximum condition, it is possible to change the section
128 APPLIED AERODYNAMICS
within moderately wide limits without producing great changes in wing
characteristics.
On the same diagram as the lift to drag curves has been plotted the
cotangent of the angle of incidence, as it brings out an interesting property
of cambered wings. For a value of lift to drag given by a point on this
curve the resultant force on the wing is normal to the chord, and both
E.A.F. 14 and R.A.F. 19 have two such points. For values of lift to drag
which lie below the cotangent curve the resultant force lies behind the
normal to the chord, whilst the converse holds for points above the curve.
It will be seen that the resultant force on the plate is always behind the
normal, whereas for R.A.F. 15 an extreme value of 7°*5 ahead of the chord
is shown. When a description of the pressure distribution round a wing
is given, it will be seen that this forward resultant is associated with an
intense suction over the forward part of the upper surface. The resultant
is of course always behind the normal to the wind direction, but in R.A.F.
14 its value has a minimum of 3°'5. The value of y shown in Fig. 65 is
then very small, and it will be understood that errors of appreciable magni-
tude would follow from any want of knowledge of the direction of the wind
relative to the wind channel balance arms. One degree of deviation would
introduce an error of 28 per cent, into the drag reading, and even with
great care it is difficult to make absolute measurements of minimum drag
coefficient to within 5 per cent. Comparative experiments made on the
same model and with the same apparatus have an accuracy much greater
than this and more nearly equal to 1 per cent. Within the limits indicated
wind channel observations are remarkably consistent.
The centre of pressure coefficient curves show that the wing forms
R.A.F. 14 and R.A.F. 19 have unstable movements, that is, the
centre of pressure moves forward as the angle of incidence increases.
The plate on the other hand has the stable condition previously
referred to.
Wing Characteristics !or Angles of Incidence outside the Ordinary Flying
Range. — In discussing some of the more complicated conditions of motion
of an aeroplane knowledge is required of the properties of wings in
extraordinary attitudes. Not only is steady upside-down flying possible,
but backward motion occurs for short periods in the tail slide which is
sometimes included in a pilot's training.
For a flat plate observations are recorded in .Table 3 for a range of angles
from 0° to 90°, and from the symmetry of the aerofoil these observations
are sufficient for angles from 0° to 360°. The values of the lift coefficient,
lift to drag ratio and centre of pressure coefficient are shown in Fig. 68 in
comparison with similar curves for R.A.F. 6 wing section. The shape of
the latter is shown in the figure and the detailed description in the height
of contours is given in Table 6 below. The numbers apply only to the
upper surface ; the small camber of the under surface is of little importance
in the present connection. A modification known as R.A.F. 6a has been
used on many occasions, and differs from R.A.F. 6 only in the fact that in
the former the under surface is flat.
The dissymmetry of the section made it necessary to test the aerofoil
DESIGN DATA FROM AEliODYNAMICS LABORATORIES 129
at angles of incidence over the whole range 0° to 360°, with the results
shown in Table 7 and in Fig. 68.
TABLE 6.
Shape of Wino Section R.A.F. 6a.
Height above chord.
Distance from leading
edge.
Distance from trailing
edge.
0000
0-007
0-004
0 003
0-001
0-000
0-007
0-000
0-003
0010
0001
0-009
0013
. 0-003
0-021
0-017
0-006
0-037
0-020
0-010
0-054
0-023
0-013
0-072
0-027
0018
0-091
0-030
0-023
0-110
0-033
0-028
0-129
0-037
0033
0-149
0 040
0-039
0-170
0-043
0-045
0-191
0-047
0053
0-215
0-050
0-061
0-238
0-063
0070
0-265
0-057
0080
0-292
0-060
0-092
0-322
0-063
0-106
0-353
0-067
0123
0-391
0070
0-146
0-436
0-073
0-181
0-493
0077
0-250
0-583
0-0776
0-320 max.
0-680 max.
0-0785
—
—
Above figures are expressed as fractions of chord.
Fig. 68 shows that the lift coefficient of the cambered wing section is
numerically greater than that of the plate so long as the thicker end or
normal front part is facing the wind, but the plate gives the greater lift
coefficients with the tail into the wind. The effect of camber at ordinary-
flying angles is seen to be greater than elsewhere, and this is emphasized
in the lift to drag curves, where the greatest value is 1 6 for the wing section
and 7 for the plate. At the other peaks of the lift to drag curve the
difference is much less marked, and with the tail first the wing section is
again seen to be inferior to the plate.
For the centre of pressure coefficient both wing section and plate have
a stable movement of the centre of pressure with angle over the greater
part of the range. The unstable movement associated with cambered wings
is confined to the region of common flying angles and is a disadvantageous
property. Judging from current practice it appears that the high ratio
of lift to drag is far more important than the type of curve for centre of
pressure, as this latter can always be corrected for by the use of a tail, an
organ which would exist for control under any circumstances.
130
APPLIED AEEODYNAMICS
The comparison between the plate and wing section shows a very-
considerable degree of similarity of form for the various curves, and indicates
the special character of the differences at ordinary flying angles which have
been developed as the result of systematic study of the effect of variation
of aerofoil section on its aerodynamic properties.
TABLE 7.
FoBOES AND Moments on R.A.P. 6..
Size 2''5 X 15'. Wind speed, 40 ft, -s.
Angle of
Lift
Centre of
Moment coefficient
incidence
Lift coefficient.
Drag coefficient.
pressure
about leading
(degrees).
Drag
coefficient.
edge.
0
+0-090
0-0152
+ 5-9
0-523
-00472
5
+0-325
00210
+ 16-5
0-346
-0-1128
10
+0-498
0-0415
+ 12-0
0-305
-0-1515
15
+0-613
00721
+ 8-5
0-279
-0-1707
20
+0-528
0-1712
+ 3-1
0-368
-0-2025
30
+0-472
0-273
+ 1-73
0-389
-0-2117
40
+0-453
0-395
+ 1-16
0-408
-0-2450
50
+0-398
0-465
+ 0-86
0-422
-0-2560
60
+0-327
0-657
+ 0-59
0-434
-0-2707
70
+0-232
0-632
+ 0-37
0-458
-0-3056
80
+0-117
0-679
+ 0-17
0-469
-0 3265
90
+0-000
0-701
0-0
—
—
100
-0-119
0-674
- 0-18
0-500
-0-3375
110
-0-268
0-625
- 0-43
0-513
-0-3460
120
-0-318
0-556
- 0-57
0-626
-0-3380
130
-0-388
0-466
- 0-83
0-645
-0-3300
140
-0-469
0-397
- 1-19
0-562
-0-3370
160
-0-479
0-278
- 1-73
0-567
-0-3130
160
-0-478
0-1700
- 2-80
0-575
-0-2920
170
-0-425
0-0605
- 8-4
0-649
-0-2760
180
-0-056
00172
- 3-1
0-089
-00049
190
+0-311
00812
+3-83
0-702
+0-2185
200
+0-320
0-1688
+2-04
0-662
+0-2185
210
+0-351
0-260
+ 1-36
0-654
+0-2840
220
+0-349
0-360
+0-97
0-638
+0-3180
230
+0-290
0-426
+0-68
0-615
+0-3140
240
+0-228
0-497
+0-46
0-698
+0-3256
250
+0-154
0-557
+0-28
0-679
+0-3290
260
+0-067
0-608
+0-11
0-545
+0-3310
270
-0-028
0-618
-0-06
0-528
+0-3145
280
-0-128
0-610
-0-21
0-478
+0-3035
290
-0-211
0-698
-0-36
0-446
+0-2716
300
-0-273
0-497
-0-66
0-428
+0-2414
310
-0-318
0-409
-0-78
0-397
+0-2047
320
—0-369
0-343
—1-08
0-377
+0-1890
330
-0-336
0-243
-1-39
0-378
+0-1555
340
-0-274
01395
-1-98
0-340
+0-1033
345
-0-245
0-1006
-2-44
0-330
+0-0866
350
-0-219
0-0649
-3-38
0-298
+0-0673
356
-0111
0-0308
-3-60
0-080
+0-0091
360
+0-090
0-0152
+5-9
0-522
-0-0472
Wing Characteristics as dependent on Upper Surface Camber. — In the
early days of aeronautics at the National Physical Laboratory a series of
DESIGN DATA FROM AERODYNAMICS LABORATORIES 181
0 6
O 5
0-4-
O 3
O 2
0-1
O
-O I
-02
-0 3
-O 4
.^•" ~*V J/ DRAG COEFFICIENT
T-l
'
"\
-•^
^ RAF6
^ 1
~<
>
•'
"s PLATE
\ 1 "^
A>
V
^
P
7\.
\
1 /
-■' \
\
\
/ /
•' >
^
/•
\
•
/ /
\
\.
/
^
\
^
1
V
\
J
V
\
V.
/
1 \/^
s.
V
t
V
•^~\jLm
-4
-6
\
—
^HBHHBl
» 1
LIFJ
• —
/
1
r
RAF6
DRAG
\
PLATE
IV
^
^
.
^■^^
■^l
\/
l)
\'
o o
Ol
0-2
0-3
Le
aoingE
DGE
CENT
PRES
COEFF
RE OF
SURE.
CIENT
i
ft
0-4
0-5
0 6
/
^
^
ft
P
LATE^
''RAF(
3
r
^
*^
^\/
■/'
V
/
^
0-7
O 8
\
/r'
>
\
X Side
CONCAV
E Side
JPPERK
OST -►
0 9
*
I
ANGLE OF INCIDENCE
1 O
270°
300°
330°
0*^
30°
60° |90° |I20° |l50°
180°
2IO°
240°
Fig. 68. — Wing characteristics at all possible angles of incidence.
132 APPLIED AERODYNAMICS
experiments on the variation of upper surface camber and upper surface
shape was carried out and laid the foundation for a reasoned choice of wing
section. Knowledge of methods of tests and particularly the discovery
of an effect on wing characteristics of size and wind speed have reduced
their value, and other examples are now chosen from various somewhat
unconnected sources. No up-to-date equivalent of these early experiments
exists, but it is to be hoped that our National Institution will ultimately
undertake such experiments with all the refinements of modern methods.
Until this series appears the results deduced from the early experiments
may be accepted as qualitatively correct, and, although not quoted directly,
have been used to guide the choice of examples and to give weight to the
deductions drawn from the study of special cases.
Aerofoils having large upper surface camber are used only in the design
of airscrews, and on pages 304 and 305 will be found details of the shapes
of a number of sections and the corresponding tables of the aerodynamic
properties. In most of these sections the under surface was flat. The
general conclusion may be drawn that a fall in the value of the maximum
lift to drag ratio is produced by thickening a wing to more than 7 or 8 per
cent, of its chord, and that the fall is great when the thickness reaches
20 per cent, of the chord. The exact shape of the upper surface does not
appear to be very important, but a series of experiments at a camber
ratio of O'lO indicated an advantage in having the maximum ordinate
of the section in the neighbourhood of one-third of the chord from the
leading edge. The position of the maximum ordinate was found to have
a marked effect on the breakdown of flow at the critical angle of lift,
but in the light of modern experimental information it appears that these
differences may be largely reduced in a larger model tested at a higher
speed. A very similar series of changes to those now under review occurred
in the test of an airscrew section at different speeds and is illustrated and
described in the chapter on Dynamical Similarity. Further reference to
the effect of size of model and the speed of the wind during the test is given
later in this chapter.
Changes of Lowdr Surface Camber o£ an Aerofoil. — It has been the
general experience that changes of lower surface camber of an aerofoil
are of less importance in their effect on wing characteristics than are those
of the upper surface. Wings rarely have a convex lower surface, but for
sections of airscrews a convex under surface is not unusual. In Table 8
and Fig. 69 are shown the effects of variation of R.A.P. 6a by adding a
convex lower surface, the ordinates of which were proportional to those
of the upper surface. The range from R.A.P. 6a to a strut form was
covered in three steps in which the ordinates of the under side were one-
third, two-thirds and equal to those of the upper surface. Inset in Fig. 69
are illustrations of the aerofoil form.
In this series the chord was taken in all cases as the under side of the
original wing, and the table shows the gradual elimination of the lift at
negative angles of incidence as the under-surface camber grows to that of
the upper surface. A distinct fall in maximum lift coefficient is observable
without corresponding change of angle of incidence at which it occurs.
ESIGN DATA FEOM AERODYNAMICS LABORATORIES 133
The minimum drag coefficient is seen to occur with a convex lower surface,
but not with the symmetrical section. Incidentally it may be noted that
a strut may have a lift-to-drag ratio of 13.
17
1
TA
^\
y
~^
\
\
RAF.6A
14-
/
\
^^
,^
— ®-
^
\
^"Ji
^
ll
1
f
N
s^
^
II
7
■\
\
.^
\
/
\
\
\\
\
\
\
8
_ L
IFp
i
I
RAG
/
/
/
\\
6
//
J
{/
' 1
//
L
/
/
/
4-
>
Y
/
7
/
1
J
/
/
//
/
2
11
/
t
/
1
0
f
X RAF 6A.
. - B ■•
f. ■■ C •••
© .. o ••
-2
/
^
/
^
-A-
>-
-6
1
,^
LIF-
r cc
J
EFFI
:iEN
L.,..
T
1
-02
-Ol
O 01 0-2 03 0-4-
Pio. 69. — Variation of lower surface camber.
0-5
0-6
The important deductions from Table 8 are more readily obtained
from Pig. 69, which shows the ratio of lift to drag as dependent on lift
coefficient. A lower surface camber of one-third of that of the upper
surface is very large for a wing, but on a high-speed aeroplane the gain
134
APPLIED AEEODYNAMICS
of 20 per cent, in lift to drag at a lift coefficient of O'l might more than
compensate for the smaller proportionate loss at larger values of the lift
coefficient. It may be observed that there is a limit to the amomit of
mider-surface camber which could be used with advantage, and reference
to the wing form of E.A.F. 15 suggests that the advantages can be
attained by a slight convexity at the leading edge only.
TABLE 8.
Effect op Variation of Bottom Camber of Aerofoil, R.A.F. 6a.
AerofoU, 3' x 18". Wind speed, 40 ft.-s.
Lift coefficient.
Drag coefQcient.
Angle
(degrees).
A
B
C
D
A
B
0
D
-6
-0149
-0-160
-0-21«
—0-283
0-0348
0-0248
0-0178
00221
-4
-0 068
-0-101
-0-151
-0-218
00224
0-0170
0-0142
0-0184
-2
+0-0126
-0-017
-0-072
-0-131
0-0164
00131
00117
00153
0
+0-106
+0-083
+0-064
-0006
0-0137
0-0106
0-0110
0-0133
2
+0-210
+0-183
+0-162
+0-127
0-0131
00126
0-0119
0-0146
4
+0-288
+0-258
+0-228
+0-218
0-0168
0-0158
0-0145
00172
6
+0-362
+0-333
+0-295
+0-280
0-0226
0-0216
0-0193
0-0208
8
+0-437
+0-406
+0-363
+0-346
0-0301
0-0284
0-0255
0-0264
10
+0-508
+0-477
+0-428
+0-395
0-0396
0-0370
00335
0-0314
12
+0-575
+0-536
+0-489
+0-441
0-0536
0-0450
0-0432
00388
14
+0-604
+0-565
+0-536
+0-476
0-0630
0-0553
0-0628
00485
16
+0-542
+0-511
+0-392
+0-392
0-110
0-1032
0-1044
0-0923
18
+0-491
+0-450
+0-367
+0-317
0-141
01386
0-130
0-129
20
+0-479
+0-422
+0-361
+0-306
0-164
0-1598
0-152
0147
Lift
Moment coefficient about leading edge.
Angle
Drag
(degrees).
A
B
G
D
A
B
C
D
-6
-4-28
- 6-47
-121
-12-8
+0-020
+0-022
+0-049
+0-081
-4
-302
- 5-9
-10-6
-11-9
+0-008
+0-010
+0-036
+0-067
-2
+0-77
- 1-3
- 5-1
- 8-6
-0-029
—0 011
+0-016
+0-043
0
7-77
+ 8-0
+ 6-8
- 0-49
-0-065
-0-043
-0-031
-0-007
2
16-0
14-6
13-6
+ 8-7
-0-084
-0069
-0-068
-0044
4
171
16-4
16-7
12-7
-0-102
-0086
-0-072
-0-068
6
16-0
15-4
15-3
13-4
-0-118
-0-102
-0-086
-0-081
8
14-5
14-3
14-2
13-1
-0-136
-0-120
-0-101
-0094
10
12-8
12-9
12-8
12-6
-0-164
-0-1.36
-0-115
-0-100
12
10-7
11-9
11-3
11-4
-0-171
-0-147
-0-130
-0107
14
8-70
10-3
10-1
9-8
-0-184
-0-163
-0-139
—0112
16
4-9
4-9
3-7
4-3
-0-159
-0-164
-0-128
-0-104
18
3-5
3-3
2-8
2-5
-0-129
-0-170
-0133
-0-111
20
2-9
2-6
2-4
2-1
-0-122
-0-167
-0-138
-0-114
Camber of upper surfaces of A, B, C and J) was that of R.A.F. 6a.
Ordinates of lower surface of A = 0, i.e. flat lower surface.
„ „ „ B = § X ordinates of upper surface of R.A.F. 6a. convex.
» ft >) '-' = f X ,, ff f, 1) tt
»» >> L* = 1 X •• t, tt >> »
k
ESIGN DATA FEOM AERODYNAMICS LABOEATORIES 135
Changes of Section arising from the Sag of the Fabric Covering of an
Aeroplane Wing. — The shape of an aeroplane wing is determined primarily
by a number of ribs made carefully to template, but spaced some 12 to
15 ins. apart on a small aeroplane. These ribs are fixed to the main spars,
and over them is stretched a linen fabric in which a considerable tension
is produced by doping with a varnish which contracts on drying. On the
upper surface the wing shape is affected by light former ribs from the
leading edge to the front spar, Fig. 1 , Chapter I., shows the appearance
of a finished wing, whilst Fig. 70 shows the contours measured in a particular
instance. From the measurements on a wing a model was made with the
full variations of section represented, and was tested in a wind channel.
Upper Surface
LEAOrNcEOGE
Lower Surface
Fig. 70. — Contoxirs of a fabric-covered wing.
After the first test the depressions were filled with wax, and a standard
plane of uniform section resulted on which duplicate tests were made.
Table 9 gives the results of both tests.
It is not necessary to plot the results in order to be able to see that the
effect of sag in the fabric of a wing in modifying the aerodynamic charac-
teristics of this wing is small at all angles of incidence. The high ratio of
lift to drag is partly due to the large model, which is twice that previously
used in illustration.
Aspect Ratio, and its Effect on Lift and Drag. — The aerodynamic
characteristics of an aerofoil are affected by aspect ratio to an appreciable
extent, but the number of experiments is small owing to the fact that the
length of a wing is fixed by other considerations than wing efficiency. One
of the more complete series of experiments has been used to prepare
136
APPLIED AERODYNAMICS
Fig. 71 ; in the upper diagram, lift coefficient is shown as dependent on
angle of incidence, and both the slope and the maximum are increased by
an increase of aspect ratio. These changes get more marked at smaller
aspect ratios and less marked at higher values, although an effect can still
be found when the wing is 15 times as long as its chord. The changes
resulting from change of aspect ratio are most strikingly shown in the
ratio of lift to drag, the maximum value of which rises from 10 at an
aspect ratio of 3 to 15 for an aspect ratio of 7 and probably 20 for an aspect
ratio of 15. The effect at low lift coefficients is small, and aspect ratio has
no appreciable influence on the choice of section for a high-speed wing.
TABLE 9.
COMPABISON BETWEEN THE LiFT AND DrAO OF AN AeBOFOIL OF UNIFORM SECTION (R.A.F. 14),
AND OF AN Aerofoil sititably grooved to represent the Sag of the Fabric of an
Actual Wing.
Aerofoil, e^xSG". Wind speed, 40 ft.-s.
R.A.F. 14 section.
R.A.F. 14 modified.
Distance of
Angle of
C.P. from
nose as a
fraction of
the ciiord.
incidence
(degrees).
Lift
coefficient
(abs.).
Drag
coefficient
(abs.).
L
D
Lift
coefficient
(abs.).
Drag
coefficient
(abs.).
L
D
- 6
-0-162
0-0363
-4-45
-0-163
0-0354
-4-62
+0-178
- 4
-0-066
00230
-2-89
-0-0682
0-0225
-3-04
-0-144
- 2
+0037
0-0133
+2-77
+0-0388
0-0125
+3-10
1-15
0
0-137
0-0096
14-30
0-134
00094
14-3
0-52
+ 2
0-214
00104
20-55
0-215
0 0102
21-1
0-413
3
0-249
00122
20-40
0-249
0-0120
20-7
—
4
0-284
00144
19-8
0-284
00143
19-8
0-37
6
0.356
00200
17-8
0-356
0-0199
17-8
0-33
8
0-423
0-0270
15-7
0-419
0-0270
15-5
0-316
10
0-485
0-0354
13-7
0-474
0-0360
13-1
0-297
12
0-521
0-0462
11-3
0-510
0-0484
10-54
0-288
14
0-5.34
0-0617
8-66
0-536
0-0753
712
0-290
15
0-544
0 0857
6-35
0-545
00957
5-68
—
16
0-542
01104
4-90
0-544
0-1140
4-76
0-324
18
0-536
01420
3-76
0-535
0-1475
3-63
0-365
20
0-504
0-1655
304
0-503
0-1720
2-92
—
Changes of Wing Form which have Little Effect on the Aerodynamic
Properties. — The wings of aeroplanes are always rounded to some extent,
and it does not appear that the exact form matters. The difference
between any reasonable rounding and a square tip accounts for an increase
of 2 to 5 per cent, on the maximum value of the lift to drag ratio and an
inappreciable change of lift coefficient at any angle.
A dihedral angle less than 10° appears to have no measurable effect
on lift, drag or centre of pressure. Its importance arises in a totally
different connection, a dihedral angle being effective in producing a correc-
tive rolling moment when an aeroplane is overbanked.
A similar conclusion as to absence of effect is reached for variations of
sweepback up to 20°. This type of wing modification is not very common,
p
ESIGN DATA PROM AERODYNAMICS LABORATORIES 137
but may be resorted to in order to bring the centre of gravity of the aero-
Iilane into correct relation to the wings. The requirements of balance and
tability do not here conflict with those of performance.
o 6
o 5
o 4-
0-3
o 2
oi
16
14-
12
10
0RA6
1
ASPP<~T J
/
RA-
no
/
~-N^
::\
>s.
k
■"V^
N^
i^
^
i
1/
^
/
\
-^
J
/
•
J
/
/
LIFT <
:OEFF
CIENT
O O I
02 03 0-4 0-5 0-6 or
Fia. 71. — Effect of aspect ratio.
Effect of the Speed of Test on the Lift and Drag of an Aerofoil.— Fig. 72
shows the lift coefficient as a function of angle of incidence and speed, and
the lift to drag ratio as dependent on lift coefficient and speed, for an
aerofoil of section R.A.F. 6a. Th6 model had a chord of 6 inches, and was
tested at speeds of. 20, 40 and 60 ft.-s., with the results illustrated. Over
138 APPLIED AEEODYNAMICS
a range of angle of incidence of 2° to 10° the effect of speed on lift coefficient
o 6
0-5
0-4
0-3
0-2
CI
O
-Ol
-0-2
-6
==:?
^ eoFT^s
A
^
^
<^F^^
LIFT
COEF
MCiEr
T
y
/
7^
20 F
^-~
y
/^
A
A
60
f/s^
^
A
^
-20 F
^
y
^
/
ANGL
1 1
E OF INCIDENCE
1
(DEGREES)
20
18
16
JJFT_
DRAG
.
^
V
.OFT/^
1
rv
,^
40f
Vs
//
20 F7
^
\
V
//
\
N
/
\
i
r
)
1
/
tl
}i
f
/
>
/
/
/
LIF
T cot
:ffici
ENT
.-Ol 0^0 Ol 02 03 0-4- , 0-5 0^6 0-7
Fio. 72. — Effect of speed of test.
is not important, but appreciable changes occur at both smaller and larger
angles. There is a tendency towards an asymptotic value at high speeds,
DESIGN DATA FEOM AERODYNAMICS LABORATORIES 203
A series of five models shows for envelope forms how the drag co-
e£Bcients vary with the fineness ratio, or length to diameter ratio. A similar
series of tests for strut forms has already been given in which the drag
coefficient on projected area was roughly 0*042. On the envelope forms
the coefficient is appreciably less and may fall to half the value just quoted.
The forms tested were solids of revolution of which the front part was
ellipsoidal ; in all cases the maximum diameter was made to occur at
one-third of the total length from the nose. The shapes of the longitu-
dinal sections are shown in Fig. 100, and have numbers attached to them
which are equal to their fineness ratio. The observations made are re-
corded in Table 41 and need a little explanation. It is pointed out in the
chapter on dynamical similarity that neither the size of the model nor the
speed of the wind has a fundamental character in the specification of
resistance coefficients, but that the product of the two is the determining
variable. In accordance with that chapter, therefore, the first column of
Table 41 shows the product of the wind speed in feet per second and the
diameter of the model in feet. Further, two drag coefficients denoted
respectively by /cq and C have been used for each model, the former giving
a direct comparison with other data on the basis of projected area, and the
latter a coefficient of special utility in airship design which is closely related
to the gross lift.
TABLE 41.
RbSISTANOK COBFFICUBINTS OF AlBSHIP EnVET,OPB FoBMS.
Vd
No. 6.
No. 4-5.
No. 4.
No. 3-5.
No. 3.
(ft. -8.).
K C
K
C
K
0
K
0
K
C
8-7
9-8
11-7
13-7
15-7
17-7
19-7
21-5
23-6
25-4
27-5
29-3
31-3
00351 0-0142
00334 0-0135
00327 0-0132
0-0323 0-0130
00322 0 0130
0-0320 0-0129
0-0330 00133
0-0331 ! 0-0133
0-0337 0-0136
0-0342 0-0138
0-0344 j 0-0139
00346 0-0139
0-0348 0-0142
0-0313
0-0305
0-0290
0 0287
0-0280
0-0269
0-0269
0-0265
0 0272
0-0270
0-0271
0-0277
0-0279
0-0149
0-0145
0-0138
00136
0 0133
0-0128
0-0128
0-0126
00129
0-0128
0-0129
0-0132
0-0132
00319
0 0298
0 0282
0 0272
0 0262
00252
00264
0-0250
0-0247
0-0255
00251
0-0249
0-0261
0-0166
00155
0-0147
00142
0-01.36
0-0131
0-0132
0-0130
0 0129
00132
00131
00130
00130
00318
0-0298
0-0292
0-0276
0-0262
0-0252
0-0249
0-0242
00246
0-0244
0 0245
0-0245
0-0245
0-0182
0 0170
00167
0-0155
0 0149
0 0143
0-0142
00138
0-0140
0-0139
00139
0-0139
0-0140
0-0323
00301
0-0287
0 0263
0-0253
0-0238
0-0238
0-0232
0 0230
0-0228
0-0228
0-0224
0 0224
0 0207
0-0192
0-0184
0-0168
0-0161
0-0152
0 0152
0-0148
0 0147
00145
0-0146
0 0143
00143
The coefficients kj) and C are defined by the equations
(20)
and drag = C/>V2 (volume) J (21)
where d is the maximum diameter of the envelope.
204
APPLIED AEEOBYNAMICS
An examination of the columns of Table 41 shows some curious changes
of coefficient which are perhaps more readily appreciated from Fig. 101,
where the values of kj) are plotted on a base of Yd. For the longest model
the curve first shows a fall to a minimum, followed by a rise to its initial
value. For the model of fineness ratio 4*5 the minimum occurs later, and
it is possible that the three short models all have minima outside the range
of the diagram. It is clearly impossible to produce these curves with any
degree of certainty. In Chapter II. it was deduced that for a rigid airship
the full-scale trials give to C a value of 0'016, and for a non-rigid, 0'03.
003
002
001
—
-^
^
■^
, — *-
6
— ^
'^
^
S&^
4-5
DRAG COEFFICIENT
1 1
■*
h^
^
•
H
^ 4-_^
^
fi
^:^
o
►— — o-
3-5
3*
PR(
30UC
T OF
DIA
1
METI
■R «
SPE
ED(i
1
1
ec)
10 20
Fig. 101. — Resistance of airship envelops models.
30
These figures contain the allowance for cars and rigging, and do not indicate
any marked departure from the figure of 0'013 given above for the envelope
alone. The comparison is very rough, but accurate full-scale experiments
of a nature similar to those on models have yet to be made.
It will be noticed from Table 41 that whilst the drag coefficient calcu-
lated on maximum projected area falls with decrease of fineness ratio, the
coefficient C which compares the forms on unit gross lift is less variable
and has its least values for the longer models. The importance of the second
drag coefficient " C " is then seen to be considerable as an aid to the choice
of envelope form.
Complete Model o£ a Non-rigid Aiiship. — A complete model, illustrated
DESIGN DATA FEOM AEEODYNAMICS LABOEATORIES 205
in Fig. 102, was made of one of the smaller British non-rigid airships, and the
)
P4
m
.a
03
•r
S
analysis of the total drag to show its dependence on main parts was carried
out. The results of the observation are shown in Table 42.
206
APPLIED AEEODYNAMICS
TABLE 42.
Resist ANOB of Non -rigid Aibship.
Drag (lbs.). Diameter of envelope, 6-65 ins. Wind speed, 40 ft.-s.
Description of Model.
Angle of incidence (degrees).
0"
4°
8°
12°
16"
20°
a
Complete airship ....
0-102
0-109
0-132
0170
0-225
0-300
h
Rigging cables removed .
0-081
—
—
—
—
—
c
Without car or rigging cables
0-066
0-073
0-092
0127
0-187
0-258
d
Without car, rigging cables
•
or rudder plane
0-052
0-068
0-078
0-115
0-171
0-234
fi
Without car, rigging cables
or elevator planes .
0051
0-054
0-061
0077
0-099
0129
f
Envelope alone . .
0-035
0-036
0-041
0-054
0-074
0101
g
Main rigging cables .
0-021
0-021
0-021
0-021
0021
0021
h
Car alone ....
0-016
0-012
0-016
0-012
0-016
0011
0-016
0-011
0016
0-012
0-OlA
i
Rtidder plane alone .
0-012
3
Elevator planes alone
0-017
0-017
0-022
0-032
0-048
0-070
k
Airship drag by addition ef
parts
0101
0-102
0-111
0-134
0-171
0-220
Each column of the table shows the drag on the model and its parts
in lbs. at a wind speed of 40 ft.-s., the maximum diameter of the model
being 6*65 inches. The rows a—f give the result of removing parts succes-
sively from the complete model, whilst rows j—j refer to the resistances of
the parts separately. At an angle of incidence of 0°, that is with the air-
ship travelling along the axis of symmetry of its envelope, the total
resistance is nearly three times that of the envelope alone. From the
further figures iu the column it will be seen that the difference is almost
equally distributed between the rigging cables {g), the car (h), -the rudder
plane {i) and the elevator plane (j). The resistance of the whole model is
very closely equal to that estimated by the addition of parts, the figures
being 0-102 as measured and 0-101 as found by addition. The agreement
between direct observation and computation from parts is less satisfactory
at large angles of incidence, an observed figure of 0-300 comparing with the
much lower figure of 0*220. The difference is probably connected with
the influence of the rudder and elevator fins in producing a more marked
deviation from streamline form than the inclined envelope alone.
Drag, lift and Pitching Moment on a Rigid Airship. — The form of the
airship is shown in Fig. 103, and the model to y^„th scale had a maximum
diameter of 7'87 inches. Forces are given in lbs. on the model at 40 ft.-s.,
whilst moments are given in Ibs.-ft. Apart from any scale effect, applica-
tion to full scale is made by increasing the forces in proportion to the square
of the product of the scale and speed, whilst for moments the square of the
speed still remains, but the third power of the scale is required. At a
value of Nd equal to 60, V being the velocity in feet per second and d
the diameter in feet, the partition of the resistance was measured as in
Table 43,
DESIGN DATA FEOM AEEODYNAMICS LABORATOEIES 207
It was noticeable that the varia-
tion of resistance coefficient " C " for
the complete model with speed of test
was much less marked than that of
the envelope alone, the coefficient
ranging from 0-0195 to 0-0210 for a
range of Vd of 15 to 50, whilst for the
envelope the change was 0-0096 to
0-0131.
TABLE 43.
Value of the drag
coefficient "C."
Complete model
. . 0-0207
Envelope alone
. . 00131
Fins and controls .
. . 0-0014
Four cars ....
. . 0-0038
Airscrew structure . .
. . 0-0024
■]
1i
In Table 44 are collected the re-
sults of observations on the model
airship for a range of angle of inci-
dence —20° to +20°, the lift and
pitching moment as well as the drag
being measured. For comparison, the
value of the pitching moment on the
envelope alone has been added. A
further table shows the variation of
pitching moment due to the use of the
elevators, and the salient features of
the two tables are illustrated in Fig.
104 (a) and (&).
Angle of incidence has the usual
conventional meaning, a positive value
indicating that the nose of the airship
is up whilst the motion is horizontal.
A positive inclination of the elevators
increases their local angle of incidence
and clearly tends to put the nose of
the airship down.
Table 44 indicates a marked m-
crease of resistance due to an inclina-
tion of 10° of the axis of the airship
to the relative wind, but a somewhat
more remarkable fact is the magnitude
of the lift, which may be 2*5 times as
great as the drag at the same angle of
incidence.
The column of pitching moment
shows a feature common to all types
if airship in the absence of a righting moment at small angles of incidence.
It does not follow that the airship is therefore unstable, since there is a
208
APPLIED AERODYNAMICS
further pitching moment due to the distribution of weight ; moreover, it
will be found that the criterion of longitudinal stability of an airship differs
appreciably from that of the existence or otherwise of a righting moment.
TABLE 44.
Dbag, Lift and Pitohing Moment on a Model of > Rigid Airship.
Maximum diameter, 7 "87 ins. Wind speed, 40 ft.-s.
Angle of
Pitching
Pitching moment
incidence
Drag (lbs.).
Lift (lbs.).
moment
envelope alone
(degrees).
(Ibs.-ft.).
(Ibs.-ft.).
-20
0-267
-0-647
-0-180
-1-065
-16
0173
—0-469
-0-238
-0-868
-12
0-119
-0-291
-0-274
-0-698
- 8
0-087
-0159
-0-266
-0-490
— 4
0-079
-0061
-0-178
-0-266
0
0083
+0019
-0-008
0
4
0-098
0-112
+0-146
0-256
8
0-130
0-240
0-220
0-490
12
0-196
0-418
0-216
0-698
16
0-301
0-621
0-180
0-868
20
0-459
0-861
0-102
1-055
At small angles of incidence the indication of Table 44 is that the ele-
vator fins and elevators neutralise only one-third of the couple on the
envelope alone, but at greater angles, where the fin is in less disturbed air,
more than 85 per cent, is neutralised. A position of equilibrium which is
stable would exist at an inclination of about 35° to the relative wind.
TABLE 45.
Pitching Moment on a Rigid Airship Model, due to the Elevators.
(Lbs. -ft. at 40 ft.-s.).
Angle
Angle of elevator (degrees).
of in-
cidence
(deg.).
-20
-15
—10
-5
0
6
10
15
20
-15
+0-004
-0-033
-0-095
-0-164
-0-250
-0-316
-0-396
-0-462
-0-606
-10
-0-020
-0-055
-0-134
-0-178
-0 270
-0-328
-0-421
-0-459
-0-520
- 8
-0-009
-0-046
-0-119
-0174
-0-257
-0-316
-0-378
-0-439
-0-486
- 6
+0-013
-0 029
-0-102
-0-146
-0-226
-0-282
-0-341
-0-390
-0-445
- 4
0-037
+0-002
-0-066
-0-106
-0178
-0-227
-0-289
-0-333
-0-372
- 2
0-078
0-048
-0-013
-0044
-0-104
-0-152
-0-202
-0-247
-0-280
0
0-164
0122
+0-076
+0-039
-0-008
-0-060
-0123
-0-156
-0-185
2
0 243
0-218
0-164
0-130
+0-097
+0-027
-0-020
-0-083
-0-103
4
0-312
0-278
0-227
0-194
0-146
0-093
+0-040
-0015
-0-044
6
0-370
0338
0-270
0-240
0-190
0133
0-070
+0-009
-0-020
8
0-405
0-364
0-314
0-270
0-220
0-165
0-096
0-028
+0-024
10
0-432
0-392
0-334
0-281
0-226
0-163
0-086
0-027
+0-011
15
0-442
0381
0-329
0-268
0-192
0-119
0-062
0-009
-0-038.
DESIGN DATA FEOM AERODYNAMICS LABORATORIES 209
0-6
LIFT & DRAG
(lbs)
0-4
PITCHING MOMENT
(lbs ft)
0-6
/
(h)
/•
LEVATOR
-20°
ANGLE
0-4
/ ^^
"
r^"^"">^_
PITCHING MOMENT
>
/^ ^
-10°
0 2
(lbs. ft.)
A
0^
"^
SINGLE OF INCIOtNC
E y
//
y^
-_«..J0°
^^^
0_
^
VX
20"
^•^v^
''^
-1
Cjr-^
^A
y^
'" 1
QO ^
..^^
-0 2
-0 4
v,^^
4/.
/
-f
'/
-0 6
A
7
--ENVE
LOPE A
LONE
-0-8
-IrO
/
/
r
/
Fia. 104. — Forces and inoraenbs on a model of a rigid airship.
210
APPLIED AEEODYNAMICS
Pig. 104 (a) shows the pitching moment on the complete model as
dependent on angle of incidence The rapid change at small angles of
incidence is followed by a falling off to a maximum at 10° and a further
fall at 20°. The lower diagram, Fig. 104 (&), shows how the couple which
can be applied by the elevators compares with that on the airship. It
appears that at an angle of 20° the maximum moment can just be overcome
by the elevators, and that a gust which lifts the nose to 10° will require ■
an elevator angle of that arnount to neutralise its effect. It is quite |
possible that most airships are unstable t>o some slight degree but are all
controllable, at low speeds with ease and at high speeds with some diffi-
culty. The attachment of fins of area requisite to produce a righting
moment at small angles of incidence is seen to present a problem of a
serious engineering character, and the tendency is therefore to some
sacrifice of aerodynamic advantage.
Pressure Distribution round an Airship Envelope. — A drawing of the
Fig. 105a.
model is given in Fig. 105b, on which are marked the positions of the points
at which pressures were measured Somewhat greater precision is given
by Table 46, the last column of which shows the pressures for the condition
in which the axis of the envelope was along the wind. Other figures and
diagrams show the pressure distribution when the axis of the airship is
inclined to the relative wind at angles of 10° and 30°. The product of the
wind speed in feet per second and the diameter in feet was 15, whilst the
pressures have been divided by p\'^ to provide a suitable pressure coeffi-
cient.
With the axis along the wind. Fig. 105a shows a pressure coefficient
of half at the nose, which falls very rapidly to a negative value a short
distance further back. The pressure coefficient does not rise to a positive
value till the tail region is almost completely traversed, and its greatest
value at the tail is only 10 per cent, of that at the nose. It is of some
interest and importance to know that the region of high pressure at the
nose can be investigated on the hypothesis of an inviscid fluid which there
DESIGN DATA FKOM AERODYNAMICS LABORATORIES 211
gives satisfactory results as to pressure distribution. The stiffening of the
nose mentioned in an earlier chapter can therefore be proved on a priori
reasoning.
When the axis of the envelope is inchned to the wind, lack of symmetry
introduces complexity into the observations and representations. By
rolhng the model about its axis each of the pressure holes is brought into
positions representative of the whole circumference ; with the hole on the
windward side the angle has been denoted by — 90°, and the symmetry of
the model shows that observations at 0° and 1 80° would be the same. The
results are shown in Table 46 and in Fig. 105b. From the latter it will be
Wind JS^^-^
Direction
Positive Values of ^ pyz
measured radially *
inwards from circumference:
& Negative values measured
outwards.
IS
Fill. lOoB. — Pressure distribution on an inclined airship model.
seen that the pressure round the envelope at any section normal to the
axis is very variable, a positive pressure on the windward side of the nose
giving place to a large negative pressure at the back. The diagrams for an
inclination of 30° show the effects in most striking form owing to their
magnitude.
Kite Balloons. — For typical observations on kite balloons the reader
is referred to the section in Chapter II., where in the course of discussion
of the conditions of equilibrium a complete account was given of the
observations on a model.
212
APPLIED AEKODYNAMICS
TABLE 46.
Pb£ssusk on a Model Aibsuif.
Inclination, 0°.
Hole.
Diameter at hole as fraction
Axial position of hole as
P/pV=
of maximum diameter.
fraction of maximum diameter.
Xd = 15.
1
0 000
0-000
+0-500
2
0-269
0117
+0-241
3
0-436
0-237
+0-073
4
0-670
0-474
-0-064 j
5
0-805
0-710
-0-112
6
0-890
0-948
-0-112
7
0-979
1-420
-0-100
8
1-000
1-895
-0-078
9
0-990
2-37
-0-068
10
0-938
2-85
-0-063
11
0-855
3-32
-0-056
12
0-756
3-79
-0 032
13
0-618
4-26
-0015
14
0-347
4-98
+0-030 ^
15
0000
5-69
+0-057 \
Values of pressure as a
iraction of pW InGlination,
0°.
Angle of
rolUdeg.).
Hole No.
1
2
3
4
5
6
7
8
+90
+0-475
+0-060
-0-091
-0-165
-0170
-0-132
-0096
-0069
+75
+0-475
+0-086
-0-086
-0-169
-0-168
-0-134
-0-100
-0-073
+60
+0-475
+0091
-0-066
-0-186
-0171
-0-144
-0-108
-0-095
+46
+0-475
+0-129
-0-050
-0-156
-0175
-0-143
-0-110
-0-088
+30
+0-475
+0-149
-0-028
-0-122
-0-210
-0160
-0-123
-0-094
+ 16
+0-476
+0-145
+0-021
-0-124
-0-156
-0-143
-0-126
-0-100
0
+0-475
+0-200
+0-046
-0-100
-0-168
-0-134
-0-130
-0-107
-15
+0-475
+0-179
+0-069
-0-077
-0-122
-0-115
-0-124
-0-105
-30
+0-475
+0-267
+0-114
-0-025
-0-120
-0-111
-0-110
-0-102
-45
+0-475
+0-300
+0147
-0-013
-0-069
-0-074
-0-085
-0-087
-60
+0-475
+0-319
+0-170
+0-017
-0-050
-0-067
-0077
-0-073
-76
+0-475
+0-354
+0-203
+0-060
-0-016
-0-038
-0-059
-0-056
-90
+0-475
+0-368
+0-218
+0-062
-0015
-0-020
-0-048
-0-057
Angle of
roll (deg.).
Hole No.
9
10
11
12
13
14
15
+90
-0046
-0-037
-0-017
+0006
+0-007
+0-023
+0052
+75
-0065
-0-032
-0 023
-0006
+0-006
+0023
+0-052
+60
-0-066
-0-037
-0-026
-0-024
-0-005
+0-016
+0-052
+45
-0-081
-0-060
-0-048
-0-020
-0-011
+0010
+0062
+30
-0-082
-0-073
-0-060
-0-034
-0-020
+0-013
+0-062
+15
-0-091
-0-081
-0-073
-0-053
-0-038
+0-005
+0062
0
-0-096
-0-086
-0-080
-0-060
-0-041
+0-005
+0-052
-15
-0106
-0-089
-0-088
-0-064
-0-048
+0003
+0-052
-30
-0-100
-0-088
-0094
-0-073
-0-053
+0-000
+0-062
-46
-0-087
—0-078
-0-091
-0-068
-0-054
+0-005
+0-052
-60
-0-073
-0-074
-0-076
-0-070
-0-053
+0-005
+0-052
-75
-0069
-0-068
-0-070
-0058
-0-045
+0-005
+0-062
-90
-0056
-0-065
-0-062
-0-060
-0-045
0000
+0-052
DESIGN DATA FKOM AERODYNAMICS LABORATORIES 213
TABLE 46 — continued.
Values of pressure as a fraction of pV^. Inclination, 30°.
Angle of
roll (deg.).
+90
+75
+60
+45
+30
+ 15
0
-15
-30
-46
-60
-75
-90
Hole No.
1
+0034
+0 034
+0-034
+0-034
+0-034
+0034
+0034
+0034
+0-034
+0-034
+0034
+0034
+0-034
-0-285
-0-275
-0-296
-0-270
-0-233
-0-221
-0-122
-0-146
+0-056
+0-208
+0-306
+0-428
+0-492
-0-340
-0-355
-0-359
-0-370
-0-376
-0-290
-0-232
-0-151
0-000
+0-139
+0-267
+0-390
+0-450
-0-290
-0-310
-0-337
-0-368
-0-380
-0-383
-0-348
-0-275
-0-148
-0-010
+0-133
+0-261
+0-324
-0-210
-0-261
-0-300
-0-368
-0-413
-0-390
-0-372
-0-314
-0-241
-0-062
+0-050
+0-175
+0-226
6
7
-0121
-0-048
-0-135
-0-081
-0-315
-0-200
-0-303
-0-312
-0-328
-0-302
-0-373
-0-332
-0-380
-0-378
-0-330
-0-370
-0-224
-0-281
-0-098
-0-150
0-000
-0-062
+0-123
+0-044
+0182
+0-120
-0-033
-0-048
-0-136
-0-272
-0-292
-0-301
-0-341
-0-336
-0-275
-0-156
-0-056
+0-045
+0-078
Angle of
Toll (deg.).
+90
+75
+60
+45
+30
+ 15
0
-16
-30
-46
-60
-75
-90
Hole No.
9
10
-0-015
-0-076
-0-168
-0-223
-0-272
-0-281
-0-298
-0-331
-0-263
-0-156
-0-061
+0-012
+0-065
-0051
-0121
-0-191
-0-238
-0-253
-0-252
-0-261
-0-310
-0-256
-0172
-0-086
-0-010
+0-031
11
-0-081
-0160
-0-240
-0-266
-0-223
-0-220
-0-226
-0-266
-0-259
-0-200
-0-100
-0-028
+0021
12
-O-IOO
-0-191
-0-256
-0-253
-0-169
-0-155
-0158
-0-181
-0-208
-0158
-0-113
-0-024
+0021
13
-0081
-0-195
-0-186
-0-175
-0-116
-0094
-0-088
-0-102
-0-122
-0-128
-0-087
-0-018
+0013
14
-0-046
-0-031
-0017
-0019
-0-012
0 000
0 000
0-000
0 000
-0-039
-0-014
+0-015
+0-032
15
+0-043
+0-043
+0043
+0-043
+0043
+0-043
+0-043
+0-043
+0-043
+0-043
+0-043
+0-043
+0-043
CHAPTEE IV
DESIGN DATA FROM THE AERODYNAMICS LABORATORIES
PAET II. — Body Axes and Non-rectilinear Flight
In collecting the more complex data of flight it is advisable for ease of
comparison and use that results be referred to some standard system of
axes. The choice is not easily made owing to the necessity for com-
promise, but recently the Koyal Aeronautical Society has recommended
a complete system of notation and symbols for general adoption. The
details are given in "A Glossary of Aeronautical terms," and will be
followed in the chapters of this book. The axes proposed differ from
others on which aeronautical data has been based, and some little care is
necessary in attaching the correct signs to the various forces and moments.
It happens that very simple changes only are required for the great bulk
of the available data.
Axes (Fig. 106). — The origin of the axes of a complete aircraft is commonly
taken at its centre of gravity and denoted by G. The reason for this
arises from the dynamical theorem that the motion of the centre of gravity
of a body is determined by the resultant force, whilst the rotation of a
body depends only on the resultant couple about an axis through the
centre of gravity. This theorem is not true for any other possible origin.
From G, the longitudinal axis GX goes forward, and for many purposes
may be roughly identified with the airscrew axis. The normal axis GZ
lies in the plane of symmetry and is downwards, whilst the lateral axis
GY is normal to the other two axes and towards the pilot's right hand.
The axes are considered to be fixed in the aeroplane and to move with
it, so that the position of any given part such as a wing tip always has the
same co-ordinates throughout a motion. This would not be true if wind
axes were chosen, and difficulties would then occur in the calculation of
such a motion as spinning. For many purposes the axis GX may be
chosen arbitrarily, whilst in other instances it is conveniently taken as
one of the principal axes of inertia.
In dealing with parts of aircraft it is not always possible to relate the
results initially to axes suitable for the aircraft, since the latter may not
then be defined. It is consequently necessary to consider the conversion of
results from one set of body axes to another. So far as is possible, the axes
of separate parts are taken to conform with those of the complete aircraft.
Angles relative to the Wind.-7-Any possible position of a body relative
to the wind can be defined by means of the angular positions of the axes.
Two angles, those of pitch and yaw, are required, and are denoted respec-
tively by the symbols a and p. They are specified as follows : first, place
214
i
DESIGN DATA FEOM AEEODYNAMICS LABOEATOEIES 215
the axis of X along the wind ; second, rotate the body about the axis of
Z through an angle j8 and, finally rotate the body about the new position
of the axis of Y through an angle a. The positive sign is attached to an
angle if the rotation of the body is from GX to GY, GY to GZ or GZ to
GX. This is a convenient convention which is also applied to elevator
angles, flap settings and rudder movements. With such a convention it is
found that confusion of signs is easily avoided.
Angles are given the names roll, pitch or yaw for rotations about the
axes of X, Y and Z respectively. It should be noticed that an angular
displacement about the original position of the axis of X does not change
the attitude of the body relative to the wind.
Forces along the Axes. — The resultant force on a body is completely
specified by its components along the three body axes. Counted positive
when acting from G towards X, Y and Z (Fig. ] 06), they are denoted by
mX, viY and mZ, and spoken of as longitudinal force, lateral force and
Pig. 106. — -Standard axes.
normal force. " w " represents the mass of an aircraft, and may not be
known when the aerodynamical data is being obtained ; the form is
convenient when applying the equations of motion.
Moments about the Axes. — The resultant couple on a body is completely
specified by its components about the three body axes. Counted positive
where they teiid to turn the body from GY to GZ, from GZ to GX and from
GX to GY, they are denoted by the symbols L, M and N and are known as
rolling moment, pitching moment and yawing moment.
Angular Velocities about the Axes. — The component angular velocities
known as rolling, pitching and yawing are denoted by the symbols p, q and
r, and are positive when they tend to move the body so as to increase the
corresponding angles.
The forces and couples on a body depend on the magnitude of the
relative wind, V, the inclinations a and j8 and the angular velocities p, q
and r. In a wind channel where the model is stationary relative to the
channel walls, p, q and r are each zero, and most of the observations hitherto
216
APPLIED AEEODYNAMICS
made show the forces and couples as dependent on V, a and ^ only. To
find the variations due to jp, q and r the model is usually given a simple
oscillatory motion, and the couples are then deduced from the rate of
damping. At the present time much of the data is based on a combination
of experiment and calculation, and discussion of the methods is deferred
to the next chapter. Examples of results are given in the chapters
on Aerial Manoeuvres and the Equations of Motion and StabiHty. In
the present section the results referred to are obtained with p, q and r
zero.
Equivalent Methods of representing a Given Set of Observations. —
Fig. 107 shows three methods of representing the force and couple on a
Fig. 107. — Methods of representing a given set of observations.
wing. The lateral axis is not specifically involved owing to the symmetry
assumed, but its intersection with the plane of symmetry at A and B is
required. An aerofoil is supposed to be placed in a uniform current of air
at an angle of incidence a. The simplest method of showing the aero-
dynamic effect is that of Fig. 107 (a), where the resultant force is drawn in
position relative to the model ; this method however requires a drawing,
and is therefore not suited for tabular presentation. Fig. 107 {b) shows the
DESIGN DATA PEOM AERODYNAMICS LABORATORIES 217
resolution into lift, drag and pitching moment ; A may be chosen at any
place, and through it the resolved components normal to and along the
wind are drawn and are independent of the position of A. The moment
of the resultant force ah about A gives'the couple M, which clearly depends
on the perpendicular distance of A from the line of action of the resultant.
Body Axes in a Wing JSection.— Keeping the point A as in Fig. 107 (b), the
axis of X has been drawn in Fig. 107 (c) as making an angle ao with the chord
of the aerofoil. The angle of pitch is then equal to a+ao, and the double
use of a for angle of incidence and angle of pitch should be noted together
with the fact that they differ only by a constant. The components of force
are now mX and wZ in the directions shown ^by the arrows, whilst M has
Arrows denote Direction
Chine Line
in which Section is
Scale for Model ! ?...! ^ i* t ? ? ^ ? ? 'P '■' '? '"ches.
N mY
YAW
PITCH
mX Z'
Wind M
Direction
Wind Direction.
'mZ
Fig. 108. — Model of a flying-boat hull ; shape and position of axes.
identically the same value as for Fig. 107 (&). To move the point A to B
without changing the inclination of the axes it is only necessary to make
use of Fig. 107 (d), where x and z are the co-ordinates of B relative to the
old axes. It then follows that
Mb = M^ — zmX + xmZ
(1)
whilst mX and mZ are unchanged. In general it appears to be preferable
to take the most general case of change of origin and orientation in two
stages as shown, i.e. first change the orientation at the old origin, and then
change the origin.
It is worthy of remark here, that although drag cannot be other than
positive, longitudinal force may be either negative or positive, and usually
bears no obvious relation to drag.
218
APPLIED AEEODYNAMICS
Longitudinal Force, Lateral Force, Normal Force, Pitching Moment and
Yawing Moment on a Model of a FIjdng Boat Hull. — ^A drawing of the model
is shown in Fig. 108, together with two small inset diagrams of the positions
of the axes. Experiments were made to determine the longitudinal and
normal forces and the pitching moment for various angles of pitch a but
with the angle of yaw zero, and also to determine the longitudinal and
lateral forces and the yawing moment for various angles of yaw j3 but with
the angle of pitch zero. The readings are given in Tables 1 and 2, and
curves from them are shown in Fig. 109.
TABLE I.
Forces and Moments on a Flying Boat Htju. (Pitch).
Wind speed, 40 ft.-s.
Angle of pitch a
Longitudinal force mX
Normal force mZ
Pitching moment M
(degrees).
(lbs.).
(lbs.).
(Ibs.-lt.),
+20
-0-067
-0-407
+0-291
15
-0-065
-0-281
+0-217
10
-0-057
-0-166
+0-153
8
-0054
-0-122
+0-130
6
-0-050
-0-084
+0-100
4
-0-047
-0-051
+0-077
2
-0-044
-0-022
+0-049
0
-0-041
+0-007
+0-024
- 2
-0-040
+0-020
+0-002
- 4
-0041
+0-043
-0-019
- 6
-0040
+0-069
-0036
- 8
-0-040
+0-102
-0066
-10
-0041
+0-142
-0-072
-15
-0-041
+0-273
-0108
-20
-0-038
+0-446
-0-148
TABLE "2.
Forces and Moments on a Flying Boat Hull (Yaw).
Wind speed, 40 ft.-s.
Angle of yaw p
Longitudinal force »iX
Lateral force mY
Yawing moment N
(degrees).
(lbs.).
(lbs.).
(Ibs.-ft.).
0
-0041
0
0
5
-0-042
0-078
+0-063
10
-0-037
0-179
+0-120
15
-0-032
0-296
+0-190
20
-0-028
0-419
+0-249
25
-0-019
0-597
+0-294
30
-0-005
0-767
+0-342
35 ,
+0-011
0-952
+0-381
Fig. 109 shows that the normal force mZ and the pitching moment M
change by much greater proportionate amounts than the longitudinal force
mX when the angle of pitch is changed, and that the lateral force wY and
yawing moment N show a similar feature as the angle of yaw is changed.
DESIGN DATA FEOM AEEODYNAMICS LABOEATOEIES 219
0-5
0-4
0-3
0-2
01
-01
-0-2
-0-3
-0-4
.
M
\
\
mZ.
y^
y
/77X« n
FORCl
(lbs.
r>2\l
s N
)
\
^^
i
\
>
x^
^
X
\,
^
my/
— ^
^
\
M, PI
M
rCHING
OMENT
ft.lbs)
\
\
\
0-3
0 2
01
01
0-2
0-3
-20
10 0 10
ANGLE OF PI TCH - (X
20-
10
0-8
0-6
0-4
0-2
^6;
/
7
mX&n
FORCf
(Lbs
s.
)
V-
/ ^
--^
N
>;
N,YA\
MO
(ft.
VINQ
^ENT.
Ubs)
y
X
:>
mY
J^
y^
^
mX^
0 4
0-3
0-2
01
-0-2
0" 10" 20" 30 40"
ANGLE OF YAW— /3
Fig. 109. — Forces and moments on a model of a ilying-boat hull.
In the latter case, Fig. 109 (6), it may be noticed that the longitudinal
force mX becomes zero at an angle of yaw of 30°. The rolling moment was
considered to be too small to be worthy of measurement.
220
APPLIED AEEODYNAMICS
For each angle of pitch it is obvious that there will be a diagram in
which the angle of yaw is varied. The number of instances in which
measurements have been made for large variations of both a and j8 is very-
small and partial results have therefore been used even where the more
complete observations would have been directly applicable. It only needs
to be pointed out that the six quantities X, Y, Z, L, M, N are needed for
all angles a, ^, for all angular velocities p, q, r, and for all settings of the
elevators, rudder and ailerons for it to be realised that it is not possible
to cover the whole field of aeronautical research in general form. For
this reason it is expected that specific tests on aircraft will ultimately
be made by constructing firms, and that the aerodynamics laboratories
will develop the new tests required and give the lead to development.
TABLE 3.
FoECEs AND Moments on an Aeroplane Body (Yaw).
Wind speed, 40 ft.-s.
Angle of
Body without fin and rudder.
Body with fin and rudder
(rudder at 0°).
(degrees).
Longitudinal
force (lbs.).
Lateral force
(lbs.).
Yawing
moment
(lb8.-ft.).
Longitudinal
force (lbs.).
Lateral force
(lbs.).
Yawing
moment
(lbs.-ft.).
0
6
10
15
20
26
30
-0-0697
-0-0740
-0-0780
-0-0827
-0-087
-0-089
-0-085
0
0-0676
0-1437
0-2481
0-390
0-660
0-764
0
0049
0-096
0131
0-153
0-163
0-140
-0-0763
-0-0780
-0-0806
-0-0811
-0-081
-0-080
-0-080
0
0-1363
0-3158
0-5347
0-768
1-027
1-307
0
-0056
-0-186
-0-330
-0-472
-0-631
-0-816
TABLE 4.
Effect of Bitddeb (Yaw).
Wind speed, 40 ft.-s.
Angle of
Body with fin and rudder
(rudder at +10°)-
Body with to and rudder
(rudder at +20").
(degrees).
Longitudinal
force.
Lateral force.
Yawing
moment.
^"'Sf"*' Lateral force.
Yawing
moment.
-30
-26
-20
-15
-10
- 6
0
+ 6
10
15
20
26
30
-0071
-0-080
-0-084
-0-087
-0-089
-0-092
-0-089
-0099
-0-107
-0-115
-0-122
-0-129
-0133
-1132
-0-845
-0-602
-0-376
-0-1691
-00021
+01330
0-2976
0-4921
0-708
0-969
1-219
1-497
+0-466
0-293
0181
+0-066
-0-058
-0-169
-0-227
-0-330
-0-483
-0-639
-0-804
-0-996
-1-160
-0091 -0-942
-0-104 -0-674
-0-108 -0-4668
-01132 -0-2270
-01224 -0-0236
-0-1270 +01364
-0-1333 0-283
-0-1547 0-460
-0-1678 0-644
-0-1785 0-852
-0-196 1-110
-0-212 1-380
-0-217 1-562
+0-163
0
-0-083
-0-177
-0-313
-0-429
-0-487
-0-631
-0-771
-0-912
-1-113
-1-320
-1-434
DESIGN DATA FROM AERODYNAMICS LABORATORIES 221
Forces and Moments due to the Yaw of an Aeroplane Body fitted with
Fin and Rudder. — The experiment on the model shown in Fig. 110 was made
Fig. 110. — Aeroplane body with fin and rudder.
02
-0-2
-0-4
-0-6
-0-8
-10
-1-2
•14
with the angle of pitch zero. For various angles of yaw the longitudinal
and lateral forces and the yawing moment were measured without fin and
rudder ; also with the fin in
place and the rudder set
over at various angles. The
results are given in Tables 3
and 4 and illustrated in Fig.
111.
The body alone, shows a
zero yawing moment with
its axis along the wind and
positive values for all angles
of yaw up to 30°. Regarded
as a weathercock with its
spindle along the axis of Z,
the body alone would tend
to turn roimd to present a
large angle to the wind.
With the fin and rudder
shown in Fig. 110, however,
a comparatively large couple
is introduced which would
bring the weathercock into
the wind. Setting the rudder
over to 10° and 20° is seen
to be equivalent to an addi-
tional yawing moment which
is roughly constant for all
angles of yaw within the range of the test. The amount of the couple
due to 10° of rudder is about twice as great as that due to an inclination
\ -*
T 1
) 1
■ BO
5 2
DY ALO
0 2
NE.
5 3
\
ANGLE
OF YA
A/, (d&^r
ees^
^^
s^
"\
\"
JDDER
ATO'
\.
"^
\,
\
\
^
\
N
)DER
TI0°
\
YAWIN
MOMEN
(fUbs.
G
T
)
K
\ RU
dderN
\
T 20'
\
\
\
s
Fig
111 .. — Yawing moments due to fin and rudder on
a model aeroplane body.
222
APPLIED AERODYNAMICS
of the body of 30°, and hence the positions of equiUbrium shown by
Table 4 at —12° for a rudder angle of 10° and at —26° for a rudder angle
of 20° must be due to the counteracting effect of the fixed fin. It will
thus be seen that the lightness of the rudder of an aeroplane depends on
the area of the fixed fin. The best result will clearly be obtained if the
fin just counteracts the effect of the body. The experiment to find this
condition could be performed by measuring the yawing moment on the
body and fin with rudder in place but not attached at ; the hinge. It
would not be sufficient to merely remove the rudder, since the forces on
the fin would thereby be affected. The possibihties of this Une of inquiry
have not been seriously investigated.
The Effect of the Presence of the Body and Tail Plane and of Shape of
Fin and Rudder on the Effectiveness of the Latter.— For this experiment the
^-4^-^
Fig. 1 12. — -Model aeroplane body with complete tail unit.
rudder was set at zero angle, and cannot therefore be differentiated from
the fin. The basis of comparison has been made the lateral force per unit
area divided by the square of the wind speed. It is found that the coefficient
so defined depends not only on the shape of the vertical surface, but also
on the presence of the body and the tail plane and elevators. The drawing
of the model used is shown in Figs. 112 and 113, the latter giving to an
enlarged scale the shapes of the fins attached in the second series of ex-
periments.
The experiments recorded in Table 5 apply to the model as illustrated
by the full lines of Fig. 112, that is without the fin marked Al. The test
leading to the second column of Table 5 was made with rudder alone held
in the wind, and will be found to show greater values of the lateral force
coefficient than when in position as part of the model. A range of angle
of pitch of 10 degrees is not uncommon in steady straight flying, and the
body was tested with the axis of X upwards (+5°), with it along the wind
DESIGN DATA FKOM AERODYNAMICS LABORATORIES 223
and with it pitched downwards (—5°), both with and without the elevators
,in position.
TABLE 5.
Effect of Body and Elevators on the Ruddek.
[Lateral forces on the rudder of Fig. 1 12 in lbs. divided by area in sq. ft. and by square of
wind speed in feet per sec.
Rudder
Angle alone (free
Rudder when attached
to body pitched
+ 6 degrees. .
Rudder when attaclied
to body in normal
flyipg position.
Rudder when attached
to body pitched
—5 degrees.
of from inter-
yaw fere nee
(dfig.). effects).
Without
tail plane
and
elevators.
With
ditto.
Without
tail plane
and
elevators.
With
ditto.
Without
tail plane
and
elevators.
With
ditto.
1
2 0-000104
4 0-000205
6 0-000315
8 0-000421
10 0-000528
0-000057
0-000114
0-000186
0-000265
0000350
0-000050
0-000104
0-000170
0-000249
0-000330
0-000071
0-000155
0-000247
0-000337
0000433
0-000063
0-000133
0-000216
0-000303
0000402
0-000083
0-000183
0-000274
0000370
0000482
0 000065
0-000143
0-000226
0-000306
0-000397
Considering first the coefiicients for the model with tail plane and
[elevators. In all cases the value is markedly less than that for the free
A.I.CB.I)
Fig. 113. — Variations of tin and rudder area.
rudder, and there is some indication of a greater shielding by the body
when the nose is up than when it is either level or down. This feature is
more readily seen from Fig. 1 14 (a), where the curves for 0° and —5° pitch are
seen to lie below those of the rudder alone, but above the curve for an angle
of pitch of +5°. Fig. 114 (6) shows, in this instance, the effect of the presence
of the elevators ; as ordinate, is plotted the lateral force coefficient with
tail plane, on an abscissa of the similar coefficient without tail plane. The
points are seen to group themselves about a straight line which shows a
224
APPLIED AEEODYNAMICS
loss of 14 per cent, due to the presence of the tail plane. A further reduction
may be expected from the introduction of the main planes in a complete
aircraft due to the slowing up of the air when gUding. On the other hand,
the influence of the airscrew slipstream may be to increase the value
materially until the final resultant effect is greater than that on the free
rudder.
The tests on the effect of shape were carried out on the same body,
but without tail plane and elevators, and the results are given in Table 6.
The fins were divided into two groups, A 1 to A 6, and B 1 to B 5, of
0 0004
0 0003
0-0002
/
00005
00004
00003
RUDC
ER AL(
INC. —
y
X
-"Yav
(With
2
Tailpla
n!>)
/(\-i
k
•"Yav
{With
2
Tailplane.)
.y
/
A
^
+ 5
/
/
'^f
y^
y
0 0001
0
/
/
A
^
f<t)
/
r
(^)
Angle of Yaw (degrees.)
00001 00002 00003 00004 OOOOS
""Vav^
(Without Tailplane.)
00004
0-0003
Set
A.
^
■"Yav
2
y
^
/^ y'
y
^
Xl
^
^
re;
00004
Set
B.
//
"■Yav
2
_/
\y/
[3 -
^
k
P^
^
^
^
W
Angle of Yaw. (degrees.)
2 4 5 S
Angle of Yaw (degrees.)
Fig. 1 14. — Effect of variations of fin and rudder area.
which A 1 and B 1 were identical in size and shape. In the A series the
forms of the vertical surface were roughly similar, the main change being
one of size. Fig. 114 (c) indicates little change in the lateral force coefficient
until the area has been much reduced. Series B, on the other hand, shows a
marked loss of efficiency due to reduction of the height of the fin (Fig.
114 {d)), and both results are consistent with and are probably explained by
a reduction in the speed of the air in the immediate neighbourhood of the
body. Experiments on the flow of fluid round streamline forms have shown
that this slowing of the air may be marked over a layer of air of appreciable
thickness.
DESIGN DATA PEOM AERODYNAMICS LABORATORIES 225
TABLE 6.
Pin Shape as afpectino Usefdlness.
Forces on the fins of Set A.
Angle of
yaw
Lateral force in lbs. per sq. ft. of fln area divided by aqaare of wind speed
(40 ft. per sec).
(degrees).
Al.
0-000063
0000131
0000214
0-000297
0-000383
A 2.
0-000065
0-000133
0-000210
0-000298
0-000386
AS.
A 4.
AS.
AC).
2
4
6
8
10
0-000062
0-000128
0-000214
0-000295
0-000385
0-000058
0000124
0000198
0-000273
0-000357
0-000046
0-000105
0-000177
0-000260
0-000333
0-000042
0-000099
0-000164
0-000219
0-000302
Forces on the fins of Set B.
Angle of
yaw
Lateral force in lbs. per sq. ft. of fln area divided by square of wind speed
(40 ft. per sec).
(degrees).
B 1 or A 1.
B2.
B3.
B4.
B6.
2
4
6
8
10
0000063
0-000131
0-000214
0-000297
0-000383
0000054
0-000114
0-000192
0-000270
0-000347
0-000041
0-000091
0-000156
0-000216
0000281
0000031
0-000064
0-000109
0000150
0-000210
0-000018
0-000037
0-000066
0-000098
0-000145
TABLE 7.
Yawikq Moments due to the Rttdders of a Rigid Airship.
Wind speed, 40 ft.-s. Model Ulustrated in Fig. 103.
Yawino Moments on Aieship (lbs .-ft. at 40 ft
-s.).
Angle
of yaw
Angle of rudders (degrees).
1
(degrees).
-20
-15
-10
-5
0 6
10
15
-0-105
20
0
0118
0-105
0-066
0-032
• 1
0 i -0032
-0-066
-0-118
2
0-230
0-210
0-171
0-140
0-107 +0-062
+0-031
-0-013
-0-035
4
0-333
0-309
0-271
0-227
0-196 0144
0-105
+0-052
+0-027
6
0-421
0-385
0-338
0-304
0-260 : 0-206
0-157
0102
0-073
8
0-478
0-448
0-395
0-360
0-309 0-245
0-202
0-137
0105
10
0-529
0-495
0-438
0-401
0-348 0-282
0-226
0-168
0-138
15
0-591
0-544
0-495
0-440
0-394 0-308
0-250
0-202
0-168
Airship Rudders. — Owing to the considerable degree of similarity
between the airship about vertical and horizontal planes, the rudders
behave for variations of angle of yaw very much in the same way as the
elevators for angles of pitch. For the airship dealt with in Part I. of this
chapter, Fig. 103, the yawingmoments on the model were measured and are
given in Table 7. The type of result is sufiBciently represented by the
226
APPLIED AEEODYNAMICS
elevators and does not need a separate figure. It should be noted that
the yawing moment is positive, and therefore tends to increase a deviation
from the symmetrical position. The effect of the lateral force which
appears when an airship is yawed tends on the other hand to a reduction
of the angle, and it is necessary to formulate a theory of motion before a
satisfactory balance between the two tendencies is obtained.
Ailerons and Wing Flaps. — The first illustration here given of the
determination of the three component forces and component moments
in which a and j3 are both varied relates to a simple model aerofoil. A
later table which is an extension shows the effect of wing flaps. The
model was an aerofoil 18 ins. long and 3 ins. chord with square ends ;
for the experiments with flaps two rectangular portions 4'5 ins. long
and 1-16 ins. wide were attached by hinges so that their angles could
be adjusted independently of that of the main surface.
TABLE 8.
Aerofoil R.A.F. 6, 3 inches x 18 inohes, with Flaps eqttal to J span. Forces and
Moments on Model at a Wind-speed of 40 feet per sec.
Both flaps at 0°.
Angle
Longi-
Lateral
Normal
Boiling
Pitching
Yawing
of
tudinal force
force
force
moment
moment
moment
pitch
mX
mY
mZ
L
M
N
(deg.)-
(lbs.).
(lbs.).
(Ibg.).
(lbs. -ft.).
(lbs. -ft.).
(lbs. -ft.).
/
- 8
-0-0222
0
+0-107
0
-0-0151
0
- 4
-0-0322
0
-0-148
0
-0-0032
0
0
-0-0267
0
-0-411
0
+0-0089
0
Angle of yaw 0° /
+ 4
-0-0030
0
-0-619
0
0-0198
0
\
8
+0-0404
0
-0-812
0
0-0288
0
12
+0-073
0
-0-873
0
00314
0
\
16
-0-027
0
-0-753
0
+0-0129
0
- 8
-0-0218
0-0043
+0-107
-0-0014
-0-0146
-0-0001
- 4
-0-0316
0-0050
-0-141
+0-0010
-0-0033
-0-0004
0
-0-0248
00053
-0-404
0-0028
+0-0085
-0-0008
Angle of yaw 10° ;
+ 4
-0-0037
0-0062
-0-603
0-0044
0-0190
-0-0014
8
+0-0270
+0-0069
-0-782
0-0075
0-0272
-0-0025
12
+0-076
-00011
-0-860
0-0318
00309
+0-0029
\
16
-0-023
-0-0029
-0-762
+0-0572
+00129
+0-0037
/
- 8
-0-0208
0-0090
+0-100
+0-0002
-0-0140
-0-0005
- 4
-0-0291
00091
-0-125
0-0016
-0-0029
-0-0009
0
-0-0242
0-0094
-0-370
0-0039
+0-0073
-0-0016
Angle of yaw 20° <
+ 4
-0-0036
0-0099
-0-556
0-0059
0-0166
-0-0029
8
+0-0338
+00132
-0-722
0-0102
0-0243
-0-0046
'
12
+0-072
-00022
-0-810
0-0496
0-0296
+0-0048
I
16
-0-013
-0-0028
-0-775
+00906
+0-0141
+0-0074
Table 8 shows that, the angle of yaw having been set at the values
0°, 10° and 20° in each series of measurements, the angle of pitch was
varied during the experiment by steps of 4° from —8° to +16°. The
origin of the axes was a point in the plane of symmetry 0*06 in. above the
chord and 1-33 ins. behind the leading edge. With the axis of X in the
direction of the wind the aerofoil made an angle of incidence of 4° when
the angle of yaw was zero : i.e. the angle ao of Fig. 107 c was —4°. With the
i
DESIGN DATA PROM AERODYNAMICS LABORATORIES 227
angle of yaw zero it follows from symmetry that the lateral force and the
rolling and yawing moments are all zero no matter what the angle of pitch.
The longitudinal force on an aerofoil appears for the first time, and a
consideration of the table shows that from a negative value at an angle
of pitch of —8° it rises to a greater positive value at +8°, and then again
becomes negative as the critical angle of attack is exceeded. The normal
force —7nZ has the general characteristics of lift, whilst the pitching moment
differs from the quantities previously given only by being referred to a
new axis.
TABLE 9.
Aerofoil with Wenq Flaps.
Flaps at ±10° (right-hand flap down, and left-hand flap up).
Angle
Longi
Lateral
Normal
RoUlng
Pitching
Yawing
of
tudinal force
force
force
moment
moment
moment
pitch
mX
wY
mZ
L
M
N
(deg.).
(lbs.).
(lbs.).
(lbs.).
(Ibs.-ft.).
(Ibs.-ft.).
(Ibs.-ft.).
- 8
-0-0312
-0-0080
+0-059
-0-0600
—0-0140
+0-0069
- 4
-0-0329
-0-0069
-0162
-0-0641
-0-0027
0-0086
0
-0-0327
-0-0094
-0-385
-0-0685
+0-0068
0-0088
Angle of yaw -20° (
+4
-0-0131
-00107
-0-580
-0-0684
0-0165
0-0093
8
+0-0302
-00118
-0-755
-0-0698
0-0244
0-0106
'
12
+0-0324
-0-0027
-0-804
-00955
0-0197
0-0049
'
16
-0-0193
-0-0054
-0-778
-0-1043
+0-0152
+0-0127
- 8
-00330
-00039
+0-069
-0-0629
-0-0152
+0-0080
- 4
-0-0357
-00036
-0-173
-0-0664
-0-0030
0-0086
0
-0-0341
-0-0052
-0-418
-0-0724
+0-0076
0-0086
Angle of yaw --10° <
+ 4
-0-0104
-0-0054
-0-629
-0-0730
0-0180
0-0084
8
+00341
-00052
-0-814
-0-0732
0-0266
0-0086
12
+00401
-00005
-0-852
-0-0767
0-0209
0-0086
\
16
-00430
+0-0018
-0-748
-0-0723
+0-0097
+0-0148
/
- 8
-0-0329
-0-0002
+0-079
-0-0654
-0-0146
+0-0076
- 4
-0-0412
0
-0-168
-0-0660
-0-0036
0-0080
0
-0-0343
+0-0005
-0-422
-0-0724
+00076
0-0075
Angle of yaw 0° /
+ 4
-0-0102
0-0012
-0-629
-00713
0-0182
0-0067
1
8
+0-0342
0-0018
-0-814
-00654
0-0268
00058
1
12
+0-0431
0-0010
-0-852
-0-0415
0-0267
0-0112
^
16
-0-0350
0-0025
-0-748
-0-0134
+0-0111
+0-0163
- 8
-00316
+0-0049
+0-093
-00654
-00125
+0-0068
/
- 4
-00405
0-0049
-0-153
-0-0632
-00040
00076
0
-0-0338
0-0058
-0-401
-0-0671
+0-0072
0-0064
Anrie of yaw 10°
+ 4
-0-0098
0-0074
-0-603
-0-0632
00175
0-0048
8
+0-0356
0-0080
-0-802
-00552
0-0260
00331
12
+00411
0-0055
-0-827
-0-0094
0-0233
0-0149
16
-0-0252
00044
-0-756
+0-0336
+0-0132
+00196
- 8
-0-0299
+0-0075
+0-096
-00628
-0-0118
+0-0051
- 4
-0-0369
0-0087
-0131
-0-0603
-0-0038
0-0058
0
-00319
0-0088
-0-363
-0-0621
+0-0060
0-0054
Angle of yaw 20°
+ 4
-00103
0-0099
-0-663
-0-0531
0-0154
0-0032
8
+00310
0-0120
-0-733
-0-0464
0-0233
0-0009
12
+0-0416
0-0064
-0-794
+0-0038
0-0222
0-0115
,
16
-00118
0-0083
-0-770
+0-0550
+00151
+0-0200
At an angle of yaw of 10° all the forces and couples have value, but
not all are large. The lateral force mY is not important as compared
with longitudinal force, whilst the yawing moment N is small compared
with the pitching moment. On the other hand, the rolling moment L
228
APPLIED AERODYNAMICS
becomes very important at large angles of incidence. This may be ascribed
to the critical flow occurring more readily on the wing which is down wind
008
0 06
004
0 02
0
-0-02
-004
-006
-0-08
/
FLAPS
o'
/•
1
r
1^
ROLLIf
MOME
(Ibs.f
IG
NT.
/
/
t
""i-i-
-1
i*-
/
/
i
1
/
J
1
X\i
/
/
/
/
/
0
-002
-004
-0 05
ROLLII
MOME
(ib$.1
NT.
V
/Tg'
1
/
+ 6^
.^d^
"T^"
-008
^
/
FLAPS
±10"
/
FLAPS
o'
002
YAW IN
MOME^
(Lbs.f
G
a.
t
n
0
0 02
\
0 02
r
-^^
0
002
YAW IN
MOMEr
{lbs. f
G
IT
+v
FLAPS
±10"
-20 ^ -10 0 10 20 -20 -10 0 10 20
ANGLE OF YAW. (degrees.) ANGLE OF YAW. (degrees.)
Si Jj u iLw ■^^' 1 IS- — Rolling and yawing moments due to the use of ailerons.
due to the yaw than to that facing into the wind. The remarks apply
with a Uttle less force when the angle of yaw is 20°. The results show that
DESIGN DATA FROM AERODYNAMICS LABORATORIES 229
side slipping to the left (+ve yaw) tends to raise the left wing (+ve roll),
and that aileron control would be necessary to counterbalance this rolling
couple. It will be found from Table 9 that the amount of control required
is considerable at an angle of yaw of 20°; and calls for large angles of jflap.
Only the quantities dealing with rolling moment and yawing moment
have been selected for illustration by diagram. Much further information
is given in Report No. 152 of the Advisory Committee for Aeronautics.
Referring to Fig. 115, it will be found that with the flaps at 0° neither the
rolling moment nor the yawing moment have large values until the angle
of pitch exceeds 8° {i.e. angle of incidence exceeds 12°). At larger angles
of pitch the rolling moment is large for angles of yaw of 10° and upwards,
i.e. for a not improbable degree of side shpping during flight. The best
idea of the importance of the rolling couple is obtained by comparing the
curves with those of the figure below, which correspond with flaps put over
to angles of ± 10°. The curves readily suggest an additional rolling moment
due to the flaps which is roughly independent of angle of yaw, but very
variable with angle of pitch. At values of the latter of —4° to +8° the
addition to rolling moment is rather more than —0*06 Ib.-ft. At an
angle of pitch of 12° the effect of the flaps has fallen to two-thirds of the
above value, whilst at 16° it is only one-fifth of it. Quite a small degree
of side shpping on a stalled aerofoil introduces rolling couples greater
than those which can be apphed by the wing flaps. The danger of
attempting to turn a stalled aeroplane has a partial explanation in this
fact.
It will be noticed that the yawing moments are relatively small, but
the rudder is also a small organ of control, and appreciable angles may be
required to balance the yawing couple which accompanies the use of wing
flaps.
The Balancing of Wing Flaps. — The arrangement of the model is shown
in Fig. 116, the end of the wings only being shown. Measurements were
made on both upper and lower flaps, but Fig. 117 refers only to the upper
at a speed of 40 ft.-sec. The model was made so that the strips marked
1, 2 and 3 could be attached either to the main part of the aerofoil or to
the flaps. The moments about the hinge were measured at zero angle of
yaw for various angles of pitch and of flap. In view of the indications
given in the last example that the flow at the wing tips breaks down' at
different angles of incidence on the two sides, it is probable that the balance
is seriously disturbed by yaw and further experiments are needed on the
point. Other systems of balance are being used which may in this respect
prove superior to the use of a horn.
The results are shown in Fig. 117, where the ordinate is the hinge moment
of the flap. The abscissa is the angle of flap, whilst the different diagrams
are for angles of incidence of 0°, 4° and 12°. In each diagram are four
curves, one for each of the conditions of distribution of the balance area.
Since in no case can an interpolated curve fall along the line of zero
ordinates, it follows that accurate balance is not attainable. In all cases,
however, an area between that of 1 and 2 leads to a moment which is nearly
independent of the angle of flap, and which is not very great. As each
230
APPLIED AERODYNAMICS
angle of incidence corresponds with a steady flight speed, large angles being
associated with low speeds, it will be seen that some improvement could
^Maximum Ordinate 0-120
Fia. 116. — The balancing of ailerons.
be obtained over the range 4° to 12° by the use of a spring with a constant
pull acting on the aileron lever.
There is, of course, no reason why this type of balance should not be
applied to elevators and rudders as well as to ailerons, and many instances
of such use exist. Owing to lack of opportunity for making measurements
DESIGN DATA FEOM AEKODYNAMICS LABORATOEIES 231
of scientific accuracy, little is known as to the value of the degrees of balance
obtained. The clearest indication given is that p'lots disUke a close
approximation to balance in ordinary flight.
0001
-0001
0-002
0002
000 I
-ooor
0-003
0002
000 I
Hinge
MOME
s.f/:)_J
Angl
NT.
E OF
4
INCIDE
0
NCE.
Hi
^
-rrr
^
.""^^
" ■
~^
«N<0
-0-00 I
^
Angl
: OF
1
iNCIDE
2°
NCE.
y
l,2«,3
HiNSE
r/6
MOME
NT
^
y
ls<2
^^
^
-^^
-^
.— -
:^
<^
v.,^
*^
-20 -10 O 10
Angle of Flap, (degrees.)
Fig. 117. — Moments on balanced ailerons.
20
Forces and Moments on a Complete Model Aeroplane.— The experi-
ments refer to a smaller model of the BE 2 than that described in Part I.,
but the results have been increased in the proportion of the square of the
linear dimensions, etc., so as to be more directly comparable. The wings
had no dihedral angle, nor was there any fin. Photographs of the model are
232
APPLIED AEEODYNAMICS
shown in Eeport No. Ill, Advisory Committee for Aeronautics. The axis
of X was taken to he along the wind at an angle of incidence of 6° and an
angle of yaw of 0°. Experiments were made for large variations of angle
of yaw and small variations of angle of pitch. Although limited in scope,
the results are the only ones available on the subject of flight at large angles
of yaw and represent one of the limits of knowledge. Application is still
further from completeness.
TABLE 10.
FoEOES AND Moments on a Complete Model Aeroplane.
Complete Model BE 2 Aeroplane,
y'jjth scale. 40 ft.-s. Angle of incidence = angle of pitch+60.
Angle
Angle
Longitudinal
Lateral
Normal
Boiling
Pitching
Tawing
of pitch
of yaw
force
force '
force
moment
moment
moment
a (deg.).
/3 (deg.)-
toX (lbs.)-
TOY (lbs.)-
toZ (lbs.).
L (Ibs.-ft.).
M (lbs.-ft.).
N (lbs. -ft).
i 0
-0-610
0
-3-23
0
+0-222
0
1
6
-0-620
0-138
-317
-0-005
0-220
-0-034
10
-0-631
0-281
-3-09
+0-002
0-209
-0-083
15
-0-618
0-464
-2-94
+0-014
0-197
-0160
20
-0-606
0-635
-2-74
+0-028
0-194
-0-217
25
-0-589
0-823
-2-60
+0-019
0-126
-0-280
30
-0-564
1-020
-2-38
-0-003
+0-039
-0-354
35
-0-535
1-175
-216
+0-002
-0-051
-0-360
\
0
-0-686
0
^413
0
0158
0
5
-0-598
0-140
-4-08
-0-003
0-146
-0-041
10
-0-593
0-286
-3-99
+0-011
0-154
-0-092
0- ,
15
-0-590
0-460
-3-81
0-031
0-157
-0-157
20
-0-564
0-633
-3-67
0-052
0-167
-0-223
25
-0-556
0-830
-3-33
0-035
0-073
-0-293
30
-0-638
1-025
-3-03
0-023
+0-002
-0-363
\
35
-0-497
1-190
-2-74
0-036
-0-096
-0-385
0
-0-506
0
-5-10
0
+0-085
0
6
-0-506
0-133
-5-06
-0-006
0-096
-0-038
10
-0-515
0-280
-4-96
+0-020
0093
-0091
+2° ,
15
-0-623
0-462
-4-70
0043
0-092
-0-159
20
-0-620
0-621
-4-44
0-071
0093
--0-226
25
-0-506
0-828
-4-14
0-062
+0030
-0-298
30
-0-489
1-020
-3-70
0-035
-0-063
-0-367
i
35
-0-464
1-200
-3-41
0-051
-0-166
-0-392
The results of the observations are given in Table 10, and are shown
graphically for zero angle of pitch in Fig. 118. The six curves, three for
forces and three for moments, are rapidly divided into two groups according
to whether they are symmetrical or assymetrical with respect to the
vertical at zero yaw. In the symmetrical group are longitudinal force,
normal force and pitching moment, whilst in the asymmetrical group are
lateral force, rolling moment and pitching moment. It is for this reason
that certain motions are spoken of as longitudinal or symmetrical, and
others as lateral or asymmetrical, and corresponding with the distinction
is the separation of two main types of stabiHty.
Up to angles of yaw of ±20° it appears that longitudinal force and
DESIGN DATA FROM AERODYNAMICS LABORATORIES 2S3
pitching moment are little changed, whilst there is a drop in the numerical
value of the normal force which indicates the necessity for increased speed
to obtain the support necessary for steady asymmetrical flight. Both lateral
force and yawing moment are roughly proportional to angle of yaw, but
the rolling moment is more variable in character. Prom the figures of
-I
-2
-3
-4
\
\
Forces (lbs.)
mX,mY8imZy
V
A--
Pil-chinq
Moment"
M
L.M&N
M.
Angle of V|aw (degre^es.)
0-3
02
01
-01
-0 2
-0 3
0-4
-30 -20 -10 0 10 20 30
Pia. 1 18. — Forces and moments on complete model aeroplane referred to body axes.
Table 10 it is possible to extract a great many of the fundamental deriva-
tives required for the estimation of stabiHty of non-symmetrical, but still
rectiUnear, flight. Before developing the formulae, however, one more
example will be given deahng with the important properties of an aerofoil
which are associated with a dihedral angle.
Forces and Moments due to a Dihedral Angle. — The aerofoil was of
8 ins. chord and 18 ins. span, with elliptical ends, the section being that of
234
APPLIED AEEODYNAMICS
R.A.F. 6. It was bent about two lines near the centre, the details being
shown in Fig. 119. The origin of axes was taken as 0'07 in. above the chord
and 1-40 ins. behind the leading edge, whilst the axis of X was parallel to
the chord. In this case, therefore, angle of pitch and angle of incidence
Fig. 119. — Model aerofoil with dihedral angle.
have the same meaning. Many observations were made, and from them
has been extracted Table 11, which gives for one dihedral angle and several
angles of pitch and yaw the three component forces and moments. Fig.
120, on the other hand, shows rolling moment only for variations of yaw,
pitch and dihedral angle.
TABLE 11.
Forces and Moments on an Aerofoil havinq a Dihedral Angle of 4°.
Angle of pitch, 0°. Wind-speed, 40 feet per sec.
/3
mX
mY
mZ
L
M
N
(degrees).
(lbs.).
(lbs.).
(lbs.).
(Ibs.-ft.).
(Ibs.-ft.).
(Ibs.-ft.).
0
-00161
0
-0149
0
-0-0002
0
5
-0-0160
0-0023
-0-148
+0-0054
-0-0001
+00003
10
-0-0159
0-0044
-0-141
0-0096
0-0000
0-0005
15
-0-0153
0-0061
-0-134
0-0142
+0-0001
0-0007
20
-0-0150
0-0080
-0-125
0-0187
0-0004
00008
25
-0-0139
0-0098
-0113
0-0224
0-0005
0-0008
30
-0-0131
0-0119
-0-098
0-0250
0-0007
0-0009
35
-00120
0-0128
-0-084
+0-0276
+0-0008
+0-0008
Angle of pitch, 5°
0
+0-0133
0
-0-440
0
+0-0136
0
5
0-0131
0-0021
-0-438
+0-0045
0-0137
+0-0003
10
0-0126
0-0040
-0-429
0-0087
0-0131
0-0007
16
0-0111
0-0055
-0-406
0-0133
0-0128
0-0010
20
0-0103
0-0075
-0-385
0-0171
0-0120
0-0013
25
0-0092
0-0093
-0-357
0-0205
0-0113
0-0014
30
0-0079
0-0101
-0-327
0-0235
0-0101
0-0017
35
0-0059
0-0115
-0-287
+0-0262
+0-0088
+0-0018
DESIGN DATA FROM AERODYNAMICS LABORATORIES 235
TABLE 11 — continued.
Angle of pitch, 10°.
0
+0-0667
0
-0-683
0
+00245
0
5
00666
00026
-0-679
+0-0052
0-0244
+0-0004
10
0-0647
0-0048
-0-664
0-0106
0-0239
0-0009
15
00613
0-0067
-0-633
00155
0-0229
0-0012
20
0-0575
0-0083
-0-601
0-0201
0-0214
0-0017
25
0-0528
0-0096
-0-560
0-0239
0-0200
0-0019
30
0-0464
0-0111
-0-505
0-0279
0-0178
0-0024
35
00404
00128
-0-454
+00306
+00157
+0-0027
Angle of pitch,
15°.
0
+01336
0
-0-821
0
+0-0329
0
5
0-1345
0-0019
-0-811
+0-0089
0-0331
+0-0003
10
0-1319
0-0035
-0-795
0-0172
0-0324
0-0009
15
0-1264
0-0050
-0-769
0-0268
0-0308
0-0011
20
0-1203
0-0057
-0-741
0-0338
0-0284
0-0015
25
0-1084
0-0068
-0-700
0-0408
0-0263
0-0022
30
0-0946
0-0075
-0-646
0-0480
0-0236
0-0029
35
0-0827
00081
-0-592
+0-0520
+0-0208
+0-0039
Angle of pitch.
20°.
0
+0-0395
0
-0-754
0
+0-0173
0
6
0-0415
00017
-0-754
+0-0194
0-0176
+0-0003
10
0-0424
0-0034
-0-765
0-0388
0-0177
00015
15
0-0466
0-0047
-0-757
0-0666
0-0180
0-0029
20
0-0603
0-0064
-0-760
0-0728
0-0172
0-0043
25
0-0642
0-0077
-0-736
0-0866
0-0186
0-0078
30
0-0673
0-0093
-0-715
00931
0-0187
0-0099
35
0-0599
0-0109
-0-687
+0-0958
+0-0188
+0-0126
The variation of longitudinal force with dihedral angle presents no point
of importance except at high angles of incidence, where as usual the flow
shows erratic features. Lateral force is, however, more regular and is
roughly proportional both to the dihedral angle and the angle of yaw, and
independent of the angle of incidence up to 20°. Its value is never so
great as to give mY any marked importance in considering the motions of
an aeroplane. Normal force shows no changes of importance even at large
angles of incidence, whilst pitching moment is not strikingly affected by
the dihedral angle except at the critical angle of incidence.
The most interesting property of the dihedral angle is the production
of a rolling moment nearly proportional to itself and nearly independent
of angle of incidence until the critical value is approached. This is most
readily appreciated from Fig. 120, ordinates of which show the rolling
moment in lbs. -feet on the model at a wind speed of 40 ft.-s. There is
a small rolling couple for no dihedral angle at angles of incidence up to
10° and a considerable couple at 15° and 20°. At very large angles
the eflect of the dihedral angle has become reversed and is not the
236
APPLIED AEEODYNAMICS
0" 5 10 15 20 25 30 35
Fig. 120. — Rolling moments due to a dihedral angle on an aerofoil.
DESIGN DATA FEOM AEEODYNAMICS LABOEATORIES 237
predominant effect. The general character of the curves will be found on
inspection to be indicated by an element theory when it is realised that
positive dihedral angle increases the angle of incidence on the forward wing.
Accompanying the rolling moment is a yawing moment of somewhat
variable character, but in all cases appreciably dependent on the value of
the dihedral angle. Much additional information will be found in Report
No. 152 of the Advisory Committee for Aeronautics.
Changes of Axes and the Resolution op Forces and Moments.
(a) Change of Direction without Change of Origin. — Referring to Fig.
121, the axes to which the forces and moments were referred originally are
denoted by GXq, GY© and GZq, and it is desired to find the corresponding
quantities for the axes GX, GY and GZ, which are related to them by the
Pig. 121. — Change of direction of body axes.
rotation a about the axis GYq and ^ about GZq. These angles correspond
exactly with those of pitch and yaw, the order being unimportant with
the definitions given. The problem of resolution resolves itself into that
of finding the cosines of the inclinations of the two sets of axes to each other,
and the latter is a direct application of spherical trigonometry. The results
are
li ^ cos XGXo = cos a cos j8 j
Ml ^ cos XGYq = cos a sin )8 ! . . . . (2)
ni ^ cos XGZq => — sin a J
I2 ^ cos YGXq = — sin )8 ]
W2^cos YGYo = cos^ ! . . . . (3)
W2 =cos YGZo=>0 )
Zg = cos ZGXo => sin a cos )8 j
wig = cos ZGYq = sin a sin /8 ! . . . . (4)
W3 ^ cos ZGZq = cos a I
288
APPLIED AERODYNAMICS
The formulae given in (2), (3) and (4) suffice to convert forces measured
along wind axes to those along body axes. In converting from one set of
body axes to another it will usually happen that ^ is zero, and the conver-
sion is thereby simphfied.
"With the values given, the expressions for X, Y, Z, L, M, and N in
terms of Xq, Yq, Zq, Lq, Mq and Nq are
X = liXo + WiYo + WiZo
Y = ZgXo + W2Y0 + ngZo
Z = ZgXo + W3Y0 + II^Zq
L = Z^Lo + miMo + ^iNq
M = ZgLo + mgMo + WgNo
N = ZgLo + W3M0 + nsNo
(5)
• (6)
(h) Change o£ Origin without Change of Direction. — If the original axes
be XflGo, YqGo and ZqGo of Fig. 122, and the origin is to be transferred
Y.
Fig. 122. — Change of origin of body axes.
from Go to G, the co-ordinates of the latter relative to the original axes
being x, y and z, then
L
M
N:
' Lo + wYo . z -
< Mo + mZo . X
mZiQ.y
- mXn . z
No + wXo . y — mYo
(7)
The forces are not affected by the change of origin. For changes both
of direction and origin the processes are performed in two parts.
DESIGN DATA FROM AERODYNAMICS LABORATORIES 239
Formulae for Special Use with the Equations of Motion and Stability.
The equations of motion in general form do not contain the angles a and
B explicitly, but obtain the equivalents from the components of velocity
along the co-ordinate axes. The resultant velocity being denoted by V
and the components along the axes of X, Y and Zhj u, v and w, it will be
seen from (2), (3) and (4) that
u — Y cos a cos j8, v = —V sin j8, w; = V sin a cos jS . (8)
and the reciprocal relations are»
a = tan
-1
w
i3=r— sin-i;^, V = \/w2 + v2_|_^2. , (9)
By means of (8) and (9) it is not difficult to pass from the use of the
variables V, a and ^tou,v and w.
StabiHty as covered by the theory of small oscillations approximates
to the value of forces and couples in the neighbourhood of a condition of
equilibrium by using a linear law of variation with each of the variables.
Mathematically the position is that any one of the quantities X . . . N
is assumed to be of the form
X =fx{u, V, w, p, q.r).
(10)
and certain values of w . . . r which will be denoted by the suffix zero
give a condition of equiUbrium. For the usual conditions applying to
heavier- than- air craft it is assumed that X can be expanded in the form
X =/x(wo, «o. w;o, VQ' %> ^o)
(11)
The quantities — , etc., are called resistance derivatives and denoted by
du
Xu, etc. As the aerodynamic data usually appear in terms of V, a and fi,
it is convenient to deduce the derivatives from the original curves, and this
is possible (for the cases in which p, q and r are zero) by means of the
standard formulae below : —
doc
du
dp
du
av
du
1 sin a
VcosjS
1 • o
— ;^ COS a sm p
— = COS a cos p
y V and w constant . (12)
240
APPLIED AEEODYNAMICS
^-^= 0
dv
dp
dv
cos jS
) u and w constant . (13)
— = —Bin B
dv ^
da. _ 1 cos a
dw~ V'cos/3
-^= — ,> sin a sin jS )u and t? constant . (14)
dw Y ^ ^
3V
dw'
sin a cos ^
From the experimental side it is known that
X = V2Fx(a.iS) . ,
and by differentiation
"~3w du "^^ da. du^ dp du
nTT n T7} TT sm a 9Fx xt • /-. oFv
= 2V cos a cos ;8 . Fx — V ^ — V cos a sm fi J
cos j8 aa ^ dp
with similar relations for the other quantities, so that '
Iv o oX sina 5Fx . oSF^N
=^ X„ =1 2 cos a cos p^f^ — -—- r. ' ^ — cos a sm j8 ^ '
L„ =
M„ =
V2
COS p
da
"-5i8
Y
5Fy
5Fy
V2
da
>)
5j8
z
aFz
5F,
V2
5a
)}
dp
L
5F^
5Ft.
V2
5a
}f
5)8
M
5F„
<^Fm
V2
5a
)>
53
N
3Fk
5F^
V2
5a
>>
dP'
(15)
(16)
. (17)
2sini8
X
V2
-cos/8^
'^ dp
1 V r, • p X , COS a 5Fx . . ^5Fx
yX.= 2smacos^^^, + — g.^ -sm a sm ^ ^^
DESIGN DATA FEOM AEEODYNAMICS LABOBATORIES 241
From the formula given in (17) it is possible to use aerodynamic data
in the form in which they are usually presented. An alternative method is
to use equations (8) to replot the observations with u, v and w as variables,
but this is not convenient except when j8=0.
For airships and hghter-than-air craft in general, the quantities have
a more complex form ; for stabiHty it is necessary to assume that
X=/x(^*, V, w, p, q, r, u, v, w, p, q, r) . . . (18)
and a new series of derivatives are introduced which depend on the
accelerations of the craft. Some little work has been carried out in the
determination of these derivatives, but the experimental work is still in
its infancy.
Examples of derivatives both for Hnear and angular velocities will be
found in the chapter on stabihty, whilst a theory of elements which goes
far towards providing certain of the quantities is developed in Chapter V.
CHAPTEE V
AERIAL MANOEUVRES AND THE EQUATIONS OF MOTION
The conditions of steady flight of aircraft have been dealt vdth in consider-
able detail in Chapter II., where the equations used were simple because
of the simplicity of the problem. When motions such as looping, spinning
and turning are being investigated, or even the disturbances of steady
motion, a change of method is found to be desirable. The equations
of motion now introduced are applicable to the simplest or the most
complex problems yet proposed. Evidence of an experimental character
has been accumulated, and apparatus now exists which enables an
analysis of aerial movements to be made. The number of records
taken is not yet great, but is sufficiently important to introduce
the subject of the calculation of the motion of an aeroplane during
aerial manoeuvres. After a brief description of these records the
chapter proceeds to formulate the equations of motion and to apply
them to an investigation of some of the observed motions of aeroplanes
in flight.
Looping. — In making a loop, the first operation is to dive the aeroplane
in order to gain speed. An indicated airspeed of 80-100 m.p.h. is usually
sufficient, but at considerable heights it should be remembered that the
real speed is greater than the airspeed. Since the air forces depend on
indicated airspeed and the kinetic energy on real speed through the air, it
will be obvious that the rule which fixes the airspeed is favourable to looping
at considerable heights. Having reached a sufficient speed in the dive the
control column is pulled steadily back as far as it wiU go, and this would
be sufficient for the completion of a loop. The pilot, however, switches
off his engine when upside down, and makes use of his elevator to come out
of the dive gently. Not until the airspeed is that suitable for cHmbing is
the engine restarted.
In looping aeroplanes which have a rotary engine it may be necessary
to use considerable rudder to counteract gyroscopic couples. The effect
of the airscrew is felt in all aeroplanes, and unless the rudder is used the
loop is imperfect in the sense that the wings do not keep level.
The operation of looping is subject to many minor variations, and until
the pilot's use of the elevator and engine during the motion is known it
is not possible to apply the methods of calculation in strictly comparative
form. A fuU account of the calculation is given a httle later in the chapter,
and from it have been extracted the particulars which would be expected
from instruments used in ffight. The instruments were supposed to con-
sist of a recording speed meter and a recording accelerometer. Both have
242
^B been n
AEEIAL MANOEUVKES AND EQUATIONS OP MOTION 24S
referred to in Chapter III., but it may perhaps be recalled here that
the latter gives a measure of the air forces on the aeroplane. The accelero-
meter is a small piece of apparatus which moves with the aeroplane ; the
moving part of it, which gives the record, has acting on it the force of
gravity and any forces due to the accelerations of its support. It is
therefore a mass which takes all the forces on the aeroplane proportionally
except those due to the air. The differential movement of this small mass
and the large mass of the aeroplane depends only on the air force along its
axis. For complete readings three acceleronleters would be required with
their axes mutually perpendicular. In practice only one has been used,
with its axis approximately in the direction of lift in steady flight, and the
acceleration measured in units of g has been taken as a measure of the
increase of wing loading.
Speed and Loading Records in a Loop. — Fig. 123 shows a record of the
true speed of an aeroplane during a loop, with a corresponding diagram
LOOP
100 -
SPEED
(MP.H)
FORCE
ON
WINGS
Fig. 123. — Speed and force on wings during a loop (observed).
for the force on the wings. The time scale for the two curves is the same
and corresponding points on the diagrams have been marked for ease of
reference. The preliminary dive from 0 to 1 takes nearly half a minute,
during which time the force on the wings was reduced because of the
inclination of the path. At 1 the pilot began to pull back the stick, with
an increase in the force on the wings to 3^ times its usual value within 5
244
APPLIED AEKODYNAMICS
or 6 sees. Whilst this force was being developed the speed had scarcely
changed. Between 2 and 3 the aeroplane was climbing to the top of the
loop with a rapid fall of force on the wings. From 3 to 4 the recovering
dive was taking place with a small increase of force on the wings, after
which, 4 to 5, the aeroplane was flattened out and level flight resumed.
The two depressions, just before 4 on the force diagram and at 5 in the speed
diagram, probably correspond with switching the engine off and on.
The calculated speed and loading during a loop are shown in Fig. 124, the
100
SPEED
M.P.H.
50 -
■•
/
1
1
1
/
/
1 ^ LOOP
» \ (CALCULATED)
-
u
'
-1- 1 .1
0-5
10 MINS
1-5
■
^
V
6
5
FORCE .
ON ^
WINGS 3
2
I
0
Fig. 124. — Speed and force on wings during a loop (calculated).
important period of ten seconds being shown by the full lines. The dotted
extensions depend very greatly on what the pilot does with the controls,
and are of Uttle importance in the comparison. The main features of the
observed records are seen to be repeated, the general differences indicating
the advantage of an intelligent use of the elevator in reducing the peaks
of stress over that of the rigid manoeuvre assumed in the calculations. It
will be found when considering the calculation that any use of the elevator
can be readily included and the corresponding effects on the loop and
stresses investigated in detail.
i
AEKIAL MANOEUVEES AND EQUATIONS OF MOTION 245
The conclusion that looping is a calculable motion within the reach of
existing methods is important, and when it has been shown that spinning
is also a calculable motion of a similar nature the statement appears to be
justified that no movement of an aeroplane is so extreme that the main
features cannot be predicted beforehand by scientific care in collecting
aerodynamic data and sufiicient mathematical knowledge to solve a number
of simultaneous differential equations.
Dive. — Fig. 125 shows a dive on the same machuie from which the loop
record was obtained. At a time on the record of about 10 sees, a perceptible
fall in force on the wings was registered due to the movement of the elevator
which put the nose of the aeroplane down. The change of angular velocity
100
SPEED
M.P.H.
50
DIVE
0-5 MINS.
10
15
DIVE
Fig. 125. — Speed and force on wings during a^dive.
near 1 was less rapid than in the loop, and the force on the wings was corre-
spondingly reduced. The stresses in flattening out were quite small, and the
worst of the manoeuvre only lasted for two or three seconds. The record
shows quite clearly the possibility of considerable changes of speed with
inconsiderable stresses, and indicates the value of " hght hands " when
flying. The pilot is a natural accelerometer and uses the pressure on his
seat as an indicator of the stresses he is putting on the aeroplane. An
increase of weight to four times normal value produces sensations which
cannot be missed in the absence of excitement due to fighting in the air.
On the other hand, incautious recovery from a steep dive introduces the
most dangerous stresses known in aerial manoeuvres.
Spinning. — To spin an aeroplane the control column is pulled fully back
246
APPLIED AEKODYNAMICS
with the engine off. As flying speed is lost the rudder is put hard over
in the direction in which the pilot wishes to spin. So long as the controls
are held, particularly so long as the column is back, the aeroplane will
continue to spin. To recover, the rudder is put central and the elevator
either central or slightly forward, the spinning ceases and leaves the
aeroplane in a nose dive from which it is flattened out.
Spinning has been studied carefully both experimentally and theoreti-
cally. It provides a simple means of vertical descent to a pilot who is not
apt to become giddy. There is evidence to show that the manoeuvre is not
universally considered as comfortable, in sharp contrast to looping which
has far less effect on the feelings of the average pilot.
Force and Speed Records in a Spin.— At "0," Fig. 126, the aeroplane
was flying at 70 m.p.h. and the stick being pulled back. The speed fell
SPIN
FORCE 4
ON 3
WINGS
Fig. 126. — Speed and force on wings during spinning.
rapidly, and at 1 the aeroplane had stalled and was putting its nose down
rapidly. This latter point is shown by the reduction of air force on the
wings. The angle of incidence continued to increase although the speed
was rising, and at 2 the spin was fully developed. The body is then
usually inchned at an angle of 70°-80° to the horizontal, and is rotating
about the vertical once in every 2 or 2| sees. The rotation is not quite
regular, as will be seen from both the velocity and force diagrams, but has a
I
AEEIAL MANOEUVKES AND EQUATIONS OF MOTION 247
nutation superposed on the average speed. There is no reason to suppose
that the period of nutation is the period of spin.
At 3 the rudder was centraHsed and the stick put sHghtly forward, and
almost immediately flattening out began as shown by the increased force
at 4. The remainder of the history is that of a dive, the flattening out
having been accelerated somewhat at 5.
It has been shown by experiments on models that stalling of an aeroplane
automatically leads to spinning, and that the main feature of the
phenomenon is calculable quite simply.
Roll (Fig. 127). — The record of position of an aeroplane shown in Fig.
127 was taken by cinema camera from a second aeroplane. Mounted
in the rear cockpit, the camera was pointed over the tail of the camera
aeroplane towards that photographed. The camera aeroplane was flown
carefully in a straight hne, but the camera was free to pitch and to
rotate about a vertical axis.
For this reason the pictures are
not always in the centre of the
film. In discussing the photo-
graphs, which were taken at
intervals of about I sec, it is
illuminating to use at the same
time a velocity and force record
(Fig. 128), although it does not
apply to the same aeroplane.
Photograph 1 shows the
aeroplane flying steadily and on
an even keel some distance
above the camera. The speed
would be about 90 m.p.h., i.e. li 12 f3 14 is
just before 1 on the speed chart. Fig. 127.— Photographic records of rolling.
The second photograph shows
the beginning of the roll, which is accompanied by an increase in the angle
of incidence. The latter point is shown by the increased length of pro-
jection of the body as well as by peak 1 in the force diagram. Both roll
and pitch are increased in the next interval, with a corresponding fall of
speed. At four the bank is nearly 90° and the pitch is slightly reduced.
The vertical bank is therefore reached in a little more than a second.
Once over the vertical the angle of incidence (or pitch) is rapidly reduced,
and as the speed is falling rapidly the total air force on the wings falls until
the aeroplane is upside down after rather more than 1| sees. At about
this period the force diagram shows a negative air force on the wings, and
unless strapped in the pilot would have left his seat. This negative air
force does not always occur during a roll, and is avoided by maintaining the
angle of incidence at a high value for a longer time. The pilot tends to
fall with an acceleration equal to g, but if a downward air force occurs on
the wings of the aeroplane it tends to fall faster than the pilot, and there-
fore maintains the pressure on his seat. This more usual condition in a
roll involves as a consequence a very rapid fall when the aeroplane is upside
rjF' <(^
4
#
\
1 2
3
4
5
^=t5 ^
/
\
\
6 7
8
9
(0
\ \.
V
^
""^^
248
APPLIED AERODYNAMICS
down. The most noticeable feature of the remaining photographs is the
fact that the pilot is holding up the nose of the aeroplane by the rudder,
a manoeuvre accompanied by vigorous side slipping. As the angle of
incidence is now normal, the speed picks up again during the recovery
of an even keel. The manoeuvres after 3, Fig. 128, are those connected
with flattening out, and occur subsequently to the roll. The complete roll
takes rather less than four seconds for completion.
The roll may be carried out either with or without the engine, and except
for speed the manoeuvres are the same as for a spin, i.e. the stick is pulled
back and the rudder put hard over. The angle is never reduced to that
for stalling, and this is the essential aerodynamic difference from spinning.
Fig. 128 • — Speed and force on wings during a roll.
The photographs show that these simple instructions are supplemented
by others at the pilot's discretion, and that the aerodynamics of the motion
is very complex.
Equations of Motion. — In dealing with the more complex motions of
aircraft it is found to be advantageous to follow some definite and compre-
hensive scheme which will cover the greater part of the problems likely to
occur. Systems of axes and the corresponding equations of motion are
to be found in advanced books on dynamics, and from these are selected
the particular forms relating to rigid bodies.
An aeroplane can move freely in more directions than any other vehicle ;
it can move upwards, forwards and sideways as well as roll, pitch and turn.
The generahty of the possible motions brings into prominence the value to
AEEIAL MANOEUVEES AND EQUATIONS OF MOTION 249
the aeronautical engineer of the study of three-dimensional dynamics, and
furnishes him with an unhmited series of real problems.
The first impression received on looking at the systems of axes and
equations is their artificial character. A body is acted on by a resultant
force and a resultant couple, and to express this physical fact with pre-
cision six quantities are used as equivalents. Attempts have been made
to produce a mathematical system more directly related to physical con-
ceptions, but co-ordinate axes have survived as the most convenient form
known to us of representing the magnitudes and directions of forces and
couples and more generally the quantities concerned with motion.
Of the various types of co-ordinate axes of value, reference in this book
is made only to rectiUnear orthogonal axes. Some use of them has been
made in the last chapter, where it was shown that experimental results are
equally conveniently expressed in any arbitrarily chosen form of such axes.
If, therefore, it appears from a study of the motion of aircraft that some
particular form is more advantageous than another, there is no serious
objection on other grounds to its use.
It happens that for symmetrical steady flight, the only point of im-
portance in the choice of the axes required, is that the origin should be at
the centre of gravity in order to separate the motions of translation and
rotation. For circling flight, in which the motion is not steady, it saves
labour in the calculation of moments of inertia and the variations of them
if the axes are fixed in the aircraft and rotate with it. A further simplifi-
cation occurs if the body axes are made to coincide with the principal axes
of inertia. Some of these points will be enlarged upon in connection with
the symbolical notation, but for the moment it is desired to draw attention
to the different sets of axes required in aeronautics and allied subjects.
Choice of Co-ordinate Axes. — The first point to be borne clearly in mind
is the relative character of motion. Two bodies can have motion relative
to each other which is readily appreciated, but the motion of a single body
has no meaning. In general, therefore, the simplest problems of motion
involve the idea of two sets of co-ordinate axes, one fixed in each of the two
bodies under consideration. The introduction of a third body brings with
it another set of axes. In the case of tests on a model in a, wind channel it
has been seen that one set of axes was fixed to the channel and another to
the model. The relation of the two sets was defined by the angles of pitch
and yaw a and ^, whilst the forces and couples were referred to either set
of axes without loss of generality. Instead of the angles of pitch and yaw
the relative positions of the axes could, as already indicated in Chapter IV.,
have been defined by the direction cosines of the members of one set
relative to the other, and for many purposes of resolution of forces and
couples this latter form has great advantages over the former. Both are
sufiiciently useful for retention and a table of equivalents was given in the
treatment of the subject of the preparation of design data.
The relation between the positions of the axes of two bodies is affected
and changed by forces and couples acting between them or between them
and some third body, and only when the whole of the forces concerned in
the motion of a particular system of bodies have been included and related
250 APPLIED AEEODYNAMICS
to their respective axes is the statement of the problem complete. As an
example consider the flight of an aeroplane : the forces and couples on it
depend on the velocities, linear and angular, through the air, and hence two
sets of axes are here required, one in the aeroplane and the other in the air.
The weight of the aeroplane brings in forces due to the earth, and hence
earth axes. In the rare cases in which the rotation of the earth is con-
sidered, a fourth set of axes fixed relative to the stellar system would be
introduced, and so on.
The statement of a problem prior to the application of mathematical
analysis requires a knowledge of the forces and couples acting on a body
for all positions, velocities, accelerations, etc., relative to every other
body concerned. This data is usually experimental and has some degree
of approximation which is roughly known. By accepting a lower degree
of precision one or more sets of axes may be ehminated from the problem,
with a corresponding simplification of the mathematics. This step is the
justification for ignoring the effect of the earth's rotation in the usual
estimation of the motion of aircraft.
A further simplification is introduced by the neglect of the variations
of gravitational attraction with height and with position on the earth's
surface, the consequence of which is that the co-ordinates of the centre of
gravity of an aeroplane do not appear in the equations of motion of aircraft
in still air. The angular co-ordinates appear on account of the varying
components of the weight along the axes as the aircraft rolls, pitches and
turns. In considering gusts and their effects it will be found necessary to
introduce hnear co-ordinates either exphcitly or impHcitly.
The forces on aircraft due to motion relative to the air depend markedly
on the height above the earth, and of recent years considerable importance
has attached to the fact. The vertical co-ordinate, however, rarely
appears directly, the effect of height being represented by a change in the
density p, and here again the approximation often suffices that p is constant
during the motions considered. Apart from this reservation the air forces
on an aircraft depend only on the relative motion, and advantage is taken
of this fact to use a special systein of axes. At the instant at which the
motion is being considered the body axes of the aircraft have a certain
position relative to the air, and the air axes are taken to momentarily
coincide with them. The rate of separation of the two sets of axes then
provides the necessary particulars of the relative motion.
The equations of motion which cover the majority of the known pro-
blems require the use of three seta of axes as follows : —
(1) Axes fixed in the aircraft. " Body axes." For convenience the
origin of these is taken at the centre of gravity, and the directions are
made to coincide with the principal axes of inertia. The latter point is
far less important than the former.
(2) Axes fixed in the air. " Air axes." Instantaneously coincident with
the body axes. In most cases the air is supposed still relative to the
earth.
(3) Axes fixed in the earth. " Earth axes."
The angular relations between the axes defined in (1) and (2) have
AEEIAL MANOEUVEES AND EQUATIONS OF MOTION 251
already been referred to (Chapter IV., page 237), as angles of pitch and
yaw; also by means of direction cosines and the component velocities
u, V and w. The corresponding relations between (1) and (3) are
required ; the angles being denoted by 6, <f) and i/j the aeroplane is put
into the position defined by these angles by first placing the body and
earth axes into coincidence, and then
(a) rotating the aircraft through an angle tji about the Z axis of
the aircraft ;
[h) then „ „ „ „ 6 about the new posi-
tion of the Yaxis of
the aircraft ;
and (c) finally „ „ „ „ <f> about the new posi-
tion of the X axis
of the aircraft.
The angles ijt, 6 and ^ are spoken of as angles of yaw, pitch and roll re-
spectively, and the double use of the expressions " angle of yaw " and " angle
of pitch " should be noted. Confusion of use is not seriously incurred since
the angles a and ^ do not occur in the equations of motion, but are
represented by the component velocities of the resultant relative wind.
That is, the quantities V, a and ^ of the aerodynamic measurements are
converted into u, v and w before mathematical analysis is applied.
With these explanations the equations of motion of a rigid body as
applied to aircraft are written down and described in detail : —
where
m{u -{- wq — vr} => mX' . . . 1w
m{v + wr — wp} =" wY' . . . Iv
m{w -\- vjp — uq} => mZ' . . . Iw
hi — rh^ -\- qh^ = L' . . . . Ip
}i2 — 'ph^-\-rhi^W . . . . \q
hs—qhi-\-ph2=>W .... If
hi = pA. — qF — rE> )
hs = rC-pE-q'D)
(1)
(2y
In these equations m is the mass of the aircraft, whilst A, B, C, D, E
an(f F are the moments and products of inertia. All are experimental
and depend on a knowledge of the distribution of matter throughout the
aircraft. The quantities mX', mY', wZ', L', M' and N' are the forces
and couples on the aircraft from all sources, and one of the first operations
is to divide them into the parts which depend on the earth and those which
arise from motion relative to the air. The remaining quantities, u, v, w^
p, q, and r define the motion of the body axes relative to the air axes. The
equations are the general series applicable to a rigid body, and only the
description is limited to aircraft.
The quantities m. A . . . F are familiar in dynamics and do not
252
APPLIED AEEODYNAMICS
need further attention except to note that D and F are zero from symmetry.
It has already been shown how the parts of X'— N' which depend on motion
relative to the air are measured in a wind channel in terms of u, v and w,
p, q and r.^
It now remains to determine the components of gravitational attraction.
A little thought will show that the component parts of the weight along the
body axes are readily expressed in terms of the direction cosines of the
downwardly directed vertical relative to the axes. Eotation about a
vertical axis through an angle «/> has no effect on these direction cosines,
and the only angles which need be considered are 6 and ^ as illustrated
in Fig. 129. The earth axes are GXq, GYq and GZq, and before rotation the
body axes GX, GY and GZ are supposed to coincide with the former.
Fig. 129. — Inclinations of an aeroplane to the earth.
Eotation through an angle 6 about GYq brings Xq to X and Zq to Zi, whilst
a subsequent rotation through an angle 0 about GX brings Yq to Y and Z^
to Z, and the body axes are now in the position defined by 6 and (f). The
direction cosines of GZo relative to the body axes are
nj^cos XGZq = — sin 0 "j
712^^ cos YGZo = COS 6 Bm<f> r
?i3^cos ZGZq =1 COS ^ cos 0 j
(3)
and the components of the weight are mg times the corresponding direction
cosines. The symbols nj, ^2 and n^ have often been used to denote the
longer expressions given in (3). The first example of calculation from the
equations of motion will be that of the looping of an aeroplane, and con-
1 The experimental knowledge of the dependence of X' — N' on angular velocities
relative to the air is not yet sufficient to cover a wide range of calculation.
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 253
siderable simplification occurs as a result of symmetry about a vertical
plane.
The Looping of an Aeroplane. — The motion being in the plane of
symmetry leads to the mathematical conditions
v=^0 r=.0 ^=0 0 = 0 . . . (4)
Y' = 0 L'=0 N'=0 (5)
and equations (1) and (2) become
u-\- wq='X' ]
w-uq=^7i' (6)
gB=M'J
Making use of equations (3) to separate the parts of X' which depend on
gravity from those due to motion through the air converts (6) to
u-\-wq='—g 9md-{-X]
w — uq= gG0s6-\-Z [ (7)
qB= M)
where X, Z and M now refer only to air forces. X depends on the airscrew
thrust as well as on the aeroplane, and the variation for the aeroplane with
u and w is found in a wind channel in the ordinary way. The dependence
of X on g is so small as to be negUgible. If the further assumption be made
that the airscrew thrust always acts along the axis of X, a simple form is
given to Z which then depends on the aeroplane only. The component of
Z due to q is appreciable and arises from the force on the tail due to pitching.
The pitching moment M also depends appreciably on u, w and q, and the
assumption is made that the effects due to q are proportional to that
quantity, and that the parts dependent on u and w are not affected by
pitching. Looping is not a definite manoeuvre until the motion of the
elevator and the condition of the engine control are specified, and more
detailed experimental data can always be obtained as the requirements of
calculation become more precise. The general method of calculation is un-
affected by the data, and those given below may be taken as representative
of the main forces and couples acting on an aeroplane during a loop.
Fig. ] 30 shows the longitudinal force on an aeroplane without airscrew,
the value of the force in pounds having been divided by the square of the
speed in feet per second before plotting the curves. The abscissa is ^,
i.e. the ratio of the normal to the resultant velocity. This ratio is equal
to sin a, where a is the angle of pitch as used in a wind channel, and no
difficulty will be experienced in producing similar figures from aerody-
namical data as usually given. The aeroplane to which the data refers
may be taken as similar to that illustrated in Fig. 94, Chapter IV. Details
of weight and moments of inertia are given later.
The corresponding values of normal force are shown in Fig. 131. The
separate curves show that aeroplane characteristics appreciably depend
on the position of the elevators. The thrust of the airscrew is given in
254 APPLIED AEEODYNAMICS
Fig. 132 and is shown as dependent only on the resultant velocity of the
/p^
001
L
^r
\
OTXi. longitudinal force in Jbs.
\/2 <:^,,„^^fXr,^^„.f^/^^
1
'r
\
n
V
"' "/"
'/"*■"
f//
\
//
i
\
■001
////
if
\
A
7
\
002
ELEV
ATOR ANGLE
y),
f
\
■'+I0''"
■^30°
■v..^^^
/^
//
0^3
==^
^
^1!!;^
■
V
^_^
•<f3F
■^
"-
/
^^
004
%
04
-03
-0-2
-01
01
0-2
0-3
Fig. 130. — Longitudinal force on an aeroplane without airscrew due to inclination to the
relative wind.
03
02
01
-01
-0-2
-0-3
-0-4
1
■
— -:z
^^^
'
~-
5^
$^.
^^
^
^.
N>
$^
^
V^
1
1
viORMAL FORCE in h
Square of speed in ft.
^
\
^^,2^:
/sec
\
^
/
V
/
^
^.
\
^
^.
^
5^
^
^
30° -—
s
^
^
"'O^^c 1
X
^
^
%
+
0»^
1
^
■
+ .
-0-4 -0 3 -0 2 -01 O 01 0-2 0-3
Fig 131 —Normal force on an aeroplane due to inclination to the relative wind.
aeroplane, and here a careful student will see that the representation can
only be justified as a good approximation in the special circumstances
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 255
The chief items in pitchuig moment are illustrated by the curves of Figs. 188
and 1 84, of which the former relates to variation with angle of incidence,
and the latter to variation with the angular velocity of pitching. Since
the couple due to pitching arises almost whoUy from the tail, a simple
approximation allows for the change of force due to pitching. If I be
800
700
600
SCO
400
300
200
100
~
THRUST /6s.
\
^
50
100
150
200
VELOCITY /.ysec.
Fio. 132. — Airscrew thrust and aeroplane velocity.
the distance from the centre of gravity of the aeroplane to the centre of
pressure of the tail, the equation
m . I .Z„=>'M.„
(8)
can be seen to express the above idea that the tail is the only part of the
aeroplane which is effective in producing changes due to pitching.
With the aerodynamic data in the form given, equations (7) are con-
veniently rewritten as
=i — wq — g am 6^
w
uq-{-g cos 6 +
Ml
V2
V2 M,
B "^ V
V2
"V2"
qY
B
V2 T
m m '
V2 M,
m"^ V
qV
ml
, (9)
(10)
(11)
256
APPLIED AEEODYNAMICS
These equations show the changes of u, w and q with time for any
given conditions of motion, and enable the loop to be calculated from
the initial conditions by a step to step process. The initial conditions
=04 -0-3 -0-2 -01 0 01 0-2 0-3
Fig. 133. — Pitching moment due to inclination of an aeroplane to the relative wind.
-10
-2 0
-30
-4 0
•5 0
-6 0
-70
-
M^'' PITCHING MOMENT oi/<? A? /O/yc/^/zj^, /^sJtT.
y^, inniilnr \/p/nrit\/ x / inpar i/p/i7r/Yv in ft'.sf
c
V
y"'""
-^
\
\
^-^
\
At
\
-02 -01 0 01 0-2 0-3
Fig. 134. — Pitching moment due to pitching of an aeroplane.
0-4
must be chosen such as to give a loop, and some further experience or trial
and error is necessary before this can be done satisfactorily. Usually,
looping takes place only at some considerable altitude, but the calculations
now given assume an atmosphere of standard density.
AEKIAL MANOEUVKES AND EQUATIONS OP MOTION 257
The weight of the aeroplane was assumed to be 1932 lbs., and other data
relating to its dimensions and masses are
m=360, B=.1500, / = 15 .... (12)
At the particular instant for which the calculation was started the
motion is specified by
6=^-20° V = 180ft.-s. ^ = -0-06)
V J . . (13)
q=:0 Elevator angle —15° J
The processes now become wholly mathematical, and the chief remain-
ing difficulty is that of making a beginning ; a little experience shows
that for the first 0*1 or 0*2 seconds certain approximations hold which
simplify the calculation. It may be assumed that in the early stages
V = constant. cos 6 => constant \
Ml J M. . mZi ,. , ^. ,w\ . . (14)
^ and ^ const. ^^-^ a Imear function of - ( ^ '
y-i V V- V;
The limitation of time to which (14) applies is indicated in the course
of the subsequent work.
Equation (11) becomes for this early period
and a solution consistent with the assumptions as to constancy of M^ and
Ml is
^--Ky-'^^ (^^)
-^ '-'o-^l'^l, (17)
These equations for q and 6 are easily deduced and verified. From the
initial data and the curves in Figs. 130-134, it will be found that for
I =-0-06
^ = 0-43, y' = -58, V = 180 and ^0 = - 0*349 radian^
and by deduction from these ) (18)
^i = -l-33 and M? = -6-96
M, B
Equations (16) and (17) now become
g = 1-33(1 -e-«»«0
^ = -0.349 + l-33i-^j
I
(19)
and from them can be calculated the various values of q and 6 which are
given in Table 1 .
B
258
APPLIED AERODYNAMICS
A similar process will now be followed in the evaluation of w for small
values of t Equation (10) may be written as
■ =Zi+(u + ^^y + gGOBd '. . . . (20)
w
w
and from Pig. 131 it is found that in the neighbourhood of ^=—0-06, and
for elevators at — 15° the value of Zj is given by
. (21)
^i = -l-59(-+0-07) I
or Zi = — 4-77m; — 60 when V = 18o) *
Inserting numerical values, equation (20) becomes
ri; = -4-77M; + 168-4g-29-7 .... (22)
The value of q previously obtained, equation (19), may be used and an
integral of (22) is
M; = Ae-*'7< + 40-8 + 102'2e-«»«' .... (28)
w
Since — = — 0-06 and V = 180 at the time f = 0, it follows that the
initial value of w is — 10"8, and the value of A in (23) is then found to be
— 153-8, so that
«; = _153-8e-*""« + 40-8 + 102-2e-6»« . . (24)
and w; = 733e-*"<-711e-6 96< ...... (26)
Values of m; and li? are shown in Table 1,
TABLE 1.
Initial Stages of a Loop.
'sec.
g-6-96«
Q
Q
e
cos 0
g-4-77«
w
M>
0
1000
0
9-26
-0-349
0-940
1-000
-10-8
22
005
0-707
0-390
6-55
-0-339
0-942
0-790
- 8-2
76
0-10
0-500
0-665
4-63
-0-312
0-952
0-621
- 3-5
99
015
0-352
0-862
3-26
-0-274
0-962
0-489
+ 1-6
108
0-20
0-249
0-999
2-31
-0-226
0-975
0-385
7-1
105
0-30
0-124
1-165
1-15
-0-117
0-992
0-239
16-7
87
0-40
0-062
1-249
0-57
+0-006
1-000
0149
24-2
65
Of the various limitations imposed by (14) the one of greatest importance
Ml
is that relating to the constancy of ~ and reference to Fig. 133 will indicate
V '
w
that this should not be pushed further than the value for ^ =
0-02.
Table 1 then shows a limit of time of 0-10 sec. before the step-to-step
method is started. The work may be arranged as in Table 2 for con-
venience. Across the head of the table are intervals of time arbitrarily
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 259
chosen ; as the calculation proceeds and the trend of the results is seen
it is usually possible to use intervals of time of much greater magnitude
than those shown in Table 2. For < => 0 a number of quantities such as
V, w, q, 6 are given as the initial data of the problem, whilst others like
mX. T
^Tf^, - , etc., are deduced from the curves of Figs. 130-134. A comparison
\^ m
between the expressions in the table and those in equations (9), (10), and
(11) will indicate the method followed. The additional equation for
finding V comes from
V2 = m2+m;2. . ,
by simple differentiation and arrangement of terms.
(26)
TABLE 2.
Beginnino of Stbp-to-Step Calotjlation.
t«^
0
0-05
0-10
0-15
<NC
'
0-05
0-10
V
180
180
1801
180-2
vq
0
70-0
115-3
u
179-6
179-8
1801
180-2
gco&d
30-25
30-31
30-6
to
-10-8
~ 8-2
- 3-1
+ 1-2
Y^ • m
- 8-64
-18-9
- 46-0
a
0
0-390
0-640
0-854
1 M, jV
15 • V * m
0
- 4-4
- 6-4
efit
9
0-039
- 0-339
0-064
- 0-310
0085
- 0-275
- 0-349
w
77-0
93-6
wSt
7-7
9-4
cos 6
0-940
- 0*342
0-942
- 0-332
0-952
- 0-305
I
sin d
1500 V2' ^
9-29
9-07
8-64
V
- 0060
- 0046
- 0-017
—
1 ^a
1500 • V '^^
0
- 2-67
- 4-00
y2
640
540
641
1-98
9-82
—
9
St
u.
m
—
6-40
0-64
4-64
-wq
0
110
3-20
10-6
,
0-46
—gsisxd
5-00
3-83
wX, V»
-15-39
-15-39
-14-53
_
yU
V« ■ m
vo .
—
— 3-51
— 1-61
T
yW
6*58
6-58
6-56
m
V
—
1-49
2-22
u
—
6-00
3-83
uht
0-50
0-38
VSi
—
0-15
0-22
The fundamental figures for f=0*05 are taken from Table 1, and using
them the necessary calculations indicated by equations (9), (10), and (11)
are made to give the instantaneous values of tt, li), g. The necessary basis
for step-to-step calculation is then complete, and many^differences of detail
260
APPLIED AEEODYNAMICS
would probably be made to suit the habits of an individual calculator.
The assumption which was made in proceeding to the next column was
that the values of it, w, q, q at t=>0'5 were equal to the average values over
the interval of time 0 to 0*10 sees. As an example consider the value of
w ; at f =0, w = —10-8. At t = 0-05, w = 77-0, and in the interval of O'lO
sec. the change of w is taken as 7"7. Adding this to the value of w at
t = 0 gives — 3'1 as the value of w at <=0-10 as tabulated. A comparison
of the values of w, w, q, q and 6 as calculated in this way with those of
Table 1 will show that the mathematical approximations of (14) had not
led to large errors. The preliminary stages of calculation for t=:0'15 are
shown, and the procedure followed will now be clear.
TABLE 3.
Details of Loop.
Time
(sees.).
Vft..a.
«'ft.-s.
«ft.-8.
9
(rads.-s.).
0
(degrees).
Angle of
incidence
(degrees).
mZi
1932
0
180
-10-8
179-8
0
-20-0
- 0-4
0-2
0-5
177-8
+24-0
176-3
0-835
+ 40
10-7
5-2
10
167-5
21-8
166-2
0-658
24-3
10-5
4-6
1-6
154-8
20-2
153-5
0-624
42-1
10-5
, 3-9
20
139-4
17-8
135-7
0-560
61-8
10-5
3-2
2-5
123-3
15-6
122-3
0-518
74-7
10-3
2-4
30
1071
12-9
106-4
0-478
89-0
9-9
1-8
3-5
92-5
100
92-0
0-451
102-3
9-2
1-2
4-0
79-6
6-9
79-3
0-435
115-0
8-0
0-8
50
61-7
0-5
61-7
0-450
1401
+ 3-5
0-3
6-0
58-5
- 5-8
58-3
0-450
165-7
- 2-7
00
6-5
62-8
- 7-0
62-4
0-488
179-0
- 3-3
-0-1
7-0
70-7
- 5-9
70-5
0-526
193-5
- 1-8
0-0
8-0
94-8
+ 2-7
94-7
0-639
226-6
+ 4-5
0-8
9-0
1251
11-8
124-5
0-642
263-6
8-4
2-2
100
151-6
17-6
150-5
0-666
300-1
9-7
3-5
The calculations were carried out for a complete loop, and Table 3
shows the variation of the quantities concerned at chosen times. At the
beginning of the loop the angle of incidence is shown as —0*4 degree,
whilst less than half a second later it has risen to 11-0 degrees. The
loading on the wings can be calculated at any time from the value of mZ
corresponding with the tabulated numbers for V and w and Fig. 131. The
maximum is 5-2 times the weight of the aeroplane, but owing to the fact
that the load on the tail is downward this does not represent the load on
the wings, which is then about 10 per cent, greater.
The shape of the loop can be obtained by integration at the end of the
calculations since the horizontal co-ordinate is
x= {u cos d^w sin d)dt
whilst the vertical co-ordinate is
z=: (usin 6~w cos 6)dt
(27)
(28)
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 261
The integrals may be obtained in any of the well-known ways, and the
results for the above example are shown in Fig. 135. It will be seen that
the closed curve is appreciably different to a circle, has a height of
nearly 300 feet and a width of 230 feet. A diagram of the aeroplane inset
to scale shows the relative proportions of aircraft and loop. The time
Fig. 135. — A calculated loop.
taken is 10 or 11 seconds, and a pilot frequently feels the bump when
passing the air which he previously disturbed.
In the calculations as made, the engine has been assumed to be working
at fall power and the elevator held in a fixed position. In many cases the
engine is cut off after the top of the loop has been passed, and the elevator
is probably never held still. In addition to the longitudinal controls, it is
262
APPLIED AEEODYNAMICS
found necessary to apply rudder to counteract the gyroscopic effect of the
airscrew and so maintain an even keel.
Failure to complete a Loop. — The calculations just made assumed an
initial speed of 180 ft.-s. in a dive at 20°, and indicated some small reserve
of energy at the top of the loop. A reduction of the speed to 140 ft.-s.
and level flight before pulhng over the control column leads, with the
same assumptions as to the aeroplane, to a failure to complete the loop.
TABLE 4.
Failtjkb to Loop.
Time
(sees.).
Besultant velocity V
(ft.-8.).
Inclination of airscrew
axis to liorizontal 9
(degrees).
Angle of incidence
(degrees).
0
140
- 1-7
4-4
0-5
138
+ 14-7
10-7
1-0
131
29-8
11-5
1-5
121
42-4
11-6
20
110
54-3
11-6
2-5
97
65-2
11-6
30
85
74-7
11-6
3-5
73
83-2
11-6
40
60
90-2
11-1
4-5
49
97-8
10-1
50
38
104-0
8-0
5-5
28
110-2
3-4
5-7
23
112-7
- 0-2
5-9
19
1150
- 5-8
6-1
16
117-3
-16-5
The figures in Table 4 are of considerable interest as showing one of
the ways in which an aeroplane may temporarily become uncontrollable
owing to loss of flying speed. Up to the end of four seconds the course
of the motion presents little material for comment ; the aeroplane is then
moving vertically upwards at the low speed of 60 ft.-s. and is turning
over backwards. The energy is insufficient to carry the aeroplane much
further, but at 5 seconds the aeroplane is 20 degrees over the vertical
with a small positive angle of incidence, but a speed of only 28 ft.-s.
In the next half -second the aeroplane begins to faU, and at the end of 6*1
sees, is still losing speed and has a large negative angle of attack, i.e. is
flying on its back, with the pilot supported from his belt. Owing to the
low speed the controls are practically inoperative, and the pilot must per-
force wait until the aeroplane recovers speed before he can resume normal
flight. If the aeroplane is unstable in normal straight flight some diffi-
culty may be experienced in passing from a steady state of upside-down
flying to one in a normal attitude.
The detailed calculations from which Tables 3 and 4 have been com-
piled were made by Miss B. M. Cave-Browne-Cave, to whom the author is
indebted for assistance on this and other occasions.
Steady Motions, including Turning and the Spiral Glide. — The equations
of motion given in (1) and (2) take special forms if the motion is steady.
AEEIAL MANOEUVRES AND EQUATIONS OF MOTION 263
(31)
Not only are the quantities u, v, io, p,.^ and f equal to zero, but there is a
relation between the quantities 'p, q and r. As the forces on an aeroplane
along its axes depend on the incHnations of the aeroplane relative to the
vertical, it will be evident that they can only remain constant if the resultant
rotation is also about the vertical. This rotation is denoted by O, and
looking down on the aircraft the positive direction is clockwise.
The direction cosines of the body axes relative to the vertical were
found and recorded in (3), and from them the component angular velocities
about the body axes are
p =. — ii sin 0 j
g = a cos ^ sin ^[ (30)
r = 12 cos 0 cos <f>)
With the products of inertia D and ¥ equal to zero the equations of
steady motion are
wq — vr='X — g sin 6 . . . (31w)\
ur — wp ^'Y -{- g cos 6 aincf) . . {21v)
vp — uq = Z -\-g cos 6 cos ^ . . {^Iw)
rq{C-B)-pqB==h ...... (dip)
^(A-C) + (p2_r2)E=M (31g)
qp{B — A) + grE = N (31r) ,
In equations (31), X, Y, Z, L, M, and N refer only to forces and couples
due to relative motion through the air. If the values of p, q and r given
by (30) are used in (31), the somewhat different forms below are obtained: —
il cos d{w sm<f> — V cos ^) =■ X — gf sin ^ . . (32w)
il{u cos 6 COS <f> -\- w sin 6) =!Y -\- g cos ^ sin ^ (32?;)
Q,{—v sin 6 — u cos 6 sin <f>) =! Z -\- g cos 6 cos <f> {d2w)
02 cos 9 sin 0{(C — B) cos ^ cos 0 + E sin ^} = L . (32^)
ii2| _(A-C) sin d cos d cos ^+E(sm2 ^-cos2 d cos2 <^)} =M (32g)
122 cos ^ sin 0{ -(B — A) sin 0 + E cos 6 cos </>} = N (32r) ^
The equations for steady rectilinear synmietrical motion are obtained
from (32) by putting 12 => 0, ^ => 0 ; they then become
X = gf sin ^ I
Z = — ^cos0 . . (33)
*'^'Y = 0 L=0 M=0 and N=o)
and the great simplicity of form is very noticeable. The solutions of (33)
formed the subject-matter of Chapter II, and cover many of the most
important problems in flying. Some discussion of the more general
equations (32) will now be given ; the process followed will be the deduc-
tion of the particular from the general case. This method is not always
advantageous, but is not unsuitable for the discussion of asymmetrical
motions.
Equations (32) contain six relations between the twelve quantities
u, V, w, d, <f}, 12, X, Y, Z, L, M, N and certain constants of the aircraft.
There are only four controls to an aeroplane and three to an airship, con-
sisting of the engine, elevator, rudder and ailerons for the former and the
(32)
264 APPLIED AERODYNAMICS
first three of these for the latter. In the best of circumstances, therefore,
only four of the quantities X, Y, Z, L, M, and N are independently variable,
but all are functions of u, v, w, 6, 0 and Q which are determinable in a
wind channel or by other methods of obtaining aerodynamic data.
Equations (32) may then be looked on as six equations between the
quantities u, v, w, 6, <f>, 12, of which four are independently variable in an
aeroplane and three in an airship.
It has already been shown in the case of symmetrical straight flight
that the elevator determines the angle of incidence, whilst the engine
control affects the angle of descent. The aeroplane then determines by
its accelerations the speed of flight. For the lateral motions the new
considera-tions show that the rate of turning and angle of bank can be
varied at will, but that the rate of side slipping is then determined by the
proportions of the aeroplane.
It follows from the equations of motion that, within the hmits of
his controls, a pilot may choose the speed of flight, the rate of chmb,
the rate of turning and the angle of bank, but the angle of incidence and
rate of side slipping are then fixed for him. A very usual condition observed
during a turn is that side slipping shall be zero, and the angle of bank
cannot be simultaneously considered as an independent variable.
A number of cases of lateral motion will now be considered in relation
to equations (32).
Turning in a Horizontal Circle without Side Slipping. — The condition
that no side slipping is occurring is shortly stated as
ij = 0 (34)
but that of horizontal flight is less direct. If h be the height above the
ground, the resolution of velocities leads to the equation
h ='U Bind — V cos 6 sia^ — w cos 6 Qoa<f> . . . (35)
and for the conditions imposed (35) becomes
w sin ^ = w cos 6 cos (f> (36)
SimpUtication of the various expressions can be obtained by a careful
choice of the position of the body axes. The axis of X will be taken as
horizontal, and therefore along the direction of flight ; this is equivalent
to d=0, 10=0, w=>V, ^=0, whilst — wZ becomes equal to the lift. mX
differs from the drag by the airscrew thrust, and will be found to be zero .
The six equations of motion now become
X = 0 (37w)
Vii cos (^ = Y + ^ sin <^ . (37??)
— Vil sin <ji = Z + gf cos 0 . . {^7w)
iP{C — B) sin (^ cos ^ = L . . . . (37^)
i22Ecos2(^ = M .... (37g)
G^E . sin 0 cos <^ = N .... (37r)
. (37)
Owing to the slight want of symmetry of the aeroplane which arises
from the use of ailerons and rudder, the lateral force iwY will not be strictly
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 265
zero. It is, however, unimportant and will be ignored ; equation (37t;)
with Y =! 0 shows that
tan <f> =1 — (38)
The angle 0 given by (38) is often spoken of as the angle of natural
bank, and is seen to be determined by the flight speed and angular velocity.
As an example, consider a bank of 45°, i.e. tan ^=>1 and a speed of 120 feet
per second. Equation (38) shows that O is then 0-268 radian per second,
or one complete turn in 23*4 sees. A vertical bank, which gives an infinite
value to tan (f>, is not within the limits of steady motion and can only be
one phase of a changing motion.
If (38) be used to eUminate VO from (37m?) the equation becomes
Lift =—mZ =wigi sec ^ (39)
and the hft is seen to be greater during a banked turn than in level flight
by the factor sec 0. For a banked turn at 45° this increase of loading is 41
per cent.
It will be noticed that the couples, L, M and N all have values which
may be written alternatively as
(C-B)Va3^ M ^ EQ V _ EVi23^
and an estimate of their magnitude depends on the moments and products
of inertia. For an aeroplane of about 2000 lbs. total weight the value of
C— B would be about 700. E is more uncertain and probably not greater
than 200. With V = 120 and 12 = 0-268, the values of L, M and N in Ibs.-
feet would be 25, 7 and 7 respectively, and therefore insignificant. It must
not be inferred, however, that the couple exerted by the rudder is in-
significant, but that it is almost wholly used in overcoming the resistance
to turning of the rest of the aeroplane. This part of the analysis, which
is of great importance, can only come from a study of the aerodynamics
of the aeroplane, and not from its motion as a whole. The difference here
pointed out is analogous to the mechanical distinction between external
forces and stresses.
Spiral Descent. — The conditions of steady motion differ from those for
horizontal turning onl}'- in the fact that equation (35) is used to evaluate h
and not to determine a relation between w and 6. It is still permissible to
choose the axis of X in such a position that w is zero, and the conditions
of equilibrium of forces are in the absence of side slipping
X=gsin0 )
VO cos ^ cos 0 = Y + gf cos ^ sin ^ | . . . (41)
— VH cos 6 mi <l> = 7i -\- g cos 6 cos 0 j
As for rectilinear flight, the inchnation of the axis of X to the hori-
zontal and, since w = 0, the inchnation of the flight path, is changed by the
variation of longitudinal force, or in practice, change of airscrew thrust.
266
APPLIED AEEODYNAMICS
The angle of bank for Y=0 is identical with that given by (38) for horizontal
turning without side slipping, whilst the normal air force is
-wZ = mg cos d sec <f>
(42)
It appears that the angle of the spiral with Y = 0 may become greater
and greater until the axis of X is inclined to the horizontal at 80° or more,
and the radius of the circle of turning is only a few feet. The following
table indicates some of the possibiHties of steady spiral flight : —
TABLE 5.
Spirals and Spins.
Angle of
descent d
(degrees).
Angle of bank ^
(degrees).
Hesultant angular
velocity a
(rads.-s.).
Resultant
velocity V
(ft.-8.).
80
Radius of plan
of spiral R
(ft.).
40
42-7
0-37
164
60
58-8
0-61
87
92
60
691
0-91
92-6
51
70
77-0
1-44
96-5
23
80
83-7
2-95
99
6
Table 5 appUes to an aeroplane at an angle of incidence of 30°, i.e. an
angle well above the critical, and is deduced from observations in flight.
The motion of wings at large angles of incidence produces remarkable
effects, and it will be seen from an experiment on a model that the rotation
about the axis of descent is necessary in order to produce a steady motion
which is stable.
Approximate Methods of deducing the Aerodynamic Forces and Couples
on an Aeroplane during Complex Manoeuvres. — A complete model
aeroplane mounted in a wind channel as shown in Fig. 136 was foimd to
rotate about an axis along the wind with a definite speed of rotation for
each angle of incidence and wind speed. The analysis of the experiment
is of very great importance, as it shows the possibiUty of building up the
total force or couple from a consideration of the parts.
If the axis of X be identified with the axis of rotation, the various
constraints introduced by the apparatus reduce the six equations of motion
to one, {Blp). Since q is zero, this equation takes the very simple form
L = 0, and one of the solutions for equilibrium is that for which the model
is not rotating. At small angles of incidence this condition is stable, and
rotation is rapidly stopped should it be produced by any means. Above
the critical angle of incidence the condition of no rotation is unstable, and
an accidental disturbance in either direction produces an accelerating
couple until a steady state is reached with the model in continuous rotation.
Figs. 137 and 138 relate to the model with its rudder and ailerons in
the symmetrical position, the direction of rotation being determined by
accidental disturbance. The speed of rotation was taken by stop-watch,
and the first experiment consisted of a measurement of the speed of rotation
at various wind speeds. As was to be expected on theoretical grounds, the
■11:1. r-us^s:^
Fig. 136. — Model aeroplane arranged to show autorotation.
I
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 267
speed of rotation was found to be proportional to the wind speed (Fig. 138).
The second experiment covered the variation of rotational speed with
18 20 22 24 26 28 30 32
Fig. 137. — Autorotation of a model aeroplane as dependent on angle of incidence.
100
50
L^
/
/
y
^^
/
/
y
WIND
SPEED
MEAN ANGLE OF INCIDENCE
20 deg.
(f/s.)
ID
20 30
Fig. 138. — Autorotation of model aeroplane as dependent on vrind speed.
change of angle of incidence, and it will be noticed that increase of the
latter leads to faster spinning, at least up to angles of 33°. The analytical
process now to be described, if carried out over the whole range of possible
angles of incidence, shows that the spinning is confined to a hmited range.
268
APPLIED AEEODYNAMICS
Over part of this range, the spinning will not occur unless the disturbance
is great, but when started will maintain itself.
Simpler Experiment which can be compared with Calculation. — Instead of
the complete model aeroplane a simple aerofoil was mounted on the same
apparatus ; a first approximation to a wing element theory was used as a
06
i \
05
0-A
LIFT
COEFFICIENT
/A
^^ *
^— ^ /
■*■«» ^ '
. ^
0-3
0-2
i
/
/
< V D
01
0
1
1
ANGLE
OF INCIDENCE
1
(degrees)
0
1
0
2
0 3
0 40
Fig. 139. — Lilt-coefficient curve for aerofoil as used in calculating the speed of
autorotation.
basis for calculation. In this illustration the difference between Hft and — mZ
is ignored, and the curve shown in Fig. 139 is the ordinary lift coefficient
curve for an aerofoil on a base of angle of incidence which has been extended
to 40°. An angle of incidence of 20° at the centre of the aerofoil was chosen
for the calculation, and is indicated by an ordinate of Fig. 139. As a result
of uniform rotation the angle of incidence at points away from the centre
is changed, being increased on one wing and decreased on the other. The
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 269
distance from the axis of rotation to an element being y, the change of
angle of incidence due to an angular velocity p is roughly equal to ^^«
Since fj is constant along the wings it may be left as indefinite temporarily ;
the lift coefiicient on the element of one wing at 14° say, will be shown by
the ordinate of the full curve. Reflecting the hft coefficient curve as shown
in Fig. 130 brings the corresponding ordinate at 26° into a convenient
10
^^ y^ (decrees)
0 \C
E
0 10
y^a\
20
-10 \
-20 \
,0
Fig. 140. — Calculation of the speed of autorotation of an aerofoil.
position for the estimation of the difference h\, and the couple due to the
pair of elements is
pY^cSkjydy (42a)
The couple on the complete aerofoil of half span 2/0. is then
L==pY^crySkj,dy (43)
.' 0
The form of (43) can be changed to one more suitable for integration
by the use of the variable ^ instead of y, and it then becomes
VoP
pV^c rv yp
'~^ I V
Jjf .Sfc..<(f) .... (44)
Since h\ is a known function of ^, the value of -^?r can be found by the
V /t>V*c
plotting shown in Fig. 1 40. For steady notion it has been seen that L =0,
270
APPLIED AEEODYNAMICS
and the curve ABCD is continued until the area between it and AE is
zero. This occurs at the ordinate ED, which then represents the value of
~^; both ?/o and V are known, and hence p is deduced from the ratio so
determined.
A more accurate method of calculation will be given later, but the
errors admitted above are thought to be justified by the simplicity of the
calculations and the consequent ease with which the physical ideas can be
traced in the ultimate motion. On one wing the angle of incidence is seen
to be increased to about 37° at the tip, whilst on the other it is reduced
to 3°, Fig. 139, before steady rotation is reached. Further, the spinning
200
100
O — Calculated Speed.
-x X — Observed Speed.
ROTATIONAL SPEED
r. p.m.
ANGLE
INCIDENCE (Degrees)
15
20
25
Fig. 141. — Comparison of the observed and calculated speeds of autorotation of an
aerofoil.
is seen to depend on the evidence of an intersection of the Hf t curve and its
image, a condition which would not have occurred had the angle of incidence
been chosen as 10°.
QuaUtatively, therefore, the theory of addition of elements agrees with
observation. The quantitative comparison can be made since the aerofoil
to which the lift curve of Fig. 139 appUes was tested in a wind channel,
and the observed and calculated curves of rotational speed are reproduced
in Fig. 141. The aerofoil was 18 ins. long with a chord of 3 ins., and the
speed of test 30 feet per sec.
The agreement between the calculated and observed values of the speed
of rotation is close, perhaps closer than would be expected in view of the
approximations in the calculation, and may be taken as strong support
for the element theory. The extra power given in the calculation of aero-
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 271
plane motion is extremely great, and will enable future investigators to
proceed to analyse in detail the motions of spinning, rolling and rapid
turning A,vithout reference to complex experiments.
Further observations in the wind channel were made on the effect of
changes of wind speed and of aspect ratio. As in the case of the complete
model aeroplane, the speed of rotation was found to be proportional to the
wind speed. Reference to (44) will show that the integral depends only
on the value of ^^ , and hence for aerofoils of greater length it would be
expected that the rate of the steady spin would be proportionately less.
The observed and calculated results are given in Table 6.
TABLE
6.
Aspect ratio.
Observed rate of spin
(r.p.m.).
Calculated rate of spin
(r,p.m.).
1^
Angle of incidence, 17° . . <6
U
4
Angle of incidence, 22° ... <6
(8
125
95
74
155
121
100
142
95
71
182
121
91
It wiU be noticed that the agreement is far less complete than was
the case for variation of angle of incidence. It is possible that the tip
effects which have been ignored are producing measurable changes in this
case, and for a higher degree of accuracy resort should be had to observa-
tions of pressure distribution on an aerofoil. It is to be expected that
future experiments will throw further hght on the possibilities of the
element theory, and probably lead to greater accuracy of calculation.
More Accurate Development of the Mathematics of the Aerofoil Element
Theory. — Any element theory can only be an approximation to the
truth, and for this reason somewhat different expressions may be equally
justifiable. On the other hand all such theories assume that the forces
on an element are determined by the local relative wind, and are sensibly
independent of changes of velocity round neighbouring elements. Further,
it is not usual to make any appHcations to small areas of a body, but only
to strips of aerofoils parallel to a plane of symmetry, and to take the x
co-ordinate of this strip as that of its centre of pressure. The last assump-
tion may be regarded as a convenient method of taking a weighted mean
of the variations over a strip, and not intrinsically more sound than the
taking of areas small in both directions and summing the results.
Usually, the aerofoils to which calculation is apphed he either in the
plane of symmetry or nearly normal to it, and consist of the fin and rudder,
tail plane and elevator, and main planes. Of these, the last provides the
more complex problem on account of the dihedral angle, and since the
treatment covers the subject a pair of wings has been chosen for illustration
of the method of calculation.
272
APPLIED AEEODYNAMICS
The relations written down will have su£ficient generaUty to cover
variations of angle of incidence and dihedral angle from centre to wing tip,
and such dissymmetry as arises from the use of the lateral controls. The
method of presentation followed is adopted as it shows with some precision
the assumptions made in applying the element theory. Axqs of reference
are indicated in Fig. 106, but the first operation in the theory uses a
new set of axes obtained by rotating the standard axes GX, GY and GZ
to new positions specifically related to the orientation of one of the elements.
Referring to Fig. 142 (a), which represents one wing of an aeroplane of which
the element at P is being considered, the axes marked GXi, GYi and GZi
have been obtained from the standard axes by rotation through an angle
a^^ about GY * and through a dihedral angle — F about GXi. The plane
XiGYi is then parallel to the plane containing the chord of the element
and the tangent to the curve joining the centres of pressure of elements
in a direction normal to the chord.
(a)
Dtrecft'on of
re/atf've w/'nd.
Z. (&)
Fig. 142. — ^Aerofoil element theory.
With the axes in their new position the aerodynamics of the problem
takes simple form. If Wj, v-^ and Wy be the component velocities of P,
whilst Ux, «?i', and Wx are the corresponding velocities of G along these
axes and 391, q^ and r\ the angular velocities about them, then
wi = Wi' + gi^i — rii/i j
«i=«i' + »'i£Ci— Pi^i (45)
w^i=w^i'+^i2/i— giiCiJ
and the angle of incidence and resultant velocity at P are defined by
tan ai = — ^ (46)
V2 = Wi2 + Vi2 + m;i2 (47)
* The angle of pitch, i.e. the inclination of the chord of an element to the axis of X as
here defined is denoted by a^ . a is used generally for angle of incidence, i.e.. the iaclination
of the chord of an element to the diiection of the relative wind as defined in (46), whilst oq
is the angle of incidence in the absence of rotations. If the axis of X coincides with the
direction of the relative wind in the absence of rotations, ax = a©.
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 278
(48)
The two quantities a and V suffice to determine the hft and drag on
an element from a standard test, preferably one in which the pressure
distribution over a similar aerofoil was determined.
Using Fig. 142 (h) as representing the assumed.resolution of forces, leads
to the force and moment equations
mdXi = (fei, sin a — fej, cos (x)pN'^cdyi
mdYi = 0
mdZi => — (fct cos a + ^u sin (x.)pY^cdyi
dhi = yimdZi
dMi = —XimdZi + ZimdXi
dNj = —yimdXi
Equations (48) complete the statement of the element theory, and will
be seen to assume that the resultant force lies in a plane parallel to X^GZ^.
In certain problems, equations (45) — (48) may be the most convenient
form of appUcation, but in general it will be necessary to resolve the
components about the original axes before integration can be effected.
The necessary relations for this purpose are given.
Forces and Moments related to Standard Axes. — It may be noticed
that the angles of rotation a^^ and r correspond closely with those of 6
and ^, as illustrated in Fig. 129. A positive dihedral angle on the right-
hand wing, however, corresponds with a negative <f>. The direction cosines
of the displaced axes relative to the original are
li ^ cos XGXj = cos a^
Wi ^ cos YGXi = 0
ni ^ cos ZGXi => — sin a^
I2 ^ cos XGYi => — sin a;^ sin F
W2 ^ cos YGYi =1 cos r
n2 ^ cos ZGYi = — cos a^ sin F
Z3 ^ cos XGZj = sin a^ cos r
m^ =: cos YGZi => sin l^
713 ^ cos ZGZi =1 cos a^ cos r
for the right-hand wing and similar expressions with the sign of F changed
for the left-hand wing.
If X, y and z be the co-ordinates of P relative to the standard axes,
Xi =3 lix + wii2/ + niz \
yi = l2pc-\-m^-{-n22\
z\ — kx + W32/ 4- n^zl
In a similar way Ui => liu + miV + Wjw;
Vi = I2U -{- W»2V + n2W
Wi = l^u + m^v + n^w
Pi =hp-{'miq + nir
2i = ZaP + ^22 -j- ?i2r
(49)
(50)
(51)
274
APPLIED AEEODYNAMICS
The relations given by (49), (50) and (51) suffice for the determination
of tan ai and V as given by equations (45), (46) and (47), and thence the
elementary forces and couples from experiment and equations (48). The
final step is the resolution from the displaced to the standard axes, which
is covered by the following equations : —
dX = kdXi + kdYi + l^dZi
dY = midXi + WgtiYi + m^dZi
dZ = fiidXi + W2dYi + n^dZi
dL = lidLi + IzdMr + ZgdNi
dM. = midhi + W2dMi + m^dKi
dN = ni^Li + n2dMi + n^d^i
. (52)
As the expressions in (52) now all apply to the same axes the elements
may be summed by integration, the element of length being
dyi — l^dx + m2dy + n2dz (52a)
where l^, m2 and ^2 are the disection cosines of the line joining successive
centres of pressure.
Examples of the Use of the General Equations. — -Two examples will be
given, one deahng with the problem of autorotation discussed earHer, and
the other with the properties connected with a dihedral angle.
1. Autorotation. — In the experiment described earUer in the chapter
it was arranged that the quantities x, z, V, v, w, q and r were all zero.
The only possible motion was a rotation about the axis of X, and the
couple L was therefore the only one of importance. Denoting the wind
velocity by Uq and using equations (45) to (52) leads to (x,^=<x.q, and
^1
= 0OJ
itCQ
Wi
= 0
ni
= —sin ao
h
= 0
W2
=-1
712
= 0
h
= sin
ao
mg
= 0
Wg
= cos ao
Xi
= 0
2/1
= y
^1
= 0
ui
= Wo
cos
ao
n'
= 0
Wl
=» Uq sin ao
Vi
= p(
30S
Ko
gi
= 0
'•l
= p sin ao
Ml
=.Mo
COS
ao
— py sin ao
Wi
^Uq
sin
ao
+ py cos ao
Therefore
ai =af
>+i^,
where
fl =
=tan"i^^
Wo
and
V2:
— uq^ + p^y^ =
= Wq^
' sec2 fx
(53)
■ . (54)
. . (55)
Finally from (52) and the values of dLi and dNj is obtained the relation
dL = — (/cj, cos /u, + ^D sin 1^1)%.-^ sec^ d{sec^ix) . (56)
2p^
Equation (56) reduces to an element of equation (44) if fi be considered
as a small quantity, i.e. if the Hnear velocity of the wing tip due to rotation
AERIAL MANOEUVRES AND EQUATIONS OF MOTION 275
is small compared with the translational velocity. The value of L is
obtained by integration as
Li = -\- 2 2 HK cos /x + /cd sin [m) sec^/x d (sec^ /x) . (57)
8 signifies the difference of the values of fej, cos /x+fep sin /* on the two
elements of the wings of the aerofoil where fx has the same numerical
value, but opposite sign.
2. The Effect of a Dihedral Angle during Side Slipping. — The simplest
case will be taken and the origin chosen on the central chord at the centre of
pressure. The wings will be assumed to be straight and of uniform chord,
and to be bent about the central chord. The mathematical conditions
are
u'l = Uq V'l =Vq w'l =0 \
Pi =0 qi =0 ri = 0 I . . . . (58)
It should be noticed that the co-ordinates are in this case taken with
respect to displaced axes, as this is convenient in the present illustration.
The direction cosines li . . . n^ are given by (49), ao and 1' are
independent of yi, and the following further relations are obtained : —
(59)
Ui=!Ui = Uq cos ao
Vi = Vi = —Uq sin ao sin I' + ^o co3 F
Wi => Wi == Wo sin ao cos r + Vq sin F
2?! = 0 g'l = 0 ri = 0
tana.=''<''^°^"'^'^+-''<'-'™f^ . . . .(60)
1*0 cos ao ^ '
V^ = {uq cos ao)^ + (— Wq sin ag sin F- + Vq cos 1')
+ (wo sin ao cos F + Vo sin F)2 (61)
Both a and V are seen from (60) and (61) to be independent of yi. From
(48) it then follows that
/7T
mdTii = — ^ = — (fcj, cos ai + Uq sin (Xi)pcYMyi
mdXi= — ^ = (fci, sin ai — k^ cos o(.i)pcY^dyi
(62)
and in these expressions k and kj) may be functions of yi, owing to
variation along the wings. Since
dL = cos ao dLi + sin ao cosF dNj (63)
the value can be obtained from (62) for the right-hand wing. A similar
expression holds for the left-hand wing if the sign of F be changed. The
important quantities Vq and 1' only appear explicitly in tan a and V^, and
V represents the quantity usually measured in a wind channel.
276 APPLIED AEEODYNAMICS
Instead of attempting to evaluate (63) in the general case, the problem
will be limited to the case of greatest importance in aeroplane stabihty by
assuming that both ^ and r are small quantities of which the squares can
Uq
be neglected. Equation (60) then becomes "
tan ai = tan an + -^ • (64)
Uq cos ao
or after trigonometrical changes
_^o
«i ~ 0^0 = I' cos ao (65)
Uq
The second term on the right-hand side of (63) becomes neghgible with
respect to the first, and for the right-hand wing dh becomes
dL = — pcY^{kj^ cos (ai — ao) + fe^ sin (a^ — ao)}2/i%i • (66)
From (65) the term in k^ is seen to be small compared with that in k^^,
whilst cos (a— ao) can be replaced by unity. Hence —
dL = -pcY^kj^y^dy^ ...... (67)
If kj^' represent the value of kj, when a = ao, it follows that
fe, = fe/+''Orcos aof-" (68)
Uq OCX.
for the right-hand wing, and
fcj^ = fei,'-^rcosao— '' (69)
Wo 5a ^ '
for the left-hand wing. The value of L then is
;'' dk
L = —2pY Vq[' cos ccq j Cj^yidyi. . . . (70)
irther ap]
yi reduces (70) to
dk
Making the further approximation that c and — ^ are independent of
h = -pcmvQGos<XQ.r^'' (71)
For comparison with tests on an aerofoil (71) may be used for a numerical
example. Since the angle of yaw ^ is equal to — sin~"^ ^ , an angle of yaw
of 10° and a velocity of 150 ft. per sec. gives
Vq = —2&'1 V = 150
For a chord of 6 feet and a length of wing of 20 feet the value of L in a
standard atmosphere for r=6° is 5600 Ibs.-ft. when ao has any small value.
From a test the couple would have been found as about 4000 lbs. -ft.,
but this includes end effects not represented in the present calculation.
AEKIAL MANOEUVEES AND EQUATIONS OF MOTION 277
Calculation of Rotary Derivatives. — It has been seen in Chapter IV.
that the rates of variation of forces and couples with variations of u, w
and V are easily determined in a wind channel, whilst variations with jp,
q and r are less simply obtained. The number of observations in the
latter case is somewhat small, and as a consequence the element theory
has been freely used in calculating the rotary derivatives required for
aeroplane stability. It is usual to consider v, p, q and r as small quantities,
and to neglect squares, the derivatives then being functions of Uq and ivq
or of V and ao-
It is now convenient to express the values of a and V in terms of u',
v', w', p, q and r instead of the corresponding variables for the displaced
axes. From the equations developed earlier it will be seen that
Ml
' ^i(w' -\-¥~ W) + ^i(''^' +rx — pz) + ni{w' -\-py — qx)
(72)
with two similar equations for Vi and Wi.
values of Ui, Vi, and Wi are
Using a shorter notation, the
where
ttittQ — riiy — TTiiZ
CiCo = n^y — m^z
With this notation
wi =^ ao(l + «i? + «23 + ^s**) ]
^1 = &o(l + hV + ^23 + &3^) • • •
Wi = Cq{1 + Cip + C2q + ^3^) '
aQ => liu' + Wi-y' + ^iw^'
&Q = I2U' + w*2^' + ^2^'
a^UQ = liZ — n^x a^ttQ => niiX — liy
&2^o =■ h^ ~ "^2^ ^3^0 =' ^2^ — hy
c^Cq = l^z — n^x C3C0 = m^x — l^y
(73)
(74)
tan ai =^^=.^^-{1 + (ci — a{)p + (cg - ag)? + (^3 - a^)r]
or ai — ao = sin ao cos ao{(ci — a{)p + (cg — a2)q + (cg — a^r] . (75)
and V2=Vo2 + 2:p(aiao2 + &iV + CiCo2)
+ 2g(a2ao^ H- &2&o2 + ^2^0^) + 2r(a3ao2 + &3&02 + C3C02) . (76)
If at be used to represent generally one of the quantities p. qor r,
d
MX,) =pV,cdy, [sin a, 1 2fc,' £ + (fc/ +^)v„ *|
and ^^MZ,) = pV„od,. [sin a„[ - 2V^-^ + (fc.' - '^)y,^^ ]
and the remaining equations are given in (48) to (50).
(77)
(78)
278
APPLIED AEKODYNAMICS
dV
Denote by /x„, the expression 2feL' —
and by v„, the expression — 2/cd' -—
act)
to reduce equations (77) and (78) to
ao + v,„ cos ao)
(79)
(80)
(81)
~-(mdXi) = pVocdijiifM,,, sin
. — (mdZi) — p^{fdyi{v,„ sin an —/a,,, cos an)
Application to Lj,, L^, N^, and N,. for a pair of Straight Wings.
Assumed conditions : —
x = 0 2/i=2/_ z = 0 \ . . . . (82)
q ^
= 0 r=o
From (49) it then follows that
l^ = cos a^ wi = 0 Wj = — sin a,
li = cos a^
^2=0
Z3 = sin a^
m2 = 1 ^2 => 0
^3 == 0 ^3 == cos a^
"3 — ""^ ^X
From (74) and the above
■ cos a^ — Wq sin a^^ = Vq cos ao
since
QiQ ^=^ Wq ^^'^ ^^y
60=0
Cq =3 Mq sin a^ + Wq sin a^^ = Vq sin uq
a. + tan-i'^"
= 0 a^ttQ = — 2/ cos a^
= 0 /) Jj. = 0
(83)
(84)
ao
ajttQ =1 — ^ sin a^ «£% == 0 a3ao = —y cos a,
bifeo = 0 &2^==0 &3&o = 0
CiCo =^ ?/ cos a C2C0 =! 0 CjiCo =! — 2/ sin a,
CjCo =^ 7/ cos a
^cosa, ^sina^ and • r « - ^^0
Vq sm ao Vo cos ao Vq^ sm ao cos ao
V 0 sm ao V 0 cos ao Vo^ sm a© cos ao
Using these expressions and equations (75) and (76)
^ (ci - ai) sm ao cos ao = ^ |
dV Vo^
da. , . . vWq
d^ = (C3 — ag) sm ao cos ao = ^^
^P"" Vo Vo
3V^ asap^ + ^3^0^ + ^3^0^ = _ ^
5r ~ Vo Vo
AEKIAL MANOEUVEES AND EQUATIONS OF MOTION 279
Since l2='0, the formulfle for dL^ and dN„ given by (52), (80), and (81)
take the forms
dh^ =^{dL) = — fi,,pYocydij
d^,=^Jd}>i)==-v^,pYocydy
and from (79) and (85) —
(86)
(87)
If the variations of hft and drag towards the wing tips be ignored the
integrals take simple form. Calling the length of each wing I, the values
are, for constant chord,
L, = - §Z3pc |-2A;,'2*o + [K + ^>oj
N, = - p3pc| -2fe„'z^o + (K' - ^g^)uo]
N, = - ll'pc^^K'uo + (h' - ^^^>o| .
(88)
(89)
(90)
(91)
Numerical values can be obtained for the condition of maximum lift
of the wings in illustration of (88) to (91). The wings being assumed of
chord 6 ft. and length 20 ft., the velocity of 150 feet per sec. will be taken
as along the axis of X. Approximate values for the aerodynamic
quantities involved are
8K
= 2-3
Skjy
= 0-1 fe/ = 0-2 and kj,' = 0*01
5a ~ " dec
and lead to Lp=-26,000 L,=4500 Np=-1100 and N,=.— 200 (92)
It was seen in connection with rapid turning that values of p in excess
of 0*5 were obtained, and it now appears that a rolling couple of more than
10,000 Ibs.-ft. would need to be overcome by the ailerons if the conditions
of (92) apphed. The angle of incidence in flight is, however, much larger
and the speed lower, both of which lead to lower values of the total couple.
In the case of the tail plane of an aeroplane the effect of dowTiwash
should be included. It is the values of the air velocities at the aerofoil
280 ^ APPLIED AEEODYNAMICS
which enter into the equations, and these are only the same as the velocity
of the centre of gravity of the aeroplane in the absence of downwash. The
difference between the two quantities introduces little further complication
into the formulae developed.
The reader who reaches this fringe of the subject will find the limits of
accuracy much wider than those admitted in dealing with steady motion.
It should be remembered that less precision is required in the treatment of
unsteady motions, and that more can always be obtained in a particular
instance of sufficient importance. It will be some time yet before the
fundamental soundness of the blade element theory is established by the
experiments of the aerodynamics laboratories to a higher degree of accuracy
than at present.
CHAPTER VI
AIRSCREWS
I. General Theory
The theory of the operation of airscrews has been made the subject of
many special experiments, and in its broad outlines is well established.
Calculation of the fluid motion from first principles is far beyond our
present powers, and the hypotheses used are justifiable only on experi-
mental grounds. Whilst frankly empiric£|,l, the main principles follow
lines indicated by somewhat simple theories of fluid motion, and in this
connection the calculated motion of an inviscid fluid most nearly approaches
that of a real fluid. The discontinuous motion indicated by a jet of fluid
resembles the motion in the stream of air from an airscrew, and W. E.
Froude has formulated a theory of propulsion on the analogy. In this
theory the thrust on an airscrew is estimated from the momentum generated
per second in the slip stream.
Another theory, not necessarily unconnected with the former, was also
proposed by Froude and developed by Drzewiecki and others. The blades
of the airscrew are regarded as aerofoils, the forces on which depend on
their motion relative to the air in the same way as the forces on the wings
of an aeroplane. It is assumed that the elementary lengths behave as
though unaffected by the dissimilarity of the neighbouring elements, and
the forces acting on them are deduced from wind-channel experiments on
the lift and drag of aerofoils.
The most successful theory of airscrew design combines the two
main ideas indicated above.
In spite of imperfections, the study of the motion of an inviscid in-
compressible fluid forms a good introduction to experimental work, as it
draws attention to some salient features not otherwise easily appreciated.
In connection with the estimation of thrust by the momentum generated,
W. E. Froude introduced into airscrew theory the idea of an actuator.
No mechanism is postulated, but at a certain disc, ABC, Fig. 143, it is
presumed that a pressure difference may be given to fluid passing through it.
The fluid at an infinite distance, both before and behind the disc, has
a uniform velocity in the direction of the axis of the actuator. At infinity,
except in the slip stream, where the velocity is ¥_«, the fluid has the
velocity V^. The only external forces acting on the fluid occur at the
actuator disc, and the simple form of Bernoulli's equation developed in
the chapter on fluid motion may be appUed separately to the two parts
of streamlines which are separated by the actuator disc.
281
282
APPLIED AEEODYNAMICS
When dealing with the motion of an inviscid fluid in a later chapter,
it is shown that pressure in parallel streams is uniform, and if this theorem
be applied to the hypothetical flow illustrated in Fig. 143 it will be seen
that the pressure over the boundary DEGF tends to become uniform
when the boundary is very large. The continuous pressure at the boundary
of the slip stream is associated with discontinuous velocity.
The total force on the block DEFG is due partly to pressure and partly
to momentum, and the first part becomes zero when the pressure becomes
uniform over the surface. The excess momentum per sec; leaving the
block is the increase of velocity in the slip stream over that well in front
of the actuator, multiphed by the area of the slip stream, its velocity
and the density of the fluid. If the thrust T applied by the actuator
Fig. 143.
is balanced by a force between the disc and the block DEFG and the latter
is to be in equilibrium, the following equation for momentuin :
T=/).7rri<„V_«(V_«,-V„) (la)
must be satisfied.
Making use of Bernoulli's equation, another expression may be ob-
tained for T which by comparison with (la) leads to the ideas mentioned
in the opening paragraphs of this chapter.
For any streamline not passing through the actuator disc Bernoulli's
equation gives
?>i+|pVi2=j,^+|pV2 ...... (2a)
where pi and Vj are the pressure and velocity of the fluid at any point
of a streamline. This equation applies to the whole region in front of
the actuator and to the fluid behind outside the slip stream. Inside the
AIESCKEWS 28
slip stream, the pressure being p2 ^^^ ^^^ velocity V2, the equation corre-
sponding to (2a) is
P2 + ipV22 = p, + |pVi« (8a)
If p^ be eliminated between (2a) and (3a) an important expression for
the pressure difference on the two sides of the actuator is obtained, as
(P2-l^i) + MV22-Vi2) = |p(V!.«-V|>) . . (4a)
Continuity of area of the stream in passing through the actuator disc
being presumed, the value of V2 will gradually approach that of V^ as
the points 1 and 2 on the streamline approach the disc. On the disc both
velocities will be the same and equal to Yq, and equation (4a) becomes
CP2-Pi)o = ip(Vi„-V2) (5a)
The right-hand side of (5a) is constant for all streamlines inside the
slip stream, and hence the pressure difference on the two sides of the
actuator is uniform over the whole disc.
A second equation for the thrust T obtained from this uniform
pressure is
T = J/>7rro2(Vi«-V2) ..... (6a)
The quantity of fluid passing through the actuator disc being the same as
that in the slip stream, it follows that Tq^Yq is equal to rLooV_», and
using this relation with (la) and (6a) shows that
Vo = i(V_oo + Voo) (7a)
The value of Vq over the actuator disc is seen from (7a) to be a mean
of the velocity of the undisturbed stream, and the velocity in the slip
stream after it has reached a uniform value.
For the purposes of experimental check it is clear that no measure-
ments far from the airscrew will be satisfactory owing to the breaking up
of the slip stream due to viscosity, and the position of least diameter of
slip stream is usually taken as sufficiently representative of parallel stream-
lines. By a modification of equations (4a) and (6a) difficulty in an experi-
mental check can be avoided. A rearrangement of terms in (4a) and (6a)
leads to the equation
-^=P2T1/>V-ST¥V ... (8a)
and the quantity p-\-^pY^ happens to be very easily measured. It is
therefore possible to choose the points 1 and 2 in any convenient place,
one in front and one behind the airscrew.
Equation (8a) is given as applied to the whole airscrew as though
Pi» P2' Vx, V2 were constant over the whole disc. More rigorously the
equation should be developed to apply to an elementary annulus, and the
T dT
expression becomes ^ — -j- ; T is then obtained by integration. With
itTq^ zirrar
this modification (8a) applies with considerable accuracy to the real flow
of air through an airscrew.
Had the actuator given to the fluid a pressure increment which was
284 APPLIED AEBODYNAMICS
inclined to the disc, a flow resulting in torque might have been simulated.
The result would have been a twisting of the slip stream, and the angular
momentum of the air when the streams had become parallel would have
been a measure of the torque. The pressure on the streams when parallel
would not have been uniform, but would have varied in such a way as
to counteract centrifugal effects.
The air near an airscrew does not, in all probability, move in stream-
lines of the kind assumed above, and only an average effect is observable.
There is, however, this connection with the simple theory, that not only
is equation (8a) nearly satisfied, but a relation similar to that given in
equation (7a) is required to explain observed results. The constant which in
(7a) is equal to | appears to be replaced by a number more nearly equal to |.
Experimental Evidence for the Applicability of Equation (8a). — ^A pitot
tube, i.e. an open-ended tube facing a current, measures the value of
^+|pV^. Within a moderate range of angle of inclination to the stream
the reading is constant, and so a pitot tube is a suitable piece of apparatus
with which to test the appUcability of equation (8a) to airscrews. A
considerable number of experiments made in a wind channel showed that
for distances of the pitot tube up to 3 or 4 diameters of the airscrew in
front of its disc no failure was observed sufficiently large to throw
doubt on equation (8a). Except for points of a streamline which lie on
opposite sides of the airscrew disc, T of equation (8a) is zero, and hence
P2 + ipV2^=j>i + JpVi^ when the two pitot tubes are both in front of
the disc or both behind it.
A typical result is given : Denoting the speed well away from the
airscrew by V, the flow was 1*22V at a chosen radius near the airscrew disc.
The change of pressure necessary to increase the velocity from V to 1 •22V
is 0-240/)V^ whilst the difference between pi + IpVi^ and p2 + iP^2^ "^^s
O-OOSpY^, or little more than 3 per cent, of the change in either p or
|pV^. A similar observation was made for the airscrew running as in a
" static " test, and equation (8a) was again found to hold with con-
siderable accuracy.
In the above experiments two pitot tubes ahead of the airscrew were
used. For a continuation of the experiment one of the pitot tubes was
moved into the slip stream, and the difference between jpi -j- ip^i^ in front
of the airscrew and ^g + ip^2^ behind was observed. Since in front of
the airscrew the value of pj + ip^i^ was everywhere the same, it was not
necessary to ensure that points 1 and 2 were on the same streamline.
In producing the results from which Fig. 144 was prepared, one pitot tube
was placed about O'lD in front of the airscrew disc and the other 0'05D
behind, D being the diameter of the airscrew. It was found that with the
second pitot tube just behind the airscrew disc the difference in total
head became very small at the radius of the tip of the airscrew, and this
showed the outer limit of the slip stream.
The speed, V, of the air past the screws and the revolutions of the screw,
n, were changed so that the'^ratio -^ varied from 0-562 to 0*922. The
nJ)
value of the thrust on an element as calculated from the difference of the
AIESCREWS
285
Ol 0.2 03 0..+ 0.5
Fig. 144. — Thrust variation along an airscrew blade (experimental).
O 4-
\
CURVE BY MEASUREMENT OF THRUST.
POINTS BY INTEGRATION OF
0.3
^
DIFFERE
MCE OF TOT
AL HEAD.
0.2
THRUST
•^v
O.I
O.O
•s^
V
n D
• ^^*»^^^^
..^
0.5 0.6 0.7 O.a 0.9 1.0
Pio. 145, — Comparison between two methods of thrust measurement.
total heads has been divided by pV^D before plotting. The reason for
this choice of variables is not of importance here and will be dealt with
at a later stage. The curves of Fig, 144 show the variation of thrust along
286 APPLIED AEEODYNAMICS
the airscrew on the basis of equation (8a), whilst the area completed by the
line of zero ordinate is proportional to the total thrust. It will be noticed
that the inner part of the airscrew opposes a resistance to the airflow, and
that by far the greater proportion of the thrust is developed on the outer
half of the blade. The total thrust as shown by the area of the curves
V V
decreases as ^ increases, and would become zero for ;f^ equal to nearly
nD nD
unity.
For comparison with the total thrust as calculated from equation (8a)
and Fig. 144 a measurement of the total thrust was made by a direct
method and led to the curve of Fig. 145. The points marked in the figure
are the result of the experiments just described. It will be noticed that
the agreement between the two methods is good, with a tendency for the
points to lie a little below the curve. The agreement is almost as great
as the accuracy of observation, and the conclusion may be drawn that in
applications of fluid theory to airscrews a reasonable application of Ber-
noulH's theorem will lead to good results. Later in the chapter it will be
shown that this theorem carried through in detail enables a designer to
calculate such curves as those of Fig. 144, and that the agreement with the
observations is again satisfactory.
Having shown that the total head gives much information on the air-
flow round an airscrew, it is proposed to extend the consideration of the
flow to the different problem of the distribution of velocity before and
behind an airscrew disc. Eeplacing the pitot tube by an anemometer,
repetition o the previous experiments provides an adequate means of
measuring the velocity and direction of the air near the airscrew.
Measurements of the Velocity and Direction of the Airflow near an
Airscrew. — ^Experiments on the flow of air near an airscrew have been
carried out at the N.P.L., and from a consideration of the results obtained
Figs. 146 and 147 have been produced. Whilst they give the general
idea of flow to which it is now desired to draw attention, it should be
mentioned that the curves shown are faired and therefore, for the purposes
of developing or checking a new theory of airscrews, less rehable than the
original observations.
It will readily be understood that measurements of velocity and
direction of the airflow cannot be made in the immediate neighbourhood
of the airscrew disc, and any values given in the figures as relating to the
airscrew disc are the result of interpolation and are correspondingly
uncertain. Qualitatively, however, the figures may be taken as correct
representations of observation, whilst quantitatively they are roughly
correct.
Each figure has been subdivided into Figs, (a), {h) and (c), which have
the following features :—
(a) The diagram shows the " streamlines " in the immediate neigh-
bourhood of the airscrew, the linear scale being expressed in terms
of the diameter of the airscrew. On each of the " streamlines "
are numbers representing the velocity of the air at several points,
whilst at a few of these points the angle of the spiral followed by
I
AIESCREWS 287
the air is indicated by further numbers. The velocity is denoted
by V, and the angle of the spiral by <f).
[b) The distribution of velocity at various radii is shown in these
diagrams. Each of the curves corresponds with a section of (a)
parallel to the airscrew disc, and the position of the section is
indicated by the number attached to the curve. The radii are
expressed as fractions of the diameter of the airscrew. If the
airscrew be not moving relative to air at infinity the velocity scale
is arbitrary, as it depends on the revolutions of the airscrew only.
Where the airscrew is moving with velocity V relative to the distant
air this is a convenient measure for other velocities connected with
the motion of the air through the airscrew.
(c) Each of the " streamlines " of {a) is a spiral, with the angle of the
spiral variable from point to point. The relation between the angle
of the spiral and the radius is shown in (c), each curve as before
corresponding with a different section of (a).
The Difference of Condition between Fig. 146 and Fig. 147.— Within
the Umits of accuracy attained the figures give a complete account of the
motion of the air over the most important region, and the two groups of
figures have been chosen to represent widely different conditions of running.
In Fig. 146 the airscrew was stationary relative to distant air, and its effi-
ciency therefore zero. In Fig. 147 the condition was that of maximum
efficiency, and was obtained by suitably choosing the ratio of the forward
speed to the revolutions.
The figures are strikingly different ; for the stationary airscrew the
streamlines converge rapidly in front of the airscrew disc, and for some
little distance behind. They are nearly parallel at a distance behind the
disc equal to half the airscrew diameter. For the moving airscrew the
most noticeable feature is the bulging of the streamlines just behind the
airscrew disc and near the axis. Outside the central region the stream-
lines are nearly parallel to the airscrew axis but show a slight convergence
towards the rear.
V . •
Had the value of -^^r been increased from 0*75 to 2*0 the airscrew would
wD
have been running as a windmill. The corresponding streamlines are more
closely related to the moving airscrew than to the stationary one, the only
simple change from Fig. 147 being a slight divergence of the streams behind
the airscrew. The bulge on the inner streamhnes tends to persist.
Stationary Airscrew, Fig. 146. —
(6) The curves of velocity show a very rapid change at radii in the
neighbourhood of 0*3 to 0*5D. These rapid changes define the edge
of the slip stream, so far as it can be defined. When the streamlines
have become roughly parallel at 0-5D (Fig. 146 a) it will be noticed
that the greater part of the flow occurs within a radius of 0'4D, and
this represents a very considerable reduction of area below that of
the airscrew disc and a consequent considerable increase of average
velocity between the airscrew disc and the minimum section. The
figure shows the velocity at the disc to be roughly 70 per cent, of
•288
APPLIED AERODYNAMICS
that 0-5D behind the disc. The curve marked 4-OD in (b) indicates
that at four times the airscrew diameter behind the airscrew disc the
mean velocity at small radii has fallen greatly, and the slip streams
must therefore have begun to widen again.
-^=0 l.e. THE AIRSCREW IS NOT MOVING RELATIVE TO
'ID THE DISTANT FLUID.
si/=009
V=0.02
-0.20 -O.ID O O.ID 0.2D 0.3D 0.40 0.50
DISTANCE ALONG AXIS OF AIRSCREW
THE NUMBERS ATTACHED TO CURVES OF FIGURES j6&C
ARE DISTANCES ALONG AXIS OF AIRSCREW
,0.5D
V
VELOCITY
THE SCALE
IS ARBITRARY
O 0.1 0 2 0.3 0.-4- 0.50
RADIUS
0.2 0.3 0.4- 0.50
RADIUS
Fig. 146. — Plow of air near a stationary airscrew.
(c) The angle of the spiral of the streamlines varies as markedly as
the velocity. In front of the airscrew disc the observed angles
never exceeded one degree. Behind it and near the centre, angles
of 25" and over were observed. On the edge of the slip stream the
AIKSCKEWS
289
v^alues are of the order of 10*' or 15°. At the airscrew disc the
interpolated curve shows angles of 10° at the centre, falling to
3° or 4" just inside the blade tip.
If the deductions from the figure be compared with those from the
w^- ^ AIRSCREW WORKING AT MAXIMUM EFFICIENCY
i[6^ OF 0.70
V'lOO V=l OO \l=\.0\ V = I02
^ ^ / / \l'\.02
V=i.oi V=i.oi
^
0=O.5
V
V=i.oi V=i oo
V=i.oi V'l.oo
V=0.97 V=0.95
a.
(p-\°o
02 \/=l.03
0 = l°5
V=l.04-
0=l?5
.04- V=l.06
0-2°O
V=l.07
ia)
V=l 04- V=l.07
0=2?O
0 = 29O
V-l GO
V=0 SO V=0.57
0=295
V=o.a4
0 = 2°5
0=5?5
-O 20
•OID O 0.1 D 0 2D 0 3D 0 4-D
DISTANCE ALONG AXIS OF AIRSCREW
O 5D
THE NUMBERS ATTACHED TO CURVES FIGURES jb&C
ARE DISTANCES ALONG AXIS OF AIRSCREW
.4 0
20
0
ANGLE OF
SPIRAL OF
SLIP STREAM
lO
4-D
0 2 0.3 0* 0.5
RADIUS
0.1
0 2 0 3 0.4- 0 5D
RADIUS I
Fig. 147. — Flow of air near a moving airscrew.
theoretical analysis given earlier, it will be seen that the ideas of trans-
lational and rotational inflow are applicable to the average motion of air
round an airscrew. Further, there is a region of roughly parallel motion
;at some moderate distance behind the airscrew in which it may be
u
290 APPLIED AERODYNAMICS
supposed that the pressure distribution adds nothing to the thrust a
calculated from pressure and momentum by the use of (8a).
Moving Airscrew (Fig. 147).
(h) The velocity does not change rapidly with the radius at large
radii, and the edge of the slip stream is not clearly defined. The most
marked changes of velocity occur at the centre and just behind
the airscrew boss. The drop of speed is there very marked. This
part of an airscrew adds very little to the total thrust or torque,
and is relatively unimportant. The velocity is unity well ahead
of the airscrew, and has added to it an amount never exceeding
7 per cent. Along each streamline, roughly half the increment
of speed is shown as having occurred before the air crosses the air-
screw disc. This condition of the working of an airscrew is of great
practical importance, and the accuracy of direct observation is
better than for the stationary airscrew. The contraction of the
stream is small, but the increment of momentum is not inconsider-
able.
(c) In front of the airscrew the twist is shown by the observations to
be small. Even behind the airscrew disc the angles are very much
smaller than for the stationary airscrew, and do not anywhere
exceed 10°.
II. Mathematical Theory of the Airscrew
The experimental work just described was necessary in order to outline
clearly the basic assumptions on which a theory of the airscrew should
rest. In the theory itself appeal is made to experiment only for the
determination of one number, which is the ratio of th^ velocity added at
the airscrew disc to that added between the parallel part of the slip stream
and the parallel streams in front. The assumption is usually made that
this number is constant, i.e. does not depend on the radius, an assumption
which is only justified by the utility of the resulting equations. ^ In the
earlier stages, in order to bring into prominence its actual character, this
assumption will not be made.
The airscrew stream is illustrated in Fig. 148 to show the nomenclature
used. The half diameter of the airscrew is denoted by Tq, whilst the half
diameter of the slip stream at its minimum section is r^^. Radii measured
at the airscrew disc are denoted by r and at the minimum section by rj.
The axial velocity of the air at the airscrew disc is V(l + ai) and at the
minimum section V(l + &i), V being the velocity in front of the airscrew
at an infinite distance; ai and &i are the "inflow" and "outflow"
factors of translational velocity.
The rotational velocity is better seen from the next diagram, which
also introduces the idea of the application of the aerofoil and its known
characteristics. Each element is considered as though independent of its
neighbour, and this involves some assumption as to the aspect ratio of
^ Later experiments are providing data for a more general assumption, but application is
as yet undeveloped.
%»
AIESCEEWS
291
the aerofoil on which the basic data were obtained and the shape of the
airscrew blade. The value taken is rather arbitrarily chosen, since real
knowledge is not yet reached.
y{\-b,)
Fig. 148.
Fig. 149 represents an element of an airscrew blade at a radius r. The
translational velocity relative to air a considerable distance away is V,
and the rotational velocity ojt, w being the angular velocity of the air-
DIRECTION OF
RELATIVE WIND
Fig. 149.
screw. Kelative to the air at the airscrew disc the velocities are V(l + a{)
and cor (1 + 02), 02 being the rotational inflow factor. These two velocities
define the angle 0, i.e. the direction of the relative wind, and since the
chord of the element makes a known angle with the airscrew disc the
292 APPLIED AEEODYNAMICS
angle of incidence, a, of the element is known when <f> has been
evaluated.
The element is considered as though in a wind channel at angle a and
velocity '\/\^{l-\- ai)^-\-o)h'^l-\- 02)^, and observations of hft and drag
determine the resultant force dH and the angle y. It is clearly necessary
to know something more about aj and 02 before the above calculation
can lead to definite results, but in order to develop the theory expressions
for elements of thrust and torque are first obtained in general terms.
Resolving parallel to the axis of the airscrew leads to
dT = dU GOB {(f> + y) (1)
for the element of thrust, whilst the element of torque is found by taking
moments about the airscrew axis, and gives the equation
dQ = dn.r. sin {(f> + y) (2)
Expressions for Thrust and Torque in Terms of Momentum at the
Minimum Section of the Slip Stream. — An alternative to the aerofoil
expressions (1) and (2) can be obtained in terms of quantities other than
tti and a2, etc., by considering the momentum in the elements of the slip
stream at its minimum section, and it is the assumptions connecting the
two points of view which are of present importance.
The elementary annulus of radius r at the airscrew disc is replaced by
an annulus of decreased radius rj at the minimum section of the slip stream.
The quantity of air flowing through each annulus being the same, the
relation between radii is expressed as
(l+ai)r^r = (l +&i)ri^ri (3)
At this point is made the important assumption on which the practicability
of the inflow theory of airscrew design depends. It is supposed that
ai = Vi (4)
where Ai is constant for all airscrews and for all the variations of condition
under which an airscrew may operate. The method of finding Aj will be
described later, but the assumption finds some rough justification in the
measurements made and described in Figs. 146 and 147.
However arbitrary the theorem may seem to be, it leads to results
far better than any other yet known to us, and at the present moment
the theory may be accepted as good.
Equation (3) becomes
{l+a,)rdr = (l+^y,dr, ..... (5)
or in its integral form
^'+^*^ (6)
and expresses the radius of the slip stream in terms of r, ai and A]
AIRSCEEWS 293
The elements of thrust and torque can now be written down. The mass
of air flowing through the annulus of the sHp stream is 27r/3V(l + &i)''id^i>
the velocity added from rest is biY, and therefore the thrust is
dT = 27r/)(l+6i)&iVVi6Zri (7)
Using equations (4) and (5) to transform (7) leads to
dT = 27Tp{l + ai)^Y^dr . . . . . .(8)
and if the momentum and aerofoil theories are to lead to identical estimates
this thrust should be the same as that given by (1). Hence
dB,cos{6-\-y) = 27Tp{l-{-ai)^^Yhdr .... (9)
In this equation every term is, by hypothesis, known im terms of a^ and 02
and equation (9) is therefore one relation between a^ and ^2. A second
relation may be obtained from the equaUty of the expressions for torque.
The element of torque is readily seen to be
dQ = lTrp{l-\-hi)b2Voyri^dri (10)
and making the corresponding assumption to (4) that
02 =-^2^2 (11)
(11) and (5) may be used to transform (10) to
dQ = 27rp(l + ai)^2Vairi2.rdr .... (12)
A2
Unhke equation (9) for the elementary thrust, which contains r only,
equation (2), for elementary torque, involves both r and ri, and the rela-
tion which is given by (6) does not lend itself to simple substitution
in (12).
Equating (12) and (2) gives a second relation between aj and a2 as
dH sin (0 + y) = 27r/3(l + ai)~^Yojri^dr ... (13)
A2
Aj and A2 being known constants, equations (9) and (13) are sufficient to
determine both Ui and a2 in terms of aerofoU characteristics.
Transformation of Equations (9) and (13) to more Convenient Form for
Calculation. — From the geometry of the airflow it follows that
, , (l+fli) V ....
tan0 = )-J — ^(.— (14)
(1 + flg) (^r
and that the resultant velocity is
(1 + a2)<^f sec 0 ...*... (15)
, is known from general wind-channel expei
dR ^pcdr{\ + a^^Yoih^ sec^ <f> ./(a) . . . (16)
The element of force, dR, is known from general wind-channel experi-
ments to have the form
294 APPLIED AERODYNAMICS
where c is the sum of the chords of the aerofoil elements at radius r, and
/(a) is the absolute coefficient of resultant force. In the same way it is
known that
tany = ^^f = F(a) (17)
The algebraic work in transforming (9) by use of (14), (16), and (17) is
simple, and leads to
«i _h ^/(a) cosec2 0 cos (<^ + y) . . . (18)
1 -f ai 27r ' r
whilst (13) becomes
r^^^^f) ' r '^^"^^ ^°®^^ ^ ^^^ ^ ^'"^ (^ + y) • • (19)
To solve in any particular case, it is most convenient to assume
successive values for a. Since ^ + a is known from the geometry
of the airscrew this fixes ^, and equations (18) and (19) then determine
ai and a^. Finally, equation (14) gives the correct value of — for the values
atr
of ^ assumed.
Example of the Calculation of ai. — The forces on an aerofoil as taken
from wind-channel experiments are most commonly given as lift and drag
coefficients h^, and fej,. In the present notation
fe,=/(a)cosy|
fe„=/(a)siny| (20)
/(a) COS (^ + y) = fc^ cos <^ - /cd sin 0|
/(a) sin {<f)-\-y)= h^ sin ^ + fei, cos ^J * ' ^ '
Take r=33-6 ins., c=2x9-65 ins., i.e. -=0-575, a+^=22°-l, Ai=0-35,
r
and proceed to fill in the table below from known data.
The tabulation starts from column (1) with arbitrarily chosen values
of a, and in this illustration a very wide range of a has been taken. Since
a + ^ = 22°-l column (2) follows immediately. The lift coefficient kj^ is
taken from wind-channel observations on a suitable aerofoil for the given
values of a, the ^ — ratio of column (4) is similarly obtained, and by
the use of trigonometrical tables leads to column (5). The remaining
columns follow as arithmetical processes from the first four columns and
equation (18).
The values found for a^ show very great variations, but discussion of
the results is deferred until ^2 has been evaluated.
Assumptions as to A2 and 02- — The assumption which has received
most attention hitherto has been that A2=0, and equation (19) then shows
that ^2 is zero. This is equivalent to assuming no rotational inflow
and other assumptions now appear to be better.
A2 plays the same part in relation to torque that Aj does to the thrust,
AIKSCEEWS
295
and it would be possible to carry empiricism one stage further and choose
Ai and A 9 so that both the thrust and torque agreed with experiment at
V
some particular value of -^r. This would lead to more difficult calcula-
tions, but not to fundamentally different ideas. A more obvious and
equally probable assumption is that the air at the airscrew disc is given
an added velocity in the direction opposite to dH, in which case
a2oyr
aiV
=— tan {<f> + y)
(22)
TABLE 1.
1
2
3
4
5
6
7
a
<<>
*L
L
y
C!os (p
Sin <^
(deg.)
(deg.)
(deg.)
-10
321
-0170
-3-1
-17-2
0-847
+0-532
- 5
271
+0-010
+0-3
• 73-0
0-890
+0-455
0
221
0195
17-5
3-3
0-926
+0-376
2
201
0-276
19-5
2-9
0-939
0-343
4
181
0-350
18-2
31
0-950
0-311
6
161
0-425
16-5
3-5
0-961
0-277
8
141
0-495
150
3-8
0-970
0-244
10
121
0-560
13-6
4-2
0-974
0-210
16
71
0-595
8-0
7-2
0-992
0124
20
21
0-645
4-3
130
0-999
0-036
*!_ COS <^ — Ad sin <P
f-0-144\
\ -0-029/"
f+0-009\
\-0015/"
/+0-180\
\ -0-004/"
/+0-258\
\-0-005r
/+0-332\
1-0-006/'
/+ 0-408 \
1-0007/"
/+0-480i
1-0-008/
|+0-545\
0 009/~
/+0-590\_
I- 0-009/ ~
/+0-545\_
\-0-005/~'
-0-173
-0 006
+0-176
+0-253
+0-326
+0-401
+0-472
+0-536
+0-681
+0-540
(7) multiplied by
^ — .cosec^A
2ir- r ^
l + Bj
-00196
-0-0009
+0-0399
+0-0691
+0-108
+0-167
+0-254
+0-390
+ 1-21
+ 10-60
10
-0-020
-0001
+0-0415
+0-0740
+0-121
+0-200
+0-339
+0-640
-5-8
-I-IO
The accuracy of this assumption is not less than that relating to aj.
The radial velocity is still ignored, and the assumption is made that Ui
and 02 are constant across the blade, which will probably be more correct
for narrow than for wide blades. Equation (18) remains as before, but
equation (14) becomes
tan<^= 1±^ ^.^ .... (23)
1 — fli tan (^ + y) —
cor
cor
— ttj tan (0 + y)
(24)
296
APPLIED AERODYNAMICS
As applied to the element considered above the calculation proceeds to
determine — from the figures in Table 1 and equation (24), which is more
conveniently written as
cor _1 -\-ai
tan <f)
TABLE 2
tti tan {<f> + y)
(25)
a
'P
V
(degrees).
(degrees).
mr
a,.
flj.
-10
321
0-633
-0020
0-011
- 5
271
0 608
-0-001
0003
0
221
0-389
+0-041
-0-008
2
201
0-337
0074
-0-011
4
18-1
' 0-288
0121
-0-014
6
161
0-236
0-200
-0-017 •
8
141
0-184
0-339
-0-020
10
121
0-128
0-640
-0-024
15
71
-0-025
-5-8
-0-037
20
21
-0-360
-1-1
-0-010
This table repays careful examination in conjunction with Table 1.
Equation (8) shows that the thrust on the element is zero when a^ is zero,
and Table 1 shows that a^ changes sign at about —5°. The thrust
is then seen to change sign at an angle of incidence rather greater
V
than that at which the lift changes sign ; the value of — is roughly 0-51.
The section considered occurred in the airscrew blade at 0-7D, where D is
V
the diameter of the airscrew, and the more familiar expression -^j ^^^ the
value 1-12 when the thrust on the element vanishes. With a =—10° the
airscrew is acting as a windmill, i.e. is opposing a resistance to motion
and is delivering power.
V
Continuing the examination, using Table 2, it will be noticed that —
changes sign at an angle of incidence of about 14°, and the aerofoil reaches
its critical angle of incidence at about 12°. The further cases in Table 2
correspond with backward movement of the airscrew along its axis. From
an angle of incidence of 14° onwards a i is negative, but
«!
l+«]
is still positive
and passes through the value oc when 0 is zero, as may be seen either from
Table 1 by interpolation or more readily from equation (18). There is
no special change in the physical conditions at this value of ^, as may be
The history
seen from the continuity of «!, which is —1 when zr-~- =cc
of further changes of — , if continued, shows continuously increasing angle
tor
AIKSCKEWS
297
of incidence up to 90° + 22°-l as a limit, as the airscrew moves back-
wards more and more rapidly.
Efficiency of the Element. — The useful work done, being measured
relative to air at infinity, is YdT, whilst the power expended is wdQ. The
efficiency is then
Substituting from equations (1) and (2) converts (26) to
(or
tan (^ + y)
and combining this with (14) leads to
__ 1 + aji tan <j>
' 1 + <*i * tan (0 + y)
(27)
(28)
TABLE 3.
a
V
Efficiency,
(degrees).
wr
n
Windmill {j^^^^^q^^ ^
-10
0-633
+2-38
No lift
- 5
0-508
-0-090
/No thrust
0
0-389
0820
Maximum efiQciency
2
0-337
0-793
Airscrew /
4
0-288
0 744
6
0-236
0-669
8
0-184
0-570
10
0-128
0-439
No translational velocity,
i.e. static test condition
15
-0025
-0 098
20
-0-360
-1-33
A word might here be said as to the meaning of efficiency and the
reason for choosing VdT as a measure of work done. Efficiency is a
relative term, as may be seen from the following example': Imagine an
aeroplane flying through the air against a wind having a speed equal to
its own. Eelative to the ground the aeroplane is stationary, but the
petrol consumption is just as great as if there were no wind. As a means
of transport over the ground the aeroplane has no efficiency in the above
instance. On the other hand, if it turns round and flies with the wind
the aeroplane would be said to be an efficient means of transport, and yet
in neither case does the aeroplane do any useful work in the sense of storing
energy imless it has happened to chmb. It is obvious that no useful
definition of efficiency can depend on the strength of the wind, and what
298
APPLIED AEEODYNAMICS
is usually meant by the efficiency of the airscrew is its value as an instru-
ment for the purpose of moving the rest of the aeroplane through the air.
The conception of efficiency is not simple and well repays special attention
during a study of aerodynamics.
From equation (28) may be calculated the values of efficiency r)
corresponding with Tables 1 and 2. The values are given in Table 3.
In interpreting Table 3 it is convenient to refer to Fig. 150, which
shows the airscrew characteristics of the element in comparison with those
-0.2
-10' -5 O 5 ID 15 20
EiQ, 150. — Comparison of characteristics of elements of aerofoil and airscrew.
of the elementary aerofoil. The characteristics are shown as dependent
V
on angle of incidence of the aerofoil, and the curves show — and efficiency
1 ' fi-
for the airscrew and lift coefficient and ^j — for the aerofoil.
drag
At an angle of incidence of —10° the thrust and torque are both
negative, and Table 3 shows the efficiency to be positive. The airscrew
is working as a windmill, the work output is codQ and not VdT, and
(26) represents the reciprocal of the efficiency of the windmill ; the value
a; = 2-38 of Table 3 represents a real efficiency of 42 per cent. At an
angle of incidence of —5° -5 the point of zero torque occurs, and the
efficiency as a windmill is zero corresponding with an infinite value in
Table 3. As the angle of incidence increases the torque becomes positive,
whilst the thrust remains negative and- the efficiency is negative. At
—4-4° the thrust becomes positive and the airscrew begins its normal
functions as a propelling agent, the efficiency being zero at this point,
f
AIBSCKEWS
299
but rising rapidly to 0*83 at an angle of incidence of about 0°'5. At
greater angles of incidence the efficiency falls to zero when the airscrew
is not moving relative to distant air. If the airscrew be moved backwards
VdT is negative and the efficiency is negative, but this condition is
unimportant and no detailed study of it is given.
The general similarity of the efficiency and ^ curves may be noticed
and suggests the importance of high ^ ratio. This is seen to be a
general property of airscrew elements by reference to equation (28).
Other things being equal, equation (28) shows maximum efficiency when
y is least, i.e. when ^r is greatest.
Relative Impoitance of Inflow Factors. — It is now possible to make a
quantitative examination of the importance of the inflow factors aj and
a2, and for this purpose Table 4 has been prepared. The first column
contains the angle of incidence of the blade element, whilst the remaining
V
columns show the values of — and -q on the separate hypotheses that
(1) both Ui and 02 are used ; (2) that neither is used, and (3) that only
Ui is used. The general conclusion is reached that aj is very important,
but that a2 may be ignored in many calculations without serious error.
TABLE 4. — Effect of Inflow Factors on the Calculated Advance per Revolu-
tion AND Efficiency of a Blade Element.
1
2
3
4
5
6
7
Y
V
V
a
—
—
q
n
(degrees).
or'
a, and a, zero.
lor
a, zero.
»i
tti and ttj zero.
ttj zero.
-10
0-633
0-627
0-642
+2-38
+2-36
+2-42
- 6
0-508
0-512
0-511
-0-090
-0-091
-0-091
0
0-389
0-406
0-390
0-820
0-865
0-821
2
0-337
0-366
0-340
0-793
0-862
0-800
4
0-288
0-327
0-292
0-743
0-846
0-753
6
0-236
0-289
0-240
0-669
0-820
0-680
8
0-184
0-251
0-188
0-570
0-777
0-582
10
0128
0-214
0-131
0-438
0-733
0-448
15
-0-026
0-125
-0-026
-0-098
0-490
-0-102
20
-0-360
0-037
-0-360
-1-34
0-137
-1-34
At the angle of no thrust, — 5°'5, the three hypotheses differ by very
small and unimportant amounts, but at an angle of incidence of 6°, which
would correspond with the best climbing rate of an aeroplane, the difference
of — - for the assumption of no inflow and that for full inflow is more than
20 per cent. If V be fixed by the conditions of flight the theory of no
inflow would indicate a lower speed of rotation for a given thrust than does
300 APPLIED AEKODYNAMICS
the theory of full inflow. This means that a design on the former basis
would lead to an airscrew which at the speed of rotation used in the design
would not be developing the thrust expected. The effect of inflow factors
on efficiency for a =6° is equally strongly marked, for in one case an
efficiency of 0-820 is estimated, whilst in the more complete theory only
0-669 is found.
General experience of airscrew design shows that the " inflow " theory
leads to better results than the older " no inflow " theory.
Although the effect of inflow factors is great, it appears that almost
the whole is to be ascribed to the effect of a^. The differences between
columns 2 and 4 and columns 5 and 7 are due to the assumption that 02
has a value in one and is zero in the other. In no case are the differences
great, and this is a justification for the fact that a great amount of airscrew
design and experimental analysis has been carried out on the basis that
a2 is zero.
It appears that a2 is never very great, and that calculation leads to
agreement with Pannell and Jones in their observation that the rotational
inflow to an airscrew is very small.
V
— and a. — Since 0 + a is constant (22"-! in our illustration)
equation (14) may be written as
constant — a = tan~iYT~^^^- — .... (26)
(22°'l) 1 + ^2 ^^
V
If tti and 02 be small — is sensibly a function of a only, and hence its
general importance as a fundamental variable in airscrew design. Fig. 150
V
shows that when inflow is taken into account the relation between and
ojr
a is linear for a large range of a. The constant in this linear relation is
15° -5 instead of the 22°-l of (26), and this is due partly to the inflow factor
aj and partly to the fact that the tangent is not proportional to the angle
over the range in question.
Approximation to the Value of «! for Efl&cient Airscrews. — An examina-
tion of column 8, Table 1, will show that the part of a^ which depends on
the drag coefficient is very small, and that
/(a) cos {(f) + y) is nearly equal to kj^ cos (f> , . . (29)
over the whole range of the example. This agreement is partly accidental,
but the expression can be examined in order to lay down the conditions
necessary for the approximation to hold.
An expansion for /(a) cos {(f> + y) is given in (21), which may be
rewritten as
/(a)cos(0 + y)=/c^cos^(l-|?tan<^)- . . (30)
and the second term inside the bracket on the right-hand side of (80) is
AIESCREWS 301
k . L .
seen to be small in comparison with unity if , ^, i.e. — is large and
tan (f> small. 77 may be as great as 20 and tan <f) = 0'5 in the parts of an
efficient aeroplane or airship airscrew which are important. Hence for the
circumstances of greatest practical importance we may use (29) as indicating
a good approximation ; over the working range aj does not exceed 0*3,
and an error of 5 per cent, in ai makes an error of 1 per cent, in the esti-
mated efficiency. At maximum efficiency the approximation is very
much closer. Instead of (18) a new approximate expression for a^ for the
ordinary design of airscrews is
— -*— =:-i..-.fe cos^ cosec2 0 .... (31)
1 + aj 2Tr r
Points of no Torque, no Thrust, and no Lift. — ^From equation (2) it
will be seen that the torque of the element will be zero if dR sin {<f> + y) =0,
and if the value of dB, from (16) be used, the condition of no torque
reduces to
dQ = 0 when /(a) sin (<^ + y) = 0
i.e. when kj^ sin <{>-{- k-o cos ^ = 0
^.e. when 1^ = - cot <^ (32)
In a similar way it may be found that
dT =^ 0 when |5 = tan i^ (33)
k
The point of no lift occurs, of course, when 7^ = 0.
In ordinary practice <f> is positive at the angle of no lift, and the positions
found from (32) and (33) are not far removed from the no-lift position.
k
For the element of the previous example j~ ——I for (32) and +2 for
ftp
(33) when the solution is obtained, the angles of incidence being
no torque — 5°'4\
no lift -5°-l (34)
no thrust — 4° '4;
This result may be taken as typical of the important sections of air-
screw blades.
Integration foi a Number of Elements to obtain Thiust and Torque
for an Airscrew. — The process carried out in detail for an element can
be repeated for other radii and the total thrust and torque obtained.
The expressions may be collected as
/•D/2
T = pc(l + a2)^(oh^ sec2 <f>{kj, cos <f> — kjy sin <l>)dr . . (35)
•^ 0
/•D/2
Q = / pc(l + a2)2a>2r3 sec^ <f>{kj^ sin <f> + kj, cos <f>)dr . . (36)
J 0
802 APPLIED AERODYNAMICS
V T
and V = -'n (^7)
cor Q ^ '
from the aerofoil side, and
'J! = r'\7rp{l-\-ai)^Yh-dr (38)
and Q = / ^Trp{l +ai)^^Y(ori^ .rdr . . . (39)
from considerations of momentum,
where [(1 + ai)rt?r = [(l + 1^ Wn (40)
defines the rj of (39).
In considering a single element it has been shown that a2 may be
taken as zero, but that r^ is finite. It has been shown that (35) and (38)
can be made to agree by suitable choice of aj, and (38) may most suitably
be used during integration to find T. As A2 may be unknown, equation
(36) is used to calculate Q.
Fio. 151, — Comparison between observed and calculated variations of thrust along an
airscrew blade.
Determination of Aj. — If various values of A^ be chosen it is obvious
that for some particular one the calculated thrust at a given advance
per revolution will agree with the observed thrust on the airscrew. It
may be supposed that this has been done in a particular case (see Fig. 151),
and that for a value of -^ of 0-645 the best value of Ai has been found to
be 0-35. Using this value of A^ for -^ =0-562 and ^ =0-726 further values
nD wD
r
AIRSCKEWS 803
of total thrust are calculable and may be compared with observation.
Curves for the blade elements may be compared by the method used by
Dr. Stanton and Miss Marshall in measuring the thrust on the elements
of an airscrew blade (see page 284), and the result of the comparison is
shown in Fig. 151. This is the most complete check of the inflow theory
which has yet been made. Generally, the agreement between calculation and
observation is very good in view of the numerous assumptions in the theory.
It will be realized that in the check as applied above, any errors in
our knowledge of the ^ — of the sections will appear as attributed to inflow
and will affect the value of A^ ; any loss of efficiency at the tip will appear
in the same way. Fage has shown, however, that for a moderate
V
range of airscrew design and for such values of -^r as are used in practice
Ai is roughly constant. The best value is yet to be determined, but is
apparently in the neighbourhood of 0-35. The comparison given in Fig. 151
showed the presence of an appreciable " end loss," the thrust observed
near the tip being less than that calculated until a reduction of lift co-
efficient had been made. At a little over 95 per cent, of the radius the
lift coefficient was apparently reduced to half the value it would have
had if far from the tip.
It will be seen that on present assumptions the value of the torque is
completely determined when A^ is known. When compared with experi-
ment the calculated values of the torque are in good agreement with
observation, the average difference being of the order of 2 or 3 per cent.
Summary of Conclusions on the Mathematical Theory. — As a result
of a combined theoretical and experimental examination of airscrew per-
formance it is concluded that rotational inflow may be neglected, and that
an average value of 0-35 may be used for the translational inflow factor
Aj. There is a tip loss which is taken to be inappreciable at 85 per cent,
of the radius, 100 per cent, at the tip and 40 per cent, at 0*95 of the
maximum radius. The values of these losses, although admittedly not
of high percentage accuracy, are of the nature of corrections, and the final
calculations of thrust and torque are in good agreement with practice.
III. Applications of the Mathematical Theory
Example of the Calculation of the Thrust, Torque and Efficiency of an
Airscrew. — In developing the method of calculation for the performance of
an airscrew opportunity will be taken to collect the formulsB and necessary
data. Following the previous part of this chapter it will be unnecessary
to prove any of the formulae in use, as they may be obtained from equations
(14), (18), (38), (36) and (37) by simple transformations where they differ
from the forms there shown.
The first step will be to collect a representative set of aerofoil sections
suitable for airscrew design, together with tables of their characteristics.
The results chosen were obtained in a wind channel at a high value of
vl, and may be used without scale correction. The shapes of six aerofoil
304
APPLIED AEEODYNAMICS
sections are shown in Fig. 152, and numerical data defining them more
precisely are tabulated below.
TABLE 5. — Contours of Six Aerofoils suitable for Airscrew Design (Aspect Katio 6).
Distance of
ordinate from
Length of ordinate above chord, expressed as a fraction of chord
leading edge,
expressed as
a fraction of
chord.
No. 1.
No. 2.
No. 3.
No. 4.
No. 5.
No. 6.
Top.
No. 6.
Bottom.
0.05
0-0510
0-0465
0-0528
0-0794
0-1167
0-1033
-0-0300
010
00651
0-0625
0-0758
0-1020
01505
01433
— 0-0383
0-20
0-0775
0-0785
0-0976
0-1218
0-1810
0-1866
-0-0476
0-30
0-0817
00816
0-1014
0-1270
0-1880
0-2000
-0-0600
0-40
0-0806
0 0790
0-0985
0-1244
0-1816
0-1933
-0-0492
0-50
0-0761
0-0711
0-0926
0-1151
0-1666
0-1758
-0-0475
0-60
0-0694
0-0631
0-0836
0-1020
0-1455
0-1525
-0-0425
0-70
0-0593
0-0531
0-0705
0-0860
0-1210
0-1233
-0-0367
0-80
0-0451
0-0410
0-0533
0-0668
0-0926
0-0883
-0-0317
0-90
0-0273
0-0266
0-0326
0-0423
0-0687
0-0600
-0-0233
Aerofoils Suitable
FOR Airscrew Design
The aerofoils Nos. 1-6 have
flat undersurfaces, whilst No.
6 has a convex undersurface.
The shape of any of the aero-
foils is easily reproduced from
the figures of Table 5, where
all the dimensions are ex-
pressed as fractions of the
chord. The table is not an
exhaustive collection of the
best aerofoils for airscrew
design, but may be taken as
fully representative.
Corresponding with the
numbers in Table 5 are values
in Table 6 of the hft coeffi-
cient, ki, and of the ratio of
lift to drag. In using the
figures for calculation it is
almost always most conve-
nient to convert them into
curves on a fairly open scale,
as the readings required
rarely occur at the definite
angles for which the results
are tabulated. Interpolation,
. „ lift .
especially on ^ — , is most
easily carried out from plotted
Fig. 162. ^^^^^^^
The aerofoil characteristics have been expressed wholly in non-dimne-
Aerofoil N^ 5. O 168
02S
AIKSCEEWS
305
TABLE 6. — Aerofoils suitable fob Airscrew Design.
Angle of
incidence,
(degrees).
Absolute lift coefficient.
No.l.
No. 2..
No. 3.
No. 4.
No. 5.
No. 6.
-20
„
_
_
_
_
-0-0390
-18
—
—
—
—
-0-0134
-16
—
—
—
■
-0-0406
+0-0193
-14
—
—
-0-192
-0-142
-0-0054
+0 0423
-12
—
—
-0-188
-0-134
+0-0257
+0-0440
-10
—
—
-0-179
-0-120
0-0389
+0-0012
- 8
—
— .
-0-131
-0-0695
0-0498
-0-0005
- f)
-0-0865
-0-1210
-0-036
+0-0099
0-0985
+0-0545
- 4
+0-0125
-0-0271
+0-047
+0-0890
0-174
+0-115
- 2
0-0935
+0-0562
0-124
+0-163
0-245
0-178
0
0167
0-1270
0-196
0-234
0-314
0-242
2
0-242
0-202
0-274
0-308
0-391
0-320
4
0-314
0-276
0-351
0-382
0-460
0-420
G
0-384
0-353
0-425
0-453
0-536
0-484
8
0-457
0-430
0-490
0-518
0-599
0-548
10
0-530
0-500
0-562
0-586
0-661
0-599
12
0-585
0-565
0-614
0-643
0-718
0-287
14
0-618
0-603
0-610
0-700
0-765
0-277
16
0-486
0-602
0-581
0-746
0-795
0-283
18
0-448
0-538
0-558
0-774
0-382
0-306
20
0-444
0-465
0-543
0-774
0-389
0-326
22
0-434
0-494
0-434
0-340
24
0-431
—
0-449
0-425
—
0-355
L
ift^
Angle of
Di
rag
incidence,
(degrees).
No. 1.
No. 2.
No. 3.
No. 4.
No. 6.
No. 6.
-20
- 0-32
-18
—
—
—
— .
.,
- 0-13
-16
—
—
—
—
- 0-45
+ 0-21
-14
—
—
- 2-4
- 1-78
- 0-07
+ 0-62
-12
—
—
- 2-7
- 1-95
+ 0-40
+ 0-68
-10
—
—
- 3-2
- 2-14
0-71
+ 0-03
- 8
—
—
- 3-3
- 1-69
1-14
- 0-03
- G
- 3-79
- 412
- 1-6
+ 0-38
2-73
+ 3-86
- 4
4- 1-08
- 1-62
+ 3-2
5-12
7-45
8-20
- 2
10-90
+ 5-45
11-8
12-0
12-25
11-60
0
18-80
14-00
17-6
16-6
14-40
13-40
2
22-00
18-80
19-7
17-5
14-70
14-30
4
19-80
20-40
18-3
17-0
. 13-90
13-30
6
17-10
18-10
16-5
15-5
13-0
12-60
8
15-30
16-10
14-8
14-1
12-0
12-00
10
13 30
14-50
13-4
12-6
11-1
11-10
12
12-00
12-80
11-8
11-3
10-2
2-85
14
10-40
1110
8-9
10-4
9-5
2-40
16
4-07
8-45
6-9
9-4
8-75
2-15
18
3-01
4-35
5-45
8-5
2-40
2-12
20
2-70
3-04
4-38
7-3
2-20
2-00
22
2-40
—
2-76
2-4
__
1-97
24
2-22
—
2-20
217
—
1-88
sional or " absolute " units, and a similar procedure will be followed for
the airscrew. The typical length of an airscrew is almost always taken
306
APPLIED AERODYNAMICS
aS its diameter, and the width of the chord of any section will be expressed
as a fraction of D. Similarly the radius of the section will be given as a
fraction of the extreme radius, i.e. of — .
An application of the principles of dynamical similarity suggests the
V '
following variables as suitable for airscrews : -.r:- , or the advance of the
nD
airscrew per revolution as a fraction of its diameter ; a thrust coefficient,
Ajj:, such that
T = hpn^D^ (41)
a torque coefl&cient, ^q, defined by
Q = k^n^D^ (42)
and the efiiciency, t].
The equations already developed are easily converted to a form suit-
able for the calculation of k^ and kq in terms of the generalised variables,
and the five equations required are
«2=-^-^'2r^-*^"(^ + >')
tan0 = --^-H — ■
TT 1 + ttp
ai Ai c D fej.
1 + Oi TT D 2r sin ^
V \2 /•!
cot j>
5)
L/
(44)
(45)
*^=4vCt))7>+'"Kb/ (*«)
The value of A^ will be taken to be 0*35.
TABLE 7.
Angle of
incidence for
Aerofoil
2r
c*
maximum
number.
J>'
D'
L
D
(degrees).
2
0-96 •
0036
3
2
0-88
0-098
3
3
0-76
0-137
2
4
0-602
0-163
2
5
0-412
0-164
2
6
0-324
0-147
2
The plan form of the blades of the airscrew is defined by Table 7, ■=:
giving the sum of the widths of the two blades for various values of
* In this example c is the sum of the chords of two blades.
2r
D'
AIESCREWS
307
Since D is not specifically defined, the shape appHes to all similar airscrews.
In addition to the blade widths, the particulars of the sections at various
2?"
values of ~ are given in the first column, the aerofoil Nos. being the same
as those of Fig. 152. The last column of the table shows the angle of
incidence of each section for which the ^j — is a maximum.
drag
The shape of the blade is not completely defined until the inclination
of the chord of each section to the screw disc has been given. This
angle, denoted by <f)Q, depends on the duties for which the airscrew is to
be designed. In general the maximum forward speed of an aircraft,
the speed of rotation of the engine, and the airscrew diameter are fixed
by independent considerations ; if the diameter is open to choice, a suitable
value can be fixed from general knowledge by the use of a chart such as
Y
that on page 319. The value of -^ fixed in this way is not sufficient to
define ^ in terms of ^ , as may be seen from (43), as. the values of ajand a2
are not known and the most convenient method of procedure is to
make a first set of calculations with approximate values and to repeat
the calculations if greater accuracy is desired. Instead of the value of
V
Y:> which is assumed known at some speed of flight, it is convenient to
guess a value for . -=: in the first approximation, and in the illustra-
tion now given it is supposed that the design requires that at maximum
efficiency
l+«i V
^ •-^=0-241
(48)
[The preliminary calculations may be made with a2 = 0 and neglecting —
I in equation (45). With these conditions the calculation for the section at
2r
j— =0'88 proceeds as in Table 8.
The first column of Table 8 contains arbitrarily chosen values of
_i_ ^ Y 27"
. -— , and since — =0*88, this leads rapidly by use of (44) to the
v nD D
alue of tan (f> in column 2. <f) is obtained from tan (f> by the use of
tables of trigonometrical functions, and the angle a is chosen as 3° when
* ^.-=r- = 0*241 . This is in accordance with the earlier analysis which
showed that the maximum efficiency of a section occurred when the
ratio of the aerofoil was a maximum. The choice of a as 3° when
[drag
W=15°-3 fixes the value of (f)Q, i.e. of the blade angle to the airscrew disc ;
rthe remaining values of a are obtained from the expression a=i^Q—<f>.
308
APPLIED AEEODYNAMICS
From the angles of incidence and Table 6 the values of the lift coe£5cient
kj^ are obtained. Using equation (45) and the values of <f>, a and kj, of
V
Table 8, — ^- was calculated, thence ai, and finally the value of ;^.
l-f-oti nD
At this stage would be introduced the second approximation if the full
accuracy were desired. Prom equation (43) it is possible to calculate
values of a2 corresponding with the values of «! in Table 8, and as a^ and
«£ then become known with considerable accuracy the table can "be re-
peated using equations (43), (44) and (45) with their full meaning. The
calculation is not made in these notes, as the first approximation is
sufficient for the purposes of illustration.
TABLE 8.
1
2
3
4
5
6
7
8
l + oi V
TT nD
chosen
arbitrarily.
tan ^
from
equation (4)
with a2 = 0
(degrees).
a, angle of
incidence
(degrees).
*l„lift
coefBclent
from
Table 2.
l + Ol
from
equation (5).
"i-
V
nD
from
columns
1 and 7.
0-319
0-287
0-256
0-241
0-223
0191
0-160
0-128
0-332
0-299
0-267
0-251
0-232
0-199
0-167
0-133
19-9
18-0
16-2
16-3
14-2
12-2
10-3
8-3
-1-6
+0-3
2-1
\^o = 18-3;
4-1
6-1
8-0
10-0
0-070
0-135
0-205
0-240
0-280
0-365
0-430
0-505
0-0070
0-0166
0-0312
0-0410
0-0558
0-0956
0-1636
0-2980
0-0070
0-0178
0-0322
0-0428
0-0591
01056
0-1965
0-4250
0-996
0-885
0-779
0-726
0-661
0-641
0-420
0-282
Thrust. — A table similar to 8 was calculated for each of the other five
blade sections of the airscrew, and the various terms give the data from
which hi is calculated. Equation (46) implies integration for a constant
V
value of -=., and the tables do not provide values of ai(l +<*i) directly
suitable for the purpose. Values of ai(l +ai) were therefore plotted for
V
each section as ordinates on a base of -^, and from these curves the
following table was prepared : —
TABLE 9.
V
0^(1 + fli)
2r
2r
2r
2r
2r
2r
D
D
D
D
B
D
= 0-96
= 0-88
-0-76
= 0-602
- 0-412
= 0-324
1-0
0-0030
0-0070
0-0080
0-0090
0-0100
0-0110
0-9
0-0064
0 0160
0-0204
0 0230
0-0225
0-0112
0-8
0-0122
0 0290
0-0389
0-0438
0 0416
0-0266
0-7
0-0215
00510
0-0692
00750
0-0704
0-0492
0-6
0 0360
0 0900
01160
0-1300
01185
0-0875
0-5
0-0620
0-1450
01950
0-2100
0-1990
0-1160
0-4
0-1100
0-2600
0-3250
0-3660
0-3490
0-0980
AIESCEEWS
309
Numbers can be deduced from Table 9 for comparison with Fig. 151.
The value of
thrust per foot run
pV^D
^.aUl+a.)jj
(49)
and values calculated by means of (49) and plotted against |^ give curves
very similar to those of Fig. 151. The central part of the airscrew has
been ignored as of little importance.
Using equation (46) in the form shown, the value of ai(l + ^i) was
plotted on a base of ( ^ j and the value of the integral obtained graphically,
the results being set out in the table below.
TABLE 10.
V
nD
Ja,(l + a,)d(|'')2
Thrust
coefficient,
10
0-9
0-8
0-7
0-6
0-5
0-4
0 0069
0-0168
0 0309
0-0544
0-0904
0-1504
0-2561
00165
0-0305
0-0443
0-0596
0-0728
0-0842
0-0918
If the values of k^ are plotted on a basis of -=r and the curve produced,
V
it will be found that JL becomes zero when -^=r- = l'l, and this number is
nD
the ratio of pitch to diameter for the airscrew in question. The pitch
here defined is called the "experimental mean pitch," and is the advance
per revolution of the airscrew when the thrust is zero.
Torque.^The calculation of torque follows from equation (47) as
below.
TABLE 11.
1
2
3
4
5
6
V
nD
from'
Table 8.
Old -f oi)
calculated from
column 7,
Table 8.
L
D
corresponding
witli the
vahies of a in
Table 8.
y-tan-'^
(degrees).
813
3-87
302
2-71
2-80
3-17
3-42
3-92
tan (<^ + y)
</) from
Table 8.
ai(l+Oi) tan (^+y)
from columns
2 and 5.
0-996
0-886
0-779
0-725
0-661
0-541
0-420
0-286
0-0071
00181
0 0333
0-0446
0-0625
01170
0-2340
0-6050
7-0
14-8
19-0
210
20-4
18-1
16-4
14-6
0-633 i 00038
0-402 i 00073
0-349 0-0116
0-326 1 00142
0-306 1 00191
0-276 0-0322
0-246 0-0574
0-216 0-1310
310
APPLIED AEEODYNAMICS
The numbers in Table 1 1 correspond with those in Table 8, and apply
to a value of ^ of 0'88. The table was repeated for other values of ^,
and the results of calculations such as are shown in column 6 of
V
Table 11 were plotted against -rz-. From the curves so plotted Table 12
was prepared by reading off values of ai(l + aj) tan {<f> + y) at chosen values
TABLE 12.
Old + Oj) tan (0 + y).
V
2f
2r
2r
2r
2r
2r
D
D
D
U
D
D
= 0-96
= 0-88
00038
= 0-76
= 0 602
= 0-412
= 0-324
10
0-0012
0 0052
0-0075
0-0146
0-0160
0-9
0-0025
0-0060
0-0100
0-0142
0-0200
0-0130
0-8
0-0040
00100
0-0160
0-0230
0-0300
0-2240
0-7
0-0060
0-0160
0-0250
0-0350
00465
0-0395
0-6
0-0095
0-0248
0-0380
0 0520
0-0710
0-0630
0-6
0-0150
0-0390
0-0600
0-0820
0-1100
0 0900
0-4
0-0240
0*0630
0-0940
0-1320
0-1740
0-0900
/2r\3
The numbers in Table 12 were plotted as ordinates with 1=^) as
V ^^^
abscissa, and curves for each value of -=- drawn through the points. The
areas of the curves obtained by planimeter gave the values of the integral
of equation (47), and from them the calculation for kq was easily completed
(see Table 13).
TABLE 13.
V
•' 0
a^) t&n CP + y)d{^Y
10
000570
0-9
0-00897
0-8
001431
0-7
0-02228 ,
0-6
0-03377
0-5
0-05311
0-4
0-0826
Torque
coefficient,
kq
0 00426
0 00543
0-00685
0-00817
0 00908
0-00992
0-00987
Efficiency,
0-580
0-806
0-825
0-814
0-765
0-676
0-591
The efficiency of the whole airscrew is
TV 1 V
Kt
(50)
' 27wQ 277 nl) kq
and the values of r) are obtained from Tables 10 and 13 and equation (50).
It will be seen that a high efficiency of 0*825 is found, and this is partly
AIKSCREWS
811
due to the fact that all elements have been chosen to give their maximum
V
efficiency at the same value of -^r .
Effect of Variations of the Pitch Diameter Ratio of an Airscrew. —
V
By choosing different values of ^=- for the state of maximum efficiency
and repeating the calculations, the effect of variation of pitch could have
been obtained. Instead of repeating the calculations, an experiment
described in a report of the American Advisory Committee on Aeronautics
will be used to illustrate the effect of variation of pitch-tiiameter ratio.
The report, by Dr. Durand, contains a systematic series of tests on 48 air-
0.8
0.7
0.6
EFFI
V
0.5
o>»-
0.3
0.2
O.I
CO
'"'^y*
K
^~
\
■
A
^
\
\
N
-V
:IENCY
A
0
/
N
1 ,v
\
^v
\
J
^
y-^
\.
A
\
\
i
V^
\
^^.
•
\
\
/
f
^N
^
^r^
\
\
\
\
/
\
\^
H
S,
\
\,
/
^
\]
\
\L
N
A
0.006
0.005
THRUST
COEFFICIENT
0.004-
^r
0.003
o.ooa
o.ooi
O O.I 0.2 0.3 0.4-0.5 0.6 0.7 0.6 0.9 1.0 I.I
Fig. 153. — ^Effect of variations of pitch diameter ratio of an airscrew.
screws of various plan forms and pitches, and the results shown are typical
of the whole. For details the original work should be consulted.
The three screws used in the particular experiment referred to, were
of the same diameter, and had the same aerofoil sections at the same
radii. The general shapes of the sections were not greatly different from
those just referred to in the calculation of the performance of an airscrew
and illustrated in Fig. 152, the lower surfaces being fiat except near the
centre of the airscrew. The chords of the sections were incHned at 3° to
the surface of a heHx, and the pitch of this hehx was 0*5D, 0-7D and 0*9D
in the three airscrews used in producing the results plotted in Fig. 153.
V
The experimental mean pitches, i.e. the values of - when the thrust is
zero, were 0*69D, 0*87D and 1 "090, and do not bear any simple relation to
the helical pitches.
312 APPLIED AEKODYNAMICS
The most interesting feature of the curves of Fig. 153 is the increase
of maximum efficiency as the pitch diameter ratio increases, an effect
which would be continued to higher values than 1"1. It is easily shown
that the greatest efficiency is obtained for any element when ^ + |y = 45°,
and as y is small for an efficient airscrew the pitch diameter ratio would
need to be tt before the maximum efficiency was reached. It is not usually
possible to rotate the screw at a low enough speed to ensure the absolute
maximum efficiency, and in addition the whole of the effective area of
the blade cannot be given the best angle on account of stresses in the
material of which the airscrew is built.
Fig. 153 can be used to illustrate the advantages of a variable pitch
airscrew, although the comparison is not exact since the screws cannot
be converted from one to another by a rotation about a fixed axis. This
latter condition is almost always present in any variable pitch airscrew,
and the details of performance may be worked out by the methods already
detailed except that in successive calculations a constant addition to
^0 is made for all sections.
Consider the medium airscrew of Fig. 153 as designed to give maximum
efficiency to the aeroplane when flying " all out " on the level, the value of
V ' .V
-=^ being then 0*6. For the condition of maximum rate of climb -^
wD wD
may be 0*4, and changing to the lower pitch increases the efficiency by
V
about 4 per cent. During a dive or glide at ^=- = 1-0 the change to the
larger pitch converts a resistance of the airscrew into a thrust, and a higher
speed is possible. Usually a dive can be made sufficiently fast without
airscrew adjustment, and for a non-supercharged engine the advantages
of a variable pitch airscrew are not very great.
For a supercharged engine the conditions are very different. The
limiting case usually presupposed is the maintenance at all heights of
the power of the engine at its ground-level value, so that at a given number
of revolutions per minute the horsepower available is independent of the
atmospheric density. For the same conditions of running, the horse-
power absorbed by an airscrew of fixed pitch is proportional to the density,
and any attempt to " open out " the engine at a considerable altitude
would lead to excessive revolutions. With a variable pitch airscrew this
excessive speed could be avoided by an increase of pitch, and Fig. 153
shows that a gain of efficiency would result. From the curves of Fig.
153 it is possible to work out the performance of the airscrew at constant
velocity and revolutions, but in the flight of an aeroplane with sufficient
supercharge the value of V would change, and hence the complete
problem can only be dealt with by some such means as those given in
the chapter on the Prediction of Aeroplane Performance.
Tandem Airscrews. — In some of the larger aeroplanes in which four
engines have been fitted, the latter have been arranged on the wings in
pairs, a rear engine driving an airscrew in the sUp stream from the airscrew
of the forward engine. It is not usual for the rear screw to be much
greater than one diameter behind the front one, and the slip stream is
AIKSCEEWS
313
still unbroken and of practically its minimum diameter. The velocity of
the air, both translational and rotational, at the rear airscrew can be
approximately calculated by the use of equations (45) and (47), and an
example of the method which may be followed will now be given.
The forward airscrew will be taken to be that worked out in this chapter
on pages 306 to 310, and of which details are given in Tables 8-13.
The first operation pecuUar to tandem airscrews is the calculation of
the details of the sHp stream from the forward airscrew. From the values
of ai(l +ai) given in Table 9 the value of (1 +ai) is calculated without
difficulty, since
(1 -f ai) = 0-50 + a/O-25 + ai(l + ai) . . .(51)
V
Taking — -=:0'6 as example, the following table shows the required
ihXJ
steps in the calculation of the radius of the slip stream : —
TABLE 14.
2r
D
0-960
0-880
0-760
0-602
0-412
0-324
l+Ol
1035
1083
1105
1116
1-107
1080
'+r;
1100
1-242
1-300
1-331
1-305
1-228
1+ai
D
0-941
0-872
0-850
0-838
0-848
0-880
0-892
0-814
0-705
0-507
0-400
0-306
The first two rows of Table 14 are obtained from Table 9, and the thiyd
row is easily obtained from the second since Ai=0'35. The figures in row
four are plotted in Fig. 154 on a base of ( =- j , a form suggested by equation
(46). The integral required was obtained by the mid-ordinate method of
finding the area of a diagram, and the result is shown in the lower part
of Fig. 154. The extreme value of the square of the radius of the slip
stream is seen to be 0*87 times that of the airscrew, and the radius of the
sUp stream 0*93 times as great as the tip radius. This value may be
compared with the direct observations illustrated in Fig. 147.
Rotational Velocity in Slip Stream. — From equation (47) the relation
is obtained by differentiation.
From equation (12) a second relation for the same quantity is obtained
in terms of the outflow factor &2- This latter expression is
dkci
<f)
7r2 , ., /2ri\2 2r
(53)
314 APPLIED ABEODYNAMICS
and a combination of (52) and (58) leads to
h,=-
7rAi"wD72ri\2
(1 + ai) tan (0 + y)
Q)
(l+ai)
(54)
and all the quantities required for the calculation of &2 have already been
tabulated.
nJ) 2ri
The rotational air velocity is 62'"" • ^ 'T)'^' ^^
2r
liV.-^tan(0 + y) (55)
D
1.2
I.O
OS
0.6
0.4
0.2
\
l+ai
.^
/
•
\
-^
-<
^^
^
^
m
L
^
^
^
^
O.I 0.2 0.3 0.-4 0.5 0.6 0.7 0.8 0.9 i.O
.2
(%7
Fig. 154. — Calculation of the size of the slip stream of an airscrew.
In this expression r^V will be recognized as the added translational
velocity between undisturbed air and the slip stream, and the factor
r-^V tan (^+y) is the component rotational velocity which would follow
from the assumption that the direction of the resultant force at the blade
is also the direction of added velocity. The remaining factor is due to
the change from airscrew diameter to slip stream diameter.
The following table shows the values of &2 and the angle of the spiral
in the sUp stream calculated from (54) and (55), and the latter can be
compared directly with Fig. 147 for observations on an airscrew : —
AIRSCREWS
815
TABLE 15.
2r
0-96
0-88
0-76
0-602
0-412
0-324
Angle of j
spiral 1
(degrees) )
-0 0060
1-5
-00166
3-2
-0-0297
4-6
-0-0476
6 0
-0-0903
8-6
-0-1095
S-l
The calculations for a second airscrew working in the slip stream of
the first can now be proceeded with almost as before. If a/ and a2
apply to the second airscrew whilst V has the same meaning as before,
then the whole of the previous equations can be used with the following
substitutions : —
Instead of Ui use v^ + ai'l
and instead of Uz use ± 62 + ^2' '
(56)
the values of aj and &2 being taken with the corresponding values of Vi
as obtained from Table 14. The ambiguity of sign corresponds with
rotations in the same and in opposite directions respectively.
If the rear airscrew runs in the opposite direction to the front one,
the existence of &2 tends to increase the efficiency, since (28) now becomes
1 ±&2 + ^' ^^^ ^
^1
(57)
and as &2 is negative the numerator of (57) is increased.
The translational inflow reduces the efficiency by the introduction of
the factor ^^ into the denominator, but as the speed of the rear airscrew
^1
relative to the air is now higher the value of
tan <f>
is increased
tan {<f> + y)
owing to the larger values of <f>.
In general it appears that some loss of efficiency occurs in the use of
tandem airscrews. The subject has been examined experimentally, and
one of the experiments is quoted below because of its bearings on the
present calculations.
The airscrews were used on a large aeroplane, and each absorbed 350
horsepower at about 1100 r.p.m., rotation being in opposite directions.
The diameter of the front airscrew was 13 feet, and that of the rear airscrew
12 feet. The maximum speed of the aeroplane in level flight was about
100 m.p.h. Models of the airscrews were made and tested in a wind
channel, and from the results obtained Fig. 155 has been prepared.
Curves for thrust coefficient and efficiency are shown for both airscrews.
In the case of the front airscrew the curves were not appreciably altered
by the running of that at the rear. An examination of the figure will
show that the ratio of pitch to diameter of the rear screw is 0*86, whilst
316
APPLIED AEEODYNAMICS
that of the forward screw is 0"80. In accordance with the experiments
on the effect of variation of pitch it would have been expected that the
maximum efficiency of the rear airscrew running alone would be higher
than that of the forward airscrew. The experiment showed an increase
of efficiency from 0*74 to 0'78 on this account.
The efficiency of the rear airscrew when working in tandem is shown
by one of the two dotted curves, and its maximum value is seen to be
0"70. In this diagram V is the velocity of the aeroplane through the air,
V
and hence -;=r has a different meaning to the similar quantity for the
forward airscrew. In the latter case the velocity of the airscrew through
0.09
0.06
0.9
0.0.2
O.O.I
0.2 0.3 0.4- 0.5 0.6 0.7
Fia. 155. — ^Tandem airscrews.
OS 0.9
the air is equal to that of the aeroplane, whilst in the former the velocity of
the airscrew through the air is appreciably greater than that of the
aeroplane. A general idea of the increased velocity in the slip stream is
given below.
TABLE 16.
Airspeed of aeroplane
(ft.-s.).
Velocity of air at rear
airscrew (ft.-s.).
Batio of speeds.
70
102
1-46
80
110
1-38
100
125
1-26
120
139
116
140
154
110
160
172
107
AIESCREWS 317
The velocity in the slip stream of the front airscrew is not uniform,
and the value as given in Table 16 is obtained by making the assumption
that the thrust coefficient of the rear airscrew when working in tandem
has the same value as when working alone, if the value of —^ is the
° nv
same in the two cases, V being the average velocity of the airscrew
relative to the air. The calculation involves the variation of engine power
with speed, and details of the methods employable are given in the chapter
on Prediction. In the present instance the object aimed at is satisfied
when the detailed theory of tandem airscrews has been developed and the
V
result illustrated. It will be noticed from Fig. 155 that for values of -y^
in excess of 0'62 the efficiency of the combination is greater than that of
two independent airscrews Uke the forward one. At the maximum speed
of an aeroplane the loss of efficiency on the tandem arrangement of airscrews
is not very great, since -=r is usually chosen a little larger than the value
giving maximum efficiency. At climbing, say at -^r =0*4, the efficiency
of the rear airscrew is 82 per cent, of the forward airscrew, and the com-
bination has an efficiency 91 per cent, of that of the front airscrew alone,
which was designed without restriction as to diameter. It may be con-
cluded, therefore, that the losses in a tandem arrangement of airscrews
may be very small at the maximum speed of flight, and that they will
become greater and greater as the maximum rate of chmb and the reserve
horsepower for climbing increase. It will, however, be the usual case
that tandem airscrews are only needed on the aeroplanes which have least
reserve horsepower, i.e. where the losses are least.
The Effect of the Presence of the Aeroplane on the Performance
OF AN Airscrew
The number of tests which relate to the effect of the presence of an
aeroplane on its airscrew are not very numerous. Partial experiments
on a combination of model airscrew and body are more numerous chiefly
because the effect of the airscrew sHp stream in increasing the body re-
sistance is very great. This increase of resistance is dealt with elsewhere
in discussing the estimation of resistance for the aeroplane as a whole
and in detail. All. the available experiments show a consistent effect of
body on airscrew, which is roughly equivalent to a small increase of effi-
ciency and an increase of experimental mean pitch. One example has
been chosen, and the results are illustrated in Pig. 156. This example is
typical of such effects as arise from a nacelle closely surrounding the
engine, and apply particularly to a tractor airscrew. Where the front of
the body of a tractor aeroplane is designed to take a water-cooled engine
the results would also apply, but it might be anticipated that the large
body required for a rotary or radial engine would have more appreciable
effects.
318
APPLIED AEEODYNAMICS
The effects of the nacelle of a pusher aeroplane are of the same general
character as for a tractor ; both the thrust and torque coefficients are
increased by the presence of the nacelle, and the efficiency and pitch are
increased. The amounts are on the whole rather greater than those
shown in Fig. 156.
Fig. 156 shows the thrust coefficient and efficiency of a four-bladed
tractor airscrew when tested alone, when tested in front of a body, and
when tested in front of a complete aeroplane. The observations were
taken on a model in a wind channel. The cross-section of the body a
short distance behind the airscrew had an area of 7 per cent, of that of
the airscrew disc. The thrust coefficient is increased by the body over the
V . . V
whole range of -=r- by an amount which increases as -^ increases. The
^ nJ) "^ wD
maximum efficiency is little affected, but the experimental mean pitch
O.I8
0.16
O.I4-
0.08
0.06
0.04-
0.02
^^>
T
1RUST
COEf
— T—
FICIENT
X
^t
El
^FICIE
NCY
,
%
h>.
thru;
T COEFFICIE
IT
y
^'
N
^^
N^EFFK
lENCY
V
prv'
U*
A
AIRS
CREW
ALONE
'<
\
^\
AIRSClijEW V\
1 NEAR B(;
/ORKING
\^>
,DY
^\\
G IN-
ANE
\\
\\
I
PLACE
ON A
EROPL
%
A'
\
N
\
ynz
^
L
0.7
0.5
0.4
0.1
O O.I 0.2 0.3 0.4- 0.5 0.6 0.7 O.S 0.9 1.0 I.I 1.2 1.3 O
Fig. 166. — ^Effect of the body and wings of an aeroplane on the thrust of an airscrew.
is increased by nearly 3 per cent. The addition of the wings and general
structure of the aeroplane brings the total effect on the airscrew to an
increase of 1 per cent, on efficiency and 5 per cent, on pitch.
On a particular pusher nacelle of greater relative body area the maximum
efficiency was raised by 3 per cent., and the experimental mean pitch by
9 per cent.
In the present state of knowledge it will probably be sufficient to
assume that calculations made on an airscrew alone can be applied to the
V
airscrew in place on an aeroplane by changing the scale of -^ by 5 per cent.
and increasing the ordinates of the thrust coefficient and efficiency curves
by 2 per cent. These changes are small, and great accuracy is therefore
not required in the practical apphcations of airscrew design.
AIESCEEWS
819
Approximations to Airscrew Characteristics
Before proceeding to the detailed design of an airscrew it is necessary
to know the general proportions of the blades, and the sections to be used.
These are at the choice of any designer, who will adopt standards of his
own, but the choice for good design is so hmited that rough generahza-
05R,
O45R.
OfeR.I
O75R.
R
OI5R .
09R.
•
0 7lC
0 87C
0 98C
0 98C
0 64C
059C
P
Fig. 157.
tions can be made for all airscrews. The plan form of the blades is
perhaps the quantity which varies most in any design, and in connection
with the approximate formulae and curves is given a drawing of the
plan form to which they more particularly refer (see Fig. 157).
50 100
HORSEPOWER | , , , , |
200 300 400 500 lOOO
1 1 ^ I ■ . ■ . I
500 600 700 800 900 1000
/f.fi^. I ■ ■ . . I .
1500 2000
I I I I 1,1 I I
REFERENCE LINE
A J
/ /
1/
/
//
50 /loo / 200
M.P.hA , . /, I / . . ■ I
DIAMETER (FEET) /
20 19 IS 17 16 15 14- 13 / 12 II 10 9
'■ '■ ' ■ I .1 J f^ U, l-T I , ■
-r — r
17 16 15 14- 13 12 II 10
NOMOGRAM.
TWO BLADES
6 7 6
r^ ^ 1
7 6
FOUR BLADES
Fig. 158. — Nomogram for the calculation of airscrew diameter.
For this shape of blade H. C. Watts has given a nomogram connecting
the airscrew diameter of the most efficient airscrews with the horsepower,
speed of translation and rate of rotation.
320
APPLIED AEEODYNAMICS
Diameter. — Example of the use of the nomogram (Fig. 168).
" What is the approximate diameter of an airscrew for an aeroplane
which will travel 120 m.p.h. at ground-level with an engine developing
400 horsepower at 1000 r.p.m. ? "
On the scales for translational and rotational speeds the numbers
120 and 1000 are found and joined by a straight hne cutting the reference
line at A of Fig. 158. The position of the 400 horsepower mark is then
joined to A by a straight line which is produced to cut the scale of diameters.
In this case the diameter of a two-bladed airscrew is given as 13 feet, and
of a four-blader as 11 feet.
In air of reduced density the ground horsepower should still be used in
the above calculation.
The nomogranf may be taken as a convenient expression of current
practice.
Maximum Efficiency. — The results of a number of calculations are
given in Table 17 to show how the efficiency of an airscrew may be expected
to depend on the horsepower, speed of translation, and diameter. As
before, the ground horsepower should be taken in all cases, and not the
actual horsepower developed for the conditions of reduced density. The
table covers the ordinary useful range of the variables. For the example
just given the table shows an efficiency of about 0'80. Interpolation is
necessary, but for rough purposes this can be carried out by inspection.
TABLE 17. — Efficiencies of Aiesoeews (Appboximate Values).
800 r.p.m.
1200 1
.p.m.
1600
r.p.m.
Aeroplane
speed,
m.p.h.
8 ft.
12 ft.
16 ft.
8 ft.
12 ft.
8 ft.
diam.
diam.
diam.
diam.
diam.
diam.
Level 1
100
0-80
0-78
_
0-74
120
0-84
0-82
—
0-79
200 ]
B.H.P.
flight j
140
0-86
—
0-84
—
0-81
Climb 1
60
0-60
,
0-55
—
0-51
80
0-68
—
— .
0-64 .
—
0-61
Level 1
flight 1
100
0-74
0-78
—
0-72
0-73
0-69
120
0-79
0-82
0-77
0-77
0-75
400
140
0-83
0-84
0-81
0-81
0-79
B.H.P.
Climb 1
60
0-54
0-56
. — .
0-60
0-50
0-46
80
0-63
0-66
—
0-60
0-61
0-56
Level
100
" 0-75
0-75
—
0-71
—
120
0-80
0-80
0-76
—
600
B.H.P.
flight
140
0-83
0-83
— .
0-79
—
Climb 1
60
.
0-53
0-52
—
0-46
—
80
■ — ■
0-63
0-63
—
0-67
~
Further Particulars o! the Airscrew, — For many purposes it is desirable
to know more about the airscrew without proceeding to full detail, and
Fig. 159 is a generalization which enables the characteristics of an airscrew
to be given approximately if four constants are determined. These
constants are the experimental mean pitch, P, the pitch diameter ratio,
AIRSCEEWS
321
^, and two others denoted by To and Qq. Tq is a number such that Tcfcr = 1
V 1 V
when ^p =0"5, and similarly Qjcq = 1 for the same value of -p . fcj and
/cq are the usual absolute thrust and torque coefficients as defined on p. 806.
To apply the curves to the example a further note is required ; it can
/
\
•
1
>
V
-\^
THRUST COEFFICIENT
1
1 o
^^
A
\
!&v «
^X TORQUE C
JOEFFIC
N
>
\
\
\
V
^
\
0.5
\
\
N
\
V
^
\
\
\
\
O
V
\
O O.I 0.2 0.3 0.4 0.5 0.6 0.7 O^S 0.9
Fio. 159. — Standard airscrew characteristics.
I.O
be deduced from Fig. 159 by calculating the efficiency. For a pitch diameter
V
ratio of 0*7 the maximum efficiency occurs at about -— = 0'6, whilst for
P V
.^ = 1*1 the value of -^ is about 0'65. In order to keep the average
efficiency of an aeroplane airscrew as high as possible the maximum speed
V
is made to occur at a value of -p somewhat greater than that giving
maximum efficiency. — will often be 0'7 or even 0*75 at maximum speed.
Y
322 APPLIED AEEODYNAMICS
Continuing the example, it is then found that
V 10-5 ^_
_=-^ =0-7, say
P 15
and therefore P = 15 feet and — = — = 1-15.
Calculation of To and Qo.— The efficiency having been found to be
0-80, the thrust is found from the horsepower available. Since
120 X 88
The thrust coefficient h, = 0-00237 X^176^ X 13^ = ^'^^^^
These figures will depend on the air density, both T and fc^ being
affected. The horsepower available for a given throttle position, etc., varies
rather more rapidly than density, and hence the thrust varies rapidly with
density, fei involves the ratio of horsepower to density, and is not there-
fore greatly altered. Ground conditions of density and horsepower may
therefore always be used in the approximate expressions for Tg and Q*'.
V
From Fig. 159 the value of Tofer at -^ = 0*7 is seen to be 0-635, and
0-635 „.Q. '^^
hence To = ^:^g = 7-95.
Similarly 400x33,000 ,,„„„ ,,
^^^q"^= 6-28 X 1000^'^''^^^-^^-
J. 2100 _n.niQi
'^«"' 0-00237 X1762 xl33
V P
From Fig. 159 the value of Qo^q at -^ = 0-7 and :=- = 1-15, is read off
0-805 ^
as 0-805. Hence Qo = QiQjgj = 61 '^^
P
Having determined P, — , To, and Qo in this way, the characteristics of
V
the airscrew at all values of -=- are readily deduced from Fig. 159.
nJ)
Use is made of these approximations in analysing the performance of
aeroplanes.
IV. Forces on an Airscrew which is not moving Axially through
THE Air
Modifications of formulae already developed will be considered in
order to cover non- axial motion of the airscrew relative to the air un-
disturbed by its presence. It is necessary to introduce a system of axes
as below.
The axis of X will be taken along the airscrew axis, and in relation to
Pig. 160 is directed into the paper. The velocity of the airscrew perpen-
AIRSCEEWS
823
dicular to the X axis is v, and the axis of Y is chosen so as to include
this motion.
The only new assumption to be made is that the component of v along
the airscrew blade is without appreciable effect on the force on it.
The velocity of the element AB due to rotation and lateral motion is
made up of the constant part wr and a variable part —v cos d, and com-
parison with Fig. 149 suggests the writing of the resultant velocity normal
to the X axis in the form
(aril cos 6)
\ car /
(58)
cos 0 now takes the place of a2. The further procedure is the
same as in the case for which v—d up to the point at which it was necessary
to make an assumption as to the value of a^,, i.e. until the completion of
7- 2
Fig. 160.
Table 1. (The rotational inflow previously included will be ignored here
as unimportant in the present connection.)
Equation (14) becomes
tan^, = ^±a-.I: (59)
1 cos d
a>Y
This equation may be written as
tan ^^
]+«!
-^-^cosfl
(60)
For given values of 0 the corresponding values of a^ have been
calculated and are given in Table 1 .
V
For the purposes of illustration, — is taken as 0*340, so that results
toT
may be compared with those for which 0=2** in the axial motion. The
824
APPLIED AEKODYNAMICS
calculations of angle of incidence, for the element to which Table 1 refers,
are now extended to cover variations of 6 during one revolution of the
airscrew. :^ is taken as 0*174, i.e. motion at 10° to the airscrew axis.
TABLE
18
0
(degrees).
^cose.
-_fcos0.
tan <i>„
ai=0074.
<t>v
(degrees).
0„+()4,-22''l
a,
(degrees)
0 and 360
20 „ 340
40 „ 320
60 „ 300
80 „ 280
100 „ 260
120 „ 240
140 „ 220
160 „ 200
180
0-174
0-164
0-134
0-087
0-030
-0-030
-0-087
-0-134
-0-164
-0-174
2-766
2-776
2-806
2-853
2-910
2-970
3-027
3-074
3-104
3-114
0-3880
0-3875
0-3830
0-3770
0-3695
0-3620
0-3555
0-3500
0-3460
0-3450
21-20
2116
20-96
20-65
20-29
19-90
19-67
19-28
19-08
19 05
0-90
0-95
1-16
1-45
181
2-20
2-53
2-82
3 02
3-05
Columns 2 and 3 need no explanation since they are calculated
values corresponding with the assumed values of 6, ^ and — . The fourth
column comes from equation (60) and the previous column. ^^ is obtained
from column 4. The last column follows from the relation that ^t, + a„
is constant and in our example equal to 22°'l. The value of a given in
Table 2 was 2°, and by interpolation will be seen to occur in the above
table when d = 90° and 270".
The first noticeable feature of Table 18 is the variation of angle of
incidence during a revolution of the airscrew. With ^=zero, the angle
of incidence is reduced by 1°-10 due to sideslipping, and with 0 equal to
1 80° the angle of incidence is increased by l°-05. With the blade vertically
downwards it may then be expected that the thrust on the element is
decreased as compared with the axial motion, when horizontal there is
no change, whilst with the blade upwards the thrust is increased. The
calculation of the elementary thrust and torque is carried out below.
TABLE 19.
Table 18.
■
0
(degrees).
(degrees).
K
K
Cos0„
Sin^,
1 OE
1 dQ
pV"cr • dr
0 and 360
0-90
0-229
0-0120
0-934
0-359
1-81
0-828
20 „ 340
0-95
0-230
0-0122
0-934
0-358
1-83
0-833
40 „ 320
115
0-237
00125
0-936
0-355
1-96
0-885
60 „ 300
1-45
0-250
0-0130
0-936
0-352
2-12
0-940
80 „ 280
1-81
0-268
0-0137
0-938
0-347
2-36
1-02
100 „ 260
2-20
0-280
00143
0-940
0-342
2-57
1-09
120 „ 240
2-53
0-295
00161
0-941
0-336
2-81
1-17
140 „ 220
2-82
0-306
00157
0-942
0-328
3 00
1-23
160 „ 200
3-02
0-314
00161
0-943
0-326
3-15
1-27
180
3 05
0-316
0-0162
0-944
0-325
317
1-29
AIESCREWS
B25
Columns 1 and 2 are taken from Table 18, and the 3rd and 4thcolumna
are then read from Fig. 161. This figure shows hj, and k^ as dependent
on angle of incidence and agrees with the values of Table 1 .
From equations (46) and (47) are deduced the expressions
1 dT / V VxVwr^^
and
^27/7^=(l-|<^os^.-)(^^) sec2^„{fe:,sin0. + /cocos<^,} (62)
and from the values in Tables (18) and (19) the right-hand sides of these
0.020
O.OI8
OEFFICIEN
O.OI5
O.OIA
O.OI2
O.OiO
0.30
0.28
/
A
r ■'
I
/ '
/K
y
/
DRAGC
0.26
LIFTCOEF
0.24.
/
A
FICIENI
i
i\
/
0.22
0.20
r
X
f
O
12 3^
ANGLE OF INCIDENCE
(DEGREES)
FlO. 161.
[expressions are evaluated to form the last^columns of Table 19. The
^comparative values of
1 dT
and
1 dQ
for the axial motion are
pN'^c* dr "'"'* pNHr' dr
'2'46 and 1*05 respectively.
The table shows that, due to an inclination of 10°, the thrust on the
; blade element varies from 74 per cent, to 126 per cent, of its value when
[moving axially. The elementary torque ranges between 79 per cent, and
123 per cent, of its axial value.
There are seen to be appreciable fluctuations of thrust and torque on
each blade during a revolution of the airscrew which need to be examined
1 /70
I further. - • ~ represents an elementary force acting on each blade of a
826 APPLIED AERODYNAMICS
two-bladed airscrew normal to the direction of motion. On the blade at
the bottom this elementary force is 0'SQ,Sp\^-dr, whilst on the opposing
blade at the top it is l-290pY^-dr. The torque on the two blades is then
I'OGpV^cdr as against the value l'05pV ^cdr for axial motion. Similar
results follow for other positions, and for airscrews with two or four blades
the variation of torque with 6 is seen to be very small.
On the lower blade the force 0'S2SpY^-dr acts in the direction of the
axis of Y, whilst on the upper blade the force is I'QidOpY^-dr in the opposite
direction. There is therefore a force dY = -~0-'22pY^cdr on the pair of
blades ; this is the same effect as would be produced by a fin in the place
of the airscrew and lying along the axis of X and Z. Such a fin would oppose
a resistance to the non-axial motion.
The thrust on the lower blade element is 1 'SlpY^~dr, and on the upper
blade is SlTpV^-dr, the resultant thrust on the two blades being 2-49pV^cdr
A
as compared with 2"46/oV^cflfr in the axial motion. As for torque, it appears
that the effect of lateral motion on thrust at any instant is very small
for two and four bladed airscrews.
On the lower blade the thrust gives a couple about the axis of Y of
c c
l'81pY^-rdr, whilst on the upper blade the couple is —d'HpY^-rdr. The
A A
resultant couple is then —0'68pVhrdr. The lower blade, as illustrated in
Pig. 160, would then tend to enter the paper at a greater rate than the
centre.
The values of the differences between axial and non-axial motion for
the element of a single blade are given below as the result of calculation
from the following formulae : —
SY= \-^ -0-525pY^crdr\cose . . . (63)
dZ = -(^-^ - 0'52,5pY^crdr) sin 6* . . . (64)
8M
rV 2
^8T
-r(^-^ -I'^^pY^cdr ) cos ^ . . . (65)
2
'8T
SN = -r(^ -l-23pV2c(?r ") sin ^ . . . (66)
These formulae assume two blades for the airscrew, and the differences
from axial motion are used instead of the actual forces during lateral
motion ; O'BIBpVhrdr and 1 '^^pY^cdr are the elementary torque and
thrust on each blade during axial motion.
AIESCREWS
327
The mean values given at the foot of Table 20 show that the average
variations of ST, 8Q, 8Z and SN as a result of non-axial motion are very-
small as compared with the average thrust and torque on the element.
The lateral force 8Y is about 4 per cent, of the thrust in this example,
whilst the pitching couple 8M is about 32 per cent, of the torque. These
mean figures apply to any number of blades. For variations on two
blades during rotation the last six columns of Table (20) should be inverted
and the figures added to those there given. For thrust and torque the
effect is to leave small differences at all angles. The same appHes to the
normal force 8Z and the yawing couple SN. For the lateral force SY and
the pitching moment SM the effect is to double the figures approxi-
mately, and these then compare with double thrust and torque.
TABLE 20.
0
degrees.
Cosfl
Sin e
J^ -1-23
,f*^ ^ -0-525
fiY
pT^edr
fiZ
pVcdr
5M
pW'CTiir
8N
pyerdr
0 and 360
1-000
0
-0-33
-0-11
-Oil
0
0-33
0
20 „ 340
0-940
0-342
-0-32
-0-11
-0-10
-0-03
0-30
-0 09
40 „ 320
0-766
0 643
-0-25
-0 08
-006
-0-05
0-19
-0-16
60 „ 300
0-500
0-866
-0-17
-0-05
-0-02
-0-04
0-08
-015
80 „ 280
0-174
0-985
-0-05
-001
-000
-0-01
0-01
-0 05
100 „ 260
-0-174
0-985
+0-05
+0-02
0-00
+0-02
001
+0-05
120 „ 240
-0-500
0-866
+0-17
0-06
-0-03
+0-05
0-08
+0-15
140 „ 220
-0-766
0-643
+0-27
0 09
-0-06
0-05
0-20
+0-17
160 „ 200
-0-940
0-342
+0-34
0-11
-0-09
0-03
0-32
0-11
180
-1-000
0
+0-35
0-12
-0-12
0
0-35
0
Mean 0-01
Mean 0-003
Mean
Mean
Mean
Mean
or less than
or about i%
-0-05
0-002
0-17
0-002
1% of 1-23
of 0-525
or about
4% of
1-23
or about
0-2o/„of
1-23
or about
32o/o of
0-525
or about
0-4% of
0-525
For a four-bladed airscrew the averaging of SY and SM would be
appreciably better, since four colunms displaced by 90° in 6 would then
be added to provide the resultant.
An angle of 10° as here used may easily occur in the normal range of
horizontal flight of an aeroplane, the displacement of velocity being then
in the vertical plane. The necessary changes of notation between Y and
Z, M and N can readily be made. For lateral stability the present notation
is most convenient.
Integration from Element to Airscrew. — The repetition of the pre-
V
ceding calculations for a number of elements and values of — provides
all the data necessary for determination of the torque, thrust, lateral force,
etc., on an airscrew.
It has been seen that for an element most of the effects of non-axial
motion are unimportant and attention will be directed to the evaluation
of Y and M. The symmetry of the figures of Table 20 and their general
appearance suggests the apphcability of simple formulae, so long as the
angle of yaw does not exceed 10°.
328 APPLIED AERODYNAMICS
Consider equation (61) when r^cos 61 — lis small enough for expansion
of the factor containing it with squares and higher powers neglected. As
^ depends on this quantity, it is necessary to expand (59) to get
"V ( V "i
tan^« = (l+ai) — \1 -] cos ^>'
oiri cor '
= tan ^o(l +^.^ cose) . . . (67)
Since as a general trigonometrical theorem
, / , , V tan <f>f, — tan ^o
tan (0^ - ^o) = T , r . . }
1 + tan ^p tan <f>Q
V
and since the numerator is seen from (67) to be of first order in — the ex-
pression becomes v « , # V
^ cos 0 . tan <f>o . —
V COT
tan i(f>v-<f>o) = (<f>v - ^o)approx. = -^Tl • (^^)
and (68) gives an expression for 8<l> due to change v cos 6.
To obtain the variations of T differentiate (61) with respect to <f),
retaining only terms of first order in :^. Substitute for 8<f), i.e. {<f>v—<l>o)
from (68).
1 -dT n^ O 9 J /7 J 7 • / \ t^*"
pyTc • ^ df "^ "~ V ^^^ ^<*^^ "^^^ "^^ ~ ° ^^^ "^^^ ' Y
V V
+( If ) 5-y ] 2 sec2 00 tan ^0(^1 cos ^o -^d sin <^o)
^ V / seo^ (pQ V.
4- sec2 0o(— kj, sin <f>o — kj, cos '^o — ^ cos <f>o + -j^ sin 0o)|
aa cos 9o *
The formula above may be apphed to the previous example where
for the element
00 = 20^ a = 2° ^
74
. . . (70)
V
— = 0-340
u)r
1=0-174
k^ = 0-275
k^ = 0-0141
^''^=2-26
da
t'^"^ 0-116
da J
AIESCBEWS 329
With these values equatiqii (69) leads to
1 rZT
pT^.-«* = -«-«'^'=»^« • • • • (")
For each blade the numerical factor should be halved before comparison
with column 4, Table 20. The simple expression, (71), gives results in
good agreement with those of Table 20.
From(69)and(65)the expression for M for theairscrewmaybewrittenas
^2
Jo
^.a-\~ fc,(cos io + ~) + Ji sin ^0 - ^ sin ^«
^.sin^K _ _ _
da cos ^0 ^
M
The average value of M is half the maximum. The value of -^
vp\
depends on the advance per revolution, chiefly because of the variation of
V
fet with — . The relation is not so simple as to be obviously deducible
cor
from (72), since the important terms change in opposite directions.
Treating the torque equation (63) in a similar way to that followed for
thrust will give the lateral force
-y2^-S^= -2|:cos^.^.sec2^o (fei sin ^0 + ^ cos ^o)
+ ^ cos ^ • ^ • ^^^{2 sec2 ^0 tan (f>Q{kj, sin <f>o + kj, cos <^o)
+ sec^ <f}Q( — ^ sin 00 — ^ cos (f>Q + kj, cos <^q — fe© sin 0o)(
(or V /)( 7 • » 7 / 1 , / \ dhf, sin^^o
= -^ . ^ cos 9] — kj, sm^o — M —jr + cos ^o ) — -5^ p
V V ^ i- rw Vcos^o -^ da cos 00
-=^ sm
da
in0o| . . (73)
With the values given in connection with pitching moment, equation (73)
leads to -, jr\
_i_.8^ = -0-22 cos « .... (74)
pW dr ^ '
and for a single blade the numerical factor should be halved. Compared
with column 5 of Table 20 the results will again be found to be in
good agreement.
The value of the lateral force Y on the whole airscrew is
D
^^ J 7, ^;^ JL 7- Z' 1 I ^^« JL \ dkj, sin^ <f>Q
.] — kj, sin 00 — h>( J- + cos <f>o) . ,
V ( NCOS 00 ^ da cos 00
-^8in0o|dr. . . (76)
and the average value is again half the maximum.
330
APPLIED AEEODYNAMICS
Experimental Determination o£ Lateral Force on an Inclined Airscrew.
— The experiments which led to the curves of Fig. 162 were obtained on
a special balance in one of the wind channels of the National Physical
Laboratory. The airscrew was 2 feet in diameter, but the results have
been expressed in a form which is independent of the size of the airscrew
in accordance with the principles of dynamical similarity.
The ordinates of the curves of Fig. 162 are the values of the lateral force
on the airscrew divided by pV^D^ except for one cur^e which shows the
thrust divided by pV^D^ to one-tenth its true scale. The number of degrees
shown to the left of each of the curves indicates the angle at which the
airscrew axis was inclined to the direction of relative motion.
D.04
nD
Fia. 162. — Lateral force on inclined airscrew.
The values of the ordinates' for the different angles of yaw will be
found to be nearly proportional to the ratio of lateral velocity to axial
V V
velocity, i.e. to - . The change of lateral force coefficient with -=r is small
Y
at high values of -^ , and in all cases the ratio of lateral force to thrust
° nD
increases greatly as -^r increases.
As an example of the magnitude of the lateral force for flying speeds
take at maximum speed
AIESCREWS 881
D = 9ft., V = 160ft..s.(109m.p.h.), -^ = 0*75 and angle of yaw = 10*
The lateral force is 48 lbs., and the thrust 655 lbs.
At the speed of cHmbing
V = 100 ft.-s. (68 m.p.h.), ^ = 0-50
nD
The lateral force is 23 lbs., and the thrust 815 lbs.
V. The Stresses in Airscrew Blades
The more important stresses in an airscrew blade are due to bending
under the combined action of air forces and centrifugal forces and the
direct effects of centrifugal force in producing tension. Both types of
stress are dealt with by straightforward applications of the engineer's
theory of the strength of beams. Recently, attention has been paid to
torsional stresses and to the twisting of the blades, but the calculations
require more elaborate theories of stress. The progress made, although
considerable, has not yet had any appreciable effect on design, and the
importance of torsional stresses is not yet accurately estimated. A further
series of calculations deals with the resonance of the natural periods of an
airscrew blade with periods of disturbance, and one general theorem of
importance has been deduced. It states that the natural frequency of
vibration of an airscrew blade must be higher than its period of rotation,
and that as a consequence resonance can only occur from causes not con-
nected with its own rotation.
The calculation of stresses due to bending and centrifugal force will
be dealt with in some detail, but torsion and resonance will not be further
treated. As a general rule, it may be said that the evidence in relation
to airscrews of normal design is that the twisting is not definitely
discernible in the aerodynamics, but appears occasionally in the splitting
of the blades. The flexure of the blade under the influence of thrust
is sufficient to introduce an appreciable couple as the result of the
deflection and centrifugal force.
Bending Moments due to Air Forces.— The blade of an airscrew is
twisted, and the air forces acting on it at various radii have resultants
lying in different planes. As each section is chosen of aerofoil form one
of the moments of inertia of the section is small as compared with the other,
and it is sufficient to consider the bending which occurs about an axis of
inertia through the centre of area of a section and parallel to the chord.
The resolution of the air forces presents no particular difficulty and the
details are given below. All the air forces on elements between the tip
of a blade and the section chosen for calculation enter into the bending
moment, and it is necessary to have a distinguishing notation for different
sections. For this purpose dashes have been added to letters to signify
use in connection with the base element for which the moment is being
calculated.
The formulae required follow in most convenient form from the ex-
pressions for thrust and torque, as these admit of ready addition for the
332
APPLIED ABEODYNAMICS
various sections. The thrust element is a force always normal to the
airscrew disc, whilst the torque element Ues in a plane parallel to the air-
screw disc.
If <f>Q be the inclination of the chord of the base element to the airscrew
disc at radius r', then the elementary addition to the moment due to the
forces at radius r is
dM.=^(dT COS <f>'o + ^ sin <l>'oyr-r') . . . (76)
and using the expressions for dTl and — which are given in equations (1)
and (2), (76) becomes
dT
dM.=-
or
cos {<f> + y)
dM dT
{cos <f)Q cos {<f> —y) + sin ^o sin (^ + y)\{r —r')
d\
Q <i)
p /2r_2r'\
2 'VD D/
2r'\ cos(0 + y-«^^)
cos (0 + y)
(77)
The value of
t?T
can be obtained by differentiation of equation (46),
(78)
and using the value so obtained, equation (77) becomes
1 J^^JL /^?r_2^L n , ^NCOs(^ + y-0^)
pV^Ds • ^/2r\ ~ 8Ai ' VD D "^^ "^ ^' cos {<f> + y)
and M is obtained by integration between the proper limits as
pV^m - SA^J^^ ^'^^ + "i\d ~ b) ' cos(0 + y) \W ^^^^
In the form shown in (79) the expressions inside the integral are easily
evaluated from the earher work on the aerodynamics of the airscrew, and
V
the important quantities for one value of -yz are collected in Table 21
below.
TABLE ai.— ^=0-60.
nD
2r
D
aid + Oi)
from Table 9.
*o
«
*> + y
Cob (0 + y)
0-960
0-0360
171
11-6
14-7
0-967
0-880
0 0900
18-3 »
13-22
15-2*
0-965
0-760
0-1160
19-6
15-6
18-7
0-947
0-602
0-1300
23-8
19-6
22-9
0-921
0-412
01186
32-3
27-1
31-4
0-854
0-324
0-0875
38-6
32-4
37-0
0-799
From Table 8.
* By interpolation in Table 8.
yfrom Table 11.
AIRSCEEWS
383
2r
For the particular values of =- chosen, the whole of column 2 will be
2r
found reproduced from Table 9. The value of (f>Q for =- = 0*880 is given in
Table 8 as 18'3 degrees, and the other values were taken from the
similar tables not reproduced. Similar remarks apply to ^ and ^ + y
as shown at the foot of the table, and the last column of Table 21 is
obtained from trigonometrical tables.
TABLE 22.
1
2
8
4
5
6
ar"
2r ar'
(degrees).
2x3x5 to give
D
Old + 0,)
D~ D
COB(0 + V-0o')
element of integral
equation (79).
00360
0-636
-23-9
0-907
0-0208
0 0900
0-556
-23-4
0-911
0-0456
0-324 .
0-1160
0-436
-19-9
0-940
00476
0-1300
0-278
-15-7
0-963
0-0348
0-1185
0-088
- 7-2
0-992
0-0103
00875
0-000
- 1-6
1-000
0
'
0-0360
0-548
-17-6
0-953
0-0188
00900
0-468
-17-1
0-956
0-0402
0-412 -
0-1160
0-348
-13-6
0-972
0-0392
01300
0190
- 9-4
0-987
0 0244
6-1185
0 000
- 0-9
1000
0
0-0360
0-358
- 91
0-987
0-0127
0-602
0-0900
0-278
- 8-6
0-989
0-0247
01160
0-158,
- 61
0-996
0-0182
0-1300
0-000
- 0-9
1-000
0
j
0 0360
0-200
- 4-9
0-997
0-0071
0-760
0 0900
0-120
- 4-4
0-997
0-0108
,
0-1160
0-000
- 0-9
1000
0
0-880 1
0-0360
0-080
- 3-6
0-997
0-0029
0-0900
0 000
- 3-1
0-998
0
The elements of the integral of (79) are calculated as in Table 22 from
values extracted from Table 21 . The processes are simple and call for no
special comment. The values in column 6 of Table 22 are plotted as
2r
ordinates in Fig. 163, with ^ as abscissae. The areas of these curves give
D
2r'
the values of the integral at the various values of -jt-, and were obtained
8Ai
M
by the mid-ordinate method. The values, which represent t)3V2'
are shown in the curve marked "moment f&ctor" in Fig. 163. Since
the air forces on the blade near the centre are small the curve tends
to become straight as the radius decreases and for practical purposes
may be extrapolated in accordance with this observation. The values
of the integral, denoted by F( ^^ ) ^^® shown in Table 24, but before
834
APPLIED AEEODYNAMICS
use can be made of them to calculate stresses it is necessary to estimate
the area, moment of inertia and distance to outside fibres for each of the
\
\
\
0.05
o.o-<»-
\
s.
MOM
lNT Fi
iCTOR
1
MOMENT F/
^
\
/
^
"""•^
8Xi
IT
ciM
\
<°
y
{)
/^
^^
\
0.03
0.02
OD^V^
\
0 4-12
c/
r
0.€02 -^y
Mi
/
A
•^v >
/ O 760-^
\i
O.OI
o
/
/
z
:::2
Aaeo^
0.03
0.02
O Ol 0.2 0.3 0.4- 0.5 0.6 0 7 0 8 09 lO
Zr
D
Fig. 163. — Calculation of bending stresses due to thrust.
aerofoil sections used. The values are given in Table 23 in terms of the
chord so as to be applicable to airscrews of different blade widths.
TABLE 23.
1
2
3
4
Number of
aerofoil.
Area of section.
Minimum moment of
inertia (axis assumed
parallel to chord).
Distance of
extreme fibre
from axis.
1
2
3
4
6
6
0-05901"
0-0590i"
0-073ci2
0-09101"
0-136Ci"
0-180ci*
O-0O0027Oi*
0 -0000270 1*
0-0000510,*
0-OOOlOOc,*
0-0003250, «
0-0007650,4
0-049Ci
0-0490i
0-061Ci
0-076Ci
0-113Ci
0-150ci
In Table 23, Cj is used to denote the chord to distinguish it from c,
which is the sum of the chords of all the blades. If the number of blades
is two, the value of M given by (79) should be halved, whilst for four blades
one quarter of the value should be taken. Using the ordinary engineer's
expression, the maximum stress due to bending is
. M
(80)
AIESCEEWS
885
where I is the moment of inertia and y is the distance to the extreme
fibre. Using (80) in conjunction with (79) and denoting the integral of
(79) by f(?^), leads to
i-k^<n^)
K
(81)
he being the coefficient of Cj in column 4 of Table 23, and ki the coeffi-
cient of Ci* in column 3. The c of equation (81) has its usual meaning
as the sum of the chords of the blades. Evaluation of (81) leads to
Table 24.
TABLE 24.
1
2
1
3
4
5
6
7
2r
D
e
D
1 ^(1)
/lbs. per sq.ft.
Compressive
stress,
lbs. per sq. in.
^pV=
Tensile stress,
lbs. per gq. in.
0-960
0-036
1
0-0000
1800
0
0
0
0-880
0-098
0-0002
1800
380
600
400
0-760
0-137
0-0016
1180
730
1160
760
0-602
0-163
'• 0-0060
760
1050
1660
1100
0-412
0-164
0-0154
288
1000
1580
1050
0-324
0147
00204
196
1260
2000
1330
2r
The values of — and ^ of Table 24 are taken directly from Table 7.
Fi
2r\
2r
. is the ordinate of the curve in Fig. 163 at the proper value of —
The fifth column of Table 24 follows
and ^ is deduced from Table 23.
K
from the figures in the previous column and equation (81). Before the
results can be interpreted numerically it is further necessary to know
pV2 and columns 6 and 7 are calculated for p=0-00237 and V=147 ft.-s.
(100 m.p.h.). For the proportions of section chosen the tensile stress due
to bending is two-thirds of the compressive stress.
The stresses increase rapidly from the tip inwards for the first quarter
of the blade, and then more slowly, the highest value shown being 2000 lbs.
per square inch for the section nearest the centre for which calculations
have been made.
It is important to note that the stress in the airscrew has been calculated
without fixing its diameter. Since, in the calculations shown, -^ is fixed
nD
by hypothesis, the choice of V is equivalent to a choice of nD, and the
stress depends on either V or nD. The latter quantity for airscrews
of different diameters is proportional to the tip speed, and hence the
conclusion is reached that for the same tip speed and value of -=- the stress
nu
836 APPLIED AEEODYNAMICS
in similar airscrews is constant. This theorem will be shown to apply in
a wider sense than its present application to bending stresses due to air
forces.
Centrifugal Stresses. — The stress due to centrifugal force depends on
the mass of material outside the section considered, on the distance to its
centre of gravity, and on the angular velocity. As most airscrews are
sohd it is convenient to use the weight of unit volume, and this will be
denoted by w. For a splid airscrew the weight of the part external to the
section at radius r' is
W = w;
r\ci^dr
J f'
./2r
D
fci is defined as the coefficient of Ci^ in column 2 of Table 23.
The centrifugal force on an element at radius r is
- . fc^ci^dr . (27m)^ (83)
and the total force at the section r' is
C.F.- (277)2- r"/c^Ci2n2^Jr (§4)
The stress on the section is
This stress can be expressed in terms of the generalised variables found
convenient in the previous calculations, and (85) leads to
Stress due to centrifugal force (lbs. per sq. ft.)
The note already made in regard to bending stresses, that the stress
depends only on the tip speed for similar airscrews working at the same
V .
value of -yv > IS seen to apply equally to the direct stress arising from
centrifugal force.
The value of the integral of (86) is obtained as shown in Table 25
and Fig. 164.
27* c
— and ^ are taken from Table 24 and k. from Table 23. Columns 4
D D ^
and 5 then follow by calculation, and feAff^ ) i^ plotted as ordinate with
AIRSCEEWS 337
"- ] as abscissa. The integral was obtained by the mid- ordinate method
of finding areas, the value of the integral being zero at the tip of the blades
0.004-
O 003
0.002
o.ooi
\]
^
^^
y-^
(£f ■
/**
4%r
\
\
^,
s
^
^
■i
a(d;
^D
'^^
^_
^
■^
.^
/€ /^
O 002
OOOI
O O.I 02 0.3 0.4- 05 0.6 O 7 O.Q 0.9 1.0
m
Fig. 164. — Calculation of centrifugal stresses.
2r
where fr = 1 • From the curve for the integral the values in column G
of Table 25 were read off.
TABLE 25.
i
1 2
3
4
5
6
7
2r
D
D
K
Mb)"
(I)'
Value of integral
of equation (86).
Stress (lbs.
per sq. in.).
0-960
0-880
0-760
0-602
0-412
0-324
0-036 0-069
0-098 0059
0-137 1 0-073
0-163 ; 0091
0-164 0-135
0-147 0-180
1
0-000072
0-000537
0-00137
0-00241
0-00363
0-00389
0-920
0-774
0-577
0-362
0-170
0-105
0-000004
0 000050
0-000220
0-000630
0-00122
0-00145
15
250
430
700
900
1000
With
V = 147 ft.-s.
V
wD
= 0-60
and
t(; = 42 lbs. per cubic foot (walnut).
the direct stress due to centrifugal force can be calculated from equation
(86) and the figures of Table 25. The stress is of course tensile, and is
additive to the stress calculated and shown in column 7 of Table 24.
The combined stress is 2300 lbs. per sq. in., and 3000 lbs. per sq. in. is
not regarded as an excessive value for walnut. This value would be
reached for a somewhat higher value of nD.
886
APPLIED AERODYNAMICS
Bending Moments due to Eccentricity of Blade Sections and Centrifugal
Force. — It will be seen shortly that as a result of centrifugal forco the
bending moments arising from small eccentricity of the airscrew sections
from the airscrew disc are of appreciable magnitude. The eccentricities
considered will be of comparable size with those produced by the deflection
of the blade under the action of thrust. The calculations are somewhat
complex, and will be illustrated by a direct example which assumes the
values of the eccentricities. The more practical problem involves processes
of trial and error for complete success.
As the area of the section of a blade at radius r' is \h'i^{c')^ the centri-
fugal force obtained from equation (86) is
D
Consider now the couples acting due to centrifugal force ; if from some
pair of fixed axes the co-ordinates of the centres of area of each section
be given as x and y, the perpendicular distance, 'p, from any one of these
centres of area on to the axis of least inertia of another is
2? = (a; — x') cos <t>Q^{y — y') sin 4>q . . .
and the resultant moment at the section denoted by dashes is
(88)
w
McF _7r2 1
pPD3~'8'p*^
(f)' r Kif"~"'^ ^^^^'o^^y-y') ^^"jo^dy (89)
The form of (89) has been chosen for convenience of comparison with
equation (79).
Given x and y as functions of ( ..^ ) , the value of Mgy can be calculated
from (89) and data previously given.
TABLE 26.
1
2
3
4
5
©■
'Aff
Cos (^Q.
X
D
V
D
0-920
0000072
0-956
0.00920
0
0-774
0 000537
0-949
000774
0
0-677
0-00137
0-943
0-00577
0
0-362
0 00241
0-915
0 00362
0
0-170
0 00363
0-845
0 00170
0
0-105
0 00389
0-782
0-00105
0
As an example the values of =- have been taken as the one-hundredth
part of ( Y^j . On a 12 ft. 6 ins. diameter airscrew the eccentricity due to
AIRSCREWS
339
design and deflection under load would be 1 '5 ins. at the tip of the blades.
Eccentricities of greater amount may easily occur in practice. The value
of y has been taken as zero everywhere. Table 26 shows the data necessary
for the calculation of moments from equation (89). The details are given
below in Table 27.
TABLE 27.
1
2
3
0-000072
4
5
6
1
Cos 0o'
. x — x'
D
Element of
integral of (89).
0-920
0-105
0-782
-000815
-0-46 X I0-«
0-774
J,
0-000537
,,
-0-00669
-2-80
0-577
»»
000137
,,
-0-00472
-5-05 „
0-362
»»
0-00241
,,
-0-00257
-4-85 „
0-170
>>
0-00363
>>
-0-00065
-1-84 „
0-105
0-00389
»»
0
0
0-920
0-170
0-000072
0-845
-000750
-0-45 X 10-8
0-774
jj
0-000537
»
-0-00604
-2-74 „
0-557
jj
0-00137
>»
-0-00407
-4-70 „
0-362
it
0 00241
»»
-0-00192
-3-90 „
0-170
0-00363
0
0
0-920
0-362
0-000072
0-915
-0-00558
-0-37 X 10-«
0-774
t>
0-000537
,,
-0-00412
-2-03 „
0-577
»»
0-00137
,,
-0-00215
-2-70 „
0-362
0 00241
»>
0
0
0-920
0-577
0-000072
0-943
-0-00343
-0-23 X 10-8
0-774
J>
0 000537
>>
-000197
-1-00 „
0-577
0-00137
>>
0
0
0-920
0-774
0-000072
0-949
-0-00146
-0-10 X 10-«
0-774
0-000537
»»
0
0
The values given in column 6 of Table 27 are plotted as ordinates in
Fig. 165 with \j-J as abscissa. For each value of ( ^ ) there is a separate
curve, the area of which is required. Found in the usual way these areas
are plotted to give the " integral " curve of Fig. 165.
To show the results in comparison with those for bending due to thrust
as shown in Table 24 the value of — ^ . ^.o^o has been calculated and
tabulated in Table 28.
TABLE 28.
2r
8Aj Mx
8A| McF
D
, ir pV-iD"
ir pV^D^
0-960
00000
-00000
0-880
00002
-0-00005
0-760
0-0016
-0 0005
0-602
0-0060
-0-0018
0-412
0-0164
-0-0041
0-324
0-0204
-0-0049
340
APPLIED AERODYNAMICS
The first two columns of Table 28 are taken from the first and third
columns of Table 24. The third column of Table 28 is calculated from
equation (89) and the integral curve of Fig. 165.
The example chosen had the tip of the airscrew forward of the boss,
and the bending moment is opposed to that due to the thrust. Roughly
speaking, the effect of the centrifugal force is one quarter that of the thrust,
and had all the values of x been increased four times, the residual bending
moment due to thrust and centrifugal force would have been very small
at all points. Appropriate variation of x would lead to complete elimina-
tion, but trial and error might make the operation rather long. It is
-6
FROM COL 6
TABLE 27
- 2
INTEGRAL OF
EQUATION 89
-a. OX ID
Fig. 165. — Calculation of bending moments due to centrifugal force.
only possible to get complete balance for moments and so eliminate
V
flexural distortion for one value of -y-, and in practice a compromise would
be necessary. It is not, however, quite clear that the possibiHty of
eliminating moments is a useful one in practice, since airscrews are built
up of various laminae with glued joints. In order to keep these joints in
compression deviations from the condition of no flexural distortion are
admitted. All that can be done in a treatise of this description is to point
out the methods of estimating the consequences of any such compromise
as is made in the engineering practice of airscrew design.
It may be noticed here that the effect of distortion under thrust is to
reduce the stress below that calculated on the assumption of a rigid blade.
The problems connected with the calculation of the deflection and
twisting of airscrew blades are more complex than those given, and have
not received enough attention for the results to be applicable to general
practice. In this direction there are opportunities for both experimental
and mathematical research.
AIKSCBEWS 341
.Formulae for Airscrews suggested by Considerations of Dynamical
Similarity
In the course of the detailed treatment of airscrew theory it has been
V
found that -=r is a convenient variable. It has also been seen that the
nD
density of the air and of the material of the airscrew are important. In
discussing the forces on aerofoils it was shown that both the viscosity and
elasticity of the air are possible variables, whilst consideration of the
elasticity of the timber occurs as an item in the calculation of deflections
and stresses.
It may then be considered, in summary, that the variables worth
consideration an
V ^ the forward velocity of the airscrew.
n ^ the rotational speed.
D ^ the diameter.
p ^ the air density.
— ^ the densitv of the material of the airscrew.
d ." . .
a ^ the velocity of sound in air as representing its elasticity.
E f^ Young's modulus for the material of the airscrew.
All the quantities, thrust, torque, efficiency, stress and strain then
depend on a function of five variables, of which
j(V ™,Y,l.f, E). .... .(00)
^nD V a p g p\^'
V
may be taken as typical. The first argument, -^r, is of great importance
and is the most characteristic variable of airscrew performance. If care
is taken in choosing a sufficiently large model aerofoil and wind speed
VD V
the variable — may be ignored. — becomes important only at tip
speeds exceeding 600 or 700 ft.-s., but complete failure occurs at 1100
1 w
ft.-s. if this variable is ignored. The argument simply states that
the ratio of the density of the material of the airscrew to that of the
air affects the performance. Since thrust depends primarily on p and
centrifugal force on -, it is obvious that moments and forces from
g
the two causes can only be simply related if be constant. A similar
^ ^^ p g
■pi
conclusion is reached in regard to ^=^
The density and elasticity of the materials of which airscrews are made
are rarely introduced into the formulae of practice. Where the material
342 APPLIED AEEODYNAMICS
is wood the choice has been between walnut and mahogany, and neither
the density nor elasticity are appreciably at the choice of the designer.
Some progress has been made with metal airscrews, and the stresses causing
greatest difficulty are those leading to buckling of the thin sheets used. In
order to reduce the weight of a metal airscrew to a reasonable amount it is
obvious that hollow construction must be used and that similarity of design
cannot cover both wood and metal airscrews. Some very special materials
such as " micarta " have been used in a few cases, and since the blades
are soHd and homogeneous, the arguments from similarity might be apphed
with terms depending on density and elasticity. (" Micarta " is a pre-
paration of cotton fabric treated with cementing material.)
The common forms of expression used are
Thrust =3 pn2D*Pi(X) (91)
Torque ^pw^D^Fsf^) (92)
Efficiency = ^^(^ (^^)
Stress = pn2D2F4(-^) (94)
Prom (94) follows the statement that for similar airscrews working at
the same value of -j- the stress depends on the tip speed of the airscrew,
and is otherwise independent of the diameter. The numerical values of
Pj and Fs are usually given under the description of absolute thrust and
torque coefficients respectively.
CHAPTER VII
FLUID MOTION
Experimental Illustrations of Fluid Motion ; Remarks on Mathe-
matical Theories op Aerodynamics and Hydrodynamics
Forces on aeroplanes and parts of aeroplanes are consequences of motion
through a viscous fluid, the air, and if our mathematical knowledge
were sufficiently advanced it would be possible to calculate from first
principles the lift and drag of a new wing form. No success has yet been
attained in the analysis of such a problem from the simplest assumptions,
and recourse is at present made to direct experiments. The viscosity of
air is always important in its effect on motion, and as the effect depends
on the size of the object it will be necessary to discuss the conditions
under which aircraft may be represented by models. The relation
between fluid motions round similar objects is so important that a
separate chapter is devoted to it under the head "Dynamical Similarity."
It will be found that for most aerodynamics connected with aeroplane
and airship motion air may be regarded as an incompressible fluid.
The present chapter contains material on fluid motion which throws
some light on the resistance of bodies. It also covers, in brief resume, the
existing mathematical theories, indicating their uses and hmitations, but
no attempt is made to develop the theories of fluid motion beyond the
earliest stages, as they can be found in the standard works on hydro-
dynamics. For experimental reasons the photographs shown will refer to
water. It will be found that a simple law will enable us to pass from
motion in one fluid to motion in any other, and the analogy between
water and air is illustrated by a striking example under the treatment of
similar motions.
Whilst it is true that the fluid motions with which aeronautics is
directly concerned are unknown in detail there are nevertheless some
others which can be calculated with great accuracy, the discussion of
which leads to the ideas which explain failure to calculate in the general
case. Fig. 166 represents a calculable motion, and when the mathematical
theory is developed later in the chapter it is carried to the stage at which
Fig. 166 is substantially reproduced. The photograph was produced by a
method due to Professor Hele-Shaw who kindly proffered the loan of his
apparatus for the purpose of taking the original photographs of which
Figs. 166, 171, 176-178, are reproductions.
The apparatus consists of two substantial plates of glass separated from
each other by cardboard one or two hundredths of an inch thick. In Fig.
343
844 APPLIED AEEODYNAMICS
166 the shape of the cardboard is shown by the black parts, the centre
being a circular disc, whilst at the sides are curved boundaries. The space
between the boundaries is filled with water, the motion of which is caused
by applying pressure at one end. To follow the motion when once started
small jets of colour are introduced well in front of the disc and before the
fluid is sensibly deflected.
Steady Motion. — ^After a little time the bands of clear and coloured
water take up the definite position shown in Fig. 166, and the picture
remains unaltered, so far as the eye can judge, although the fluid continues
to flow. When such a condition can be reached the final fluid motion is
described as "steady." The point of immediate interest is that the shape
of all the bands can be calculated (see p. 355). The mathematical analysis
of the problem of flow in these layers was first given by Sir George Stokes,
and an account of the theory will be found in Lamb's "Hydrodynamics."
Except for a region in the neighbourhood of the disc and boundaries the
accuracy of calculation would exceed that of an experiment. Near the
solid boundaries, for a distance comparable with the thickness of the film,
the theory has not been fully applied.
It is, of course, perfectly clear that there is nothing in the neighbour-
hood of the wheel axle of an aeroplane, say, which corresponds with the
two plates of glass, and Fig. 166 cannot be expected to apply. It is difficult
to mark air in such a way that motion can be observed, but it is possible
to make a further experiment with water by removing the constraint of
the glass plates. Even at very low velocities the flow is " eddying "' or
" unsteady," and a long exposure would lead to a blurred picture. To
avoid confusion a cinema camera has been used, and the life-history of an
eddy traced in some detail in Figs. 167-170. The colouring matter in
Fig. 167 is Nestle's milk, and the flow does not at any stage even faintly
resemble that shown in Fig. 167. With eddying motion the colour is
rapidly swept out of the greater part of the field of view, and only
remains dense behind the cyhnder where the velocity of the fluid is very
low. The eddjdng motion depicted in Fig. 167 is yet far beyond our
powers of mathematical analysis, but a considerable amount of experi-
mental analysis has been made, and to this reference will be made almost
immediately.
The water flows from right to left, and the cylinder is shown as a circle
at the extreme right of each photograph. The numbers at the side represent
the order in which the film was exposed, and an examination shows a
progressive change running through the series of photographs. Starting
from the first, it will be seen that a small hook on the upper side grows
in size and travels to the left until it reaches the limit of the photograph
in the sixteenth member of the series. By this time a second small hook
has made its appearance and has about the same size as that in 1. Some
of the more perfect photographs occur under the numbers 18-24, and show
clearly the simultaneous existence of four hooks or " eddies " in various
stages of development and decay. The eddies leave the cylinder alternately
on one side and then on the other, growing in size as they recede from
the model.
Fig. 166. — Viscous flow round disc (Hele-Shaw).
Fig. 171. — Viscous flow round strut section (Hele-Shaw).
Fig. 172. — Viscous flow round strut section (free fluid).
FiQ. 167.— Eddies behind cylinder (N.P.L.).
FLUID MOTION
846
Unsteady Motion. — The root ideas underlying the unsteady motion of
a fluid are far less simple than those for steady motion. Figs. 167-170 all
refer to the same motion, and yet there is little evident connection between
the figures. An attempt will now be made to trace a connection, and we
start with the definitions suggested by the illustrations.
Stream Lines. — In an unsteady motion the position of each stream line
depends on the time. In all cases with which we are concerned in aero-
dynamics the position of the stream lines in the. region of disturbed flow
repeats at definite intervals, i.e. the flow is periodic. The period in Fig. 167
can be seen to extend over 13 or 14 pictures. In producing Fig. 168 the
flow was recorded by the motion of small oil drops, and no less than eighty
periods were observed. The cinematograph picture for the beginning of
each period was selected and projected on a screen whilst the lines of flow
^ FLOW
Fig. 168. — Instantaneous distribution of velocity in an eddy (N.P.L.).
were marked, and Fig. 168 is the result of the superposition of 80 pictures.
Had the accuracy of the experiment been perfect none of the lines so plotted
would have crossed each other. As it is, the crossings do not confuse the
figure until the eddies have broken up appreciably.
If now one proceeds to join up the lines so that they become continuous
across the picture, the result is the production of stream lines. Stream
lines have the property that at the instant considered the fluid is every-
where moving along them.
Fig. 169 shows the general run of the stream lines at intervals of one-
tenth of a complete period. Only five diagrams are shown, since the
remaining five are obtained by reversing the others about the direction of
motion ; Fig. 169 (/) would be like Fig. 169 (a) turned upside down, and soon.
Most of the stream lines follow a sinuous path across the field, but occasion-
ally bend back upon themselves (Fig. 169 (a)). Two partsmay then approach
346
APPLIED AEEODYNAMICS
each other and coalesce so as to make a closed stream line. The bend of
Fig. 169 (b) is seen in Fig. 169 (c) to have become divided into a small
CO
FLOW
(d)
Fig* 169, — Stream lines in an eddy at different periods of its life (N.P.L.).
closed stream line and a sinuous line through the field. The process is
continued between Figs. 169 (d) and 169^(e), where two closed streams are
FLUID MOTION
347
shown, and so on. These closed streams represent vortex motion, and as
the vortices travel down-stream they are somewhat rapidly dissipated.
Fig. 168 shows that the velocity inside the vortex is small compared
with that of the free stream.
Paths of Particles. — Fig. 170 shows the paths followed by individual
particles across the field of view. Unlike " stream lines " " paths of
particles " cross frequently. Some of the particles were not picked up
by the camera until well in the field of view. In one case (the lowest of
Fig. 170) a particle had entered a vortex and for four complete turns travelled
slowly against the main stream, which it then joined. The upper part of
Fig. 170 shows a series of paths varying from a loop to a cusp, for particles
all of which had passed close to the cylinder.
Fig. 170. — Motion of particles of fluid in an eddy (N.P.L.).
To produce these curves it was only necessary to expose the plate in a
camera during the passage of a strongly illuminated oil drop across the
field. Since observation of all oil drops across the field gives both stream
lines and paths of particles, one set of pictures must be deducible
from the other. Before paths of particles can be obtained by calculation
from the stream lines of Fig. 169 the velocity at each point of the stream
hnes must be deduced. Draw a line AB across Fig. 169 as indicated; the
quantity of fluid flowing between each of the stream lines being known,
the number representing this quantity can be plotted against distances of
the stream lines from A. The slope of the curve so obtained is the velocity
at right angles to AB. Since the resultant velocity is along the stream
line the component then leads to the calculation of the resultant velocity.
The calculation is simple, but may need to be repeated so many times as
to be laborious in any specified instance of fluid motion. For the present
we only need to see that Fig. 169 gives not only the stream lines but the
velocities along them.
348 APPLIED AERODYNAMICS
From Fig. 169 we can now calculate the path of a particle. Starting
at C, for instance, in Fig. 169 (a), a short line has been drawn parallel to the
nearest stream line. This line represents the movement of the particle in
the time interval between successive pictures. In the next picture the
point D has been chosen as the end of the first and another short line
drawn, and so on, the whole leading to the line CG of Fig. 169 (e). Further
application of the process would complete the loop. The line CG is illus-
trative only, since the velocity along each of the stream lines was not
calculated ; it is sufficient to show the connection between the lines of
Fig. 169 obtained experimentally and those of Fig. 170, also deduced from
the same experiment.
There are two standard mathematical methods of presenting fluid
motion which correspond with the differences between " stream lines "
and " paths of particles."
Filament Lines. — Filament lines have been so called since they are the
instantaneous form taken by a filament of fluid which crosses the field of
disturbed flow. They are the lines shown in Fig. 167. The colouring matter
of Fig. 167 was introduced through small holes in the side of the cylinder. •
The white lines therefore represent the form taken by the line joining all
particles which have at any time passed by the surface of the cyhnder.
They could be deduced from the paths of particles by isolating all the
paths passing through one point, marking on each path the point corre-
sponding with a given time and joining the points.
In experimental investigations of fluid motion it is important to bear
in mind the properties of filament lines when general colouring matter is
used. The use of oil drops presents a far more suitable line of experimental
research where attempt is made to relate experimental and mathematical
methods.
Although eddying motion is very common in fluids, it is not the
universal condition in a large mass. Two examples will be given of a com-
parison between steady free flow and the flow illustrated by Prof. Hele-
Shaw's experiments. The question will arise, does the method of flow
between plate glass surfaces indicate the only type of steady flow ? There
is, of course, no obvious reason why it should. As a further example of
Prof. Hele-Shaw's method of illustrating fluid motion, the case of a strut
section will be considered (Fig. 171 opposite p. 344). It will be noticed
that the streams were quite gently disturbed by the presence of the
obstruction. If we consider the fluid moving between the stream lines
and the side qf the model, it will be noticed that the streams, which are
widest ahead of the model, gradually narrow to the centre of the strut and
then again expand. The fact that the coloured bands keep their position
at all times means that the same amount of fluid passing between any
point of a stream line and the strut must also pass inside all other points
on the same stream line, and because of the constriction the velocity will
be greatest where the stream is narrowest and vice versd.
It is interesting to compare Fig. 171 with another figure illustrating the
flow of water round a strut of the section used for Fig. 171, the flow not being
confined by parallel glass plates. The stream lines in Fig. 172 are shown as
Fig. 173. — Eddying motion behind short strut (N.P.L.).
EiG. 174. — Eddying motion behind medium strut (N.P.L.).
Fig. 175. — Eddying motion behind long strut (N.P.L.).
FLUID MOTION 849
broken lines, the lengths of which represent the velocity of the fluid. The
flow will be seen to consist of streams with the narrowest part near the
nose, and from that point a steadily increasing width until the tail is reached.
The gaps in the stream lines are produced at equal intervals of time, and
their shortening near the strut shows the effect of the viscous drag of the
surface.
The general resemblance between Figs. 171 and 172, which relate [to
struts, is in marked contrast with the difference between Figs. 166 and 167
for cylinders. When measurements are made in a wind channel of the forces
acting on struts and on cyhnders, it is found that to this difference in the
flow corresponds a very wide difference in resistance. A cylinder will have
10 to 15 times the resistance of a good strut of the same cross- sectional area.
On examining the photographs given in Fig. 167 a region will be found
immediately behind the cylinder which is not greatly affected in width
during the cycle of the eddies. Just behind the body the water is almost
stationary and is often spoken of as " dead water." In the case of the
cylinder illustrated, the dead water is seen to be somewhat wider than the
diameter of the cylinder itself. Figs. 173-175 show further photographs
of motion round struts in free water; in Fig. 173 the "dead water "is shown
to be as great as for a cylinder, the strut being very short. The longer
strut of Fig. 174 is distinctly less liable to produce the dead water, whilst
a further reduction is evident on passing to the still longer strut. Fig. 175.
The photographs were taken in water, and it does not necessarily follow
that they will apply to air without a discussion which is to come later,
but it is of immediate interest to compare between themselves the resistances
of a cylinder and three struts under conditions closely approaching those
of use in an aeroplane. The relative resistances are given in Table 1 .
TABLE 1. — ^The Resistance of Cyondees and Steuts.
Model.
Belative resistance.
Cylinder, Fig. 167 .
Strut, Fig. 173 . .
„ Fig. 174 . .
„ Fig. 175 . .
6
3-5
1-2
10
The general connection between the size of the " dead-water " region
and the air resistance is too obvious to need more than passing comment.
The more aerodynamic experiments are made, the more is it clear that
high resistance corresponds with a large dead-water region, and perhaps
the most satisfactory definition of a " stream-line body " is that which
describes it as " least liable to produce dead water."
If, now, a return be made to Fig. 166 — Prof. Hele-Shaw's photograph
of flow round a cylinder — it will be seen that there is neither " dead water "
nor " turbulence," and the mathematical analysis leads to the conclusion
that if the plates be near enough together no body would be sufficiently
360 APPLIED AEEODYNAMICS
blunt and far removed from " stream line " to produce eddying motion.
The influence at work to produce this result is the viscous drag of the
water over the surface of the two sheets of plate glass. It is obvious
without proof that this viscous drag will be greater the closer the surfaces
are to each other, and that on moving them far from each other this essential
constraint is reduced. It is not equally obvious that an increase of velocity
of the fluid between the plates has the effect of reducing the constraint,
but on the principles of dynamical similarity the law is definite, and ad-
vantage is taken of this fact in producing Figs. 176 and 177, which show
different motions for the same obstacle.
The photographs, taken by Professor Hele-Shaw's method, show the
flow round a narrow rectangle placed across the stream in a parallel-sided
channel. The thickness of the water film was made such that at low
velocities it was only just possible to produceFig. 176, which shows streams
behind the rectangle which are symmetrical with those in front. Without
changing the apparatus in any way the velocity of the fluid was very
greatly increased and Fig. 177 produced. In front of the obstacle careful
examination of the figures is necessary in order to detect differences between
Figs. 176 and 177, but at the back the change is obvious. The first points
at which the difference is clearly marked are the front corners of the rect-
angle. The fluid is moving past the corners with such high velocity that
the constraint of the glass plates is insufficient to suppress the effects of
inertia. The fluid does not now close in behind the obstacle as before,
and an approach to " dead water " is evident. There is a want of definition
in the streams to the rear which seems to indicate some mixing of the
clear and coloured fluids, but there is no evidence of eddying. We are thus
led to consider three distinct stages of fluid motion.
(1) Steady motion where the forces due to viscosity are so great that
those due to inertia are inappreciable.
(2) Steady motion when the forces due to viscosity and inertia are both
appreciable; and
(3) Unsteady motion, and possibly steady motion, when the inertia
forces are large compared with those due to viscosity.
The extreme case of (3) is represented by the conventional inviscid fluid
of mathematical theory where the forces due to viscosity are zero. It is
not a little surprising to find that the calculated stream lines for the steady
motion of an inviscid fluid are so nearly like those obtained in Professor
Hele-Shaw's experiments as to be scarcely distinguishable from them. It
needed a mathematical analysis by Sir George Stokes to show that the very
different physical conditions should lead to the same calculation. The
common calculation illustrates the important idea that mathematical
methods developed for one purpose may have applications in a totally
different physical sense, and the student of advanced mathematical
physics finds himself in the possession of an important tool applic-
able in many directions. This is, perhaps, the chief advantage to be
obtained from the study of the motion of a conventional inviscid fluid.
Before considering the theory, one further illustration from experiment
will be given.
Fig. 176. — Viscous flow round section of flat plate (Hele-
Shaw). Low speed.
Fig. 177. — Viscous flow round section of flat plate (lleie-
Shaw). Highspeed.
Fig. 178. — Viscous flow round wing section (Hele-Shaw).
FlQ. 179. — Viscous flow round wing section (free fluid).
'SI
FLUID MOTION 851
Wing Forms. — The motion round the wing of an aeroplane probably
only becomes eddying when the angle of incidence is large, and the re-
sistance is then so great as to render flight difficult. At the usual flying
angles, there is some reason to believe that the motion is " steady." Two
further photographs. Figs. 178 and 179, one by Prof. Hele-Shaw's method
and the other by the use of oil drops, show for a wing section two steady
motions which differ more than appeared for the struts.
If Fig. 178 be examined near the trailing edge of the aeroplane wing, it
will be noticed that the streams close in very rapidly. At a bigger angle
of attack it would be obvious that on the back there is a dividing point in
one of the streams. At this dividing point the velocity of the fluid is zero,
and such a point is sometimes called a " stagnation point." A second
" stagnation point " is present on the extreme forward end of the wing
shape.
In the freer motion of a fluid, such as that of air round a wing, the
forward " stagnation " point can always be found, but the second or rear
"stagnation point" is never recognisable. The effect of removing the
constraint of the glass plates will be seen by reference to Fig. 179, although
this does not accurately represent the flow at high speed on a large wing.
The slowing up of the stream by the solid surface, which was noticed for
the strut, is again seen in the case of the wing model.
Elementary Mathematical Theory of Fluid Motion
Frictionless Incompressible Fluid. — In spite of the fact that other and
more powerful methods exist, it is probably most instructive to start the
study of fluid motion from the calculations relating to " sources and sinks."
In his text- book on Hydrodynamics, Lamb has shown that the more
complex problems can all be reduced to problems in sources and sinks.
The combinations may become very complex, but methods relating to
complex sources and sinks are developed in a paper by D. W. Taylor,
Inst. Naval Architects, to which reference should be made for details.
A " source " may be defined as a place from which fluid issues, and
a " sink " a place at which fluid is removed ; either may have a simple
or complex form.
Consider Fig. 180 (a) as an illustration of a simple source, the fluid from
which spreads itself out over a surface parallel to that of the paper. The
thickness of this fluid may be conveniently taken as unity, the assumption
being that it forms part of a stream of very great thickness.
If m be the total quantity of fluid coming from the source, the "strength"
is said to be m. A corresponding sink would emit fluid of amount —m.
Since the fluid is equally free in all directions, symmetry indicates a
continuous sheet of fluid issuing from the centre, and ultimately passing
through the circular section CPG. Whether on account of instability the
flow would break into jets or not we have no means of saying at present,
but it should be remembered that the " continuity " of the fluid
throughout the region of fluid motion is definitely an assumption. Such
a physical phenomenon as " cavitation " in the neighbourhood of
862
APPLIED AEEODYNAMIOS
propeller blades in water would be a violation of this assumption. As
cavitation arises from the presence of points of very low pressure, it
is clear that even in a hypothetical fluid no solution can be accepted
for which the pressure at any point is required to be enormous and
negative. An instance of this occurs in relation to one of the solutions
for the motion of an inviscid fluid round a plane surface.
Assuming continuity and incompressibility for the fluid, it is obvious
Fig. 180. — Fluid motions developed from sources and sinks in an inviscid fluid.
that the velocity of outflow across the circle CPG will be uniform, and
calling the velocity v we have
27Trv=im (1)
m
or
V =
2rrr
(la)
so that the velocity becomes smaller and smaller as the distance from the
source increases.
For the motion of any inviscid incompressible fluid whatever, there is
a relation between the pressure and velocity at any point of a stream line.
The equation, proved later, is extremely useful in practical hydrodynamics,
and is one particular form of BernoulK's equation. It states that
P + Ip^^ = const (2)
where p is equal to the mass of unit volume of the fluid. We have seen that
the stream lines in Fig. 180 (a) are radial lines, and from (la) it appears that
FLUID MOTION 353
V ultimately becomes zero when r becomes very great ; this is true for all
the stream lines impartially. If in (2) the value of v be put equal to zero
when r is very great, it will be seen that the " const." on the right-hand
side is the pressure of the stream a long way from the source, and since
this is the same for all stream lines it follows that (2) gives a relation
between p and v for any point whatever in the fluid. The same proposition
is true for all motions of frictionless incompressible fluids if the " const."
does not vary from one stream to the next. Most problems come within
this definition. Equation (2) is only true for an inviscid incompressible
fluid, and cannot be applied with complete accuracy to any fluid having
viscosity.
Stream Function. — It has been shown that the total quantity of fluid
moving across the circle CPG is m. The same quantity obviously flows
across any boundary which encloses the source. It is convenient to have
an expression for the quantity of fluid which goes across part of a
boundary. The " stream function " which gives this is usually repre-
sented by tp. It is clear that the same quantity of fluid flows across any
line joining two stream lines, and the change of «/» from one stream line
to another is therefore always the same, no matter what the path taken.
It follows from this that along a stream line tp = const.
In arriving at this conclusion, it will be remarked that the only assump-
tions made are that the fluid fills the whole space and is incompressible.
It need not be inviscid.
In the particular case of the source of Fig. 1 80 (a) it is immediately obvious
g
that the amount of fluid flowing across the line CP is equal to jr-wi, and
it is usual to write
^=-2^" •('"
for the value of «/> which corresponds with a source of strength m, the
negative sign being conventional. If m be suitably chosen, the diagram
of Fig. 180(a) maybe divided up by equal angles such that t/i=0 along OG,
^=1 along OP, 0=2 along OC, and so on. Any line might have been called
that of zero i/j, as in all calculations it is only the differences between the
values of i// which are of importance.
Fig. 180 (&) shows the drawing of stream lines for a combination of simple
source and sink. Two sets of radial lines, similar to Fig. 180 (a), are drawn,
and these produce a series of intersections. For the case shown, equal
angles represent equal quantities of flow for both source and sink. If the
strengths had been unequal, the angles would have been proportioned so
as to give equal flow, i.e. the lines are lines of constant ^ differing by
equal amount from one line to its successor.
If lines be drawn from O to Oi through the points of intersection of the
stream lines in the way OAOi and OBOi are drawn, the lines so obtained
are the stream lines for a source and sink of equal strength. Lines drawn
through the points of intersection along the other diagonals of the ele-
mentary quadrilaterals would give the stream lines for two equal sources.
2 A
354 APPLIED AEEODYNAMICS
The assumption has here been made that the effect of a sink on the
motion is independent of the existence of the source, and vice versa. The
assumption is legitimate for an inviscid fluid, but does not always hold for
the viscous motions of fluids ; it is proved without difficulty that any
Mumber of separate possible inviscid fluid motions may be added together
to make a more complex possible motion.
Addition of Two Values of i/r. — The construction given in Fig. 180 (b) can
be seen to follow from the statement that two separate systems may be
added together to produce a resultant new system. The group of radial
lines round 0 is numbered in accordance with the scheme of Fig. 180 (a), and
represents values of 0 for a source. For the.sink-a set of numbers are
arranged round Oj, the sink being indicated by the fact that the numbers
increase when travelling round the circle in the opposite way to that for
increasing numbers round the source. If we call j/r^ the value for the source
and 02 the value for the sink, the addition gives t/ji + tjjo for the combina-
tion, or
«A = 0i + 'A2 (5) .
As a stream line is indicated by ^ being constant, we may write
j^^ -|- jjtg = const (5«)
and by giving the " const." various values the new stream lines can be
drawn. As an example, take " const." =^ 31, and consider the point A of
Fig. 180 (h) ; the line from the source through this point is 0i=25, and from
the sink 02='6, or i/fi+j/r2=31. At E, 0i=26, ip2—^> 0i-|-^2=31- Hence
both A and E are on a stream line of the new system. The advantage
of the method lies in the ease with which it can be extended, and to one
such extension it is proposed to call immediate attention.
A steady stream of fluid will be superposed on the source and sink
of Fig. 180 (h). The stream lines for this are equidistant straight lines, and
they will be taken parallel to OOi. It can easily be shown that the curves
of Fig. 180 (b) are circles, but this would only be true for a simple sourceand
sink, and not for a case presently to be discussed. The method of procedure
is not confined to such a simple source and sink. If parallel lines be drawn
on a sheet of tracing paper which is then placed over the lines from source
to sink, a set of intersecting lines will again be formed of which the diagonals
may be drawn to form the new system ; the result is indicated in Fig. 180 (c).
The result is interesting ; an oval-shaped stream in the middle of the
figure separates it into two parts. Inside there are stream lines passing
from source to sink, and outside streams passing from a great distance on
one side to a great distance on the other. As the fluid is frictionless, the
oval may be replaced by a solid obstruction without disturbing the stream
lines, and the method of sources and sinks may then be used to develop
forms of obstacles and the corresponding flow of an inviscid fluid round
them.
By the addition of the velocities of the fluid due to source, sink and
translation separately by the parallelogram of velocities, the resultant
velocity of the fluid at any point round the oval can be obtained. The
FLUID MOTION
365
direction of this resultant must be tangential to the oval at the point
because it is a stream line. Once the magnitude of the resultant velocity
has been obtained, equation (2) will give the pressure at the point. From
the symmetry of Fig. 180 (c), back and front, it is clear that the pressures will
be symmetrically distributed, and there will be no resultant force on the
oval obstacle. The theorem is true that no body in an inviscid fluid can
experience a resistance due to a steady rectilinear flow of the fluid past it,
unless a discontinuity is produced.
Flow of an Inviscid Fluid round a Cylinder. — It has already been
remarked that the stream lines in Professor Hele-Shaw's method can
be calculated, and it is proposed to make one calculation (graphically).
The method of sources and sinks is used, not because the fluid is inviscid
Fig. 181. — Calculated flow round circular disc for comparison with Fig. 166.
but because the equation of motion in Professor Hele-Shaw's experiment
happens to agree with that for an inviscid fluid.
If the source and sink of Fig. 180 (h) be brought nearer and nearer together,
the circles showing the stream lines will become more and more like the
larger ones there shown, and ultimately when the source and sink almost
coincide the circles will be tangential to the line joining them. They then
take the form shoA\Ti in the lower half of Fig. 181, the radii being inversely
proportional to the value of tjt.
On to these stream lines superpose those for a uniform stream and
draw the diagonals. Instead of the oval of Fig. 180 (c) the closed curve
obtained is now a circle, and three of the stream lines have been drawn
on the lower half of the figure. The upper half of the figure was completed
in this way with a larger number of stream lines, and alternative streams
356
APPLIED AEKODYNAMICS
were filled in so that the figure might bear as much resemblance as possible
to the photographs shown by Professor Hele-Shaw. The result is some-
what striking.
The Equations of Motion of an Inviscid Fluid. — Eeaders are referred
to the text-books on Hydrodynamics for a full treatment of the subject as
applied to compressible fluids and the effects of gravity, and attention will
be limited to the c'ases outlined in the previous notes.
Suppose that Fig. 182 represents a steady motion in the plane of the
paper. Isolate a small element between two stream lines and consider the
forces acting on it, which are to be such that it will not change its position
with time although filled with new fluid. The force on the elementary
block is due to pressures over its four faces and the difference between
Fro. 182.
the momentum entering by the face AD and that leaving by BC. If the
block is not to move the resultant of these two must be zero.
Forces in the Direction of Motion. — If 'p be the pressure on AD. that
on BC will be p 4- ~ds, and along the faces AB and DC the pressure will
ds
be variable. The resultant of the uniform pressure f over all' the faces is
zero, and the total force against the arrow is therefore
d'p
ds'
ds . dn
(4)
if we neglect quantities of relatively higher order. The mass of fluid
passing AD and BC per unit time is the same and is equal to pvdn, where p
is the density of the fluid and v its velocity. The momentum entering is
then pv^dn, and that leaving is pvdn(v -\ — ds\ and the difference is
FLUID MOTION 357
pv—.dsdn (5)
ds ^
in the direction of the arrow, and therefore exerting a force in the opposite
direction on the element. The force equation is made up of (4) and (5),
and is
l+^vH •. («)
Equation (6) is easily integrated and gives
P + Ip'^^ = const (7)
Equation (7) is very important, and often applies approximately to
the motion of real fluids.
Forces Normal to the Direction of Motion. — If r be the radius of the
path, the centrifugal force necessary to keep the block from moving out-
wards is p— . dnds, whilst the difference of pressure producing this force is
r
— . dnds, and hence the equation of motion at right angles to the direction
dr
of motion of the fluid is
p'-^^. .' (8)
^ r dn ^ '
Substitute from (7) for -^, and (8) becomes
dn
^ + ^ = 0 (9)
In dealing with sources and sinks equation (9) was assumed to hold,
and it is now seen that the assumption was justified, since r is infinite and
-- is zero along each of the stream lines.
dn ^
dv
If the radius of stream lines be infinite, equation (9) shows that ^
must be zero, i.e. the velocity must be uniform from stream to stream.
Equation (7) then shows that jp is constant. The converse is of course
true, that uniform pressure means uniform velocity and straight stream
lines.
Comparison of Pressures in a Source and Sink System with those on
a Model in Air. — The calculations and experiments to which reference
will now be made are due to G. Fuhrmann working in the Gottingen
University Laboratory. The general lines of the calculations follow those
outlined, but the source and sink system is not simple. The models,
instead of being long cyhnders as in the cases worked out in previous
pages, were solids of revolutions, but the transformations on this account
are extremely simple. The complex sources and sinks are obtained by
858
APPLIED AERODYNAMICS
integration from a number of elementary simple sources and simple sinks
and present little difficulty. For details, reference should be made to the
original report or to the paper by Taylor already mentioned.
The original paper by Fuhrmann contains the analysis and experi-
mental work relating to six models of the shape taken by airship envelopes.
Some of these shapes had pointed tails, whilst one of them had both pointed
head and tail. The investigation was carried out in relation to the
development of the well-known Parseval airship, and the model most
like the envelope of that type of dirigible is chosen for the purpose of
illustration. Starting with various sources and sinks the flow was calcu-
lated by method? similar to those leading to Fig. 180, but needing the
Fig, 183. — Calculated flow of inviscid fluid round an airship envelope.
application of the integral calculus for their simplest expression. The
type of source chosen for the model in question is illustrated by the sketch
above Fig. 183. The sink begins at C, gets stronger gradually to D and
then weaker to B ; at this latter point the source begins and grows in
strength to A, when it ceases abruptly.
The complex source and sink so defined are reproduced in Fig. 183, the
upper half of which shows the stream lines due to the system. The resem-
blance to circular arcs is slight. Superposing on these streams the
appropriate translational velocity Fuhrmann found the balloon-shaped
body indicated, together with the stream lines past it. These stream lines
are shown in the lower half of Fig, 183. The model has a rounded head
a little distance in front of the source head A and a pointed tail, the tip
of which coincides with the tip C of the sink.
Having obtained a body of a desired character, Fuhrmann proceeded
FLUID MOTION 359
to calculate the pressures round the model in the way indicated in relation
to Fig. 180 (c), using the formula j9 + |p«2 — const. ; the results are shown
plotted in Fig. 184, and are there indicated by the dotted curve. The
pressure is highest at the extreme nose and tail, and has the value ^fyv^,
where v is the free velocity of the fluid stream far from the model. Near
the nose the pressure falls off very rapidly and becomes negative long before
the maximum section is reached, and does not again become positive until
within a third of the length of the model from, the tail. The calculated
resistance of the balloon model is zero.
For comparison with the calculations, experiments were made. Models
were constructed by depositing copper electrolytically on a plaster of Paris
mould, the shape being accurately obtained by turning in a lathe to care-
fully prepared templates. The modds were made in two or three sections^
these being joined together after the removal of the plaster of Paris. As
a result a light hollow model was obtained suitable for test in a wind tunnel.
To measure the pressures, small holes were drilled through the copper, and
the pressure at each hole
measured by connecting
the interior of the model
to a sensitive manometer.
Finally, the total force on
the model was measured
directly on an aerody-
namic balance. For the
elaborate precautions
taken to ensure accuracy
the origmal paper should j^q^ 184.— Comparison of calculated and observed
be studied ; SOmerecent re- pressure on a naodel of an airship envelope.
searches suggest a source
of error not then appreciated, but the error is of secondary importance,
and the results may be accepted as substantially accurate.
The observed pressures are plotted in Fig. 184 in full lines, the black
dots indicating observations. The first point to be noticed is that at the
extreme nose the maximum pressure is ^pv^ as indicated by the calculation,
and that good agreement with the calculation holds until the pressure
becomes negative. From this point the observed negative pressures are
appreciably greater than those calculated, whilst at the tail the positive
pressure is not so great as one-tenth of that calculated. The total force
due to pressure now has a distinct value, which Fuhrmann calls " form
resistance." The method of pressure observation does not, of course,
include the tangential drag of the air over the model. The total resistance,
including tangential drag or " skin friction " and " form resistance," was
found by direct measurement, and it was found that " skin friction "
accounted for some 40 per cent, of the whole, and " form resistance "
for the remainder.
The effect of friction in the real fluid is therefore twofold : in the first
place the flow is so modified that the pressure distribution is altered, and
in the second the force at any point has a component along the surface
360
APPLIED AEEODYNAMICS
of the model ; both are of considerable importance in the measured total
resistance. From the analogy with flat boards towed with the surfaces
in the direction of motion, so that the normal pressures cannot exert a
retarding influence, the tangential drag is generally referred to as " skin
friction." It will be seen that appreciable error, 50 per cent, or 60 per
cent., would result if the pressure distribution were taken to be that of an
inviscid fluid.
Six models in all were tested in the air-channel at Gottingen, and the
results are summarised in the following table : —
TABLE 2. — ^The Form Resistance and Skin Friction of Airship Envelopes.
Number of model.
Fraction of resistance
caused by the change
of pressure distribution
arising from the viscosity
of the fluid
(form resistance).
Fraction of resistance
caused by the tangential
forces arising from tlie
viscosity of the fluid
(skin friction).
Relative total
resistances.
1
2
4
6
6
0-67
0-53
0-53
0-63
0-59
0-69
0-43
0-47
0-47
037
0-41
0-31
1
1-22
1-20
0-79
0-87
0-81
The general conclusion which might have been drawn is that for forms
of revolution of airship shape the resistances are more dependent on form
resistance than on skin friction. This conclusion should be accepted with
reserve in the light of more recent experiments.
The experiments referred to above were all carried out at one speed.
Measurements were made of the total resistance at many speeds, but there
are no corresponding records of pressure measurements. A series of tests
on a model of an airship envelope has been carried out at the N.P.L. at
a number of speeds with the following results : —
TABLE 3. — ^Variation of Form Resistance and Skin Friction with Speed
Speed (ft.-s.) . . .
20
■ 30
40
50
60
Form resistance
I
0-90
0-61
0-59
0-56
Skin friction . . .
I
0-89
0-89
0-84
0-84
Form resistance
0-23
0-23
017
017
0-16
Total resistance
f
From the last row of Table 3, it will be seen that the form resistances
are far smaller fractions of the total at all speeds than those given in
Table 2. Further examination of the original figures shows that the
measurements of total resistance at the N.P.L. are very much the
same in magnitude as those at GGttingen. No suggestion is here put
forward to account for the difference, the experiments at various speeds
having an interest apart from this. It will be noticed that both the
FLUID MOTION 361
" form resistance " and the " skin friction " vary with speed, and in the
particular illustration the variation of the pressure is the greater. This
evidence is directly against an assumption sometimes made that the
pressure on a body varies as the square of the speed whilst the skin
friction increases as some power of the speed appreciably less than two.
There is certainly no theoretical justification for such an assumption,
as will be seen later, and many practical results could be produced to
show that experimental evidence is against such assumption.
One other illustration -of the variation of pressure distribution with
speed, may be mentioned here. A six-inch sphere in a wind of 40 ft.-s.
has a resistance dependent almost wholly on the pressure over its surface,
but this resistance is extremely sensitive to changes of speed ; the curious
result is obtained that for certain conditions a reduced resistance accom-
panies an increase of speed. A corresponding effect is produced by covering
with sand the smooth surface formed by varnish on wood. At about the
speed mentioned the resistance may be decreased to less than half by such
roughening. The general aspects of the subject are dealt with under the
heading of Dynamical Similarity. For the present it is only desired to
draw attention to the fact that the law of resistance proportional to* square
of speed is not accurately true for either the pressure distribution on a body
in a fluid or for the skin friction on it. The departures are not usually so
great that the v^ law is seriously at fault if care is taken in application.
A fuller explanation of this statement will appear shortly, when the
conditions under which the v^ law may be taken to apply with sufficient
accuracy for general purposes will be discussed.
Cyclic Motion in an Inviscid Fluid. — In the fluid motions already dis-
cussed, the flow has been obtained from a combination of a motion of
translation and the efflux and influx from a source and sink system. The
initial assumptions involve as consequences —
(a) Finite slipping of the fluid over the boundary walls ;
(h) No resultant force on the body in any direction ;
(c) A liability to produce negative fluid pressures.
No theory has yet been proposed, and from the nature of an inviscid fluid
it would appear that no theory could exist which avoids the finite
slipping over the boundary. It appears to be fundamentally impossible
to represent the motion of a real fluid accurately by any theory relating
to an inviscid fluid. It is not, however, immediately obvious that such
theories cannot give a good approximation to the truth, and as claims in
this direction have often been made, further study is necessary before any
opinion can be formed as to the merits of any particular solution.
The difficulties (h) and (c) can be avoided by introducing special
assumptions ; two standard methods are developed, one involving " cyclic "
motion and the other " discontinuous " motion.
Leaving the second of these for the moment, attention will be directed
to the case of " cyclic " motion of an inviscid fluid. A simple cyclic
motion can perhaps best be described in reference to a simple source. In
the simple source the stream lines were radial and the velocity outwards
varied inversely as the radius. In a simple cyclic motion the stream lines
362
APPLIED AEEODYNAMICS
are concentric circles, the velocity in each circle being inversely proportional
to the radius.
From the connection between pressure and velocity it will be seen that
the surfaces of uniform pressure in a cyclic motion and in motion due to
a simple source are the same.
As in the case of sources and sinks, complex cyclic motions could be
produced by adding together any number of simple cyclic motions. Cyclic
and ^n-cyclic motions may also be added.
€ nsider the effect of superposing a cyclic motion on to the flow of an
inviscid fluid round a body, say a cylinder placed across the stream ; before
the cyclic motion is added the stream lines are those indicated in Fig. 166 ;
add the cyclic motion as in Fig. 185.
The angles AOP, DOP, BOQ and COQ having been chosen equal, the
symmetry of Fig. 166 shows that the velocities there will be equal
for the upper and lower parts of the cylinder. These velocities are
indicated by short lines on the circle, the arrow-head indicating the
direction of flow.^ Since the
pressure in an inviscid fluid
is perpendicular to the sur-
face it can easily be seen that
the pressures, all being equal
and symmetrically disposed,
have no resultant. Superpose
a cyclic motion which has its
centre at 0, and which adds
a velocity at the surface re-
presented by the lines just
outside the circle ABCD. On
the upper half of the cylinder,
the cyclic motion adds to the
velocity and adds equally at A and B, Below, the velocity is reduced or
possibly reversed, but the resultant has the same value at C and D.
From the relation between pressure and velocity given in equation (2)
the deduction is immediately made that pa and pi, are less than pa and p^,
and a simple application of the parallelogram of forces then shows that a
resultant force acts on the cylinder upwards. The result is somewhat
curious, and may be summarized as follows : if a cylinder is moved in a
straight line through an inviscid fluid which has imposed upon it a cyclic
motion concentric with the cylinder, there will be a force acting on the
cylinder at right angles to the path, but no resistance to the motion.
If the body had been a wing form, it appears that the resultant force
would not then have been at right angles to the line of motion, and there
would have been a resistance component.
Kutta in Germany and Joukowsky in Eussia have developed the
mathematics of cyclic motion in relation to aerofoils to a great extent.
Starting from a circular arc, Kutta calculates the lift and drag for various
angles of incidence, and compares the results with those obtained in a
wind tunnel. Before giving the figures it is desirable to outline the basis of
Fig. 185. — Cyclic flow round cylinder.
FLUID MOTION
863
the calculation a little more closely. If by the source and sink method the
flow round the circular arc ABC (Fig. 186) is investigated when the stream
comes in the direction PQ, it is found that one stream line (shown dotted)
coming from P strikes the model at D where the velocity is zero and there
divides, one part bending back to A and then round the upper surface to F,
whilst the other part takes the path DCF round the aerofoil. The point
F where the two parts reunite is a second place of zero velocity, and from
F to Q the speed increases, ultimately reaching the original valuf The
Fig. 186. — Cyclic flow round circular arc
points D and F have been referred to previously as " stagnation points."
The velocity of the fluid at A and C is found to be enormous, and so to
require negative fluid pressure. This violates one of the conditions im-
posed by any real fluid.
By adding a cyclic motion it would appear to be possible to move the
stagnation points D and F towards A and C, and if this could be done
completely the fluid would come from P^, strike the arc tangentially at A,
and there divide finally leaving the arc tangentially at C. All very great
velocities would then be avoided.
Kutta showed that it is always possible to find a cyclic motion which
will make F coincide with C, no matter what the inclination of the chord
Fia. 187. — Cyclic flow round wing section.
of the arc might be. He did not, however, succeed in making D coincide
with A as well as F with C except when a was equal to zero. In that
particular case the aerofoil, according to calculation, gave lift without drag
just as we have seen was the case for a cylinder. To meet the difficulty
as to enormous velocity of fluid at A, Kutta introduced a rounded nose-
piece ; Joukowsky by a particular piece of analysis showed how to obtain
a section having a rounded nose and pointed tail which solved the mathe-
matical difficulties and made it possible to find the cyclic flow round a body
of the form shown in Fig. 187, such that the stream leaves C tangentially.
There is then no difficulty in satisfying the requirements as to absence
864
APPLIED AERODYNAMICS
of negative pressure at any angle of incidence whatever for a limited range
of velocity.
TABLE 4. — ^Ktttta's Table a Oompabison of Calotilatbd and Meastjebd Forces.
Measured lift
Calculated lift
Drag
Excess of drag
per unit area
over that at 0°.
Lllienthal
(kg/m^).
Calculated drag
Inclination of
per unit area.
per unit area.
per unit area.
per unit area.
cliord a.
Lllienthal
Kutta
Lllienthal
Eutta
(kg/m'').
(kg/m2).
(kg/mO.
(kg/m^).
- 9°
0-20
0-72
0-90
0-60
0-78
- 6°
1-74
2-45
0-54
0-24
0-36
- 3°
3-25
4-30
0-36
0-06
0-09
0
4-96
6-23
0-30
0 00
0 00
+ 3"
7-27
8-21
0-37
0-07
0-10
+ 6°
9-08
10-20
0-70
0-40
0-39
+ 9°
10-43
1216
1-12
0-82
0-88
+ 12°
11-08
14-06
1-51
1-21
1-56
+ 15°
11-52
15-86
1-95
1-65
2-44
The table of figures by Kutta is given above. The experiments referred
to were probably not very accurate, and the disagreement of the calculated
and observed values of lift and drag is not so great as to discredit the
theory. It may be noticed that the calculated drag has been compared
with the excess of the observed drag above its minimum value, and so
throws no light on the economical form of a wing. The theory cannot
in its existing form indicate even the possibility of the well-known critical
angle of an aeroplane wing. It is not possible to justify the assumptions
made, and the result is a somewhat complex and not very accurate
empirical formula.
Discontinuous Fluid Motion. — The simplest illustration of the meaning
of discontinuous motion is that presented by a jet issuing into air from an
orifice in the side of a tank of water. If the orifice is round and has a sharp
edge the water forms a smooth glass-like surface for some distance after
issuing. After a little time the column breaks into drops, and Lord
Rayleigh has shown that this is due to surface tension ; further, if the jet
issues horizontally the centre line is curved due to the action of gravity,
whilst if vertical an increase of velocity takes place which reduces the
section of the column.
Neglecting the effects of gravity and surface tension, a horizontal jet
would continue through the air with a free surface along which the
pressure was constant and equal to that of the atmosphere. The
method of discontinuous motion is essentially identified with the mathe-
matical analysis relating to constant pressure, free surfaces. The
examples actually worked out apply to an inviscid fluid and almost
exclusively to two-dimensional flow. Lamb states that the first example
was due to Helmholtz, and it appears that the method of calculation was
made regular and very general by Kirchhoff and Lord Eayleigh. The
main results have been collected in Report No. 19 of the Advisory Com-
mittee for Aeronautics by Sir George Greenhill, and since that time ex-
tensions have been made to curved barriers.
p
FLUID MOTION 866
It is not proposed to attempt any description of ihe special methods
of solution, but to discuss some of the results. The first problem examined
by Sir George Greenhill is the motion of the fluid in a jet before and after
impinging on an inclined flat surface. The jet coming from I, Fig. 188,
impinges on the plate AA' and splits into two jets, the separate horns of
which are continued to J and J'. One stream line IB comes up to the
barrier at a stagnation point B, and then travels along the barrier A'A in
the two directions towards J and J'. Finite slipping is here involved, and
the analysis must therefore be looked on as an approximation to reality
only. In the case of jets it appears to be justifiable to assume that the
effect of viscosity on the fluid motion and pressures is very small compared
with that arising from the usual resolutions of momentum, and so far as
experimental evidence exists, it suggests that the motiqp of jets worked
out in this way is a satisfactory indication of the motion of a real fluid
such as water, when issuing into another much less dense fluid such as air.
Fio. 188. — Discontinuous motion of a jet of fluid.
From I to J, from I to J', and from A' to J', A to J the fluid is bounded
by free surfaces along which the pressure is constant. From equation (2)
this will be seen to imply the condition that the velocity is constant ;
further, if the free surface extends to great distances from the barrier, the
velocity all along it must be that of the fluid at such great distances.
Solutions of discontinuous motions almost always involve the assumption
that the velocity along the free surfaces is that of the stream before dis-
turbance by the barrier.
Fig. 168, already referred to, shows behind a cylinder a region of almost
stagnant fluid the limits of which in the direction of the stream are very
sharply defined, and it is clear that in real fluids, in addition to the periodi-
city, there is indication of the existence of a free surface. Direct experi-
ments show that inside such a region the pressure is often very uniform,
but appreciably below that of the fluid far from the model.
Assuming a free surface enclosing stagnant fluid extending far back
from the model the whole details of the pressure, position of centre of
pressure, and shape of stream lines for an inclined plate have been worked
366 APPLIED AERODYNAMICS
out. In addition to finite slipping at the model, there is now also finite
slipping over the boundary of the stagnant fluid and objections on the score
of stability have been raised, notably by Lord Kelvin. The following
summary of the position is given by Lamb : —
" As to the practical value of this theory opinions have differed. One
obvious criticism is that the unlimited mass of ' dead-water ' following
the disk implies an infinite kinetic energy ; but this only means that the
type of motion in question could not be completely established in a finite
time from rest, although it might (conceivably) be approximated to asymp-
totically. Another objection is that surfaces of discontinuity between
fluids of comparable density are as a rule highly unstable. It has been
urged, however, by Lord Rayleigh that this instability may not seriously
affect the character of the motion for some distance from the place of origin
of the surfaces in question.
"Lord Kelvin, on the other hand, maintains that the types of motion
here contemplated, with surfaces of discontinuity, have no resemblance
to anything which occurs in actual fluids ; and that the only legitimate
application of the methods of von Helmholtz and Kirchhoff is to the case
of free surfaces, as of a jet,"
With the advance of experimental hydrodynamics, and since the
advent of aviation particularly, the position taken by Lord Kelvin has
received considerable experimental support ; one instance of the difference
between the pressure of air on a flat plate and the pressure as calculated
is given below. It is clearly impossible to make an experiment on a flat
surface of no thickness, and for that reason the experimental results are not
strictly comparable with the calculations • in addition, the conditions were
not such as to fully justify the assumption of two-dimensional flow. Never-
theless the discrepancies of importance between experiment and calculation
are not to be explained by errors on the experimental side, but to the
initial assumptions made as the basis of the calculations.
The experiments were carried out in an air channel at the National
Physical Laboratory, and are described in one of the Reports of the Ad-
visory Committee for Aeronautics. The abscissae, representing points at
which pressures were observed, are measured from the leading edge of the
plane as fractions of its width. The scale of pressures is such that the
excess pressure at B over that at infinity would just produce the velocity
V in the absence of friction. It appears to be very closely true, whether
the fluid be viscous or inviscid, that the drop of pressure in the stream line
which comes to a stagnation point is ^pv^. There are other reasons, which
will appear in the discussion of similar motions, for choosing pv^ as a basis
for a pressure scale.
In the experiment the pressure of +ip^^ is found on the underside of
the inclined plane, very near to the leading edge ; this is shown at B in
Fig. 189. Travelling on the lower surface towards the trailing edge, the
pressure at first falls rapidly and then more slowly until it changes sign
just before reaching the trailing edge. The whole of the upper surface is
under reduced pressure, the variation from the trailing edge to the leading
edge being indicated by the curve EFGHKA.
u
ifV'
H
/
r^
"^^----^
PRESSURE ON UPPER SURFACE
^y
A
/
PRESSURE ON LOWER SURFACE
0|e_
— D
5V«
/
'
EXPERIMENT
NO PRESSURE ON UPPER SURFACE. F
DISCONTINUOUS MOTION
({PRESSURE
f\ NEGATIVE AVERY GREAT
I V AT LEADING EDGE
• \
t V
I \
♦- 0^6 oa ijo
■*■ >^ PRESSURE ON
^ UPPER
F ^^URFACE*
\
\
Et
CONTINUOUS MOTION
FiO. 189. — Observation and calculation of the pressure distribution on a
flat plate inclined at 10° to the current.
368 APPLIED AEEODYNAMICS
The area inside the curve ABC . . . HKA gives a measure of the force
on the plate due to fluid pressures. At an inclination of 10° it appears
that more than two-thirds of the force due to pressure is negative and
is due to the upper surface. The same holds for aeroplane wing sections
to perhaps a greater degree, the negative pressure at H sometimes exceeding
three times that shown in Fig. 189.
Fig. 189 shows for the same position of a plane the pressures calculated
as due to the discontinuous motion of a fluid. On the under surface the
value of ^pv^ at B is reached very much in the same place as the experi-
mental value. Travelling backwards on the under surface the pressures
fall to zero at the trailing edge, but are appreciably greater than those of
the experimental results. On the upper surface there is no negative
pressure at any point. The total force is again proportional to the area
inside the curve ABC . . . HKA, and is clearly much less than the area
of the corresponding curve for the experimental determination. The
degree of approximation is obviously very unsatisfactory in several respects,
the only agreement being at the ^pv^ point.
For the sake of comparison, the pressure distribution corresponding
with the source and sink hypothesis is illustrated in Fig. 189. As before,
starting at the leading edge A and travelling on the under side, the ^pv^
point at B occurs in much the same place as before, but from this point the
pressure falls rapidly and becomes negative just behind the centre of the
plane ; proceeding further, the pressure continues to fall more and more
rapidly until it becomes infinitely great at the trailing edge. Exactly the
same variations of pressure are observed on returning from the trailing edge
to the leading edge vid the upper surface as have been described in passing
in the reverse direction on the lower surface.
The total area is now zero, the convention in the graphical construction
being that when travelling round the curve ABCD . . . EFGHKA areas
to the left hand shall be counted positive and areas to the right hand
negative. It is clear, however, that the moment on the aerofoil is not zero,
and the centre of pressure is therefore an infinite distance away ; the couple
tends to increase the angle of incidence, and further analysis shows that the
couple does not vanish until the plate is broadside on to the stream, i
It will be noticed that the edges of the plate are positions of intense
negative pressure, such as we have seen no real fluid is able to withstand.
This brief summary covers in essentials all the conventional mathe-
matical theories of the motion of inviscid incompressible fluids, and will;
it is hoped, have shown how far the theories fall short of being satisfactory
substitutes for experiment in most of the problems relating to aeronautics.
Motions in Viscous Fluids
Definition o£ Viscosity. — OOi, Fig; 190, is a flat surface over which a
very viscous fluid, such as glycerine, is flowing as the result of pressure
applied across the fluid at AB . . . F. By direct observation the
velocity is known to be zero all along 00 1, and to gradually increase as
the distance from the flat surface increases. If the velocity is proportional
FLUID MOTION
869
to y the definition of viscosity states that the force on the surface OOj is
given by the equation
F = Area X ti X -
y
(7)
In this equation v is the velocity of the fluid at a distance y from the
surface, and " Area " represents the extent of the surface of OOi on which
the force is measured ; (x is the coefficient of viscosity.
If the fluid velocity is not proportional to y but has a form such
as that shown by the dotted line of Fig. 190, the force on the surface is
In exactly the same way the force acting on a
dv\
AreaX/xx(-^)
^dy surface
fluid surface such as DDi is Area Xixxl—] . The definition is
equivalent to the statement that the forces due to viscosity are pro-
0 X
Fia. 190. — Laminar motion of a viscous fluid.
portional to the rate at which neighbouring parts of the fluid are moving
past each other.
Experimental Determination]||o£ /x. — If the motion of a viscous fluid as
defined above be examined in the case of a circular pipe, pressure being
apphed at the two ends, it is found that under certain circumstances the
motion can be calculated in detail from theoretical considerations. More-
over, the predictions of theory are accurately borne out by direct experiment.
Only the result of the mathematical calculation will be given, as it is desired
to draw attention to the results rather than to the method of calculation.
The quantity of fluid flowing per second through a pipe of length I is
found theoretically to be
vol.
per sec.
128/A* I
(8)
where d is the diameter and pi and p2 ^^® pressures at the ends of the
length I. The calculation assumes that /a is constant and that the motion
satisfies the condition of no sUpping at the sides of the tube.
When the corresponding experiment is carried out in capillary tubes of
different diameters and different lengths, it is found that the law of varia-
tion given by (8) is satisfied very accurately. Lamb states that Poiseuille's
experiments showed that " the time of efflux of a given volume of water is
2 B.
370 APPLIED AEEODYNAMICS
directly as the length of the tube, inversely as the difference of pressure at
the two ends, and inversely as the fourth power of the diameter. Formula
(8) then gives a practical means of determining /x which is in fact that almost
always adopted in determining standard values for any fluid.
As indicated by (8) it is easily seen that the skin friction on the pipe is
equal to (pi — P2) • — » or the product of the pressure drop and the area
., ., 1 -i • vol. per sec. x* -c^ •
of the cross-section. Also the average velocity is ^^j^ — . li i^ is
(11)
the total force and v the velocity, we then have
vol. per sec. = « -—
4 )
Substituting torjpi — p2 and vol. per sec. in (8) the values given by (11),
we have
. ■nd^_d^ F
or ¥ = i-fi.v.l (12)
from which it appears that the force is proportional to the coefficient
of viscosity fj,, the velocity v, and to the length of the tube I. The variation
of force as the first power of v appears to be characteristic of the motion
of very viscous fluids.
If the experiment is attempted in a large tube at high speeds the resist-
ance is found to vary approximately as the square of the speed, and it is
then clear that equation (8) does not hold. The explanation of the difference
of high-speed and low-speed motions was first given by Professor Osborne
Eeynolds, who illustrated his results by experiments in glass tubes. Water
from a tank was allowed to flow slowly through the tube, into which was
also admitted a streak of colour ; so long as the speed was kept below a
certain value, the colour band was clear and distinct in the centre of the
tube. As the speed was raised gradually, there came a time at which,
more or less suddenly, the colour .broke up into a confused mass and became
mixed with the general body of the water. This indicated the production
of eddies, and Professor Osborne Eeynolds had shown why the law of
motion as calculated had failed.
Carrying the experiment further, it was shown that the law of
breakdown could be formulated, that is, having observed the break-
down in one case, breakdown could be predicted for other tubes and for
other fluids, or for the same fluid at different temperatures. Denoting the
mass of unit volume of the fluid by p, Osborne Eeynolds found that break-
down of the steady flow always occurred when
^ (13)
FLUID MOTION 371
reached a certain fixed value. This result indicates some very remarkable
conclusions. It has been shown that /x. is usually determined by an
experiment in a capillary tube where d is very small. (13) indicates that
if V be very small the result might be true for a large pipe, or that if /u, is
very large both v and d might be moderately large and yet the motion
would be steady. As an illustration of the truth of these deductions, it
is interesting to find that the flow in a four-inch pipe of heavy oils
suitable for fuel is steady at velocities used in transmission fronj the
store to the place of use.
The expressions p and fx both express properties of the fluid, and it is
only the ratio with which we are concerned in (13) ; as the quantity occurs
repeatedly a separate symbol is convenient, and it is usual to write v for
-. The quantity - , which now expresses (13), is of considerable im-
portance in aeronautics. Before proceeding to the discussion of similar
motions to which this quantity relates, the reference to calculable viscous
motions will be completed.
The motions shown experimentally by Professor Hele-Shaw comprise
perhaps the greatest number of cases of calculable motions and these have
already been dealt with at some length. The experiments of Professor
Osborne Keynolds indicate that the flow will become unstable as the vis-
cosity is reduced, and it seems to be natural to assume that the inviscid
fluid motion having the same stream lines would be unstable. If this
should be the case, then the function of viscosity in such mobile fluids as
water is obvious.
Very few other calculable motions are known : the case of small spheres
falling slowly is known, the first analysis being due to Stokes, and is
applicable to minute rain- drops as they probably exist in clouds. Another
motion is that of the water in a rotating vessel, the free surface of which is
parabolic, and which can only be generated through the agency of viscosity.
Although the number of cases of value which can be deduced from
theory are so few, there are some far-reaching consequences relating to
viscosity which must be dealt with somewhat fully. Up to the present it
has only been shown that for steady motion fi is sufficient to define the
viscous properties of a fluid. It will be shown in the next chapter that /x
is still sufficient in the case of eddying motion, and that results of apparent
complexity can often be shown in simple form by a judicious use of the
function — .
CHAPTER VIII
DYNAMICAL SIMILARITY AND SCALE EFFECTS
Geometrical Similarity. — The idea of similarity as applied to solid objects
is familiar. The actual size of a body is determined by its scale, but if
by such a reduction as occurs in taking a photograph it is possible to
make two bodies appear alike the originals are said to be similar. If one
of the bodies is an aircraft or a steamship and the other a small-scale
reproduction of it, the smaller body is described as a model.
Dynamical Similarity extends the above simple idea to cover the motion
of similarly shaped bodies. Not only does the theory cover similar motions
of aeroplanes and other aircraft, but also the similar motions of fluids.
It may appear to be useless to attempt to define similarity of fluid motions
in those cases where the motion is incalculable, but this is not the case.
It is, in fact, possible to predict similarity of motion, to lay down the
laws with considerable precision and to verify them by direct observation.
The present chapter deals with the theory, its application and some of the
more striking and important experimental verifications.
A convenient arbitrary example, the motion of the links of Peaucellier
cells, leads to a ready appreciation of the fundamental ideas relating to
similar motions. A Peaucellier cell consists of the system of links illus-
trated in Fig. 191. The four links CD, DP, PE and EC are equal and
freely jointed to each other. AD and AE are equal and are hinged to
CDEP at D and E and to a fixed base at A. The link BC is hinged to
CDEP at C and to the same fixed base at B. The only possible motion
of P is perpendicular to ABF. The important point for present purposes
is that for any given position of P the positions of D, C and E are fixed
by the links of the mechanism.
Consider now the motion of a second cell which is L times greater than
that of Fig. 191, and denote the .new points of the link work by the same
letters with dashes. The length AN will become A'N'=LxAN. Put P'
in such a position that P'N'=LxPN, and the shape of the link work will
be similar to that of Fig. 191. A limited class of similar motions may now
be defined for the cells, as being such that at all times the two cells have
similar shapes.
An extension of the idea of similar motions is obtained by considering
the similar positions to occur at different times. Imagine two cinema
cameras to be employed to photograph the motions of the cells, the images
being reduced so as to give the same size of picture. Make one motion
twice as fast as the other and move the corresponding cinema camera
twice as fast. The pictures taken will be exactly the same for both cells,
and the motions will again be called similar motions. We are thus led to
372
f
DYNAMICAL SIMILARITY AND SCALE EFFECTS 373
consider a scale of time, T, as well as a scale of length, L. All similar
motions are reducible to a standard motion by changes of the scales of
length and time.
If the links be given mass it will be necessary to apply force at the
point P in order to maintain motion of any predetermined character, and
this force, depending as it does on the mass of each link, may be different
for similar motions. The study of the forces producing motion is known as
" dynamics," and " dynamical similarity " is the discussion of the conditions
under which the external forces acting can produce similar motions.
Still retaining the cell as example, an external force can" be produced
by a spring stretched between the points P and F. The force in this spring
depends on the position of P, and therefore on the motion of the cell. It
may be imagined that a spring can be produced having any law of force
Fio. 191. — Peaucellier cell.
as a function of extension, and if two suitable springs were used in the
similar cells it would then follow that similar free motions could be
produced, no matter what the distribution of mass in the two cases.
Particular Class of Similar Motions
At this point the general theorem, which is intractable, is left for an
important particular class of motions exemplified as below. In the cell
of Fig. 191 the distribution of mass may still be supposed to be quite arbi-
trary, but in the similar mechanism a restriction is made which requires
that at each of the similar points the mass shall be M times as great as
that for the cell of Fig. 191.
For similar motions of cell any particular element of the second cell
moves in the same direction as the corresponding element of the first. It
moves L times as far in a time T times as great. Its velocity is therefore
- times as great, and its acceleration ^^ times as great. Since the force
producing motion is equal to the product " mass X acceleration," the ratio
of the forces on the corresponding elements is —^. This ratio, for the
374 APPLIED AEKODYNAMICS
limited assumption as to distribution of mass, is constant for all elements
and must also apply to the whole mechanism. The force applied at P'
will therefore be ~ times that at P if the resulting motions are similar.
The constraints in a fluid are different from those due to the links of
the Peaucellier cell, but they nevertheless arise from the state of motion.
The motion of each element must be considered instead of the motion
of any one point, and the force on it due to
^B pressure, viscosity, gravity, etc., must be
j estimated. If the fluid be incompressible,
C the mass of corresponding elements will be
proportional to the density and volume.
Consider as an example the motion of
similar cylinders through water, an account
of which was given in a previous chapter.
The cylinders being very long, it may
^^ U be assumed that the flow in all sections is
the same, and the equations of motion for
Fig. 192. the block ABCD, Fig. 192, confined to two
dimensions. The fluid being incompressible
and without a free surface, gravity will have no influence on the motion,
and the forces on ABCD will be due to effects on the faces of the block.
These may be divided into normal and tangential pressures due to the
action of inertia and shear of the viscous fluid.
From any text-book on Hydrodynamics it will be found that the appro-
priate equations of motion of the block are
D^ _ _^ /dhi ^^u\^
Du dp , (Bh) , bH\ \ ,. .
'^r-Ty-^A^-^^M • • • • ^1)
and ^-V^"=0
bx dy
It is the solution of these equations for the correct conditions on the
boundary of the cylinder which would give the details of the eddies shown
in a previous chapter. With such a solution the present discussion is not
concerned, and it is only the general bearing of equations (1) which is of
interest. Equations (1) are three relations from which to find the quanti-
ties u, V and p at all points defined by x and y. If by any special hypo-
thesis u and V be known, then p is determined by either the first or the
second equation of (1). Consideration of the first equation is all that is
required in discussing similar motions.
Define a second motion by dashes to obtain
,Dm' ___dv' J BH' . d^u' \ „.
^W-' W^^VW? W?^ . . . . (^)
As applied to a similar and similarly situated block there will be certain
DYNAMICAL SIMILAEITY AND SCALE EFFECTS 375
relations between some of the quantities in (2) and some of those in (1).
The elementary length dx' will be equal to lidx, where L is the rat.o of
the diameters of the two cylinders. Similarly dy' — Ldy. The element
of time Dt' will be equal to T . Di, and u' = -u. Make the suggested
substitutions in (2) to get
,L ^^ _\ ¥ fi^/dhj^ dH\
^ T2 • D« L* 3a; ~^LTl\dx^ "^ dyV • • • • ^»;
and now compare equation (3) item by item with the first equation of (1).
The terms on the left-hand sides differ by a constant factor - . ^^, whilst
p T^
the second terms on the right-hand sides differ by a second constant factor
— . =-^^. In general, if L and T are chosen arbitrarily it is not probable
that the equation
P^.L^I^.L (4)
p T2 /M LT ^^
will be satisfied.
Law of Corresponding Speeds. — Since L is a common scale of length
applying to all parts of the fluid, it must also apply to those parts in
contact with the cylinder, and L is therefore at choice by selecting a
cylinder of appropriate diameter. Similarly T is at choice by changing
« the velocity with which the cylinder is moved. For any pair of fluids
equation (4) can always be satisfied by a correct relation between L and
U, that is, by a law of corresponding speeds.
To find the law, rearrange (4) as
p:.^=i . . .' (5)
and multiply both sides by DU, the product of the diameter of the
standard cylinder and its velocity. Equation (5) becomes
^i.D'U'=^DU (6)
Since - = v, the kinematic viscosity of the fluid, equation (6) shows
that the multipliers of the terms in equations (1) and (3) not involving ;/
become the same if — j— is equal to — ,
V V
With the above relation for , equations (1) and (3) give the con-
nection between the pressures at similar points. They can be combined
to give
^P'=f> (^)
876
APPLIED AEEODYNAMICS
and between corresponding points in similar motions the increments of
pressure dp vary as pU^.
This case has been developed at some length, although, as will be
shown, the law of corresponding speeds can be found very rapidly
without specific reference to the equations of motion. It has been shown
on a fundamental basis why a law of corresponding speeds is required in
the case of cylinders in a viscous fluid, and that the pressures then calcu-
lated as acting in similar motions obey a certain definite law of connection.
The result may be expressed in words as follows : " Two motions of viscous
15
rt
—
—
—
■"
"
-■
■
■-
■-
—
—
"
—
—
-
"
—
■
)
y
RESISTANCE OF SMOOTH WIRES
\
\
10
V
s.
V
pD^U'
\
1
\
tf
x
V,
•i.
•~i
•
•^
>^
'^
,
\f
r*
r-
' —
05
.^
k;
5 ,
■ ,
}
^
^
►<
y
•J
'—
77"
»J
^
0
^
1^
^
10
2 0 LOG,n-?H
V
30
40
Fig. 193. — Application of the laws of similarity to the resistance of cylinders.
fluids will be similar if the size of the obstacle and its velocity are so related
UD . ...
to the viscosity that — is constant. The pressures at all similar points
of the two fluids will then vary as pU^."
Since the pressures vary as /oU^ at all points of the fluid, including those
on the cylinder, the total resistance will vary as /oU^D^ and it follows that
E' . , , UD .
^.„^^ vanes only when — varies.
The law is now stated in a form in which it can very readily be sub-
mitted to experimental check. Smooth wires provide a range of cylinders
DYNAMICAL SIMILAKITY AND SCALE EFFECTS 377
of different diameters, and they can be tested in a wind channel over a
considerable range of speed. Two out of the three quantities in — are
then independently variable, and the resistance of a wireO'l in. in diameter
tested at a speed of 50 ft.-s. can be compared with that of a wire 0'5 in.
in diameter at 10 ft.-s.
The experiment has been made, the diameter of the cylinders varying
from 0*002 in. to 1*25 ins., and the wind speeds from 10 to 50 ft.-s. The
number of observations was roughly 100, and the result is shown in Fig. 193.
Instead of — as a variable, the value of log — has been used, as the result-
^ ^ E
ing curve is then more easily read. The result of plotting ^ ^ as
UD . . . .^
ordinate with log — as abscissa is to give a narrow band of points which
V
includes all observations * for wires of thirteen different diameters.
The rather surprising result of the consideration of similar motions is
that it is possible to say that the resistance of one body is calculable from
that of a similar body if due precautions are taken in experiment, although
neither resistance is calculable from first principles. The importance of
the principle as applied to aircraft and their models will be appreciated.
Fuithei Illustrations of the Law of Corresponding Speeds for Incompressible
Viscous Fluids
A parallel set of experiments to those on cylinders is given in the
Philosophical Transactions of the Boyal Society, in a paper by Stanton and
Pannell. These experiments constitute perhaps the most convincing
evidence yet available of the sufficiency of the assumption that in many
applications of the principles of dynamical similarity to fluid motion,
even when turbulent, v and p are the only physical constants of importance.
The pipes were made of smooth drawn brass, and varied from 0*12 in.
to 4 inches. Both water and air were used as fluids, and the speed range
was exceptionally great, covering from 1 ft.-s. to 200 ft.-s. at ordinary
atmospheric temperature and pressure. The value of v for air is approxi-
mately 12 times that for water.
The curve connecting friction on the walls of the pipes with or —
V V
was plotted as for Fig. 193, with a result of a very similar character as to the
spreading of the points about a mean line. The experiments covered not
only the frictional resistance but also the distribution of velocity across
the pipe, and showed that the flow at all points is a function of — . The
original paper should be consulted by those especially interested in the
theory of similar motions.
In the course of experimental work a striking optical illustration of
similarity of fluid motion has been found. Working with water, E. G. Eden
'*' Further particulars are given in B. & M. No. 102, Advisory Committee for Aeronautics.
378 APPLIED AEEODYNAMICS
observed that the flow round a small inclined plate changed its type as the
speed of flow increased.
In one case the motion illustrated in Fig. 194 was produced ; the
coloured fluid formed a continuous spiral sheath, and the motion was
apparently steady. In the other case the motion led to the production
of Fig. 195, and the flow was periodic. The flow, Fig. 194, is from left
to right, the plate being at the extreme left of the picture. The stream
was rendered visible by using a solution of Nestl6's milk in water, and the
white streak shows the way in which this colouring material entered the
region under observation. At the plate the colouring matter spread and
left the corners in two continuous sheets winding inwards. The form of
these sheaths can be realised from the photograph.
For Fig. 195 the flow is in the same direction as before, and the plate
more readily visible. Instead of the fluid leaving in a corkscrew sheath,
the motion became periodic, and loops were formed at intervals and suc-
ceeded each other down- stream. The observation of this change of type of
flow seemed to form a convenient means of testing the suitability of the
law of similarity thought to be proper to the experiment. To test for this
a small air channel was made, and in it the flow of air was made visible
by tobacco smoke, carefully cooled before use. The effects of the heat
from the electric arc necessary to produce enough light for photography
was found to be greater than for water, and equal steadiness of flow was
difficult to maintain.
In spite of these difficulties it was immediately found that the same
types of flow could be produced in air as have been depicted in Figs. 194
and 195. Variations of the size of plate were tried and involved changes of
speed to produce the same types of flow. Two photographs for air are
shown in Figs. 196 and 197, and should be compared with Figs, 194 and
195 for water. The flow is in the same direction as before, and the smoke
jet and plate are easily seen. The sheath of Fig. 196 is not so perfectly
defined as in water, but its character is unmistakably the same as that
of Fig. 194. Fig. 197 follows the high-speed type of motion found in
water and photographed in Fig. 195.
To make the check on similarity still more complete, measurements
were taken of the air and water velocities at which the flow changed its
type for all the sizes of plate tested Taking three plates, J in,, | in.
and I in. square, all in water, it was found that the speeds at which the
flow changed were roughly in the ratios 3:2:1 respectively. Using a
plate 1| ins, square in the air channel, the speed of the air when the flow
changed type was found to be 6 or 7 times that of water with a |-in, plate,
vl
This is in accordance with the law of similarity which states that —
should be constant ; if for instance the fluid is not changed, v remains
constant and v should vary inversely as I. If both I and v are changed
by doubling the scale of the model and increasing v 12 or 14 times, clearly
V must be 6 or 7 times as great. The experiments were not so exactly
carried out that great accuracy could be obtained, but it is clear that
great accuracy was not needed to establish the general law of similarity.
Fig. 194. — ^Flow of water past an inclined plate. Low speed.
Fig. 195. — ^Flow of water past an inclined plate. High speed.
I
Fig. 196. — Flow of air past an inclined plate. Low speed.
FiQ. 197. — Flow of air past an inclined plate. High speed.
DYNAMICAL SIMILAEITY AND SCALE EFFECTS 379
The Principle of Dimensions as applied to Similar Motions. — All
dynamical equations are made up of terms depending on mass M, length
L and time T, and are such that all terms separated by the sign of addition
or of subtraction have the same " dimensions " in M, L and T.
As examples of some familiar terms of importance in aeronautics
reference may be made to the table below.
TABLE 1.
Quantity.
Dimensions.
Angular velocity . .
Linear velocity
Angular acceleration
Linear acceleration
Force ....
1
T
L
T
1
Ta
L
ML
M
Density ....
LT*
M
Kinematic viscosity
L"
L«
T
In order to be able to apply the principle of dimensions, it is necessary
to know on experimental grounds what quantities are involved in producing
a given motion. Using the cylinder in an incompressible viscous fluid as an
example, we say that as a result of experiment —
The resistance of the cylinder depends on its size, the velocity relative
to distant fluid, on the density of the fluid and on its viscosity, and so far
as is known on nothing else. The last proviso is important, as a failure in
application of the principles of dynamical similarity may lead to the
discovery of another variable of importance.
Expressed mathematically the statement is equivalent to
B =f{p, l,u,v) (9)
As the dimensions of E and/ must be the same, a little consideration will
^' show that the form of/ is subject to certain restrictions. For instance,
examine the expression
E
_pH^v
(10)
which is consistent with an unrestricted interpretation of (9). The
ML pHH M^ L T . M*
dimensions of E are -n^, whilst those of - — are t-« .L3.=,-t-5» ^'^^ tT»
T2 V L* T L^ L*
I and the dimensions of the two sides of (10) are inconsistent.
380 APPLIED AEKODYNAMICS
It is not, however, sufficient that the dimensions of the terms of an
equation be the same ; the equation
Il=pZV (11)
has the correct dimensions, but clearly makes no use of the condition
that the fluid is viscous, and the form is too restricted for valid application.
It will now be appreciated that the correct form of (9) is that which has the
correct dimensions and is also the least restricted combination of the
quantities which matter.
The required form may be found as follows : —
Assume as a particular case of (9) that
R=p«ZVt;'* (12)
and, to form a new equation, substitute for E, p, I, v, and v quantities
expressing their dimensions :
Equate the dimensions separately. For M we have
M = M'' (14)
and therefore a — \. For L the equation is
L = L(-8« + * + 2c + d) ...... (15)
and with a = 1 this leads to
& + 2c + d = 4 (16)
The equation for T is
rji— 2 __ rji— c-d
or c + d = 2 ...... . (17)
From equations (16) and (17) are then obtained the relations
and &Id~1 ^^^^
and with a = 1 equation (12) becomes
=pv<tr (19)
The value of d is undetermined, and the reason for this will be seen if
the dimensions of — are examined, for they will be found to be zero. It
is also clear that any number of terms of the same form but with different
values of d might be added and the sum would still satisfy the principle
of dimensions. All possible combinations are included in the expression
B=pv2f(^-) (20)^
/ \^^)
V
where F is an undetermined function.
* This formula and much of the method of dealing with similar motions by the principle
of dimensions are due to Lord Rayleigh, to whom a great indebtedness is acknowledged by
scientific workers in aeronautics.
r
DYNAMICAL SIMILAEITY AND SCALE EFFECTS 381
Equation (20) may be written in many alternative forms which are
exact equivalents, but it often happens that some one form is more con-
venient than any of the others. In the case of cylinders the resistance
varies approximately as the square of the speed, and y ) fA j is written
in=?tead of F( ) to obtain
R=/)i2|,2Fi(-) ...... (21)
-^^C") (2ta)
E
ptV
A reference to Fig. 193 shows that the ordinate and abscissa there used
are indicated by (21a). An equally correct result would have been obtained
by the use as ordinate of —^ as indicated by (20), but the ordinate would
vl
then have varied much more with variation of — , and the result would have
V
been of less practical value.
After a little experience in the use of the method outlined in equations
(12) to (19) it is possible to discard it and write down the answer without
serious effort.
Compressibility. — If a fluid be compressible the density changes from
point to point as an effect of the variations of pressure. It is found ex-
perimentally that changes of density are proportional to changes of pressure,
and a convenient method of expressing this fact is to introduce a coefficient
of elasticity E such that
^-t • ■ -^''^
P
where E is a constant for the particular physical state of the fluid. E
has the dimensions of pressure and therefore ol pv^, and hence the quantity
^ is of no dimensions.
E
If the viscosity of the fluid does not matter, the correct form for the
resistance is
B=pW¥lP^) (23)
where F2 is an arbitrary function. It is shown in text-books on physics
that -y/ - is the velocity of sound in the medium, and denoting this quantity
by "a," equation (23) becomes
Rr^piVEs^-) (24)
Whilst equation (24) shows that the effect of compressibility depends
on the velocity of the body through the fluid as a fraction of the velocity
382 ^ APPLIED AERODYNAMICS
of sound in the undisturbed fluid, it does not give any indication of how
resistance varies with velocity.
The knowledge of this latter point is of some importance in aeronautics,
and a solution of the equation of motion for an inviscid compressible
fluid will be given in order to indicate the limits within which air may safely
be regarded as incompressible.
In developing Bernoulli's equation when dealing with inviscid fluid
motion the equation
dp+pvdv = 0 (25)
between pressure, density and velocity was obtained and integrated on the
assumption that p was a constant. The fluid now considered being com-
pressible, there is a relation between p and p, which depends on the law of
expansion. Assuming adiabatic flow the relation is
V-^P'- (26)
Po
where Pq and pq refer to some standard point in the stream where v is
uniform and equal to Vq and y is a constant for the gas. Differentiating
in (26) and substituting for dp in (25) leads to
— ^.^.f^y =- 1^2 + constant . . . (27)
y — lpo Vpo^
and the constant is evaluated by putting v==Vq when p = pQ. The value
of -^^ = — and is equal to the square of the velocity of sound in the
Po Po
undisturbed fluid. Equation (27) becomes
-^^=0+^-'^>"' (^«)
and since —=(—), a new relation from (28) is
■Da VDa''
Po ^Po'
. K'^'^-'^T' (-)
The greatest positive pressure difference on a moving body occurs
at a " stagnation point," i.e. where v = 0. Making i; = 0 in (29) and
expanding by the binomial theorem
Po" "^2-a-^+8-a"4+ ^^^^
Denoting the increase of pressure p — Po hj 8p leads to the equation
or smce ^ = po
« /I «,2
Sp=|poi'§(l+^.|° + . . .) .... (32)
DYNAMICAL SIMILARITY AND SCALE EFFECTS 383
If — be small the increase of pressure at a stagnation point over that
a
,2
I
of the uniformly moving stream is IpqVq, and this value is usually found
in wind-channel experiments. For air at ordinary temperatures the
velocity of sound is about 1080 ft.-s., and the velocity of the fastest
aeroplane is less than one quarter of this. The second term of (32) is
then not more than 1 '5 per cent, of the first. As the greatest suction on
an aeroplane wing is numerically three or four times that of the greatest
positive increment the effect of compressibility may locally be a little
more marked, but to the order of accuracy yet reached air is substantially
incompressible for the motion round wings.
The same equation shows that for airscrews, the tips of the blades
of which may reach speeds of 700 or 800 ft.-s., the effect of compressi-
bility may be expected to be important. At still higher velocities it appears
that a radical change of type of flow occurs, and when the tip speed
exceeds that of half the velocity of sound normal methods of design
need to be supplemented by terms depending on compressibility.
Similar Motions as affected by Gravitation. — An aeroplane is supported
against the action of gravity, and hence gr is a factor on which motion
depends. Ignoring viscosity and compressibility temporarily, the motion
will be seen to depend on the attitude of the aeroplane, its size, its velocity,
on the density of the fluid and on the value of g. The principle of dimen-
sions then leads to the equation
R=/>?VF4(^). . (33)
For two similar aeroplanes to have the same motions when not flying
steadily the initial values of — must be the same. For terrestrial purposes
g is very nearly constant, and the law of corresponding speeds says that
the speed of the larger aeroplane must be greater than that of the smaller
in the proportion of the square roots of their scales. This may be recognised
as the Froude's law which is applied in connection with Naval Architecture.
The influence of gravity is there felt in the pressures produced at the base
of waves owing to the weight of the water.
Combined Effects of Viscosity, Compressibility and Gravity. — The
principle of dimensions now leads to the equation
^-^-^'K'l'i) («*)
and a law of corresponding speeds is no longer applicable. It is clearly
not possible in one fluid and with terrestrial conditions to make—, - and =—
V a Ig
each constant for two similar bodies. It is only in those cases for which
only one or two of the arguments are greatly predominant that the prin-
ciples of dynamical similarity lead to equations of practical importance.
Static Problems and Similarity of Structures. — The rules developed for
dynamical similarity can be applied to statical problems and one or two
384 APPLIED AEEODYNAMICS
cases are of interest in aeronautics. Some idea of the relation between
the strengths of similar structures can be obtained quite readily.
Consider first the stresses in similar structures when they are due to
the weight of the structure itself. The parts may either be made of the
same or of different materials, but to the same drawings. If of different
materials the densities of corresponding parts will be assumed to retain
a constant proportion throughout the structure. Since the density appears
separately, the weight can be represented by pl^g, and if the structure be
not redundant it is known that the stress depends only on p, I and g. If /,
represent stress, the equation of correct dimensions and form is
Is = P9l (35)
This equation shows that for the same materials, stress is proportional
to the scale of structure, and for this condition of loading large structures
are weaker than small ones. It is in accordance with (35) that it is found
to be more and more difficult to build bridges as the span increases.
The other extreme condition of loading is that in which the weight of
the structure is unimportant, and the stresses are almost wholly due to a
loading not dependent on the size of the structure. If w be the symbol
representing an external applied load factor between similar structures, the
principle of dimensions shows that
/.=^ (36)
If the external loads increase as P, i.e. as the cross- sections of the similar
members, equation (36) shows that the stress is independent of the size,
The weight of the structure, however, increases as l^ if the same materials
are used.
In an aeroplane the conditions of loading are nearly those required by
(36). If the loading of the wings in pounds per square foot is constant, the
total weight to be carried varies as the square of the linear dimensions.
Of this total weight it appears that the proportion due to structure varies
from about 26 per cent, for the smallest aeroplane to 33 per cent, for the
largest present-day aeroplane. The change of linear dimensions corre-
sponding with these figures is 1 to 4, but it should not be forgotten that
the principles of similarity are appreciably departed from. In building
a large aeroplane it is possible to give more attention to details because
of their relatively larger size, and because the scantlings are then not so
frequently determined by the limitations of manufacturing processes.
Since small aeroplanes have been used for fighting purposes where they
. are subjected to higher stresses than larger aeroplanes, a lower factor of
safety has been allowed for the latter. The margin of safety for small
present-day aeroplanes would be almost twice as great as for the larger
ones if both were used on similar duties.
In the case of engines the power is frequently increased by the multi-
plication of units and not by an increase of the dimensions of each part.
The number of cylinders may be two for 30 to 60 horsepower, 12 for 300
horsepower to 500 horsepower, and for still higher powers the whole engine
DYNAMICAL SIMILAEITY AND SCALE EFFECTS 385
may be duplicated. For 1500 horsepower there will be say 4 engines with
48 cylinders, each of the latter having the same strength as a cylinder
giving 30 horsepower. The process of subdivision which is carried to great
lengths in the engine is being applied to the whole aeroplane as the number
of complete engines is increased, and this tends to keep the structure weight
from increasing as the cube of the linear dimensions of the aeroplane as
would be required in the case of strict similarity. In the extreme case two
aeroplanes may be assumed to flly independently side by side, and some
connecting link used to bind them into one larger aeroplane. In principle
this is carried out in the design of big. aeroplanes. The weight of the con-
necting mechanism appears to be of appreciable magnitude, since with
advantages of manufacture and factors of safety the structure weight
as a fraction of the whole shows a distinct tendency to increase. There
is, however, no clear limitation in sight to the size of possible aeroplanes.
The position with regard to airships is of a very similar character, and
the structure weight will tend to become a greater proportion of the
whole as the size of airship increases, but the rate is so slow that again
no clear limitation on size can be seen.
Aeronautical Applications of Dynamical Similarity. — ^Fig. 193 may be
used to illustrate an application to aeronautical purposes of curves based on
similarity. As an example suppose that a tube containing engine control
leads is required in the wind, and that it is desired to know how much
resistance will be added to the aeroplane if the tube is circular and unf aired.
The diameter of the tube will be taken as 0*5 in. (0*0417 ft.), and its length
6 ft. or 144 diameters. At or near ground-level the density (0-00237)
and the kinematic viscosity (0*000159) may be found from a table of
vl
physical constants.* At a speed of 100 ft.-s. the value of log — will
* Kinematic Viscosity, v. — If fi is the coefficient of fluid friction, v = yifp.
Am.
Ft. -lb. -sec. units.
Temp.
<Tv in sq. ft. per sec.
where 000237<r = p.
0°C.
15°
100°
0-000152
0-000169
0000194
Water.
Ft. -lb. -sec. units.
Temp.
V m sq. ft. per sec.
6°C
8°
10"
15°
20°
0-0000159
0-0000150
0-0000141
0-0000123
0-0000108
2 0
386 APPLIED AEEODYNAMIGS
easily be found from the above figures to be 4-42, and Fig, 193 then
shows that
E being the resistance of a piece of tube of length equal to its diameter.
The resistance of the whole tube is then
144 X 0-595 X 0-00237 X (0-0417)2 x 100^ = 3-54 lbs.
At 10,000 ft. the resistance will be different. The density is there equal
vl
to 0-00175, and the kinematic viscosity to 0*000201. The value of log -
is 4-32, and that of -j^—^ is 0-592. Finally the resistance is 2-60 lbs.
In this calculation no assumption has been made that resistance varies as
•R
the square of the speed, and the fact that -jg-^ has changed is an indication
E
plH^
of departure from the square law and strict similarity. The value of ^ ^
has only changed from 0-595 to 0-592 as a result of changing the height
from 1000 ft. to 10,000 ft. Most of the change in resistance is due to change
in air density. It might have happened that the curve of Fig. 193 had been
"P
a horizontal straight line, and in that case the resistance coefficient -^^„
vl P^^
would not have changed at all, and motions at all values of — would have
V
been similar. We may then regard the variations of the ordinates of Fig. 193
as measures of departure from similarity. It does not follow that similar
E
plH''
being that — is constant.
flow necessarily occurs when ,„ „ has the same value, the correct condition
V
vl
If such curves as that of Fig. 193 do not vary greatly with - the fluid
motions might be described as nearly similar, and with a certain loss of
precision we may say that the resistance of the cylinders does not depend
vl
appreciably on -. In many cases our lack of knowledge is such that
much use must be made of the ideas of nearly similar motions, and this
applies particularly to the relations between models of aircraft and the
aircraft themselves. Fortunately for aeronautics, most of the forces for
a given attitude of the aircraft or part vary nearly as the square of the
speed, and - is only of importance as a correction. The law of resista^bc©
given by (21), ^.e. . ,.
R=pZ2^2ji(^M (37)
is worth special attention in its bearing on the present point. Both model
DYNAMICAL SIMILAEITY AND SCALE EFFECTS 387
and aircraft move in the same medium, and therefore v is constant. If
vl .
- is also to be constant it follows that vl is constant, and equation (37)
then shows that E is constant. This means that similarity of flow can
only be expected on theoretical grounds if the force on the model is as
great as that on the aircraft. Stated in this way, it is obvious that the
law of corresponding speeds as applied to aerodynamics is useless for
complete aircraft. For parts, it may be possible to double the size for
wind channel tests, and so get the exact equivalent of a double wind speed.
This is the case for wires and struts, and the law of corresponding speeds
is wholly satisfied.
For aircraft as a whole and for wings in particular it is necessary to
vl
investigate the nature of Fj over the whole range of ~ between model and
full scale if certainty is to exist, and, if the changes are great, the assistance
which models give in design is correspondingly reduced, since results are
subject to a scale correction.
Aeroplane Wings. — The scale effect on aeroplane wings has received
more attention than that of any other part of aircraft for which the range
vl
of - cannot be covered without flight tests. It has been found possible
in flight to measure the pressure distribution round a wing over a wide
range of speeds. For the purposes of comparison a complete model structure
was set up in a wind channel and the pressure distribution observed at
corresponding points. The full-scale experiments are more difficult to
carry out than those on the model, and the accuracy is relatively less. It is,
however, great enough to warrant a direct comparison such as is given in
Fig. 198. The abscissae of the diagrams represent the positions of the
points at which the pressures were measured, whilst the values of the latter
divided by pv^ are the ordinates in each case. The points located on the
upper surface will be clear from the marking on each diagram. The
curves represent the extreme observed angles of incidence for the lower
and upper wings of a biplane, the continuous curves being obtained on the
full scale and the dots on the model.
The general similarity of the curves is so marked that no hesitation
will be felt in saying that the flow of air round a model wing is nearly
similar to that round an aeroplane wing.
A close examination of the diagrams discloses a difference on the lower
surface of the upper wing which is systematic and greater than the acci-
dental errors of observation. It is difficult to imagine any reason why this
difference should appear on one wing and not on the other, and no satis-
factory explanation of the difference has been given. It must be concluded
from the evidence available that the model represents the full scale with
an accuracy as great as that of the experiments, since it is not possible
to give any quantitative value to the difference. It follows from this that
until a higher degree of accuracy is reached on the full scale the character-
istics of aeroplane wings can be determined completely by experiments
on models.
388
APPLIED AERODYNAMICS
It is not possible from diagrams of pressure distribution alone to
determine the lift and drag of a wing. An independent measurement is
necessary before resolution of forces can be effected, and on the full scale
Pressure
COMPARISON OF PRESSURE DISTRIBUTION ON WINGS
MODEL & FULL SCALE. rrrrr. ''^^VdIl'^
LOWER WING.
o.a
ANGLE OF
INCIDENCE
o°
0.4-'
0
W
• »
-**'
/W^
UPPER
.'^>
0.4-
L
SURFACE
f
0.4-
^
W
III"
O
"*~-~-S*-i-i
-rt
•
y
Pres
sure
^^j^
/^
V2
04-
X
UPPER
,^
SURFACE
/••
0.6
f
>
1
-1.2
-1.6
20
4-0
60
DISTANCE FROM LEADING EDGE
(INS, ON FULL SCALE)
UPPER WING.
Pressure
/3 V^
o^i
ANGLE OF
INCIDENCE
.*■ — —
°r
^
\^
UPPER
SURFACE
0.4-
•
^""
121
^*"^».^^
•
^l^ta.,,^ •
O
Pressure
•
-0.4-
/^ t
/
UPPER
/
SURFACE
p
-1.2
-1.6
f
3 20 4-0 60
DISTANCE FROM LEADING EDGE
(INS,ON FULL SCALE)
Fig. 198. — Comparison of wing characteristics on the model and full scales,
this measurement involves either a measure of angle of incidence, of gliding
angle or of thrust. Of these the determination of gliding angle with air-
screw stopped gives promise of earliest results of suflficient accuracy. For
DYNAMICAL SIMILAEITY AND SCALE EFFECTS 389
drag an error of 1° in the angle of incidence means an error of 30 per cent.,
and a sufficient accuracy is not readily attained ; a reliable thrust meter
has yet to be developed. As the resultant force is nearly equal to the lift,
this quantity can be deduced with little error from the pressure distribution
and a rough measure of the angle of incidence, and the model and full scale
agree. This is not, however, a new check between full scale and model.
TABLE 2. — Changes of Lift, Deao and Moment on an Aerofoil over the Model
Range of vl.
Centre of gravity at 0-4 chord.
Liftco-
eflicienT,
8*1,
8*L
Drag
coeffi-
cient,
&ko
8*D
Angle
of inci-
Moment
dence.
*L
vl= 20
vl=>10
Ad
vi = 20
vl= 10
coefficient,
Sku
6*M
vl = 30
fi = 30
*M
vl = 20
vl =10
vl = SO
-6°
-0152
-0-003
-0-007
0-0352
0-0003
0-0016
-0-0710
+0-0002
0-0004
-4°
-0-047
-0-016
-0025
0-0208
0 0004
0-0017
-0-0.330
0-0010
0-0020
-2°
+0-062
-0 022
-0-054
0-0124
0-0008
0 0022
-0-0250
0-0015
0-0030
0»
+0-144
-0-005
-0-044
0-0099
0-0006
0-0023
-00143
0-0004
00010
2°
0-216
-0-002
-0-019
0-0113
0-0004
0-0021
-0-0028
-0-0001
-0-0002
4"
0-290
-0-002
-0014
00146
0-0004
0 0020
+0-0109
-0-0001
-0-0002
6«
0-362
-0 002
-0-014
0 0206
0-0004
0-0020
+0 0225
-0-0001
-0-0002
8°
0-440
-0-003
-0018
0-0279
0 0004
0 0023
0 0350
-0-0001
-0-0002
10°
0-512
-0 004
-0-022
00365
0 0004
0-0026
0 0441
-0 0001
-0-0003
12°
0-584
-0007
-0 030
0 0456
0 0005
0 0030
0-0542
-0 0002
-0-0005
14°
0-630
-0-020
-0-050
0-0562
0-0006
0 0035
0-0628
-0-0003
-0-0007
16°
0-618
-0 025
-0-059
0 0742
0 0011
0 0043
0-0625
-00004
-0-0010
18°
0-576
-0 033
-0-067
01008
great
great
0-0267
—
—
20°
0-520
-0-025
-0060
0-1475
great
great
0-0092
~
~
It is easily possible in a wind channel to make tests on wings of different
sizes and at different speeds, but the tests throw little light on the behaviour
of aeroplane wings since the variations of vl which are possible are so small.
The smallest aeroplane is about five times the scale of the largest model,
and travels at speeds which vary from being less than that of the air current
in the channel to being twice as great. For very favourable conditions the
range of vl from model to full scale is 4:1. Table 2 shows roughly how
the values of the various resistance coefficients of a wing are affected by
changes of vl over the wind channel range. The wing section had an
upper surface of similar shape to that shown in Fig. 198, but had no
camber on the under surface.
The table shows the lift, drag and moment coefficients for vl = 30 for
a range of angles of incidence together with the differences in these quanti-
ties due to a change from vl = 30 to vl = 20 and vl = 10. An examination
of the table will show that for the most useful range of flying angles, i.e.
from 0° to 12°, the variations with vl are not very great, the minimum drag
coefficient being the most seriously affected. At angles of incidence less
than 0° the lift coefficient is affected appreciably, whilst at large angles
of incidence, 14°-20°, the effect of changing vl is appreciable on both the lift
and drag coefficients. It is in the latter case that recent extensions of
model experiments will be of great value.
390 APPLIED AEEODYNAMICS
Judging from these results alone it might be expected that for efficient
flight the model tests would be very accurate, but that at very high and very
low speeds of flight, scale factors of appreciable magnitude would be neces-
sary. At the present moment all that can be said is that full-scale experi-
ments have not shown any obvious errors even at the extreme speeds.
Something more than ordinary testing appears to be required if the correc-
tions are to be evaluated, and for the present, wind channel tests at vl = 30
{i.e. 6" chord and a wind speed of 60 ft.-s.) may be applied to full scale
without any vl factor.
Variation o£ the Maximum Lift Coefficient in the Model Range of vl. —
The variation of lift coefficient in the neighbourhood of the maximum
varies very greatly from one wing section to another. For the form shown
in Pig. 198 the changes are appreciable but not very striking in character.
Changing to a much thicker section such as is used in airscrews the effect
of change of speed is marked, and shows that the flow is very critical in
the neighbourhood of the maximum lift coefficient. Fig. 199 shows a good
example of this critical flow. The section is shown in the top left-hand
corner of the figure, and the value of vl is the product of the wind velocity
in feet per second and the maximum dimension of the section in feet. With
vl='5 the curve for lift coefficient reaches a maximum of 0'41 at an angle of
incidence of 8°, and after a fall to 0-32 again rises somewhat irregularly
to 0*43 at an angle of incidence of 40 degrees. At the other extreme of
vl, i.e. 14*5, the first maximum has a value of 0*60 at 12°'5, followed by a
fall to 0*45 at 15° and a very sharp rise to 0*78 at 16° '5. For greater angles
of incidence the value of the lift coefficient falls to 0-43 at 40°, and agrees
for the last 1 0 degrees of this range with the value for vl = 5. Intermediate
curves are obtained for intermediate values of vl, and it appears probable
that at a somewhat greater value of vl than 14*5 the first minimum would
disappear, leaving a single maximum of nearly 0'8. The drag curves show
less striking, but quite considerable, changes with change of vl.
The curves for all values of vl are in good agreement from the angle of
no lift up to 6 or 8 degrees, and for the higher values of vl the region of
appreciable change is restricted to about 4°. If the experiments had been
carried to vl = 30, it appears probable that substantial independence of vl
would have been attained. It is to this stage that model experiments
should, if possible, be carried before application to full scale is made. There
is, of course, no certainty that between the largest vl for the model and that
for the aeroplane some different type of critical flow may not exist. There
is, however, complete absence of any evidence of further critical flow, and
much evidence tending in the reverse direction.
There are no experiments on aeroplane bodies or on airships and their
models which indicate any instability of flow comparable with that shown
for an aerofoil in Fig. 199. In all cases there is a tendency to lower drag
coefficients as vl increases, the proportionate changes being greatest for
the airship envelopes. Table 3 shows three typical results ; in the first
column is the speed of test, whilst in the others are figures showing the
change of drag coefficient with change of speed, or, what is the same thing
so long as the model is unchanged, with change of vl. The first model was
DYNAMICAL SIMILAEITY AND SCALE EFFECTS 391
comparable in size with an aeroplane body, but its shape was one of much
lower resistance for a given cross-section. The change of drag coefficient
over the range shown is aboiit 8 per cent. Comparison with actual airships
is difficult for lack of information, but it is clear that this rate' of change is
(^iniosgv) siN3DiJd3oo 9vya qnv un
not continued up to the vl suitable for airships, and it is probable that the
rate of change is a local manifestation of change of type of flow from which
it is impossible to draw reliable deductions for extrapolation. As applied
to aeroplane bodies however, the range of vl covered is so great that the
392
APPLIED AERODYNAMICS
slight extrapolation required may be made without danger. This con-
clusion is strengthened by the last two columns, which show that when
rigging, wind screens, etc., are added to a faired body the drag coefficient
changes less rapidly with vl, and the usual assumption that the drag coeffi-
cient of an aeroplane body is independent of vl is sufficiently accurate for
present-day design.
TABLE 3. — Scale Effect on Aeroplane Bodies and Airship Models,
Ratio of drag coefficients at various speeds to tlie
drag coefficient at 60 ft.-s.
Velocity
(ft.-s.).
Model of. rigid
airship envelope,
1"6 ft. diameter,
15 ft. long.
Model of non-rigid
airship envelope
and rigging,
06 ft. diameter,
3 ft. long.
Model of aero-
plane body,
1-5 ft. long.
40
50
60
70
80
105
101
100
0-99
0-97
102
101
100
100
100
100
100
100
100
0-99
The Resistance of Struts. — In describing the properties of aerofoils it
was shown that the thickening of the section led to a critical type of flow
O 5
028
026
024
022
0-2
R O'le
p(l.v
O 16
OI-+
O 12
O-IO
O08
006
O04-
0-02
0
1
■"
1
^^^p
^^^^^^
f^^^^^H
''^^m^
1
■
v
V
O , 1
i-*^
±j5_
^-il-
8_j9_
10 ,11
12 ,13
14 ,|5
16 ,17
18 ,19
20 ,21
22 ,23
Fig. 200. — Scale effect on the resistance of a strut.
R = resistance in lbs.
I == smaller dimension of cross-section in feet.
L = length of strut in feet.
V = speed in feet.
at certain angles of incidence. A further change of aerofoil section leads
to a strut, and experiment shows that the flow is apt to become extremely
critical, especially when the strut is inclined to the wind. Even when
DYNAMICAL SIMILARITY AND SCALE EFFECTS 393
symmetrically placed in the wind the resistance coefficient of a good form
of strut changes very markedly with vl for small values (Fig 200)*
Consider a strut of which the narrower dimension of the cross-section
is 1| ins. or 0*125 ft. At 150 ft.-s. the value of vl is nearly 19 and the
drag coefficient is 0*040. It is obvious from Fig. 200 that the exact value
of hi is unimportant. Even had vl been as small as 6 the drag coefficient
would still not have varied by as much as 20 per cent. If, on the other
hand, the test of a model at 75 ft.-s. is considered, the scale being 2\ith, the
value of i^l is about 0*5, and the corresponding resistance coefficient is 0*15.
The variation from constancy is then great, and for this reason it is usual
when testing complete model aeroplanes to cut down the number of inter-
plane struts to a minimum and to eliminate the effect of the remainder
before applying the results to full scale. The same precaution is taken in
regard to wires.
Wheels. — The resistance of wheels varies very accurately as the square
of the speed over the model range, and there is no difficulty in getting
values of vl approaching those on the full scale. There is an appreciable
mutual effect on resistance between the wheels and undercarriage and
between the struts at the joints, and except for wires the complete under-
carriage should be tested on a moderately large scale if the greatest accuracy
is desired.
Aeroplane as a Whole. — It was shown when discussing the resistance
of an aeroplane in detail that the whole may be divided into planes,
structure, body, undercarriage and tail, and the resistance of these parts
obtained separately ; the results when added give a close approximation
to the resistance of the whole. It may therefore be expected from the
preceding arguments that the aeroplane as a whole will show the same
characteristics on lift as are shown by the wings alone, and will have a less
marked percentage change in drag with change in vl. The number of ex-
periments on the subject is very small, but they fully bear out the above
conclusion.
To summarise the position, it may be said that a model aeroplane
complete except for wires and struts, having a wing chord of 6 ins., may be
tested at a speed of 60 ft.-s., and the results applied to the full scale
on the assumption that the flow round the model is exactly similar to that
round the aeroplane.
Airscrews. — The airscrew is commonly regarded as a rotating aerofoil,
and there is no difficulty on the model scale in obtaining values of vl
much in excess of 80. The possibiHty of experiments by the use of a
whirling arm also makes more full-scale observations available. Although
the number of partial checks is very numerous, accurate comparison has
not been carried out in a sufficient number of cases to make a quantitative
statement of value. For normal aeroplane use the general conclusion
arrived at is that the agreement between models and full scale is very close.
It has been pointed out that the compressibility of air begins to become
evident at velocities of 500 or 600 ft.-s., and airscrews have been designed
and satisfactorily used up to 800 ft.-s. At the higher speeds empirical
correction factors were found to be necessary which had not appeared at
394 APPLIED AEEODYNAMICS
lower speeds. One experiment, a static test, has been carried out at speeds
up to 1150 ft.-s. In the neighbourhood of the velocity of sound the type
of flow changed rapidly, so that the slip stream was eliminated and the main
outflow centrifugal. The noise produced was very great and discomfort
felt in a short time. It is clear that no certainty in design at present
exists for tip speeds in excess of 800 ft.-s.
Summary of Conclusions. — This resume of the applications of the prin-
ciples of dynamical similarity will have indicated a field of research of which
only the fringes have yet been touched. So far as research has gone,
the result is to give support to a reasonable application of the results of
model experiments. This conclusion is important since model results are
more readily and rapidly obtained than corresponding quantities on the
full scale, and the progress of the science of aeronautics has been and will
continue to be assisted greatly by a judicious combination of experiments
on both the model and full scales.
CHAPTEE IX
THE PBEDICTION AND ANALYSIS OF AEROPLANE PERFORMANCE
The Performance of Aeroplanes
The term " performance " as applied to aeroplanes is used as an
expression to denote the greatest speed at which an aeroplane can fly
and the greatest rate at which it can climb. As flight takes place in the
air, the structure of which is variable from day to day, the expression
only receives precision if the performance is defined relative to some
specified set of atmospheric conditions. As aeroplanes have reached
heights of nearly 30,000 feet the stratum is of considerable thickness, and
in Britain, aeronautical experiments and calculations are referred to a
standard atmosphere which is defined in Tables 1 and 2.
TABLE 1. — Standard Height.
The pressure is in multiples of 760 mm. of mercury, and the density of 0"00237 slug jier cubic ft.
Standard
height
(ft.).
Belative
density.
Relative
pressure.
P
Temperature
Absolute
temperature
Aneroid height
(ft.).
0
1026
1000
9
282
0
1,000
•994
•964
75
280 5
1,000
2,000
•963
•929
6
279
2.010
3,000
•932
•895
46
277-5
3,020
4,000
•903
•861
3
276
4.040
5,000
•870
•829
16
274-6
5,070
6,000
•845
•798
0
273
6,100
7,000
•818
•768
-15
271-5
7,130
8,000
•792
•739
-3
270
8,180
9,000
•766
•711
-45
268^6
9,230
10,000
•740
•684
-6
267
10,290
11,000
•717
668
-8
265
11,360
12,000
•696
•632
-10
263
12,440
13,000
•673
•607
-12
261
13,520
14,000
•652
•683
-14
259
14,600
15,000
•630
•560
-16
257
15,700
16,000
•610
•638
-18
256
16,800
17,000
•690
•616
-20
253
17,900
18,000
•671
, -496
-22
251
19,010
19,000
•653
•476
-24
249
20,140
20,000
•636
•456
-26
247
21,270
21,000
•616
•437
-28
245
22,410
22,000
•498
•419
-29-5
2436
23,560
23,000
•481
•402
-316
241 ^6
24,720
24,000
•464
•385
-33
240
25,890
25.000
•448
•369
-35
238
27,060
26,000
•432
•353
-37
236
28,240
27,000
•417
•338
-38^5
2345
29,430
28,000
•402
•324
-40 5
2326
30,640
29,000
•388
•310
-42
231
31,860
30,000
•374
•296
-44
229
33,100
395
39G
APPLIED AEEODYNAMICS
TABLE 2. — ^Aneroid Height,
The pressure is in multiples of 760 mm. of mercury, and the density of 0*00237 slug per cubic ft.
Aneroid
Belative Belative
Temperature
Absolute
Standard
height
(ft.).
pressure. density.
p a-
Temperature
height
(ft.).
0
1
1000 1
025
9
282
0
1,000
•964
994
7-5
280-5
1,000
2,000
•929
962
6
279
1,990
3,000
•896
933
4-5
277-5
2,980
4,000
•863
904
3
276
3,960
6,000
•832
876
1-5
274-5
4,940
6,000
•802
849
0
273
5,900
7,000
•773
822
-1-5
271-5
6,870
8,000
•745
796
-3
270
7.830
9,000
•718
771
-4
269
8,780
10,000
•692
747
-55
267-6
9,730
11,000
•667
724
-75
263-5
10,670
12,000
•643
703
-9
264
11,600
13,000
•620
683
-11
262
12,620
14,000
•697
663
-13
260
13,440
15,000
•576
644
-14-5
258-6
14,360
16,000
•555
626
-16-5
256-5
15,270
17,000
•535
607
-18-5
254-6
16,180
18,000
•516
589
-20
253
17,090
19,000
•497
671
-22
251
18,000
20,000
•480
554
-24
249
18,880
21,000
•462
537
-25-5
247-5
19,760
22,000
•445
521
-27
246
20,650
23,000
•429
506
-29
244
21,520
24,000
•414
491
-30^6
242-5
22,380
25,000
•399
477
-32
241
23,240
26,000
•384
462
-33-5
239-5
24,110
27,000
•370
448
-35
238
24,960
28,000
■357
435
-36-5
236-5
25,800
29,000
•344
422
-38
235
26,650
30,000
•332
410
-39-5
233-5
27,480
The tables show the quantities of importance in the standard atmo-
sphere with the addition of a quantity called " aneroid height." The
term arises from the use of an aneroid barometer in an aeroplane, the
divisions on which are given in thousands of feet and fractions of the
main divisions. As a measure of height the instrument is defective, and
it will be noticed from the table that an aneroid height of 33,100 feet
corresponds with a real height of 30,000 feet in a standard atmosphere.
In aeronautical work of precision the aneroid bat-ometer is regarded solely
as a pressure indicator, and the readings of aneroid height as taken, are
converted into pressure by means of Table 2 before any use is made of
the results. The term " aneroid height " is useful as a rough guide to
the position of an aeroplane, and for this reason the aneroid barometer
has never been displaced by an instrument in which the scale is calibrated
in pressures directly.
The first column of Table 1 shows for a standard atmosphere the real
height of a point above the earth (sea level), whilst the others show relative
pressure, relative density and temperature, both Centigrade and absolute.
PKEDICTION AND ANALYSIS FOR AEROPLANES 397
16.000
In trials, temperature is observed by reading a thermometer fixed on one
of the wing struts, and the density is calculated from the observed tem-
perature and the pressure deduced from the aneroid height.
An illustration is given in Fig. 201 of variations of temperature which
may be observed during performance trials. The curves cover the months
May to February, and contain observations for hot and cold days. Whilst
the general trend of the curves is to show a fall of temperature with height
there was one occasion on which a temperature inversion occurred at
about 3000 feet. The extreme difference of temperature shown at the
ground was over 25° C, and at 12,000 ft. the difference was 10° C. It
will be noticed that the curve for aneroid height which would follow from
Table 2 would fall amongst the curves shown, roughly in the mean position.
There are some atmospheric variations which affect performance, but
of which account can-
not yet be taken. If
the air be still no diffi-
culties arise, but if it
be in movement — ex-
cept in the case of
uniform horizontal
velocity — errors of ob-
servation will result.
To see this it is noted
that the natural ghding
angle of an aeroplane
may be 1 in 8, i.e. the
effect of gravity at such
an angle of descent is
as great as that of the
engine in level flight.
Suppose that an up-
current of 1 in 100
-20
TEMPERATURE (CENTIGRADE)
Fio. 201. — Atmospheric changes of temperature.
exists during a level flight, the aeroplane will be keeping at constant height
above the earth by means of the aneroid barometer, and consequently will
be descending through the air at 1 in 100. This is equivalent to an 8 per
cent, addition to the power of the engine and an increase of 3 miles per
hour on the observed speed. The flight speed being 200 ft.-s. the up-
current would have a velocity of 2 ft.-s., an amount which is much less
than the extremes observed. It is generally thought that up-currents are
less prevalent at considerable heights than near the ground, but no regular
means of estimating up-currents with the desired accuracy is available for use.
A variation of horizontal wind velocity with height introduces errors
into the observed rate of climb of an aeroplane due to the conversion of
kinetic energy of the aeroplane into potential energy. If, in rising 1000 ft.,
the wind velocity increases by 30 per cent, of the flying speed of an aero-
plane, the error may be ± 8 per cent, dependent on whether flight is into
the wind or with the wind. This error can be eliminated by flying back-
wards and forwards over the same course.
398 APPLIED AEEODYNAMICS
Special care in regulating the petrol consumption to the atmospheric
conditions is required ; without regulation the petrol-air mixture tends
to become too rich as the height increases, with a consequent loss of engine
power, and an increased petrol consumption. The following figures will
show how important is the regulation of the petrol flow.
In a particular aeroplane the time to climb to 10,000 feet with un-
controlled petrol was 25 mins., and this was reduced to 21*5 mins. by
suitable adjustment. The increase of speed was from 84 m.p.h. to 91
m.p.h., and although this is probably an extreme case, it is clear that the
use of some form of altitude control becomes essential for any accurate
measurements of aeroplane performance. The revolution counter and the
airspeed indicator afford the pilot a means of adjusting the petrol- air
mixture to its best condition.
The prediction and reduction of aeroplane performance proceeds on
the assumption that all precautions have been taken in the adjustment
of the petrol supply to the engine, and that during a series of trials the
prevalence of up-currents will obey the law of averages, so that the mean
will not contain any errors which may have occurred in single trials.
The question of the calibration of instruments is not dealt with here,
but in the section dealing with methods of measurements of the quantities
involved in the study of aerodynamics.
Prediction of Aeroplane Performance
When the subject of prediction is considered in full detail, taking
account of all the known data, it is found to need considerable knowledge
and experience before the best results are obtained. A first approximation
to the final result can, however, be made with very little difficulty, and
this chapter begins with the material and basis of rapid prediction, and
proceeds to the more accurate methods in later paragraphs.
Rapid Prediction. — An examination of numbers of modern aeroplanes
will indicate to an observer that the differences in form and construction
are not such as to mask the great general resemblances. Aeroplane bodies
and undercarriages present perhaps the greatest individual characteristics,
but a first generalisation is that all aeroplanes have sensibly the same
external form. Aeroplanes to similar drawings but of different scale
would be described as of the same form, and the similarity is extended to
the airscrew. Even the change from a two-bladed airscrew to one with
four blades is a secondary characteristic in rapid prediction.
The maximum horizontal speed of which an aeroplane is capable, its
maximum rate of climb and its " ceiling," are all shown later to depend
only on the ratio of horsepower to total weight, and the wing loading, so
long as the external form of the aeroplane is constant. The generalisation
as to external form suggests a method of preparing charts of performance,
and such charts are given in Figs. 202-204.
Maximum Speed (Fig. 202). — The ordinate of Fig. 202 is the maximum
speed of an aeroplane in m.p.h., whilst the abscissa is the standard horse-
power per 1000 lbs. gross load of aeroplane. The standard horsepower
is that on the bench at the maximum revolutions for continuous running.
PREDICTION AND ANALYSIS FOR AEROPLANES 399
M
r
T
\
\
l\
To allow for loading, unless 7 lbs a' .
Divide Standard B.H.Ra\<ailable by('^y*
and read oFf the velocity on the diagram.
Multiply the velocity so read b>j(M^)i to
get the correct speed.
W = Wt in lbs.
a/ IS loading in lbs/a
i\
\
i\
\
\v
\
1
\
\
1
A
\
l\
\
V
I
\
\
o
o-
o
\,
J
1
\
^
^
\
i]
s.
\
\
\
\
\
\
in
S^
J
\^
\
V
\
O
O
o
IN
3
\
^
;
o
\^
\
1
\
J
\
V
1
o -■
00 o
o
o
Q. -g
o
<
V)
o or
w a uj
~ O uj
i-as
o
q: 2
400 APPLIED AEEODYNAMICS
A family of curves relating speed and power is shown, each curve of
the family corresponding with a definitely chosen height. The curves
may be used directly if the wing loading is 7 lbs. per sq. foot ; for any
other wing loading the formula on the figure should be used.
Example 1. — Aeroplane weighing 2100 lbs., h.p. 220. Find the probable top speed
at the ground, 6500 ft., 10,000 ft,, 15,000 ft., and 20,000 ft., assuming that the engine
may be run " all out " at each of these heights. The wing loading is to be 7 lbs. per
sq. foot.
h.p. pet 1000 lbs. = 105
and from Fig. 202 it is found that —
At ground Top speed = 124 m.p.h.
„ 6,500 ft. „ =123 „
„ 10,000 ft. „ ==121 „
„ 15,000 ft. „ =117 „
„ 20,000 ft. „ =103 „
This example illustrates the general law, that the top speed of aeroplanes
with non-supercharged engines, falls off as the altitude increases, slowly
for low altitudes but more and more rapidly as the ceiling is approached.
Example 2. — The same aeroplane will be taken to have increased weight and horse-
power, the wing loading being 10 lbs. per sq. foot instead of 7 lbs. per sq. ft., but the
horsepower per 1000 lbs. as before.
By the rule on Fig. 202 find _12L., i.e. 88. :
VT
On Fig. 202 read off the speeds for 88 h.p. per 1000 lbs. weight.
Ground Speed for 88 h.p. per Speed for 105 h.p. per
1000 lbs. and 7 lbs.
1000 lbs. and 10 lbs.
6,500 ft.
10,000 ft.
15,000 ft.
20,000 ft.
per sq. ft. .
= 117
= 115-5
= 114
= 109
= 88
per sq. ft. .
= 140 m.p.h.
= 138 „
= 136 .,
= 130 „
= 105 „
by>/"
To get the real speed for 105 h.p. per 1000 lbs. multiply the figures in the second column
The results are given in the last column of the table, and the point of interest
is the increase of top speed near the ground due to an increase in loading. The penalty
for this increase in top speed is an increase in landing speed in the proportion of -v/lO to
-y 7, i.e. of nearly 20 per cent. There are also losses in rate of climb and in ceiling.
Maximum Rate of Climb (Fig. 203). — The ordinate of the figure is the
rate of climb in feet per minute, whilst the abscissa is still the standard
horsepower per 1000 lbs. gross weight. The same aeroplanes as were used
for Examples 1 and 2 will again be considered.
Example 3. — Find the rate of climb of an aeroplane weighing 2100 lbs. with an engine
horsepower of 220, the loading of the wings being 7 lbs. per sq. foot.
The standard h.p. per 1000 lbs. is 105, and from Fig. 203 the following rates of climb
are read off :• —
Ground Rate of climb = 1530 ft.-min.
6,500 ft. „ =1120 „
10,000 ft. „ =890 „
15,000 ft. „ =580
20,000 ft. „ =270
PKEDICTION AND ANALYSIS FOE AEEOPLANES 401
The rapid fall of rate of climb with altitude is chiefly due to the loss
of engine power with height, and it is here that the supercharged engine
would make the greatest change from present practice. The ceiling, or
I
1800
r-
i
r
1600
/
J
f
J
1400
RATE OF
CLIMB
Ft MIN.
1200
/
i
/
/
f
/
/
/
f
/
f
/
/
f
1000
/
/
/
V
#
/
/
/
,/
800
(
f
o.
/
/
J
/
/
f
i/
/
/
600
/
/
/
f
i
^A
/
/
/
f
/
i
/
/
/
f
/
^
oC
r
r
I
/
/
/
/
/
/
^
/
r
/
/
/
/
J
/
/
ZOO
/
/
/
J
/
/
n
u
f
(z
r
z
z
20
40
140
60 80 100 120
Standard H.P./IOOO lbs.
Fig. 203. — ^Rate of climb and horsepower chart for rapid prediction.
To allow for loading, unless 7 lbs. /ft. ^.
Multiply Std. H.P./1000 lbs. when climb is zero by / ~ j , then subtract the excess of this
over the value when w = ^ from the Std. H.P./1000 lbs.
W = wt. in lbs. w = loading in Ibs./ft.*.
height at which the rate of climb is zero, is seen to be just below 25,000 ft.
A further diagram. Fig. 203a, is drawn to show this point more simply'
and from it the ceihng is given as 24,000 ft.
Example 4.— Conditions as in Example 2, where the loading is 10 lbs. per sq
foot
2 D
402
APPLIED AEEODYNAMICS
The rule on Fig. 203 is applied below.
(1)
(2)
(3)
(4) (5)
Std.
h.p. at zero
i^)W?
(2)-(l).
105— numbers Rate of climb from
rate of climb.
in (.3). (4) and Fig. 203.
Ground
26
31
5
100 1350
6,500 ft.
36
43
7
98 940
10,000 ft.
45
54
9
96 700
15,000 ft.
60
72
12
93 380
20,000 ft.
83
99
16
89 60
Ceiling
—
—
/iO 0-, 0 . . . ceiling
-105x>/y = 88 21,000 ft.
The effect of increasing the loading in the ratio 10 to 7 is seen to be a
reduction in the rate of chmb of nearly 200 ft. per minute, and a reduction
of the ceiHng of about 3000 ft.
The four examples illustrate a general rule in modern high-speed
30,000
Ceiling
^^
>
•^
y
y^
/
/^
/
/
,
/
/
/
/
1
J.
20,000
HEIGHT
(Feet)
10.000
20
40
60
80 100 120
Standard H.P./IOOO lbs.
140
Fig. 203a. — Ceiling and horsepower chart for rapid prediction.
The curve applies at a loading of 7 Ibs./ft.^.
An approximate formula which applies to all loadings is
At ceiling, (j^^ f(h) = O-QIO
W
W = wt. in ]
Std. B.H.P.
., (T — relative density, w — loading in Ibs./ft,-.
aeroplanes, that high speed is more economically produced with heavy
wing loading than with light loading, whilst rapid chmb and high ceiHng
are more easily attained with the light loading. The reasons for this
appear from a study of the aerodynamics of the aeroplane, which shows
PEEDICTION AND ANALYSIS FOE AEEOPLANES 403
that the angle of incidence at top speed is usually much below that giving
best lift/drag for the wings, so that an increase of loading leads to a better
angle of incidence at a given speed. For climbing, the angle of incidence
is usually that for best lift/drag for the whole aeroplane, and the horse-
power expended in forward motion (not in climbing) is proportional to
the speed of flight. To support the aeroplane, this speed of flight must
be increased in the proportion of the square root of the increased loading
to its original value. It is not possible in climbing to choose a better angle
of incidence.
Rough Outline Design for the Aeroplaneof Example 1.— In estimating the
approximate performance the data used has been very limited, and no
indication has been given of the uses to which such an aeroplane could
be put. How much of the total weight of 2100 lbs. is required for the
engine and the structure of the aeroplane ? How much fuel will be
required for a journey of 500 miles ? What spare load will there be ?
Structure Weight. — The percentage which the structure weight bears
to the gross weight of an aeroplane varies from 27 to 32 as the aeroplane
grows in size from a gross load of 1500 lbs. to one of 15,000 lbs. The
smaller aeroplanes usually have a factor of safety greater than the large
ones, and so for equal factor of safety the difference in the structure weights
would be greater than that quoted above. For rough general purposes,
the structure weight may be taken as 30 per cent, of the gross weight.
Engine Weight. — The representative figure is " weight per standard
horsepower," and for non-supercharged motors the figure varies from
about 2*0 lbs. per h.p. for a radial air-cooled engine to 3-0 lbs. per h.p.
for a light water-cooled engine. For large power, water-cooled engines
are the rule, whilst the smaller-powered engines may be either air-cooled
or water cooled. As a general figure 3 lbs. per h.p. should be taken as
the more representative value.
Weight of Petrol and Oil. — An air-cooled non-rotary engine or a water-
cooled engine consumes approximately 0*55 lb. of petrol and oil per brake
horse-power hour when the engine is all out.
The consumption of petrol varies with the height at which flight takes
place roughly in proportion to the relative density o-. The general figure
for fuel consumption is then
0-55(T lb. per standard h.p. hour.
Example 5. — Estimates of weight available for net load can now be made.
Total weight of aeroplane 2100 lbs.
Structure 2100 x 0-30 630 lbs.
Engine 220 x 3 660 lbs.
Fuel for 500 miles, i.e. 4 hrs. at a height of 10,000 ft.
4 X 0-55 X 0-74 X 220 360 lbs.
For pilot passenger and useful load 450 lbs.
Out of this 450 lbs. the pilot and passenger weigh 180 each on the
average, leaving about 90 lbs. of useful load in a two-seater aeroplane, or
270 lbs. of useful load in a single-seater aeroplane.
In this way a preliminary examination of the possibilities of a design
to suit an engine can be made before entering into great detail.
1650
404 APPLIED AEEODYNAMICS
More Accurate Method of predicting Aeroplane Performance
In the succeeding paragraphs, a method of predicting aeroplane
performance will be described and illustrated by an example. At the
present time, knowledge of the fundamental data to which resort is
necessary before calculations are begun has not the accuracy which makes
full calculation advantageous. Simplifying assumptions will be introduced
at a very early stage, but it will be possible for any one wishing to carry
out the processes to their logical conclusions to pick up the threads and
elaborate the method. Another reason for the use of simpHfying assump-
tions is the possibility thereby opened up of reversing the process and
analysing the results of a performance trial. It appears in the conclusion
that the number of main factors in aeroplane performance is sufficiently
small for effective analysis of aeroplane trials, with appeal only to general
knowledge and not to particular tests on a model of the aeroplane.
In estimating the various items of importance in the design of an aero-
plane as they affect achieved performance, it is convenient to group them
under four heads : —
(a) The estimation of the resistance of the aeroplane as a glider
without airscrew,
(b) The estimation of airscrew characteristics.
(c) The variation of engine-power with speed of rotation.
(d) The variation of engine power with height.
It is the connection of these four quantities when acting together which
is now referred to as prediction of aeroplane performance. In the example
chosen the items (a) to (d) are arbitrarily chosen, and do not constitute
an effort at design. It is probable that the best design for a given engine
will only be attained as the result of repetitions of the process now developed,
the number of repetitions being dependent on the skill of the designer.
Of the four items, (a) and {b) are usually based on model experiments,
of which typical results have appeared in other parts of the book. The
third item is obtained from bench tests of the engine, whilst the fourth
has hitherto been obtained by the analysis of aeroplane trials with support
from bench tests in high-level test houses.
It has been shown that the resistance of an aeroplane may be very
appreciably dependent on the slip stream from the airscrew, and for a
single-seater aeroplane of high power the increased resistance during chmb,
of the parts in the slip stream may be three times as great as that when
gliding. One of the first considerations in developing the formulae of
prediction relates to the method of dealing with slip-stream effects.
Experiments on models of airscrews and bodies at the National Physical
Laboratory have shown certain consistent effects of mutual interference.
The effect of the presence of a body is to increase the experimental mean
pitch and efficiency of an airscrew, whilst the effect of the airscrew shp
stream is to increase the resistance of the body and tail very appreciably.
The first point has been dealt with under Airscrews and the latter when
dealing with tests on bodies. It is convenient to extract here a typical
PEEDICTION AND ANALYSIS FOE AEROPLANES 405
instance of body resistance as affected by slip stream because the formulae
developed depend essentially on the observed law of change.
For a single-engined tractor aeroplane the total resistance coefficient
has a minimum value at moderately high speeds, say 100 m.p.h. near the
ground, and of this total roughly 40 per cent, is due to parts in the slip
stream. If R^ be the resistance of the parts in the slip-stream region,
but with zero thrust, and R/ the resistance of the same parts when the
airscrew is developing a thrust T, then
f = 0-85 + 1-2^-^,. (1)
is a typical relation between them. Without exception an equation of
the form of (1) has been found to apply, variations in the combination of
airscrew and body being represented by changes in the numerical factors.
Using this knowledge of the generahty of (1) leads to simphfied formulae
in which the airscrew thrust and efficiency have somewhat fictitious values
corresponding with an equally fictitious drag for the aeroplane. It will
be found that the efficiency of the airscrew and the drag of the aeroplane
so used are not greatly different from those of the airscrew and aeroplane
when the effects of interference are omitted.
A more detailed statement will make the assumptions clear. If T be
the thrust, V the forward speed, W the weight of the aeroplane and Vg its
rate of climb,
T = R + W^" (2)
on the justifiable hypothesis that the thrust is assumed always to act
along the drag axis. The hypothesis which is admitted here is not admis-
sible in calculations of stabihty because the pitching moment is there
involved, and not only the drag and lift. Another assumption which will
be made is that the inchnation of the flight path is so small that the cosine
of the angle is sensibly equal to unity.
The resistance R depends appreciably on the shp stream from the
airscrew, but that fraction which is in the slip stream is not greatly affected
by variations of the angle of incidence of the whole aeroplane. The part
of the resistance which arises from the wings and generally the part not
in the slip stream, is appreciably dependent on the angle of incidence and
is related to the lift coefficient, hj^.
R may therefore be written as
R = Ro + R/ (3)
where Rq represents the resistance of parts outside the slip stream, and
Ri' the resistance of the parts in the slip stream. Equation (1) is now
used to express R/ in terms of the resistance of the parts in the absence of
shp stream. If R^ be the glider resistance of the parts,
R/ = R,{a + fc(^) \] (4)
The value of ^o^g is not strictly equal to fe^ on account of the load on the
406 APPLIED AEKODYNAMICS
where a and h are constants, and k^ is the thrust coefficient defined by
'^^^ (^^
" a " is usually less than unity apparently owing to the shielding of
the body by the airscrew boss. Its value is seen to be 0-85 in equation (1),
and this is a usual value for a tractor scout. " & " is more variable, and the
tests on various combinations of body and airscrew must be examined in
any particular case if the best choice is to be made.
Using the various expressions developed, equation (2) becomes
T = Eo + E,{a + <^^)~\| + W^'' .... (6)
Equation (6) will now be converted to an expression depending on
fej, kjy, and fcj, by dividing through by pSV^ where S is the wing area.
tail, but the approximation is used in the illustration of method as suffi-
ciently accurate for present purposes. With these changes equation (6)
becomes
{^-'-KAb)^|(J))~\ = Wo + «W^ + A;LY" ... (7)
D^
The factor ^, — h(kj,)i inequation (7) will now be recognised as a constant
for all angles of incidence, and it is convenient to introduce a fictitious
thrust coefficient defined by •
h'=^[^-^AK)i]h (8)
The curve representing this overall thrust coefficient as a function of
advance per revolution differs from that of the airscrew in the scale of
its ordinates. To estimate the value of the multiplying factor for the
new scale the following approximate values may be used : —
^-, = 5, b = l% (U = 0-01 (9)
and the coefficient of /c^ in (8) is 0-94. The new ordinate of thrust is then
6 per cent, less than that of the real thrust. As the effect of the body is to
increase the airscrew thrust, it will be seen that the fictitious thrust co-
efficient is within 5 per cent; of that of the airscrew alone over the useful
working range.
The term (/cd)o + a{ho)i may be regarded as a fictitious drag coefficient
for the aeroplane as a glider. The correct expression for the glider drag
coefficient being (kjy)o + {kB)iy the departure of the coefficient "a" from
unity is a measure of the difference between the fictitious and real values
of the drag coefficient. Prom the numerical example quoted it will be
found that the difference is 6 per cent, of the minimum drag coefficient
PREDICTION AND ANALYSIS FOR AEROPLANES 407
of the whole aeroplane. It has been previously remarked that this
difference arises from the shielding of the body by the airscrew boss,
and in any particular case the effect could be estimated with fair accuracy
if required as a refinement in prediction.
The equation for forces which corresponds with (7) is
T'^D' + W^" . . . ; . . (10)
where T' and D' may be regarded provisionally as the thrust of the air-
screw and the drag of the aeroplane estimated separately.
Since D' depends only on the air speed of the aeroplane, it is possible
to deduce from (10) a relation of a simple nature between thrust and climb,
if flying experiments be made at the same air speed but at different throttle
positions. The relation is
8r=^8\\ (11)
where 8Ye is the increment in rate of climb corresponding with an increase
W
of thrust ST'. Since ^ and SV^ are measured during performance,
equation (11) can be used in the reverse order to deduce ST' from a trial.
The treatment of slip stream given above completes the special
assumptions ; at various places assumptions have been indicated which
may become less accurate than the experimental data. The more accurate
algebraic work which would then be required presents no serious difficulty.
Details of a Prediction Calculation
Calculations will be made on assumed data corresponding roughly
with a high-speed modern aeroplane ; although the actual numbers are
generally representative of an aeroplane they have been taken from
various sources on account of completeness, and not on account of special
qualities as an efficient combination in an aeroplane.
Data required.
(1) Drag and lift coefficients of the aeroplane as a glider, corrected for
shielding of airscrew boss.
V
(2) Thrust and torque coefficients of the airscrew as dependent on -=-•
V
(For general analysis -^ has been preferred ; if P and D be known
V V
the variables -— and ^ are easily converted from one to the other.)
nD nP *^ '
The correction for slip-stream factor indicated in (8) is supposed to have
been made.
(3) Engine horsepower as dependent on revolutions at standard density
and temperature.
(4) Engine horsepower as dependent on height. A standard atmo-
sphere is used.
408
APPLIED AERODYNAMICS
The brake horsepower of the engine under standard conditions will be
denoted by " Std. B.H.P.," whilst the factor expressing variation with
height will hefQi). At any height in the standard atmosphere the brake
horsepower at given revolutions will be
(B.H.P.)a=/W xStd. B.H.P.
(12)
H
,^
^
--.
N,
\
\
•^
K
^
\
w^Ff
ICIEN
:y
t
/'
\
V
\
0012
>^
\
\
\u*^
>
^
' s
A
\
V
OOIO
TORQ
\\
v
\
JECOE
FFICIEI
IT ^
^
\j'
\
<THRl
tSr COEFFICI
:NTjy
f
^
\,
S
\
10008
10
■"■
\
\
s.
\
0006
\
\
\
\
X
\
\
V
H^
Pt
\
\
\
\,
\
\
01X)2
\
\
\
\
k
V^^
\
s,
000
^s
^
^
80
70
60
PERCENTAGE
EFFICIENCY
50
40
30
20
10
0-4 0-5 0-6 0-7 0-8 0-9 10
yhP i.e. ADVANCE PER REVOLUTION AS A FRACTION
OF THE EXPERIMENTAL MEAN PITCH
Fio. 204. — ^Airscrew characteristics used in example of prediction.
From the ordinary definition of torque, Q, and torque coefficient,
/cq => ;;;;;^-g, the expression
is deduced.
'^=2-^5g-St<l.(B.H.P.) ,
. (18)
It should be noticed from (13) that the value of ,,^: is independent of
PEEDICTION AND ANALYSIS FOE AEEOPLANES 409
the aerodynamic properties of the aeroplane, and the revolutions of the
engine and airscrew are therefore calculable for various speeds of flight
240
'
220
^^
^
^
"^
200
STAN
B.
y
^
}ARD
H.P.
y
^
180
y
y
/
/
(«)
160
/
/^
/
140
/
700 800 900 1000 1100 1200 1300 1400
ENGINE R.P. M
ru
\
\,
08
\
\
\
(6)
06
HEI6
FORH
s
HT FA
ORSEP
CTOR
OWER
X
\
0 4
^
^^
0-2
^^
0
O • lO.OOO 20,000 30.000
HEIGHT (Ft.)
Fig. 205. — (a) Engine characteristics used in example of prediction.
(6) Variation of engine power with height used in example of prediction.
without knowledge of the drag and lift of the aeroplane. This is the first
step in the prediction.
Airscrew Revolutions and Flight Speed. — The data required are
given in Pigs. 204 and 205, to which must be added the diameter.
410
D = 8-75 feet, and the pitch, P
leads to
APPLIED AEEODYNAMICS
10 feet. For these data equation (13)
a (r.p.m.)3
(14)^
The relative density, o-, is unity in a standard atmosphere at a height
of about 800 feet, this value having been chosen to conform with the
standards of the Aerodynamics laboratories throughout the world and
with the average meteorological conditions throughout the year*
The following table is compiled fromFigs. 204 and 206 and equation (14).
TABLE 3.
B.p.m.
Std. B.H.P.
Std. B.H.P.
(r.p.m.)''
K
Ground.
6000 ft.
10,000 ft.
15,000 ft.
20,000 ft.
1400
1350
1300
1250
226-0
223-4
220-0
216-5
8-23 X 10-*
9-10 X 10-"
10-00 X 10-*
1110 X 10-*
001245
0-01380
001510
0-01215
0-01345
0-01475
001175
0-01305
0-0145
0-01125
0-01245
0-01360
0-015Q5
0-01080
0-01195
0-01315
0-01460
V
From the values of Aiq and the curves of Fig. 204 the values of -p
can be read off and the value of V calculated, leading to Table 4.
TABLE 4.
Ground.
5000 ft.
10,000 ft.
15,000 ft.
20,000 ft.
B.p.m.
V
Vft.
V
Vft.
V
Vft.
V
Vft.
V
Vft.
nP
per sec.
nP
per sec.
mP
per sec.
nP
per sec.
nP
per sec.
1400
0-692
161-5
0-705
165
0-728
170
0-750
175
0-768
179
1350
0-611
137-5
0-635
143
0-660
148-5
0-692
156
0-717
161-5
1300
0-420
91-0
0-496
107-5
0-545
118
0-622
135
0-652
141-5
1250
—
—
—
—
—
0-430
89-5
0-520
108-5
Table 4 shows the relation between the engine revolutions and the
forward speed of airscrew for all altitudes, the engine being " all out."
The relationship is shown diagrammatically in Fig. 206. The corresponding
relation between -^ and the forward speed of the airscrew is also shown
nr
in Fig. 206.
The fall of revolutions with height which is observed in level flights
* Throughout the theoretical part of the book the units used have been the foot and second
with forces measured in pounds. The unit of mass is then conveniently taken as that in a
body weighing g lbs., and has been called the " slug." Common language has other units,
speeds of flight being in miles per hour, rate of climb in feet per min., and rotation in revolu -
tions per minute. Where the final results are required in the common language, early adoption
often leads to a saving of labour.
PEEDICTION AND ANALYSIS FOE AEEOPLANES 411
is deducible from these observations and the properties of the aeroplane
as below: —
The expression for lift coefficient in terms of weight is
W 1
W
S >V2'
and in the example the loading ^ will be taken as 7 lbs. per square foot.
R.P.M
140 160
SPEED \^(f/s)
Fig. 206. — Calculated relations between forward speed, engine speed, and advance
per revolution as a fraction of the pitch.
Converting to common units and particular values for the aeroplane leads
to
1372
^' ^(V„..p.h.)^
(16)
The quantity aiV is important and has been called indicated air
412
APPLIED AEEODYNAMICS
speed. Equation (16) shows that kj^ depends on the indicated air speed
and not on the true speed.
Fig. 207 shows the value of drag coefficient for a particular aeroplane
0-14
0-12
010
0 06
006
004
002
■-
PRAG
COEFF
ICIENT
1
1
/
/
X
>-
N
/
/
/
N
v
/
/
/
\
/
/
f
\
1
/
/
>
\
^
y
I
T
7
L
10
8
6
01
02 0-3 0-4
LIFT COEFFICIENT
0-5
0 6
Fig. 207. — Aeroplane glider characteristics used in example of prediction.
glider as dependent on lift coefficient, and hence with (16) leads to a
knowledge of drag coefficient for any value of the indicated air speed.
The equivalent of equation (10) as applied to the relation between
thrust, drag and lift coefficients is
PREDICTION AND ANALYSIS FOR AEROPLANES 413
T)2x y .-2 Y
HId) *^=«=»+4 • • • • ■ c^)
and the determination of kjy and /cj, for any angle of climb together with
V
equation (17) leads to the estimation of -^ and kj,, since the latter is
V
known when -=r is known. For convenience of use in connection with
wD
(17) a new curve has been prepared in Fig. 204, which shows the value of
-(— j fcr as dependent on -^, and equation (17) may be rewritten as
w
It is now necessary to fix the area of the wings, S, or since -^ has
b
been taken as 7, the weight of the aeroplane. The value of S will be taken
as 272 sq. feet, giving a gross weight of 1900 lbs. With P=10 and D=8-75,
equation (18) becomes
i(«T)"''^ = ^'"('=" + '='v) <'">
Level Flights. — For level flying V^ is zero, and the calculation of per-
formance starts by assuming a value of indicated air speed CT^Vm.ph., and
calculating the corresponding value of fej, from (16). From the lift coefficient,
the value of the drag coefficient is obtained by the use of Fig. 207, and
1/ V \-2
equation (19) leads to the calculation of tI -^ ) ^t- One of the curves
V l/V\-2 V
of Fig. 204 gives ^r; for any value of -( -= ) km, and from ^^i and the curve
wP 4\nP/ wP
V
for torque coefficient /cq is obtained. From -^ the value of o*(r.p.m.) is
calculated since a^Y and P are known. Finally from equation (14) and
Std B IT P
the known values of ko and <Ti(r.p.m.), the value of '- — '- — '—^f(h) is
\ r- /7 r.p.m. •'^ ^
obtained as dependent on the airscrew.
A second value for this same quantity is obtained from the engine
curve, and the indicated air speeds for which the two agree are those for
steady horizontal fhght. The detailed calculation is carried out in the
table below.
For the values of indicated air speed chosen in column 1, Table 5,
equation (16) has been used to determine the lift coefficient of column 2.
The rest of the table follows as indicated above.
Columns 6 and 8 of Table 5 give a unique curve, PQ of Fig. 208, between
oi. J 1-? TT P
<T*r.p.m. and J{h) — '- — '- — '—^ for level flights. The relation between the
two quantities has been derived wholly from the aerodynamics of the aero-
plane, and will continue to hold if the engine be throttled down.
414
APPLIED AERODYNAMICS
TABLE 5.
1
2
3
4
5
6
7
8
Indicated
air speed.
Lift
coefflcient.
Drag
coefflcient.
From equa-
tion (19) and
col. 8.
V
mP
o-J . r.p.m.
K
/«^*^|iP
<r»Vin.p.h.
0-550
From col. 4
and curve
for airscrew.
From
cols. 1
and 5.
From col. 5
and curve for
airscrew.
From col8. 6
and 7 and
equation (14).
60
0-120
0 1390
0-457
962
00153
0 0903
62
0-507
0079
00915
0-535
856
00148
00696
53
0-489
0070
00811
0-558
837
00142
00651
54
0-470
00644
00747
0-577
826
00141
00626
66
0-437
0-0553
00642
0-606
815
00139
00598
58
0-408
00492
00570
0-627
815
00136
0 0580
60
0-381
00448
00520
0-646
818
00134
00578
65
0-325
00370
0-0429
0-682
838
00124
00560
70
0-280
0 0320
00371
0-707
872
00122
00595
80
0-214
00270
00313
0-733
960
00117
0 0691
90
0169
0-0250
00290
0-747
1060
00115
00827
100.
0-137
0-0244
00283
0-757
1163
00113
00990
110
0113
0 0240
0-0278
0-759
1280
00113
01180
120
0095
0 0240
0-0278
0-759
1395
00113
0-1405
130
0081
00240
0-0278
0-769
1510
00113
0-1660
To calculate the top speed, use is made of a graphical method of finding
when the engine horsepower is that required by the aerodynamics.
TABLE
6.
Relative
density,
Horse-
power
factor.
(T* .(r.p.m.).
Std. B.H.P. f,,^^ f^^^
engine.
Height
r.p.m.
(ft.).
■ ■
/(*).
r.p.m.
r.p.m.
r.p.m.
r.p.m.
r.p.m.
r.p.m.
= 1500
= 1400
1
= 1300
= 1600
= 1400
= 1300
Ground
1025
1038
1520
'■ 1418
1314
01582
0-1680
01760
5,000
0-874
0-842
1402
1309
1216
0-1284
0-1365
0-1435
10,000
0-740
0-686
1290
1204
1118
0-1046
01112
01165
15,000
0-630
0-558
1190
1111
1032
00861
00902
0-0948
20,000
0-635
0-446
1097
1024
950
0-0680
00722
0-0758
25,000
0-445
0-352
1000
933
866
00537
0 0670
0 0599
The curves connecting o*. r.p.m. and f{h)
Std. B.H.P.
r.p.m.
as deduced
solely from the engine are plotted in Fig. 208 from the numbers of Table 6.
The necessary calculations are simple, the necessary data being contained
in Figs. 205 {a) and 205 (&).
The curve PQ of Fig. 208 is that obtained from aerodynamics alone,
and applies at all heights. The separate short curves marked with the
height are from the engine data. An intersection indicates balance
between power available and power required for level flight. At 10,000 ft.
the balance occurs when <t« . r.p.m. = 1227. . Since o- = 0*74 this gives the
r.p.m. as 1427.
PREDICTION AND ANALYSIS FOR AEROPLANES 415
A further unique relation independent of the position of the engine
throttle is given by columns 1 and 6 of Table 5. For any value of <r* . r.p.ih.
vcu
■""-v^d.B.H.R
FROM ENGINE CURVE
>v^M
018
\
N,
s
OGROU
NO
016
A'
N
N
r
ES
TIMA
TOPS
TION
PEED
OF
/
N.
k
014
Std
v^oo
0 FT
/
B.H.P
P.M. ^
C/Ll
\
H
R.
01?
/
K
XK)0C
\t
010
/
^
\
Q.
J500C
/
0-08
/
^
7
/
^1°
)00 F'
'.
006
L
N
*
CEIL
ING 2
5800
^
^50G
0 FT.
004
002
0
400
600
1200-
1400
1600
800 ^ 1000
(J ^ f R P M )
Fig. 208. — Calculated relation between horsepower and revolutions for steady horizontal flight.
the value of a^Vm.p.h. is known (see Fig. 209). Full particulars of top
speed of aeroplane are now obtained from the intersections of Fig. 208 and
416
APPLIED AERODYNAMICS
the above relation between indicated air speed and revolutions. The results
are collected in Table 7.
GROUND
1400-
A
ysoooFt
1200-
A
^00 F
t.
/
y^5.000 f
t.
J-
y
/
^
i^-^^
^
/^20,
)00 F
:
cr^R
.P.M.
v|^
25,00
0 Ft.
STA
SP
.LING
:eo
c
\
EILIN(
400-
cOO
20
40
60 ** ^ 80 100
120
Fig. 209. — Calculated relation between forward speed and revolutions.
TABLE 7.
Height (ft,).
o-i . r.p.m.
«'*Vm.p.h.
^m.p.h.
r.p.m.
Ground
1492
129
127-6
1472
5,000
1352
117
125
1448
10,000
1227
105-5
122-5
1427
15,000
1108
95
119-5
1398
20,000
/996
I (901)
83-5
114-2
1361
(50-5)
(69)
(1232)
25,000
/872
\ (820)
69-6
104-2
1307
(65)
(82-6)
(1230)
Fig. 209, which shows the relation between indicated air speed and a*
PREDICTION AND ANALYSIS FOR AEROPLANES 417
r.p.m., indicates the most direct comparison between prediction and
observations in level flight.
Maximum Rate of Climb. — It has already been shown that for the
condition of " engine all out " there is relation between the speed of flight
V
V and the quantity — . For certain specified heights this relation is
shown in Fig. 206. Using this relation and the values of lift and drag
coefficients of Fig. 207 it is possible to calculate the rate of climb Vg from
Rate of
Climb
(FT/Sj
20000FT.
25000 FT.
lOO 120
True Speed ^(f/s)
Fio. 210. — Calonlated rate of climb.
equation (19) for any assumed value of V. The procedure followed is the
calculation of the rate of cHmb for assumed air speeds and by the plotting
of the results finding the condition of maximum cUmb. A sample table
for ground level is given below.
After plotting in Fig. 210, the maximum rate of chmb was found to be
30*2 ft.-s. or 1815 ft.-min. The speed was 110 ft.-s. or an indicated
air speed of 76 m.p.h. The airscrew revolutions were 1320 p.m.
The calculation was repeated for other heights, and the results obtained
are shown in Table 9 and Fig. 210.
2 E
418
APPLIED AEEODYNAMICS
TABLE 8.
1
2
3
4
5
6
7
V
ft.-s.
K
K
V
fieu'np/ ^'j
Column 6
minus column 3.
ft.-8.
90
100
110
120
0-366
0-288
0-238
0-200
00412
0 0328
0-0288
00264
0-414
0-459
0-501
0-642
01531
01180
00940
0-0760
0-1119
0 0852
0-0652
0-0496
28-3
29-6
30-2
29-8
TABLE 9.
1
2
3
4
Height
(ft.).
Best indicated
air speed
(m.p.h.).
Maximum rate
of climb
(ft.-min.).
Airscrew
revolutions
(r.p.m.).
Ground
6,000
10,000
15,000
20,000
25,000
760
73-3
700
66-1
61-2
60-0
1815
1400
1020
690
400
40
1320
1310
1300
1286
1270
1245
ao 4.0 3PE£0(M.P.ri) 80 lOO 120 140
Fig. 211. — Detailed results of performance calculations.
PEEDICTION AND ANALYSIS FOB AEKOPLANES 419
The well-known characteristics of variation of performance with height
are shown in this table. The maximum rate of climb decreases rapidly
with height from 1815 ft.-min. near the ground to zero at a little more
than 25,000 feet. The best air-speed and airscrew revolutions both fall
off as the height increases.
The results of the calculations of top speed and rate of climb are
collected in Fig. 211, and illustrate typical performance curves. As the
data were not representative of any special aeroplane it is not possible
to make a detailed comparison with any particular trials, but within the
limits of general comparison the accuracy of the method of calculation
is amply great.
Theory of the Keduction of the Observations of Aeroplane
Performance from an Actual to a Standard Atmosphere
The problem is to find how to adjust observations under non-standard
conditions so that the results will represent those which would have been
obtained had the test been carried out in a standard atmosphere. General
theoretical laws govern the aerodynamics of the problem, and a relation
between the power required by the airscrew and that available from the
engine must be satisfied.
As in most aeronautical problems, the assumption is made that over
the range of speeds possible in flight the resistances of the aeroplane for
a given angle of incidence and advance per revolution of the airscrew
vary as the square of the speed. With the possible exception of airscrews
having high tip speeds the assumption has great practical and theoretical
sanction.
To develop the method, consider the forces acting on an aeroplane
when flying steadily. The weight is a force which, both in its direction
and magnitude, is independent of the motion through the air. The
resultant air force must be equal and opposite to the weight if the flight
is steady, but the magnitude and direction are fixed solely by motion
relative to the air. Fig. 212 helps towards the mathematical expression
relating the weight and resultant air force.
A line, assumed parallel to the wing chord for convenience, is fixed
arbitrarily in the plane of symmetry of the aeroplane. The direction of
motion makes an angle a with this datum line, and the velocity is V. The
airscrew revolutions are n, and if similarity of external form is kept and the
dimension of the aeroplane defined by I, it is known experimentally that
R and y, the resultant force and its angular position, are dependent on
a, V, n, I and the density of the air. As was shown in discussing dynamical
similarity, a limit to the form of permissible functions of connection is
easily found.
The variable I will be departed from at once and will be replaced by
1)2
two variables, S for l^ and D for I. The quantity — must be kept
420
APPLIED AERODYNAMICS
constant, but otherwise the use of the two leads to expressions of common
form more readily than I. The functional relations required are
R=pV^Sy(a,^^) (20)
the first giving the magnitude of R and the second its direction.
The conditions of steady motion are seen from Fig. 212 to be R=W and
y = 6, and equations (20) and (21) become
'A^'ud) ^^^)
^^{a,^} ...... (23)
S .p\2
' d
These equations contain the fundamental formulae of reduction and
are of great interest. It will be noticed that the important variables are
W
the loading per unit area, -^, the air speed, <t*V, the angle of chmb
b
(^— a), the angle of incidence of the wings, a, and the advance per revolu-
V
tion as a fraction of diameter, -=-.
nD
Level Flight. — As the angle of climb is zero, 6 is equal to a, and
equation (23) shows that ^ is a function of a only. Equation (22) then
PEEDICTION AND ANALYSIS FOE AEROPLANES 421
shows that the angle of incidence is determined by the wing loading and
air speed. For an aeroplane S is fixed and W varies so little during trials
that it may be considered as constant, and the important conclusion is
reached that the angle of incidence in level flight depends only on the
air speed. No assumption has been made that the engine is giving full
power.
For the same aeroplane carrying different loads inside the fuselage,
equation (22) shows the relation between loading and air speed which
makes flight possible at the same angle of incidence, and, given a test at
W
one value of -^, an accurate prediction for another value is possible. It
b
is only necessary to introduce consideration of the engine for maximum
speed. The details are given a little later in the chapter.
Climbing Flight. — For a given loading and air speed, equation (22)
V
shows a relation between a and -^ which, used in equation (23), deter-
nv
mines 6 and hence the angle of climb, 6 — a. Unless another condition
be introduced, such as a limit to the revolutions of the engine or the
V
knowledge that the throttle is fully open, both air speed and ^=- can be
varied at the pilot's wish. Before the subject can be pursued, therefore,
the power output of the engine must be discussed.
Engine Power. — The engine power depends on many variables, but
the only ones of which account is taken in reduction are the revolutions
of the engine and the pressure and density of the atmosphere. The
particular fuel used is clearly of great importance, as is also the condition
of the engine as to regular running and efiicient carburation. These
points may be covered by bench tests, using the same fuel as in-flight and
by providing a control for the adjustment of the fuel-air mixture during
flight. This latter adjustment can be used to give the maximum airscrew
revolutions for a given air speed.
Unless the points mentioned receive adequate attention during test
flights it is not possible to make rational reductions of the results.
At full power the expression
¥=<f>{n,p,p) (24)
is used to connect the power, revolutions and atmospheric pressure and
density. The form of <f> is determined by bench tests where the three
variables are under control.
The torque of the engine, Q^, is readily obtained as
and this must balance the airscrew torque, which by the theory of dimen-
sions has the form
Qa=pn2D6^a, ^) (26)
422 APPLIED AEEODYNAMICS
Eor known values of p and p the equality of the two values of Q gives
V
a relation between n, a, -^ and D. In the early part of this chapter,
when dealing with prediction the detailed interpretation of this relation
was given, D being constant and tp independent of a. Theoretically the
present equations are more exact than those used before, but they are
not yet in their most convenient form. Equating the two values of Q
leads to
<^(n,y,/>)=/,n3D«.|^.^(a,^)
YnD\3 D2 27r / V\
_(^VW.D^ 2,r /nD\3 / V\
The next step is to use equation (22) to substitute for pV^S in terms of
W, and equation (27) becomes
W3 D2 27r /nD\3 ^x'nD/
W
If the loading per square foot, i.e. -^, be denoted by w, and
b
\{a, -yr) be written for the quantity beginnmg with -q-, equation ^
(28) reduces to the important relation
|V^x(.l) (-) I
The result of the analysis has been to introduce a variable which
contains as a factor the horsepower per unit weight, a quantity well known
to be of primary importance in the estimation of the performance of an
aeroplane.
A combination of equations (22) and (29) shows that the angle of
incidence and advance per revolution of the airscrew are fixed for all
P / ~ j«
aeroplanes of the same external form if the quantities vt,\/- and -^^rs
W ^ m; pV'*
are known. In level flight it has been seen that the angle of incidence is
w
a function of the advance per revolution and it now follows that -^
is a function of TT-r\/ ^ • The angle a is rarely used in reduction, but -^
YS ^ w ° "^ nD
is of importance. The power P as used, has been the actual power and is
equal to/ . Pg, where Pg is the standard horsepower and/ the power factor
which allows for changes of pressure and temperature from the standard
condition.
PEEDICTION AND ANALYSIS FOB AEKOPLANES 423
A figure illustrating the relation between the quantities of import-
ance in level flight is shown (Fig. 213). The units are feet and sees.
where not otherwise specified. For international comparisons p would
be better than o-, as the dimensions of the quantities are then zero and
consequently the same for any consistent set of dynamical units.
For chmbing flight, the form adopted needs development ; since
V / p w . V
^^r^\/ - and -^ determine both a and -^, it follows from equation (23)
that they also fix 6— a, the angle of chmb. The value of -^ is equal
10 20 30
1000. f. p^y^
40
Fig. 213. — Fundamental curves of aeroplane performance.
to sin 6, and hence an equation for the rate of climb may be written as
V
(30)
or, multiplying by \/- on both sides,
^ w
Vo\/^ = Vv^^F/a, I-) (31)
Equation (22) shows that V\/ - is a function of a and -^, and hence it
^ w nD
424
APPLIED AEEODYNAMICS
follows from (31) that \o\/ - is also a function of a and
^ w
seen above
'^^wn/It,
nD
, or as was
w
The results obtained from a climbing test on an aeroplane are shown
in Piff. 213, which now connects the variables ^,\/-, V\/-, Yq\/ -
Y \\ ^ w ^ w ^ w
and -^ for both level and climbing flights. The condition that the
rate of climb is to be a maximum converts V\/ - from an independent
to a dependent variable. For a complete record of aeroplane performance
Yx/ - and :^j\/ - would need to be considered as independent variables,
making an infinite series of curves of which the figure illustrates the two
most important cases.
The general theorem has important applications in which all the
variables are used. For the reduction of performance simplifications can
be made, since in the process W, w and D are constant.
Application of the Formulae of Reduction to a Particular Case
Observations on a high-speed scout taken in flight are shown in Table 10.
TABLE 10.— n) Climb.
Aneroid height,
feet.
Time,
min. sec.
Temperature,
Indicated air
speed (m.p.h.).
B.p.m.
0
27
75
1490
4,000
0
18
75
1495
6,000
1 28
16
75
1500
8,000
3 12
11
75
1500
10,000
5 7
7
75
1505
12,000
7 4
3
70
1480
14,000
9 22
- 1
70
1485
15,000
10 41
- 2
70
1485
16,000
12 3
- 4
70
1485
17,000
13 38
- 6
70
1480
18,000
15 18
- 8
70
1485
19,000
17 4
-10
70
1480
20,000
18 60
-10
70
1480
(2) Levd Speeds.
Aneroid height,
feet.
Temperature,
Indicated air speed
(m.p.h.).
Il.p.m.
20,000
-10
87
1565
18,000
- 8
91
1680
16,000
- 4
98
1610
14,000
- 1
101
1620
12,000
3
107
1636
10,000
7
111
—
PKEDICTION AND ANALYSIS FOE AEEOPLANES 425
After preliminary tests to find the best air speed, the aeroplane was
climbed to 20,000 feet, readings being taken of time, temperature of the
air, indicated speed and engine revolutions at even values of height as
shown by the aneroid barometer. The level flights with the engine all
out were then taken at even values of the aneroid height by stopping at
each height on the way down.
The bench tests of the engine are shown in Figs. 214 and 215, the first
showing and the horsepower at standard pressure and temperature and
the second the pressure and temperature factor for variations from the
standard.
Aneroid Height. — The aneroid barometer is essentially an instru-
ment for measuring pressure, the relation between the two quantities
aneroid height and pressure being shown in columns 1 and 2 of Table 2.
The aneroid height agrees with the true height only if the temperature be
10° C. Since the difference of pressure between two points arises from
the weight of the air between them, i.e. depends on the relative density,
it will follow that at any other temperature than 10° C. the relation
between real height H and aneroid height h will be obtained from the
equation
dH. 273 + f
dh^^mr ^^^^
where t is the temperature Centigrade. This gives a relation which, in
conjunction with the measurement of t, enables the real height H to be
calculated for actual conditions. For present purposes this would not
be important unless the day happened to be a standard day.
The pressures as shown in Tables 1 and 2 are based on a unit of 760 mm.
of Hg at the ground, a temperature of 15°'6 C, and a relative density of
unity. The relation between p, t and a- is then
288-6 ,„„,
^=^2-73Tt-^ ^^^^
From the observations and figures. Table 11 is now prepared.
TABLE 11.
1
2
3
4
5
6
7
Aneroid
height
(ft.).
Relative
pressure,
(atmos.).
Tempera-
tnre,
°C.
Height factor
for iMJwer,
Relative
density.
<ri
<ri r.p.m.
20,000
18,000
16,000
14,000
12,000
10,000
0-480
0-516
0-655
0-597
0-643
0-692
-10
- 8
- 4
- 1
3
7
0-489
0-525
0-565
0-605
0-646
0-695
0-530
0-564
0-600
0-637
0-677
0-719
0-728
0-761
0-775
0-798
0-822
0-847
1140
1185
1247
1292
1344
The second column of Table 11 is obtained from the first by the use
of the'^relation between aneroid height and pressure shown in Table 2.
426
APPLIED AERODYNAMICS
270
260
250
240
230
220
2IO
200
>-
^
?.B.H.P.
^^
y^
7
FT
i
f
y^
^ —
• 200 I300 I4-00 I500 I600
ENGINE REVOLUTIONS R.P.M
Fig. 214. — Standard horsepower and revolutions.
I70O
0-4
OS
0-6
0'7
0-8
0-9
10
^
'
HORSEPC
fWER FAC
TOR
r
/
^
■
^
^
7
10
0-9
0-8 0-7 06
PROPORTIONAL PRESSURE.
OS
0-4-
Fig. 216. — Variation^^of horsepower with pressure and temperature.
PEEDICTION AND ANALYSIS FOR AEROPLANES 427
Column 3 was observed, and 4 then follows from Fig. 215. The relative
density a was calculated from columns 2 and 3 by use of equation (88),
and the last column follows from column 6 and the observations of revolu-
tions.
Further calculation leads to the required fundamental data of
reduction.
TABLE 12.
standard
V
Aneroid height
B.p.m.
horsepower,
/.PsV<r
Viii.p.b.
(ft.).
P.
r.p.m.
20,000
1565
257
91-5
00766
18,000
1580
258
101-7
0-0768
16,000
1610
260-5
114-2
0-0786
14,000
1620
261
1260
00782
12,000
1635
262-6
139-2
0-0798
10,000
—
264?
151?
—
The first two columns of Table 12 are observations ; the third is
obtained from the second and Pig. 214, and the fourth and fifth are calcu-
/Ps^
I50
200
Fig. 216. — Standard curves of performance reduction.
lated using the figures in Tables 10 and 11. The results are plotted in
Fig. 216, and are now standard reductions of maximum speed.
To find the performance in a standard atmosphere the process is
reversed as follows. From the definition of a standard atmosphere and
the law of variation of horsepower with pressure and temperature as
given in Table 1 the calculation proceeds as for Table 11, except for the
last column.
428
APPLIED AEEODYNAMICS
TABLE 13.
1
2
3
4
5
6
standard
Relative
Temperature,
Height factor
Belative
height
pressure
for power,
density,
ai
(ft.).
(atmos.).
/
<r
20,000
0-456
-26
0-470
0-535
0-732
18,000
0-496
-22
0-512
0-571
0-756
16,000
0-538
-18
0-549
0-610
0-781
14,000
0-583
-14
0-595
0-652
0-808
12,000
0-632
-10
0-642
0-695
0-834
10,000
0-684
- 6
0-693
0-740
0-861
From the standard curves of Fig. 216 are then obtained the following
numbers : —
TABLE 14.
<TiV
V
r.p.m.
/.PaV<r
85
0 0756
88-6
90
0-0767
98-4
95
00777
108-6
100
0-0786
120-5
105
0-0790
133-6
110
0 0795
148-5
113
00796
158-4
The final figures for performance in a standard atmosphere are obtained
by finding that solution of Tables 13 and 14 which is consistent with full
power of the engine. The calculation is simple, and at 10,000 ft. is found
by assuming values of 110 and 113 for aiV and calculating the values of
r.p.m. and Pg.
a*V = 110,
(T*V = 113,
r.p.m. = 1605,
r.p.m. =1640,
Pg = 248
P, = 265
(34)
These figures are readily obtained by calculation from numbers already
tabulated. The two values of r.p.m. and Pg are then plotted in Fig. 214
and joined by a straight line. The intersection with the real horsepower
curve occurs where the revolutions are 1635, and the real speed in m.p.h.
is 1635 X 0-0795 = 130 m.p.h. By a repetition of the process the final
performance during level flight in a standard atmosphere is found, see
Table 15.
TABLE 16.
1
Standard height
(ft.).
Maximam true speed
in level flight
(m.p.h.).
Engine speed
(r.pjn.).
20,000
18,000
16,000
14,000
12,000
10,000
113
120
125
127
129
130
1500
1660
1600
1615
1625
1635
PEEDICTION AND ANALYSIS FOE AEKOPLANES 429
Maximum Climb. — ^The observations are the times taken to climb to
given aneroid heights, and the times depend on the state of the atmo-
sphere at all points through which the aeroplane has passed. The quantity
which depends on the local conditions is the rate of climb, and it is necessary
to carry out a differentiation. The accuracy of observation is not so great
that special refinement is possible, and a suitable process is to plot height
against time on an open scale and read off the time at each thousand feet.
The rate of climb at 10,000 feet say, may then be taken as the mean
between 9000 and 11,000 feet. In this way the observed results give the
second column of Table 16 for the aneroid rate of climb. To convert to
real rate of climb these figures must be multiplied by -— as given by
equation (32) and tabulated in column 4. The relative density, o-, is
obtained from equation (33). The last column is calculated from the
two preceding columns.
TABLE 16.
Aneroid
Aneroid rate of
(
dH
Beal rate of
Iieight (ft.).
climb (ft.-m.).
°C.
dh
climb (ft.-m.).
(r*V(j (ft.-m.).
4,000
1370
18
1015
1390
0-860
1290
6,000
1230
15
1010
1240
0-808
1116
8,000
1120
U
1-000
1120
0-760
975
10,000
1020
7
0-990
1010
0-719
865
12,000
935
3
0-975
910
0-677
750
14,000
815
- 1
0-960
780
0-637
620
16,000
660
- 4
0-950
625
0-600
485
18,000
690
- 8
0935
550
0-564
415
20,000
530
-10
0-930
490
0-530
355
The rest of the calculation for climb follows exactly as for level flying,
and the table of results is given without further comment.
TABLE 17.
Indicated
Standard
'in.p.h.
r.p.m.
Aneroid
height (ft.).
Height factor
for power, /.
air speed.
K.p.m.
horsepower.
/.P,V<r
4,000
0-869
75
1495
260-5
193
0 0542
6,000
0-800
76
1500
251
180
00556
8,000
0-745
75
1600
261
163
0-0573
10,000
0-695
75
1605
251
148
0-0688
12,000
0-646
70
1480
249
132-5
00676
14,000
0-605
70
1485
249-5
120
0 0590
16,000
0-565
70
1485
249-5
109
00608
18,000
0-525
70
1485
249-5
98
00627
20,000
0-489
70
1480
249
88-5
0-0650
The results are plotted in Fig. 216 together with those for level flights.
The procedure followed in calculating the rate of chmb in a standard
atmosphere is exactly analogous to that for level flights until the engine
revolutions and horsepower have been found. After this the values of
/ . PgVff are calculated, and VVa and Y^Va read from the standard
430
APPLIED AEEODYNAMICS
curves of Pig. 216. V and Vo are then readily calculated. The results
are shown in Table 18.
TABLE 18.
standard height
B.ate of climb
Time to climb
Indicated air speed
Engine revolutions
(ft.).
(ft.-min.).
(mins.).
(m.p.h.).
(r.p.m.).
0
1740
0
76-6
1385
2,000
1570
1-21
76-5
1416
4,000
1430
2-54
76
1435
6,000
1295
4 02
76-6
1460
8,000
1160
6-75
74-6
1475
10,000
1020
7-49
73-5
1490
12,000
865
9-63
72-5
1495
14,000
735
1215
71-6
1490
16,000
615
1512
70
1480
18,000
505
18-70
69
1460
20,000
370
23-30
68
1430
The third column of the above table is obtained by taking the reciprocals
of the numbers in the second column and plotting against the standard
height, i.e. plotting -5=^. against H. The integral, obtained by any of the
standard methods, gives the value of t up to any height H.
Remarks on the Reduction. — The observations used for the illus-
trative example were taken directly from a pilot's report. In some respects,
particularly for the indicated air-speed readings, the analysis shows that
improvement of observation would lead to rather better results. On the
other hand, it is known, both practically and theoretically, that the best
rate of climb is not greatly affected by moderate changes of air speed and
the primary factor is not thereby appreciably in error.
The procedure followed is very general in character, and may be applied
to any horsepower factor which depends on pressure and density, no
matter what the law. It is shown later in the chapter that flying experi-
ments may be so conducted that a check on the law of variation with
height is obtained from the trials themselves, the essential observations in-
cluding a number of flights near the ground with the engine *' all out," the
conditions ranging from maximum speed level to maximum rate of climb,
As the flight experiments can only give the power factor for the particular
relation between p and t which happens to exist, it is still necessary to
appeal to bench tests for the corrections from standard conditions, hut
not for the main variation.
The standard method of reduction of British performance trials has
up to the present date been based on the assumption that the engine
horsepower depends only on the density. Questions are now being raised
as to the strict vaUdity of this assumption, and the law of dependence of
power on pressure and temperature is being examined by means of specially
conducted experiments. The extreme differences from the more elaborate
assumption do not appear to be very great, and affect comparative results
only when the actual atmosphere differs greatly from the standard atmo-
PEEDICTION AND ANALYSIS FOR AEROPLANES 431
sphere. It appears that a stage has been reached at which the differences
come within the limits of measurement, and the rather more complex law
will then be needed.
If the horsepower depends on the atmospheric density only, the
reduction of observations is simplified, for the height in the standard
atmosphere is then fixed by the density alone and all observations of speed
and revolutions apply at this standard height irrespective of the real
height at the time of observation. For level speeds only the 1st, 2nd, 3rd
and 5th columns of Table 11 are required. From the values of o- and
Table 1 the values of the standard height are obtained, and using these as
abscissae the indicated air speeds and the revolutions of the engine are
plotted. This is now the reduced curve, and at even heights the standard
values of air speed and revolutions are read from the curve.
For climbs the first six columns of Table 16 are required, and the real
rate of climb is then plotted against the standard height as determined by
(T. The remaining processes follow as for level flights.
By whatever means the calculations are carried out, the results of the
reduction of performance to a standard serves the purpose of comparison
between various aeroplanes and engines in a form which is especially
suitable when their duties are being assigned.
For some purposes, such as the calculation of the performance of a
weight- carrying aeroplane or a long-distance machine in which the weight
of petrol consumed is important, the standard reduction is appreciably
less useful than the intermediate stage represented by Tables 12 and 17,
or preferably by curves obtained from them and the loading to give the
form of Fig. 213. The loading, w, was 8*5 lbs. per square foot.
Examples of the Use of Standard Curves of the Type shown m Fig. 213
Aerodynamic Merit. — The first point to be noticed is that the curves
are essentially determined by the aerodynamics of the aeroplane and air-
screw, and do not depend on the engine used. This will have been appre-
ciated from the fact that a special calculation was necessary to ensure
that the engine was giving full power in any particular condition of
flight.
The variables Y\/A V /v/^ against ,^.\/-.ie. f/.^fx/-)
are
non-dimensional coefficients which for the aeroplane and airscrew play the
same part as the familiar lift and drag coefficients for wing forms. Using
either a or p, two sets of curves for different aeroplanes may be superposed
and their characteristics compared directly. If for a given value of
i^jXy- one aeroplane gives greater values of V\/- and Vnx/ *^ than
W^m; oo ^ w ^ w
another, the aerodynamic design of the former is the better. In this
connection it should be remarked that the measure of power is the torque
dynamometer on the engine test bed, and that the engine is used as an
intermediary standard. It is unfortunately not a thoroughly good inter-
432
APPLIED AEEODYNAMICS
mediary, and the accuracy of the curves is usually limited to that of a
knowledge of the engine horsepower in flight. All aeroplanes give curves
of the same general character, the differences being similar in pro-
portionate amount to those between the Hft and drag curves of good
wing sections.
Change of Engine without Change of Airscrew. — Since the aero-
dynamics of the aeroplane is not changed by the change of engine, it '
follows that the standard curves are immediately applicable. The only
effect of the change is to introduce a new engine curve to replace the old
one in order to satisfy the condition that the engine is fully opened up
during level flights or maximum climb.
Change of Weight carried. — Again the aerodynamics is not changed,
and the curves are applicable as they stand. As an example, consider the
effect of changing the weight of an aeroplane from 2000 lbs. and a loading
24-0
220
STANDARD
BRAKE HORSEPOWER
200
I60
reo
14-0
1300 14-00 1500 1600 1700 I800 1900 2000 2100 2200
ENGINE SPEED R.P.M.
Pio. 217. — Balance of horsepower required and horsepower available when
the gross load is changed.
of ^8 lbs. per sq. foot to a weight of 2500 lbs. E^nd a loading of 10 lbs. per
sq. foot, the height being 10,000 ft.
The value of <t at a height of 10,000 ft. in a standard atmosphere is
0'740, and the horsepower factor will be taken as / = 0-68. The engine
curve of standard horsepower is shown in Fig. 217.
To begin the calculation, two values of standard horsepower, Pg,
are assumed, and the curve of Fig. 217 shows that 160 and 220 are
reasonable values. Greater accuracy would be attained by taking three
values.
Taking one loading as example, the procedure is as follows : —
(1) P8 = 220,
/•^«
J w
^ w
22*7 from the data given.
(2) From the standard curves of Fig. 213 read off, for the above value
of 22*7 as abscissae, the ordinates to get
PREDICTION AND ANALYSIS FOR AEROPLANES 433
X = 0-736, and V\/- = 56-4 for level flight ;
nD ^ w , o
and -=r => 0-548, and V\/ - = 38-7 for maximum rate of climb
nD ^ w
With D = 7-87 feet and the given values of a and w the values of n x 60
from the above are 1975 and 1820r.p.m., it being noted that the standard
figure uses V in ft.-s. and n in revolutions per second.
(3) For P8=>160, /.:rj^.'\/- = 16-5, and proceeding as before the
revolutions are found to be 1744 r.p.m. for level flight and 1655 for
climbing flight.
The two values of Pg and r.p.m. are plotted in Fig. 217 and the
points joined by a straight line (or curve if three values were used). The
intersection of the line with the standard horsepower curve gives
the condition that the engine is developing maximum power for the
assumed conditions. The results for both loadings are
loflHintT ^^^^- f Ps = 217 and r.p.m. ^ 1980 for level flight,
loaamg ^^ ^^ ^p^ =, 190 and r.p.m. = 1700 for climbing flight.
loaHi-ncr ^Q ^^^- JPs = 208 and r.p.m. = 1870 for level flight,
loaamg ^^ ^^ <^^^ ^ ^^^ ^^^ ^^^ ^ ^^^^ ^^^ dimbing flight.
The balance of engine and airscrew having been found, J-^. \/ - can
be calculated, and the corresponding values of Y\/ - and Vn\/ - read
. ^ w °^ w
from the standard curves. Fig. 213. The results, converted to speeds in
m.p.h. and rates of climb in feet per min. are
ft ^V\a ^
loading — ^ I Maximum speed 129 m.p.h.
• 1.. ^n^n\l I Maximum rate of climb 575 ft.-min. at A.S.I, of 75 m.p.h.
weight 2000 lbs. j ^
loading ^ (Maximum speed 119 m.p.h.
• ^..c..^5\l (Maximum rate of climb 230 ft.-min. at A.S.L of 75-5 m.p.h.
weight 2500 lbs. j ^
The result of the addition of 500 lbs. to the load carried is seen to be a loss
of 10 m.p.h. on the maximum speed at 10,000 feet, and a loss of nearly
350 ft.-min. on the rate of climb.
The point should again be noted here, that although the rate of cHmb
calculated for the increased loading is a possible one, it does not follow
that it is the best except from the general knowledge that rate of climb
when near the maximum is not very sensitive to changes of air-speed
indicator reading. The necessary experiments for a more rigid appUcation
can always be made when greater accuracy is desired.
2 F
434
APPLIED AEEODYNAMICS
Separation of Aeroplane and Airscrew Efficiencies
In the previous reduction and analysis of aeroplane performance no
separation of the efficiencies of the aeroplane and airscrew has been
attempted, and the analysis has been based on very strong theoretical
ground. The proposal now before us is the reversal of the process followed
in the detailed prediction of aeroplane performance, and in order to proceed
at all it is necessary to introduce data from general knowledge. In the
chapter on Airscrews it was pointed out that all the characteristics of air-
screws can be expressed approximately by a series of standard curves
applicable to all. The individual characteristics of each airscrew can be
represented by four constants, and the analysis shows how these constants
may be determined from trials in flight. The determination of these four
constants also leads to the desired separation of aeroplane and airscrew
efficiencies.
The principles involved have been dealt with in the earlier section on
detailed prediction where the fundamental equations were developed.
The analysis will therefore begin immediately with an application to an
aeroplane.
The aeroplane chosen for illustration was a two seater-aeroplane with
water-cooled engine. The choice was made because the flight observations
available were more complete than usual. The observations reduced to
a standard atmosphere are given in Table 19 below, whilst the standard
engine horsepower as determined on the bench will be found in a later
table.
TABLE 19.
Level flights.
'Maximum climb.
Relative
Speed
Engine
Relative
Speed
Engine
Rate of
climb
(ft.-min.).
density.
(m.p.h.).
(r.p.m.).
density.
(m.p.h.).
(r.p.m.).
0-833
134
1935
0-963
77-6
1700
1265
*
116
1700
0-903
78-0
1700
1145
*
98
1500
0-845
79-0
1700
1025
*
80
1300
0-792
80-2
1695
905
,
0-740
81-4
1690
780
0-717
132-5
1910
0-695
81-6
1685
660
*
115
1720
0-652
81-7
1675
640
*
100
1565
0-610
82-0
1660
420
*
80
1360
0-611
126
1860
*
100
1595
*
80
1400
0-740
133
1915
0-673
130-5
1895
0-630
128
1870
0-600
125
1855
'
* These level flights were made with throttled engine.
PKEDICTION AND ANALYSIS FOE AEROPLANES 436
TliG revolutions of the airscrew were less than those of the engine,
the gearing ratio being 0*6 to 1. Further particulars are :
Gross weight of aeroplane : . . 3475 lbs. ]
Wing area 436 sq. ft. . . (35)
Airscrew diameter 10-13 ft. j
It will be found that it is possible to deduce from the data given —
(1) The pitch of the airscrew.
(2) The variation of engine power with height.
(3) The efficiency of the airscrew.
(4) The resistance of the aeroplane apart from its airscrew.
Determination of the Pitch of the Airscrew. — The pitch of the airscrew is
deduced from the torque coefficient of the airscrew as shown by the standard
1-2
\^^c'v\
\
\\
^
10
^\^\>
^Qc
K
08
07
06
0-5
0-4
0-3
\ \
s ^^
^
\
"^
\
i(nH
X-fcr^
^
\,
^
\\
^D=0-8
N
^
ALL VALUI
5^\
\>
Nj^ioN
V
^,
\s
\VK
01
s
\^
\
Vy\;l4
l-6\
^^-\
x^
\\
■3 0
4 0
5
0
6
0
'y<;p°
•8 0
■9 1-
0 1
1 1-
Fio. 218. — Standard airscrew curves used in the analysis of aeroplane performance.
curves of Fig. 218 and the bench tests on the power of the engine as
foUows. From the numbers in Table 19 and equation (13) the value of
k
^ can be calculated from bench tests of the engine. The speed of the
JW
aeroplane, the engine revolutions and gearing and the airscrew diameter
V
being known, the value of -^p. as shown in Table 20 is easily calculated.
Using equation (13) and putting in the numerical values of the example
feo ^.. Std. B.H.P.
-^ = 341,000 . --
. (36)
486
APPLIED AEKODYNAMICS
and the values given in the last column of Table 20 are calculated from
this formula. Table 20 shows that at the same height two values of
k
~r are obtained, one from the maximum level speed and the other
from the test for maximum rate of climb. The particulars in Table 21
were extracted from columns 1, 6 and 7 of Table 20.
TABLE 20.— ExpEBTMENTS WITH Engine "all oirr.
Height
Relative
Speed
Engine
Standard
V
Aq
(ft.).
density.
(m.p.h.).
(r.p.m.).
(B.H.P.).
nD
f(h)
6,500'
0-833
134
1935
354
1-005
0 0201
11,000
0717
132-6
1910
353
1004
0-0242
16,000
0-611
126
1860
351
0-978
0-0304
10,000
0-740
133
1915
354
1005
0 0232
13,000
0-673
130-6
1895
353
1000
00263
15,000
0-630
125
1855
351
0-978
0-0297
16,500
0-600
128
1870
351
0-994
0-0306
2,000
0-963
77-6
1700
338
0-661
0-0244
4,000
0-903
78-0
1700
338
0-664
0-0260
6,000
0-845
79-0
1700
338
0-672
0 0278
8,000
0-792
80-2
1695
338
0-687
0 0299
10,000
0-740
81-4
1690
338
0-700
00324
12,000
0-695
81-6
1685
337
0-701
0 0345
14,000
0-652
81-7
1676
336
0-706
0-0373
16,000
0-610
820
1660
334
0-716
00408
TABLE 21.
Level flight.
Maximum climb.
Height
(ft.).
V
nD
/(»)
V
nD
/(A)
6,000
10,000
14,000
1-006
1-005
0-998
0-0198
0-0232
0-0278
0-672
0-700
0-706
00278
0-0324
00373
For each row of the table f{h) is constant, and a relation between k^
This relation is sufficient to determine the pitch of
V
and -^is obtained
nD
the airscrew if use be made of the standard curves of Fig. 218. As
shown in the chapter on Airscrews the ordinates and abscissae of these
curves are undetermined, but the shape is determined when the pitch
P
diameter ratio j. , is known.
The value of =:r is found as follows.
PEEDICTION AND ANALYSIS FOR AEROPLANES 437
P V
Assume ^ = 10. From Table 21, this leads to -^=.1-005 at 6000 ft.
D nr
for level flight. The value of Qq/^q from the standard airscrew curves is
0-365, and by combination with Table 21 Qo/(/i) is found as 18-6. For the
climbing trial the corresponding number is 30*6 ; had the assumed value
P
of jT been appropriate to the experiment this latter number would have
agreed with that deduced from level flights. To attain the condition of
p
agreement the calculation is repeated for other values of ^ with the
results shown in Table 22,
TABLE 22.
Pitch diameter
ratio,
P
D
Qo/(*)
Height.
Level flight.
Climbing flight.
6000 ft.
rlO
1-2
(1-4
18-5
30-6
38-9
30-6
36-8
36-5
Inspection of the figures in the two last columns will show that equahty
P
occurs at ^ equal to about 1*3. The actual value was obtained by plotting
p
the two values of Qc/(/i) on a base of =^ and reading off the intersection.
p
In this way a number of 1*32 was found for =-. Repeating the process for
observations at 10,000 feet gave 1-30, and at 14,000 feet, 1-33.
It will thus be seen that the observations give consistent results, and
that the analysis is capable of giving full value to the observations,
p
The mean value of — being 1-32 and the diameter 10*13, the pitch is
13-3 feet.
Variation o! Engine Power with Height and the Value of Qo.—
In calculating the pitch of the airscrew it was also shown incidentally
how the value of Qo/(/t) could be determined, and an extension of the
calculations is all that is necessary to determine both quantities when
once it is noted that J{h) is unity when o- is unity. The values of QofQi)
if plotted against a will give a curve which can be produced back to unit <t
with accuracy, and the value of Qo is thereby determined. Since Qo is
independent of height the value oifQi) is then readily deduced. The calcu-
lations for all observations with the engine all out are given in Table 23.
The first, second and fourth columns of Table 23 are takqn from the
first, second and last columns of Table 20. The value of -^p. is obtained
wP
438
APPLIED AERODYNAMICS
from -=; of Table 20 by the use of the pitch diameter ratio, 1 "32, already
found. Qo^Q is read from the standard curves for airscrews for the values
dn
^
"in
■h)
.
^^'^"'^
^
9n
^»
^
0-5 06 07 0-8 0-9 1-0
RELATIVE DENSITY (T
Fia. 219. — Calculated variation of horsepower with height from observations in flight.
V P
of -^= in column 3, the particular values for^ = l-32 being interpolated
wP P P
between those for = = 1 '2 and — = 1 -4. Column 6 follows bv division of
D D ^
the numbers in column 5 by those in column 4.
TABLE 23.
1
2
3
4
5
6
7
Height
(ft.).
Relative
density,
a
V
wP
*Q
/(A)
Qo*Q
1=1-32
Qo/(A)
/(A)
6,500
0-833
0-761 '
0-0201
0-715
35-6
0-83
11,000
0-717
0-761
0-0242
0-715
29 6
0-69
16,000
0-611
0-740
0-0304
0-747
24-6
0-57
10,000
0-740
0-761
00232
0-715
30-8
0-72
13,000
0-673
0-758
0 0263
0-720
27-4
0-64
15,000
0-630
0-753
0-0297
0-728
24-5
0-57
16,500
0-600
0-740
00306
0-747
24-4
0-57
2,000
0-963
0-501
00244
1-000
410
0-95
4,000
0-903
0-503
0-0260
0-999
38-4
0-89
6,000
0-845
0-509
0-0278
0-995
35-8
0-83
8,000
0-792
0-520
0-0299
0-988
33-0
0-77
10,000
0-740
0-630
0-0324
0-982
30-3
0-70
12,000
0-695
0-631
0-0345
0-982
28-4
0-66
14,000
0-652
0-534
00373
0-978
26-2
0-61
16,000
0-610
0-542
0-0408
0-972
23-8
0-66
PEEDICTION AND ANALYSIS FOR AEROPLANES 439
The values of Qo/(/t) in column 5 are then plotted in Fig. 219 with <t as
a base. The points lie on a straight line which intersects the ordinate at
<T =3 1 at the value 43. Since f{h) is then unity, this value determines Qq
for the airscrew, column 7 of Table 23 is obtained by division and shows
the variation of engine power with height.
The law of variation as thus deduced empirically may be expressed as
m
(T-O-12
0-88
(37)
and shows that the brake horsepower falls off appreciably more rapidly
than the relative density.
In the course of the calculation oif{h) it has been shown that
Qo-43
(38)
TABLE 24.
Speed
fielative
v
kj,
*4°
(m.pJi.).
density.
nP
134
0-833
0-761
0-105
0
*115
jj
0-744
0142
0
* 98
!•
0-718
0-196
0
* 80
»»
0-676
0-296
0
132-5
0-717
0-761
0125
0
*116
»»
0-736
0156
0
*100
»>
0-703
0-219
0
* 80
»»
0-647
0-343
0
126
0-611
0-740
0-162
0
*100
»t
0-689
0-257
0
• 80
>f
0-628
0-402
0
133
0-740
0-761
0120
0
130-5
0673
0-758
0137
0
128
0-630
0-753
0152
0
125
0-600
0-740
0168
0
77-6
0-963
0-601
0-270
0-0600
78*0
0-903
0-603
0-286
0-0476
79-0
0-845
0-509
0-297
00440
80-2
0-792
0-520
0-308
0-0395
81-4
0-740
0-530
0-320 1
0-0350
81-6
0-695
0-631
0-339 !
0-0313
81-7
0-652
0-634
0-361
0 0272
820
0-610
0-542
0-382 ,
00224
Determination of the Aeroplane Drag and the Thrust Coefficient
Factor, Tq. — To determine the aeroplane drag and thrust coefficient
V
factor To, use is made of equation (18), two values of -= for the
same air speed being extracted from the observations, so that the drag
coefficient may be eliminated as indicated in producing equation (11). The
♦ Engine throttled^
440
APPLIED AEEODYNAMICS
lift coefficient, fe^, is now an important variable, and giving the particular
values of the example to the quantities of equation (15), shows that
K
1570
(tV2
'in.p.h.
(39)
With this formula and the rates of cHmb given in Table 19 the values of
fci, and kj,-^ can be calculated. The results are given in Table 24.
V
From the numbers in Table 24, -— for level flight is plotted on a base
V
of kj, in order that values of — =- may be extracted for values of the air
0-1 02 0-3
LIFT COEFFICIENT A^i.
Fig. 220.
0-4
speed intermediate between observations. The condition required is that
V
values of -^^ from the curve for level flights shall be taken at the same
nr
air speed as for climbing. Constant air speed means constant kj^. From
Fig. 220, Table 25 is compiled, part of the data being taken directly
from Table 24.
TABLE 25.
Lift
coefficient,
V
nP
*l,
Climbing flight. ' Level flight.
0-270 00600
0-320 00350
0-382 0-0224
0-601
0-530
0-542
666
The formula which leads to the thrust coefficient factor, Tc, is obtained
from equation (18), and may be written as
4VnP/
1 P^S,
Vo
•lo"T — 7 • "tTTJ Ic\«^ + n^L y
')•
• (40)
PEEDICTION AND ANALYSIS FOE AEEOPLANES 441
ion
The left-hand side of (40) is known for any value of -^ from one of
nr
the standard airscrew curves, Fig* 218. For each value of kj^ in Table 25
sufficient information is now given from which to calculate Tq and kj^.
1 P^S
The particular value of -.-^r^ for the example is 1*85, and for level
flight with ^=-0-680 the value of -(^) \k^ = 0'^6G, and equat
(40) becomes
0-366 = l-85To(A;i,) (41)
For climbing at the same value of kj^ the resulting equation is
1 -000 = l-85To(/(b + 0-050) (42)
From the two equations Tq is found as
1-000-0-366
Tc
1-85 X 0-050
= 6-86
(43)
TABLE 26.
1
2
3
4
5
6
V
nP
-i(^)^A
*»+*,;Y
*Ly
*»
*L
0-761
0 189
0-0146
0-0146
0106
0-744
0-244
00188
0-0188
0-142
0-718
0-290
0-0224
0-0224
0-196
0-676
0-372
0-0288
—
0-0288
0-296
0-761
0-189
0-0146
0-0146
0-126
0-735
0-260
00201
00201
0-166
0-703
0317
00245
00245
0219
0-647
0-440
0-0340
—
00340
0-343
0-740
0-260
00193
_
00193
0-162
0-689
0-347
0-0268
00268
0-267
0-628
0-490
0-0378
0-0378
0-402
0-761
0-189
00146
00146
0-120
0-758
0-222
00164
00164
0137
0-753
0-230
00178
0-0178
0152
0-740
0-260
0-0193
—
00193
0-168
0-601
1-000
0-0773
0-0500
0-0273
0-270
0-603
0-985
00761
0 0476
00285
0-286
0-509
0-963
0 0736
0 0440
0 0296
0-297
0-520
0-896
0 0692
0-0395
0-0297
0-308
0-530
0-845
0-0652
0-0360
0 0303
0-320
0-631
0-840
0-0649
00313
00336
0-339
0-534
0-825
0-0637
0-0272
0 0365
0-361
0-642
0-793
00613
0-0224
00389
0-382
442
APPLIED AEEODYNAMICS
The other values of Zci, yield To = 6-72 and To = 7-56, and the consistency
of the reduction is seen to be only moderate. An examination of equation
(40) shows why, the differences on which To depends being smaller and
smaller as the rate of climb diminishes. In meaning the observations, due
weight is given to the relative accuracy if the numerators and denominators
of the fractions for Tq be added before division. The result in the present
instance is to give
To = 7-0
(44)
In tests carried out with a. view to applying the present line of analysis
the evidence of glides would be included, and the accuracy of reduction
appreciably increased.
Aeroplane Drag. — To having been determined, equation (40) is a
0-05
004
003
002
OOI
CUR
EXAMPLE
^E USED FOR
IN PREDICTIC
»N — *y
/
/
D
DRAG ,
COEFFICIENT tC
D ^^„^
Q
8
o ee
01
0-4
02 0-3
LIFT COEFFICIEIMT ^^
Pig. 221. — Aeroplane glider drag as deduced by analysis of performance trials.
relation between the drag coefficient h^ and known quantities. The
calculation is given in Table 26, using figures from Table 24 as a
basis.
Column 1 is taken from Table 24, and column 2 is deduced from it by
PEEDICTION AND ANALYSIS FOE AEEOPLANES 443
use of one of the standard airscrew curves, Fig. 218; column 3 then follows
from equation (40). The fourth and sixth columns are also taken from
Table 24. whilst the fifth column is deduced from columns 3 and 4.
The curve showing /cu as dependent on kj^ is given in Fig, 221, together
with the curve which was previously used in the example of prediction.
For values of the hft coefficient below 0*15 the calculated points fall much
below the curve drawn as probable. A discussion of this result is given
a little later ; as an example of analysis the drag as deduced will be
found to represent the observations.
Airscrew Efficiency. — The analysis is practically complete as
already given, but as the airscrew efficiency is one of the quantities used
in describing the performance of an airscrew its value will be calculated.
The formula in convenient terms is
V Tah
(jA-X
or, in the example
27r"D*To*nP'Qo^Q
V ToK
07 = 1-29
nP ' Qo/cq
(45)
(46)
From the standard airscrew curves the efficiency at various values of -^
V ^^
(or — if required) is easily obtained as in Table 27.
TABLE 27.
V
nP
Toi:,
Qoto
n per cent.
0-5
1-000
1000
64-6
0-6
0-823
0-922
691
0-7
0-629
0-804
70-6
08
0-421
0-662
66-5
0-9
0-209
0-472
61-5
10
0
0-2^3
0
The maximum airscrew efficiency is seen to be 70-5 per cent.
Remarks on the Analysis. — T'he analysis should be regarded as a
tentative process which will become more precise if regular experiments
be made to obtain data with the requisite accuracy. The standard air-
screw curves may need minor modification, but it is obvious that a further
step could be taken which replaces them in a particular instance. From
the drawings of the airscrew the form of the standard curve could be
calculated by the methods outlined in the chapter on Airscrews. It is not
then necessary that the calculations of efficiency, thrust or torque as made
from drawings shall be relied on for absolute values of the four airscrew
constants determined as now outlined, but only for the general shape of
the airscrew curves.
Both the drag of the aeroplane and the efficiency of the airscrew as
444
APPLIED AERODYNAMICS
deduced by analysis are less than those used in prediction in an earher
part of the chapter, and the differences are mutually corrective. The
actual values depend primarily on Tq, and for this purpose large differences
of rate of climb are required if accuracy is to be attained. This object
can be achieved by a number of judiciously chosen glides.
The Shape of the Drag Coefficient — Lift Coefficient Curve at Small
Values of the Lift Coefficient. — The difference between the result of
analysis and that of direct observation on a model is, in the example,
so striking that further attention is devoted to the point. The model
curve as "used injprediction, Fig. 207, shows a minimum for /Cp at about
2,000
-" // :
-
aK
pm.
-ENGINE
AEROPLANE B
THREE ENGINES*
AIRSCREWS lU y
1o/y'^
-
^^^y^^y^
1.000
-
^^^^^^Y
A
-Jl^^ AEROPLANE A
•^ THREE ENGINES*
AIRSCREWS.
0
■/.
1
20 30 4-0 50 60
INDICATED AIRSPEED
Fia. 222.
70 80
90 100 110 120
cr^^V
m.p.t
fci,= 0'10, and no great increase in value occurs up to /(;i=0*15. It is
possible to make a very direct examination for the constancy of k^ over a
limited range of /Ci,,which is independent of the standard curves for airscrews.
It has been shown in equation (20) that the drag coefficient of an aeroplane
V
is dependent on a and -=- only, and the new limitation removes the
dependence on a. Similarly the thrust coefficient of an aeroplane is fixed
V
by — and is not appreciably dependent on a. It then follows that
constant drag coefficient involves constant advance per revolution for the
airscrew. Advantage is taken of this relation in plotting Pig. 222. The
ordinates are the values of <t* r.p.m. for the engine, and the abscissae are
PEEDICTION AND ANALYSIS FOE AEEOPLANES 445
the air speeds for the aeroplane. A line from the origin to a point on
any of the curves is inchned to the vertical at an angle whose tangent
is — , and if such a Une happens to be tangential to the curves, — is
n n
constant, and hence h^ is constant by the preceding argument.
Experiments for two aeroplanes were chosen. In aeroplane A the
airscrew speed was that of the engine and about 1250 r.p.m. With an
airscrew diameter of 9 feet the tip speed is nearly 600 ft.-s. Aeroplane
B was fitted with engines of different gearing and engine speed, and the
tip speeds of the airscrew were roughly 650 ft.-s., 600 ft.-s. and
700 ft.-s. for the curves h, c and d of Fig. 222.
An examination of the curves of Fig. 222 shows that in three out of the
four, lines from the origin through the points for high speeds lie amongst
the points within the Hmits of accuracy of the observations for an ap-
preciable range. Curve d is a, marked exception. Taking the values of
indicated air speed from the parts of the curves which coincide with the
lines shows the values below.
Aeroplane A. Aeroplane B,
Loading 6 lbs. per sq. ft. Loading 7 lbs. per sq. ft.
(r*V varies from 90 m.p.h. to (t*V varies from 90 m.p.h. to
107 m.p.h. in curve a. 103 m.p.h. in curve b, and
from 100 m.p.h. to 115
m.p.h. in curve c.
hj, varies from 0*10 to 0*15 fci, varies from 0"13 to 0-17 for
for curve a. curve h, and from 0*10 to
0'14 for curve c.
The values of fej, as calculated from the observed air speeds for which
— is sensibly constant are in very good agreement with observations on
n
models, a range of fe^ from 0*10 to 0*17 being indicated over which the drag
coefficient varies very little.
Since curves h, c and d all refer to the same aeroplane, it is not
permissible to assume that the drag coefficient can sometimes depend
appreciably on air speed and at other times be independent of it over the
same range. The figures given for the tip speeds of the various airscrews
show that they are above half the velocity of sound, and that the greatest
discrepancy occurs at the highest tip speed. In the example for which
detailed analysis was given the tip speed was about 600 ft.-s., and the
ratio of the tip speed to the velocity of sound varies Httle at high speeds,
since the velocity of sound falls as the square root of the absolute tempera-
ture and tends to counteract the fall of revolutions with height. The
evidence for an effect of compressibility is therefore very weak.
A more probable source for the difference is the twisting of the airscrew
blades under load. An examination of the formulae for thrust and lift
coefficients will show that for a constant drag coefficient (or advance per
446 APPLIED AEKODYNAMICS
revolution of the airscrew) the thrust is inversely proportional to the lift
coefficient. Between fej^=0'15 and ]cj^=0'10 there is a 50 per cent, increase
in force, and if the blade is liable to twist under load the result will be a
change in experimental pitch and a departure from the assumption that
an airscrew is sensibly rigid.
It may then be that failure to obtain a standard type of curve as a
result of analysis is an indication of twisting of the airscrew blades. At
any rate, the result has been to suggest further experiments which will
remove the uncertainty. It will be appreciated that the sources of error
now discussed do not appear in the test of an aeroplane which is gliding
down with the airscrew stopped. The analysis of such experiments may
be expected to furnish definite information as to the constancy of Jtjy at
high speeds. Flying experiments will then give information as to the effects
of twisting and compressibility, and the advantages of research in this
direction do not need further emphasis.
CHAPTEE X
THE STABILITY OF THE MOTIONS OF AIRCRAFT
PAET I.
General Introduction to the Problems covered by the term Stability. —
The earlier chapters of this book have been chiefly occupied by considera-
tions of the steady motions of aircraft. This is a first requisite. The
theory of stabihty is the study of the motions of an aeroplane about a
steady state of flight when left to its own devices, either with controls
held or abandoned
Figs. 223 and 224 show observations on two aeroplanes in flight, the
speeds of which as dependent on time were photographically recorded.
One aeroplane was stable and the other unstable, and the differences in
record are remarkable and of great importance. The flights occurred in
good ordinary flying weather, and no serious error will arise in supposing
that the air was still.
Stable Aeroplane (Fig. 223). — A special clutch was provided by means
of which the control column could be locked ; the record begins with the
aeroplane flying at 62 m.p.h., and the lock just put into operation. As
the steady speed was then 73 m.p.h., the aeroplane, being stable, commenced
to dive and gain speed. Overshooting the mark, it passed to 83 m.p.h.
before again turning upwards : there is a very obvious dying down of the
oscillation, and in a few minutes the motion would have become steady.
The record shows that after a big bump the aeroplane controlled itself
for more than two miles without any sign of danger.
Unstable Aeroplane. — The next record, Fig. 224, is very different and was
not so easily obtained, since no pilot cares to let an unstable aeroplane
attend to itself. No positive lock was provided, but by gently nursing
the motion it was found possible to get to a steady flying speed with the
control column against a stop. Once there the pilot held it as long as he
cared to, and the clock said that this was less than a minute. After a few
seconds the nose of the aeroplane began to go up, loss of speed resulted and
stalling occurred. Dropping its nose rapidly the aeroplane began to gather
speed and get into a vertical dive, but at 80 m.p.h. the pilot again took
control and resumed ordinary fhght. The aeroplane in this condition is
top heavy.
A stalled aeroplane has been shown, Chap. V., to be liable to spin, and
the ailerons become ineffective. Near the ground an accidental stalling
may be disastrous. The importance of a study of stability should need
no further support than is given by the above illustration.
447
448
APPLIED AEEODYNAMICS
In all probability difficulties in respect to stability limited the duration
of the early flights of Santos Dumont, Farman, Bleriot, etc. It may be
said that the controls were imperfect before the Wright Bros, introduced
their system of wing-warping in conjunction with rudder action, and that
this deficiency in control would be sufficient to account for the partial
failures of the early aviators. Although this objection may hold good, it
is obvious that a machine which is totally dependent on the skill of the
80
M.PH
.1. 1
f A
A
L ' i
A ,/\ ,.
70
60
j 10 \ 20 /30\ 4oy
A^ CONTROL LOCKED
^50'
\^
los^l \y
Fig. 223. — The uncontrolled motion of a stable aeroplane.
pilot for its safety is not so good as one which can right itself without the
pilot's assistance.
Definition of a Stable Aeroplane. — A stable aeroplane may be defined
as one which, from any position in the air into which it may have got either
as the result of gusts or the pilot's use of the controls, shall recover its
correct flying position and speed when the pilot leaves the machine to
choose its own course, with fixed or free controls, according to the character
of the stability.
Sufficient height above the ground is presumed to allow an aeroplane
to reach a steady flying state if it is able to do so. The more rapidly
the aeroplane recovers its flying position the more stable it may be said
to be. If a pilot is necessary in order that an aeroplane may return to its
normal flight position, then the aeroplane itself cannot be said to be stable
STABILITY
449
100
-M.RH
^A CONTROL LOCKED
B
AEROPLANE
STALLS
VEtmCAL
NOSEDIVE
Fig. 224. — The uncontrolled motion of an unstable aeroplane.
2 a
450 APPLIED AEEODYNAMICS
although the term may be appHed to the combination of aeroplane and
pilot.
A subdivision of stability is desirable, the terms " inherent " and
" automatic " being already in use. An aeroplane is said to be " inherently
stable " if, when the controls are placed in their normal flying position
whilst the aeroplane is in any position and flying at any speed, the result
is to bring the machine to its normal flying position and speed. " Auto-
matic stabihty " is used to describe stabihty obtained by a mechanical
device which operates the controls when the aeroplane is not in its correct
flying attitude.
Although the subject of stabihty may be thus subdivided, it will be
found that the methods used for producing inherent stability throw light
on the requirements for automatic stability devices. Before a designer
is in a completely satisfactory position he must have information which
will enable him to find the motion of an aeroplane under any conceivable
set of circumstances. The same information which enables him to calculate
the inherent stabihty of an aeroplane is also that which he uses to design
effective controls, and the same as that required for any effective develop-
ment of automatic stability devices.
A designer cannot foretell the detailed nature of the gusts which his
aeroplane will have to encounter, and therefore cannot anticipate the
consequences to the flying machine. In this respect he is only in the usual
position of the engineer who uses his knowledge to the best of his abihty
and, admitting his hmitations, provides for unforeseen contingencies by
using a factor of safety.
Effect of Gusts. — The aeroplane used as an indication of what may be
expected of an inherently stable machine had the advantage of flying in
comparatively still air.' It is not necessary during calulations to presume
still air and neglect the existence of gusts. For instance, the mathematical
treatment includes a term for the effects of side shpping of the aeroplane.
Exactly the same term applies if the aeroplane continues on its course but
receives a gust from the side. A head gust and an upward wind are simi-
larly contemplated by the mathematics, and even for gusts of a comph-
cated nature the mechanism for examining the effects on the motion of an
aeroplane is provided.
Before entering on the formal mathematical treatment of stabihty
a further illustration of full-scale measurement wiU be given, and a
series of models will be described with their motions and their peculiarities
of construction. The series of models corresponds exactly with the out-
standing features of the mathematical analysis.
The Production of an Unstable Oscillation. — An aeroplane has many
types of instabihty, one of tha more interesting being illustrated in Fig. 225.
which incidentally shows that an aeroplane may be stable for some con-
ditions of flight and unstable for others. The records were taken by the
equivalent of a pin-hole camera carried by the aeroplane and directed
towards the sun. In order to record the pitching oscillations the pilot
arranged to fly directly away from the sun by observing the shadow of the
wing struts on the lower wing. The pilot started the predominant
STABILITY 451
oscillations by putting the nose of the aeroplane up or down and then
•15
INCLINATION.
(degrees.)
Control
9bandoned.
(Degrees)
-15 r
lo.ooo Ft loo M.P.H
(Degrees.)
-15 r-
-10
-5
0- .^,
5 -
10
15 -
4.000 Ft 9o M.P.H.
20
(Degrees)
-15 f-
-10
-5
10 -
15
O MiNS 2
_l 1 1 20
MiNS, 3
_i I I
4.000 Ft. 7o MPH
Fro. 225.— The uncontrolled motion of an aeroplane," showing that stability depends
on the speed of flight.
abandoning the control column. A scale of angles is shown by the side
of the figure. The upper diagram shows that at a speed of 100 m.p.h. and a
452 APPLIED AERODYNAMICS
height of 10,000 ft. the aeroplane was stable. During the period " a "
the pilot did his best to fly level, whilst f or " fe " the aeroplane was left to
its own devices and proved to be a good competitor to the pilot. At the
end of " b " the pilot resumed control, put the nose down and abandoned
the column to get the oscillation diagram which gives a measure of the
stability of the aeroplane. At a speed of 90 m.p.h. at 4000 feet one of the
lower diagrams of Pig. 225 shows an oscillation which dies down for the first
few periods and then becomes steady. The stabihty was very small for
the conditions of the flight, and a reduction of speed to 70 m.p.h. was
sufficient to produce an increasing oscillation. Two records of the latter
are shown, the more rapidly increasing record being taken whilst the aero-
plane was climbing shghtly.
The motions observed are calculable, and the object of this chapter is
to indicate the method. The mathematical theory for the aeroplane as
now used was first given by Professor G. H. Bryan, but has since been
combined with data obtained by special experiments. The present limita-
tions in appUcation are imposed by the amount of the experimental data
and not by the mathematical difficulties, which are not serious.
The records described have been concerned either with the variation
of speed of the aeroplane or of its angle to the ground, i.e. with the longi-
tudinal motion. There are no corresponding figures extant for the lateral
motions, and the description of these will be deferred until the flying
models are described in detail.
Flying Models to illustrate Stability and Instability
Model showmg Complete Stability (Fig. 226). — The special feature of the
model is that, in a room 20 feet high and with a clear horizontal travel of
30 feet, it is not possible so to launch it that it will not be flying correctly
before it reaches the ground. The model may be dropped upside down,
with one wing down or with its tail down, but although it will do different
manoeuvres in recovering from the various launchings its final attitude is
always the same.
The appearance of the little model is abnormal because the stability
has been made very great. Recovery from a dive or spin when assisted
fully by the pilot may need 500 feet to 1000 feet on an aeroplane, and
although the model is very small it must be made very stable if its
characteristics are to be exhibited in the confines of a large lecture hall.
Distinguishing Features on which Stability of the] Model depends.— In
a horizontal plane there are two surfaces, the main planes and the tail
plane, which together account for longitudinal stabihty. The angle of
incidence of the main planes is greater than that of the tail, and the centre
of gravity of the model lies one-third of the width of the main plane from
its leading edge.
In the vertical plane are two fins ; the rear fin takes the place of the
usual fin and rudder, but the forward fin is not represented in aeroplanes
by an actual surface. It will be found that a dihedral angle on the wings
is equivalent in some respects to this large forward fin.
Fig. 226. — Very stable model.
(1) Main plauo. (2) Elevator fin. (3) Rudder 6n. (4) Dihedral tin.
Fig. 227.— Slightly stable model.
Centre of pressure changes produce the effects of fins.
STABILITY 453
All the changes of stabihty which occur can be accounted for in terms
of the four surfaces of this very stable model. The changes and effects
will be referred to in detail in the succeeding paragraphs.
A flying model may be completely stable with only one visible surface,
the main plane. Such a model is shown in Pig. 227. It has, however,
properties which introduce the equivalents of the four surfaces.
The simplest explanation of stability applies to an ideal model in which
the main planes produce a force which always passes through the centre
Fig. 228,
of gravity of the aeroplane model. In any actual model, centre of pressure
changes exist which complicate the theory, but Fig. 228 may be taken to
represent the essentials of an ideal model in symmetrical flight.
In the first example imagine the model to be held with its main plane
horizontal just before release. At the moment of release it will begin to
fall, and a little later will experience a wind resistance under both the
main plane and the tail plane. Two things happen : the resistance tends
to stop the falHng, and the force F2 on the tail plane acting at a consider-
able distance from G tends to put the nose of the model down.
Now consider the motion if the model is held with the main plane
vertical just before release. There will be no force on the main plane due
to the fall, but as the tail plane is inclined to the direction of
motion it will experience a force F2 tending to put the nose of the
model up. The model cannot then stay in either of the attitudes
illustrated. Had there not been an upward longitudinal dihedral
angle between the main plane and tail plane there would have
been no restoring couple in the last illustration, and it will be
seen that the principle of the upward longitudinal dihedral angle
is fundamental to stability. It is further clear that the model
cannot stay in any attitude which produces a force on the tail,
and ultimately the steady motion must lie along the tail plane,
and since the angle to the main planes is fixed, the angle of
incidence of the latter must be a^ when the steady state of
motion has been reached.
From the principles of force measurement, etc., it is known ^m 229
that the direction of the resultant force on an aerofoil depends
only on its angle of incidence, and as the force to be counteracted must be
the weight of the model, this resultant force must be vertical in the
steady motion. This leads directly to the theorem that the angle of glide
is equal to the angle whose tangent is the drag/lift of the aerofoil.
Although the direction of the resultant force on an aerofoil is determined
solely by the angle of incidence, the magnitude is not and increases as the
454 APPLIED AERODYNAMICS
square of the speed. In a steady state the magnitude of the resultant
force must be equal to the weight of the model, and the speed in the glide
will increase until this state is reached. The scheme of operations is now
complete, and is
(a) The determination of the angle of incidence of the main planes
by the upward setting of the tail-plane angle.
(h) As a consequence of (a) the angle of ghde is fixed.
(c) As a consequence of (a) and (b) the velocity of ghde is fixed.
Further appUcation of the preceding arguments will show that any
departure from the steady state of flight given by (a), [b), and (c) intro-
duces a force on the tail to correct for the disturbance.
Degiee of Stability. — No assumptions have been made as to the size
of the tail plane necessary for stabiHty, nor of the upward tail setting.
In the ideal model any size and angle are sujB&cient to ensure stabiHty. It
is, however, clear that with a very small tail the forces would be small and
the correcting dive, etc., correspondingly slow ; such a model would have
small stabiHty. If the tail be large and at a considerable angle to the
main plane, the model will switch round quickly as a result of a disturbance
and will be very stable. It wiU be seen, then, that stabiHty may have a
wide range of values depending on the disposition of the tail.
Centre of Pressure Changes are Equivalent to a Longitudinal Dihedral
Angle. — ^Fig. 227 shows a stable model without a visible tail plane. In the
case just discussed the force Fj on the main planes was supposed to act
through the centre of gravity at ah angles of incidence. This is equivalent
to no change of centre of pressure on the wings, a case which does not
often occur. The model of Fig. 227 is such that when the angle of
incidence falls below its normal value the air pressure acts ahead of the
centre of gravity, and vice versa. The couple, due to this upward air
force through the centre of pressure and the downward force of weight
through the centre of gravity, tends to restore the original angle of
incidence. The smaU mica model has an equivalent upward tail-setting
angle in contradistinction to most cambered planes, for which the
equivalent angle is negative and somewhat large. Tail-planes are therefore
necessary to balance this negative angle before they can begin to act as
real stabilising surfaces. The unstable aeroplane for which the record is
given in Fig. 224 had either insuflficient tail area or too smaU a tail
angle.
The equivalent tail-setting angle of an aeroplane is not easily
recognisable for other reasons than those arising from changes of the centre
of pressure. Tail planes are usually not flat surfaces, but have a plane
of symmetry from which angles are measured. The Hft on such a
tail plane is zero when the wind blows along the plane of symmetry. The
main planes, on the other hand, do not cease to lift until the chord is inclined
downwards at some such angle as 3°. If the plane of symmetry of the
tail plane is parallel to the chords of the wings there is no geometrical
dihedral angle, but aerodynamically the angle is 3°.
A complication of a different nature arises from the fact that the tail
plane is in the downwash of the main planes.
STABILITY
455
Although all the above considerations are very important, they do not
traverse the correctness of the principles outlined by the ideal model.
Lateral Stability. — Suppose the very stable model to be held, prior to
release, by one wing tip so that the main plane is vertical. At the moment
of release there will be a direct fall which will shortly produce wind forces
on the fins, but not on the main plane or tail plane. On the front fin the
force F3, Fig. 230, in addition to retarding the fall, tends to roll the aeroplane
so as to bring A round towards the horizontal. The air force F4 on the tail
fin tends to put the nose of the aeroplane down to a dive and so gets the
axis into the direction of motion. Both actions continue, with the result
that the main planes and tail plane are affected by the air forces and the
longitudinal stabihty is called into g
play. It is not until the aeroplane
is on an even keel that the fins cease
to give restoring couples. Any
further adjustments are then covered
by the discussion of longitudinal
stability already given.
Lateral stabihty involves rolling,
yawing and side shpping of the
aeroplane, all of which disappear
in steady flight. The mica model
Fig. 227 has rolling and yawing
moments, due to centre of pressure
changes when side shpping occurs. yiq. 230.
The equivalent fins are very small,
and the stabihty so shght that small inaccuracies of manufacture lead to
curved paths and erratic motion.
The large central fin of the very stable model is never present ui an
aeroplane, as it is found that a dihedral angle between the wings is a more
convenient equivalent.
P'ig. 231 shows a model which flies extremely well and which has no
front fin. The dihedral angle between the wings is not great, each of them
being inclined by about 5° to the Ime joining the tips. The properties of a
lateral dihedral angle have been referred to in Chaps. IV. and V.
Unstable Models. — Two cases of unstable aeroplanes have been men-
tioned, and both instabilities can be reproduced in models. The tail plane
of the model shown in Fig. 232 will be seen to be small, whilst the balancing
weight which brings the centre of gravity into the correct place is small
and well forward, so putting up the moment of inertia of the model for
pitching motions. ^
To reproduce the type motion of Fig. 224 the tail plane would be set
down at the back to make a shght negative tail-setting angle and the model
launched at a high speed. It would rise at first and lose speed, after which
the nose would fall and a dive ensue ; with sufficient height the model
would go over on to its back, and except for the lateral dihedral angle
would stay there. The righting would come from a rolhng over of the
model, and the process would repeat itself until the ground was reached.
456 APPLIED AEKODYNAMICS
As illustrated the tail plane is set so that the model takes up an nicreas-
ing oscillation similar to that shown in Fig. 225. The rear edge of the tail
plane is higher than for the nose dive, and there is a small upward angle
between the main plane and the tail plane, which tends to restore the
position of the model when disturbed Owing to the smallness of the
restoring couple, the heavy parts carry the wings too far and hunting
occurs. About an axis through the centre of gravity the model would
exhibit weathercock stabiUty, whilst with the centre of gravity free the
motion is unstable.
If the tail plane be further raised at its rear edge the model becomes
stable, and if launched at a low speed would take a path similar to that
of the aeroplane for which the record is given in Fig. 223.
Lateral Instability. — A model which illustrates three types of lateral
instability is shown in Fig. 233. As illustrated the model when flown
develops a lateral oscillation as follows : the model flies with the larger
fin forward, because the distance from the centre of gravity is less than
that of the rear fin, but if held as a weathercock with the axis through
the centre of gravity there will be a small couple tending to keep the
model straight. Due to an accidental disturbance the model sideslips to
the left, the pressure on the fins turns it to the left, but since the centre
of the fins is high there is also a tendency to a bank which is wrong for the
turn. This goes on until the lower wing is moving so much faster on the
outer part of the circle as to counteract and overcome the direct rolling
couple, and the model returns to an even keel, but is still turning. Over-
shooting the level position the sideslipping is reversed and the turning
begins to be checked. As in the longitudinal oscillation, hunting then
occurs.
A second type of instabihty is produced by removing the front fin, the
result being that the model travels in a spiral. Suppose that a bank is
given to the model, the left wing being down ; sideslipping will occur to the
left and the pressure on the rear fin will turn the aeroplane to the left and
tend to raise the left wing. On the other hand, the outer wing will be the
right wing, and as it will travel faster than the inner wing due to turning
the extra lift will tend to raise the right wing still further. There is no
dihedral angle on the main plane and the proportions of the model are such
that the turning lifts the right wing more than the sideslipping lowers it.
The result is increased bank, increased sideslipping and increased turning,
and the motion is spiral.
The third instability is shown by the model if the front fin be replaced
and the rear one removed. The model does not then possess weathercock
stability, and in free flight may travel six or ten feet before a sufficient
disturbance is encountered. The collapse is then startlingly rapid, and the
model flutters to the ground without any attempt at recovery.
Remarks on Applications. — Aeroplanes are often in the condition of
gliders, and their motions then correspond with the gliding models. When
the airscrew is running new forces are called into play, and the effects on
stabihty may be appreciable. The additional forces do not in any way
change the principles but only the details of the application, and the
Ftg. 231. — Stable model with two real fins.
The dihedral tin is not actually present, but an equivalent etfect is
produced by the dihedral angle between the wings.
Fig. 232. — Model which develops an unstable phugoid oscillation. Large
moment of inertia fore and aft with small restoring couple.
Fig. 233. — Model which illustrates lateral instabilities.
(1) With front fin removed: spiral instability. (2) As shown: unstable
lateral oscillation. (3) With rear fin removed : spin instability.
STABILITY
457
description of stable and unstable motion just concluded applies to the
stable and unstable motions of an aeroplane flying under power.
From the short descriptions given it will have been observed that the
simple motions of pitching, falling, change of speed are interrelated in the
longitudinal motions, whilst the lateral motions involve sideslipping,
rolling and yawing. The object of a mathematical theory of stabihty is
to show exactly how these motions are related.
Mathematical Theory of Stability
The theory will be taken in the order of longitudinal stability, lateral
stability, and stabihty when the two motions affect each other.
Longitudinal Stability
The motions with which longitudinal stabihty deals all occur m the
plane of symmetry of an aircraft. Changes of velocity occur along the
HORIZONTAL LINE
T
Fio. 234.
axes of X and Z whilst pitching is about the axis of Y. Axes fixed in
the body (Fig- 234) are used, although the treatment is not appreciably
simpler than with fixed axes, except as a link with the general case.
The equations of motion are
u-\- wq = X'l
io — uq^Z'l (1)*
qB =m)
* The group of equations shown in ( 1 ) has valid application only if gyroscopic couples due
to the rotating airscrew are ignored ; the conditions of the mathematical analysis assume
that complete symmetry occurs in the aircraft, and that the steady motion is rectilinear
and in the plane of symmetry. This point is taken up later.
A point of a different kind concerns the motion of the airscrew relative to the aircraft,
and would most logically be dealt with by the introduction of a fourth equation of motion —
2^In + Q, = Q,
(la)
where I is the moment of inertia of the airscrew, Q„ is the aerodynamic torque, and Q, is the
torque in the engine shaft. All present treatments of aeroplane stability make the assump-
tion, either explicitly or implicitly, that I is zero.
Mathematically this is indefensible as an equivalent of (la), but the assumption ia
468 APPLIED AEKODYNAMICS
The forces niX' and wZ' depend partly on gravitational attraction and
partly on air forces. M, the pitching moment, depends only on motion
through the air.
Gravitational Attractions. — The weight of the aircraft, mg, is the
only force due to gravity, and the components along the axes of X and
Z are
— ^ sin ^ and g cos 6 (2)
Air Forces. — Generally, the longitudinal force, normal force, and
pitching moment depend on u, w and q. An exception must be made
for lighter- than-air craft at this point, and the analysis confined to the
aeroplane. The expressions for X, Z and M are
X =/x(w, w q)]
Z =U^^,w,q)\ (3)
Restatement of the Equations of Motion as applied to an Aeroplane-
Substituting for X, Z and M in (1) leads to the equations
w 4- w;g = —g sin d + /x(m, w,q)\
w — uq= g cos 6 +fz{u, w, q) \ .... (4)
qB = fj^{u, w, q) ]
In the general case, which would cover looping, these equations cannot
be solved exactly. For such solutions it has been customary to resort to
step-to-step integration, an example of which has been given in Chapter V .
The particular problem dealt with under stability starts with a steady
motion, and examines the consequences of small disturbances.
If Ug, Wo, Bq be the values of u, w and 0 in the steady motion, equations
(4) become
0 = — gf sin do +fx{Uo, Wo, 0) )
0 = ^ cos do +fz{Uo, Wo,0)\ (5)
0 = /m(Mo, Wo, 0) )
Since q — ^, it follows that q must be zero in, any steady longitudinal
motion, 6 being constant.
The third equation of (5) shows that the pitching moment in the
steady motion must be zero. The first two equations express the fact that
the resultant air force on the aeroplane must be equal and opposite to the
weight of the aeroplane. There is no difficulty in satisfying equations (5),
and the problems relating to them have been dealt with in Chapter II.
nevertheless satisfactory in the present state of knowledge. The damping of any
rotational disturbance of an airscrew is rapid, whilst changes of forward speed of an
aeroplane are slow and are the only changes of appreciable magnitude to which the airscrew
has to respond.
The extra equation of motion does not lead to any serious change of method, but it adds
to the complexity of the arithmetical processes, and the simplification which results from the
assumption 1=0 appears to be more valuable than that of the extra accuracy of retaining it.
A little later in the chapter, is given a numerical investigation of the validity of the
assumption, but it is always open to a student to recast the equations of stability so as to
use the variables u, w, q and n instead of confining attention to u, w and q only.
STABILITY :' 459,
■ " '■• i , -
Small Disturbances. — Suppose that u becomes u,, + 8m, w becomes
Wo + 8w, 6 becomes dg + 86, and q becomes 8q instead of zero. Equations
(4) will apply to the distmbed motion so produced. If 8u, 8w, 8q be made
very small, equations (4) can be modified very greatly, the resulting forms
admitting of exact solution. To find these forms the new values of u, etc.,
are substituted in (4), and the terms expanded up to first-order terms in
8u, 8w, etc. In the case of the first equation of (4) the expanded form is
8u + Wo8q = —g sin d„ -\-jyiUo, w^, 0)
-i,cos»..89+».f + 8„f + 8g|. .(6)
From the conditions for steady motion, equation (5), the value of
—g sin $0 +/x(Woj ^0) ^) is seen to be zero, and (6) becomes
8u + wM = -g cos ^,8^ + 8u^ + 8w^-^8q^J^ . . (7)
Resistance Derivatives. — The quantities ^^, •'^, -^, etc., are called
all aW dq
resistance derivatives, and as they occur very frequently are written more
simply as X„, X„, Xg, etc.
A further simpHfication commonly used is to write u instead of 8u.
With this notation the equations of disturbed motion become
jM + Woq = —g cos-^o • ^ + ^X„ + w^„ + gXg j
iD — Uoq = —g sin Oo-O + uZ„ + wZ^t + gZg | . . . (8)
B^ = mM„ + M?M«, + gMg j
In these equations q = 0, and the equations are hnear differential equa-
tions with constant coefficients. Between the three equations any two of
the variables m, w and q may be ehminated by substitution, leading to
an equation of the form
F(D).M-=0 (9)
where F(D) is a differential operator. For longitudinal stability F(D)
contains all powers of D up to the fourth.
The standard solution of (9) is
u = Mie^i' + 1626^2' -j- WgC^a* -|- w^eV .... (10)
where A^, A2, A3, and A4 are the roots of the algebraic equation F(A) =0,
and Ml, M2, M3, and M4 are constants depending on the initial values of the
disturbance. There are similar relations for w and q with the same values
for Ai, A2, A3, and A4.
For each term of the form ue^*, etc., the value of u is Am, w = Xw, etc.,
where A may take any one of its four values, and in finding the expansion
for F(A) this relation is first used to change equations (8) to
(A-X> -X^iv -\-{Wo\-X^-\-gco^^o)0 = 0]
- Z„M + (A - T'Jw + (-M^ - ZgA + ^ sin 60)6 = 0 . (11)
- M„M - M^m; + (BA2 - M^)d = 0 J
460
APPLIED AERODYNAMICS
and the elimination of any two of the variables u, w and 6 leads to the
stability equation
F(A) =
X,
-X.
WqX — XjA + g cos Bq
Z„ A — Z^ — MoA — Z5A + gf sin B^
M„ - M„ BA2 - M^
= 0
(12)
The coefficient of the highest power of A, i.e. A*, is B, and in order to
arrive at an expression for which the coefficient is unity it is convenient
M
to divide through everywhere by B. This is effected if ~ is written instead
B
M M
of M„, ~ instead of M^, and -^ instead of Mg in a new determinant other-
B B
wise the same as (12).
The expansion of — ^ in powers of A is easily achieved, and the results
are given below.
Coefficient of A*, 1
Ai = Coefficient of A^, _ X« - Z„ - ^M^
Bi = Coefficient of A^, —
B
M.
W0+Z3 i +
M„'
Xj<
M.
Ci ^ Coefficient of A^,
X,
Z.
Xm,
z,,,
Di=Coefficientof AO, |
B
X„
M.
M„ M,
Xm,
Z..
M„
W0+Z5
M„
-Wo-
fX,
+
x« x„,
M,
Za Z^
B
M„
— sin Bo
XJ
M.
COS^o
COS Bq
sin Bq
0
(13)
The conditions for stability are given by Routh, and are that the five
quantities Ai, Bi, Cj, Dj and AjBiCi— Ci^— Ai^Di shall all be positive.
Emample I. — For an aeroplane weighing about 2000 lbs. and
122-4
-4-3
= — 2°-C
the following approximate values of the derivatives may be used :—
X„ = -0-169 X„ = 0-081 X, =
Z„ = -0-68 Z„ = -4-67 Z„ =
0
-0-60
(14)
B "
-0-0047
^M„= -0130 . iM„= -9-8
B B
Substituting the values of (14) ia (13) leads to
Ai = 14-8, Bi = 62-0, Ci = 9-80, Di = 2-16
All these quantities are positive.
AiBjCi - Ci^ - Ai^Di = 8420
and the motion is completely stable.
STABILITY 461
Some further particulars of the motion are obtained by solving the
biquadratic equation in A.
The equation A* 4: 14-8A3 + 62-0A2 + 9-80A + 2-16 = 0 )
has the factors [. . (15)*
(A + 7-34 ± 2-45i)(A + 0-075 ± 0-170i) = 0 j
All the roots are complex. A pair of complex roots indicates an oscil-
lation. The real part of a complex root gives the damping factor, and the
imaginary part has its numerical value equal to 2tt divided by the periodic
time of the oscillation. In the above case the first pair of factors indicates
an oscillation with a period of 2*57 sees, and a damping factor of 7*34,
whilst the second pair of complex factors corresponds with a period of
37*0 sees, and a damping factor of 0-075.
The meaning of damping factor is often illustrated by computing the
time taken for the amplitude of a disturbance to die to half magnitude. If
u will be half the initial value Ui when
Taking logarithms,
-Ai<= — loge2 = -0-69
and •'• t to half amphtude = -y—
Ai
In the illustration the more rapid oscillation dies down to half value
in less than f^oth second, whilst the slower oscillation requires 9*2 seconds.
It will be readily understood from this illustration that after a second
or so only the slow oscillation will have an appreciable residue. The
resemblance to the curve, shown in Fig. 223, of the oscillations of an
aeroplane will be recognised without detailed comparison.
V - -
Airscrew Inertia as affecting the Last Example
A numerical investigation can now be made of the importance of the assumption
that the motion of an aeroplane is not much affected by the inertia of the airscrew.
Corresponding with the data of the example are the two following equations for aero-
dynamic torque and engine torque : —
Q„=l-004w2- 0-0018^' (15a)
ft
and Q, = 875 - 14-6% (156)
Solving the equation, Q^ = Q,, for m = 122-4 leads to the value n = 25-2.
Substituting %„ + n for n and m<, + w for « in equations (15a) and (156) and separating
the parts corresponding with disturbed motion from those for steady motion converts
equation la into
A + Aa6 + O-16()». + O00O143^A'\n==2:?2^??^'M . . . (15c)
or with u„ = 122-4 and «„ = 26-2
n + 5-59» = 0-256M (I5d)
* For method of solution, see Appendix to this chapter.
462 APPLIED AEEODYNAMICS
Before any solution of {I5d) can be obtained u must be known as a function of n and t.
In equation (15) a value of u was found of the form Wje^i', but this assumes a definite
relation between n and u for all motions whether disturbed or steady. The value
u^e 1^ so found may be used in {15d) and the result examined to see whether any funda-
mental assumptions on which it was based are violated. A solution of (I5d) is now
n = n^'" + 0;?^ (15e)
Aj -j- 5 oy
except in the case where ^i = — 5*59, when the solution is
n = e~^'^^{ni + 0-25&Uit} (15/)
A.J is frequently complex, and following the usual rule h + ik is written for Aj and h—ik
for the complementary root A,, and the two roots are considered together. For an
oscillation, equation (15e) is replaced by
+ -^^=E^E^I=- {«! cos (M + y) + u^ sin {kt + y)} ■ (ISgr)
V A -|r 5"59 + k^
ft + 5-59 ■ . -k
where cos y = , and sm y = ,
V ;i + 5.592 + p Vh + 5'59^ + k^
The first term of (15e) and {15g) is reduced to 1 per cent, of its initial value in less than
one eecond. In the case of (15/) the maximum value of the second term occurs at
i=0'18 sec, and is 0'125tti, and like the first term becomes unimportant in about a
second.
Had the inertia of the airscrew been neglected the relation obtained from (156^)
would have been
0-256M ,,-,.
n = (15ft)
5-59 - ^
Instead of which the more accurate equation (15e) gives after 1 sec.
TO= (15*)
Aj + 5-59
and it is seen immediately that if A^ be real, equation {15h) may be used instead of
(15i) if A 1 is small compared with 5-59. If 50 per cent, of the motion is to persist after
1 sec, Aj cannot exceed 0*69, and in the more important motions of an aeroplane Aj is
much less. In such cases the assumption is justified that the relation between airscrew
revolutions and forward speed is substantially independent of the disturbance of the
steady motion.
In the case of an oscillation the motion shown by {I5g) involves both a damping
factor and a phase difference. The damping factor corresponding with (15i) is
Vh + 5-59^ + k^
whilst the phase difference is
tany = , ~^ - (15*)
^ h + 5-59
Applied to Example I. (15j) and (15*) give
Rapid oscillation h = — 7'34 k = 2*90
n = -0-075% and y = 240°
whilst the approximate formula (15^) gives
n — 0*046% and y = 0
STABILITY 463
It is clear, therefore, that the approximation 1=0 must not be applied to the first
second of the motion without further consideration.
Phugoid oscillation h = —0-075 and )fc = 0*170 •
n = 0046zt and y = l°-8
whilst the approximate formula gives
n = 0'046tt and y = 0
In this case it is equally clear that the approximation 1=0 is quite satisfactory.
It may therefore be concluded that any investigation of the early stages of distm-bed
motion should start with the four equations of motion, whilst any investigation for the
later periods can be made by the use of three only.
Variation of Thrust due to Chakge of Forward Speed
Whilst dealing with the subject of the airscrew it may be advantageous to supplement
the equation for Q^ by the corresponding expression for the thrust, viz.
T = l-25»a - 0*0222^2 (15i)
Using equation {15h) and remembering that n and u are small quantities, the change
of revolutions with change of forward speed is
3!*=::^' =0-0458
du u , ■
Differentiating in equation (15Z) leads to
dT dn
^- = 2-50n„^-0-0444«„ (15m)
With 7^0=25-2 and Wo=122*4, the value of ~ is —2-64 lbs. per toot per sec. The mass of
the aeroplane being 62-1 slugs, the contribution of the airscrew to the value of X„ is seen
2-64
to be — «o-rj i-e. —0-043. This is rather more than one-quarter of the total as shown
in (14).
Effect of Flight Speed on Longitudinal Stability.— The effect of varia-
tion of flight speed is obtained by repeating the process previously
outlined, and as there are many common features in aeroplanes a set of
curvQg is given showing generally how the resistance derivatives of an
aeroplane vary with the speed of flight.
The stalling speed assumed was 58'6 ft.-s. (40 m.p.h.), and it will be
noticed that near the stalling speed most of the derivatives change very
rapidly with speed. For lateral stability as well as longitudinal stability
it will be found that marked changes occur in the neighbourhood of the
stalling speed, and that some of the instabihties which then appear are
of the greatest importance in flying.
The derivatives illustrated in Figs. 235-238, correspond with an aero-
plane which is very stable longitudinally for usual conditions of flight.
Not all the derivatives are important, and X^ is often ignored. The
periods and damping factors corresponding with the derivatives are of
interest as showing how stability is affected by flight speed. A table of
results is given (Table 1).
\
\
Xj/,
\
■ —
______
50 60 70 60 90 100 110 120 130 140
SPEED FT/S
0
-0.2
-0.4
-0 6
/
.
-0.6
-1.0
-1.2
-1 '^
Iv,
\
50 60 70 80 90 100 110 120 130 140
SPEED FX/S
0.10
i
J
E
^u
s.
"x
____
■-
0.00
50 60 ?0 80 90 100 lip 120 130 140
SPEED FT/S
Fig. 236. — Resistance derivatives for changes of longitudinal velocity.
0.2
^
-^
O
■■^~-
-^
.^
0 ?
OA
/\'U/
0.6
0.6
■1.0
50 60 70 SO 90 100 110 130 130 KO
SPEED FT/9
1
0
-2
\
V^
z^
^,
-*
\
.
"^
-6
so 60 70 80 90 too 110 \20 130 140
SPEED FT/S
—
■^
b'^
-0.3
50 60 70 80 90 100 110 120 I30 I4-0
SPEED FT/S
FiQ. 236. — Resistance derivatives for changes of normal velocity.
2u
Xo-
50 60 70 80 90 100 110 (20 130 140
SPEED FT/S
(
s
V
V
"~"
"^
-^
50 60 70 60 SO 100 110 120 130 140
SPEED FT/S
-2
(
\
B"y
\
^
X
nJ
s.
N
bJ
so 60 70 80 SO 100 110 120 130 l4-'0
SPEED FT/s
Fig 237. — Resistance derivatives for pitching.
STABILITY
467
PO"
^
IO°
\
\ v.
>
■^"
^LEOFlh
ClOENC
:OFMA
N PLANES
0
"S. ,
te ^^
^-^.
• «. ^
^oFOR
LEVEL FLIGHT
OR A
NGLEOF PITCH,
TAN'w/y
50
60
70
80
SO
100
no
izo
130
140
SPEED FT/S
Fig. 238. — Angle of pitch and flight speed.
TABLE 1.
Flight
speed
ft.-s.
Bapid oscillation.
Phugoid oscillation.
Periodic
time
(sees.).
Damping
factor.
Time to half
disturbance
(sees.).
Periodic
time
(sees.).
Damping
factor.
Time to half
disturbance
(sees.).
69-2
58-6
60-0
70-0
80-0
90-0
1000
122-5
247-0
1-67
2-05
2-22
2-21
2-25
2-23
2-16
2-57
5-24
-0-01
+0-99
1-46
2-97
3-86
4-66
6-34
7-34
16-27
(Doubles in
70 sees.)
0-70
0-47
0-23
0-18
0-15
0-13
0-096
0-042
8-5
8-3
9-3
13-3
17-0
21-1
26-4
37-0
Aperiodic
0-170
0-061
0-031
0-037
0-044
0-061
0068
0076
0-272
and 0-028
4-0
13-6
22-2
18-6
16-7
13-6
11-9
9-2
2-64
and 24-6
Throughout the range of speed possible in rectilinear steady flight the
disturbed motion naturally divides into a rapid motion, which in this case
is an oscillation, and a slower motion which is an oscillation except at a very
high speed. This latter motion was called a '* phugoid oscillation " by
Lanchester, and the term is now in common use.
At stalHng angle the short oscillation becomes unstable, and a critical
xamination will show that the change is due to change of sign of Z^ and
X^. Physically it is easily seen that aboye the stalling angle falUng
468 APPLIED AEBODYNAMICS
increases the angle of incidence, further decreases the hft, and accentuates
the fall.
At the higher speeds the damping of the rapid oscillation is great, and
in later chapters it is shown that the motion represents (as a main feature)
the adjustment of angle oi incidence to the new conditions.
The slow oscillation in this instance does not become unstable, but is
not always vigorously damped ; at 60 ft.-s. the damping factor is only 0-031.
A modification of aeroplane such as is obtained by moving the centre of
gravity backwards will produce a change of sign of this damping factor,
and an increasing phugoid oscillation is the result.
At high speeds the period of the phugoid oscillation beconres greater,
and ultimately the oscillation gives place to two subsidences. In a less
stable aeroplane the oscillation may change to a subsidence and a divergence,
in which case the aeroplane would behave in the manner illustrated in
Fig. 224.
All the observed characteristics of aeroplane stability are represented
in calculations similar to those above. Many details require to be filled
HORIZONTAL
DIRECTION OF
MOTIONOFG.
Fig. 239.
in before the calculations become wholly representative of ^ the disturbed
motion of an aeroplane. The details are dealt with in the determination of
the resistance derivatives.
Climbing and Gliding Flight.— The effect of cutting off the engine or
of opening out is to alter the airscrew race effects on the tail of an aero-
plane. The effects on the steady motion may be considerable, so that each
condition of engine must be treated as a new problem. The derivatives
are also changed. The effect of cHmbing is to reduce the stabihty of an
aeroplane at the same speed of flight if we make the doubtful assumption
that the changes of the derivatives due to the airscrew are unimportant.
There is not, in the analysis so far given, any expression for the in-
clination of the path of the centre of gravity, G. Eeferring to Fig. 239, it
is seen that the angle of pitch a is involved as well as the inclination of the
axis of X to the horizontal. The angle of ascent 6 is — a + tf, or in terms
of the quantities more commonly used in the theory of stabihty
e=^-tan-i'^
u
STABILITY 469
In level flight 0 is zero, and the value of 6 differs from the angle of
incidence of the main planes by a constant.
Whether chmbing, flying level or ghding, the angle of pitch, i.e. tan^
w
u
is almost independent of the inchnation of the path ; it is markedly a
function of speed. The curve in Fig. 238 marked " ^o ^^^ level flight or
w . . .
angle of pitch, tanr^ -" is most satisfactorily described as "angle of
Xv
pitch."
Variation of Longitudinal Stability with Height and with Loading. —
When discussing aeroplane performance, i.e. the steady motion of an aero-
plane, it was shown that the aerodynamics of motion near the ground could
be related to the motion for different heights and loadings if certain
functions were chosen as fundamental variables. In particular it was
shown that similar steady motions followed if
/w
were kept constant for the same or for similarly shaped aeroplanes, ^-q-
is now used for the wing loading, to avoid the double use oiwm the same
formula.) It was found to be unnecessary to consider the variation of
engine power with speed of rotation and height, except when it was desired
to satisfy the condition of maximum speed or maximum rate of chmb.
In order to develop the corresponding method for stabihty it is
necessary to examine more closely the form taken by the resistance deriva-
tives. In equation (3) the forces and moments on an aeroplane were
expressed in the form
X =/x(m, w, q)
with n a known function of u, w, and q. No assumption was made that
for a given density, attitude and advance per revolution, the forces and
moments were proportional to the square of the speed.
If appeal be made to the principle of dynamical similarity it will be
found that one of the possible forms of expression for X is
™X = ,!W||,|,^) (16)
where p is the density of the fluid, V is the resultant velocity of the aeroplane,
and I is a typical length which for a given aeroplane is constant.
The arguments ^, ^, — are of the nature of angles ; ^ is a measure of
the angle of incidence of the aeroplane as a whole, ^ represents local changes
nl
of angle of incidence, and ^ defines the angle of attack of the airscrew
blades.
470 APPLIED AEEODYNAMICS
Since V is the resultant velocity,
Y^ = u^ + w^ (17)
BY u ^ BY w
and ^ "V ^ W~ ~v
"Uw = const. ' "^M = canst. ♦
Proceeding now to find one of the derivatives by differentiation of X
with respect to u, whilst w and q are constant, leads to
"~"m 3m ^""'^"^ m 3i^ Y^w Y^' Ig^ dY\Y )la\ (18)
"~ OT Y w Y Iq ' dY\YJ nil Y
^Y ^V ^V
(19)
w
If now, during changes of p and m, ^ = const., equation (17) shows
that ;^== const. Further, make :^= const., and ^^ = const., and ex-
amine (19). The partial differential coefficients—,-^ and -— have the
w Lq nl
dy dy dy
same value for variations under the restricted conditions. The outstanding
term which does not obviously satisfy the condition of constancy is
VA(^). ...... .(20)
and this must be examined further ; it will be found to vary in a more
complex manner than the other quantities.
The airscrew torque may be expressed as
^. = p™^i=xj|.|4^| (21)
and the engine torque as
<f>,,==<l>{h).tfs{n) (22)
In a standard atmosphere p is a known function of the height h.
Equating (f>o and <f) , putting (f>{h) =<f>,{p) gives
P^'l'x^y%^\-<f>.{pmn) (23)
Differentiating partially with ~ and ^ constant leads to
STABEjITY 471
nl
Since in changes the — = const, there is a relation between the changes
of n and V given by
dn _ n
3V~V
and equation (24) becomes after rearrangement of terms
ylC^)^—: PIl!— (25)
nl
ai
X and -*• are both constant during the changes of density and load, and
4
the complex expression
—pnl^ ....... (26)
is the only one requiring further consideration.
Equation (23). shows that ^'^^^'^^ ' is constant for our restricted con-
m
Oth L
ditions, and again utilising the condition that and l^ are constant,
° m
t/f'{n) const. ,^.
ijj{n) ~ n ^"^
is an equation to be satisfied by the torque curve of an engine if the value
d / Til\
for V^^;r^ ) is to take simple form. This equation can be integrated
to give
^(n) = AnB (28)
where A and B are constants. The only member of this family of curves
which approaches an actual torque curve for an aero-engine is with B = 0,
and this assumption is often made in approximate calculations. A more
usual form for ^(n) is
ij,{n) = a-hn (29)
where the approximation to a torque curve can be made to be very good
over the working range, and where hn will not exceed ^rd of a. Using
(29) the value of ^'^P'^J^'^' may be estimated as compared with — 2x, for
equation (23) gives
472 APPLIED AEEODYNAMICB
2(a — bn)
(80)
The second term in the bracket is seen to be one-quarter of the first
in the extreme case.
It may then be taken, as a satisfactory approximation, that V-v^(^^ j is
constant for the conditions of similar motions, and the resistance de-
rivative X„ varies with weight {mg) and density (p) according to the law
X„ a ^-^ (31)
The same expression follows for the other force derivatives. For
the moment derivatives,
mm a m k^
where h is the radius of gyration. The necessary theorem for the relation
between stability at a given height and a given loading and the stability
at any other height and loading can now be formulated.
Wo
Let pq, Vq and -~ be one set of values of density, velocity and load-
ing for which the conditions of steady motion have been satisfied and the
resistance derivatives determined.
For another state of motion in which the density, velocity and loading
Wi
are pi, Vi and ^ , the conditions for steadiness will be satisfied if
b
"W^^o^ ^ ^
and the advance per revolution of the airscrew be made the same as before
by an adjustment of the engine throttle.
The derivatives in the new steady motion are obtained from the values
in the original motion by multiplying them by the ratio ^^^ . -^^ for
Wi po\o
forces and by - ^^ for couples. The first ratio is equal to =^ or to
/Wn Pi
^.— , as may be seen by use of (32).
Wn
If the derivatives of (13) be identified with density po and loading-^
b
a new series of coefficients for the stabihty equation can be written down
Wi
m terms of them, but for density pj and loading — ^ . They are
b
STABILITY
473
Coefficient of Ai*, 1
A-i' ^ coefficient of Xi^,
Bi' =. coefficient of Ai^,
7 ,, ^1 ^Oj_7
'■to ^"-q
Ci' ^ coefficient of A^i,
+
1¥
B/c,2
B/ci^
M.
M„
+
bWi
X„ Xto
Zf, Zj
M„ M,.
... WiPo, X
WoPi
M„
M.
z« z
-sin ^0
cos ^0
Dj' ^ coefficients of AjO,
Z„ Z„ sin^o
M„ M«, 0
(33)
It will be seen from (33) that several modifications are introduced into
the stabiHty equation by the changes of loading and density.
For changes of density only, ki = /cq- If the weight of an aeroplane
be changed it will usually follow that the radius of gyration will be changed,
as the added weight will be near the centre of gravity. If the masses are
so disposed during a change of loading that ki = /cq, and the height is so
chosen that ^. — = 1, (33) leads to the simple form of equation >
Wi Po
^i* + AiAi3 + BiAi2 + CiAi + Di=0 . . . (34)
and the stability is exactly that of the original motion. The condition
=y5, — =1 is not easily satisfied, since the heavy loading in one case
Wi Po
may involve the use of too great a height in the corresponding hghtly loaded
condition.
The factor
M„ —sin ^0
M„ cos ^0
M
which occurs in (33) represents the quantity
'^ , i.e. the change of ^ due to change of flight speed at constant altitude.
B B
Apart from the airscrew this quantity would always be zero since M is
then zero for all speeds. For an aeroplane with twin engines so far apart
M
that the tail plane does not project into the tail races the value of -^
will be very small.
474
APPLIED AEEODYNAMICS
k W
As an example of the use of (33) it will be assumed that r^ = 1 , ^
p «o Wo
=1-20, and — = 0-74, i.e. the loading has been increased by 20 per cent.
Po
and the flight is taking place at 10,000 ft. instead of near the ground. The
least stable condition of the aeroplane has been chosen. Table 1 shows
that it occurs for Vq = 60 ft.-s. The conditions lead to
y:
and Vi = l-27Vo = 76-4ft.'S.
Wo Pi
In the original example, page 467, the values of the coefficients of the
stabihty equation were
Ai = 2-80, Bi = 10-0, Ci = 1-86 and Di = 4-39
With Wo = 60, Wq^=12 and the values of the derivatives given in Figs.
235-237, the new equation for stability becomes
A4 + 2-21 A3 + 7'00A2 ^ 0'96A + 2-82 = 0 ]
and a solution of it is \. . . (35)
(A + 1-105 ± 2-3K)(A + 0-001 ± 0-655i) = 0 J
The second factor shows that the motion is only just stable.
The new and original motions are compared in the Table below.
TABLE 2.
Flight speed
Period of rapid oscillation ,
Damping factor
Time to half disturbance .
Period of phugoid oscillation
Damping factor
Time to half disturbance .
Original motion near
the ground.
New motion at 10,000
feet witli an increase of
20 per cent, in the
load carried.
60 ft.-s.
76-4 ft.-s.
2-22 sees.
2-72 sees.
1-45
110
0-47 sec.
0-63 sec.
9 '3 sees.
9-6 sees.
0031
0001
22 sees.
700 sees.
The general effect of the increased loading and height is seen to be
an increase in the period of the oscillations and a reduction in the damping.
The tendency is clearly towards instability of the phugoid oscillation.
Approximate Solutions o£ the Biquadratic Equation for Longitudinal
Stability. — ^If the period and damping of the rapid oscillation be very
much greater than those of the phugoid oscillation, the biquadratic can be
divided into two approximate quadratic factors with extreme rapidity.
The original equation being
A4 + AiA3 + BiA2 + CiA + Di = 0
the approximate factors are
A2 + AiA + Bi = 0)
and
A2+(^i
Bi
STABILITY 475
An example, see (15), gave
(A + 7-34 ± 2-45i) = 0
and (A + 0-075 ± 0-1 70i) = 0
as a solution of
A4 + 14-8A3 + 62-0A2 ^ g.goA + 2-16=0
Applied to this equation (36) gives one factor as
A2 + 14-8A + 62-0 = 0
or (A + 7-4 ± 2-68i) = 0
which substantially reproduces the more accurate solution for the rapid
oscillation.
The factor for the phugoid oscillation is
A2 + 0-150A + 0-0332 = 0
or A + 0-075 ± 0-165i = 0
a factor which again approaches the correct solution with sufficient
closeness for many purposes.
A second example is provided in (35), the approximate factors being
(A + 1-105 ± 2-40i) and (A + 0-005 ± 0-635i)
instead of the more accurate
(A 4- 1-105 ± 2-35i) and (A + 0-001 ± 0-655i;)
of (35). The approximation is again good.
Lateral Stability
The theory of lateral stability follows hues parallel to those of longi-
tudinal stability, and some of the explanatory notes will be shortened
in developing the formula?.
The motions with which lateral stabihty deals are asymmetrical with
respect to the aeroplane. Side shpping occurs along the axis of Y, whilst
angular velocities in roll and yaw occur about the axes of X and Z. Axes
fixed in the aeroplane are again used.
The equations of motion are —
«j-fwr==Y')
pA-fE=L (37)
fC-pE= Nj
The force mY depends partly on gravitational attraction and partly on
air forces. The rolling moment L and the yawing moment N depend
only on the motion through the air.
In the steady motion each of the three quantities Y, L and N is zero.
Vq, Vo and Tq are also zero.
476 APPLIED AEEODYNAMICS
Gravitational Attraction
The component of the weight of the aeroplane along the axis of Y is
mg cos ^0 • sin ^ (38)
where ^ is a small angle. The approximation sin <f> = <f> will be used.
Air Forces
Generally, the lateral force, roUing moment and yawing moment
depend on v, p and r. With a reservation as to lighter-than-air craft,
Y, L and N take the forms
Y=Mv,p,r)]
L=A(^,^r) . (39)
N =/>, p, r))
There are no unsteady motions exclusively lateral, such as that of
looping for longitudinal motion. Such motions as turning and spinning,
although steady, cannot theoretically be treated apart from the longitudinal
motion. For these reasons Y, L and M do not contain terms of zero
order in v, p and r, and expansion of (39) leads immediately to the deriva-
tives. Expanding by Taylor's theorem,
Y = s/^-? + 8p^-^^-+87-^ (40)
dv ^ dp dr ^ ^
etc., or with a notation' similar to that employed for longitudinal
derivatives
Y = vY,-\-pYp + rYr (41)
with similar expressions for L and N.
Forming the equations for snaall oscillations from (87) and (41) leads to
« + wor = fif cos ^0 • 9^ + vY^ + pYp + rY? ]
pA - fE = vL^ + pLp + rL, . . .. (42)
f C — j)E = vN^ + j)Np + rN, j
Before equations (42) can be used as simultaneous equations in v, p
and r, it is necessary to express <f) in terms of p and r.
To obtain the position denoted by Oq, <f), ifs the standard method is
to rotate the aeroplane about GZ through ^, then about GY through
^0, and finally about GX through ^. The initial rotation about GZ has a
component about GX (Fig. 240), and consequently ^ is not equal to p.
The two modes of expressing angular velocities lead to the relations —
p=^-^^md,l (43^
r = 0 cos ^o3
Combining the two equations, we have
<f)=^p -\-r tan ^o (44)
STABILITY
477
Equations (43) might be used to convert equations (42) to the variables
V, <f> and tp. The alternative and equivalent method is to use the know-
ledge that ^ = X<f) in order to express <f) in terms of y and r. Equations
(42) become
V - uor =. y cos ^o| + i) ^in ^o^ + ^Y, 4- V^p + rY,\
^A-rE= vL^+pLp + rhl . (45)
f C - pE = »N, + pNp + rNj
The solution of (45) is obtained by the substitutions
V = Xv, J) = Xp, r = Xr (46)
where Uj, pi and fi are the initial values of the disturbance.
X
Equations (45) become-
y^),-|.(^^o
(A
-Y^> + (
—g sin ^0
^+Moy=0
-M+( XA-h,)p
-N,u + (-AE-N>
+(-AE-L>
+( AC-N,)r
= 0 (47)
= o)
The elimination of any two of the quantities v, p and r leads to the
equation from which A is determined, i.e. to
X-Yv -
g cos ^0
A
AA-Lj
-AE - N„
-Y„ -
AE-L,
AC-N,
= 0
(48)
If the first row be multipHed by A to clear the denominators the equation
will be seen to be a biquadratic in A, the coefficient of the first term being
AC - E2.
For the purposes of comparison of results it is convenient to divide
all coefficients of powers of A by AC by dividine the second row by A and
the third by C. The coefficients obtained, after these changes, by ex-
pansion of (48) in powers of A are
478
Coefficient of A^, 1 —
APPLIED AERODYNAMICS
E2
A2 ^ coefficient of A3,
AC
Y„-!L.-iN,
A
E
+ ^(EY,-L,-N^)
B2 ^ coefficient of A2,
1
C2 = coefficient of A,
i- Y
AC "
Lj, Lp
N„ N,
— Uo+Yr
AC
N,
A sin ^0
C cos ^0
gE
AC
D2 = coefficient of Ao,
X
AC
— cos ^0
sin 60
N.
cos ^0
sin ^0
(49)
It is clear that (49) is greatly simplified in form if the axes of X and Z are
chosen so as to coincide with principal axes of inertia, since E is then
zero. It appears from a comparison of the magnitudes of the various
terms that those containing E as a factor are never important for any
usual choice of axes.
The terms of (49) which do not contain E show a strong general
similarity of form to those for longitudinal stability.
The conditions for stability are that A2, B2, C2, D2 and A2B2C2
— C22 — A22D2 shall all be positive.
Example
0°'9,
Yp = - 0-90,
1>
^^Np = - 0-032,
Mo ==90 ft. -s., ^0
Y, = - 0-105,
fL„= -0-051, xL„=-8-
N„ = 0-0142,
C
E
= 0
Y,
= 15
i-'
= 3-40
e^'
= - 0-40
(50)
Substituting the values of (50) in (49) leads to
A2 = 9-10, Bg = 5-52, C2 =11-26, Dj = - 0-960
Dgis negative and indicates instability.
STABILITY
479
The equation
has the factors
A* + 9-10A3 ^ 5.52A2 + 11-26A - 0-960 :- 0
(A + 8-60)(A2 + 0-570A + 1-36)(A - 0-082) = 0
. . ■ (51)
The roots are partly real and partly complex, and this is the common
case. The instability is shown by the last factor, and it will be seen
later that the aeroplane is spirally unstable. The first factor repre-
sents a very rapid subsidence, chiefly of the rolling motion. The remaining
factor has complex roots and the corresponding oscillation is weU damped.
The time of reduction of the rolUng subsidence to half its initial
value is 0'08 sec, whilst the instability leads to a double disturbance
in about 8| sees. The period of the oscillation is 5| sees., and damps
down to half value in 2-| sees.
Effect of Flight Speed on Lateral Stability
The procedure followed for longitudinal stability is again adopted
and typical curves for lateral derivatives are given (Figs. 241-243). The
stalUng speed has been kept as before, and the values of ^0 ^^^Y be taken
from Fig. 238.
UnUke the longitudinal motion, which was usually very stable, the
illustration shows instability to be the common feature, and later this
will be traced to the choice of -L^ and T^Np, which are largely at the
designer's disposal.
The periods and damping factors at various speeds corresponding with
the derivatives of Figs. 241-243 are given in Table 3 and are of great
interest.
TABLE 3.
Flight
speed
(ft.-s.).
Boiling subsidence.
Lateral oscillation.
Spiral subsidence.
Damping
factor.
Time to half
disturbance
(sees.).
Periodic
time
(sees.).
Damping
factor.
Time to half
disturbance
(sees.).
Damping
factor.
Time to half
distiurbance
(sees.).
59-2
58 -r.
60
70
80
90
100
122-5
140
0-652
2 0
3-07
6-60
7-50
8-60
9-60
11-81
13-60
10
0-35
0-22
0-11
0-09
0-08
0 07
0-06
0-05
6-25
6-48
6-41
7-00
6-25
5-55
4-91
3-95
3-46
-1-31
-0-48
+0-19
0-35
0-31
0-28
0-28
0-31
0-35
-0-53
-1-4
+ 3-6
2-0
2-2
2-5
2-5
2-2
2-0
+ 1-53
+0-12
+0-03
-0-16
-012
-0-08
-0-05
-0-01
+0-003
0-45
1-6
230
-4-3
-5-7
-8-6
-14-0
-700
+ 230
Negative values occurring in the above table indicate instability,
and the expression " time to half disturbance " when associated with
a negative sign should be interpreted as " time to double disturbance."
Throughout the speed range of steady flight the stability equation
480
APPLIED AERODYNAMICS
for the lateral motion has two real roots and one pair of complex roots.
When the aeroplane is stalled or overstalled the oscillation becomes
very mistable, and stalling is a common preUminary to an involmitary
spin. For speeds between 70 ft.-s. and 100 ft.-s. the oscillation is very
stable, and neither the period nor the damping shows much change.
The damping of the rolling subsidence is compared below with the
value of jhp on account of the remarkable agreement at speeds well
above the minimum possible.
TABLE 4.
Flight speed
Damping factor of
1,
(ft.-8.).
rolliiig subsidence.
-jU>
59-2
0-65
-1-6
58-6
20
+0-6
60
3 07
2-7
70
6-50
6-0
80
7-50
7-5
90
8-60
8-6
100
9-60
9-6
122-5
11-8
11-8
140
13-5
13-4
The agreement suggests that (A + -L^) is commonly a factor of the
A.
biquadratic for stability except near stalling speed. The motion indi-
cated is the stopping of the downward motion of a wing due to the increase
of angle of incidence. This is the nearest approach to simple motion
in any of the disturbances to which an aeroplane is subjected. It is
possible that the first two terms entered under spiral subsidence really
belong to the rolling subsidence, as the analysis up to this point does not
permit of discrimination when the roots are roughly of the same magni-
tude. In either case the discrepancy between — ^ and the damping
factor at 59*2 ft.-s. is great, and in itself indicates a much less simple motion
for an aeroplane which is overstalled and then disturbe'd.
Over a considerable range of speeds (70 ft.-s. to 130 ft.-s.) instabiHty is
indicated in what has been called the " spiral subsidence." This is not a
dangerous type of instability, and has been accepted for the reason that
considerable rudder control has many advantages for rapid manoeuvring,
as in aerial fighting, and the conditions for large controls are not easily
reconciled with those for stability.
For navigation, such instabiHty is undesirable, since, as the name
implies, the aeroplane tends to travel in spirals unless constantly cor-
rected. This motion can be analysed somewhat easily so as to justify
the description " spiral."
As was indicated in equation (51), spiral instabiHty is associated with
a change in sign of D2 from positive to negative, whilst C2 is then
0.0
Xr
-0.1
«
'*-
^^
-_
-0.2
481
so 60 70 80 90 100 110 120 130 KO
SPEED FT/S
0
0.02
0.04
/•
^
0.06
0.08
0.10
0.12
0.14.
"
f
\K
""^
"-^
50 60 70 80 90 too IIQ 120 130 1-4-0
SPEED PT/S
0.02
_^
hK
^
O.OI
*?:-'
^
-^
c
0
50 60 70 60 90 100 110 120 130 KO
SPEED FT/S
FlO. 241. — Resistance derivatives for sideslipping.
2 I
482
\ ^
^s ■■
50 60 70 60 90 100 JIO 120 130 KO
SPEED FT/S
N
'VJ^ — .
so 60 70 ao 90 100 110 120 130 140
SPEED FT/S
0
•0.2
•0.4
/-
■^
^^
-^
^-.^^^
/^
i^r
■ 1
\
10
'50 60 70 80 90 100 HO 120 130 140
SPEED FT/S
FiQ. 242 — Resistance derivatives for rolling.
/
N^
Y
p
^
\.^
^"->
^--
483
50 60 70 80 90 100 1 10 120 130 140
SPEED'FT/S
1
0
/
'■""
\
•5
\
^^
\
\s
10
>^
^\
.^
•IJ5
"^
50 60 70 SO 90 100 110 120 130 140
SPB^D FT/S
0.1
H
0
1
\
"^^
0.1
so 60 70 ao 90 100 MO 120 130 140
SPEED FT/S
Fio. 243. — Resistance derivatives for yawing.
484 • APPLIED AERODYNAMICS
moderately large. If D2 is very small the root of the biquadratic cor-
responding with the spiral subsidence is
A + ^^^-0 (52)
^0 is zero between 90 ft.-s. and 100 ft.-s., and equation (49) shows that
when ^0 is zero
^ AC N, N,
(53)
and D2 depends on the roUing moments and yawing moments due to
sidesUplping and turning, and changes sign when N^L, is numerically
greater than L^N,.
Consider the motion of the aeroplane when banked but not turning :
the aeroplane begins to sideshp downwards, and the sideslipping acting
through the dihedral angle produces a rolling couple L„ tending to reduce
the bank. At the same time the sideslipping acting on the fin and
rudder produces a couple N^ turning the aeroplane towards the lower
wing. The upper wing travels through the air faster than the lower
as a result of this turning, and produces a couple L, tending to increase
the bank. The turning is damped by the couple N,.
There fire then two couples tending to affect the bank in opposite
directions, and the aeroplane is stable if the righting couple preponderates.
If, on the other hand, the aeroplane is unstable it overbanks, sideslips
in more rapidly, and so on, the result being a spiral. There is a limit to
the rate of turning, but the more formal treatment of disturbed motion
must be deferred to a later part of the chapter. Enough has been said
to justify the terms used.
Climbing and Gliding Flight
Owing to the twist in the airscrew race the effect of variation of
thrust on the position of the rudder may be very considerable. The
derivatives also change because of the change of speed of the air over
the fin and rudder. An airscrew which has a velocity not along its axis
experiences a force equivalent to that on a fin in the position of the
airscrew. Yawing and sideslipping produce moments as well as forces,
and the calculation of stability must in general be approached by the
estimation of new conditions of steady motion and new derivatives.
Variation of Lateral Stability with Height and Loading
The derivatives change with density and loading according to the
law already deduced for longitudinal stabihty, where it was shown that
the force derivatives and the moment derivatives divided by the mass
of the aeroplane varied as — , if the quantities ^^r?r^ and ^ft were kept
^ m W nD ^
STABILITY
485
. Wn
constant in the steady motions. If -~ and po correspond with loading
Wi
and density for one steady motion and -^ and pi with loading and
density for another, then the force derivatives in the second motion
are obtained from those in the first by multiplying by
the moment derivatives the multiplying factor is
conveniently
nil
Wo
/Wo Pi
V Wi po
is /^i.'^i
^/ Wo Po'
W] Po
or more
In writing down the coefficients of the biquadratic for stability it
will be assumed that the axes of X and Z have been chosen to be principal
axes of inertia, so that E is zero. The coefficients are :
Coefficient of Aj*, 1
., - coefficent of A,, ( W,.P.y| _y„ - ^(g)^L. - ^<g) N,
B2^ ^ coefficient of A^^^
L,
N.
02^ ^ coefficient of A^,
Wp PiV
Wi'po
'^Ac[ki^)Mi^\
Wi Po
ij, L,
) Y„ Yp — ^OTTT * ~ ~f" Jf^r
^ I Wo Pi
Lj, hp
N„ cos ^0 1 1
D2^ = coefficient of Aj",
N„
0
N,
cos ^0 sin ^0
(54)
If ^.^ - 1, (^) = 1 and (^:) = 1, the stabiHty
is again the same
as the original stabiUty.
It has been pointed out that spiral instabiUty occurs when D2 changes
sign, and from (54) it is clear that the new factors will not change the
486
APPLIED AEKODYNAMICS
condition altliougli they may affect the magnitude. It follows that
spiral instability cannot be eliminated or produced by changes of height
or loading.
Example. — Increase of loading 20 per cent, and the height 10,000 feet, where
f"^ = 0-740. Speed 60 ft.-s., (^J) - 1, (\\\ ^1,6^^ 11"
Pa ^.1^1 /a \*i /o
v/
^'•^=1-27 and Vi = l-27Vo
"0 Px
76-4 ft.-s.
For the loading Wq and Pq the values of the coefficients of the biquad-
ratic which correspond with Table 3 are
A2 = 3-48, B2 == 2-33, C2 = 3-12, D2 = 0-104
and from (54) the values for the increased loading and height are found as
A2' = 2-74, B2' = 1-45, C2' = 1-83, Dg' = 0-0645
The biquadratic equation with these coefficients has been solved,
the factors being
(A + 2-45)(A2 + 0-255A + 0-728)(A + 0-0362) = 0 |
(55)
or (A + 2-45)(A + 0-127 ± 0-852i)(A + 0-0362) = Of* '
The new and original motions are compared in the Table below
TABLE 5.
Original motion near
' ground.
New motion at 10,000 ft.
with an increased
loading of 20%.
Plight speed
Damping factor of rolling subsidence
Time to half disturbance
Period of lateral oscillation
Damping factor .
Time to half disturbance
Damping factor of spiral subsidence .
Time to half disturbance
60 ft.-s.
3-07
0-22 sec.
6-41 sees.
019
3-6 sees.
0 03
23 sees.
76-4 ft.-s.
2-45
0-28 sec.
7-37 sees.
0-063
11 sees.
0 036
19 sees.
The rolling subsidence is somewhat less heavily damped for the
increased loading and height, whilst the spiral subsidence is more heavily
damped. The period of the lateral oscillation is increased and its
damping much reduced.
In both longitudinal and lateral motions the most marked effect of
reduced density and increased loading has been the decrease of damping
of the slower oscillations.
Stability in Circling Flight
The longitudinal and lateral stabihties of an aeroplane can only be
considered separately when the steady motion is rectilinear and in the
plane of symmetry, and it is now proposed to deal with those cases in
which the separation cannot be assumed to hold with sufficient accuracy.
STABILITY 487
The analytical processes followed are the same as before, but the quantities
involved are more numerous and the expressions developed more complex.
In order to keep the simplest mathematical form it has been found advan-
tageous to take as axes of reference the three principal axes of inertia of
the aeroplane.
The equations of motion have been given in Chapter V., and in refeifencft'
to principal axes of inertia take the form —
u-{-wq~vr =i'X.\
v-\-ur — wp='Y\
w-\-vp — uq =■ 2 \
'pA-rq(B~C)=-h( ^^^^
qB-pr{C-A)==M\
The axes are indicated in Fig. 106, Chapter IV., whilst in Chapter V.
various expressions are used for the angular positions relative to the
ground. Of the alternatives available, the expressions in terms of direction
cosines n^, 11.2 and n^ for the position of the downwardly directed vertical
relative to the body axes will be used.
Gravitational Attractions. — The values of X, Y and Z depend partly
on the components of gravitational attraction and partly on motion through
the air. The former are respectively
n-^g, n^g and n^g (57)
Air Forces. — In an aeroplane the forces and moments are taken to be
determined wholly by the relative motion, and each of them is typified by
the expression
X ==h{u, v,w,p,q,r) (58)
Before the stability of a motion can be examined, the equations of
steady motion must be satisfied, i.e.
Woqo — Wo =Xo^
UoTq - WoPo = Yo
f^oPo-^oqo =Zo\
-Mo(B-C)=^Lo'' ^^^^
-Po^o(C-A)-Mo
-?oPo(A— B)=No,
must be solved. It has already been "pointed out (Chapter V.) that steady
motions can only occur if the resulting rotation of the aircraft is about the
vertical, in which case
Po=>niQ, q^^n^H r^ — 712,0. .... (60)
where 12 represents the resultant angular velocity. Some problems
connected with the solution of equations (59) have been referred to in
Chapter V.
Small Disturbances. — As in the case of longitudinal stability, the
quantities ~J~ , 1^ ^ etc., are spoken of as resistance derivatives, and their
488
APPLIED AEKODYNAMICS
values are determined experimentally. The shorter notation X„, X^
introduced by Bryan is also retained. If Uq-\-u be written for u, Vq-\-v
for V, etc., in equations (56) and the expansions of X ... N up to first
differential coefficients used instead of the general functions, the equations
can be divided into parts of zero and first order. The terms of zero order
vanish in virtue of the conditions of steady motion as given by (59), and
there remain the first-order terms as below :—
u + wqQ-{-Woq — vTq — ^o** ■■
V + urQ-{- UqT— wpQ — WqP
w-\-vpQ + VqP - uqQ — u^q ■■
C-B
V
{'^%+m)
Bj?+ g (l>ro + Por)\
Y+ (-Y {qpo+qoPYi
In these equations u, v, w,
turbances, whilst the same letters
. (61)
will be written down in terms of
motion.
= gdni + uXu + vX/ + wXJ\
+2?X/ + 5X;+rX/
= gdn2 + uYu + ^Y^' + wYJ
+^Y/ + gY;+rY/
= gdn^ + uZ,/ + vZJ + wZJ
,+pZ;+qZ,' +rZr'
uL^ + vLJ + wlij
+pV+5L/+rL/
wM„'+vM/+w;M^'
+i5M/+^M/+rM/
+:pN/+gN;+rN/ /
p, q and r represent the small dis-
with the suffix zero apply to the steady
motion, and are therefore con-
stant during the further cal-
culations. The dashes used to
the letters X . . . N indicate
that the parts due to air only
are involved; the derivatives
are all experimentally known
constants.
Evaluation of dn^, dn^ and
d7i^ in terms of p, q and r. —
Before progress can be made
with equations (61 ) it is necessary
to reduce all the quantities to
dependence on p, q and r. In
developing the relation, three
auxiliary small angles a, ^ and
y are used which represent dis-
placements from the original
position, and expressions for
p, q and r and d7ii, dn^ and dn^
a, fi, y, and the rotations in the steady
STABILITY
489
If GP of Fig. 244 represent the downwardly directed vertical defined
by the direction cosines n^, n^ and W3 before displacement and by ni-\-dni,
etc., afterwards, it is readily deduced from the figure that
Ui -j- drii = 111 ~ ^3iS + '^27
(62)
with similar expressions for n^ and n^. The changes of direction cosines
are therefore
dn^ = — fiiy -f- n^cc I
(63)
The resultant velocity being made up of il about the vertical and
a, ^ and y about the axes of X, Y and Z, the changes from Pq, % and Tq
can be obtained by resolution along the new axes, and hence
r = — g-oa+^o^ + y
■,\
(64)
In the case of small oscillations it is known from the general type of
solution that
rAa
i8 = Ai8
Ay
(66)
and using these values in (64) reduces the equations to simultaneous linear
form for which the solution is
-'0
A
%
-Po
A
-p
y _
^ % V
. ^ -^0 v\
^0 -Po q
To X q
-% A r
-% Po 'T
1
A -ro
fn A
Po
-%
%
-Po
A
(66)
The determinant in the denominator of the last expression is easily evaluated
and found to be A(i22 _^ ^2^^ ^nd from (63) and (66) it can be deduced that
\ Til p I
dn,^ To n^ q /(il^ + A^) .... (67)
-qo ^3 r ^
1
02 + A2
{(1
Wi2)i2p — (nigo+W3A)g — (niro — rt2A)r} . (68)
Similar expressions for dn2 and dn^ follow from symmetry by the ordinary
laws of cyclic changes.
It is convenient to make temporary use of a quantity (x defined by
ix =
_ 9
122 + A2
(69)
With the aid of the relations developed it is now possible to rewrite
equations (61) in more convenient form as
490
APPLIED AEKODYNAMICS
(X„' - X)u + (X/ + ro)v + (XJ - qo)w >,
+ {Xp'+/xn(l-Wi*)}2>+ {X/- Wo-M(»ii5'o+Aj«3)}^+ {X,'+«„-M(Wiro-A/i2)}r = 0
(Y„' - ro)u + (Y/ - X)v + (YJ + Po)w
+ {Yp'+M;o-M(w2Po-A«8)}2'+{Yg'+/^ii{l-«2^)}'i+iY/-«o-M(w2ro+Awi)}r=0
(Z„' + fi„)M + (Z»' - Po)i' + (Z«,' - A)u;
A
L '
(¥-A> H
■^ A
+ * A + A '
+
hui'lO
')' Hi
L/ , B-G
+
-2oj^
= 0
M„'m
B
+
M/v
+
M„'w;
B
,/M„',C-A \ , /M/ ,\ i/*l''j_^~A^ \
C
+
N'v
+
C
N/
= 0
(70)
,/Np' A-B \ /N/,A-B \„,/Nr \
An examination of the equations will show that certain constants may
be grouped together and treated as new derivatives. The table below will
be convenient for reference to the equivalents used.
1
u
X
X„'
Y
Y„'-r„
Z
Z„'+<7o
L
w
A
M
M„'
B
N
N„'
C
X„'+ro X„'— ^o
Y/ Y«,'+po
Z/-PO Z„'
A
M/
B
N/
C
V
A
B
C
x^;
Yp + Wo
Zp' — Wo
V
A
M/ , C-A
B ■*" B ''»
Np^ A - B
x/-
Y/
Z/ +
A '
M,'
B
N/
B-C
C
+
Po
\
X/ + Wo
Y/ - «o
Z/
A + A ^^
B ^ B
Nr'
/
Table (71) needs little explanation ; it indicates that in the further work
an expression such as X^ is used instead of the longer one X„' +^0' ^^^ ^^ ^'^•
If now the variables p, q, r, u, v and w be ehminated from equations (70),
the stabihty equation in A is obtained, and in determinantal form is given
by (72).
x„-.^
x»
X
Y„
Y„-A
Y
z„
z„
z.
L„
L„
L«
M„
M„
M,
N«
N„
N,
Xp+/in(l-wi2) X,-^(nigo+'^3A) X,-M(mi»-o-»2A) j
Yp-MwaPo-w-sA) Y4+Mn(l-V) Yr-Mw2»-o+™iA)
Zp-MC^aPo+WgA) Z^— Ai(w3g'o-WiA) Z^+M^ll-ns^)
Lp— A Lj L,
Mp M,-A M,
Np N, N,-A
=(0)
(72)
The further procedure consists in an appHcation of (72), and the point
at which analytical methods are used before introducing numerical values
is at the choice of a worker. The analysis has elsewhere been carried to
the stage at which the coefficients of A have all been found in general form,
STABILITY
491
but the expressions are very long. It would be possible to make the sub-
stitution in (72) and expand in powers of A by successive reduction of the
order of the determinant, and from the simphcity of the first three columns
it would be expected that this would not be difficult. The presence of /x
is a comphcation, and perhaps the following form, in which it has been
eliminated, represents the best stage at which to make a beginning of the
numerical work : —
A2
-gil
-gil
-gil
+Q=
x„-
-A
x„
Xm,
X.
Y gns
X,- ^
Y„-
-A
Y.
z.-
-A
Y,+
gn-s
A
Y.
^«+ A
^' A
z.
z.
z^-
gn2
A
Lp-
A
M,-A
Mp
N„
N,
N„,
N.
N«
N,-A
x„-
-A
X,
Xm,
-(l-ni2)
X,
X,
Y„
Y.-
-A
Y.
tHn^
Y.
Y.
Zu
z.
z.-
A
niUs
z.
z.
L„
L.
L«,
0
L.
L.
M„
M,
M.
0
M,-A
M,
N„
N„
N,
0
N,
N,-A
x„-
-A
X,
Xw;
X.
nin2
X.
Y„
Y.-
-A
Y.
Y,
(1-V)
Y,
z„
z«
z.-
-A
Zp
W2W3
z.
L„
L„
L.
Lp-
-A
0
L,
M„
M„
M.
Mp
0
M,
N„
N„
N^
N,
0
N,-A
x„-
-A
x«
x„,
Xp
X,
Wing
Y„
Y„-
-A
Y.
Y.
Y.
W2W3
z«
z.
'^w-
-A
z.
z.
-(1-^3^)
L„
L.
K
L,-
-A
L,
0
M„
M,
M,
Mp
M,-A
0
Nu
N.
N.,
N,
N,
0
x„-
-A
X.
Xjt,
Xp
X,
X.
Y„
Y„-
-A
Y.
Y.
Y,
Y,
z«
z.
K-
-A
z.
z.
z.
L„
L.
Lw
Lp-
-A
L.
L,
M„
M,
M,,
Mp
M,-A
M.
N„
N.
N„.
N.
N,
N,-A
. (73)
= 0
492
APPLIED AERODYNAMICS
The equation proves to be of the eighth degree, the term which appears
to be of order X'^ having a zero coefficient. The expressions which occur
when the longitudinal and lateral motions are separable are underlined in
the first determinant of equation (73)) which therefore contains the octic
(A4 + AiA3 + BiA2 + CiA + Di)(A4 + A2A3 + B2A2 + C2A + D2) • (74)
If Ci _ be written for Ci when the g terms are neglected, it is obvious
that the second determinant pontains a term
i22(A3 + AiA2 + BiA + Ci^^^(A3+A2A2 + B2A + C2^^o) • C^^)
From the third and fifth determinant can be obtained the term
L,, no I \
iO gui gn^ —gX
!L. L^ l I
iN^NpNj,
The fourth determinant furnishes a similar term :
W2i2(A3 + A2A2 + B2A + C2^^o)
N„— W]
X„ X^ grii
-gX
n^ — ni
2„ Zj, gus
M„M„0
M„ M«,
(76)
(77)
The remaining terms of (73) are too complicated to analyse in a general
way, but from one or two numerical examples it would appear that the more
important items are shown in (74) . . . (77).
The factors of (74) are exactly those which would be used if the motions
were separable, but with the derivatives having the values for the curvi-
linear motion.
Example of the Calculation of the Stability of an Aeroplane when turning during hori
zontal flight.
Initial conditions of the steady motion : —
m = 0 ^2 = 0-707 ns = 0-707
i.e. the axis of X is horizontal and the eieroplane banked at 45°.
Uf) — 113*5 ft.-s. , Vq = 0 Wq — O
i.e. the flight speed is 113-5 ft.-s., and there is no sideslipping or normal
velocity. The last condition constitutes a special case in which the re-
sultant motion has been chosen as lying along one of the principal axes of
inertia
il = 0-284 rads.-8ec.
i.e. one complete turn in about 22 sees.
^0 = 0 ^0 = 45° as deduced from Wj, Wj *^d n^
The only condition above which requires specific reference to the equations of motion
for its value is that which gives n. The second equation of (59) is
^o'*o — M'o^o = Yo (79)
and for the condition of no sideslipping Yq depends only on gravitational attraction
and is equal to n^ ; since rQ—n^^, whilst Wq and p^ are zero, equation (79) becomes
(78)
UnO.
(80)
STABILITY
493
a relation between n and quantities defined in (78) which must be satisfied. The
other equations of (59) must be satisfied, and the subject is dealt with in Chapter V.
Since there are only four controls at the disposal of the pilot, some other automatic
adjustment besides (80) is required, and is involved above in the statement that «(,= 11 3*5
ft.-s. when Wq=0. The state of steady motion is fixed by equations (59), and the small
variations of «... r about this steady state lead to the resistance derivatives. In
the present state of knowledge it is apparently sufficient to assume that derivatives are
functions of angle of incidence chiefly and little dependent on the magnitude of Vq, p^,
q^ and r^. Progress in application of the laws of motion depends on an increase in
knowledge of the aerodynamics.
With these remarks interposed as a caution, the derivatives for an aeroplane of about
2000 lbs. weight flying at an angle of incidence of 6"^ may be typically represented by the
fol owing derivatives.
Resistance Derivatiws (see Table (71)).
u
V
u>
P
■
9
r
X
—0111
0-201
—0-020
0
0
0
Y
—0-201
—0-128
0
— 107
0
— 109-8
Z
— 0T>98
0
-2-89
0
102-6
0
L/A
0
— 0-a333
0
—7-94
-0-088
2-48
M/B
0
0
-0-1051
0-088
-8-32
0
N/C,
0
+0-0145
0
0-694
0
-1023
/
(81)
The values of A, B and C occur only in the derivatives, and the use of -, _ and -
in (73) does not affect the condition for stability. The whole of the quantities in (81)
are essentially experimental and must therefore be obtained from the study of design
data. When the effects of airscrew slip stream are included the deduction from general
data is laborious and needs considerable experience if serious error is to be avoided.
The numerical values of the derivatives as given in (81) can be substituted in (73)
and the determinants reduced successively until the octic has been determined. It is
desirable to keep a somewhat high degree'of accuracy in the process in order to avoid
certain errors of operation which affect ^the solution to a large extent. The- final
result obtained in the present example is
A» + 20-4A7 + 151 3A« + 490A5 + 687A* + 719A3+150A2+109A + 6-87 = 0 . (82)
This equation has two real roots only, which can be extracted if desired by Homer's
process. A general method for all roots has been given by Graeffe, and as this does not
appear in the English text- books an account of its application to (82), is given as an
appendix to this chapter. By use of the method it was found that equation (82) has
the factors
(A2+ll-25A+35-l)(A2-0-006A+0-171)(A+7-79)(A+0-067)(A2+l-33A+2-19)=0 . (83)
and the disturbed motion consists of three oscillations, one of which is unstable, and two
subsidences.
A careful examination of (83) in the light of the separable cases of longitudinal and
lateral disturbances shows that the factors in the order given correspond with (a) Rapid
longitudinal oscillation ; (6) Phugoid oscillation (imstable) ; (c) Rolling subsidence ;
(d) Spiral subsidence ; and (e) Lateral oscillation. It appears from further calculations
that at an angle of incidence of 6° the effect of turning shows chiefly in the phugoid
oscillation and in the spiral subsidence, the former becoming less stable and the latter
more stable. At or near the stalling angle changes of a completely different kind may
be expected, but the motion has not been analysed.
494
APPLIED AERODYNAMICS
Comparison of Straight Flying and Circling Flight. — For reasons given
earlier as to the inadequacy of the data for calculating derivatives, too
much weight should not be attached to the following tables as repre-
sentative of actual flight. They do, however, illustrate points of
importance in the effect of turning on stability. Four conditions are
considered : —
(1) Horizontal straight flight.
(2) Ghding straight flight.
(3) Horizontal circling.
(4) Spiral gliding.
The data is based on the assumption that the airscrew gives a thrust
only, and therefore ignores the effects of slip stream on the tail which modify
the moment coefficients in both the longitudinal and lateral motions. A
recent paper by Miss B. M. Cave-Browne-Cave shows that our knowledge
is reaching the stage at which the full effects can be dealt with on
somewhat wide general grounds. The tables are based on flight in all
cases at an angle of 6°, and the speed has been varied to maintain that
condition.
The angle of bank in turning has been taken as 45°.
' Rapid longitudinal oscillation. —
Horizontal
straight.
Gliding
Btraiglit.
Horizontal
clroling.
Spiral '
gilding
Damping factor
Modulus
Damping factor -i- velocity .
Modulus -f velocity
4-71
4 '97
0-0495
0-0521
4-67
4-92
0-0494
0-0520
5-62
5-92
0 0496
00622
5-53 )
5-82
0-0494
0-0520
(84)
■ The damping factors for curvihnear flights are both appreciably greater
than those for rectihnear flight, and it will be seen from the third row
of the table that the increase is entirely accounted for by the change of
speed.
Phugoid oscillation. —
>
Horizontal
straight.
Gliding
straight.
Horizontal
circling.
Spiral
gliding.
0026
0-41
0 000201
0 0037
Damping factor . . .
Modulus
Damping factor -=- velocity.
Modulus -^- velocity . .
-
0-0465
0-28
0-000488
0-0029
00666
0-28
0-000586
0-0030
-0-003
0-41
-0-00003
0-0036
(85)
The damping factors for curvilinear flight are very much less than
those for rectilinear motion, whilst the moduli are greater. The oscillation
is, therefore, rather more rapid, but less heavily damped, whilst the effect
of descending is of the same character for both straight and curved flight
paths, and descent gives increased stabihty in all cases.
STABILITY
495
Rolling subsidence. —
! Horizontal
straight.
Gliding
straight.
Horizontal
circling,
Spiral
gliding.
Damping factor ... 6*55
Damping factor -^ velocity . 0 -0686
i •
6-50
0 0687
7-79
0 0686
7-76
0-0694
>
(86)
As in the case of the rapid longitudinal oscillation, the changes in the
damping coefficient of the rolHng subsidence are accounted for by changes
of speed, as may be seen from the second row of (86).
Spiral motion. —
Horizontal
straight.
Damping factor
Damping factor -
... -00069
velocity. -0 00007
Horizontal
circling.
Spiral
gliding.
0-067
0-00059
0 092
0-00082
(87)
The effect of the turning has been to increase very considerably the
damping factor of the spiral motion, and the change appears to be closely
associated with the opposite change noted in connection with the phugoid
oscillation. Here, as in that case, the changes of speed do not account for
the changes of damping factor.
Lateral oscillation. —
Damping factor
Modulus
Dam ping factor -i- velocity .
Modulus -^ velocity
Horizontal
straight.
0-550
1-27
0 00576
0 0133
Gliding
straight.
0-525
1-23
0-00555
00130
Horizontal
circling.
0-665
1-48
0-00586
0-0130
Spkal
gliding.
0-633
1-43
0-00666
0-0128
(88)
J
The changes of modulus are seen to be almost entirely accounted for
by the changes of speed. A considerable part of the change in the damping
factors is also accounted for in the same way, although in this case the
influence of other changes is indicated.
General Remarks on the Tables. — So far as the oscillations are involved,
the tables indicate a tendency for the product of the velocity and the
periodic time to remain constant. The rapid lateral and longitudinal
oscillations remain practically independent of each other. An important
interaction, which probably occurs in the circular flight of all present-day
aeroplanes, connects the spiral and phugoid motions. It appears that
turning increases the damping factor of the spiral motion whilst simul-
taneously reducing the stability of the phugoid oscillation. In one of
the examples here given, the motion has changed from a stable phugoid
oscillation and an unstable spiral motion in horizontal straight flight to
an unstable phugoid oscillation and a stable spiral motion for a horizontal
banked turn.
496
APPLIED AEKODYNAMICS
Effect of Changes of the Important Derivatives on the Stability of
Straight and Circling Horizontal FUght. — The derivatives considered were
M^, L„ and N„ with consequential changes of M^ and N^, and are important
in different respects. M^, can be varied by changing the position of the
centre of gravity and the tail-plane area, L„ by adjustment of the lateral
dihedral angle, and N^ by change of fin and rudder area. All are appreci-
ably at the choice of a designer, and the following calculations give some
idea of the possible effects which may be produced. At a given angle of
incidence resistance derivatives are proportional to velocity, and simplicity
of comparison has been assisted by a recognition of this fact.
Variations of M^. L^ and N„ constant.
Rapid longitudinal oscillation. —
100 -7- VLJ'R X velocity.
-0-264
H)176
-0093
—0-042
0
0044
00884
10*x damping j
factor 1
4- velocity
10* X modulus^ \
-f- velocity /
Horizontal straight
Horizontal circling
Horizontal straight
Horizontal circling
6-24
6-23
7-02
6-98
5-58
5-66
6-17
616
4-95
4-95
6-21
5-20
/5 04
14-07
(4-91
14-08
(4-53)*
(4-46)
5-92
2-52
5-43
3-07
(3-86)
(3-90)
6-55
1-27
6-27
1-51
(2-89)
3-08)
6-60
0-57 \
6-56
0-42
(89)
The range given to M^ is particularly large, and. the most noticeable
feature of (89) is the small effect of turning on the rapid longitudinal oscil-
lation. The figures in brackets correspond with a pair of real roots, viz.
(4-53)2 =: (5-04 X4-07), and it will be seen that the motion represented is
always stable but not always an oscillation. For a very unstable aero-
plane as represented oy the lowest value of M„ there is some indication of
a complex interchange between the longitudinal and lateral motions, which
would need further investigation before its meaning could be clearly
estimated.
Phugoid oscillation,-
100 M„/B X velocity.
10* X damping
factor
-^Velocity
10* X modulus
^velocity
Horizontal straight
Horizontal circling
Horizontal straight
Horizontal circling
— 0-264 —0-176 —0098 —0042 0 0-044 00884
4-3
0-39
3-66
3-82
4-5
4-9 5-6
0-14 -0-26 -0-
3-40 2-921 2-26
3-74 3-64| 3-56
/14-5
I 0-0
-1-26
3-34
56-0
-24-4
-2-78
2-94
A
56-8
-46-9 (90)
- 9-48'^
211
The differences for stabiUty between straight and circling flight are
here very marked. The former shows stabihty at all positive values of
M^, and the change from stabihty of the oscillation to instabihty in a nose-
dive occurs without the intermediate stage of an unstable oscillation. In
circling flight, however, the general result of a reduction of M.^ is to
produce in increasing oscillation. In all cases' the damping is very
small in circling motion at an angle of bank of 45° as compared with that
in straight flying, and a greater value of Mj<, is needed for stability. In
STABILITY
497
straight flying there is indicated a hinit to the degree of damping of the
phugoid oscillation which can be attained.
Spiral vMtion. — ■
100 M^B X velQcity.
-0-264
10* X damping
factor
-h velocity
Horizontal straight —0*72
Horizontal circling 4*78
—0176
-0-72
518
—0098
-0-72
5-93
—0042
-0-72
6-89
•fO-044
-0-72 -0-72
8-52 j 13-8
(91)
In rectilinear flight the spiral motion is miaffected by changes of M^,
and the negative value indicates instabihty. The effect of turning is to
convert a small instability into a marked stability which is dependent for
a secondary order of variation on the magnitude of M^,.
Rolling Subsidence and Lateral Oscillation. — It appears that neither
of these quantities is appreciably affected by either the variation of Mj<,
or of circling, beyond the changes which are proportional to the velocity
of flight. The expressions corresponding with those used in (90) are then
constants for the conditions now investigated. For the rolling subsidence
" damping factor/velocity " has the value 0-0686, whilst for the lateral
oscillation "damping f actor/ veloci ty " is equal to 5-85 X 10~3^ whilst
"modulus/velocity" has the value I'Bl X 10^2^
Variations o! L,, and N„. Ma, Constant. — The changes of rapid longi-
tudinal oscillation due to change of lateral derivatives are inappreciable,
and the differences between straight flying and circling are produced only
by the changes in the velocity of flight. Similar remarks apply to the rolUng
subsidence, as might have been expected from the very simple character
of the motion and the fact that the only important variable of the motion,
i.e. lip, has not been subjected to change.
Phugoid osciUation. Circling flight. —
N /C X lOVveloclty.
L^/Ax velocity I -0-6 | 0 j +0-5 j +1-28 \ -0-5
0 +0-5
+ 1-28
Damping factor x 10*/velocity
— 216 -3-88 I -3-34
1-54 4-84 0-86 I -025
29-9 120 7-4 4-71
-0-0002935
-0 001
Modulus X lO'/velocity
— ! 2-75 I 3-36 , 360
4-16 I 3-64 I 3-56 I 3-64
3-71 3'73 3-61 i 3 62
(92)
and
Straight flight.
Damping factor X 10*/velocity =4*9
Modulus X lO'/velocity = 2*92 for all values of L^ and N,
For the numerically smallest values of L„ and N, the centrifugal terms
introduced by turning, convert a stable phugoid to an unstable one.
Increase in the dihedral angle has a counterbalancing effect, and the phugoid
becomes stable over the range of N„ covered by the table. The longi-
tudinal stability of rectihnear motion is of course unchanged by a dihedral
angle or by the size of the fin and rudder, which are the parts of the
aeroplane which primarily determine L„ and Np.
2k
498
Spiral motion. —
APPLIED AERODYNAMICS
Damping factor X 10* /velocity. \
Ne/CX 10* /velocity.
£e/A X velocity
-0-5
0
+0-5
+ 1-28
+ 2-00
» {
-U 0002935 J
-0-00076 j
-0-001 1
Horizontal straight
Horizontal circling
Horizontal straight
Horizontal circling
Horizontal straight
Horizontal circling
Horizontal straight
Horizontal circling
127
106
69-2
54-8
1-44
0
10-7
26-9
401
32-8
31-7
2-57
-8-33
7 05
6-61
5-56
17-7
20-65
3 93
-9-70
6-70
-0-72
5-94
9-06
12-72
4-69
-1015 \
- 315
5-40
9-06
/
(93)
The value of N„ changes sign when the aeroplane, regarded as a weather-
cock rotating about the axis of Z, just tends to turn tail first. In the ab-
sence of a dihedral angle the steady state is neutral in straight flight, but
becomes stable on turning. For both straight flying and turning, stabihty
may be produced in an aeroplane showing weathercock instability by the
use of a sufficiently large dihedral angle. It is not known how far this
conclusion may be applied at other angles of incidence.
Lateral oscillation. —
Lb/A X velocity
N,/C X lOVvelocity.
—0 0002935
-0 00075
-0 001
Horizontal
straight
Horizontal
circling
Horizontal
straight
Horizontal
circling
Horizontal
straight
Horizontal
circling
Horizontal
straight
Horizontal
circling
- 0-5
0 0-5 1-28 2-00-0-5
0-5
1-28
2-00
Damping factor
X lO'/velocity.
Modulus X 10*/velooity.
- 1-76113
+ 12-7 8-83,
— 107 5-76
- 0-982-90
+ 10-6 4-69
4-47
2-65
5-80 6-50 7-10; —
6-36 — -i —
5-76 6-48> —
5-90, — i —
4-86 5-61' —
+ 0-92
- 1-44
+ 1-26
2-48
0-26
3-26!
4-79
5-23
3-80
3-42
4-05
4-455-21! 0-828
4-91
— 0-680
— 0-836
i
— 0-850
— 0-946
1
0-5740-946
1-24
1-223
1-326
1-304
0-906 1-170,1-486
1066 1-286
0-9671-216
1-576
1-62
1-514
\ (94)
1-59
1-724
1-80
The figures in (94) show that the lateral oscillation is very dependent
on the size of the dihedral angle and little dependent on the rate of turning
except when the aeroplane is devoid of weathercock stabihty, i.e. N^ > 0.
General Remarks on the Numerical Results. — Although all the calcu-
lations refer to one angle of incidence (6°) and to circling at an angle of
bank of 45° when turning is present, they have nevertheless shown that the
STABILITY 499
stability of the slower movements of an aeroplane, i.e. the phugoid oscilla-
tion, the spiral motion and the lateral stability, is markedly affected by
the details of design and by the centrifugal terms. The theory of stabihty
in non-rectiUnear flight is therefore important, and methods of procedure
for further use should be considered. It was found that the approxima-
tion indicated in (74) . . . (77) sufficed to bring out the sahent changes
in the examples tried, and it may be permissible to use the form generally
if occasional complete checks be given by the use of (73). The reduction
of labour is the justification for such a course. Such indications as the
change from spiral instabihty to stabihty by reason of turning can be de-
duced in more general form from the approximation, since it is only neces-
sary to discuss the change of sign of the term independent of A.
Further data relating to the above tables may be found in the " Annual
Eeport of the Advisory Committee for Aeronautics," pages 189-223
1914-15, by J. L. Nayler, Eobert Jones and the author.
Gyroscopic Couples and their Effect on Straight Flying. — If P be the
angular velocity of the rotating parts of the engine and airscrew, and the
moment of inertia be I, there will be couples about the axes of Y and Z
due to pitching and yawing which can be deduced from the equations of
motion as given in (56). There are certain oscillations which occur with
two blades which are not present in the case of four blades, but the average
effect is the same. Putting A = I, B => C => 0, and taking the steady effects
of rotation only, leads to
M=+I.P.r (95)
N = -I.P.^ (96)
for the couples needed to rotate the airscrew with angular velocities r and
q. There will therefore be couples of reversed sign acting on the aeroplane
which may be expressed in derivative form by
M,=--I.P, N,=:I.P (97)
and. these are the only changes from the previous consideration of the
stabihty of straight flying. Equation (73) takes simple form since the last
four determinants disappear-when O is zero, whilst in the first determinant
only the terms underlined together with M^ and N^ have any value, and the
equation becomes
= 0
(98)
A2
x„-
-A
0
^w
0
Y 9'^3
0
0
Y„-
-A
0
0
^' A
z„
0
Z.-A
0
7 , S'Wl
0
0
L.
0
L^-A
0
L.
M,
0
M,
0
M,-A
M,
'
0
N.
0
N^
N.
N,-A
500 APPLIED AEKODYNAMICS
This determinant is easily reduced to
(A4 + AiA3 + BiA2 + CiA + Di)(A4 + AgAS + B2A2 + C.A + D^)
-M,N,A2
X« — A Xjp
Z„ Z,„ — A
X
L«) Lp— A
. (99)
= 0
where the quantities A^ . . . Di, A2 . . . D2 are those for longitudinal
and lateral stability when gyroscopic couples are ignored.
An examination of (99) in a particular case showed that the coefficients
of powers of A in the gyroscopic terms were all positive and small compared
with the coefficients obtained from the product of the biquadratic factors.
The rapid motions, longitudinal and lateral, will therefore be little affected.
It appears, further, that the change in the phugoid oscillation is a small
increase in stabihty. Since the gyroscopic terms do not contain one in-
dependent of A, the above remark as to signs of the coefficients shows that
a spirally stable or unstable aeroplane without rotating airscrew will remain
stable or unstable when gyroscopic effects are added. In any case of
importance, however, equation (99) is easy to apply, and the conclusion
need not be relied upon as more than an indicative example.
The Stability of Airships and Kite Balloons
The treatment of the stabihty of hghter-than-air craft differs from that
for the aeroplane in several particulars, all of which are connected with the
estimation of the forces acting. The effect of the buoyancy of the gas is
equivalent to a reduction of weight so far as forces along the co-ordinate
axes are concerned, but the combined effect of weight and buoyancy
introduces terms into the equation of angular motion which were not
previously present. The mooring of an airship to a cable or the effect of
a kite wire introduces terms in both the force and moment equations.
The mathematical theory is developed in terms of resistance derivatives
without serious difficulty, but the number of determinations of the latter
of a sufficiently complete character is so small that the applications cannot
be said to be adequate. This is in part due to the lack of full-scale tests on
which to check calculations, and in part to the fact that the air forces and
moments on the large bulk of the envelopes of lighter-than-air craft depend
not only on the linear and angular velocities through the air, but also on
the linear and angular accelerations. In a simple example it would appear
that the lateral acceleration of an airship is httle more than half that
which would be calculated on the assumption that the lateral resistance is
determined only by the velocities of the envelope.
The new terms arising from buoyancy will be developed generally and
the terms arising from a cable, left to a separate section, since they do not
affect the free motion of an airship. Tho separation into longitudinal and
lateral stabilities will be adopted, and the general case left until such time
as it appears that the experimental data are sufficiently advanced as to
permit of their use.
STABILITY 501
Gravitational and Buoyancy Forces. — If the upward force due to
buoyancy be denoted by F, the values of the component forces along the
axes are
mX^ni{mg — ¥) mY — n^img — Y) mYj~n^{mg —Y) . (100)
For an airship in free flight w^— F is zero and the component forces vanish.
In the kite balloon reserve buoyancy is present and is balanced by the
vertical component of the pull in the kite wire.
Gravitational and Buoyancy Couples. — The centre of gravity of lighter-
than-air craft is usually well below the centre of buoyancy, i.e. below the
centre of volume of the displaced air. The latter point will vary with the
condition of the bftlloonets and must be separately evaluated in each case
as part of the statement of the conditions of steady motion. Both the
centres of gravity and buoyancy will be taken to lie in the plane of symmetry,
and the co-ordinates of the latter are denoted by x and z relative to the
body axes through the centre of gravity. The buoyancy force F acts
vertically upwards, and the components of force at (x, o, z) are therefore
— niF — naP and -WgF .... (101)
Taking moments about the body axes shows that on this account the
components are
L = n2F.0, M = {n3X — niz)F, N^-ngF.x . (102)
Air Forces and Moments. — To meet the new feature that the forces
and moments depend on accelerations as well as on velocity, it is assumed
that in longitudinal motion the quantities X, Z and M have the typical
form
.X=fy^{u,w,q,u,w,q) (103)
as a result of motion through the air ; following the previous method X is
expanded as
X =/x(mo, Wq^ qo) + uXu + wX^ + ^X^ + wX^ -{-wX^ + qX^ . (1 04)
The number of derivatives introduced is twice as great as that for the
longitudinal stabihty of an aeroplane.
Changes of Gravitational and Buoyancy Forces and Couples. — These
changes depend on the variations of the direction cosines Wj, 112 and W3
arising from displacements of the axes, and may be determined directly or.
from the general form given in (68) by putting 12, /),), qQ, Tq and tio equal
to zero. The changes of the direction cosines are
d», = - M, dn, = -";'• + "f ' ^n, = "^ . (1 05)
A A A A
of which the first and last refer only to longitudinal stability and the second
to lateral stability.
502
APPLIED AEKODYNAMICS
Division ol (56) into Equations of Steady Motion and Disturbed Motion.
— Using the separate expressions for forces due to gravity, buoyancy and
air, equations (56) become
/ F \ . . . ^
u-\-wq== 7ii\^g — - j +/x (w, w, q, u, w, q)
Tjl
w — uq = nj^g — )+/z(w, w, q, u, w, q)
(106)
qB = {n^x — niz)¥ +/m(w, w, q, ii, io, q) )
In steady motion, u, w, q, and q are zero, and hence (106) becomes
F
0 = ni(^g - - ) +f^{vQ, Wq, qo)
/ F\
0 = ns(g - - ) +/z(mo' %' So)
(107)
0 = (n^x — niz)¥ +/m(wo, Wq, qo)
ii + M?o3 = — Hg — W + wX„ + wX^ + qXg
W
u^q.
(108)
If in (106) Uq-\-u is written for u, etc., ni-\-dni, for n^, etc., the equations
of disturbed motion are obtained, the terms of zero order being those of
(107), and therefore independently satisfied ; the first-order terms are
F
m
+ iiKu + wK;„ + qX^
gB = F(wia: + n^z)\ + wM„ 4- wM.^ + gM,
A
+ iMi, + ioM^ + gM^ j
Collecting these terms in accordance with the note made in (10) and
carried out for the aeroplane in equations (11) and (12) leads to
(X„+AXi-A)7.+(X«,+AXi)2^+|x,+AXi-w)o-^/(^-^)]g==0^
(109)
(Z„ + AZii)w+(Z«,+ AZi,- X)w + |Z, + AZ,+ u^-^'^^i^g-^y^Q
(M„+AMi)M+(M«,+AM^)w;+|M5+AM4-BA+^(nia;+n30)|g=O
Comparing (109) with (11) shows that the changes consist of the writing
of gf — F/w for g, X„ + AX^ for X^, etc., except that in the case of M^ the
expression M^ -|- ^^q + (^i^; + ^3'2')F/A is written instead of M^.
Ehminating u, w and q from the three equations of disturbed motion
leads to an equation in A which is of the fourth degree as in the case of the
aeroplane. Except for the term independent of A the coefficients in the
STABILITY
503
equation contain terms depending on accelerations. In particular the
coefl&cient of A* is made up of the moment of inertia B and acceleration
terms ; the first two lines are most easily appreciated by multiplying by m,
when it is seen that wXjj, wZ^, etc., are compared directly with the mass m.
This analytical result is the justification for a common method of expressing
the results of forces due to acceleration of the fluid motion as virtual ad-
ditions to the mass of the moving body.
One type of instability may be made evident by a change of sign of the
last term of the biquadratic equation for stability, but this is not so likely
to occur in longitudinal as in lateral motion. The criterion for this type
of stability is independent of the acceleration of the fluid motion, as may
l)e seen from the coefficients of the biquadratic equation given below.
Coefficient of A*,
■^M — 1 ^w X^
iMi M-
Mi-B
Coefficient of A^,
X,;— 1 X^ Xq—WQ\
M;, Mi M I
Coefficient of A^,
+
XJ.-1
M-
X,<,
Zm>
M„,
Xa
M^-B
M„
X«
M,;
Xi 1
Zi j
M^-Bl
x„
X.
X5
H-X,
X,;,
X^— w;o
z«
z«,
Zi
iZ«
z^-i
Z«+Wo
M„
M„
Mi-
-B
:M„
MeJ.
M,
'
+ !Xi-
-1 X,
X,-«;o +!Xi-
-1 x«,
Zi
K
Z5+M0
K
Z^-
-1
m;
M^
M,
Mi,
M.
Coefficient of A,
X„ X«, 'X^—Wq I + X„ X^;
Z„ Z^ Z,+2^o I Z„ Zj,
M„ M«, M, 1 M„ M;
Coefficient independent of A,
X„ X«,
M„ M„,
-n^{g--Flmy + \Xl-
1 niig-^lm) \ 7^
X«
Zw
M..
-n^ig-F/m)
ni{g-F/m)\
-n^ig-F/m) 1
ni{g-F/m)
¥{niX-\-n2z) j
(110)
Lateral Stability. — The results of the steps only will be given, since
the method has been illustrated previously. After substitution in (56) for
the parts due to gravity and buoyancy and those arising from motion
through the air, the equations of lateral motion become for principal axes
of inertia
fi' — -)+/y(v, p, r, v,j),r)
pA =dn2Fz +Mv, p, r, v, p, r)
rC =—dn2Fx +/n(«, p, r, v, p, f)
(111)
504
APPLIED AERODYNAMICS
where dn2 only appears because 7^2 is zero as a condition of the separate
consideration of the lateral and longitudinal motions. Similarly Vq, p^,
and Tq are zero, and the equations of equilibrium are automatically satisfied
by the forces and couples due to the air being also zero from the symmetry
of the motion. The value of d7i2 has already been given, and the three
equations of disturbed motion in terms of v, p and r are
pK =(
rC =.{'f~'f)Fx
+ i)Y, + :pY-+fYf
-\- vL^ -\- j)L^ +rL;.
+ vN,+^Np+rN,
+ ?JN^ + pN|, + fNf.
(112)
Arranging the terms as factors of v, p and r leads to
(Y,+AY,-A)i;+|Y,+AY.+^^(^-^)|p+|Y,+AY,-Ko-^^(^-^)|r=0|
(L, + ALj,)^+ (l^+AL^-AA+^3F^)p+ (L, + AL; - ''^iF0)r=O Wb)
(N„+AN,)^+ (N, + AN--^^?Fa:)2)+ (n, + AN;- AC + -^iFa;)r=0
The ehmination of v, pa.nd r from equations (113) gives a biquadratic
equation with the following coefficients : —
Coefficient of A*,
Yi
L^
N-
Li,-A
N-
Y.
Lf
N.
C
Coefficient of A^,
Y.
Y- Y
+
Y,
L.
Lp — A Lj.
L^
N„
N^ N;-C
N;,
Coefficient of A^,
Y.
Lp— A
N-
Yy— M(-
N,
+ni
Y,T
U
N-
+
■1 Y.
N;
Y Y
hp hi.
Np N--C
+
N.
Yr 1 +
N^-C
N-
(^-F/m)
F0
+^3
Fa;
Y^
L^-A
Y,-
L.
N,
1 Y.
N.
Y,— Wo
L.
N,
-1 g—¥/m
. F^
-Fa:
Y.
L;
N;-C
STABILITY
505
Coefficient of A,
Y Y
ip Ijf
K ^p K
L„ L
+nj
+ni
Y„
L«
-{g-Flm)\-\-ns:Y,
Fa; ' 'N„
?— F/w Y^
F2 L^
-¥x N--C
Y„-l Yp _(^_F/w) 1+713
N.
Coefficient independent of A,
Y.
N,
-(9-
'Fx
■F/m)
-Fz
Fx
+%
Yv— 1 g—F/m Y^—Uq
hi Fz L,
N,; -Fa; N,
-F/w Y,
F^ L,
-Fa; N,
-Mo
. (114)
The formula of which most use has hitherto been made in airship
stabiHty is deduced from (Il4) by considering horizontal flight with the
axis of the envelope horizontal ; ni is then zero. The reserve buoyancy
is zero, i.e. g—F/m='0, and the centre of buoyancy is vertically above the
centre of gravity so that x is zero. The coefficient independent of A is
then
ngF^'Y, Y,
IN, N,
(115) •
and if this quantity changes sign there is a change from stability to insta-
bility, the latter corresponding with a positive sign under usual conditions.
For an airship to be laterally stable the condition becomes
Y,N, >(Y, - Wo)N,
Examples o! the use of the Equations of Disturbed Airship Motion. —
The further remarks will be confined to horizontal flight, in which case
ni==0. The numerical data are not all that could be desired, and use must
as yet be made of general ideas.
Remarks on the Values of the Derivatives. — For an airship of any type
in present use, there is approximate symmetry not only about a vertical
plane, but also about a horizontal plane through the centre of buoyancy.
There are then some simple relations between the forces and couples due to
rising and falling and those due to sideslipping. It may be expected that
the forces on an airship will not be affected appreciably by a slow rotation
about the axis of the envelope, and if this assumption be made it is easily
seen that the relationship between derivatives due to rolling and derivatives
due to sideslipping is simple. The relations which may be simply deduced
as a result of the above hypotheses are : —
X,,-X, = 0 (116)
i.e. there is no change of resistance for slight inclinations of the axis of
the airship to the wind.
Z„, = Y« (117)
This relation expresses the fact that the lift and lateral force on the airship
506 APPLIED AEKODYNAMICS
have the same value for the same inclinations of the axis of X to the wind
in pitch and yaw respectively.
X, = X,-^XJ = -^X„'. . . . .(118)
where X^, the variation of resistance due to pitching, differs from X^., tlic
variation of resistance due to yawing, because the axis of X lies at a distance
z below the centre of buoyancy whilst the axis of Z passes through that
point. Symmetry about a vertical plane is sufficient to ensure that X^ is
zero.
As the car and airscrew are near the centre of gravity, X'„ will be
almost wholly due to the resistance of the envelope in its fore and aft
motion due to pitching about the axis of Y. The change of resistance
of the whole airship due to a change of forward speed u will be greater
than X'„, partly on account of the additional resistance of the car, but
also because of reduced thrust from the airscrew.
Z.--Y, (119)
The variations of normal force due to pitching and lateral force due to
yawing will be roughly related as shown in (119). Both are associated
dii^ctly with Wq in the conditions of stabiHty, and their value is not known
with any degree of accuracy. There is a possibiHty that Y^ may be half as
great as Uq.
Z„ = 0 . (120)
Since the hft due to wind is zero, the rate of change of Z with change
of forward speed will also be zero.
The pitching moment due to change of forward speed, i.e. M„, may
not be zero. If the airscrews are at the level of the car, and therefore
near the C.G., it would appear that the change of airscrew thrust with
change of forward speed will not greatly affect M. It can then be stated
as probable that
M„ = m^X'„
and M5 = N, + m^2X'„ (121)
Equation (121) gives a relation between the damping derivatives in
pitch and yaw, assuming equal fin areas horizontally and vertically ; the
term mz^l^ ^ occurs because the axis of X is below the centre of buoyancy.
M,. = -N, (122)
is a further relation which assumes equal fin areas. Both M«, and N^ are
greatly dependent on the area and disposition of the fins, and are two of
the more important derivatives.
The approximate relation
L, = -m^Y, (123)
can be deduced from the consideration that the lateral force on the car is
unimportant compared with that on the envelope, and that the rotation
STABILITY 507
of the envelope about its own axis produces no lateral force. Further
relations of a similar character are
Y.=
-zY,
Lp =3
— zL^
N,=
-^N„
(124)
The rolling moment due to yawing will often be small, and the derivative
Ly may be negligible in its effects as compared with the large restoring
couple in roll due to buoyancy and weight.
Of the three moments of inertia the order of magnitude will clearly be
A, C, B.
The value of X;, has been determined in a few cases and appears to
range from — 0'15 to —0-25. Z^„ and Y; both have values of the order of
—1. Up to the present the other derivatives have not been determined,
and in calculations their existence has been ignored.
Approximate Analysis of Airship Motions. — Using all the simplifications
indicated previously, the equations of disturbed motion given in (108) and
(111) take simple form, and the results of an examination of them are
useful as a guide to the importance of the terms involved. The longitudinal
group becomes
{X^-X{\-Xi)]u-zX\q -0 )
{Z, - A(l - Z,)}2/; + (Mo + Z,)q =0 . (125)
mzX.'uU + M^w + {(Mj - BA) +F0/A}g = 0 )
whilst the lateral group is —
{Y, - A(l - Y,)}^ - zY.'p - (Wo - Y.)r = 0 ]
- mzY^v + {-zL^ - AA + F^/A)p =0 . (126)
N^v - ^N^^ + (N, - AC)r = 0 )
In this form the dissimilarity of the longitudinal and lateral disturbances
is shown, and since the derivatives have been based on symmetry of the
envelope the conclusion may be drawn that the difference is due to the
fact that the axis of Z passes through the centre of buoyancy, whilst
the axis of Y is some distance below that point.
Critical Velocities. — It has already been shown that for a given attitude
of a body in the air the resistance derivatives due to change of linear or
angular velocity are proportional to the wind speed. A similar theorem
shows that the acceleration effects are independent of the wind speed so
long as the resistance varies as the square of the speed. Without using
any approximations, therefore, it will be seen from (110) that for longi-
tudinal stability the biquadratic equation takes the form
fciA'-f fe2VA3 + (fc3V2+fe4)A2 + (fc5V2 + fc6)VA + fe7V2=0 . (127)
whilst the lateral stabihty leads to an exactly similar form. Instability
occurs when any one of the coefficients changes sign, and equation (127)
shows two possibiUties with change of speed if h^ and h^ or k^ and Uq
happen to have opposite signs. There is the further condition given by
508 APPLIED AERODYNAMICS
Routh's discriminant which might lead to a new critical speed, but the
further analysis will be confined to an examination of the approximate
equations (125) and (126). The first of these has the stability biquadratic
{X„-A(1-X„)}IZ,-A(1-Z,) 2*0 + Z. I
I M^ (M,-BA)+P^/A I
-{Z,-A(l-Z-)}m^2(X'„)2 = o . . . .(128)
It is not strictly legitimate to say that resistance derivatives due to
changes of velocity vanish when V=0, since slight residual terms of higher
order are present, but in accordance with the theory of small oscillations
as developed this will be the case, and with the airship stopped, equation
(128) reduces to
A2B-P^ = 0 (129)
Since z is negative, whilst B and F are positive, this is the equation of an
undamped oscillation of period —
W-l, 030)
If, as appears probable, we may neglect mz^iX.'y)'^ in comparison with
Xj^Mj, equation (128) has one root given by
^ = f^^ ....... (131)
which indicates that a variation of forward speed is damped out aperiodi-
cally. The neglected terms are those arising from changes of drag of the
envelope due to pitching about an axis below the centre of figure.
Approximate Criterion for Longitudinal Stability. — Equation (128) now
takes the form
|Z,-A(1-Z^) % + 2, 1 = 0
|M„ (M,-BA) + F^/A| • • ^^^^^
and by a consideration of the terms, using the theory of equations, an
important approximate discriminant for longitudinal stability is obtained.
The equation is a cubic in A, and must therefore have at least one real
root. The product of the roots is ^,v- J"., the value of which is essenti-
B(l— Z^j
ally negative and important. This follows from general knowledge, for z is
negative, P positive, Z^, and Z^ negative and B positive in all aircraft
contemplated. If only one real root occurs it must therefore be negative,
whilst if all the roots are real they must either all be negative or two
positive and one negative. A change of sign of a real root can only occur
by a passage through zero, and in the present instance this does not occur
since the product of the roots cannot be zero. The cubic may represent
a subsidence and an oscillation, and the only possibility of instability arises
from an increase in the amplitude of the latter.
STABILITY
509
The condition for change of sign of the damping coefficient of the
oscillation can easily be deduced, for the sum of the roots is
M,/B+ZJ(l-Zej,) (133)
and the damping of the oscillation will be zero if the real root is equal to
this value. Making the substitution for A in equation (132) leads to the
criterion for stability :
Uq + '^q
■Z^y/B
>0
(134)
M,(l-Z^)/B + Z,
The periodic time of the oscillation at the critical change is found from
the product and sum of the roots, and is
T
=^277/s/-
B( M,(l-Zi),
Fzl '^ BZ„, .'
(135)
Since the second term in the bracket is always positive, comparison of
(135) with (130) shows that the oscillation in critical motion is slower
than that at rest. The critical velocity above which the motion is unstable
is easily determined from (134), and a knowledge of the manner of variation
of the derivatives with change of speed. If w^, be the critical velocity and
Uq the velocity for which the derivatives were calculated, the expression for
(Mc/Mo)'
IS
u^) -f Z,^
z.
/i*of F^(l-Z^)2/B
\uj M,(l-Z^)/B + Z,
'0
(136)
From equation (.136) can be seen the condition given by Crocco (see
page 41 , " Technical Keport of the Advisory Committee for Aeronautics,
1909-10") for the non-existence of a critical velocity, i.e. Uc^=^^. Con-
verted into present notation, Crocco's condition is
M„
v I
(137)
except that Crocco assumed that Zi^ was neghgible in comparison with Uq.
His expression for lateral stabiHty has an exactly analogous form.
Uq -j- Zq is positive, whilst M^, Z^, and z are negative, and the remaining
terms are positive with the exception of M^. If M^ be negative,. -i.e. if a
restoring moment due to the wind is introduced by angular displace-
ment, expression (136) shows that the airship's motion is stable at all
speeds. It will be seen, however, that stability may be obtained with M^
positive, and this is the usual state owing to constructional difficulties in
attaching large fins.
Approximate Criterion for Lateral Stability. — The biquadratic equation
for stability which is obtained from equation (126) is
lY,-A(l-Y^) -zY, -UQ + Yr=0
\ -mzY^ -zL^-XA + Yz/X 0 . (138)
510 APPLIED AEEODYNAMICS
If the motion through the air is very slow, the derivatives due to
changes of velocity become isero, and (138) reduces to
A2-F5;/A=0 ...... (189)
and arising from the expression containing p clearly refers to an oscillation
in roll. It appears that no airship is provided with controls which affect
the rolling, and an oscillation in roll may be expected at all flight speeds.
This suggests that the term —mzY^ is usually unimportant, in which case
the oscillation in flight is given by
A2-|-^i^-:^=0 (140)
A A.
and is seen to have a damping term due to the motion. The remaining
factor of the stability equation is then
Y,-A(l-Y.) -Wo + Y,
N, N,-AC
-^ . . . (141)
N Y
The sum of the roots of equation (141) is -7^ + ^ — ^^, and is negative
in each term for ah airships. If equation (141) has complex roots, there-
fore, the real part must be negative and the corresponding oscillation
stable. Lateral instability can then only occur by a change in the sign
of the term independent of A, and the criterion for stabiHty is
|Y« -wo + YJ>0
|n, N, I (142)
As Y„ and N^ are both negative, it is an immediate deduction from
(142) that a restoring moment about a vertical axis through the C.G. due
to wind forces, i.e. a positive value for N^, is not an essential for stability.
Moreover, combined with the condition that equally effective fins be used
both vertically and horizontally, (142) is sufiicient to ensure the complete
stabihty of an airship at aU speeds.
In the criterion of stability, Y^ and Y^ are inversely proportional
to m, the mass of the airship, and it is interesting to examine the
possibilities of variation of Y^, and Y^ at various heights, i.e. when m
varies. It has been assumed in the preceding analysis that the mass of
the airship included that of the hydrogen, i.e. that the hydrogen moved
as a solid with the envelope. This is obviously only an approximation
to the truth, as internal movements of the gas are clearly possible, but,
so far as it holds good, the mass concerned in the motion is that of the air
displaced by the airship. This mass is independent of the condition of
the hydrogen or the amount of air in the balloonets ; on the other hand,
it is proportional to the density of the air and therefore varies with height.
The forces on the airship at the same velocity also vary directly as the
air density, and hence Y^ and Y, are independent of height. The stabilty
of an airship is not affected by height, at least to a first approximation.
STABILITY
611
Illustration in the Case of the N.S. Type of Airship of the Values of the
Derivatives with Different Sizes of Fin. — Photographs of this typo of
airship are shown in Fig. 9, Chapter I. ; Fig. 245, showing the dimensions
of the model tested and the fins used, is given in connection with the
(lorivatives. The figures should be regarded only as first approximations
to the truth, to be replaced at a later stage of knowledge by data obtained
under more favourable conditions than those existing during the war.
They however served their purpose in that the fins selected as a result of
these calculations were satisfactory in the first trial flight, and so aided
in the rapid development of the type. The airship was designed at the
Fig. 245. — ^Model of a non-rigid airsJiip used in th6 determination of resistance derivatives.
E.N. Airship Station, Kingsnorth, and the model experiments were made
at the National Physical Laboratory. The data obtained were
Symbol in calculations.
Volume 360,000 cubic feet . . . —
Length 260 ft —
Speed 75 ft.-s uq
Total lift 23,500 lbs F
Wt. of hydrogen .
Wt. of displaced air .
Mass
Height of centre of buoy-
ancy above the centre
of gravity ....
Moments of inertia — •
About longitudinal axis
About lateral axis .
About normal axis .
2,500 lbs.
26,000 lbs
26,000
32-2
10 ft.
= 800 slugs approx.
m
— z
4 X 105 slug-ft.2
2-1 X 106 ,,
1-9x106 „
A
B
C
612
APPLIED AERODYNAMICS
Horizontal fins are denoted in Fig. 245 by a and b.
Vertical fins are denoted in Fig. 245 by c, d, e, f, g and h.
Of the vertical fins /, g and h were arranged as biplanes. The presence
of the horizontal fins was found not to affect appreciably the forces on the
vertical fins, and vice versa.
Derivatives.
No fins.
Fins h.
Fins a.
w»X'„
- 32
- 32
- 32
mXu
- 60
- 50
- 50
inZ„
-300
-390
-490
mZ,
—
- 36
- 35
M„
4-7 X 10*
2-5 X 10*
2-1 X 10*
M,
+ 2-0 X 10*
-7-5 X 10«
-7-9 X 10*
x;
-0-25
-0-25
-0-25
Zi,
-10
-10 5
-10
No fins.
Monoplane fins.
Biplane fins.
c
iL
e
/
V
A
mY,
wY,
-300
-4-7 X 10*
-360
+ 40
-2-7x10*
-7-lxlO«
-260
+ 35
-3-3x10*
-5-9 X 10*
-220
+ 35
-3-8x10*
-5-4xl0«
-450
+ 40
-1-7x10*
-7-8xlO«
-360
+ 40
-2-5x10*
-7-3xlO«
-290
+ 40
-3-3x10*
-e-gxio"
Owing to the shielding of the fins by the body of the envelope the
numerical value of Y^ is less for symmetrical flight than when yawed.
It is probable, therefore, that turning tends
to produce greater stabihty as it introduces
sideslipping. The additional terms can be
introduced as required, and some discussion
of the subject has already been given by
Jones and Nayler. The mathematical theory
is well ahead of its applications, and no difii-
culty in extending it as required have as yet
appeared.
Forces in a Mooring Cable or Kite Balloon
Wire due to its Weight and the Efect of the
Movement of the Upper End. — The axes 0^,
Or; and 0^ (Fig. 246) are chosen as fixed
relative to the earth, the cable or wire being
fixed at 0. The point of attachment of the
cable to the aircraft is P, and may have
movement in various directions.
Forces at P due to the Wire. — If the stiffness of the wire and the wind
forces on it be neglected, the form of the wire will be a catenary, and it is
Fig. 246.
STABILITY 513
clear that the forces in it will not be affected by rotations about the axis
of (,. The problem, so far as it affects the forces at P due to the kite wire,
can then be completely solved by considering deflections of P in a plane.
In any actual case it is certain that waves will be transmitted along the
wire, but the above assumptions would appear to represent those of
primary importance.
If h be the horizontal component of the tension in the wire (constant
at all points when wind forces are neglected), the equation to the catenary
can be shown to be
t, = - cosh ^(^ + ^o) - - cosh ^^0 . . . (143)
w is the weight of the wire per unit length, and ^q is a constant of
integration so chosen that ^ => 0 when ^ =i 0. From the geometry of
the catenary it will readily be seen that ^q is the distance from the point
of attachment of the wire to the vertex of the catenary, the distance being
measured along the negative direction of ^. This follows from the fact
that ^ = 0 when^ = -^o-
It is convenient to use, as a separate expression, the length of wire
from the point P to the ground. If s be used to denote this length,
then
Equations (143) and (144) define k, the horizontal component of the
tension in the wire, and the length s, in terms of the position of the point
P and the weight of unit length of wire. In the case of an aircraft the
co-ordinates of P may be changed by a gust of wind, and it is now
proposed to find the variations of k which result from any arbitrary
motion of P in the plane of the wire. A further approximation wiU
be made here in that the extensibiHty of the wire will be neglected.
As the problem mathematically wiU be considered as one of small
oscillations, this assumption falls within the limitations usually imposed
by such analysis.
Since the length of the wire is constant in the motions of P under
consideration, it follows by differentiation of (144) that
+ d^cosh|(^ + ^o)+^^o'f . . . (146).
It will be obvious from the definition of Kq given previously that any
variation in P will produce a corresponding change in ^q, and although
a constant of integration when P is .fixed, its variations must be included
in the present calculations.
2l
514 APPLIED AERODYNAMICS
Differentiating equation (143) gives an expression corresponding to (145)
+ dK sinh I (^ + ^o) +dKoj ... (146)
Eliminating d^Q between equations (145) and (146) the relation
t,dt + ^smh'^dK
<^fc = 2fc7 "l^x — ! IJJ^ • • • ^^^^^
,( 1 — cosh -T^ I + - sinh -^
'\ k ' w k
w
is obtained, which gives the variations of horizontal force dk in terms of
the movements of the upper end of the wire.
To find the variation of the vertical component of the tension of the
wire as a consequence of changes d^ and dZ, in the position of the point P,
it is useful to employ equations (147) and (145). The slope of the wire
at the point P can be obtained from equation (143) by differentiation,
giving
| = sinh|(^ + ?o) (148)
and therefore the vertical component of the tension Ti is
Ti-fesinh|(| + ^o) (149)
In the displaced position of the point P the vertical component of the
tension, T2, will be given by
T2 = Ti + dfc|sinh I (^ + So) - "^^^^ cosh ^ (^ + ^0) ]
11)
+ w;cosh~(S + ^o)-(^S + ^So) (150)
Using the value of d^Q, which may be obtained from (145), equation
(150) becomes
T2 - Ti + d/i;[sinh I (S + So) + w cosh | (S + So)
+ dS.w;cosh|(S+So)|l-^ ■cosh|(^ + ^o)| • (151)
Substituting in equation (151) the value of dh obtained from equation (147)
an expression for T2 — Ti in terms of d^ and dt, is obtained.
STABILITY
515
The forces acting on the aircraft at P along the axes of ^ and t, are
-(fc+M + vd^) (152)
(and
where
•1'
w
—]k sinh ^ (s + y +/^MC -\ (t'/^i + i'l)^^!
(153)
fX =
2k
I
r
i^^
(l — cosh =^ n + - sinh ->
k . , wK
- smh -
1^ A;
2A;
Vl - cosh ^^)+^ sinh '^^
2V k ' w wt,
w
' . (154)
^1 = ^ cosh I (^ + s-o) cosh I ^0 - 4 sinh ^
.1=2. COEh |(^ + y |l - ^^ cosh ^ (^ + Q| ^
The expressions /x,./xi and v are always positive, and vj negative.
All the above .relations have been developed on the assumption that
the rope lies entirely in the plane ^0^. In the case of the disturbed position
of an aircraft, especially if more than one wire is used, it will be necessary
to consider the components of the tension along the axes of K, r) and ^ when
the plane of the wire makes an angle 6 with the plane ^0^.
If ^, 7) and ^ are the co-ordinates of P, then the angle 6 is such that
tan 6 =
(155)
The values of ^, ^q and d^ in the previous expressions must now be replaced
by ^ sec 6, ^q sec 6, and dE, sec 9 respectively. If the point of attachment
of the wire for a second rope is not at the point (o, o, o) but at the point
{Ki, rji o), then instead of (155) there will be the relation
tan di
rj — 7)1
(156)
^-^1
and the values of E, ^q and d^ in (152), (153) and (154) will need to be
replaced by
{K — Ki) sec ^1, (^0 — ^oi) sec ^i and d^ sec di . . (157)
By means of (156) and (157) any number of wires connected together
at P can be considered. The conditions relating to equilibrium will
indicate some relation between the angles 6, 6i, 62, etc., since the force
on the aircraft along the axis of 17 must then be zero from considerations
of symmetry.
If the wires are not all brought to the same point P, the relations given
above can be used if, instead of the co-ordinates of P (^, rj, Q, the co-
ordinates of Q, the new point of attachment, are used. In the case of more
516 APPLIED AEEODYNAMICS
than one point of attachment at the aircraft, it will be possible to have
equilibrium without having the plane of symmetry in the vertical plane
containing the wind direction. If, however, symmetry is assumed, it will
be necessary to arrange that the moment about ,any axis parallel to 0^
shall be zero.
With the aid of the above equations it is possible to determine both the
conditions of equilibrium for a captive aircraft and the derivatives due to
the swaying of the rope.
CHAPTEK X
THE STABILITY OF THE MOTIONS OF AIRCRAFT
PART 11. —The Details of the Disturbed Motion of an Aeroplane
In developing the mathematical theory of stability it was shown that
the periods and damping factors of oscillations could be obtained together
with the rateg of subsidence or divergence of non-periodic motions. It
was not, however, possible by the methods developed to show how the
resultant motion was divided between forward motion, vertical motion
and pitching fop»longituduial disturbances, or between sideslipping, rolling
and yawing for lateral disturbances.
It is now proposed to take up the further mathematical analysis in
the case of separable motions and to illustrate the theory by a nilmber of
examples, including flight in a natural wind." The subject includes the
consideration of the effect of controls and the changes which occur as an
aeroplane is brought from one steady state to another. It is possible that
the method of attack will be found suitable for investigations relating to
the hghtness of controls and the development of automatic stability
devices.
Reference to the equations of disturbed motion, (8) and (45), will show
that three equations are defined for longitudinal and three for lateral
motion, and that in each case a combination of them has led to a single
final equation for stability. There are left two other relations which can
be used to find the relative proportions in the disturbance of the various
component velocities and angular velocities.
Longitudinal Disturbance. — The condition for stabihty was obtained
by eliminating u, w and q froiri the equations of motion and determined
values of A from which the periods and damping factors were calculated.
The method of solution of the differential equation depends on the know-
ledge of the fact that
u =1 ae^< w = he^* q = ce^' . . . (158)
are expressions which when introduced into the differential equations of
disturbed motion reduce them to algebraic equations, a, b and c are the
initial values of the disturbances in u, w and q which correspond with
the chosen valXie of A. An examination of the stability equation shows
that there are four values of A in the case of an aeroplane, some of which
are complex and others real. Using (158) the equations of disturbed
motion become
517
518
APPLIED AERODYNAMICS
(X„ - X)a + X,,h + {X, - ivo - g'^y = 0 ^
Z„a + (Z - X)h + (Z, + Wo + g''^} = 0
(159)
M„a + M> + (M^ - BA)c = 0
Since A is known from one combination of these three equations, only
two of them can be considered as independent relations between a, h and c,
and choosing the first two, a solution of (159) is
& . c •
a
Xm,
X^~Wq
-^"/
2m,-
-A
Zg+Uo
+^"x
X
ng
q-t^0-9j
X„ — A
rii
'^,+H+9w 2
■A Xm,
Z„, — A
(160)
and the ratios b/a and c/a are determined. This is essentially the solution
required, and for real values of A the form is suitable for direct numerical
application. If, however, A be complex, it is necessary to consider a pair
of corresponding roots and to separate the real and imaginary parts of (160)
before computation is possible.
If the roots be Xi=h-\-ik and Ag^i/t— i/c, the two values of such a term
as u group together as
w=e*«(aie'*' + a2e-»*«) . '. . . . (161)
or in terms of sines and cosines instead of exponentials,
u = e^^Utti -\- a2) cos kt -\- i{ai — a2) &m kt\ . . (162)
and from equations (160) it is desired to find the values of aj +a2 and of
i{ai — a^ in order to give u the real form of a damped oscillation.
On substituting h-\- ik for A in equations (160) the expressions become
complex and of the form
ai{i^i+ivi)=hi{ix2-\-iv2)=ci{tM^-\-iv2). . . (163)
with a corresponding expression for the root h — ik, which is
a2{iii—ivi)=h2{fJ'2 — iv2)=(^2{f^3 — '^^3)- • • (1^4)
where the values oi ni, 1^2, /X3, v^, V2 and v^ are found from
A*i
V/Xi2 + v{'
- x„
Xq — WQ —
h Zg + UQ +
gn^h
W+k^
gn-Ji
W+k^
gn^k'^
^i
V/ii2 +
Vi'
■h +
^^3 j_/y ^^, gnsh Vi
9^1
/t24-fe2|
(165)
STABILITY— DISTUEBED MOTION
519
/f2 __ ^
^« + Wo +
gn^h
h-^ + /c2
giiih
z„
-h +
gtiik-
V/X22 +
'^2-
-1
+
gn-i
x„-.:-(z,+.o+,f:;:t)l
) . (166)
/^3
^3
h2 + /c2
z«
(167)
\//X32 + V32
and are directly calculable from known values of h, k and the derivatives
of the aeroplane. From the expressions connecting a, b and c with /m and
V it is easily deduced that
«i + 02 — 21 ,,"2 V«i + «2; ,, 2_i 1. 2 ^^ ~ 2)
/*2 i^ ^2 /^2 "T 1^2
and i{bi
(168)
with similar expressions for c^ + Cg and 'i(ci — C2).
If A and B be used instead of aj + a2 and %{ai — 02) the expressions
for a disturbed oscillation become
e''«(A cos kt-\-B sin U)
,ht
1^2" + »'2
2\y i-{^\i^2 + »^1»'2)A — (/X1V2 — fA2»'l)B} COS fci
+ {(^i»'2 — /^2»'i)A + (^1/^2 + »^i^2)B} sin fef]
[{(^1/^3 + nv^)^ — (/Ai»^3 — /^3»'i)B} cos kt
+ {(Mil's — /^si'OA 4- (M1M3 + i'i»'3)B} sin U] (169)
In actually calculating the motion of an aeroplane the integrals of m,
i^ and g may be required. From (I61) it will be seen that
/*3^ + »'3^
y h 4-%k h — ik
+
Expressed in terms of sines and cosines, (170) is
(170)
/
udt
.-7=_|^-^«os(to-y)+-,^sin(fa-r)
where sin y
k
V/i^ -{-k^^ being always taken.
, and cos y
h
Vh^ + k^
(171)
, the positive value of
520 APPLIED AERODYNAMICS
Similar expressions follow for w and q. In the rase of rectilinear
motion in the plane of symmetry and in still air, q—6, a^ .i hence integration
gives the value of 6, i.e. the inclination of the axis of X to the horizontal.
Equal Real Roots. — It appears that it may be necessary to deal with
equal or nearly equal roots, and the method outlined above then breaks
down. Following the usual mathematical method, it is assumed that
w = (C + Dfy (172)
Erom (160) b =^ a^{A) and the solution for w is
M; = {(C4-D0e^'<^(A)+DeV(A)} . . . (173)
It is therefore necessary in the ease of equal real roots to find the value
of <f>'{X) as well as that of ^(A). The differentiation presents no serious
difficulties and does not occur sufficiently often for the complete formulae
to be reproduced.
Example. — ^The derivatives assumed to apply in a particular case are : —
(174)
^1 = 0 «8^1 Wg^O Uo = 80 . . . . (175)
From (175) it will be seen that flight is horizontal with the axis of X in the direction
of flight. Proceeding to the biquadratic for stability and its solutions, shows that
Ai = — 5-62 A2 = - 5-62 A3 and A4 = - 0-075 ± 0-283i . (176)
Applying the formulae of (165) . . . (167) leads to
^j= +0000639 vi= -0-00313]
^2 =-0-00396 1^2 =+0-0143 I . . . . (177)
/X3= 0-350 ,.3= -1-12 J
and ^i(A)=177 ^/(A)=204 \ ,,„«.
^,(A)=-6-92 ^/(A)=-5-63/ ^^'^>
and by substitution in equations (158) and (169)
u = e-» 07S'{A cos 0-283< + B sin 0-2830} + c-5«'^(C + DO
where A, B, C and D are arbitrary constants to be fixed presently by the initial conditions
of the motion.
to = e-o 07«{ ( — 0-214A + 0-0147B) cos 0-283< - (0-0147A + 0 214B) sin 0-2830 }
+ e-''«2'(i77c + 204D + 177D0
q = ''-''■''■''''{(0-00274A - 0-000277B) cos 0-283< + (0-000277A + 0-00274B) sin 0 283<}
_g-5-62//g.92c + 5.530 + 6-92D0
g— 0-07fi«
e = '^r^^{iO-00214:A - 0-000277B) cos (0-283« - y)
+ (0-000277A + 0-00274B) sin (0-283< -y)}
x„= -
-014
Z„ = - 0-80
M„== 0
x„ =
0-19
Z„. = - 2-89
-M«, = - 0-106
X, =
0
Zj = -9
^M,^-8-40
f,— 5-621 C / 1 \) '
where cos y = — 0*257 and sin y = 0-966.
STABILITY— DISTURBED MOTION
521
vVj
L
(181)
Iviluil oMulilimis. — Let Ui, Wi, qi, aaid Oi be the values of u, w, q and 0 when <— 0, then
ui=^ A + C )
Wi ^ - 0-214A f 00147B + 177C + 204D (]r()\
qi= 000274A - 0000277B - 6-92C - 5-53D * " ' ^ '
9i==- 0-00333A - 000883B |- 1-23C + 1-204D |
and four linear equations are produced to give A, B, C and D in terms of the initial
values of components of the disturbance. Illustrations of the motion are given in Fig. 247
for the four simple initial disturbances in.u,to,q and 6. For the first of these, where
«i = % Wi — 0 qi — Q and dj—0,whent — 0
the values of A ... D are -^
A= lOOlWl ] ': '
B = - 0-265mi f
C = - 000147wi
D = 000235wi )
The substitution of these in (179) gives the analytical expressions for the disturbed
motion due to meeting a head-on gust. The completed formulae are shown in (182), and
the curves of Fig. 247 (a) were obtained from them.
u = ■Mie-'>o"'(l-001 cos 0-283< - 0-265 sin 0 2830 + Uie-^'^^(-0-00U7 + 000235«)
w = wic-o <»7'*'( - 0220 cos 0-283< -|- 0-042 sin 0*2830 + Uie-^'^'"{0-220 + 0-4160
q ^ tt^e-* »7*'^0-00281 cos 0-283« - 0-00045 sin 0-2830 - i^ie^^ °2<(0-00281 + 001630
e ^ ■Mie-o"^5'(- 0-00104 cos 0-283« + 0-00970 sin 0-2830
— Wie-s-62'(- 0-00104-0-00290 . • • (182)
Effect of the Movement of a Control. — If an aeroplane be flying steadily
under given conditions and the elevator be moved or the engine throttle
adjusted, it will begin to move to some new condition of equiHbrium if
the aeroplane is stable. The disturbances of motion may then be regarded
as the differences between the original steady motion and the final steady
motion, and if small can be covered by the theory of small oscillations. A
movement of the elevator will be denoted by fi and a change of thrust by
V ; the changes in the forces and moments which result will be assumed to
be proportional to/x and v. Equation (5) then becomes —
0 =—g sin ^o+AK, Wq, 0)—g cos dQ.d+uXu+wXu,+tJiX^i-vXy (183)
where Uq, Wq and 6q still apply to the original motion and the first two
terms are therefore zero, whilst u, w and 6 are the changes in the steady
motion which arise from the elevator movement fj, and the thrust change v.
Three equations are obtained which define the disturbances u, w and Q in
terms of /x and v, and are — , " - 3
— gcQ&BQ.d + uXu, +m;X«; -f/xX^ 4-i^X„ = 0]
-^si^^o-^ + ^Z„ 4-w;Ze* +/^Zm -fJ'Z, =0 . .(184)
The solution of these equations presents no difficulty and leads to
6 -M'^ 7^^ -^-^
|X„ X^ y^fj,-\rvXv
|Z„ Z^ [lly.-^vZy
— gooffdo X^'-fiX-n-i-vXy
gsi:i:dQ Z^ ixZ^ + vZ^
0 M«, /xM^ + vM.
-1 ^
— gr cos ^0 X„ jLtX/* + vXy
—
g cos ^0 X„
^u
gai^do Z^,i[xZix + vZy
g siii.l^o '^u
'^w
0 M„ fiM^ + vMy
0^ M„
M„,
-^>
/
(185)
y
522 APPLIED AEEODYNAMICS
The motion of the aeroplane is found for changes of elevator and thrisi
on the assumption that the old steady conditions persisted whilst X^, T'- ,
etc., were measured, this being the usual assumption underlying t\. i
calculation of derivatives.
Example : Use of Elevator only. — In addition to the derivatives previously given
in this section it is necessary to have the values
X^ = 0 Zm=-^2 b^^--3 (186)
in order to calculate the elementary disturbances in u, w, q and 6 which are equivalent
to a movement of the elevator. Equations (185) then lead to
ui = 95-9fi Wi=-28-2ix qi = 0 6 ^ -O'SSlfi . .(187)
and these values together with equations (180) serve to determine the values of A, B, C
and D, which are
A=95-7/x B=28-5/x C=0-221;u D=-0-231/i . . (188)
and suffice to determine the whole motion from equations (179). As calculated, the
values of «, etc., refer to the final steady motion ; they can be used relative to the
original steady motion by adding constants to make the initial disturbances in u, w, q
and 0 zero. This was the procedure followed in producing Fig. 248 from the analytical
expressions.
Change of Airscrew Thrust only. — In order to give X^ a value it would be necessary
to define v as some quantity depending on the position of the throttle, viz. the revo-
lutions of the airscrew. If, however, a simple example be taken, it is permissible to write
+mvX„-=ST Z, = 0 M, = 0 (189)
where 8T represents the increment of thrust which constitutes the disturbance. Since
the original steady motion was horizontal, ^q— 0, and the component disturbances are
8T
u, = 0 ^1 = 0 g-i-O ^=- .... (190)
and a reduction of thrust leads primarily to a descent and not to a change of speed.
fK^'ifi ■ /.The diagram corresponding with (190) is the typical simple disturbance shown in
" ' -'Fig. 247 (rf).
Description of Figs. 247 and 248 illustrating the Kesults of the
Calculations of Longitudinal Disturbances
Gust in the Direction o£ Flight. — The result is shown in Fig. 247 (a), the
ordinates of which are proportional to the magnitude of the increase of
wind speed, u^, and the abscissae the times in seconds after entering the gust.
(Variations of gust with time are dealt with later.) The speed through the
air is seen to fall rapidly from m=Mi at <=0 to zero in less than five seconds,
and to continue its falltow— — 0"5wi in nearly 10 sees. The record is that
of a damped oscillation of insignificant amplitude at the end of one minute.
The value of w at first falls rapidly, showing a rapid adjustment of angle of
incidence to the new conditions, and is accompanied by a very similar but
oppositely disposed curve for the angular velocity g. The inclination of
the aeroplane axis to the ground is seen to vary considerably, and to have
its maximum and minimum nearly a quarter of a period later than the
velocity, whilst that of i^ is a half-period later and g almost in phase. This
relation constitutes a characteristic of the phugoid oscillation, and applies
to the later parts of all the diagrams. The fact can be deduced from
Stability— DiSTUEBED motion
523
the analytical expressions by any one used to the manipulation of the
formulae relating to damped oscillations.
(a) (b)
\ou.
0-52^
-0-5U,.
1
DISTURBANCE
\
kIN
UONLY
V
k-U
\
<~>
^9*100
i ^
\
X
\
1
\
\
rk
\
\
^
1
w
-?
'100
>
\.
/
/
i
N
^'
\
"^
(^
—
—
y\
y
'\ '>
\.
\
y
/
-•-.
->
fl ^
X
^-
\
f
•
/
1
/
^
^'
/
V
/
\
\-^
/
0-5 Itf
10 SECONDS 20
-o-57<;u;
100(J.
1 1 1
DISTURBANCE
100 6,
50 0,
0
-500,
■^ —
\
DISTURBANCE
IN (^ ONLY
\
IN 9 ONLY
k
\
\
.
V
K^
'/e?o
iu<y,
)
*^
<^xy
00
\
'
,/^
"V
\
/
S
\
.^"
100
,'-
-\
"1
w-
/_
7
s
0
■-^
£:
.
^
/
^
.-^
s
>
~—
/
^
3=4
--
j-~
\
/
/ ^
^-
.-.
f-
'-
^
^
u
t
V
M
t
/
ufioi
\
J
/
K
l/
\
J
1 '
■—
r
50^,
—
\
/
I
u
\
^
(c) {d)
Fig. 247. — The effect of simple longitudinal disturbances. (Aeroplane stable.)
Up-current.— The first effect of running into an up-current is an
increase of normal velocity and therefore of angle of incidence. The
524
APPLIED AEEODYNAMICS
aeroplane gettj an angular velocity very rapidly and loses it almost equally
rapidly, so that the angle of incidence has adjusted itself at an early stage
to the value suitable for the residual phugoid oscillation.
Disturbance o£ Angulax Velocity. — A horizontal whirlwind is the only
means of producing such an effect, and could not continue without producing
permanent inclination. The only valid deduction to be drawn from a
disturbance in q is that it will bo taken up with extreme rapidity and will
leave a phugoid of small amplitude.
/
N,
n
^^
^^
^IM
■""
/
s
\.
/
\
WOju.
/
V
__
^
f
\
s
^
^
~-~
"~~d
-
—
/
\
^
^
IL
SQ/l
t
h-^
[9
K/d
0
/
y
r
^
•^
^.
__
__,
__.
__
V^
/
--
— 1
*^
"'
"'
I
V
►
y
\
N
/
ur
,^.
—
— ,
.—
"■-.
1
V
^
__'
Jt'
^^
'•"■
--,
"""
\
/
i
/
>
«
,•*
— •
-•
•«.
. ...
\
/
0X
fOC
N
N>
.
■•»'
'''
"^
\
y
\
/
V
/
\
/
^^
10
so
20 SECONDS SO 40
Fig. 248.— Disturbance due to the movement of an elevator. (Aeroplane stable.)
Disturbance o£ Path. — Change of thrust is the nearest equivalent to
this disturbance, which is mainly phugoid. Since the value of e-5-62« ig
small at the end of half a second, the analytical expressions show that the
subsequent motion in all cases consists of the phugoid oscillation with an
amplitude which depends both on the magnitude of the disturbing cause
and on its type.
Movement of the Elevator.— Fig. 248 shows the result of giving a positive
movement to the elevator ; this corresponds with the control column
forward and the elevator down. The result is a rapid angular velocity
which raises the tail and reduces the angle of incidence ; the aeroplane
STABILITY— DiSTUEBED MOTION
525
dives and gains speed. After the damping of the oscillation is complete,
the aeroplane has no angular velocity, a reduced angle of incidence, an
increased speed and a downward path. The final motion was indicated
by the simpler methods of Chapter II., but the present result shows exactly
how the new state is reached.
Disturbances of Lateral Motion
The arguments followed are those already dealt with, and much of the
detail will therefore be omitted. If the disturbances be
oX«
— r»/)^t
. (191)
V =3 oe'" p = &e^' r = ce"
the values of a, h and c are given by the relations
a he
\\ L,-AA; (192)
|Lp — AA
iN«
L. j
N,-AC I
In,
AC N,
N„ N.
where principal axes of inertia have been chosen so that E is zero. If A
be real, the values of h/a and cja are obtained from (192), whilst if complex,
the procedure is that followed in connection with longitudinal stability.
The expressions for v, p and r for complex roots and therefore for
oscillations, will be given somewhat different form, but are essentially
similar to those given in (169).
V =>(xje^* cos (to + jS)
V=
/^2^+»'2^
e'^'{0^i/*2+^i»'2) COS (/rf+^)+(/AiV2— i:i2»'i) sin (to+^)}
= -2Z~^^'"((/*i/*3+»'i»'3) COS (to+^)+(/AiV3-/i3Vi) sin (to+j8)}
f*3 "r»'3
Values of ^ and xjf as required are determined from the relations
^ =3 ^ + »• tan ^0
^ = f sec ^0
The values of ii\, /*£, etc., are given below as —
_„ /xi _|Lp-/iA L, |-fe2AC ^
N.-ci
clLp-feA 0
1|Np -C
I L,. L, I __^^^^^^^^
(198)
(194)
h\
/^
\//>t? + V2^ I N,
Vix^^v^ N,
-A(N,-feC)i
»'2
feCL„
> (195)
Ml
vz
N,
V>3^ + 1^3^
-feAN.
626
APPLIED AERODYNAMICS
Example. — The derivatives assumed to apply in a particular case are
Y„ = -0-25
Y„= 1
Y,= -3
L„ = -0-0332
L„=-8-0
N„ = 00154
^N,
0-80
L, = 2-60
C
tt>n = 0
Nr= -1-06
80
(196)
n-i — O Wg = 0 «3 = 1
The solution of the biquadratic for stability gives to A the values
Ai = 0-0157 Ag = -8-26 A3 and A4 = -0-526 ± 0-984* /
The first value of A is positive, "tuid the aeroplane is therefore unstable.
For Ai = 0-0157 equations (192) We
\6/a = 0-OOCfe9 c/a=— 0-0149
whilst for Ag = —8-26 ^- --
hja = —2-24 cja = -0-246
For the complex roots A3 and A4
/*! = 0-0000170
Ma = 0-0164
/*3 = 0-0125
vj = -0-000144
vg = 0-0238
V. = —0-00213
(197)
(198)
Using expressions corresponding with those for longitudinal disturbances,
V = e~^'^'^'{A cos 0-984^ + B sin 0-9840 + Ce"oi"" + I)e~*''^'
p = e-»-526<| ( -0-00376A - 0'00332B) cos 0-984^ \
+ (0'00332A - 0-00376B) sin 0-9840} + O-OOOebCeO^^"/ _ 2-24De-S'2«'
r = e-o 52«| (0-00S24A - 0-01 lOB) cos 0-984^ J
+ (0-OllOA + 0-00324B) sin 0-984«} + 0-01496eO"i"« + 0-246De-8-26/
<f> = fpdt since ^0 — ^
. = e-o'526.{(-0-00338A - 0-00298B) cos (0-984« - y)
+ (0-00298A - 0-00338B) sin (0-984< - y) + 0-0440Ce'>»i"/ _|_ o-271De-8-2«'
where cosy =—0-472 sin y =0-882 (199)
Initial conditions. — Let Vj, pi, r^, and ^j be the values of v, p, r and ^ when ^=0,
then ?;i = A + C + D )
p^ = _000376A - 0-00332B + 0-00069C - 2-24D .._.,
rj= 000324A - 0-OllOB +001490 + 0-246D ( ' * ^"^""^
,^1 = -0-00103A + 0-00439B + 0-0440C + 0*27 ID J
Illustrations of the four simple types of initial disturbance are given in Fig. 249. For
the first of these
p^=0 ri=0
and the values of A
^1=
D are
^1=0
(201)
A = 0-992vi ]
B= 0-258^1 I
C= 0-0104vi (
D = -0-00205i;i )
The analytical expressions for the disturbance can be obtained by using these values
in equations (199).
Effect oi the Movement of a Control. — If ^ and tj be used to denote the
angles through which the ailerons and rudder are moved and these angles
be restricted to be small quantities for which the moments and forces are
proportional to the angle of aileron and rudder, the effect on the motion
STABILITY— DISTUEBED MOTION
527
can be represented by derivatives X^, X^, etc., and the equations of
small oscillations applied. Following the same procedure as for elevators
and thrust shows that the equivalent elementary disturbances are —
—<f>ig cos ^0 ^1
+ r)L„
Y.
Y, - Yp tan Oq -
-Uq SY<
L„
Lf — Lp tan ^o
^Ls
N„
N, — Np tan Oq
^N
L^ — Lp tan Oq
N^ — Np tan Bq
^Lj + r}L^
I
-^■l
1 .
~ 1 Lj, L, — Ls tan 6q
'N, N,-Njtan^o
(202)
The denominator of the last of these expressions is equal to D2/g'C0S Oq,
where D2 is the coefficient independent of A in the biquadratic equation
for lateral stabihty. Sensitivity of control is then seen to be greatest when
the aeroplane is just not spirally unstable.
Example : Use of rudder only. — The additional quantities required are dependent
on the area of the rudder, but it may be taken that
Y„ = -0 06N^ and that L„ = 0 (203)
in which case the curves can be drawn to an arbitrary ordinate as shown in Fig. 250
where t^Ni has been chosen to give w„ some arbitrary value.
If the conditions of turning require that there should not be any sideslipping,
equations (202) show that
|L, -Lptan^o ^U + '?L„j=0 (204)
|N,-N„tan^o ^Nj + ^N J
must be satisfied, and this cannot be the case for rudder only unless
^ = ^ (205)
L, — Lp tan 0^ N, — Np tan 9q
This condition varies with the inclination of the axis of X to the horizontal.
The ratios depend on the design of the aeroplane, and it would not appear to be possible
to approximate to the condition of turning without sideslipping by the use of the rudder
only, since L„ is always very small. In general the effect of turning without sideslipping
is produced by using the ailerons in conjunction with the rudder, and the practical
possibilities are known to cover the requirements of lateral balance.
Description op Figs. 249 and 250 illustrating the Eesults of the
Calculations op Lateral Disturbances
Lateral Gust. — Fig. 249 (a) shows that the preliminary adjustments to
new conditions occur very rapidly. A positive value of v corresponds with
sideslipping to the right or a gust from the right. The gust starts a turn
to the right temporarily with the wrong bank, but after about 3 sees, this
is corrected. At the end of 10 sees, the conditions are those for a right-
hand turn with sideslipping inwards, and due to instability the turn gets
more rapid. It has been seen from the treatment of the stability of circling
flight that this type of instability ultimately disappears and a steady turn
results. Here is one of the known limitations to the application of the
theory of small oscillations.
528
APPLIED AEEODYNAMICS
-ST
o \ o
X 1 X
1 ■T
8
-or
o
U) .5
Q "
Z ^ "S
o e -g
Hi m
ji
p
STABILITY— DISTURBED MOTION
529
Rolling due to Up-gust striking the Left Wing. — The rolling is stopped
with great rapidity (Mg. 24!> {b)), and leaves the aeroplane with, a small
bank ; sideslipping then occurs, and the aeroplane finishes with a spiral
turn as for a lateral gust.
Yawing due to a Gust which strikes the Tail from the Right. — The effect
of turning is to increase the velocity of the left wing and increase its lift,
causing a bank for a right-hand turn. The bank reaches its maximum
in about 2 sees. Under the action of centrifugal forces the aeroplane
begins to sideslip to the left, and until the bank is sufficient to reverse this
effect the changes are rapid. The aeroplane reaches an unstable turn
of appreciable magnitude.
Elffect of a Sudden Bank. — The preliminary rapid movements are
towards the final spiral turn of large magnitude ; the aeroplane with its
left wing up, begins to sideslip inwards and to turn to the right.
9x100
^
10 20 SECONDS 30 40
Fig. 250. — ^Disturbance due to the movement of a rudder. (Aeroplane unstable.)
In all cases the only important disturbances existing at the end of
10 sees, are those of the unstable spiral turn.
The Effect of a Movement of the Rudder is seen from Fig. 250 to be
the initiation of a spiral turn, and the control of such an aeroplane involve s
the pilot as an essential feature. The ordinate of the curves is arbitrary,
but is proportional to the movement of the rudder.
The Mathematical Theory of Disturbed Motion and Control when
THE Disturbing Causes are Variable with Time
The general theory of the solution of differential equations of the type
met with in problems of the stability of small oscillations shows that the
effects of two simple disturbances coexist as though independent of each
other. It is therefore permissible to regard the new problem as a search
for a method of adding a number of elementary disturbances due to gusts
or to movements of the controls which may occur at any time.
For any aeroplane subject to disturbance it has already been shown
that the motion can be expressed in a number of terms of the type
u = Ae^i' + Be^a' + Ce^a' -(- De^*
Kt
. (206)
2 M
530 APPLIED AERODYNAMICS
where u has its usual meaning of change of velocity along the axis of X.
The coefficients A, B, 0 and D are, in general, linear functions of the
disturbing causes. If, for instance, a horizontal gust of velocity Ui is
encountered. A, B, etc., are all proportional to w^. It is convenient to
have a shortened notation for (206) and it is proposed to denote it by
u='Ui{uu)i (207)
Similarly an expression
w='Ui{uw)t (208)
is written for the effect of a horizontal gust of magnitude Mj at t=0 on
the normal velocity w. Similar expressions follow for q and 6.
{uu)t, {uw)t, {uq)t; and {u6)i are all definite functions of t for a given
aeroplane, and examples have been given in Fig. 247 (a).
It is now necessary to define a second timer and to explain its relation
to t. The expressions {uu)t, etc., give the magnitude of a disturbance t
sees, after the disturbing cause operated. It will be evident that in a
succession of gusts the final disturbance at time i wiU be the sum of the
effects of disturbing causes at all previous times, and t is used to distinguish
the time at which the disturbance occurred. The expression {uu)t_^
represents the disturbance factor at time t from a disturbance at r, the
magnitude of which can be represented by the element /'(t) . dr. By
means of this definition it will be seen that /(t) represents a continuous
disturbing cause over any range of time whatever, and that the residual
disturbance in w is
u=^ (\uu)t_J'{r)dT (209)
J Q
f{t) may be a record of horizontal velocity as obtained from an anemometer,
and in that case the differentiation to fiiid f{t) becomes laborious. The
necessity for the operation may be avoided by a partial integration which
leads to
w = - {uu)J{0) + {uu)oM - \ ^^{uu\_J{r)dr . . (210)
Jo
The differentiation of the known algebraic expression luu)^_^ presents no
serious difficulty, and this latter form is in many ways preferable to the
former.
In the case of w, q and 6 the integrals at the limits are zero for a
horizontal gust, whilst for u the quantity {uu)q=1 and /(o) is zero if con-
tinuity in u is assumed as an initial condition. In the latter case, as
J{t) is the change in velocity of the wind over the ground at time t and u
is the change of velocity of the aeroplane relative to the air in the same
interval of time, it can be seen that
-Cl{uu),_J{r)d^ ..... (211)
is the change of the velocity of the aeroplane relative to the ground.
f
STABILITY— DISTURBED MOTION
5^1
The evaluation of (211) is easily carried out graphically, and one method
'> of arranging the work is shown in Fig. 251. The full curve of the upper
Fia. 251. — Disturbances due to a natural wind. (Aeroplane stable.)
diagram being taken to represent an anemometer record,* shows wind
* The record is prepared from one reproduced by Dr. Stanton in a paper on wind
pressure, Proc. Civil Eng., vol. 171, 1907-8. It is one of the few records with a sufficiently
532 APPLIED AEEODYNAMICS
velocity as ordinate on a time base and covers a period of about 40 sees.
d
The dotted curve represents — {uu)i^^, except that the scale of t has been
OT
reversed for convenience. When f = 49 sees., i.e. at M, Fig. 251, the value
of /(t) is MQ, whilst the value of (ww)«-t is MP owing to the special
OT
arrangement. The product of MQ and MP then represents an element of
the integral of (211), and is plotted as QiMj in the figure below. Points
at other times similarly obtained complete the lower curve, the area of
which represents the total disturbance "at 60 sees." due to passage
through gusty air. Further curves are shown for other times to emphasize
the fact that the effect of a gust depends markedly on the time which has
elapsed since it was encountered. A large effective increase shown by F
at 60 sees, becomes zero at 56*7 sees, at H, and has a considerable negative
value at K at 51 '7 sScs. The total areas show somewhat similar
characteristics.
A repetition of the calculations, using {uw)t, {uq)t and (w^)^ would lead
to the determination of w, q and 6 as functions of time. In effect (see foot-
note) * this has been done, and the more important items are shown in
open time scale, and was made at Kew Observatory, using a Dines Recording Anemometer.
The part reproduced on a time base of 60 sees, occupied 240 sees, in the taking, the average
wind speed being 20 ft.-s. Since no better information is available it has been assumed
that the gusts met by an aeroplane travelling 80 ft. through the air are the same as those
registered during the passage of 80 ft. of air over tlie observatory instrument.
d
* The dotted curve of Fig. 251 was not equal to ^^{uu)t—^, but to a portion only of it, since
a saving of labour was thereby effected. It represents the value of
e-ooTw cos 0-283« (212)
By a change of the time epoch to the point at which this dotted line cuts the axis of time,
it is clear that the multiplying factor
i;e-oo75/sinO-283f
is simultaneously applied, k being a known constant equal to the value of e— ""^s^ at the
expiration of a quarter of a period. Two curves representing
/;
^(T)e-oo75«-T) COS 0-283(« - T)aT
0
and //(T)e-«o76«-T 8in0.283(«-T)aT (213)
were obtained as continuous functions of t in this way.
In addition calculations were made for the rapid oscillations, and it was found that to a
sufficient degree of accuracy the integrals
J 0
/(T)e-5'62«-'^>('-'>dT
0
and f /(T)e-5«2(«— )(< _ r)dT ....... (214)
J 0
were proportional to/(T), the values being 0'177/(t) and 0 0308/(t) respectively. Physically
this means that the motion of the aeroplane represented by (214) is so rapid that the dis-
turbance is a comparatively exact counterpart of the gust.
The curves of disturbance in v, w;, q and d can be obtained from the above curves by
addition in various proportions determined by the arbitrary constants A, B, C and D.
STABILITY— DISTUEBED MOTION 533
Fig. 252. One of the quantities estimated^ has been the variation of height
above the ground, and involves the relations
h^uod-w ) .
h=^fhdt f ^^^^^
the form having a special character since in the steady motion the axis of
X is along the direction of flight and is horizontal.
Description of Fig. 252. — The upper curve shows the wind record from
0 to 60 seconds. The aeroplane was supposed to have a flying speed of
80 ft.-s. in its steady state, and to have this velocity relative to the air
at <=!0. Its velocity over the ground was then 60 ft.-s., and the wind
speed of 20 ft.-s. against the aeroplane's motion. The full curve
marked " variation of air speed " was calculated at the points indicated,
whilst the dotted curve shows the variation of ground speed. The differ-
ence between the curves shown by the shaded area is equal to the variation
of the wind from 20 ft.-s., i.e. the ordinates are equal to those of the
upper diagram. It will be noticed that the inertia of the aeroplane is
great enough to average out all the more rapid changes of wind speed, and
shows the advantage of speed of flight as a means of producing average
steadiness. It will be seen that variations of speed of ±12 ft.-s. are
indicated, and these may be considered too large to come within the defini-
tion of small oscillations. Certain other approximations will have been
noticed by a careful student which could be met by more rigorous treat-
ment if desired. The advantages of the present methods are however
thought to be sufficiently great to warrant their use.
The curve marked UqO represents the rate of climb due to inclination
of the axis of X, whilst w shows the change of normal velocity. The
ordinates of the shaded area then represent the total rate of chmb h.
Integration with respect to t then leads to the last curve for " rise and fall."
On the whole the aeroplane gains height, the maximum gain being 40 feet ;
in one place a fall below the original level of about 15 feet is shown. It is
possible that the aeroplane shown would just be able to land itself, as the
vertical downward velocity due to the gusts is not more than 5 ft.-s.
The Effects of Continuous Use of Controls, and the Calculation
OF the Movements necessary to counter the Effects of a Gust
The first problem to be attacked will be the finding of an elevator
movement of a continuous character which will eliminate the effects of
an isolated gust. By a method analogous to that following in adding
the effects of gusts it is clearly possible to calculate the motion of an
aeroplane which results from a prescribed motion of the elevator. The
problem now considered is the converse of this, since it is proposed to find
the elevator movement which corresponds with a prescribed aeroplane
motion.
It has been seen in the discussion of disturbed motion that changes
in u, w, q and d which arise from isolated disturbing causes in^any of the
534
APPLIED AEEODYNAMICS
0 10 20 30 SECONDS 40 50 60
Fio. 252. — Uncontrolled flight in a natural wind (Aeroplane stable )
STABILITY— DISTUBBED MOTION 585
quantities have a common analytical form with four or sometimes five
constants peculiar to the disturbance considered. The same analytical
form applies to disturbances of v, p, r and <f), and the disturbances produced
by the controls. The general problem can therefore be approached by the
consideration of any quantity S defined by
S = Ae^'' + Be^2' + Ce^s« -f DcV + E . . .(216)
where S may be u, v, w, p, q, r, 6 or (f), and suffixes such as /x, v, ^ and r)
may be used in addition to ?^ ... 0 to signify the initial disturbing cause.
As an example H^ would be a disturbance due to change of elevator
position, whilst Su would be a disturbance due to a gust in the direction
of the axis of X.
It is now proposed to show that any disturbance such as S can be
eliminated (practically if not theoretically) by the use of controls. Stability
is a means of reduction of a disturbance to zero, but needs time for its
operation. Whereas a phugoid oscillation may take one or two minutes
for extinction by inherent stability, the use of the controls can reduce
the time required to a few seconds.
In order to assist the explanation of the method it will be supposed, in
the first instance, that S is the vertical velocity of an aeroplane, and is
produced by the addition of elementary disturbances S^ occurring at
different times from the progressive movement of the elevator. The
disturbance of vertical velocity due to a horizontal gust at time f^O is
Su, and it is desired to make the continuous use of the elevator eliminate
Hu- S is of course equal to {u^d — w) by the definition of vertical
velocity.
By the theorem in addition developed in connection with a variable
wind it will be found that the resultant vertical velocity due to the
elevator, i.e. S, is made up from its parts by the integral.
A (continuous use of
eIevator)=/jF'(r)(Su),_.C?r ..... (217)
where ¥{t) is the angular position of the elevator at time t.
By a proper choice of F{t),S can be made to have almost any desirable
form. It is convenient to integrate by parts to avoid infinite differential
coefficients due to a sudden change of elevator position, and the expression
becomes
3 (continuous use of elevator) =|F(t)(5?m)«— I — ^{t) -J Sfi)t— r^T . (218)
If before the elevator is put over the aeroplane is flying steadily, the
value of (S/x)o is zero since there is no immediate change in U, w, q or 6 by
reason of the putting over of the control. Also if movements of the ele-
vator are measured from the position in the steady motion, F(o) is zero,
and the partial integration leads to
^ (contlnnous use of elevator) = ~ I •F(7')"(S^)<— r"'' • • • • {^iJ)
586 APPLIED AERODYNAMICS
If the rate of change of the particular disturbance due to putting over
the elevator is zero when t — 0, then the rate of change of H must also be
zero. In any practical case the rate of change may be small, and it is then
necessary to determine some limit to the elevator movement which is
permitted. This imposes some slight limitation to the values of S which
can be produced.
If, due to a horizontal gust, a disturbance S^ has been given, the final
motion is
r d ^
^ (continuous use of elevator = S„ — F(t) „ {'A[^t—,^'^ .... (220)
and isolated horizontal gust) Iq "^
and the condition of no residual motion is clearlv that the right-hand side
(220) shall be zero.
Solution of Equation (220). — A solution of this equation can always
be found by a process of trial and error, even when limitations are given
to the motions of the controls. In some cases an analytical expression
can be found, for an examination of (220) when ^ = 0 and S'm and S;^
have the form of (216) shows that F(i) must be a sum of exponential
terms of the typap--. -^
' - 'F(oiAieKie + BieK*« + (!ri^''+'I>i' . . .(221)
Assuming the result it will be shown that the expression is correct,
and that Ai . . . Di and K^, K2 and K3 can be determined.
Writing down the expressions for ^^ and S'j, in accordance with
(216) gives—
/
c- -• ^ — -^ St
AaeM + Bge^''* + Cge^"' + Da?^*' . • • (222)
for the rate of change of vertical velocity at time t due to an elevator
moment at time 0, and
^.^-AgeV + BgeM + CseV + DseM + Es . . (223)
for the vertical velocity at time t due to a horizontal gust at time f=0
(actually E3 is zero in the case supposed, but would have had a value if
the disturbance had been due to a change in engine speed).
The terms of equation (220) are now specified, and the integration
presents little difficulty. The result is to obtain
Agc^i' + 636^2+ Cse^s* + DjeM + Eg ^
-^^^'[\ K,-Ai K,-Ai K3-AJ A," Ki-A/ ^K^-A, ^K,-A,
+ ^^^\X, K,~\, K,-X, K3-AJ A2 ^Ki-A/
+ (224)
as an identical relation. The coefficient of e^i' in the identity gives the
equation
'iTABILITY— DISTURBED MOTION 587
and two similar expressions follow for K2 and K3. It will then be seen that
"Ki, K2 and K3 are determined by the solutions of the cubic equation in K —
()_ ^2 _j B2 I ^2 ■ D2 ^ ^ ^ /22g\
K — Aj K — A2 K — A3 K — A4
Equating the coefficients of e-^i', e^-^^ . . . (224) gives
Ai Bi
A —A /%^ -^1 ^1' ^i'j '\
B
^9(f K1-A2 K2-A2 I^^/
-^<^
^2 j^ B2 I Cg _|_ D2
A2 A3 A4
E3 = -Di(?-2 + ?-' + ?+^M (227)
or apparently five equations from which to determine the four quantities
Ai, Bj, Ci and Di. Adding the equations together, however, it will be
seen that there is obtained the expression
A3 + B3 + C3 + D3 + E3 = 0 (228)
and this is often intrinsically satisfied. In that case Ai . . . Di can be deter-
mined. If E 3 be zero then Df is zero, and the addition of the equations
still requires that
A3 + B3 + C3 + D3 = 0 (229)
if a solution is to be possible.
The values of K^, K2; and K3 must all be negative if the aeroplane is to
be permanently controllable, otherwise the elevator angle will increase to
the permissible limit and failure will then occur.
Some physical ideas illustrate the value of the restrictions (228) and
(229) . It is, for example, not possible to ehminate all the effects of a head-on
gust of initial amplitude Ui, for E is then zero and the value of A+B+C+D
is unity. It is clear that nothing short of an infinite force could neutralise
the assumed instantaneous increase in Ui- The same objection does not
apply to w, q and 6, but for the former of these it is readily seen that the
amplitude of the elevator movement would be prohibitive in the initial
stages. The operation attempted would be that of producing a change
of down load on the tail equal to the change of up load on the wings, q and
q are both zero as a result of the gust, whilst q may be given a large value
by use of the elevator. Either q ov 6 is therefore a suitable quantity
for complete ehmination by the use of the elevator. In relation to the
figures previously given for an elevator it appears that a solution which
leads to the elimination of 6* is .—■- ^'-' -
Ki =0 K2 - -0-192 K3 =• -2-62^ ..
A, =-Bi Ci=0 Di=0 / • • ^^^^^
so that the elevator starts from its zero position at time t and returns there
when i is great. The condition Ki=0 introduces some Httle difficulty in
538
APPLIED AEEODYNAMICS
interpretation and appears to involve the condition Ai =— Bj as a funda-
mental consequence,, so that in the case for which A2+B2+C2+])2— ^' J^
is probable that a further identical relation holds.
Approximate Solution of Equation (220). — The necessity for dealing
with partial elimination leads to a substitution of a graphical method
for the exact analytical expressions just given. The process of finding
0-5
X
^,
/
s
'^
Ver
rical
velocity due toguG
t
\
>
V
^^
Ni.
\
-■- ~J\
/'
s
\,
0
Pt-Residual ' \
K^ Disturbance.
^
V
\J
^
"^
-^
'^^
^
/
\
/
/
\
-^
■^^
^
\
/
/
i
1
"
^
/
- n.t^
I
:/
/r~
Vert
ical
veic
city
due
to
elev
a tor
V
(a)
TIME 10 SECONDS 20
30
\
\
1
\
\
y^
'El
eva
:or mov
erne
nt
\
S^
^s
\
"v.^
-^
—
. —
0
(6)
0 TIME 10 SECONDS 20 30
Fig. 253. — Use of the elevator to eliminate a disturbance. (Aeroplane stable.)
the elevator movement is one of trial and error, but presents no serious
difficulties. It has already been pointed out that equation (217) provides
the means of calculating the disturbance due to an elevator when its
movement is known ; the process of trial and error assumes a curve
for a short time, beginning at some arbitrary limit (1-5 in Fig. 253 (b)) and
this suffices for a calculation of the motion. The result is compared with
the motion it is desired to repeat and corrected accordingly. If small steps
of time are taken at the beginning, the calculation will be well established
and can proceed more rapidly in the later stages. The resulting movement
f
STABILITY— DISTURBED MOTION 539
is shown in Fig. 253 (b), the scale being arbitrary and proportional to the
elevator angle. The vertical velocity corresponding with this elevator
movement is shown in Fig. 253 (a), together with the vertical -velocity
which would result from a horizontal gust in the absence of control.
The two curves are seen to be almost exact mirror images about the line
of zero ordinates, and the small residual disturbance when the two are
combined is due to the limitation placed on the initial angle of the
elevator. The ordinates of the curves are proportional to the strength of
the gust in one case and to the angle of the elevator in the other. The
actual values of Wj and /ii are of little present interest.
Examination of the Residual Disturbances in ?*, w, § and 6 when
Equation (220) has been satisfied for Vertical Velocity
It has been seen that the elevator can be used so as to make {uqO—w)
very small in the case of meeting a horizontal gust. It is necessary to
consider what is the effect of the same elevator movement on the flight
speed and angle of path. An equation similar to (217) shows that all
the motions are fully determined, and it will appear that they are of
reduced magnitude. The general investigation leads to apparently
complex expressions, but it is hoped that an illustration will make it clear
that the residual errors in u, w and q can be obtained with little diflficulty.
Some of the equations previously used will be collected here in the form
most suitable for the present purpose.
It has been shown previously that all possible disturbances of the
motion of an aeroplane can be expressed by exponential and trigono-
metrical functions, rigidly so in the case of small oscillations, and very
nearly so for the real motion of an aeroplane in approximately rectilinear
flight. The following particular forms are equivalent to those given on
p. 510 :—
M = ae*<cos (/i;f + ^)+(7 + 8^)eV + ^i . . . .(231)
w = -2- — - e^{ (jiiiJ,2 + vi vg) cos (kt + P)
iJ^O — — ^9
+ (^1^2— fX2'^i) sin {kt + P)]
5 =" 2I 2 ^^'^(^1/^3 + yivs) cos {kt + ^)
''' ' + (H'lr^s -/^3Vi) sin {kt + P)}
+ ^2(y + 80«''' + ^2S«''' + ^3 • • . (2B3)
d = (qdt= _2-^2«*H(/^ii^4 + nv^) cos {kt + P)
^' ' +{liiv,-ii,v{)mn{U + P))
+ t?3(y + S0^''* + ^38e^>' + ^4 . . . (234)
In these expressions the arbitrary constants are a, j8, y and 8, whilst
j^i>f*2' • ' . V\,V2. . . . 171, 'q2' etc., are known from equations (165) . . . (1G7).
540 APPLIED AERODYNAMICS
If the disturbance contemplated (at present all disturbances are considered
as isolated and sustained) is a change in the wind, then Ci, ^2^ ^3 ^.nd ^4
are all zero, and the aeroplane ultimately settles down to the same steady
motion relative to the new wind. If the disturbance is due to an internal
cause, such as a movement of the elevator, we have u=0, w=0, q=0, 6=0,
when t=>0. The values of a, j8, y and 8 are found as indicated in equa-
tions (180), and finally, to satisfy the initial conditions just given,
^1 = — (a cos p + y) \
^,=_(«^i/i2±^ij:2 ,,,p^,lH^^l^.^i sin ^ + ,,y + .^,8) (^35)
etc.
Unless rotations in the wind are assumed to occur ^3 will be zero.
It will be seen from equations (231) . . . (234) that the phase differences
between* the oscillations in u, w, q and 6 are independent of the values of
a and j3, and no matter what the nature of the disturbance, the motions
in u, w, q and 6 will follow each other in the same order, with the same
phase differences and the same relative amplitudes. This relation between
the oscillations can be clearly seen from the curves of Pig. 247, since the
other terms are vanishingly small at the end of 2 sees.
The terms in e^i* can be divided into two parts, the relations between
the disturbances in u, w, q, and 6 for the parts being independent
of y and 8.
The relations between the oscillations just referred to will be found
later to simplify the analysis of motions due to an elevator. It is clear
that to an exceedingly high degree of approximation, all simple disturbances
rapidly tend to become similar damped oscillations of phugoid type,
and it appears that only the dissimilar portions, limited to a time of
about 5 sees, in the worst case, need special treatment in passing
from variation of vertical velocity say, to that of variation of flight
speed, etc.
The elevator movement F{t) having been chosen so that there is no
resultant vertical velocity, it is now desired to find the change of flight speed
due to gust and elevator. The value is
u^+fr{T){u;)t_4r (286)
but for the general problem w„ will be replaced by S'^, where !S' is any
linear combination of the variations u, w, q and 9.
From equations (231) . . . (234) the values of S'„ and S^ may clearly
be expressed respectively as
a'u = a^e^{a' cos {U + ^,) + h' sin {kt + ^,)] + 7,\(y, + 8,t)e''i* + ^\8,e^i* . (237)
and B'i^=a2^^{a' cos {kt+P2)+b' sin {kt+^^)\+ri\(y2+8J)e^i*+^\Sze^i*+^\ . (238)
where ajo' cos ^g + 036' sin ^2 + Tj'iyj + I182 + ^i = 0
;S' differs from !S in being some different linear function of u, w, q
I STABILITY— DISTURBED MOTION 541
and 6 ; the values of a^, ^i, yi and S^ are therefore the same for both S'
I^Hpid S, but a, b, r], etc., have been changed to a', h', tj', etc.
It r^=^^ri„ — cos n , ,■ „ , = —sin M and ^ / ^JLIl ™ /o^Sra^
the value of SV becomes
-V == /„?!^ f^/* ^°« (^'' + i^i^ + '^') + ^'i(n+ S J)e^'' + I'xS^e^i' + l\ . (239)
with a somewhat similar expression for S'u-
The value of the integral / Y{T){';s:,tj)t-,dT, which is the disturbance
of vertical velocity due to the elevator, has already been determined
(Fig. 253). It is now proposed to make the integral / W {r){'B' fj)t-4'^
Jo
depend on the known integral, which for brevity will sometimes be
referred to as /(f). Its value in the complete notation above is
f(l) =j*^ ^'^'\^,^T+ b.e'^'-^^ «o« {^(' - -) +i8, + n}
+ Vi{y2 + S^e - r) }e^i^«-^) + |i82e^2<'-^> + ^Adr . . (240)
and t cannot be negative in this integral.
It has already been pointed out that after the lapse of a short interval
of time all curves for isolated sustained disturbances are similar in general
character, but differ in the phase and amplitude of the oscillation. Changes
in phase and amplitude will therefore be introduced successively into
the expression (240). The phase difference between the oscillations
in S' and in S is seen to be n' — n, or is equivalent to a time phase of
^ — r- — - sees. If then t + — r — is substituted for t in (240), a suitable
K K
correction will have been made for the phase difference. As a result of
this substitution, (240) becomes
f(t+'^\= r^V(r)r -°' /(^-^•^"^jt') <^°« {Ht-r)+P,+n'}
\ J Jo l\/a^+b'
+vAyi+5,[t-r+^^Me'\ * ^+i^S,e'\ * f+C^^dr . (241)
The integral required for the disturbance of flight speed, S', is
j^ F'(t)(H'^)<_A- / I"(t)[;^7^^!^/ cos {k{t-T)+fi,+n'}
+ Vi{7.+5.(<-t)}c'^'+|'x /'^'"^^CxjdT . (242)
and on comparison with (241) it will be seen that the phases of the
oscillations of S' and S which refer to flight speed and vertical velocity
respectively, due to use of the elevator only, have been brought into
542 APPLIED AEEODYNAMICS
agreement, but the amplitudes are still different. Agreement of ampli-
— fe(n'— n'
tude can be produced by multiplying both sides of (241) by me ''
and the equation then becomes
—Mn'—n) n'—n\ /' ""~" ««„
me *~ i(t+ -5.-U P+ i F'{t)-^^/('-^)cos {^(<-T)+)3,+K'}dT
/•« / n'-n\r -Hn'-n) ,, . :^h(n^^
*"
In this expression the second integral has been obtained by a substitu-
tion of T-\-{n'—n)k for t in that arising from the separation of the integral
of (241) into two parts.
From (242) and (243) an expression can be written down for
/ F'(t)(S';u)<-t^''" in terms of /(<) and residual integrals.
Jo
ft -^<*|'-'L) . n'-n.
J t ^/a^+b^
. n'—n / , —h(n'—n) —h(n'—n) ■.
+ r~* r'(T+-pj me '* ivi{y2+h{t-r)} + ^,S,}e^'^^-'^+mC,e * jdr
/•( / n'~n\( —h(n'—n)
-Jo ^V+ nr Jl^"*"^" * ^ -';'i){72+8.(i-T)}e'^('-^>
+ (m|ic F~-I'i)s./'^' '^+mCie * -C'lj^^ • (244)
By a few simple transformations, such as the substitution of t—r for t in
the first integral of the right-hand side, (244) becomes
ft ^M^) „._
j^ F'(r)(EV),_dr=me * ^0+~T")
n— n
n— n
+ / F'{t-r)^j~^/' cos (A;T+/3,+«')|dT
/■ k~ I n' — n\i —h(n'-n) xui—r) -h(n'-n)^
rt ' r —h(n'—n)
—h(n'—n) -h(n'—n) ^
+(mJie * -riJ5/'^'~'^+("^Cie ^* -^'ijjdr . . . (245)
The interpretation of (2 5) is not difficult. The first term on the right-
hand side, i.e. me * /( t-\ j-— j shows that part of the disturbance in S'
(or flight speed) is proportional to the disturbance in ^ (or vertical
velocity) and differs in phase by a time — r — . This part of the
rC
STABILITY— DISTURBED MOTION 543
itegral is then easily evaluated from the previously determined vertical
lotion. The two integrals first occurring on the right-hand side have
lefinite limits, and the whole of the integration involved is limited to a
'time equal to the time difference in phase, which can be arranged so
^ as not to exceed one-half the period of the oscillation, in this case about
11 sees. The last integral is taken between more general hmits, but
the terms involving e-^iC'-^) as a factor rapidly become negligibly small,
say at the end of 2 sees., whilst the remaining terms are easily integrated
and give the value of a motion which is proportional to the elevator
onovement.
Compaiison between the changes of flight speed and vertical velocity
due to a gust and the corresponding changes due to continuous use of the
elevator.
Considering the change of flight speed during a gust and the change
of vertical velocity due to the gust, it will now be shown that a somewhat
similar relation holds between S'^ and S„ as between S' and S. P(0
having been chosen so as approximately to satisfy equation (220), that
equation can be modified by the use of /(t), giving rise to
/(0 + A/(<) + S„ = O (246)
where Af{t) is a small term depending on the degree of approximation to
an exact solution of (220). Substitutuag for S„ from the equivalent of (237)
and using (238a),
f(t) = - A/(<) - ,"f - ,.,008 {kt +p^+n)- e^^'{7h(yi + 8,t) + i,h,) . (247)
and further, directly from (237) and (238a)
-H'« =- ,^=reht cos (kt + p, + n') - e^i^{7,\(y, + 8,t) + i\8,} . (248)
An expression is easily obtained which differs from (247) in phase and
amplitude, giving
The form of the term involving cosines is now identical with the eorre-
sponding term m (248). Equation (249), however, apphes until t -f - — =- —
is zero, whilst (248) is limited to positive values of t. If — |- — is positive,,
the part of (249), which corresponds with negative values of t in the cosine
term, wiU not be ehminated by subtracting (249) from (248), and there will
be a corresponding remainder in S'u for positive values of t when
544 APPLIED AERODYNAMICS
. is negative. If we call this difference ^^ then by combination of (249)
and (248) ' ,
—Mn'n) , '—h(n'—n) ,
_ A^/-«. (Ai—^Xw— n)
'f^m[^\y. + 8,(. + '^)]+ IA] . (250)
n — n .,
Ji 1— ^i^tj irom u 10 —
n' —n .,
where ^ represents the value of (— S„) from 0 to — ^ ^^ if n'— n is positive
ki
and S'^ from 0 to — if w' — n is negative.
The motion of the aeroplane in S' , i.e. the variation of flight speed,
as the result of both the'wind and elevator movements is
>-,' / /"'
'^ (continuous use of elevator and =^ '* M 4" ( ^ (T)(3r ju)<_t<*^ .... (251)
isolated horizontal gust) JC
an exactly similar expression to that for S from which F(0 was originally
found. By an examination of the terms in (245) and (250) it will be seen
that the term depending on /(f) vanishes from the expression for S'. The
remainder of the terms in ;h;'„ are easily evaluated, me * A/(^H r. — )
can be plotted from the known value of A/(f), whilst the remaining terms
involve only calculations for times up to 2 sees., when they become
negligible.
Adding (245) and (250) together the value of S' is seen to be
n— n
—Hn'-n) -hjn'-n)^
jo Va' + 6'
When n' => ?i, to=1, 17/ = 17, etc., (252) reduces to
s:« + /Je'(t)(£:^),_A + a/(o=o . . .(253)
In any case in which ?i' — n exceeds n it would be advantageous to take
(n' — n) as the excess over 77 instead of over zero. This is equivalent to
changing the sign of m and proceeding as before.
p
STABILITY— DISTUEBED MOTION 545
Illustration o£ the Mathematical Processes of Equations (241) to (252) by
Reference to a Particular Case. — The assumed elevator movement will be
that indicated in Fig. 258, and is one which practically eliminates varia-
tions of vertical velocity of the aeroplane when moving through an
isolated horizontal gust. In this case S = UQd — w, and the varia-
tion of vertical velocity due to the gust whilst the elevator remains
fixed is {uQd—w)u=i=^u ^^^ is shown in one of the curves of Fig. 253.
The almost similar motion produced by the movements of the elevator
alone with no wind is also shown in the same figure and is the value of
,-t
¥'(r)(S,fi)i.rdr. The difference between the two curves has been
Jo
indicated in the same figure and corresponds with the mathematical
expression A/(i) of equation (246). The variation of vertical velocity due
to a horizontal gust, i.e. {uQd—w)^ is identical with/(i) of the same equation.
It is now desired to find the variation of forward speed of the aeroplane
relative to the air. 3' of the mathematical equations is then u. The
variation of the velocity of the aeroplane due to the wind alone is mainly
a damped phugoid and is shown in Fig. 247 {a). The variation of velocity
will be modified by the elevator movement, and the value of SV> i-^- the
variation of velocity due to a single sudden movement of the elevator is
represented by the curve marked u in Fig. 254 (c) ; since the elevator
movement is known,
:S'„+£f'(t)(3:V).-A- . . . (254)
and the integral for u which gives the curve of Fig. 254 could have been
determined without any reference to the fact that F(r) was chosen to
ensure the ehmination of {u^d-^-w). It is, however, already clear that
some such relationship exists, and in finding the value of u from (254) the
results were so arranged as to indicate this relationship.
In the actual working it was found to be convenient to transform
(254) by a partial integration, obtaining
S =u
= S„+/;F()^^(3:V).-A . . . (255)
The curve (m^) is drawn in Fig. 254 (a), and from this ciirve and the
elevator movement in Fig. 253 the value of the integral of (255) has been
found. The ordinate SQ of the curve KQMNP, Fig. 254 (a), is plotted
in a manner similar to that previously adopted for determining the
elevator movement F(<), i.e. the elevator position at 12 sees, obtained
from Fig. 253 is multipUed by the ordinate of the curve % of Fig. 254 (a)
for a time (15-8— 12) sees, to obtain the element SQ of the integral repre-
senting the disturbance at 15-3 sees. The area of the figure KQMNP is
then the disturbance at 15'3 sees, which results from the movement
of the elevator alone, and the complete disturbance at any time due
to elevator alone is represented by the curve ABCD of Fig. 254 (6).
The curve u^^ of the same figure is the disturbance due to wind alone, and
the disturbance of the machine as a consequence of both wind and elevator
2n
546
APPLIED AEEODYNAMICS
movement is obtained by simple addition of the separate effects and is
given by the curve EBF. The cm-ve has the general characteristics of
the curve in Pig. 253, i.e. the variations in u are roughly proportional to
the elevator movement in this case of simple initial disturbance.
(a)
(b)
1
B
/
1
\
\
1
b
.
V
/
N^J
/
Ril
v
J
\
\
K
/
M
/a
I
At
V
p
s
\,
\
J
/
\
\
/
/
/
v^
^0
/
-0-5
^
/
"^
^
1 1 1 1 I 1
tt
y
\
Motion in u due to Assumed
—
\
^
b/-
^
\
/Motion
of Elevator
1 1 ,
/
\
)
1
'N
\'
V
'
B^
*t^
\
Residual Motion
nU
A
,/
\
^
;/ . ' '
J
/
kC,
-^^
"p~
JN
y
^
/
w
V
t
\
/
\
\
^
/
---.
V
/
\
S,
/
k
I
D.
V
A
Uu.
\
^
/
-Motion in 11 due to gust.
1
_
J
10
Time (Seconds).
20
10 20
Time (Seconds).
30
Resultant Disturbance of an Aeroplane after choosing
Elevator Movement to Eliminate Variations of Vertical Velocity
wtiicti would otherwise be caused by a sudden HorizontalGush
10
\
\
\
\
\
sfj
■u
N
«-
q.x
100
'N
k
^
— '
-^
^
\
N
%
^
:::
^
:<
100
^
==
~
^^
;^
—
—
\~_
b
^^
=
-f
~~-
L_
_
~-~
^^
—
_
~
^^^^
^ 20. , ^50
Time (SecotxJs)
(f)
50
Fig. 254.
-Eesidual disturbances of a controlled aeroplane when full use of the elevator
is made in maintaining level flight.
Eeturning to a consideration of Fig. 254 (a), it will be seen that a curve
ABCED has been drawn which coincides with Uy, for all points after B. The
part ABCED is areproduction to another scale of thecurvefor {uqQ—w)^., but
has been moved along the axis of time so that the point K of Fig. 254 (&)
coincides with the point E of Fig. 254 (a). The change of time is equivalent
to substituting i-\ — ^^for i as was done to obtain equation (241), whilst
STABILITY— DISTURBED MOTION 547
the multiplication by 0*940 to make the ordinates agree corresponds with
-h(n'-n)
the multipUer me * used in obtaining equation (240). If now the
curve ABCBD of Fig. 254 (a) be used instead of PBCE, a new curve
TEiMNP is obtained, which encloses part of the area RQMNP. This
area is proportional to (mq^— w?) as plotted in Fig. 253 (a). The actual
values of all such areas are plotted in Fig. 253 (a) as AiBiCiDi, which
then represents
—h(n'-n , —h(n'-n) ,
of equation (250). A reference to Fig. 254 (6) shows that for times greater
than 6 sees, this curve is almost symmetrical with m„, and hence, on adding
the disturbances, the oscillation after about 6 sees, is completely eliminated.
The elimination of disturbance here indicated corresponds with the
t-\- , - j by the addition of equations (245)
and (250).
The residual disturbance is seen to depend to an appreciable extent on
the integral corresponding with the area RQEiTj, Fig. 254 (a), and since
the elevator movement is approximately an exponential curve, the residual
motion on this account is after a few seconds nearly proportional to the
elevator movement. This follows from the figures, since the effect of multi-
plying the ordinates of EQRiT, Fig. 254 (a), by an exponential is to pro-
duce figures which differ from one another only in the scale of the ordinates.
This does not apply for times less than 6 sees., i.e. for times less than the
base of the figure RQRiT. The curve obtained from the values of the
areas of RQRiT at all times is shown in Fig. 254 (&) as ABBgF, and is
the graphical representation of the integrals on the right-hand side of (252).
There remains to be considered the portion of the curve u^ which was
not eUminated by subtracting AjBiCiDi. The part from K onwards with
increasing time has been already dealt with as A/u -\ =- — jto a new scale.
The part from < = 0 up to K, Fig. 254 (&), i.e. the value of m„ up to K corre-
sponds with the 1^ of equation (252), and x is thus seen to be of some
appreciable importance in certain cases, but ceases to increase in value
after an interval of time which need not exceed one-quarter of the phugoid
period.
There is then, for disturbances in u,w,q and 6, a period of not more than
8 or 10 sees., during which the character of the motion is somewhat com-
plicated owing to its being made up of two or three relatively important
components. After this, however, the motion in each case is very nearly
a copy of the elevator movement, and it appears that the motion is not
oscillatory.
Curves corresponding with those of Fig. 248 are drawn in Fig. 254 (c)
for the continuous movement of the elevator given in Fig. 253 instead of
for a single sudden movement. The ordinates are given as fractions of Ui,
the variation of speed in the gust.
548 APPLIED AERODYNAMICS
Extension to Motion in a Natural Wind.— The process to be followed
from this point onwards is identical with that described in the previous
section, on the disturbed longitudinal motion of an aeroplane flying in
a natural wind (pages 529-533). The difference in the initial assumptions is
covered completely if the curves of Pig. 254 (c) are used in the new calcula-
tions in every case in which the curve of Fig. 247 (a) were used in the
earlier calculations.
A brief reference to the results and a comparison with those previ-
ously obtained for an uncontrolled aeroplane will show how different
the disturbances may be. The importance of the results appears to He
not in the demonstration that such reduction of disturbance is possible,
but in indicating a method for the systematic investigation and design of
automatic devices for aeroplanes. The results also show that there is a
possibility of getting more and more advantage from the use of inherent
stability without the attendant disadvantages of violent motion in winds,
if in addition some mechanical device can be invented which will operate
the controls so as to reduce the disturbances which the inherent stability
has to eHminate.
In considering the results of the calculations referring to the longitudinal
motion of an aeroplane in a natural wind, it should be remembered that the
calculation has been carried out on the assumption that a perfect pilot has
instantaneous knowledge of the variations in the wind and is able to make
the necessary correct movement. An actual pilot would produce a less
exact approximation, and in particular would probably not attempt to give
such complicated movements to his elevator. This introduces a further
modification, and it will be interesting later to find the effect of a slow
elevator movement which averages out rapid fluctuations ; it is, however,
a question of order of approximation and not of principle.
The elevator movement requisite to cut out the variations of vertical
velocity due to the wind described in the previous section is given in Fig.
255, together with the anemogram.
Its general characteristics foUow those of the wind somewhat closely.
There is, however, a superposed variation which does not bear any
simple relation to the wind at the instant and is, in fact, dependent to a
large extent on previous history during the .last minute.
The residual variation in vertical velocity has been plotted to ten times
the scale of the corresponding diagram for the uncontrolled machine, as
otherwise it would have been too small to see clearly. The residual vertical
velocity is shown in Fig. 255. together with the vertical velocity of the
uncontrolled aeroplane. The maximum vertical velocity when the aero-
plane is controlled as assumed is only a fraction of a foot per second instead
of the 10 ft.-s. previously found at the end of a minute. This indicates
the practical elimination of the vertical velocity and is the best which can
be done under any circumstances.
In a similar way, the elevator movement might have been chosen so as
practically to eHminate the variation of speed over the ground or the
inclination of the axis of the aeroplane. With the elevator movement
assumed, which was not primarily arranged to reduce anything but the
STABILITY— DISTUEBED MOTION
549
0-2^ Variation of Vertical Velocity Relative to the Ground when controlled. i
-0-2
10 T
8 +
6
4 +
2--
0-
-2-
-4-
-6 t
-8
-10
2-n
0
2-I-
4I
/Sec
Variation of Speed of Stable Aeroplane
over Iground when controlled.
20
30, , 40
TIME (Seconds)
50
60
Pig. 255. — (JontroUed flight in a' natural wind'( aeroplane stable) compared with
uncontrolled flight of same aeroplane.
550 APPLIED AEBODYNAMICS
vertical velocity, the variations of horizontal speed are greatly reduced,
as will be seen by reference to Fig. 255.
One of the curves of this figure shows the variation of horizontal
velocity of the aeroplane in the natural wind for the elevator movements
shown above, i.e. when the aeroplane is controlled. The variations of
speed are not great, and vary between an increase of 2 ft.-s. and a
decrease of 4 ft.-s. The comparative curve of velocity is reproduced from
the previous section, and shows a speed of flight over the ground varying
from an increase of 10 ft.-s. to a decrease of 12 ft. s.
I
APPENDIX
The Solution op Algebraic Equations with Numerical Coefficients in
THE Case where Several Pairs of Complex Eoots exist
Introduction. — ^The conditions for the stability of an aeroplane in the general
case involve as part of the analysis the solution of an algebraic equation of the
eighth degree. The roots may commonly consist of two real roots and three
complex pairs, and it was found on reference to the English text-books available
that no general method of solution was indicated which did not involve almost
prohibitive labour in the arithmetical calculations. In a book by H. von Sanden,
issued in Germany last year, a method of solution is described which appeared
to have the required generality ; information on one point of importance in
relation to complex roots being missing, reference was made for fuller informa-
tion to a paper by C. Runge in the Ency. Math. Wissen., 1898-1900. It was
there found that a method of solution of a completely general character was
devised by G-raefEe in 1837 and described by Runge as the best method known
at the time of writing the above article (at least, when all the roots are required) ;
Runge further describes a method of dealing with complex roots, developed by
Encke.
Another paper, by Jelinek, referred to by Burnside and Panton, deals with
the same problem, and, although written in German in 1869, is not mentioned
by Runge. As a method of finding approximate values of the roots, that pro-
posed by Jelinek appears to be much less useful than GraefEe's, but may possibly
be of considerable value in obtaining a continuous increase in the accuracy of
any root, i.e. the method may bear somewhat the same relation to complex
roots that Horner's process does to real roots. Graeffe's method appears to
be much more convenient than Sturm's for finding the approximate position of
real roots, and has the further advantage of giving approximate values for
the complex pairs of roots.
As the solution of equations is of some importance in calculations of
stability, and as the methods mentioned above have not yet appeared in the
English text-books, it has been thought advisable to give an account of them
in some detail, and in particular to show how they have been applied.
Solution of a Biquadiatic Equation.* — ^Taking the example given in longitudinal
stability which lead to equation (15), p. 461, the detailed arithmetical process
of finding the roots will be shown. The equation is
A4-fl4-8A3 + 62-0A2 + 9-80A + 216=0 . . . . (1)
and for the purposes of explanation the coefl&cients will be defined by writing
the equation as
a4A* + a3A3 + a2A2 + aiA-f-ao=0 .(2)
* The more oomplete explanation of the method occurs in the illustration of the Holution
of an equation of the eighth degree.
551
552 APPENDIX
• The process followed isthat of multi plying two equations together, the second
of which differs from the first in that the sign of A has been changed. The
work is simply arranged as
+ - +
+ 202^4 — Sajtts + 2aoa2
2aQa4
(3)
and the addition of the results leads to the product required. In a numerical
example the process is repeated until ^a^a^ is very small and until the second line
is devoid of consecutive terms. Separation of the roots has then been effected.
If a^a^ becomes small whilst a2a4 and ^0^2 ^^^ not negligible, the presence of two
pairs of complex roots is indicated. If a^a^, on the other hand, is left as important
two real roots are indicated and one complex root, and so on.
In the numerical example above the process is carried out as below : —
1+11-48 + 61-20 + 9-80 +2-16
- + - +
1-22-19 +33-84 -91 -60 +4-66
+ 1-24 —0-29 +26-8
0-00
2ndpowerof roots 1—91-5 +33-55 +12-72 +4-66
+ + - +
1-93-02 + 17-260 -2*96 +21-17
+ 7-10 +0-000 +3-30
The separation is proved by the last line to have been complete at the second
/4.*fifi
power, and the moduli of the roots are found as 1/33-55 and \J w^,^' The
convenient arrangement for working is
Log. Dlff. of logs. Diff./2. Antilog.
33-53 3-550 3-550 1-775 59-6 r-^
4-66 0-668 3-118 . 2-559 0-0362 . . . rg
If pi and 2)2 I>® t^® coefficients of A in the quadratic factors they can be obtained
from the formula
V^ f (4)
03-
r
a4-
r2
The values are jpi = 14*64 and 0-158, and the factors of (1) are
A2 + 14-64A + 59-6=0 and A2 + 0-158A + 0-0362=0 . .(5)
These factors differ a little from those given in equation (15), p. 461, but,
as will be shown later, an accurate answer can always be obtained from any
APPENDIX 553
approximation whenever it is required, and for many purposes high accuracy is
not required.
Illustration of the Solution of a Numerical Equation of the Eighth Degree by
Graeffe's Method. — -The equation to be solved will be taken as equation (82),
p. 493, and is
A8+20-4A7-f-151-3A6+490A5+687A4+719A3+150A2+109A+6-87=0 . (6)
The roots are known to be partly real and partly complex, but this knowledge
is not of assistance in the application of the method. GraefEe forms the equation
whose roots are the squares of the roots of (6), and treating the new equation
in tlie same way, forms the equation whose roots ate the fourth power of those
of (6). After continuing the process for a number of times (n) the roots will
have been raised to the (2n)th. power, and it is almost obvious without formal
proof that this will lead to a separation of the roots, at any rate when they are
real and unequal. One point is, however, worthy of notice here, and that is,
the suppression of sign which takes place on squaring. This leads to no great
difficulty when taking the (2n)th root of a real quantity, but introduces the
necessity for special consideration of complex roots. In the case of real roots
the signs must be found by trial if necessary, but the use of Descartes' rule of
signs may render trial unnecessary.
To find the equation whose roots are the squares of (6) it is only necessary to
change the sign of A to form a new equation and then to multiply this new equa-
tion by equation (6). The method of arranging the multiplication is of some
importance, and the form adopted by GraefEe is as follows : —
Suppose the original equation is
a„a;" + a„_ia;''-^+a„_2a;"-2 + a„_3a;"-=' ... =0 . . . (7)
Write down only the coefficients, and beneath them the signs of the new co-
efficients formed by changing the sign of x. The multiplication process is then
readily seen to follow as below : —
+ - + -
2(a„)(a„_2) — 2(a„_i)(a„_3) -f2(a„_2)(a„_4)
2(a„)(a„_4) — 2(a„_i)(a„_6)
-f 2(a„)(a„_6) .... (8)
The products are continued in successive rows as far as possible, and the sums
of the columns give the new equation whose roots are the squares of the roots
of equation (6). After repetition it will be noticed in a numerical example
that the terms in the lowest rows soon become very small, and if all the roots
are real and unequal, the process rapidly leads to all the terms in the second
and succeeding rows becoming negligible, and the separation of the roots is
then complete. If a complex pair occurs, one of the products in the second
row will not become unimportant, and the calculation is stopped when the terms
immediately to the right and left of it become negligible. More than one
complex pair leads to more than one important term in the second row, but in
the absence of repeated roots these terms can never be contiguous.
The process presumes the existence of a limit of accuracy of calculation
554
APPENDIX
and in the work which follows, this limit will be taken to be that which can be
obtained by a 20-inch slide-rule. The exact limit taken afiects the accuracy
of determination of the roots, but it would probably not be advantageous to
get high accuracy directly, but to do this as a second and entirely separate
calculation. One other point of convenience remains: it will readily be seen
that the raising of numbers to high powers will lead to the introduction of
extremely large and extremely small numbers in the final equation. It is a
convenience therefore to have some means of readily indicating powers of 10.
The notation used by Von Sanden will be adopted, and this expresses a number
such as r323 X 10^ by 1*323^. The notation is not unobjectionable, but no
better alternative suggests itself. Proceeding now to solve equation (6) the
first step corresponding with (8) is
x" X'
x«
x^
x*
a;8
a;2
X
1 2-042
1-5132
4-9012
6-872
7-19«
l-60«
1-092
6-87
+ -
+
—
+
—
+
—
+
1 -4-1612
+2-290*
-2-402''
+4-719«
-5-168«
+ 2-250*
-1-188*
+4-7191
+3-026
-2-000
+2-080
-7-050
+2-060
-15-67
+0-206
+0-137
-0-293
+0-003
+0-454
-0-044
+0-001
-1-068
+0-021
+ 0-94
2nd power —
1 -1-135''
+4-27"
-6-12*
-1-925"
-4-1556
-1-2486
-9-82»
+4-7191
=0 . . (9)
Equation (9) is the equation whose roots are the squares of the roots of
equation (6), the powers of x at the head of each column being supposed to
apply all down the column. The numbers in the fourth and fifth rows are
small, and in continuing to the fourth powers of the roots it will be found that
they become negligible. The sequence of signs due to changing x to —x in
(9) is given as the next row, and the multiplication is then continued for two
sequences : —
+
+
+
+
+
+
+
1 -1-288*
+0-854
+ 1-823'
-1-389
-0-038
-3-7450
-1-644
-0-094
+3-7051"
-6-083
-0107
-1-726"
+0-480
+0-011
+ 1-6671'
-0-816
-0-002
-9-642'
-1-177
+2-2278
4th power —
1 -4-348
+ +
+3-968
+
-5-483»
+
-1-4851°
-1-23611
+
+7-399
+
-1-0828
+
+2-2278
+
. (10)
1 -1-883'
0-792
+ 1-56818
-4-760
-0 003
-3-00819
-0012
*
+2-20529
-3-55
*
-1-52522
-0 022
*
+5-46119
-2-991
*
-1-17118
+0-003
+4-959«
8th power —
1 -1091'
-3-1951'
-3-02019
-1-^13521
-1-54722
+2-47019
-1-16818
+4-959«
. (11)
In the process of finding the equations with roots of the 8th power from
that with roots of the 4th, it will be noticed that only one term, and that a very
small one, occurs in the third row. In the second row the 3rd, 5th and 7th terms
are very small and will disappear in the process of finding the 16th powers. This
indicates that a considerable degree of separation of the roots has already been
f
APPENDIX 555
effected, although the occurrence of two important terms in the second and third
columns shows that the separation is not complete. Replace (11) by a general
equation as in (8) and proceed to the next step, putting zero for the terms which
are seen to vanish from a consideration of (11).
+ - + - + _ + _-{-
The sequence in the second row now indicates that separation has not been
obtained for the terms bracketed (4) whilst separation has been obtained for
the rest, there being complex pairs to be obtained from the quadratic factors
bracketed (3) and (2), and a real root to be obtained from the linear equation (1).
After separation to the extent shown has been obtained the further arithmetic
can be confined to the first four columns. Reverting to (11), the next sequence
of signs, etc., is
+ +
—
+
1 -119114
-0-639
+
1-02027
0-659
-9-12038
16th power 1 -1-83014
+ +
+
3-6126
+
-9-12038
+
' 1 -3-35028
+0-072
+
1-30353
3-337
-8-3277
32nd power 1 —327828
—
2-03453
-8-3277
(13)
(14)
46 4a
The separation of (4) is indicated when obtaining the equation of 32nd
powers from the 16th. The first term on the second row is seen to be small in
comparison with that above it, whilst the second term of the row is important.
At the 32nd power the process has been carried far enough, and (4) has been
divided into a linear and a quadratic factor. The original equation has then
two real roots and three pairs of complex roots.
The larger of the two real roots is now obtained from 4&, (14), since the root
of the equation
a; = 3-27828 (15)
is the 32nd power of the larger real root of the original equation. The 32nd
root is easily extracted, leaving an ambiguity of sign which can be removed by
a trial division of the original equation. Since all the signs of the original
equation are positive, it follows in this case from Descartes' rule of signs that
the root is negative.
The largest pair of complex roots is obtained from the solution of —
-3-27828a;2-2-034"a;- 8-3277 = 0 .... (16)
556 APPENDIX
and by making use of De Moivre's theorem the 32nd root can be taken. There
are now, however, 32 ambiguities corresponding with the 32 roots of unity, and
as all the roots are complex, no simple means of determining the final answer
is immediately apparent. The method proposed by GraefEe in such a case is to
solve (16) as a quadratic equation, obtaining one ambiguity only, i.e. with a
positive or negative sign for the real part of the root. By trial in (13) one of
these will be found to be inadmissible. Extracting the square root a further
ambiguity occurs which can be removed by trial in (11), and so on.
The method will be seen to be perfectly general, but it is not the only
method by which the complex factors can be extracted. It will be noticed that
directly from (16) the value of the modulus of the root of the original equation
can be obtained as the 32nd root of ?>'^'^'^IZ'21^^^, and the sign to be taken is
necessarily the positive one. There is then a factor of the original equation of
the form {x^-\-px-\-r), where r is known but jp is to be found. If the original
equation is divided by this factor a remainder which is linear in x will be left,
and since r is known, the coefficients of x and the coefficient independent of x
give two equations from which to determine f. This may be effected by the
process of finding the cominon factor. It will be shown later that much of the
division can be carried out generally, and in any particular case the highest
power of f to be dealt with in the detailed arithmetical division is not greater
w 4- 1 . .
than — ^ — > where n is the highest power of x in the original equation. For
equations up to and including the 6th order the whole division has been carried
out generally, giving the following formulae for f.
For a cubic the value can be obtained from the sum of the roots and the
value of the real root.
If the coefficients have the significance given to them in equation (7), the
formulae for f are —
ag - -
Biquadratic jj= (17)
.Quintic « = — ; ' (18)
Sextic : — write j8 f or ag 1 ' y f or ag ^ ; 8 for ai^ ^
then p=— \ i 8~H • ■ • ^-^^^
|( «5-a6 p - \r{^ - 2^6) - «6 ^ + «4|
For equations of higher degrees the formula gets appreciably longer and may
not be advantageous. The formulae given above cover the usual cases occurring
for the stability of an aeroplane, as in general two of the roots of the octic equation
are real.
To apply the foregoing analysis to equation (6) the moduli are required, and
these can be obtained at the same time as the numerical value of the real roots.
The further calculations are given below in a form suggested by H. von Sanden,
APPENDIX 557
the numbers in the first column being obtained directly from equations (14)
and (11).
3-27828 28-516 28-516 0-892 7-80 real negative root.
8-3277 77-920 49-404 1-545 3507 r^ modulus of complex root. •
^
I
3-02019
19-480
—
DifE./8
1-54722
22-189
2-709
0-3385
2-18 rg
l-168i«
16-067
7-878
1-2348
0-172 rg
4-9596
6-695
10-628
2-8285
0-0674 real negative root
(20)
The process does not need detailed description, as it is the same as that
followed in extracting the wth root by means of logarithms.
The original equation will now be reduced to one of the 6th degree by dividing
through by the factors A + 7-80 and A +0-0674 obtained from the values of the
real roots. The original equation is represented by its coefficients in the first
line, and the second and third give the figures obtained for the successive quotients
and remainders when dividing by A + 7*80.
6-87 . (21)
1 2-041 1-5132
4-9012
6-872 7-192
1-502
1-092
1 1-259 0-531
0-765
0-900 0-1746
0-1386
0-00881
7-81 9-821 41362 5-972 7-015^ 1-3612 I-O8I2
The terms underlined give the seventh degree equation required, and the
first term in the third row, viz. 7-81, shows that the root is approximately correct.
It is essential for success that the division by large roots should begin from the
term independent of A, and for small roots should begin at the term containing
the highest power of A.
Dividing by (A + 00674) and working in the ordinary decimal notation »
1 12-59 53-1 76-5 90-0 17*46 13-86 0-881=0(22)
0-07 0-84 3-53 " 49 5-74 0-79 0881
A6 + 12-52A5 + 52-3A4 + 72-97A3 + 851A2 + ll-72A + 13'07 =0 (23)
The last line is a sextic with three pairs of complex roots. Using the formula
given in (19) with ri = 35-07, the value of p is obtained as below : —
^ = 1 A = 12-52 8 = 49-9
72-97-12-52 X 3507 -067 „ _ ,_.,
« = : — — =11-2J . . . (24)
^ -35-07-49-9+52-3 ^ '
One quadratic factor is therefore A2 + 11-22A + 3507 (25)
Divide out by this factor, remembering that it is a large root —
1 12-52 52-3 72-97 851 1172 1307
1 r322 2;350 0-2150 0370 408 ... . (26)
11-20 49^95 72-96 84-73 7*54
14-83 26;38 ^;31
35-12 46-38 8242
The quotient is indicated by the figures which are doubly underlined, and
the approximate correctness of the factor is indicated by the agreement of the
first numbers in the third and fifth rows with those in the quadratic factor.
p58
APPENDIX
The biquadratic (26) can now be solved, using formula (17).
r2=2-18
and /-g = 0172
P =
1-32 X 2-18 -0-215
218
P =
0-370
218
1-32 X0172-0'215
-201
1-325
-0-0059
. (27)
• (28)
(29)
(30)
and the two remaining quadratic factors are
(A2 + 1-325A + 218)
and (A2-0-0059A + 0172)
The whole calculation to this stage can be carried out to slide-rule accuracy
by two computers in about 3 hours. It is necessary to work independently
for successive steps and to make comparisons at the end of each step. When
the powers of 10 in the later stages become great, considerable discrepancies in
the significant figures occur and seem to indicate want of accuracy. This is not
usually the case, and the roots ultimately deduced by both computers will be
found to agree even when the discrepancies mentioned above appear to be very
great. The reason for this is obvious when column 2 in table (20) is examined.
Method of obtaining Any Result more accurately. — In examining the stability
of a particular aeroplane it is probable that the roots thus obtained are sufficiently
accurate for all practical purposes. In an investigation concerning the efEect
of certain modifications of detail, higher accuracy is desirable, and methods will be
described for increasing the accuracy of any complex root progressively, without
the necessity for a knowledge of the remaining roots.* The procedure is as
follows : Divide the original equation by the approximate quadratic factor,
obtaining a remainder of the form B,jX + Ro> ^^^ ^ quotient. Again divide
this quotient by the approximate quadratic factor, leaving a remainder R3a;+E2.
If the approximate quadratic factor be x^-'rpx-\-r, then the corrected quadratic
factor is x^-\-(p-}-Sp)x-\-r-\-hr, where
8p =
Rq
R3
R2
R3
R2
pB,2 — R2
rRs
Ri
yRg — R2
Rq
rRg
R3
pRs — R2
R2
rRg
(31)
(32)
The process can be repeated to give any desired degree of accuracy. It is
probable that the values of R2 and R3 once obtained will be sufficiently accurate
for use in several successive divisions.
Numerical Illustration of the Use of the Above Method of Successive Approxima-
tion.—The factor given by (29), i.e. A2 + 1-325A + 2-18, is known to be an
approximate solution of equation (123). It is desired to find a factor which is a
more accurate solution. The slide-rule is here replaced by a calculating
machine on which both the multiplications and subtractions required are
* For real roots, Newton's and Homer's methods as described in English text-books are
available.
APPENDIX
559
carried out, and some of the steps in a division such as that in (26) do not appear
in the working, the calculating machine rendering the writing of them unnecessary.
The full working is given below : —
1 20-4 151-3 490 687 719 150 109 6-87
19075 U912 448-42 41702 99182 62163 9-166 3*634
123-85 284-32 40292 45-796 14844 7*199
.md continuing to a second division by the same factor —
17-750 121-67 245-62 -173-67 -206*17 71393
98-15 115-58 -326-81 +226-86 ...... (33)
The remainders are : —
Ro - 3-634 ; Ri = 7 199 ; Rg = 713*93 and Rg = 22686 . (34)
and from these it is found that
Rj Rg --4315, ; Ri />R3-R2 =5060, i Rs
and
Rr
rR.
i Rq R2 i
Using these values in (31) and (32)
R9
jfyRg— R2
8p = + 0-01060 and 8r = + 0-01243
and a new approximation to the quadratic factor is
A2 + 1-3356A + 219243 . .
=407,300
• (35)
. (36)
(37)
Repeating the process, keeping more figures in the calculation, the following
numbers are obtained : —
20-4
19 0644
From which
151-3
149-10757
123-64516
490
448-20264
283-06216
109
4-11022
-0-02973
687
415-91664
37-85882
6*87
0-07415
719
98-40603
47*84179
150
66-99719
3-09969
(38)
R(, = 007415 and Ri = — 0*02973 . . .
In the calculation of 8p and 8r the values of R2 and R3 will be taken to be
those in (151)
I Rj Rg = - 38*04 Ri ^Ra - R2 | =15*69
I Rq R2 -^0 *'-^8 I
and 8p=— 0*0000934, 8r = 0*0000385 . . . (39)
The new approximation to the quadratic factor is
A2 + r3355066A + 2*1924685 (40)
The degree of accuracy is now approaching that with which the calculating
machine can be used directly, and one further calculation brings the numerical
values correct to eight significant figures.
20*4 151-3 490 687 719
19-0644934 149-1075315 4482017 415-90833 98*3766
123*6467755 2830706 3786566 47-8068
50
109
6*87
66-98073
4-18518
-0*002266
3*13449
-0*000949
560
APPENDIX
Ko is now —0-002266 and Rj is —0-000949 as compared with the -3-634 and
7-199 of (34).
Rj Rg =-0-164 Ri PR3-R2 =-1-404
Ro
R2
and Sp
R(i
/R,
0-00000040, 8f=: -0-0000034
- • (41)
The quadratic factor is
A2 + 1-3355062A + 2-1924651 (42)
with an accuracy which probably extends to the last digit.
The method of approximation will be seen to correspond with Newton's
method of approximation to the value of real roots, and it is greatly assisted by
the use of a calculating machine. No counterpart of Horner's process for real
roots is known, the nearest approach to it being one described by Jelinek. All
such methods require special consideration in the case of repeated roots, but
such cases are not of sufficiently common occurrence to make a detailed discus-
sion necessary.
INDEX
Accelerated fluid motion, 112, 501, 507
Accelerometer, 83, 243
Actuator, 281, 282
Admiralty Airship Department, 7, 17
Advisory Committee for Aeronautics, 7, 75,
96, 116, 152, 192, 229, 232, 237, 364, 366,
377, 499
Aerial manoeuvres, 242-280
Verodynamic merit, 431
Aeroplanes, 9-13
models of, 101, 182, 231, 232
drag and speed, 23, 33
efficiency and gliding angle, 36
horsepower and speed, 27, 28, 31, 41, 44,
399
maximum speed, 27, 41, 44, 398, 416, 421,
433
performance of, 395-419
scale efifect on, 393
Aerofoil. See " Wings "
airscrew and, 298, 304-305
camber, 130-134, 304-305
contours of, 125, 129, 135, 160, 304
definitions, 118-120
dihedral angle on, 234
element theory, 271-274, 290-301
geometry of, 117
measurement of forces on, 97
Ailerons and wing flaps, 226-228, 230-231
Air-cooled engines, 14, 182
Airscrews, 281-342
aerofoil and, 298
airflow near, 286-290
bending moments in, 331
blade element theory, 291-295, 297, 302
body resistance and, 105, 177, 318, 404
centrifugal stresses in, 336
characteristics, 319-321
diameter : nomogram for, 319-320
drag at high speed, 33
effect of aeroplane on, 317-318
efiiciency of, 302, 310-311,315-320, 408,
434, 443
horsepower and speed, 25, 27
inclined, 322-331
inflow factors, 291, 294, 299
pitch, 24, 311, 435
revolutions and speed, 25, 26, 409, 415-
418, 424-430
slip stream, 178, 290, 291, 313-314, 405
tandem, 312
theory of, 290-303
thrust and torque, 23-26, 33, 292, 293,
301-309
variable pitch, 311-312
Airships, 5, 7, 15, 201, 241, 500
envelopes of, 5, 6, 100, 113, 201, 358, 369,
360
non-rigid, 5, 6, 16, 17, 64-66, 204-206,
358
pressure distribution on, 210
rigid, 15, 65, 206-209, 225
Altitudes, flight at, 42, 399-^02, 407-419,
420-445
American Advisory Committee for Aero-
nautics, 8, 311
Analysis of aeroplane performance, 434—446
resistance, 191
Aneroid barometer, 13, 81, 396, 426
height, 395-396, 425
Anemometers, 13, 74, 75, 77, 80, 286, 632
Angles, 19, 117, 118
dihedral, 118, 233-237, 275, 453
down wash, 193-197
gliding, 28, 36
incidence, 19, 118, 120, 121
pitch, 214, 237, 467
stagger, sweepback, 118
tailsetting, 48
Aspect ratio, 117, 135, 137
Atmosphere, standard, 395-396
Autorotation, 266-271, 274
Automatic stability, 450
Axes, body, 214-215, 217, 250, 273
change of, 237
forces along and moments about, 215
B
Balance, aerodynamic standard, 96-98
of an aeroplane, 46
of an airship, 65
Bank, 37, 87, 529
Bernoulli's equation, 281, 282, 352, 382
Biplane, 10, 139-151, 157-168, 182-192,
232
forces on separate planes of, 157
gap, stagger, angle of chords, 140-147
monoplane and triplane, 141
pressure distribution, 159
Bleriot, 4, 448
Body axes, 214-241, 250
drag or resistance, 21, 105, 175, 177
318, 404
forces and moments on model aeroplane,
233
Booth, Harris, 114
Bramwell, F. H., 75
British performance trials, reduction of, 430
Bryan, Prof. G. H., 4, 452
Busk, E., 5
661
2 0
662
INDEX.
c
Cables, struts and wires, 168
Camber, variation of, 124, 130-133, 148, 304-
305
Cameras, 91
Cave-Browne-Cave, Miss, 262, 494
Cavitation, 361
Ceiling and horsepower chart, 402
Centre of buoyancy, 66-67, 501
Centre of pressure, 45, 98, 452-464
Chanute, 3
Chattock, Prof., 76, 107, 108
Chord, definition of, 117
Climb, maximum rate of, 31, 400, 401, 417
Climbing flight, 28-31, 421, 468
Coeiiicient, centre of pressure, 119, 122, 144,
164, 168
drag and lift, 119, 121, 122, 123,
163, 167, 203
moment, 119, 122, 145, 154
thrust torque, 306, 316, 321
factors of thrust and torque, 321,
437, 439
viscosity, 369, 386
Compass, 13, 87
Compressibility, 381
Control stick, effects of movement of, 621-
622, 633
forces on, 63, 200
Convective equihbrium, 60
Corresponding speeds, 377
Critical velocities, 507
Crocco, Capt., 7, 509
Cyclic flow, 361-363
D
Darwin, Sir Horace, 87
Density, atmosphereic, 396-396
Derivatives. See " Resistance derivatives "
Discontinuous fluid motion, 364-368
Disturbed aeroplane motion, 617-550
Dive, 32, 245
Downwash, 47, 193-197
Drag, aeroplane, 94, 439, 442
airship, 64, 100, 206-208
body, 21, 168-190
seaplane, 67
Drzewiecki, 281
Durand, Dr., 311
Dynamical similarity, 372-394
aeronautical applica-
tions, 383, 385
corresponding speeds,
375, 377
principle of dimen-
sions, 379
E
Eddies, 346-348
Eiffel, 7, 96, 125
Elevator area, variation of, 198
hinge moment and effort to move,
62, 53, 200
motion of aeroplane due to, 524,
540, 543
Engine, 14-15, 181-182
power at height, 43, 409, 421, 426
weight, 403
Equations of motion, 251
Experimental mean pitch of airscrew, 309
F
Fabric of wing, sag of, 135
Fage, A„ 75, 303
Farman, H., 4, 448
Filament lines, 347
Fin shape and usefulness, 222-226
Flappiing flight, 8-9
Fligi i at altitudes, 42, 415-418, 424-430
controlled and uncontrolled in wind,
634, 649
circling, 262, 494
straight, 18-72
speed and airscrew revolutions, 409-
411, 416
Floats and flying-boat hulls, 54, 55
Flow of air near airscrew, 288-289
inclined plate, 379
inviscid fluid round cylinder, 366
water near inclined plate, 378
cylinder, 345-350
Fluid motion, 343-371
discontinuous, 364
elementary theory, 361
sources and sinks, 352
steady and unsteady, 344-
345
stream lines and stream fimc-
tion, 353
viscous and inviscid, 366, 368
Flying-boat hulls, 54-68, 110, 217-219
Form resistance and skin friction, 359-360
Formulas for aeroplane performance, 419-
, . 424
airscrews, 341
airship resistance, 65
stability derivatives, 239
Frictionless fluid, 351
Froude National Tank, 55, 79
Fronde's law, 110, 383
Fuhrmann, 109, 367-359
G
Gap, 118
biplane, 143
triplane, 162
Gas containers, 58, 62
General description of aircraft, 1-17
Geometrical similarity, 372
Greometry of wings, 117
Glide, angle of, 28, 35
spiral, 262
Glider drag, 404-442
Gottingen University, 7, 109
Graeffe, 651, 653
Gravitational attraction, 252
Greenhill, Sir George, 364, 365
Gusts and aeroplane motion, 450-550
Gyroscopic couples and flight, 499
INDEX.
563
H
Height, variation of engine power, iS, 409,
421, 426
l£ele-Shaw,343, 348, 349, 351, 355, 356, 371
Flelium, 58
llelmholtz, Von, 364, 366
Hill, G. I. R., 91
Hinge moment on elevator, 52-53, 199-200
Hydrodynamics, 351
I
Indicators, aerodynamic turn, 87
airspeed, spring-controlled, 13, 81
gravity-controlled, 88
revolution, 13, 83
Inflow factors for airscrews, 290-300
Inherent stability, 450
Instruments —
accelerometer, 83, 243
anemometers, 13, 74, 75, 77, 80, 286, 532
aneroid barometer, 13, 81, 396, 425
cameras, 91
levels, 73, 84
' manometer, 90
thermometer, 81
Italian airships, 7
J Olios, R., 499
Joukowsky, 362, 363
K
Kelvin, Lord, 366
Kew Observatory, 532
Kinematic viscosity, 385
Kirchhoff, 364, 366
Kite baUoons, 17, 66-72, 98, 211, 500, 512
Kutta, 362, 363, 364
Laboratories, aerodynamic, 5, 7, 8, 109, 357
Laboratory apparatus, 94-96, 108
Lamb, H., 351, 364, 366, 369
Lanchester, 467
Landing speed, 121, 123
wheel, 180
Langley, Prof., 3, 18, 38
Lateral stability, aeroplanes, 475-480
airships, 503-512
resistance derivatives, 481,
483, 512
speed and, 479
height and loading and, 484
Laws of corresponding speeds, 375-377
Level flight, 413-420
Lilienthal, 3
Logarithmic decrement, 104
Longitudinal stability, aeroplanes, 457-475
airships, 500-503,
505-512
resistance derivatives,
464-466, 512
Longitudinal stability — continued.
speed and, 463
height and loading
and, 469^74 ,
Looping, 243, 253-261
M
Manoeuvres, aerial, 242-248
Marshall, Miss, 303
Martlesham, 7, 91
Maxim, 3, 18
Maximum rate of cUmb, 400, 429
speed, 27-41, 44, 398
Methods of measurement in aerodynamics,
73-115
Model aeroplane, 101, 102, 182-184, 231
airship, 205, 207, 359, 392
body and modifications, 176
Monoplane, 11, 117-139
N
National Physical Laboratory, 5, 7, 94, 98,
100, 107, 108, 109, 130, 152, 168, 182, 286,
360, 366, 511
Nayler, J. L., 499
Newton, 18, 558, 560
Nomogram for airscrew diameter, 319
Non-rectilinear flight, 214-241, 243-280,
486-498
Non-rigid airships, 5, 6, 16, 17, 204
O
Observations, methods of representing, 118-
120, 216
Oscillations, lateral, 92, 495-498, 528
longitudinal, 92, 448-451, 467,
494-497, 523-524
unstable, 494
Parseval airship, 7
Particles, paths of fluid, 347
PeauceUier ceU, 372-373
Performance, aeroplane, 398-419, 423
airscrew, 319-321
airship, 64-65
Petrol and oil, 403
Photomanometer, 90
Pilcher, 3
Pitch, angle of, 218, 237
airscrew, 24, 309, 435
diameter ratio, 311
Pitching moment, aeroplane, 199, 256, 466
airship, 208
Pitot static-pressure head, 75, 177, 284
Poiseuille, 369
Porpoising, 55, 112
Potential temperature, 60
Prandtl, Prof., 7
Prediction of aeroplane performance, 398-
419
Pressure, atmospheric, 395, 396
2 0 2
664
INDEX.
Pressure distribution, flat plate, 3(J7
airship envelope, 210-
211
wings, 159-1 G8
gauges, 81, 107, 108
Principles of flight, 18-72
dimensions and similarity, 379
Pusher body, 179
R
Radiator and engine coohng, 181
R.A.F. 6 biplane, 140, 143, 144, 145
triplane, 152, 156
wing, shape of, 129
R.A.F. 15 wing, 125
R.A.F. 19 wing, 125
Rapid prediction of aeroplane performance,
398-402
Rayleigh, Lord, 18, 38, 364, 366, 380
Reduction of aeroplane performance, 418-432
Relf, E. F., 75
Resistance derivatives,
lateral, aeroplane, 239-241, 277-280, 481-
483
longitudinal, aeroplane, 239-241, 256,- 464-
466
circling, aeroplane, 490, 493
lateral, airship, 503, 505-512
longitudinal, airship, 501, 505-512
Resolution of forces and moments, 237
Revolution counters and indicators, 13, 83
Reynolds, Prof. Osborne, 370, 371
Riabouchinsky, D., 8
Rigid airships, 15 .
Roll, 215, 247
Rotary derivatives, 277
Royal Aeronautical Society, 117, 214
Royal aircraft factory, 5, 7,81, 83, 91,92,94
Rudders, 220, 223, 529
Runge, C, 551
S
Santos Dumont, 2, 448 ^
Scale effects and dynamical similarity, 372-
394
aeroplane, bodies and wings,
etc., 387-394
airship envelope, 392
airscrews, 393
Searle, Dr., 83
See, A., 4
Skin friction, 359-360
Slip stream of airscrew, 178, 314, 404
Slug, 119, 410
Soaring, 37-38
Sources and sinks, 351-363
Spiiming, 245
Spiral glide, 262, 265
Stability, 447-516
circling, 486
lateral aeroplane, 475
airship, 503
longitudinal aeroplane, 457
airship, 501
Stability, disturbed motion, 517-650
effect of gusts, 522-533
effect of controls, 526, 549
Stagger, angle of, 117, 139
biplane, 146
triplane, 155
Stalling speed, 90, 94, 123, 247, 447
Stanton, Dr., 303
Static pressure tube, 78, 177
problems and similarity, 383
Steady motion of fluids, 344
Stokes, Sir George, 350, 371
Stream function, 353, 354
Streamline body, definition, 349
wires, 173
Streamlines in eddy, 346
Stresses in airscrew blades, 331-337
Structure weight for aeroplanes, 403
Struts, drag of, 21, 168-174, 392
Submarine, 9
Sweepback, angle of, 118, 136
T
Tail plane, 48-51, 186
Tandem airscrews, 312
Tayler, D. W., 351, 358
Temperature, atmospheric, 397
Thrust of airscrew, 30, 33, 255, 285, 291-293,
318
coefficient factor, 321, 439^42
Torque, 292-293, 303
coefficient factor, 321-322, 437
Torres Quevado, 16
Triplane, gap and stagger, 152-155
monoplane and biplane, 139
Turning and spiral glide, 262
U
Uncontrolled flight in wind, 534
Undercarriages, drag, 21, 178, 180, 188
Unstable aeroplane, 447, 449
models, 455
Unsteady fluid motion, 345
Velocity, critical, of airship, 507
measurement of, 74
Viscosity, coefficient of, 368, 369, 385
Viscous fluid motion, 368-369
law of corresponding
speeds, 377
Vortex motion, 347
W
Water ballast, airship, 66
Water-cooled engine radiators, 181
Water resistance of flying-boat hull, 56, 1 10
Watts, H. C, 319
Wheels, landing, 180, 393
Whirling arm, 24, 105
Wind channels, 7, 94, 139, 330. 366
Wind, natural, 534, 549
INDEX.
565
Wings, biplane, 10, 139-151, 157-168, 182-
192 232
camber, 124, 130-132, 148, 304-305
downwash, 47, 193-197
drag. 22, 23, 118-170
gap, 118, 143, 152
lift, 19-23, 97, 118-170
moments, 46, 118-170
pressure distribution, 159-168
scale effect on, 387-391
section, 123
stagger, 117,139,146, 155
Triplane, 139, 151-155
variable camber, 148
Williams, 4
Wires and cables, 70-72, 168-174, 376
Wright Bros., 4, 448
Yaw, angle of, 215, 237
Yawed, forces and moments when, 218-236
Yawing moments, 221-225
Zeppelin, Count, 6
THE END
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