NAVY DEPARTMENT
THE DAVID W. TAYLOR MODEL BASIN
WASHINGTON Zi DG:
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ANALYSIS OF WAVE RESISTANCE ___
WHO?
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by
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Georg P. Weinblum, D. Eng.
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( ae September 1950 Report 710
INITIAL DISTRIBUTION
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1 Dr. J.H. McMillen, U.S. Naval Ordnance Laboratory, White Oak,
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1 Commanding Officer, Naval Torpedo Station, Design Section,
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4 U.S. Maritime Administration, Washington, D.C.
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1 Senior Naval Architect
1 Supervisor, Hydraulic Laboratory
2 Director, Experimental Towing Tank, Stevens Institute of Technology,
711 Hudson St., Hoboken, N.J.
1 Dr. Hunter Rouse, Director, Iowa Institute of Hydraulic Research,
State University of Iowa, Iowa City, Iowa
10
Dr. Robert T. Knapp, Director, Eydrodynamic Laboratory, California
Institute of Technology, Pasadena 4, Calif.
Dr. Lo.G. Straub, Director, St. Anthony Falls Hydraulic Laboratory,
University of Minnesota, Minneapolis 14, Minn.
Director, Experimental Naval Tank, Department of Naval Architec-
ture and Marine Engineering, University of Michigan, Ann Arbor,
Mich.
Dr. V.L. Streeter, Illinois Institute of Technology, 3300 Federal
Street, Chicago 16, Ill.
Head, Department of Naval Architecture and Marine Engineering,
Massachusetts Institute of Technology, Cambridge 39, Mass.
Director, Applied Physics Laboratory, Johns Hopkins University,
8621 Georgia Avenue, Silver Spring, Md.
Prof. W.S. Hamilton, Technological Institute, Northwestern
University, Evanston, Ill.
Prof. Garrett Birkhoff, Harvard University, Cambridge, Mass.
Prof. K.& Schoenherr, School of Engineering, Notre Dame Univer-
sity, South Bend, Ind.
Prof. W. Spannhake, Armour Research Foundation, 35 West 33rd
St., Chicago 16, Ill.
Dr. M.S. Plesset, California Institute of Technology,
Pasadena 4, Calif.
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Commodore H.A. Schade, USN, (Ret.) Department of Engineering
Research, 201 Building T-3, University of California, Berkeley 4,
Calif.
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PrP wD
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Canada
Prof. T.H. Havelock, 8 Westfield Drive, Gosforth, Newcastle-
on=Tyne 3, England
Mr. C. Wigley, 6-9 Charterhouse Square, London EC-1, England
Prof. G. Schnadel, Ferdinaudstr 58, Hamburg, West—Germany
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TABLE OF CONTENTS
ABSTRACT .
INTRODUCTION .
Wo
‘oO CO MN OO
THE REPRESENTATION OF SHIP FORMS :
1.1. The Geometry of Ships
1.1.1. Graphical Method . Ht till eredacrsh AIA
1.1.2. Description of Coefficients and Lines
1.1.3. Equations of Ship Surfaces .
RESISTANCE .
GENERATION OF BODIES BY SINGULARITIES AS BASIS FOR CALCULATING
WAVE RESISTANCE .... f
WAVE RESISTANCE Peete sts sa
4.1. Calculated and Measured Wave Resistance
WAVE RESISTANCE AS A FUNCTION OF THE SHIP FORM
5.1. Proportions and Shape Bie i mvetien Ree cccte haha Arash Merah
5.2. Wave Resistance as a Function of Principal Dimensions
5.2.1. Variation of Beam for Constant Draft H
5.2.2. The Effect of Variation of Draft for Constant Beam B
5.3. The Wave Resistance as a Function of the Hull Shape
5.3 1. General Remarks Ne Maesteg caret aegis
5.3.2. Longitudinal Distribution of Displacement
5.4. The Influence of the Vertical Distribution of Dashiaeenent
on the Wave Resistance :
5.4.1. The Influence of the CERES Section Coefficient .
5.4.2. Shape of Sections, Load Water Line and Sectional-
Area Curve . Been Vea Arca ater Ha Mie Me ea fen Joh Gh VIVE see fet
5.4.3. Bulbs and Cruiser Sterns .
5.4.4. Ships of Least Resistance
INFLUENCE OF VISCOSITY .
WAVE PHENOMENA DUE TO THE PROPELLER ACTION .
RESISTANCE IN RESTRICTED WATER .
EXTENSION OF THEORY
10. WHOLLY SUBMERGED BODIES .
SUMMARY
ACKNOWLEDGMENT .
APPENDIX 1 - THE EQUATION OF THE SHIP SURFACE
APPENDIX 2 - FORMULAS FOR THE WAVE RESISTANCE
A. The Wave Resistance of a Source or Sink
B. Michell's Integral
C. Sretensky's Formula for the Wave Resistance in Shallow Water
D. The Wave Resistance of Ships in Rectangular Canals
APPENDIX 3 - THE VISCOUS PRESSURE RESISTANCE .
REFERENCES
6
Page
74
76
80
80
89
89
91
ok
2p
95
a7
NOTATION
A Area
A(x) Sectional area curve
at(é) Dimensionless sectional area curve
A as index Afterbody
B Beam
C A constant, coefficients
E(7,H/L,F) Resistance function
F = Val Froude's number
F as index Forebody
G Resistance function
H Draft
I Resistance function
J Resistance function 1
ape Dimensionless moment of inertia J, = f &@ndé
K with sub- Bessel function of third order ‘
script
L Length
M Resistance function
P Resistance function
Q Resistance function
R Resistance
Ry Wave resistance
S Wetted surface 4
Se Dimensionless static moment Ss, = fénae
S(y) Resistance function i“
U Velocity in the X-direction
xX Axis
X(€) Dimensionless equation of water line
X(x) ; Equation of water line
Y Axis
Z Axis
Z(z) Equation of midship section
Z(¢) Dimensionless equation of midship section
; EHP x 427.1 V - knots
© . Resistance coefficient = ee A= $B.
a Longitudinal axis of a body of revolution
a Coefficients of polynomials
b = B/2 Half-beam, radius
b Width of a tank
c Velocity
e Eccentricity
e = x,/L Ratio of distance of CB from midsection
f Depth of immersion :
g Gravity acceleration
h Finite depth of water
k Integer, coefficient
k(x) Equation of centerplane contour
js bye Half-length
Integer, exponent
Intensity of doublet distribution
Variable of integration
Integer, especially exponent
Intensity of source-sink distribution (output)
Resistance coefficient
Taylor's tangent value
Speed
Coordinate
X Coordinate of a centroid
ted tab GP fe (ey ta} EP =} E}
oO
Coordinate
= y(X,z) Equation of surface
= y(x) Equation of water line
= y(z) Equation of section
Coordinate
Water line coefficient
= ¢ Midship section coefficient
Variable of integration
=A Block coefficient
Variable
Coordinate
~NOoee DBRNSG GY Se
iT]
Q
3
nT
> poftdfed |e
Coordinate Dimensionless
g => Coordinate
n = n(&,6) Dimensionless equation of surface
Dimensionless equation of water line
Dimensionless equation of section
3
nou
Sas
no
Symmetrical \ parts of 7
Asymmetrical
Wave length
Moment of doublet
Strength of source
Density
Prismatic coefficient
Ratio of slenderness
Weight displacement
Volume displacement
Resistance function
yyy rn ;
ANALYSIS OF WAVE RESISTANCE
by
Georg P. Weinblum, D.Eng.
ABSTRACT
The purpose of the report is to summarize the contributions made by hydro-
dynamics to solve problems of wave resistance of ships moving with a constant speed of
advance. A necessary condition for dealing with the problem is the introduction of mathe-
matically defined ship lines. The difficulties in determining the wave resistance experi-
mentally and theoretically are discussed. Endeavors are made to work out separately the
influence of proportions and the dimensionless shape on wave resistance; as a most im-
portant result, from the viewpoint of shivbuilding practice, the proof is obtained that under
certain conditions small continuous changes of longitudinal-displacement distribution in-
volve large cheupes in wave resistance. The influence of the vertical-displacement dis-
tribution and of special features such as the hull are investigated. Consideration is given
to the propeller action and to resistance in shallow water and in canals (basins). Wholly
submerged bodies of revolution are dealt with.
Appendices 1 and 2 contain a list of formulas used and Appendix 3 some remarks
on viscous pressure resistance.
INTRODUCTION
The problem of wave resistance is perhaps the most interesting if
not the most important problem in theoretical naval architecture. Our great
ship-research laboratories were founded and originally developed for the pur-
pose of investigating essentially this part of ship resistance, and a tre-
mendous amount of experimental data on models have been collected. Neverthe-
less, the long nourished aim to represent the resistance of a ship as a
function of her form has not been solved in a general manner by such experi-
mental methods, although systematic model tests have been by far the most
important means of improving the shape of hulls up to the present time.
Concurrently theoreticians, beginning with Lord Kelvin,2*” have
tried to solve the problem of the wave resistance of floating bodies by classi-
cal hydrodynamic methods. Starting with the simplest abstractions, which are
suitable to describe the phenomena only in the most general and tentative way,
they developed theories which are claimed to be useful for practical design
work. No clear conclusion has yet been reached as to how far this claim is
justified. Some scientists who have made important contributions to the sub-
ject hold the opinion that the existing theories give only a good qualitative
picture of the actual facts, while other authors claim that close quantitative
agreement can be reached.
In 1925, Professor Havelock delivering a lecture on "Some Aspects
of the Theory of Ship Waves and Resistance" ”’ quoted a paragraph from Kelvin's
famous paper on ship waves ("I made it a condition to the Council that no
practical results were to be expected from it. I explained that I could not
say one word to enlighten you on practical subjects...... ") and appropriated
this statement to his own work. Havelock could well afford to underestimate
the practical importance of his work because it is in fact his paper, to-
gether with a contemporary article by C. Wigley, ** that marks the beginning
of the application of theory to practical discussions on wave resistance The
lecture mentioned and Wigley's reviews on the subject are recommended for
study to all students of the subject. °*» 8°
The present paper is to be judged from the viewpoint of how far it
is able "to enlighten on practical subjects..;" its purpose is to show to
what extent theory has succeeded in furnishing valuable practical results and
how the scope of its applications can be extended. It is not a treatise on
hydrodynamics dealing with problems of free-surface effect; rather an attempt
to reconsider a basic phase of theoretical naval architecture in the light of
hydrodynamics. It bears by necessity a somewhat programmatic character be-
cause the tedious computations connected with the evaluation of the theoreti-
cal work have been initiated but not yet completed.
There is common agreement that theory has furnished a valuable de-
scription of general phenomena; it is less well known that it also has given
us the proof of considerable practical value of how sensitive wave resistance
can be to changes, even small changes, in ship form.
Experienced experimenters are often somewhat bewildered by the fact
that the wave resistance may vary appreciably for different but reasonable
types of lines, although all the form parameters generally considered as de-
cisive are identical. From a theoretical viewpoint this appears to be quite
*References are listed on page 97.
natural, since the wave resistance depends to a first approximation upon a
complicated function of the surface slope in the longitudinal direction, i.e.,
on derivatives. On the other hand, the most commonly used coefficients are
integrals, which even when kept constant still admit of very wide variations
of the slopes. We realize now why the solution of the: basic problem of the
model tanks mentioned above—to establish the resistance as a function of the
form—remains almost hopeless as long as the ship surfaces (or at least their
most important features) are not defined in a rigorous way by mathematical
expressions. Hence, our first task must be to find equations for the ship
surface, continuing the work of D.W. Taylor.*5 This phase of Taylor's re-
search has been more or less neglected by later investigators.
The well known resistance phenomena are briefly reviewed herein, and
the formal procedures are set out by which hydrodynamics leads to a computa-
tion of wave resistance.*
The main purpose of the report is to establish simple functional re-
lations between resistance and form of "normal" ships, based on Michell's
theory and on a comparison of calculated and measured resistance values. This
includes a discussion of the effects of viscosity and of some proposals in-
tended to improve the theoretical procedure.
' It was further thought useful to mention the interaction between
ship and propeller and to study the effects of restricted water, especially
with respect to the influence of the finite dimensions of model basins, on
model wave resistance.
Finally a synopsis of current knowledge regarding the wave resist-
ance of wholly submerged bodies is included.
Wave resistance phenomena in a seaway have not been treated to avoid
making the report too long, although theory has recently made valuable contri-
butions to this thrilling subject. For the same reason the problem of nonuni-
form motion has been omitted.
When emphasizing the importance of theoretical work, it is necessary
to state that the theory is based on many abstractions and that the combina-
tion of theoretical and experimental work is a necessary condition for reach-
ing useful practical results.
*The subject of the report is the analysis of wave resistance. Therefore for brevity "resistance"
is often used instead of "wave resistances" when other "kinds" of resistance are discussed the fact will
be clearly stated.
Mm
1. THE REPRESENTATION OF SHIP FORMS
1.1. THE GEOMETRY OF SHIPS
Any treatise on theoretical naval architecture should begin with
(or at least include) a chapter on the representation of ship forms. This
could be called "The Geometry of Ships," although this terminology has some-
times been used for discussions of problems of statical stability.
1.1.1. Graphical Method
The generally used graphical method of representing ship surfaces
by "fairing" the lines is efficient from some viewpoints, as it leads within
a reasonable time to solutions which comply with the necessary conditions of
buoyancy, stability, etc. The resulting surfaces (lines) have a high degree
of smoothness within the practical degree of accuracy required when "spline
curves" (battens) are used. A curve is called "smooth" when the first deriv-
ative is continuous (the curve itself has no corners); we define the "order
of smoothness" of a curve as the order of the highest derivative which is con-
tinuous. A curve in which there is a continuous radius of curvature (a con-
tinuous second derivative) is smooth to the second order. Spline curves drawn
in the proper way should generally be smooth to a still higher order. This
can be easily understood since the curvature of the elastic line of a batten
is proportional to the bending moment; the graph of the bending moment remains
continuous, even if we apply to the batten horizontal concentrated loads by
weights—a procedure contrary to the idea of fairing under normal conditions.
On the other hand, discontinuities in the curvature of ship lines are admitted,
for instance, when using a combination of a straight line and a circular arc
for sections.
These points will be considered later;* at the present it is suffi-
cient to state that a definite order of smoothness may be a necessary or de-
sirable condition for a ship line, but it is not a sufficient condition as is
weil known from experiments and will be proved by theory. It is the purpose
of resistance research to develop criteria for good ship lines—lines deter-
mined by minimum resistance qualities. Because of the lack of rigorous re-
sults, however, earlier practice—guided by experience and other considera-
tions, some of arbitrary character—has introduced a working concept of "fair-
ness of lines," with which hulls should comply.
*See Appendix 4, where it is shown that modern hydrodynamics supports the wisdom of artisanship in
shipbuilding.
It is not easy to give a reasonable definition of the real meaning
of "fairness." As applied to a curve some features characterizing this prop-
erty are: (a) Conditions of monotonic increase or decrease, (b) avoidance of
flat parts, (c) lack of abrupt change in the curve and its first and second
derivative, and (d) the possession of a finite number of points of inflection.
The basis of the idea of fairness is the fact that a trained human
eye is very sensitive to any peculiarities of curves. It will be proved later
that fairness has not given us reliable indications as to the resistance qual-
ities of forms, although the concept can be useful from resistance and struc~
tural viewpoints. The well known resistance qualities of bulbs and "swan-
necks" furnish a good illustration of how far the idea of fairness must be mod-
ified in the light of present knowledge.
The disadvantages of the graphical method of ship design are; (a)
The necessity of laying off to a large scale, and (b) the unsuitability of
graphical representations for establishing general laws for wave resistance,
behavior in a seaway, and so on.
_ 1.1.2. Description by Coefficients and Lines
The "Explanatory Notes for Project Number 2"°7 issued by The Society
.; of Naval Architects and Marine Engineers contain a consistent system of form
parameters and lines which are assumed to be known.
Use is made of dimensionless representation: Dimension factors are
separated from the pure form or shape defined as the ratio of hull ordinates
y to half-beam B/2, etc. The dimensions can be expressed by the absolute
length L and two ratios as L:B,B:H; the shape can be approximately described
by suitable integral or differential properties (coefficients of fineness,
tangents, curvatures). Thus two sets of important parameters are obtained
which, within a certain range, can be treated independently, as will be
demonstrated. In the light of this tendency there does not seem to be any
advantage in the use of such expressions as Sa where the pure form con-
stants and the parameters derived from dimensions are mixed, except for a
first orientation; for practical work the use of ratios, like L/B, B/H, and
CR = § (which are all known) is preferable.* Our purpose is to approximate
the ship form by as many characteristic parameters as possible, not to merge
several known parameters into one.
*As a matter of courtesy, symbols like Cae » proposed by the Sub-committee of the SNAME, are men-
tioned here beside the well established Greek jettors which are used throughout the report. For actual
work Greek letters have a definite advantage over the C coefficients by reducing the prohibitive* number
of subscripts.
It is definitely advantageous to use fore and afterbody coefficients
like
pon peal iH
Ca = 6 =5 (Cop + Coa) = (6p + 4) [1]
The differences (Cop - BA)? (Cr - Coa) can be connected to a good approxi-
mation with the ratio *o, where X5 denotes the distance of the L.C.B. from the
midships section. (It is an offense against the spirit of approximate calcu-
lus to refer the longitudinal position of the CB to the fore or aft perpen-
dicular.) The normal ship form consists of a main part symmetrical with re-
spect to the midship section roughly characterized by CR: Cp and an asym-
metrical part described by
SBR BA oS BN 5 [2]
6 L
B
where k is a factor depending on the fineness and the form of the sectional-
area curve. Values of coefficients depending on the longitudinal distribution
of volume, etc., are useless if the reference lengths (Low, Lop ) are not
clearly stated. Where Lowy # Lpp» both values like Coy (5) and Capp) should
be given.
The basic form coefficients have been developed with respect to prob-
lems of buoyancy and stability; it is a furtunate coincidence that they are
characteristic for resistance problems too. Parameters introduced from hydro-
dynamic considerations are:
a. t=etany=¢ |B , (Taylor's tangent value) [3]
Re Ss
2
b. Length of parallel body.
c. Position of the point of inflection.
d. Curvature at the midship section.
e. Position of the centroid of the forward and aft parts of a
waterline (sectional-area curve).
f. Bulb-area ratio f defined by Taylor as the ratio of intercept of
the sectional-area curve at the bow to the maximum ordinate.
Items c to e have not been much used. The list may be increased by using:
g. Moments of inertia, or
h. Higher moments with respect to suitable axes.
However, graphical procedures become cumbersome when all these coefficients
have to be considered, so that it is preferable to introduce mathematically
defined ship lines.
A great success has been achieved in naval architecture by the use
of integral curves of which the sectional-area curve is perhaps the most
important. : ‘
1.1.3. Equations of Ship Surfaces
Quite a number of attempts have been made to define ship lines by
mathematical equations, with different purposes in view. There is no need to
dwell on Chapman's parabolas, which can be found in every handbook. Obviously
they are almost impractical, as they contain only one parameter, i.e., their
shape is fixed by the area coefficient alone. One should therefore eliminate
them as a method for design, notwithstanding the fact that they have formed
the starting point for all further work on the exact representation of ship
form.
According to D.W. Taylor's own statement,*> he developed "Mathemat-
ical formulae not with the idea that they give lines of least resistance but
simply to obtain lines possessing desired shape." This statement is important;
contrary to some attempts to ascribe magic properties to certain analytically
defined curves like trochoids, sine curves, etc., the principle of systemi-
zation is put forward as the decisive argument for their adoption. Formulas
are given for waterlines and sections separately, no parallel body having been
apparently envisaged, although there is no difficulty in inserting this fea-
ture. Fore and afterbodies are treated separately; the origin is situated at
the bow or stern, respectively. The family of curves intended to represent
waterlines and sectional-area curves is given by a fifth-degree polynomial
with three arbitrary parameters; the discussion of the properties of this fam-
ily is performed in classical style and is perhaps the most refined contribu-
tion to our problem. The lines are perfectly suitable for practical use.
Unfortunately, Taylor's work is not very suitable for theoretical
investigations since the equations of waterlines and sections are not linked
together into equations of surfaces and fore and afterbody are treated separ-
ately. Nevertheless, it is a great loss to the profession that his basic pub-
lication*® has remained almost unknown; otherwise model research would have
obtained a more systematic character, as earlier stated. 3
A system of ship-hull equations has been developed for broader pur-
poses?°S,1°* in which expressions were sought which would:
a. Be suitable for fixing and systematizing ship forms.
b. Give a basis for investigating systematically wave-resistance qual-
ities of ships and eventually lead to forms of least (low) resistance.
c. Forma more general foundation for research on other problems in
naval architecture such as motions in a seaway, stability, etc.
d. Reduce the work in the mold loft. (The author has made some contri-
butions to this idea without being able to claim much practical success. )
The formal development is briefly reviewed in Appendix 1. Although the
analytical approximation of a given hull may be a tedious problem, for some
scientific purposes quite simple expressions prove to be valuable.
We locate the origin at the midship section, and the planes of ref-
erence are given by the vertical-center plane (X,Z), the load-waterline plane
(X,Y) and the midship-section plane (Y,Z); see Figure 1. In dealing with re-
sistance problems to a first approximation, we confine ourselves to the under-
water part of the hull.
Figure 1 - Axes of Reference
We denote as an "elementary" hull a form defined by a rectangular
longitudinal contour and an equation of the type
y = B x(x) 2(z) [4]
where X(x) and Z(z) are the dimensionless equations of the load waterline and
midship section. It can be easily seen that for such hulls:
a. Ch = 6 = Cy xCy = ab [5]
b. The sectional-area curve is affine to the load waterline (their
equations differ only by a constant factor)
Cp=¢=Cy-@ [6]
ec. The sections are affine to the midship section and the waterlines
to the load waterline.
It is astonishing that such an elementary equation leads to quite
reasonable ship forms as long as the Cry = 6 relationship is low; it is appro-
priate for investigating the influence on wave resistance of the longitudinal
and, within certain limits, of the vertical distribution of displacement. By
introducing additional functions we are able to represent changes in different
sections (for instance, inclination to the vertical, variation of fineness,
etc.) and waterlines.*
Analytical representation of ship forms, developed for the purpose
of eliminating the mold loft, meets with two essential difficulties:
1. The longitudinal profile is generally not a rectangle but a curve,
often with corners and discontinuities. This complication probably can be
overcome by a method already proposed.?°?
2. The representation of full sections. Here the use of high-degree
parabolas leads to regions with high curvature, which are detrimental from a
hydrodynamic as well as a practical viewpoint. Some improvements can be made,
but as long as this difficulty is not overcome, it is not suggested that the
formulas be applied for construction. However, readers interested in the prob-
lem may refer to a paper by Childsky,° where a criticism of the present meth-
ods and indications of some further development may be found.
An important question arises: Should the usual method of ship de-
sign, based on the sectional-area curve A(x), design waterline X(x), longi-
tudinal contour C(z) and midship section Z(z), be altered when using analyti-
cal expressions? Within the present range of application there seems to be
no need: for a radical change. Of course, the polynomial A(x) must have a suf-
ficiently high degree to comply with all functions of x representing the
ship's surface, and a further difficulty arises when the longitudinal contour
departs from a rectangle, because then the equation of the hull cannot be
easily expressed by a polynomial.
The use of algebraically defined surfaces enables us to fit ship
forms in a rigorous manner, and especially to describe in a definite way
changes and variations in the forms. In applying the equations to problems of
resistance, motion in a seaway, etc., we expect to deduce results capable of
generalization and to establish parameters which are characteristic for the
problems involved. An interesting, if not very important, application of
mathematical ship lines is the development of reliable approximate formulas
for ship design (position of centroids, moments of inertia) as already indi-
cated*°* and extended by Sparks.°°
*Examples of waterlines and sectional-area curves are given in Figures 15, 16, and 17 on pages 55, 36,
and }1,
10
2. RESISTANCE
When analyzing the wave resistance, some remarks on ship resistance
in general are necessary. We ignore the air resistance, which may be quite an
important item but is outside the scope of these considerations.
Froude developed his well known method of using model experiments by
starting from the idea that it is comparatively easy to calculate the fric-
tional drag and impossible to compute the wave resistance. He tacitly assumed
that the different "components" of resistance can be superposed with reason-
able accuracy, i.e., that their mutual influence is relatively the same for
the model and the ship. The "eddy" or "separation" resistance, although ob-
viously due to viscosity, is merged into the residual resistance and its co-
efficient considered to be independent of scale effect; it was assumed to be
small for good ship forms.
These assumptions have led to a practical technique used all over
the world. From the viewpoint of hydrodynamics, however, matters are somewhat
different. Here the frictional resistance due to viscosity effects is in prin-
ciple a more complex problem (involving the solution of Navier-Stokes equa-
tions) than the wave making force which arises in an ideal fluid. The gener-
ation of surface waves in a viscous medium is a problem which has not yet been
investigated by theory, so that the hypothesis of the mutual independence of
the resistance components from scale effect (model and ship) cannot be quan-
titatively discussed in the light of rational mechanics.
The "eddy" resistance due to viscosity is a function of some appro-
priate Reynolds number. From the reasoning given in Appendix 3, a sericus
scale effect can be expected when model data are converted to the ship, al-
though it is assumed that the character of separation does not change when
full turbulence has been reached over the model surface. In aerodynamics,
contrary to Froude's procedure, frictional and "eddy" resistance are treated
together as viscous drag.
These undetailed remarks indicate now difficult it is to analyze
rigorously the wave resistance of the ship using model experiments. For most
types of ships wave resistance is only a small part of the total resistance
and as the amount of viscous drag is known only to a relatively low order of
accuracy, the wave-resistance values obtained by subtraction are liable to be
quite erroneous.
To put things on a more rigorous foundation we define the following
kinds of resistance:
a. Frictional resistance is the tangential resistance due to viscous
forces. It differs in principle from the resistance of the equivalent plate
having the same wetted surface, this difference being denoted by frictional
11
form resistance. So long as no separation occurs, a lower limit for this
frictional form resistance can be estimated from Millikan's formula for bodies
of revolution,°® which gives an increase of the order of 4 to 6 percent over
the corresponding plate values. A further effect is the change in the wetted
surface due to the changed attitude of the model and to the wave flow. The
latter is important for high Froude numbers only; ®* it becomes decisive for
vessels with large hydrodynamic lift.
b. Viscous normal or pressure resistance; This resistance is caused
by the pressure changes due to viscosity, especially in the afterbody. It
includes separation effects and loss of pressure due to finite boundary thick-
ness. This definition is broader than "separation" or "eddy" resistance; the
addition "viscous" becomes necessary to avoid confusion with wave resistance
which can also be derived from pressure measurements. Frictional and normal
resistances—a and b above—can be separated by pressure measurements when
wave making is absent or negligible.
Throughout this report the denotation "eddy resistance" will be used
in the same sense as viscous normal resistance. Sometimes the sum of friction-
al-form resistance plus viscous-pressure resistance is referred to as viscous-
form resistance; the adjective viscous should not be dropped . *®
ec. Wave resistance: Items a and b give the total viscous drag. Wave
resistance can be computed as the difference between the total normal and the
viscous normal resistance, or between the total resistance and the total vis-
cous drag. Surface models and wholly submerged double models have been used
for this purpose under the assumption that for the same Reynolds number the
wave flow does not influence a and b.7’ 1*°°8 One can dispense with the dou-
ple models if pressure measurements on large simple models admitting of high
Reynolds numbers are made at low Froude numbers giving a negligible wave pat-
tern. A more direct approach would be to calculate the resistance from actual
wave patterns (stereo-photogrammetric pictures) which, however, never has been
successfully tried. Most of these methods are suitable for research work
only.
d. Spray or jet resistance: This becomes important for very-high-
speed length ratios, such as are found in gliders and seaplanes. It must be
listed here as a surface effect sui generis which is not described by the the-
ory of surface waves; in fact its generation does not depend on gravity. In
normal vessels, sprays can be prevented by avoiding blunt surfaces normal to
the direction of advance.
WZ
e. Thrust deduction or suction force due to the propeller: Generally
this force is not considered when dealing with the resistance, since it de-
pends on the interaction between propeller and hull. However, it seems to be
appropriate to mention it, as it influences decisively the form of the after-
body, especially for single-screw ships, by requiring the creation of a wake
axially symmetrical with respect to the propeller shaft.
f. Resistance in a seaway: The increase of resistance in a seaway is
mainly due to the oscillatory motions (pitching and heaving),®°? and, judging
from results for a fixed ship/* the reflection of waves is responsible for
only a small part of the resistance increase. The damping of oscillations is
primarily caused by wave making.
Older experiments led to the belief that ships with good resistance
qualities in calm water keep their superiority in regular waves. There are,
however, some indications that ships with a steep rise of the calm-water re-
sistance curve near the service speed are liable to high resistance in rough
water. This remark applies especially to hulls with high prismatic coeffi-
cients.
Summarizing the state of knowledge on some problems of ship resist-
ance—so far as it is needed for analyzing wave resistance—we may say:
a. The residual resistance derived by Froude's method is highly arti-
ficial, including as it does items which do not follow Froude's law of simili-
tude. For small Froude numbers the "residual" resistance may not yield even
an approximate estimate of wave resistance.
b. No reliable results are known for the viscous pressure resistance;
2, 14 From
even contradictory statements are to be found as to its range.
measurements on bodies of revolution fully submerged, we can guess that the
total viscous drag of models up to moderate prismatics corresponds closely to
the frictional drag including the frictional form effect.* —
¢. An accurate estimate of the total resistance of slow ships depends
more on the reliable computation of frictional resistance including the rough-
ness and form effect than on accurate values of the residual resistance de-
rived from model results. Thus the routine towing experiments would not seem
to be indispensable for this most important class of ships as long as the com-
putation of absolute value of resistance is the main problem; nevertheless,
*Special investigations should be made as to the dependence of the eddy resistance (viscous pressure
resistance) upon the form of the ship, especially the afterbody (see Appendix 53). Various rules have
been made for the avoidance of appreciable eddymaking; 4 unfortunately they are more a matter of opinion
than of real knowledge. Some deductions are derived from inadequate experiments, for instance as to
bodies of revolution where the influence of the free surface was not eliminated, etc.
13
tests are valuable from the viewpoint of obtaining reliable comparative re-
sults for various forms. Accurate absolute values of wave resistance cannot
be obtained by Froude's method for low speeds, but we can hope to obtain (by
using great care) consistent differences in wave resistance corresponding to
definite changes in form; much is achieved when we can rely on the sign of the
result. Unfortunately, the same doubts as to the absolute values apply to the
calculated wave resistance in this range. Our main task will be, therefore,
to establish whether changes in the calculated wave resistance do correspond
to the experimental values with respect to the general trend and range, not so
Much with respect to the absolute values, although the agreement in the latter
sometimes is also very satisfactory.
3. GENERATION OF BODIES BY SINGULARITIES AS BASIS FOR
CALCULATING WAVE RESISTANCE
The representation of ship forms by algebraic expressions enables
us to calculate the wave resistance comparatively easily for the limiting case
of a slender body, the so-called "Michell's ship." The assumption is made
that the slope of the hull to the vertical center plane expressed by oy, Sy.
is everywhere small.
While the condition for the longitudinal slope $¥ is not unreason-
able for most normal ship forms, the condition = is very seldom satisfied.
Let us investigate, at least superficially, the assumptions on which the the-
ory is based.
Lord Kelvin2* substituted for the ship a pressure concentrated at
a@ point and applied to the surface which he designated by "forcive." He suc-
ceeded in explaining the main features of the wave pattern created by a ship.
His work was widely extended by E. Hogner’® who gave a general formula for the
wave resistance of pressure systems, which enables us to calculate the resist-
ance of bodies with negligible draft, especially gliding craft, when the pres-
sure distribution is known.
For normal ship forms the approach by the usual singularities—
sources, sinks and doublets—has proved to be more appropriate. The genera-
tion of bodies by sources and sinks (doublets), proposed by Rankine,°° should
‘always form a basic study in theoretical naval architecture. Before discus-
sing bodies floating on the surface, we will investigate wholly submerged bod-
ies in an unbounded fluid (the free surface is very far away from the body).
It is well known that under appropriate conditions, closed slender
bodies of revolution are generated by sources and sinks (doublets) located
14
along a straight line in a uniform stream. Weinig's important approximation
holds: That the intensity of the doublet distribution is roughly proportional
to the sectional-area curve of the body.*°
We now distribute surface singularities over an elongated region of
a vertical plane (Figure 2). Again in a horizontal uniform stream a closed
surface is created, if the integral of the strength of the source and sink
system, taken over the given region of the plane, is zero. One can prove that,
within certain limitations, the surface ordinate ve of the resulting body at
a point Xo Zo? is roughly proportional to the doublet strength at the point,
and that the sectional area curve A(x) is approximately proportional to the
Figure 2 - Generation of Bodies by Singularities
integral of the doublet intensity taken over z. Under the same conditions the
strength of the sources and sinks is proportional to the slope of the surface
in the horizontal direction.
Let the distribution over the plane be symmetrical with respect to
the x-axis, then the resulting body is also symmetrical with respect to the
xy plane. Within reasonable assumptions as to its character we can assert
that the maximum beam 2b is smaller than the height 2H. This is easily under-
stood in the limiting case of a body of revolution when the singularities are
concentrated on the axis; by definition 2b = 2H. By displacing an element of
the distribution away from the x-axis in the direction of the z-axis we ob-
viously increase the vertical dimension of the resulting body.
To generate bodies characterized by 2b > 2H one must obviously dis-
tribute singularities in a horizontal (x,y) plane. Some quantitative rela-
tions may be estimated from the known results for a general ellipsoid; Figure
3. Assuming a >c>b, i.e., the vertical axis greater than the horizontal
axis in the y direction, it has been proved that the ellipsoid is generated by
a doublet system M(x,z), which is distributed over the focal ellipse in the
xz plane
ee eee [7]
with the moment
[8]
where C is constant. The ellipsoid is a simple example of a "double" model.
This somewhat lengthy reasoning on submerged bodies has been made
with the purpose of obtaining an approximation for surface ships. It is as-
sumed that the underwater part of the ship is one half of the "doubled" sub-
merged body and is generated by half the corresponding singularity distribu-
tion, an assumption which is rigorous only when the wave making can be ne-
glected. The approximate relations between the body form and the image system
are also applicable to the ship under this condition.
A closer approximation can be obtained by distributing singularities
over the surface. As no useful practical results have been obtained in this
way we shall not pursue this method at present.
"Michell's ship" corresponds to an image system distributed over a
vertical plane. Actually ships with b = £ >H should be approximated in an-
other way, which has been initiated by Havelock” and used by Lunde. °! Never-
theless, our reasoning will be based on Michell's formula®= which is valid
under the following conditions:
_a. Ideal fluid.
b. Small slopes of the surface.
c. small wave aianee.
d. No change of model attitude.
16
In using this reasoning, we can expect good agreement between theory and ex-
periments for slender bodies only and depend upon experimental checks and cor-
rections for ships of normal proportions.
Originally Michell developed his resistance formula by computing the
pressure exerted on the body.°* Other methods are based on the computation
of the energy of the wave system caused by the motion of a body and on the
dissipation of energy calculated by means of an artificially introduced van-
ishing viscosity term introduced by Havelock.?° °° A fourth approach used
by Havelock’? is the so-called method of singularities, which we shall review
briefly because of its advantages for calculating forces when the image sys-
tems involved are known.
The method of singularities may be simply explained as the law of
attraction applied to sources and sinks: Two point sources (or sinks) with
the output Q,Q, (-Q,, -Q,), attract each other with a force
Q,Q
oS fas [9]
where r is the distance between the two singularities, while a source and a
sink experience a repulsion of the same absolute value.°* It is quite aston-
ishing that no broader use has been made of this formula in hydrodynamics,
which when applied to electricity is familiar to any student.*
The formula [9] can be rewritten and generalized to give the force
experienced by a source Q due to the velocity v of the stream at the location
of the source
K = -pQv [10]
where v can vary throughout space, but is steady at any given point. The mi-
nus Sign indicates that a source is pulled by the stream in the opposite di-
rection of v. (This equation [10] known as Lagally's formula is as important
as Kutta-Joukovsky's formula for a flow with circulation. >:27)
When the velocity potential corresponding to a source-sink distri-
bution is known, the horizontal velocity is also known, and the resistance X
can be written down as the integral of the product of the distribution and
the horizontal velocity over the region of the distribution. The method can
be generalized for calculating the mutual interaction between bodies (ships)
advancing with constant speed in the same direction in tandem or for any other
arrangement. The influence of fixed walls can be treated as a special case
of this problem.
*Note the difference in the sign of the force due to charges of the same kind, when dealing with
electric and with hydrodynamic phenomena.
17
A most important application was made by Dickmann to the problem of
ship and propeller interaction.°®’?°»1++ By replacing the latter (only when
considering the mutual interaction of the system) by a sink or a sink distri-
bution a comprehensive theory of thrust-deduction phenomena was developed.
Further interesting results can be obtained by applying this method
to the calculation of forces and moments due to the motion of a wholly but not
too deeply submerged body moving steadily parallel to the free surface; in
principle this solution is given when the vertical velocity at any point of
the image system representing the body is known.
4, WAVE RESISTANCE
Many curves representing wave resistance as a function of Froude
number deduced from theory and experiment have been published; they are char-
acterized by "humps" and "hollows" over definite ranges of speed-length ratios
due to interference effects of different wave systems. The experimental curves
reveal a smaller fluctuation than the theoretical because of the influence of
viscous forces.
Various attempts have been made to calculate the wave resistance
from the wave profiles; the most valuable proposal is due to R. Guilloton.*’
A great difficulty in using wave profiles for resistance computation is the
loss in accuracy due to difference errors. Tnis difficulty is increased by
the fact that the vertical pressure distribution due to waves does not follow
closely the simple exponential law. We shall consider here the wave patterns
only with the purpose of obtaining some general ideas about interference ef-
fects. The basic work on this subject is due to Wigley yen S wand
Havelock. °” °*’®
A distinction must be made between a "wave" and "non-wave" portion
of the profile; the latter does not contribute anything to the resistance in
an ideal fluid and is, therefore, less important from the present viewpoint.
Following Wigley we consider a cylinder (vessel of infinite depth) whose water-
line is a parabola with some parallel middle body.
The wave part of the profile consists of:
a. Wave systems due to the finite angle of entrance and run (the sys-
tems at bow and stern are identical when the angles are equal).
b. Systems due to the curved parts of entrance and run.
e. systems due to the shoulders if these are pronounced.
However, the definition of such systems is made to a great extent by consider-
ations of easy integration, so that different interpretations and divisions
are possible.
18
The resistance curve can be split up into a monotonically increas-
ing part and fluctuating components. Neglecting the parallel middle body, we
obtain the resistance of the parabolic cylinder as the algebraic sum of 5
terms, due to:
a. Bow and stern patterns (as if each existed alone).
b. Curved sides (entrance and run).
ce. Interference of bow and stern.
d. imrenterenee of bow or stern with entrance or run.
e. Interference of entrance and run.
Patterns a and b are not oscillatory, being proportional respective-
ly to the 6th and 8th power of Froude's number F. For low values of F the
finite angle at bow and stern is more important, but with increasing F the
second term gains in value. Terms c, d, and e give fluctuating resistance
curves.
The influence of the various terms depends on the speed and the form
of the ship. Results obtained by Wigley for a prismatic pile with a trape-
zoidal half waterline differ widely from those corresponding to the parabolic
waterline. The very pronounced shoulders (corners) in the former cause a
strong interference effect between the bow and the shoulder system, while the
influence of the shoulders is of secondary importance for the parabolic lines
with parallel middle body.*°
Many discussions have been devoted to the length of separation on
wave-making distance. Even the definition of this concept is not unique.
Following D.W. Taylor,*? the most pronounced interference effects are due to
the first crest just abaft the bow and the first crest of the sternwave sys-
tem somewhat abaft the stern; hence, the distance between these crests may be
considered as length of separation. On the other hand, in the opinion of the
Froudes, this length should be defined as the distance between the bow crest
and the trough caused by the after-shoulder. In the light of the preceding
remarks and more detailed investigations by Wigley and Havelock, we quote the
latter:
"Although simple empirical formulae for so-called wave-making dis-
tance may be of some use it is doubtful whether they are worth the time in
inventing them, or in proving or disproving them..... 11 90
*When there are no corners in the waterline—as nearly always in actual practice—the concept of
shoulder wave system becomes somewhat arbitrary.
19
Thus, it may be recommended to drop entirely the concept of wave-
making length and to estimate the regions of high and low wave making, as
functions of Froude number, from suitable experimental or theoretical curves.
Particulary, it should be emphasized that the so-called ®)"theory," based on
similar reasoning and used to prophesy the positions of humps and hollows on
the resistance curve, is at best an interpolation formula valid for a limited
range of ship forms and Froude numbers, ”° and that the use of
©-Venxv [11]
as abscissa when plotting resistance curves should be abandoned.
A great deal of confusion has been caused by substituting for the
ship simple-pressure systems. Commonly, a positive and a negative pressure
system are assumed to illustrate the action of the bow and the stern. This
choice is based on the doubtful assumption that bow- and stern-wave patterns
are similar in character except that crests and troughs are interchanged. The
argument is not consistent, however, since the wave profiles due to the finite
angle of stern and bow have the same sign, i.e., they both start with crests.
While two positive pressure systems are sometimes substituted for
the ship, this also leads to an erroneous concept as the contributions of the
curved sides of the hull to the wave pattern generated at the bow and stern
have opposite signs. This is clearly brought out by Havelock®’ and Wigley, ”°
whose work should be consulted by anybody interested in the subject.
4.1. CALCULATED AND MEASURED WAVE RESISTANCE
Various papers have been written with a similar purpose as the pres-
ent one: a. To give a synopsis of the more important results obtained by the
evaluation of existing theories, b. to describe the results of comparisons be-
tween theory and experiments, and c, to decide how far theory is able to help
in the solution of practical problems. We begin with a synopsis of a recent
publication by Giers and Sretensky" ° which represents the most general attempt
to answer the three points mentioned although we do not agree with the basic
results reached.
Figures 4 and 5 have been recalculated from this paper.13* They rep-
resent values of the coefficient r = mt as a function of F, where R is the
wave resistance calculated by Michell's integral for a family of elementary
ships:
n= X(&) Z (¢) [12]
*The accuracy of the curves, Figures 4 and 5, may be sufficient for qualitative estimates only.
20
The dimensionless form depends upon a single parameter Cp =, which appears
as the parameter of the resistance curves r =r(F). For all curves: L = const.
B/H = 3, B = 0.95. Two displacements, characterized by w = (@) = 8,(Figure 4)
and w =4(Figure 5), are investigated.
0.07
50
920
0.06 w=80
yo
0.05 4-02
62
622 40
2950.18
R 004 ¢
iy
0.20 030. 040. 0.50 060° «070 «0800.80 100
PEE
Figure 4 - Calculated Wave Resistance Ww = 8 (Sretensky)
ozal
¢=0.50
w=4.0
¢=0.55
0.20+
¢=0.65
0.16 + : ¢=0.70
$=0.75
$=0.85
B ouep /
$=0.70 UA
¢=0.75
0.08 ORD
¢=0.85 J
NO f;
0.04 [ae>
F \Y Wi $=0.50
LY $=0.55
Hr.
$=0.65
iia l | j
0.20 030 O40 0.50 0.60 070 080 0.90 1.0
Figure 5 - Calculated Wave Resistance w = 4 (Sretensky)
21
Within any one diagram W = const. L/B varies as Vé since, from the
condition, v is constant it follows that ¢(B/L)* is constant; the beam of the
finer forms is larger than that of the fuller ones.
The curves "¢ = const." on the two diagrams 4 and 5 are connected
by the relation L/B = w*”.
Although the equation for the hulls is so restricted that it can
represent useful ship lines only within a limited range of prismatics and
Froude numbers, and the representation of results is somewhat unfortunate, due
to the use of W, the two diagrams give a useful general idea of resistance
properties of ship forms over a wide region of prismatics and speeds.
A weak point of the investigation lies, however, in the deductions
made from Figure 6, reproduced from the paper, which represents a comparison
of the wave resistance calculated for the restricted mathematical lines and
the residual resistance derived from Taylor's Standard Series. (For some un-
known reason, another resistance coefficient
is used.) Thus the comparison is based on the equality of prismatics alone
neglecting the difference between the actual shape or equations of the two
series. This coarse approach yields a reasonable approximation within the
region of the first and (in a lesser degree) of the second hump; it can, how-
ever, become inconsistent for lower Froude values. Checks indicate that in
this region the calculated resistance of some of Sretensky's ill-chosen forms
can be twice or three times as high as the calculated resistance of Taylor's
models. Hence some important differences stated are essentially due to dif-
ferences in forms used and have nothing to do with discrepancies between the-
ory and facts.
Giers and Sretensky state: "The use of wave resistance coefficients
calculated by theory appears to be inadmissible for drawing quantitative con-
clusions in the present state of knowledge." Especially, according to the
author, these coefficients cannot be used for obtaining relations dependent
upon the variation of one parameter while others are kept constant.
If these final conclusions—arrived at by one of the greatest. author-
ities on the subject—were true, the theory of ship wave resistance would be
useless from a point of view of naval architecture. We have shown that the
conclusions are based on an erroneous procedure and cannot be upheld. More
reliable, if less general, comparisons between calculated and measured resist-
ances are due to Wigley and to an extent to the author, who came to much more
favorable results.
22
Calculated
— Experimental
0.20 0.30 040 0.50 0.60 0.70 0.80
0.04
F Calculated
\ A — Experimental
\/
| Seeley aac a te
0.03 4
ie}
0.20 0.30 0.40 -0.50 0.60 0.70 0.80
pt
Figure 6b
Figure 6 - Comparison of Calculated and Measured Resistance (Sretensky)
23
The question may be raised as to why Sretensky's analysis based on
the prismatic coefficient alone has failed, while the application of Taylor's
Standard Series results, based on the same parameter, has proved to be ex-
tremely successful in practice except for full forms. The answer is that
Taylor's forms are advantageous or reasonable; they have not been derived from
a narrow family which may yield extremely bad forms as some admitted by
Sretensky. 0.3 by second hump, etc. contrary to the habit in naval architecture, by which the hump
at the highest speed is called "last" hump. This change appeared to be necessary since from a mathe-
matical viewpoint we have an infinite number of humps between 0.5>F>0.
24
ama
Dobie
CALCULATED WITHOUT
ee
—_|,-©
8 | VISCOSITY CORRECTION.
We sue
Ce ee RECTION.
Y2 ANGLE OF ANGLE OF ENTRANCE
ENTRANCE OW L.W.LJ COEFI=/ TAN Of /4
ee
Hh
HES
P
Aas
{tt EC
ee | LP none e221, EO Hine =a ©w CALCULATED WITHOUT.
ees a lial Ze SA alee edie
ESS Lf L Ls4
me, 2/3,
the fineness coefficient of the resulting body ¢ decreases with increasing
d/L (example: Rankine's oval),b.and when bq < 2/3, the fineness coefficient
increases with d/L. Thus one must infer that the image system suitable to
generate a ship form depends on the ratio of principal dimensions. Hence,
even the shape (and not just its magnitude) of a resistance curve derived for
a dimensionless form varies in principle with these ratios.
Some additional information is given in the next chapter. We antic-
ipate the conclusion: The assumption that the wave resistance of hulls can
be treated independently as a function of the dimensionless form and the
27
proportions, means a rough if necessary approximation which must be corrected
especially when the ship forms differ widely from "Michell's ship."
5.2. WAVE RESISTANCE AS A FUNCTION OF PRINCIPAL DIMENSIONS
If n is the equation of a hull and L is kept constant, the basic
variations in beam and draft* are given by:
a. H = const., B variable, i.e., an affine distortion in the direction
of the y-axis.
b. Be= const., H variable, i.e., an affine distortion in the direction
of the z-axis.
Two further variations are popular with experimenters:
ec. BH = const., i.e., an affine distortion along y and z.
d. B/H = const., i.e., similarity distortion.
5.2.1. Variation of Beam for Constant Draft H
From Michell's integral or the symbolic expression [13], it follows
immediately that the wave resistance varies with the square of the beam for
all Froude numbers, thus
R = R(B) ~ B® or R ~ (B/H)? or R ~ (B/L)? [14]
where H and L are constants. (As Michell's integral is valid only when L/B> | ,
2H/B > 1 and st are small, this simple result should be checked.)
The total of relevant experiments is astonishingly small; of these
the most important measurements are due to Wigley. &4 He found that within
the region 16 2 L/B ? 8 agreement between theory and experiment is reasonably
good, and that an exponent n somewhat smaller than 2 in the formula
R(B) > Bo [15]
is more in keeping with experimental results.
From Taylor's experiment, an exponent n ~ 1.6 can be derived. An
empirical curve n = n(F) was given by Mumford®> (Figure 9 on page 29).
Theoretical estimates of the validity of the law R ~ B® were made
using limiting conditions, as follows:
a. Comparing a spheroid with an ellipsoid of twice the width it was
found for F = 0.226 and F = 0.50 that the law R ~ B® holds within the accuracy
of computation. A similar result was found using a theorem due to Lamb .7°
*Cf., Figure 8, page 28.
28
(1) (2)
fei te eras eae
| |
|
[aide i, ig
|
! |
H = Const |
ee a eel
B= Const
Affine distortion in the direction Affine distortion in the direction
of the y-axis of the z-axis
(3) (4)
| |
|
|
|
| BH=Const
Unda eal ars cay
Affine distortion along y and z Similarity distortion
B/H = Const
Figure 8 - Variations of Proportions
b. Another limiting condition may be derived from Hogner's resistance
formula [18] for a pressure system. Using, for simplicity, a constant pres-
sure over a rectangular region of the free surface, it was found that n(F) in
the formula
R ~ pn(F) [16]
is always smaller than 2. For example, n(F) = 1.5 for F = 0.25; in principle
n(F) depends upon B/L. Although this application is rather tentative, we may
infer that for surface ships n is smaller than 2.
ec. Finally, some information may be obtained from more advanced theo-
ries mentioned later, but no numerical computations have so far been performed.
Thus, the results of all the endeavors made to date is rather meager.
However, from the known form of the solutions, we can infer that within the
range of normal ship forms the wave resistance is a smooth function of B, and
that small proportional changes in the beam cause correspondingly small
changes in the wave resistance. When model experiments sometimes yield an
abnormal result (as for instance in the case of the "NORMANDIE"), it is possi-
ble to assert that such a pathological behavior is not significant as to the
wave resistance of the ship, whatever the reasons may be.
29
5.2.2. The Effect of Variation of Draft for Constant Beam B
In this case the volume is proportional to draft.
Contrary to its dependence upon beam, the wave resistance cannot be
expressed in an explicit manner as a function of the draft in a general way.
However, approximate formulas can be derived from Michell's integral; they can
be treated as special cases of the symbolic expression
R = pg ae gE, nF) fen
Only a small number of such calculations has as yet been made 22878
The form of the hull (7),especially the vertical distribution of displacement,
influences to some extent the relation between resistance and draft, but even
the longitudinal distribution can have some bearing on the problem.
For simplicity the influence of 7 on the function E(H/L,7,F) can be
neglected. Then E is a function of H/L and F only. Attempts have been made
to approximate H°E by a power relationship, R ~ C yn F) where C is a constant
[16] (Figure 9).
Rea Bie q”2
Figure 9 - Mumford's Exponent Curves
Obviously, such a simple approximation can be expected to hold only
for a limited range of H/L which again is dependent upon F.* Some values of
n are given below:
*In the limiting case of H/L>O0, a quadratic law n = 2 results as an asymptotic value, which, how-
ever, is of no practical use because of the breakdown of the theory.
30
a. From resistance curves calculated by Sretensky®®
Vo i 2 Voth Reie OoH0) > IP S O.235, sical
rn
2. n#1.6 for higher Froude numbers 0.7 > F > 0.47; both are
valid in the neighborhood of H/L = 1/20.
87
b. Using Wigley's paper the following data are obtained:
Vo | ie) EOS} store O23) > i! S Oni Gi,
2. n= 1.3 to 1.5 for the second hump 0.32 >F > 0.26.
Ben = WleSitonlinecorethe! tanstehunplOsom she nOns or
It seems that on the average the exponent n is higher for finer
ship forms. In Wigley's case the range of H/L is
Mumford's curve agrees reasonably with theoretical computations for
the first hump; in the range of lower Froude numbers empirical values are
smaller than the theoreticai ones.
Using Michell's integral, both of the following cases, ec and d, can
be easily derived from the two basic ones (a and°b).
ec. A number of well known experiments can be classified under the con-
ditions: BH = const, Any = const and displacement constant.
These are the sets of Taylor's Standard Series*? B/H = 2.25 and
B/H = 3.75, Ackerson's Series,? Rota's,*? and Kent's°’ experiments.
Obviously, the wave resistance R increases with increasing B/H, as
the beam contributes more to the drag than the draft, although the total re-
sistance may change only slowly within a length/beam ratio 10 > L/B 28. No
theoretical analysis has been applied to this case.
d. In this case B/H = const; L/B and L/H are variable (similarity
transformation) and displacement varies with B“. This problem has been studied
most thoroughly by D.W. Taylor,*? Ackerson,? and Bragg.° The displacement-
length ratios of the Taylor Standard Series models were varied by similarity
"transformations. A comparison has been made between calculated and measured
resistances for sets of models defined analytically and run in the Berlin
Tank.°®
The wave resistance grows very rapidly with increasing B, amd the-
oretically with the fourth power for extreme values of F. For L/B = 6 the
theoretical values are excessive.
31
In agreement with Wigley's conclusion as to the dependence of the
resistance upon beam, we infer that theory overestimates the absolute value
for low values of L/B and relatively high prismatics at moderate Froude num-
bers (below the first hump).
Figures 10, 11, and 12 show an attempted comparison by Antimonoff
of calculated and measured resistance coefficients
as functions of L/B and B/H.
Figure 11 - Calculated and Measured
Resistance as Functions of
Principal Dimensions
Figure 10 - Calculated and Measured
Resistance as Functions of
Principal Dimensions
The analytical ship forms
for which the computations were made
do not correspond very closely to
those of the models tested. Because
the theoretical values were primarily
Figure 12 - Calculated and Measured
Resistance as Functions of
based on the present author's work Principal Dimensions
Bi
while the model results were taken from those of Taylor*® and Kent 5” only ¢
and 6 were identical. This coarse procedure may invalidate the comparison of
absolute values for moderate and low Froude numbers but not the general trend
as a function of the principal dimensions.
For F = 0.387 (Figure 10) the agreement between theory and experi-
ment is good as to the character of the curves and reasonable with respect to
absolute values above an L/B of about 8.
Figure 11, valid for F = 0.24, shows a complete failure of the com-
parison. Allowing for the difference in forms mentioned above, it can be said
that:
a. The values of the calculated wave resistance are much exaggerated
for smaller L/B ratios.
b. Even the trend in the experimental residual-resistance curves and
calculated wave-resistance curves as functions of B/H does not agree. This
indicates that the "residual resistance" does not furnish any information
about the actual wave phenomena in the present case because of the presence
of viscous drag. An increase in this resistance with increasing B/H has been
found by experiments with double models.?*
c. The diagram (Figure 12) representing conditions for slow vessels
does not show such pronounced anomalies, but supports the impression that the
model results in question do not contribute to the analysis of wave resistance.
To summarize, we may say that theory has contributed some rough es-
timates of the relations between principal dimensions and wave resistance;
their validity is limited mainly by the L/B ratio. Only a small number of ex-
periments exist which are reliable enough to check the theory and to deduce
simple empirical laws for the basic cases discussed. The presence of viscous-
form drag and other viscosity effects have so far seriously hampered the study
of wave resistance of slow full ships.
5.3. THE WAVE RESISTANCE AS A FUNCTION OF THE HULL SHAPE
5.3.1. General Remarks
The restrictions under which the concept of dimensionless form can
be used in resistance research have already been enumerated. The present task
is: a) To analyze theoretically the resistance properties of different forms,
b) to deduce some general rules from this analysis, and c) to report on
experimental checks. The influence of the longitudinal distribution of dis-
placement on resistance is the most important problem, both from the viewpoint
of theory and practice.
33
Tt was found that in principle resistance effects due to the verti-
cal and longitudinal distribution of displacement cannot be separated; fortu-
nately, however, it is possible to derive a great number of characteristic
properties of ship forms without considering the shape of sections and the
draft, which together determine the vertical distribution.
In the light of this knowledge, the basic item is the sectional-area
curve; it is therefore natural to begin with the well studied elementary
shapes given by
n= X(€) 2(¢) [12]
which embody a given sectional-area curve in the simplest way. Later, more :
complicated forms will be investigated.
The present most important chapter of the paper is perforce present-
ed in a rather rudimentary state, since numerous lengthy computations needed
for a rigorous discussion are not yet completed. Thus, instead of calculated
resistance curves some simple intermediate functions are used as a basis of
our analysis.
This analysis uses systematic geometrical variations of ship lines.
In principle, a more elegant way would consist in minimizing the resistance
integral. The latter procedure yields forms of least resistance which are
basic results in themselves. Besides, when trying to establish simplified
useful relations between important form parameters and resistance properties
it is generally favorable to investigate good forms, since because of the pos-
sibly complicated actual relations the results may depend upon the selection
of the forms. However, the method of minimization has not yet lead to a satis-
factory solution; hence, the approach of geometrical variation adopted here
appears to be the only one suitable.
It is pertinent to mention here why so many "mathematical models"
have been tested which bear little resemblance to actual ship forms, since the
erroneous opinion is widely held that theory can deal only with oversimplified
models:
a. Simplicity of mathematical expressions was aimed at in earlier work
when methods of computation were not developed and the physical bearing of the
theory was unknown.
b. Variations in form for basic research must sometimes be made not
with the purpose of improving forms, but of obtaining pronounced changes in
resistance.
344
o Taylors Standard Series ¢ = 0.56
1 - 2.02542 + 1.55&* - 0.5258
= 1 - 2.762 + 2.445 - 0.764
Oa Oc2 Os OS O55 0.6 Gor 9:8
-9800 .9215 .8300 .7315 .5825 .4475 .3180 .1990 .
) .9934 .9554 .8716 .7431 .5860 .4222 .2762 .1630 .
) -9753 .9101 .8162 .7037 .5813 .4557 .3321 .2117 .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
§
Figure 13 - Sectional Area Curves Re Models Berlin 133/, 13/0
Expressed by the Equations (2,3,4; 0. 56;1) (2,4,6; 0.56;1)
Various attempts have been made, using Michell's integral, to per-
form wave-resistance calculations for "actual" ship forms, i.e., forms defined
graphically by means of the normal-lines plan. No advantage is gained by such
methods,* as it is easier to "mathematize" the ship form and then perform cal-
culations. This leads to an easier method of comparing forms and improving
their resistance properties, which in the present state of the theory is a
much more valuable achievement than the possibility of calculating the re-
sistance for an individual form represented graphically.
To get an idea of what actually can be reached by application of
theory we refer to Figure 7 on page 24, discussed earlier, and to Figure 14
where the differences between the resistances of two models obtained from ex-
periment and theory are compared. The results are impressive. In the light
of these and other investigations it is impossible to question the practical
value of the present theory.
5.3.2. Longitudinal Distribution of Displacement
Outline of a General Procedure: A survey of the resistance proper-
ties of ship forms must be based upon sufficiently general equations; the
failure by using forms with a single arbitrary parameter has already been dis-
cussed. Basic families of ship lines studied earlier admit of sufficient var-
iations (see Figures 15 and 16). Although each set contains only two param-
eters gd, t, a third one can be introduced by immediately "mixing" two families
or by adding an appropriate polynomial.
*This criticism does not apply to a method proposed by Guilloton.
35
3000
@ 1337 A/2 A/2 je
© 1370 A/2 As2 7@
7
2500 A O
7e2 8
E Pp Froude's Frictional 9
2 /e OS Resistance 4
oa 700 Wy
ec 2000 é y
£ $9 i :
in 29 y
Efe 1337 cS
5 ty
a ‘Ss 1370
1500 “P 4 _P
rd o
Ja O xg Se ;
pV a
1000 a a
I
1.5 2.5 v-@
0.2 0.3
v
F=
Vgl
Figure 14 - Models Berlin 1337, 1370, Curves of Total Resistance
a
1.0
$$
(3,4,6,¢,0)
Lg
\
0.8
0.4
0.2
(eo)
3
2)
o
Son
°
Mw
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 15 - Ship Lines, Basic Family (3,4,6; ¢;t);t =0
Taylor's curvature value « can serve as such a third parameter. It
will be shown that it is a useful concept, but that even three parameters may
not be sufficient to fix the resistance properties of ship lines.
The process of "mixing" or adding form polynomials can be extended
resulting in a polynomial with an increasing number of powers and arbitrary
coefficients. In addition, a parallel middle body can be inserted or a bulb
n= No + 10¢ Agn + tA,n
mn =) 1 = 526" tab See ied
Agn = 4 [26° - 36% + |
an = &[-869 + 1564 - 72°]
O52. Oops cOpulb O55}
4gn .0040 .0264 .0709 .1290 .1822 .2098 .1944u .1337 -0492
ies Hogi 210068 "01 66-0901 0286 6 1eI-s oosT 0355-0487 ~_
added. Thus various form parameters mentioned earlier (page 6) can be intro-
duced; new parameters may be found by analyzing calculated resistance curves.
The calculation of wave resistance for sets of lines which cover the
whole range involves a considerable but not prohibitive amount of work. The
procedure adopted is outlined below.
As the first step, elementary ships symmetrical with respect to the
midship section are analyzed. Further steps will deal with asymmetry and gen-
eral forms of hulls.
In the equation
n = K(€) Z(¢)
let X(é) represent the longitudinal distribution of displacement and contain
a great number of members with arbitrary parameters, Z(¢) may be of simpler
form.
Intermediate functions based on the slope ox(§) are calculated: To
each power £” in ote) corresponds a function a
1
M,(7) = Je” sin rede [17]
Grouping products of these functions M(y) in an appropriate manner
and performing a further integration of these products multiplied by some
other factors, some fundamental values are obtained and tabulated. From
37
these tables the wave resistance of practically any assumed normal elementary
form with simple midship section can be derived by multiplication with the
parameters involved and subsequent summation. A first contribution in this
direction has been made by Wigley:*” He performed calculations for the family
(2,4,6,¢,t) based on a slightly different form of the polynomial. Although
the author has himself used this type of equation in earlier work, he prefers
the expanded form
n=1-a,@? -a,6* - (1 - a, - a,)6°— [18]
or in Taylor's notation
n= f,(é) taf, (€) + tf,(6) | [19]
Wigley's work covers only one basic family of ship lines, but it in-
cludes a check of calculations by experiments.
Method of Approximate Discussion; Pending the computation of tables
of complete resistance integrals, a simple if rather coarse procedure has been
developed which allows comparison of the relative merits of forms. It is
based on the discussion of the integrand of Michell's formula, or rather on
only one part of it, which depemis on the longitudinal displacement distribu-
tion and is formally handled by the Me) functions previously mentioned.
This integral is written in the symbolic form
r= | S?(y) 62(y) t(y)ay [20]
Yo
where y, the variable of integration varies for a given Froude number F
between ves Bee and infinity,
f(y) is a simple algebraic function,
@7(y) is a function dependent upon the vertical distribution of displace-
ment, and the product
$*(y)f(y) ensures a rapid decrease of the integrand with increasing 7.
+1
S(y) = iy = sin yédé [a]
is an oscillating function dependent upon the longitudinal displacement dis-
tribution. S*(y) > 0 represents the most important factor of the integrand.
38
The longitudinal function S(7) can be easily computed from tabulated
intermediate functions M(y) for a given X(é).
Basic families of S(y) or S?(y) functions are plotted in the same
way as families of ship lines, with which they are associated, for instance by
keeping t = const and varying ¢ by steps of 0.02. (Figures 24 to 28) S(y) is
a linear function of ¢, t, when the associated ship line (water line) is also
a linear function of ¢, t. Corresponding to difference curves for the ordi-
nates (n non, 5 021) 10) Wand (n non, ; 0; 1), difference curves of S(y) can be
plotted which we denote by a,S(7) and a,S(y)* These permit a whole set of
S(y) curves to be developed for various ¢ and t values, when one curve S(y¥)
for Po? to is known; they are also a valuable help in various discussions.
Using the squared values S*(y), the resistance of different ship
lines can be compared as follows for a given Froude number F:
a. Calculate Us Only S?(y) values to the right of Y, are to be
BAU
Oe |
considered for a given F.
b. The area enclosed between the S*(y) curve (multiplied by the mono-
tonic decreasing function ¢(y) f(y), the ordinate at ye and the y-axis is
_ proportional to the resistance. Taking into account all conditions, generally
the first waves of the S*(y) are decisive for the determination of resistance.
For a first orientation, comparisons can be made without multiplying S?(y) by
¢*(y) f(y).
c. Values of y, for which S(y) or S*(y) are close to zero characterize
the position of a hollow in the resistance curve, a Ue value just to the left
of a crest of the S(y) or S*(y) lines, corresponds to a hump. Thus we can
estimate regions of low and high resistance by simple inspection.
Results of Investigations on Longitudinal Distribution of Displace-
ment: DW. Taylor's experiments have revealed fundamental relations between
the resistance and the standard parameters ¢, t.** It has been shown that
theory has succeeded in obtaining results which agree closely with these ex-
perimental data. The following discussions will, therefore, deal preferably
witn more refined form effects which are not so universally known. The suc-
cess of these investigations depends to a high degree upon the use of mathe-
matical lines.
*Some examples of these curves can be derived from Figures 2h and 25.
**Exceptions will be mentioned later.
BY)
Let us study the S(y) function for a given polynomial, say (2,4,6;
¢@;t) Figure 24 and (3,4,6;¢;t) Figure 25; the parameter t = t, = const.
Curves for different prismatics intersect at fixed points; the abscissas of
these points are given by zero values of the curves AgS(y), and are constants
for a basic family; the ordinates vary with varying t, but are constants for
5 = ty = const.
When all maxima of an S(y) curve coincide with such fixed points, we
obviously obtain a curve with small amplitudes and good resistance qualities
over the whole range of Froude numbers. When such a coincidence occurs only
for one or several points the particular curve may be advantageous over a lim-
ited speed range. For to = const and a given Froude number HRS is a value
of the prismatic which corresponds to a minimum resistance; a further reduc-
tion of @ means a deterioration of the resistance properties. This agrees
with experiments. From Taylor's Standard Series the prismatic of least spe-
cific resistance appears to be something like 0.52 for moderate Froude numbers
(second hump); following theory for hollow forms with t = 0, the minimum is
somewhat lower as can be easily inferred from the S(y) curves.
We investigate now the influence of the curvature parameter K on the
resistance. We compare for this purpose the family (2,4,6;¢31) with the re-
lated (2,3,4;¢;1), keeping for simplicity the tangent value t constant and
equal to one. The curvature at the midsection is K = ar:
rene ue OL6r) ira" (20-4), (6 TOLG-) ire) 1 eee Obes [22]
is common to both families. If we let ¢<0.6,it can be shown that for equal val-
ues of @ the coefficient a, is higher in (2,3,4) than in (2,4,6), hence the
form (2,3,4,) has, ceteris paribus, a higher curvature at the midsection.
When ¢ > 0.6, the reverse holds.
Take the curves corresponding to ¢ = 0.56; it is seen from Figure
13 that they differ only slightly. However, the functions S(y) or S?(y) dif-
fer very much in the region 11 > ¥ > 8,and we must expect that the resistance
associated with (2,4,6; 0.56; 1) will be much lower over an extended range of
Froude numbers.
A crucial test with rather wide consequences therefore appeared to
be possible; it was performed at the Berlin Tank’®’ and yielded a beautiful
agreement between calculation and experiment, see Figure 14. Model 13570 was
developed from the sectional-area curve (2,4,6; 0.56; 1) and Model 1337 (2,3,
Me” GOS 7) s Unfortunately, because of a widely spread but ill advised thrift,
the original readings have not been published, but the author testifies that
every effort was made to obtain reliable results. Furthermore, the results
4O
were indirectly checked later by fitting equal bulbs to the two models. Further
comparative calculations were made for sectional-area curves of Taylor's Stand-
ard Series.” From Figure 14 it follows that Taylor's form for ¢ = 0.56 is
very near to the good one (2,4,6; 0.56; 1). The associated S(y) and S?(¥7)
curves indicate clearly the excellent resistance properties of Taylor's forms
for low and moderate Froude numbers, when ¢ = 0.52 and 0.56.
Important conclusions can be drawn from these investigations, which
have been anticipated to some extent in our introductory remarks:
a. Small deviations in form may cause appreciable differences in wave
resistance; the "fairness of lines" does not give the slightest indication as
to wave-resistance properties. For instance the line (2,3,4; 0.56; 1) is
"fairer" than (2,4,6; 0.56; 1) since it has only one point of inflection.
b. Every ship form is a unique problem; one must be very cautious in
extra- or interpolating resistance properties when the decisive parameters of
the problem are not known.
c. Theory gives powerful, if not thoroughly reliable, means of investi-
gating even fine peculiarities of from with respect to their wave-resistance
properties.
d. Changes in wave resistance due to deformations of models (for in-
stance of wax models by high temperature) may account in some cases for incon-
sistencies in model results, especially in cases where repeated experiments
do not agree with the original ones.
In the present case the parameter K proved to be significant as re-
gards wave resistance. As an example, the resistance properties of two ship
lines were investigated, which were derived by adding to a given line two
polynomials expressed symbolically by (2,3,4,6; 0; 0) and (2,4,6,8; 0; 0)
Appendix formulas [37] and [38]. The resulting lines have the same parameters
@, t, K; nevertheless the resistance functions S(y) differ appreciably. Other-
wise expressed, in such cases it is not possible to fix the resistance proper-
ties of lines even by three parameters.
The two families discussed characterized by K < 0 are typical for
hulls run at F < 0.25. The absence of a parallel middle body is an important
feature.
An inspection of the S(y) or S#(y) curves, Figure 24, page 45, ex-
plains formally why in the ascending branch of the first hump higher pris-
matics and t values are beneficial: Both tend to shift the steep rise of the
curves to the left towards higher Froude numbers. This property of shifting
is valid for smaller Froude numbers too, but here the effect mentioned is
canceled by others.
44
Low prismatics ¢ are a necessary condition for good resistance qual-
ities below the first hump, F < 0.32. Normally considerations of resistance
are so decisive for vessels operated within the range 0.32 > F > 0.25 that
only small ¢ values should be used. Grave mistakes have been committed when
designing liners with rather high prismatics, although the basic facts could
have been easily ascertained from Taylor's Standard Series. For ship types
run at F < 0.25, it may be more advantageous to compromise between resistance
and carrying capacity, since the absolute value of the wave resistance de-
creases. Thus, moderate prismatics become a reasonable solution.
At the same time, ship lines with vanishing curvature K gain in im-
portance, as may be guessed from Figures 26 to 28 representing S(y) curves for
g@ and t = const corresponding to different equations. The family (3,4,6) ap-
pears to be useful over a certain range; with increasing ¢@ and reduced speed
ratios, ship lines corresponding to higher degree polynomials become advan-
tageous. The influence of K = 0 or, more generally, of a parallel middle body
expressed by an increase of degree in a polynomial on the resistance is oppo-
site to the increase of ¢ and t: The humps in the function S(y) are shifted
to the right (smaller F). Our analysis shows that Taylor's Standard forms
¢= 0.6 and ¢ = 0.64 are quite successful, but not outstanding within their
useful speed ranges; it is possible to obtain better results when bulbs are
fitted to some of the good normal forms here discussed (Figures 19, 20, 26,
27).
eg ts
8
5 , 105
Sa aiid
=] - ap - a&
5
~ 315 * 315
) = 1.322587(1 - 4°)?
= -0.375(¢7 - S f+ te)
Figure 17 - Ship Lines - Basic Family (2,4,6;¢,t) t = 1
42
33) (4,6,8)
(2,3,4) ES
4ino 4 aN ;
ee
(6,8,10)
1.0
o Taylor's Standard Series ¢ = 0.60
Of) —— +—
06
Ui
penal
(2,4,6) = 1 - 1.562 + 0
(3,4,6) =1 - 18,9 +
0.24 |
al 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 19 - Examples of Lines
VSO, weil
An example has been given of how to find an economical lower limit
of prismatics from resistance considerations (page 39). Still more important
is the problem of finding a corresponding upper limit for a given Froude num-
ber. Discussing the family (4,6,8) the result was obtained that for F = 0.25,
@ = 0.68 is a reasonable value which cannot be exceeded without loss in effic-
iency (Figures 21 and 28). To check the deductions the curve (4,6,8; 0.68;2)
was used as sectional-area curve of the forebody in a model; the results of
T T
©. Taylor's Standard Series ¢ = 0.64
Figure 20 - Examples of Lines
C= Ok, = 1
Figure 21 - Examples of Lines
¢?=0.68, t=1,t=2
towing experiments with this model supported calculations in a very satisfac-
6 The shape of the curve denoted as "small swanneck" is closely
tory manner.?°
related to a bulb form. However, the "small swanneck" remained superior to
normal lines fitted with a bulb; it is definitely superior to Taylor's Stand-
ard form for ¢ = 0.68. We dwell at some length on this range of speed since
it acquires increased interest in connection with fast cargo ships.
yyy
eS
Beers |
SE 1O16 | AO? SEO OpummnO®
0)
fo) O.l 0.2 0.3 0.4 O. 9 1.0
§
Figure 22 - Examples of Full Lines
e eae
1.0 piles aa A
0.8 ah
n |
: a
‘ a
0.2 ine
) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
§
Figure 23 - "Pathological" Sectional-Area Curve ¢ = 0.82, t = 2
45
(L *9G°0 ‘n°¢‘2) 09 pue (9‘4‘z) ATTWeY
2Uuy JO souTT ewog 04 SuTpucdseds0g (4)S UOTZOUNY SsoUeASTSEeY - he OMIT A
& ;
Z 9 S
(a)s
46
‘qinq 4jTM suIOy TINY ZuTpuodser109 944 IOJ uoTJOUNZ (4)G ey} ST
uoT}OUNF Q{ng oy} pue SeUTT SNONuUT}UOD 244 USEeM}eq edUeTESITpP SUL
[eTwoukTog qtng e pue (1 ‘ P'g‘H‘¢) soz (4)g - Ge eunstTy
| GEx =
iN a Abe
See cee eae Zee = 10-
eras SA pe 2S ra MEN Ah
0.4
5 ]
sen |
3,4,6;0.6;!)
; 0.58; 1)
Te 8 9 10. Wl 12 13 14 15 16 17 18 19 20 2l
5 6 iG 8 9 10 I 12 13 14 15 16 17 18 19 20 a!|
v
Figure 27 - S(y) for Some Lines ¢ = 0.64
Extended investigations were made on fuller forms following the same
lines. However, less success was reached with respect to the applicability of
results;°® Figure 22.
We were not able to develop forms with prismatics of 0.7/2, which
have actually a low resistance at a Froude number above about 0.22. Up to
this limit an equation (6,8,10; 0.72; 2) yields theoretically excellent re-
sistance qualities; unfortunately, experiments have not corroborated the cal-
culations.
The degree of the polynomials used (sectional-area curves) which is
indicative of the amount of parallel middle body, is sometimes more decisive
than the choice of t and, within narrower limits, even ¢g. Thus, it can happen
that for low Froude numbers the resistance may not be very sensitive to
changes in ¢ provided an upper limit is not exceeded.
48
With decreasing F, the optimum t values decrease. Also on the aver-
age, it depends upon the prismatic and the degree of polynomial used. While
very fine ships yield good results with t = 0 up to F = 0.26, full vessels
need t values > 0 for F = 0.18 and even less. Thus, charts indicating optimum
t values must be prepared for several prismatics.
Absolute values of calculated resistance are not reliable when deal-
ing with high ¢ and low F; the results due to pronounced interference effects
are especially doubtful. Investigations on a pathological model, Figure 23—
whose resistance qualities according to theory should be outstanding—are very
Significant: Experiments did not agree at all with these deductions up to a
Froude number of about 0.19, but for some higher values of F the form was
efficient.
Thus, even a qualitative agreement is sometimes lacking when dealing
with full hulls run at speeds below F about 0.19. For this reason, a more
thorough discussion of this most important subject is delayed until further
research has been done.°°» °°»1°© This work must be based on a closer investi-
gation of the different parts of wave phenomena constituting the total wave
resistance, of wave patterns due to bow and stern angles and curved parts and
the mutual interference of the systems mentioned. No decisive attempt has
been made so far because of the lack of appropriate tabulated functions.
The following suggestion may be of interest to the experimentor when
investigating the frictional resistance of plates. It is well known that, at
some Froude numbers, wave phenomena can influence results. Havelock has given
a first estimate on the subject ’ using a parabolic form and an arbitrary val-
ue for the width of the plate. A closer approximation is reached when the
6 M4 8 9 10 if 12 i) 14 ihe) 16 I7 18 19 20 2i
0.10
0.9
0.08
0.07
0.06
R 0.05
0.04
0.03
0.02
0.01
49
——--— (3,4,6; 0.64;1)
—— -—— (3,4,6;0.60;1)
(2,3, 4; 0.60)
eee (283 4\-10'56))
(2,4,6 50.56)
Figure 28b
Figure 28 - S(y) for Some Lines ¢ = 0.68
50
equation of the actual water line of the plate or at least its area coeffic-
ient (prismatic) is considered. Within the region of the second hump the wave
resistance of a "plate body" with a prismatic ¢ ~ 0.9(which is nearer to actual
practice than 2/3}can be 10 times higher than that of a parabolic form ¢ = 2/5).
A special investigation was made into "hollow versus straight lines".?°% From
this work it follows that theory tends to overestimate the optimum t values.
Finally, two asymptotic laws have been derived:+++
a. For vanishing Froude numbers, the wave resistance becomes propor-
tional to t®
Re [23]
b. For very high Froude numbers
Ri~ 6° [24]
The latter law can be easily derived, also, from an asymptotic formula due to
Lamb .7°
Although the practical significance of these relations is not very
great, two important remarks should be made:
a. The S(y) curves here reproduced give only a rough picture. For
actual computations the functions must be extended to higher values of the
variable y; some tabulated values are published” and more comprehensive
tables are available at the Taylor Model Basin.
b. Theory deals only with that part of the hull which is submerged at
zero speed. Obviously the hull above the load water line at rest must have
some influence on the resistance. This point is strongly supported by Eggers'
28 However, resistance measurements made on models with different
experiments.
section forms above the load water line (see Figure 29) did not disclose ap-
preciable differences in resistance results.°*® Of course two experiments can-
not disprove the consistency of the preceding reasoning, and much remains to
be done on the subject following an outline given by Guilloton.
5.4. THE INFLUENCE OF THE VERTICAL DISTRIBUTION OF DISPLACEMENT ON THE
WAVE RESISTANCE
5.4.1. The Influence of the Midship Section Coefficient
The problem may be stated as follows: In the equation 7 = X(é) Z(¢),
the function X(&) is kept the same; variations in resistance have to be inves-
tigated for various midship sections defined by the function Z(¢).
51
Figure 29 - Changes in Bow Sections Above the D.W.L.
Within the range of normal midship section forms, similar to parab-
olas of higher degree, Z(¢) = 1 - ze", it is found that the wave resistance is
not sensitive to changes of pure shape of section when the area is kept con-
stant. Thus, contrary to the conditions valid for water lines, the resistance
qualities of the midship section can be determined in normal cases by the
single area coefficient 8. Under more general assumptions, the following
asymptotic laws are valid:
a. When F becomes infinite;
R ~ p* [25]
b. When F tends towards zero,
R is independent of 8 [26]
For finite F, R depends upon 8, on the ratio = #2 and on the function X(é).
Thus no simple rule for calculating R as function of B can be given. Neglect-
ing, however, the dependence upon X(é), as in the case of draft, interpolation
formulas of the type
R ~ pP(F) [27]
NZ
can be deduced. For normal midship sections, the relations between resistance
R andfare similar to the corresponding ones between resistance and draft. Un-
fortunately, no numerical results of calculations are available at present and
no experiments are known which correspond closely to this problem; therefore,
we must confine ourselves to qualitative estimates which will be replaced by
more adequate data as soon as the pertaining calculations are completed.
In the region of the first hump, the exponent p in Equation [27] is
nearer to 2 than to one; hence, small midship coefficients are used in actual
design. In the neighborhood of the second hump, high @ values are common;
this is reasonable although the influence of the midship coefficient on resist-
ance is not negligible, since the use of a large @ value admits of the choice
of low prismatics without reducing unduly the block coefficient 6.
D.W. Taylor's famous experiments dealing with the resistance R as a
function of the midship-section shape** refer to a rather complicated case.
They prove, however, that for high Froude numbers the influence of the ratios
B/L, H/L and 8 upon R is similar; for moderate and low F-values, the effect
of the ratio B/L becomes predominant. More definite information will follow
from the systematic resistance evaluations mentioned earlier.
When the shape of the midship section departs from the normal, spec-
ial investigations become necessary. These can be performed comparatively
easily, however.
In many cases a rule of thumb method proves useful: By shifting
some of the submerged volume near the surface vertically downwards the resist-
ance is decreased. The effectiveness of such a change increases with the ra-
tio of the distance d by which the volume is moved to the length of the free
wave.
Ro SE [28]
i.e., decreases with increasing Froude numbers.
5.4.2 Shape of Sections, Load Water Line and Sectional-Area Curve
Let the sectional area curve A (é) be given.
The problem of developing good sections can then be replaced to some
extent by the simpler one of developing a suitable load water line X(é), at
least from the point of view of minimum wave resistance.
Within the range of validity of the present theory it can be
inferred that reasonable changes in the shape of the sections have only a
secondary influence on the wave resistance once the A*(é) and X(é) are fixed.
33
Thus, comparatively simple types of surface equations appear applicable (see
Appendix 1, Equation [41]).
As a starting point, the fact is used that parts of the displacement
near the surface offer more resistance than those near the bottom of the ship.
Hence, U-shaped sections appear to be superior to V-shaped sections as far as
wave resistance is concerned. Since for moderate Froude numbers the forebody
produces more waves than the afterbody, R.E. Froude's rule* for design can be
deduced: U-shaped sections for the forebody, V-shaped sections for the after-
body (the latter with the purpose of reducing "eddy resistance").
Actual calculations prove the superiority of U-forms in almost all
cases; however, under exceptional conditions a V-shaped forebody may be super-
ior. Experimental results of this kind were found by Lindblad@® for a fast
cargo ship form.
A physical explanation can be given, which will be applied later to
bulb forms: Although the wave resistance generally decreases by increasing
the depth of immersion of displacement, favorable interference effects may be
reduced by shifting some parts of the volume vertically downwards.
In general, agreement has been found between theory and facts, al-
though important examples are known where V-shaped forms were superior to U-
shaped forms, contrary to theoretical deductions.**’1°7 The following effect
has been stated experimentally for Froude numbers F > 0.33: V-shaped models
with low or moderate t-values prove to be better than U-shaped models with
high t-values although both features (U-shape and large t) are advantageous in
the light of theory. In such cases the resistance qualities cannot be pre-
dicted from the sectional-area curve alone; Taylor's well known diagrams rep-
resenting R/a as functions of t and v/VL are no longer applicable. No theo-
retical explanation of these anomalies has so far been found, so presumably
the attitude of the model has an important bearing on the subject. When hull
shapes depart appreciably from the double wedge form, as for instance in the
afterbodies of destroyers, we can no longer rely on the results of resistance
calculations, especially in the range of high speeds, without introducing more
consistent physical concepts of the phenomena. Consequently, one must be
cautious in applying the simple rule of thumb given for the influence of verti-
cal displacement distribution in complicated cases.
*Obviously, considerations of propulsion can change this rule!
54
5.4.3 Bulbs and Cruiser Sterns
Actually the sectional-area curves of most present day ships are
characterized by a condition t ~ 0, due to the more or less heavy rake of the
stem. As long as the rake is small we suppose that its influence on the re-
sistance is not large. In such cases, we therefore determine t as though the
normal form extended up to the forward perpendicular; the actual rounding off
at the stem also tends to compensate the error committed. The resistance cal-
culations can thus be performed in the usual way.
Matters become more complicated for heavily raked stems like the
well known Maier form. Here the generating singularities should be distrib-
uted within a triangular contour at the bow. Although in principle, calcula-
tions of wave resistance do not present difficulties even in this case they
become rather tedious, so that the properties of such distributions have not
been rigorously investigated. Actually one substitutes a distribution limited
by a rectangle, keeping the intensities approximately equal (Figure 30). Even-
tually a more elaborate calculation will be made instead of this rather coarse
Figure 30 - Pronounced V-sections at the Bow
op)
procedure, although we cannot expect that Michell's theory will give a com-
plete answer for forms with a steep inclination of the sections at the bow.
Still more serious objections can be made when applying the theory
to hulls with bulbous bows; but nevertheless the resistance properties of
bulbs have been studied with success.
The "bulb effect" is a wave-making phenomenon. In principle the
pearlike shape of the bow sections is not a necessary attribute of a bulb form;
the latter is defined by the shape of the water lines or of the sectional-area
curve, for instance by the ratio f proposed by Taylor (Figure 30a).
However, the pearlike form has resulted from the necessity of avoid-
ing spray formations, which arise when the load water line is rounded off by
a large radius.
Replacing the bulb by a sphere (doublet) located at the bow (and
later at different distances from the bow) Wigley explained very clearly how
the wave trough which generally starts just abaft the sphere diminishes the
bow wave of the normal ship and thus also the resistance. He further succeed-
ed in demonstrating that the most advantageous position for the bulb was, gen-
erally speaking, just at the bow over the whole useful range of speeds. °?
For practical work it is preferable to express the bulb by a high-
power polynomial;*°® the resistance effects of this bulb and of any normal
form can be easily combined using the appropriate S(y) functions. Figure 25
indicates how the S(y) function corresponding to a normal form is favorably
influenced by a bulb of definite shape and strength. The bulb shape is fixed
by the equation of the polynomial used; the strength of the bulb is denoted by
wou
a constant factor a by which the polynomial is multiplied. Obviously for a
given hull, bulb shape and Froude number, the strength "a" will have an opti-
mum value, which theory seems to overestimate.
By the method proposed, we get a much closer description of the bulb
form than by Taylor's rather summary procedure:
a. It is easy to show that the efficiency of a bulb depends both on its
own shape and upon the character of the ship lines. Generally speaking, the
bulb is more advantageous for hollow than for straight lines.
b. The advantage of the bulb disappears at low speed-length ratios;
its lower limit of effectiveness depends on the shape of the normal ship form.
For hollow lines it. is about F ~ 0.2 or even less, while for straight lines
it may be as high as F ~ 0.24 or 0.26. Theory indicates as upper limit for
the application of a bulb a Froude number of approximately 0.6, the exact val-
ue depending somewhat upon the form of the bulb and the ship; in fact the lim-
it is somewhat lower.
56
c. Generally a suitable bulb improves bad forms by a greater absolute
amount than it does good ones, except in cases where the inneficiency of the
form is caused by features like an exaggerated t value. Reference is made to
Figure 15; there the good model 13/0 could not be improved by a bulb, which,
applied to the bad model 1337, proved to be quite effective. Theory suggests
the use of a bulb when rather full forms are driven beyond their limit of
economical speed.
d. Generally a bulb uniformly distributed over the draft influences the
wave resistance of a normal form more than a submerged bulb (Figure 30a) pro-
vided the sectional-area curve of both bulbs is the same. This can be easily
understood by the fact that wave effects, hence interference effects too,
are stronger at the surface than near the bottom of the ship. In Wigley's
paper °! a different statement is made; however, it seems to be due to the
rather abstract shape of the bulb used. Since the uniformly distributed bulb
cannot be used because of the spray formation, the submerged bulb is the only
practical solution. Probably the loss of efficiency mentioned above can be
compensated by a higher strength factor a
e. In an- ideal fluid the optimum solutions are symmetrical with respect
to the midship ratios, i.e., bulbs should be fitted both at the bow and at the
stern. Obviously this deduction may be wrong for viscous flow. It has been
tried*®® on different models with the result that: 1) For higher Froude num-
bers, say F 2 0.3 the symmetrical combination had nearly double the effect of
a bow bulb alone, and 2) a model with a stern bulb alone did not show any ad-
vantages compared with the normal form. The idea of a stern bulb may have
some merits with respect to the interaction between propeller and ship.
In summary, the application of theory to bulb forms means a serious
violation of the assumptions on which Michell's integral is based. Neverthe-
less results are obtained which are useful for guidance in research and design
work.
As for the cruiser stern, by fitting this type of afterbody the
total resistance of a model may be reduced at some Froude numbers while, for
other ones, the success may be negligible.*’ This dependence upon speed indi-
cates that at least one part of the beneficial effect is due to wave
interference.
The phenomena involved can be investigated using a method proposed
by Havelock for calculating the influence of viscosity on wave resistance
(page 63). Numerical computations are under way. Similar calculations can be
made when dealing with a possible influence of bossings on the wave resistance.
DF
5.4.4 Ships of Least Resistance
Ship-resistance research will have reached its practical goal when
we will be able to indicate the form of least total resistance for any given
conditions of speed, displacement, etc. Theory emphasizes the well-known fact,
often forgotten by inventors, that there does not exist one ship form of least
resistance, but that optimum forms vary with Froude numbers and with other con-
ditions. So called optimum shapes like "pisciform" (fish form) which have
been derived from considerations valid for an unbounded fluid lack any serious
background for surface ships.
The problem of finding ships of least total resistance can be formu-
lated analytically; however, this formulation does not seem to be helpful as
long as no analytical expression for the viscous drag is known. The friction-
al resistance may be assumed with reasonable accuracy to be proportional to
the wetted surface.
Thus, the problem of calculating ships of least wave resistance ap-
pears to be the appropriate first step towards the solution of the more gen-
eral task. In fact, the wave resistance is the "component" most sensitive to
changes in form and is responsible for the dependence of optimum form upon
Froude number. We can hope, therefore, to obtain the most essential informa-
tion on ships of least resistance by solving the problem for the wave resist-
ance—an assumption which underlies Froude's method.
From the form of the resistance integrals for ships, submerged bod-
ies of revolution and pressure systems, it follows that the forms of least re-
sistance are symmetrical with respect to the midsection, since then the term
I? (see Appendix 2, Equation [12]) becomes zero.
Another .deduction which will be needed later is that the resistance
of an asymmetric body is the same when moving ahead or astern. These results,
which are contrary to our general experience, are caused by the assumption of
an ideal medium; at high Froude numbers, however, the effect of viscosity is
small, so that for a restricted class of bodies symmetry may become an approx-
imate condition of least wave resistance as has been shown by experiments.
Actual ship forms suitable for very high speeds, however, do not comply with
the condition of symmetry, one reason for the departure from the results of
simplified hydrodynamic theory being the influence of the changed attitude of
the ship at such speeds (trim and bodily rise).
Keeping in mind that the following results must be applied with
caution to actual conditions, we discuss methods used, the difficulties met
with, and the practical information obtained when trying to find ships of
least wave-making resistance.
58
To eliminate the trivial answer that the wave resistance vanishes
for vanishing displacement some additional "restraint" must be introduced; the
most important condition is to assume a constant displacement. Also, the con-
ditions of a fixed midship section Ay = BxBxH = const, and of least specific
resistance are of theoretical interest. Thus, a number of isoperimetric
problems in wave resistance are formulated.
From an inspection of Michell's integral as well as from physical
reasoning, it is obvious that a rather trivial answer exists as to the best
vertical distribution of the displacement. Since the influence of the wave
making decreases with Orta the displacement should be concentrated as far
below the water line as possible or the draft should be infinite. Thus, even
when the volume is fixed, additional restrictions on the draft and the shape
of the transverse sections are necessary. The essential remaining problem to
be solved is the optimum longitudinal distribution of displacement.
A plausible simplification is to substitute an infinite draft as
long as only general information is desired; but such an approach is not suit~-
able when detailed practical results are needed.
Some results have been found by applying Ritz's method which, how-
ever, are valid only for the restricted kind of functions used (Figure 31).
To quote from an earlier paper+°? "As a further difficulty it may be mentioned
that the assumption of the type of surface equation involves a highly arbi-
trary element, and very advantageous forms can remain outside the scope of our
considerations by lack of knowledge of their analytical representation; "
Figure 31 - Sectional Area Curves for Ships of Least Resistance
39
"Considering all the circumstances" (the approximate character of
the hydromechanic theory and the choice of the hull equation in order to use
Ritz's method) "one should not expect to obtain a final solution of such a
difficult problem by heaping up approximations..."
Later it was shown by Pavlenko,*’ von Kdrmén®° and Sretensky®’ that
the solution of the problem was hampered by serious mathematical difficulties.
All three authors used infinite draft for their final deductions. The con-
dition v = const reduces, in the case of infinite draft, to Ay ="CONSitay eller
the final answer represents the best shape of the water line, which can also
be interpreted as the shape of the sectional-area curve.
Following von Karman, an exact solution of the problem of calculus
of variations exists only for a limited range of medium Froude numbers. This
statement agrees to some extent with Pavlenko's analysis. L. Sretensky, how-
ever, denies the existence of any solution by square integrable functions over
the whole speed range. Because of the fundamental theoretical importance of
the problem, at present Wehausen of the Taylor Model Basin is reconsidering
the matter.
The naval architect's point of view is somewhat different from that
of the mathematician's. Michell's integral is only an approximate solution
even in the case of an ideal fluid, therefore we must check experimentally any
optimum form derived from it. The physical meaning of such results is deci-
Sive; exact solutions of the integral (if they exist) may be less valuable
than approximations which yield results within the important range of the
theory.
The practical results reached may be summarized as follows (see Fig-
ures 31 and 32):
The optimum longitudinal and (within restricted limits) vertical dis-
tribution of displacement agree well with experimental work, the latter due
mainly to Taylor.*? In particular some optimum values of g@=C_, t, B= Cu
found experimentally were in agreement with theory. "Swanneck" forms were
rediscovered and some new features like "small swannecks" (indication of a
bulb) found when studying moderately full vessels. Only restricted use can be
made of Pavlenko's forms (Figure 32); the extremely blunt sectional-area
curves (or water lines) probably indicate that the theory has been over-
stressed. On the whole, theory has lagged behind experiment—partly because
of the difficulties in principle, partly because of the inadequate effort ap-
plied to the subject. The methods of computation have hitherto admitted the
use of only two arbitrary parameters, and more widely applicable approximate
results can be expected by improving the methods of calculation.
60
Figure 32 - Sectional-Area Curves for Ships of Least Resistance and
Infinite Draft (Pavlenko)
A similar investigation on wholly submerged bodies of revolution
has proved rather fruitful, particularly because of a lack of any earlier work
on this subject .??+
Optimum sectional-area curves (optimum doublet distributions) are
similar to corresponding curves of surface ships, but peculiar form effects
seem to be more pronounced with submerged forms.
Finally, we must return to our point of departure—ships of least
total resistance. Adding a frictional-resistance term to the wave resistance
is = Nile a2 Re [29]
where Re is assumed proportional to the wetted surfaces, and applying Ritz's
method to Ry; a closer approximation can be obtained. Within the range of
examples investigated, no great differences were found between the forms cal-
culated from conditions of minimum wave and minimum total resistance as given
by [29]. However, it is expected that such a procedure may eliminate some
61
mathematical difficulties. A further step can be made by using some correc-
tions for viscosity, as shown later, but taking into consideration the some-
what arbitrary character of these corrections no results of general value can
be achieved.
6. INFLUENCE OF VISCOSITY
So far emphasis has been put on the wave resistance of symmetrical
ships, since in an ideal fluid the even terms of the surface equation contrib-
ute the principal part of the resistance as long as normal forms are consid-
ered. The resistance "component" corresponding to the antisymmetrical part
is computed in the same way as for the main part by slightly different func-
tions; the two components can be simply superposed. As mentioned before, in
an ideal fluid the wave resistance of asymmetrical bodies is the same when
moving either ahead or astern.
Obviously, this paradox does not hold for a real liquid. When asym-
metrical variations of the surface are made, resistance calculations valid
for an ideal fluid in many cases do not agree with the facts even as to sign.
Thus, the classical problem of determining the optimum longitudinal position
of the center of buoyancy involves the consideration of viscous effects.
Because of our restricted theoretical knowledge, we have to rely
upon experimental data; but attempts have been made to deduce from such data
more general results. The phenomenological approach used is based mainly on
measuring the resistance of asymmetrical models run in both directions. The
most important experiments are due to Wigley.
Some practical deductions can be obtained from some earlier work at
the Berlin Tank.+°* By using four symmetrical models appertaining to the family
(2,4,6; 6; 1) with ¢ = 0.52, 0.56, 0.60, 0.64 and two asymmetrical models
@ = 9.56, 0.60 run in both directions (p= 9% = 0.04), the following deductions
were obtained:
a. For the fuller model ¢ = 0.60 the shifting of the center of buoyancy
aft proved to be advantageous except for F > 0.4.
b. For the finer model ¢ = 0.56 the aftward movement of the LCB gave
a mcedel worse than the symmetrical basic model.
c. Thus the influence’ on the resistance of the shifting of the LCB for
given Froude numbers depends on the form (equation) of the original surface
and the method of moving the LCB (equation of odd members added). As a rule
resistance properties of asymmetrical ships cannot be described by the posi-
tion of LCB (or the difference op - $y) alone.
62
d. For moderate F the resistance properties of the forebodies (judged
by the resistance qualities of symmetrical models) yield an estimate of the
success obtained by moving the CB.
Conditions for forms with higher prismatics are especially compli-
cated, since here the viscous-pressure drag may become as sensitive to form
variations as the wave resistance. These two resistance components influence
the total resistance in opposite ways. For low wave making we need a fine
forebody; for low viscous-pressure drag a fine run. The LCB of full slow
ships lies normally forward of the midsection, although the wave resistance
is increased by this location, since in this case, the viscous-pressure drag
is more important than the wave resistance.
An additional complication is caused by a possible instability of
flow, to which this kind of model is especially subject. Thus, the analysis
is peculiarly unsatisfactory when dealing with the important class of slow and
moderately fast cargo ships, because:
a. The theory of wave resistance can be applied only with great re-
strictions on account of their full forms and low Froude numbers.
b. Our knowledge of viscous-pressure resistance and frictional-form
drag is completely inadequate.
c. Numerous earlier experiments are questionable as to their accuracy.
Various attempts have been made by Havelock to estimate the influ-
ence of viscosity on wave phenomena. In the first place, Havelock pictured
the process phenomenologically in terms of a friction belt whose effect may
be equivalent to reducing the slope of the ship towards the stern; the equiv-
alent shape of the hypothetical body was rather arbitrary.°* The correspond-
ing resistance curve showed much less waviness; the greatest part of this ef-
fect can be explained by the heavy reduction of the prismatic due to the vir-
tual lengthening of the form.
The next step consisted in introducing a correction factor [see to
allow for a decrease in efficiency of the elements of the ship's surface in
going from bow to stern;’+ the frictional effect is treated as a diminution
in the effective relative velocity of the model and the surrounding water.
This method was developed by Wigley, who applied the same factor B,,* for the
decay of the bow waves.* He deduced the reduction factor B,,* from resistance
curves by finding the values necessary to give reasonable agreement between
theory and experiment. Important practical results thus obtained are dis-
cussed later.
63
From a theoretical point of view, the procedure leaves much to be
desired. The factor Bee is assumed to be constant over the whole after half
of the ship, while the influence of the viscous phenomena is undoubtedly con-
centrated at the extreme end of the run, where the generation of waves is most
heavily rescued by its effect. Following Havelock, this fact alone explains
why the influence of viscosity on the wave pattern is so much more pronounced
for low Froude numbers.”
Havelock calls a third most promising step "an illustration of the
possible effect of boundary layer on wave resistance;" Small modifications
of the lines near the stern are made so as to obtain the required kind of
change in the calculated resistance curve.”
The displacement thickness* of the frictional layer( something of the
order of one-tenth of the boundary-layer thickness) is inappreciable, except
at the stern of the ship, where, because of the reduced girth and eventual
separation, a wider wake is created.
Quantitative estimates of viscous effects on wave patterns appear
to be possible when an appropriate singularity distribution is found which
takes into consideration the influence of viscosity.
With the kind permission of the Institution of Naval Architects, two
figures (Figures 33 and 34) are reproduced from a paper by Wigley, which sum-
marize some important resistance results. The hulls investigated belong to a
family (2,4,6); asymmetry is produced by adding a term a,(é° - Be).
The investigation is similar in purpose to the corresponding one in
the first part of this chapter, but more elaborate computations have been per-
formed. Instead of the resistance curves, differences in the resistance be-=-
tween appropriate models are plotted and experimental values are compared with
results of calculations with and without viscosity correction. In Figure 33
the symmetrical basic Teddington Model 1970B sectional-area curve (2,4,6;
0.7; 2) is compared with an asymmetric model 2170A derived from 1970B by
shifting the center of buoyancy by 0.02L.
When the full end of 2130A is leading, a reasonable agreement is ob-
tained between measured and calculated resistance without any viscosity cor-
rection. However, when the fine end is leading the concept of ideal fluid
breaks down, while the semiempirical viscosity correction yields at least a
qualitative agreement.
*The displacement thickness 6* is the amount of displacement by which the main stream is thrust away
from the body due to the slowing down in the boundary layer. The mathematical expression is
ea U f
é -fa pie
64
0-3
a © MODEL 2130, LESS © MODEL 1970,
= FULL END LEADING
za
w 0:2
3 CALCULATED WITH
re VISCOSITY CORRECTION
= rs
a +t DIRECTION OF MOTION
© : ra LW.Ls. OF MODELS
ra re
[o)
WwW
2 See CRESS
oO
172)
MODEL Recreate COEFFICIENTS RUGEROF ENTRANCE
Laser FULL END] FINE END] TOTAL [FULL END|FINEEND:
742 GEES EC EOE |
Foo [-700 [700] 2-0 | 2-0 |
+
2
© DIFFERENCES
SCALE. OF
———
DIRECTION OF MOTION
SCALE OF VV (V IN KNOTS EWts. OF MODELS
08 10 +2
SCALE OF f - V/VGL
Figure 33 - Effect of Asymmetry on Resistance (Wigley)
(Courtesy of Institution of Naval Architects)
The effect on resistance of a reversal in the direction of motion
is given by Figure 34; only calculated curves with viscosity correction can be
compared as without correction the difference is obviously zero.
Interesting results have been obtained for the influence on resist-
ance of changes in the forebody only, the afterbody being kept constant, and
vice versa.
For variable bow shape, calculations without viscosity correction
are in good agreement, on the average, with experimental data within the range
of forms investigated.
65
°
nS
MODEL 2130g (©)FULL END LEADING LESS © FINE END LEADING
MODEL PRISMATIC COEFFICIENT:
FUL alae eao|roTaL POP ERAS
Sa. cavcucaren FEN SC
[21308] 763 |:637 | 700] 35 | 0-5 _|
a
DIRECTION OF MOTION
LW.LS. MODEL 21306
°
SCALE OF © DIFFERENCES
Ww
)
2)
2
w+ 0:
(4
7)
&
3
+ e
—_—_—_—_—_oO——
©) DIRECTION OF MOTION
o LWALS. MODEL 2130,
C =
3 SCALE OF V/VT (VIN KNOTS)
2 0:8 10 1-2 4 2:0
0-3 0-4
SCALE OF f= U/VgL
Figure 34 - Effect on Resistance of Reversal of Direction
(Courtesy of Institution of Naval Architects)
For variable stern shape, the results depend upon the type of hull.
By fining the stern of a full symmetrical vessel, only negligible or moder-
ate reductions in resistance were obtained up to a Froude number of F = 0.3
(thus "Froude's rule" proved to be consistent), while by increasing the after-
body displacement of a finer ship a more pronounced increase was found.
7. WAVE PHENOMENA DUE TO THE PROPELLER ACTION
The interaction between ship and propeller can be successfully stud-
ied by the method of singularities. In particular, the mutual forces between
hull and propeller can be calculated by Lagally's theorem as soon as appro-
priate images are known. Source and sink distributions which picture the hy-
drodynamic properties of a ship have been already discussed. As far as the
66
problem of interaction is concerned, a suitable "model" for the propeller is
furnished by a sink distribution over its dise and for greater distances even
by a single sink only. The "input" is given by
Q = Ave [30]
where 9 is the ratio of the slip stream to the propeller speed, and A the pro-
peller disc area. Denoting the flux through the propeller disc by Q, the
thrust is written
T = pQv(1 +3) [ei
From Lagally's formula a force of attraction is obtained between the hull and
the working propeller, since the afterbody represents a sink system which is
closer to the propeller sink than the positive forebody system. The internal
force is the thrust deduction or resistance augmentation in an ideal fluid.
One can further compute the wave resistance due to the working propeller alone.
Dickmann found that for normal conditions of depth of immersion and loading
factor this resistance is a small fraction of the thrust, say of the order of 1
percent ;it depends on a Froude number v/ygh (h immersion) and can reach great-
er values for high loading factors and small immersions. The interference be-
tween ship and propeller waves causes more important effects.
Following F. Horn??? the propulsive efficiency depends on the posi-
tion of the propeller relative to the wave created by the hull; in a wave
crest the orbital motion is directed horizontally forward, thus generating 2
favorable positive wake, while in a trough conditions are just the opposite.
Dickmann °’?°°11 has proved Horn's reasoning in a rigorous manner by means of
wave theory; he has further shown that beautiful experiments by Yamagata sup-
port his conclusions. In the light of these ideas wave patterns at the loca-
tion of the propellers should be more carefully studied.
Thus, the theory developed by Dickmann has succeeded in revealing
the mystery which for a long time enveloped the intricate problem of inter-
action between hull and propeller; but one is far from a complete quantitative
solution. We may note some facts where no complete agreement between theory
and experiment has been reached.
The thrust deduction (contrary to the wake) should be fairly inde-
pendent of wave phenomena or, otherwise expressed, the part of the thrust de-
duction coefficient due to waves t is very small. Some experiments, however,
indicate that the thrust deduction can be influenced by variations in the en-
trance of a hull which to our knowledge affect measurably only the wave
67
pattern.4+* This discrepancy in the behavior of t,, introduces an element of
uncertainty in our reasoning which should be eliminated as far as possible by
reliable experiments. Another disturbing fact is the bad quantitative agree-
ment between the values of the suction force derived from the difference be-
tween thrust and resistance and those evaluated from pressure measurements;
this discrepancy is not due primarily to wave phenomena.
The fact that the wave formation around a hull is only slightly in-
fluenced by the propeller action gives a valuable support to our present
technique of model towing phenomena; a model test without screws discloses 4
basic property of the hull which normally remains unchanged in a self-
propelled condition. Additional suggestions are made by theory:
a. To locate the propeller in a region of a high wave wake.
b. To change the wave pattern of the hull itself in such a way that a
high wake may be created at the propeller disc. This viewpoint has not hither-
to been applied in model research; it is even doubtful if it would lead to
practical results.
Dickmann has extended his analysis to actual flow. He has explained why non-
uniformity of the wake increases the thrust deduction.*
In the light of this knowledge and our remarks on stern bulbs an
idea due to E. Hogner and G. Kempf?2® may be reconsidered. These authors pro-
posed to create an axially.symmetrical wake by giving an appropriate shape to
the run. Unfortunately, according to van Lammeren, gains in propulsive ef-
ficiency so reached are counterbalanced by increased resistance.
However, it is worth while investigating whether with such a run a
stern bulb effect can be obtained which combined with a bow bulb may reduce
the wave resistance. The remark applies to high-speed single-screw ships as
weli as for triple-screw vessels.
Finally the possible influence of propeller suction on separation
over full sterns has been often mentioned. Unfortunately this idea cannot be
dealt with in a reliable way until adequate research is carried out intothe
flow patterns at the stern.
8. RESISTANCE IN RESTRICTED WATER
Two types of problems are met with: Motion in shallow water, in
which only the depth is limited, and motion in canals.
*van Lammeren has deduced this statement from experiments .°*
68
Only a short enumeration of the most important facts is possible.
The wave pattern of a ship advancing with a constant speed v in shallow water
differs from the corresponding pattern in infinite depth by an increase in
the wave length A and by a change in the configuration of the waves, more en-
ergy being stored in the echo waves.
The existence of a critical velocity is due to the fact that for
v > Vgh the transverse waves must disappear.
The phenomena are complicated, but useful results have been reached
by a simple method devised by Schlichting~? which yields an estimate of the
resistance in shallow water when a resistance curve for infinite depth is
given. Schlichting introduces the hypothesis that the resistance in deep and
shallow water is the same when the length of the free wave corresponding to
the ship speed is the same. Thus differences in height and configuration of
the wave pattern are neglected. Using well known formulas for the wave (ship)
= Vex = Ves émh
We = &5,-(deep water) and Vv, = Vas, tanh => [32]
(shoal water) it is easy to calculate the shallow-water speed Vn from the con-
velocity
dition A = const, when Vy and h are given. Experiments give considerable sup-
port to this rather bold idea, but the method fails at speeds above the
critical.
A hydrodynamic solution for shallow-water resistance has been given
by Sretensky;°® it is valid approximately under the same assumptions as
Michell's formula (see Appendix 2). Sretensky's integral has been used to
demonstrate that Schlichting's hypothesis has some theoretical foundation in
that, within the subcritical range, the most important phenomena can be deduced
from it.?°
We summarize briefly some important points:
Vv ° — —
a. An additional Froude number F = Veh is useful; Fi = "alee hor Ve = Vgh,
the critical wave speed.
b. Shallow water effects become appreciable only when Fh > 0.7; gen-
erally below this limit the water can be considered as infinitely deep (in so
far as wave resistance is concerned).
ec. Obviously the common Froude number F = ya's connected with Fa by
the depth-length ratio e
69
For general analysis of resistance properties in shoal water, the
ratio h/L is more characteristic than the draft-depth ratio H/h. As usual,
Taylor's representation of the matter*+ based on F and h/L is superior to
others.
Within the validity of the theory (H/h small) H/h is a parameter
whose changes cause only moderate deviations from corresponding deep water
phenomena when h/L is kept constant. Hence, it is in principle disadvantage-
ous to link up wave effects with the draft H instead of the length L.
d. Besides the critical speed defined by te: Vgh, another critical
speed v,, (the speed of maximum resistance) is sometimes used; we prefer to re-
strict the terminology "critical speed" to Vig = Vgh.
The maximum wave resistance occurs at a speed ve < Vgh, the differ-
ence Veh - Ve increasing with increasing h/L.
e. The maximum wave resistance in shoal water for a given displacement
does not seem to depend much on the ship form. This can be inferred (within
the validity of the theory) as follows: The speed of maximum resistance in
shoal water corresponds to an extremely high speed in deep water; but in the
latter case, displacement is decisive and the resistance is nearly independ-
ent of form.
f. When h/L becomes large, say of the order of unity, so that Fy crit-
ical corresponds to F = 1, the difference between deep and shallow water re-
sistance is not too large. This is the reason why resistance tests with mod-
els of high-speed ships yield useful results for deep water, although they may
run in the range of the critical speed.
The motion of a ship in a canal can be treated by the methods of
engineering hydraulics. A clever application was made by Kreitner;}* he ex-
plained supercritical conditions by analogy with the hydraulic jump and gave
information on the average speed of flow of the water around a ship in a canal.
The study of motions in a canal,besides its immediate practical application to
canal and river shipping, also has a bearing on foundations of model testing.
In a canal the speed for maximum resistance occurs somewhat earlier
- than in shoal water of unlimited breadth. 0. Mueller tries to explain this
fact by introducing an empirical Froude number~*
Vv V: h V h
2S aR aan | elie 25+ 1 [33]
Substituting the hydraulic radius
a for the depth h.
—+ 1
(0
Theoretical solutions of the problem are due to Keldysh and Sedow>®
and Sretensky.°° Using the latter's formula, some estimates were made of the
effect upon the wave resistance of different size models with a parabolic
waterline in rectangular tanks.1°° When properly extended, such evaluations
should lead to correction factors for the influence of the limited cross sec-
tion. Obviously the task is much more complicated than the corresponding one
in wind tunnels. To cover adequately the whole field we need results for:
a. The whole useful range of Froude numbers.
b. Different ratios h/b.
ce. Different ratios L/b.
d. Different characteristic ship forms.
Preliminary calculations indicate that we have to distinguish roughly between
deep- and shallow-water conditions (say h/b ~ 0.5 and h/b < 0.1). For deep-
water tanks we distinguish between conditions far from the critical speed
(FL = 1) and near to it. When Fh < 0.7 and L/b < 0.5 the limited breadth of
tanks does not seem to influence the wave resistance appreciably, although a
recent publication by Wigley does not support even this assumption.”
A wave resistance increase up to 15 percent has been found at high
Froude numbers when using a model length ratio L/b = 1 instead of L/b = 0.5.
Generally, an increase in resistance due to tank walls is found, when the re-
Sistance curve for infinite liquid has a tendency to rise. At some Froude
numbers the resistance in the canal is smaller than in open water. Models
with a length L = b are liable to furnish totally "wrong" resistance results
at some Froude numbers when applied to deep-water conditions.
When the intended correction factors will be available we shall be
able to indicate upper limits of model sizes as given by wave phenomena, while
lower limits are fixed by conditions of viscous flow.
Somewhat surprising results are found for shallow-water conditions,
Fy = 0.9: Keeping the canal breadth b constant but doubling both the size
of the model (II) and the water depth (h/L = const) the calculated wave-
resistance coefficient of the large model was nearly twice that of the small
one, Figure 35. The values were
I II
h/b = 0.05 h,/b = 0.1
L/b = 0.5 L/b = 1
h/L = 0.1 remaining the same in both cases. But even for Arrangement I the
resistance coefficient is some 35 percent higher than for the corresponding
case where b > o and h/b > o.
11
Near the critical speed, model results must be converted cautiously
into full-size data valid for unlimited breadth. On the other hand, at
Fh = 0.8 the resistance of the models discussed is practically uninfluenced
by the finite breadth b.
UM
Figure 35 - Models in Shallow Water Basins
Having in mind the oscillatory character of wave effects, one cannot
expect too much from the Equation [32]; perhaps at the best it can explain the
earlier rise of the resistance curve due to the finite cross section near the
critical speed. Some experiments made at the Hamburg Tank appear to support
the theoretical reasoning. >* Comstock and Hancock's paper? will furnish val-
uable checks, although it is thought that the influence of the finite cross
section on wave resistance has in principle a more complicated character than
represented by the curves on Figure 18 of the paper quoted.
9. EXTENSION OF THEORY*
Not ali shapes of bodies of revolution can be represented by singu-
larities distributed over the axis. A fortiore, not all ship forms can be gen-
erated by sources and sinks distributed over the center plane; even the H/B
ratio attainable by this procedure is limited to H/B > 0.5. Furthermore, it
is impossible to expect that highly curved parts of the hull such as the
bilges can be obtained by a plane distribution.
A first attempt to escape from this restriction was the introduction
° His "interpolation formula"
of volume singularities proposed by Hogner.*
agrees in the limit of vanishing draft or vanishing beam with his integral or
Michell's integral, respectively.
Havelock has criticized Hogner's attempt from theoretical considera-
tions;°? nevertheless, a closer agreement between calculation and experiment
can be expected by its use. Some numerical work based on the "interpolation
formula" showed results which, although not unfavorable, were not conclusive °°
Unfortunately, we do not know what kind of body results from the singulari-
ties assumed. Another ingenious proposal was linked by Hogner ;7? who sug-
gested applying influence lines to resistance research.
*This chapter can be omitted by readers interested only in practical results.
{2
The most general solution consists of a distribution of singulari-
ties over the ship surface itself.
For this case a resistance integral has been given by Havelock. °°
However, he himself adds somewhat resignedly: "it is not likely that it would
give any better agreement with experimental results; for the more we depart
from the simple narrow ship the more necessary it is to take into account the
effect of wave motion upon the distribution of fluid velocity around the ship."
Thus three steps can be listed which may lead to a really compre-
hensive theory:
1. Determination of the distribution of surface singularities corres-
ponding to a given ship form. A solution by an integral equation has been
indicated by Kotchine,°* and has been discussed for very low Froude numbers,
i.e., actually for a deeply submerged doubled body. Here, however, the most
important problem consists of finding the changes of singularities generating
a given form with speed (Froude number). No solutions so far are known except
an investigation of a submerged cylinder by Havelock.” It appears therefore
to be appropriate to start with the simplest bodies.
2. The calculation of the real attitude of models due to changes in
hydrodynamic-pressure distribution and of the resulting resistance. A general
approach by Hamilton's integral is imaginable in principle, but nothing can
be said as to the practical value of such an attempt. Probably it is more
reasonable trying to find corrections for simple bodies as under 1.
3. The representation of the viscous flow by appropriate systems of
Singularities as a base of resistance calculations.
The difficulties in dealing with this general approach are very ser-
ious. Therefore, some simpler methods of improving Michell's integral have
been sought.
Havelock recommended for high-speed vessels a method based on con-
centrated singularities; as originally applied by him, it means even a simpli-
fication of Michell's theory—the substitution of concentrated sources and
sinks along the center plane for plane surface singularities.** In so far as
this method leads only to simplified computations, we are not interested in it
from our present viewpoint; but it can be generalized by locating these
sources outside of the plane.
Appropriate formulas have been developed by Lunde for this case.**
When applied to a model investigated by Wigley, Lunde's calculations showed
a somewhat better agreement with experiment than computations based on
Michell's integral. It is expected that Lunde's method will be a useful means
———————
13
of research. Again, the difficulty arises of how to find the shape of the
body generated by the assumed concentrated singularities.
A rather general method due to Kotchine>® may be mentioned, devel-
oped with the idea of obtaining all the force components due to free surface
effects. Kotchine's elegant expressions have not yet led to new results for
the wave resistance, but they have proved to be useful when calculating verti-
cal forces.
Finally, reference must be made to the work of R. Guilloton. Two
new basic ideas may be distinguished in his publications (References 15, 16
and 17):
a. The improvement of resistance calculations over those based on
Michell's potential only, mentioned before.
b. The use of resistance calculations based on a second approximation
to the velocity potential valid for forms with fuller section (flat bottom).
Both methods of calculation are characterized by the use of wave
profiles and of the pressure distribution due to wave motion. An original
approach based on finite differences leads to solutions for an arbitrary ship
form with finite draft.
Figure 36:‘indicates how far Guilloton's method is superior to
Michell's integration of the pressure
distribution from a physical point of
view; while the latter is confined to
the underwater part of the form only,
Guilloton extends it over the proper
limits. Objection can be made that
Guilloton's procedure is liable to lead
to errors of computation, since the
result is arrived at by evaluating quan-
tities which are relatively small dif-
ferences of other quantities, themselves
known with only limited accuracy. This
question can only be decided by actually
performing calculations, and the present h :
author has not to date had the opportun- Lae So cabhesuneupesurtuuri oy
ity of carrying out such work. However, of a Ship, Following Michell
Guilloton's method of evaluating wave and Guiltoton
profiles and the wave pressure exceeds as
to rigor all the proposals hitherto made; hence, it must be considered as a
fundamental work intended to base the computation of wave resistance on an
evaluation of the complete flow pattern.
74
10. WHOLLY SUBMERGED BODIES
The wave resistance of wholly submerged bodies deserves a short
treatment even within the scope of the present report. We mentioned the prob-
lem when treating ships of least resistance and the influence of beam on the
resistance of ships with the purpose of elucidating the conditions valid for
surface vessels.
More generally, the theory of wave resistance of wholly submerged
bodies moving horizontally near to the surface represents an interesting
study in hydrodynamics capable of wide applications; it has a fundamental
bearing on problems of submarines and torpedoes.
An important problem in tank work is to reproduce conditions for a
body moving in an unbounded fluid. In this case we try to establish the min-
imum depth necessary to avoid wave phenomena or to reduce them so drastically
that their influence can be eliminated by rather small correction factors,
similar to the well known procedure in aerodynamics. Such considerations were
sometimes neglected in earlier experiments and led to doubtful results.
From Michell's (Havelock's) integral a resistance formula can be im-
mediately written down which is valid for a totally submerged system of singu-
larities distributed over the vertical centerplane and which is suitable to
generate a submerged body like a submarine, etc., provided the total output
is zero. Particularly, distributions symmetrical with respect to a horizontal
plane can picture double models which are valuable for resistance research.
For a first orientation we may confine ourselves to bodies of revolution,
whose image system is’given by a line distribution.
The theory is based on the assumption that the depth of immersion f
is great compared with the radius b of the resulting body; f/b>>1. Under
this condition the wave resistance of a very elongated body of revolution
gives a fair approximation to the resistance of more general bodies of the
same length and sectional-area curve, provided the vertical and horizontal
maximum dimensions (height H and beam B) do not differ too much. This state-
ment is supported by some calculations made earlier and by a remark due to
Havelock ®® and to some extent by Lamb's formula.?® From a formal point of
view the calculation of wave resistance for a body of revolution is simpler
than for a surface ship, the same auxiliary functions being used. A matter
of primary importance for practical work is to determine the limiting value of
the ratio f/b above which the theory may be expected to yield a reasonable
result.
15
Calculations have been checked by experiments for three bodies of
revolution over a range of medium Froude numbers (up to 0.42).+?° The quanti-
tative agreement was not satisfactory for F = 0.26, and some unexpected shift
of phase was found between calculated and measured resistance curves. However,
a first orientation as to the relative properties of different models can be
obtained even for the case f = b, i.e., when the backs of the bodies touch
the surface. Generally, a closer agreement is found between computations and
experiments for submerged than for surface ships as regards the form of humps
and hollows in the resistance curve. Some astonishing results were found:
For instance due to pronounced interference effects at some Froude numbers
> 0.35 the total resistance of a full body (¢= Ce = 0.80) is lower than that
of a very fine one (¢ = C_ = 0.546) having the same principal dimensions but
some 30 percent less volume.??°
One must be cautious in applying results obtained from blunt bodies
like a sphere or circular cylinder to elongated bodies. For the former the
speed v = Vet is a critical value since the resistance curve has a maximum
value at that speed (comparable to the case of finite depth h); hence, the use
of a Froude number Fe = vV¥ef is advisable in plotting results. But resistance
curves of bodies having ratios of slenderness comparable to those of ships,
a/b = L/VA.,» do not show any peculiarity at v = Vef; therefore the dimension-
less number Fe = v/Vgf cannot be recommended when investigating the resistance
at constant immersions. On the other’ hand the parameter v/Vet is appropriate
when investigating the resistance as a function of the immersion.
Finally, reference may be made to the problem of bodies of least
wave resistance. It was mentioned that results are similar to those for sur-
face ships, but peculiarities of form are still more pronounced. For a given
Froude number the optimum form varies slightly with the depth of immersion.
Figure 3/7 indicates the shape of some forms derived under rather special con-
ditions. In the light of the remarkable qualitative agreement between calcu-
lation and experiments it appears legitimate to develop optimum forms by
calculations.
So far no attempt has been made to treat bodies having blunt noses,
which cannot be represented by a line distribution.
76
topt-o F=0.25
O Of O02 03 0.4 05 06 O07 08 09 10
§
Figure 37a
Popt=0.838 F=+0.408
Poot =0.703
ie) Ol 0.2 03 0.4 oe 06 O7 O08 O9 1.0
Figure 37b
Figure 37 - Bodies of Revolution of Least Wave Resistance
m(é) is the doublet distribution (approximate sectional area curve).
SUMMARY
Pending a thorough tabulation of Michell's integral, the relations
between ship form and wave resistance are of necessity discussed here only in
a rather broad way.
1. The relations involved are generally complicated. Thus it is not
surprising that two seemingly contrary basic properties can be stated:
a. Under certain conditions the wave resistance can be very
sensitive to changes in lines; and
b. Widely differing hulls can yield equivalent resistance re-
sults for given Froude numbers. Under the headings 2 to 8 below.
some theoretical results are summarized and under 9 to 16 the re-
sults of experimental checks are discussed.
1
2. From the form of the wave integral it follows immediately that for
research the appropriate independent speed variable is Froude number referred
to the length of the ship. This rule holds for all slender bodies.
3. When investigating the wave resistance as a function of the ship
form, two working hypotheses are widely used: '
a. A more basic division, in which the resistance is con-
sidered as being a separate function of the principal dimensions
(or their ratios) and cf the dimensionless shape and,
b. A splitting up of the resistance due to the dimensionless
shape into functions dependent upon the longitudinal and vertical
distribution of displacement. More general and more rigorous in-
vestigations are required to consider the mutual interdependence
of most of the factors concerned.
4, The wave resistance R depends upon the square of the beam R ~ B®
and in a more complicated manner upon H. Relations R = R(H) can be estab-
lished by somewhat cumbersome but not difficult computations. However, the
limits of validity of R(B) and R(H) derived from Michell's integral are
restricted.
5. The concept of the sectional-area curve, which embodies the longi-
tudinal distribution of displacement, proves to be fruitful both from a the-
oretical as well as from a practical viewpoint. It has been shown that the
basic parameters ¢ and t are indispensable for any research work on the sub-
ject; in addition it has been demonstrated that Taylor's curvature parameter
x is valuable when dealing with fine lines. However, even the three param-
eters ¢, t and « may not be sufficient to fix the wave resistance properties
of a ship line. For fuller shapes the length of paralleled middle body re-
places the curvature x as the third parameter.
6. From this fact it follows that any systematic research on ship forms
should be based on analytically defined lines and surface equations. The most
suitable expressions are polynomials.
(. By evaluating the resistance integral for a sufficient number of
"longitudinal" polynomials, the resistance properties of the whole field of
normally shaped sectional-area curves can be derived. This may lead to the
use of further parameters as enumerated in the present report. Besides, import-
ant results for such peculiarities as bulbs and cruiser sterns already have
been obtained.
78
8. The influence on wave resistance of the vertical distribution of the
displacement (within a dimensionless shape) can be investigated with less ef-
fort. It leads to the simple rule that the displacement should be arranged
as deep as possible. Thus, U-shaped sections, from the viewpoint of wave re-
sistance, are as a rule superior to V-shaped sections; possible exceptions are
mentioned.
9. The experimental check of theoretical results at low speed-length
ratios is seriously hampered by the principle on which Froude's method is
based as well as by experimental inconsistencies. No reasonable experimental
analysis of the wave resistance of full slow hulls can be made without an
additional research on viscous-form drag, especially viscous-pressure drag.
However, keeping in mind these restrictions, the following conclusions can be
drawn from comparisons between calculated wave resistance and measured re-
sidual resistance values 10 to 16.
10. The trends of the calculated and measured resistance curves gener-
ally agree well, but the interference effects (humps and especially hollows)
are exaggerated by the theory.
11. The absolute values of the curves agree reasonably within certain
ranges of L/B and ¢g; for low L/B ratios and high prismatic coefficients at
small Froude numbers the discrepancies are large. Michell's theory clearly
overestimates the effects of increasing beam; unfortunately experimental data
dealing with the dependence of resistance upon beam and draft are astonish-
ingly scarce, so that neither theory nor experiment yields accurate data on
resistance effects due to the most elementary changes of a hull.
12. The relative merits of different forms as established by theory are
in many cases supported by experiments. However, theoretical deductions are
liable to lead to exaggerations and even to errors when they are based on
interference effects which can be affected by viscosity. Theory may fail com-
pletely when phenomena are discussed which depend essentially upon viscosity,
such as the relative efficiencies of fore and afterbodies, unless some addi-
tional corrections are introduced.
13. The close coincidence between theoretical and experimental deduc-
tions on the effect of small changes in shape is stressed as a fact of primary
importance.
14. Although experiments generally corroborate theoretical results as to
the influence of the vertical distribution of displacement, there are excep-
tions where V-sections in the forebody are superior to U-sections at high
Froude numbers contrary to calculated results. The actual optimum t values
19
are much lower than following from theory or Taylor's experiments. Thus the
Vv
universal validity of Taylor's a ty charts (Reference 42) must be
questioned.
15. The omission of the actual model attitude in the theory causes er-
rors at high Froude numbers.
16. The application of Michell's theory to ships with a flat bottom, al-
though contrary to the conditions for which the integral is valid, still
yields useful results. There are theoretical and experimental indications
that the concept of dimensionless shape may be overstrained when it is applied
to larger values of B/L or B/H.
17. Shallow-water effects can be investigated by a formula due to Sreten-
Sky. The basic parameters are a Froude number F, = v/Vgh and the ratio of
depth of water to model length h/L; the ratio H/h is less characteristic as
long as it is not close to unity.
18. A further integral valid for the wave resistance of ships moving in
a rectangular canal yields information on the permissible model sizes for dif-
ferent towing basins. Earlier data based only on the ratio of the cross sec-
tion of the models to the cross section of the basins are generally insuffic-
ient. Correction factors can be derived for converting model results to full
size; besides F, and h/L, the ratios of model length to basin width, L/b, and
of basin depth to basin width are characteristic parameters.
19. The simultaneous treatment of the ship and the propeller leads to
important results, on the interaction between the hull and the propeller, in-
cluding wave phenomena.
20. The resistance of wholly submerged bodies, especially bodies of rev-
olution, when running close to the surface can be treated on similar lines to
those of ships.
21. A short synopsis of methods is given which have been proposed by
different authors with the intention of improving Michell's theory of wave
resistance. The direction of these aims is given by the serious restrictions
of Michell's theory:
a. Assumption of a frictionless liquid.
b. Assumption of a wedgelike form.
80
ec. Assumption of a fixed position (neglect of bodily rise
or sink and trim).
d. Assumption of a small height-length ratio of waves created
by the ship including neglect of the ship form above the water when
computing pressure.
ACKNOWLEDGMENT
This report presents only a skeleton of the subject; by studying the
numerous references which, however, do not include important experimental work,
it is possible to fill up to some extent the rather sketchy picture. Further,
it is hoped that some of the projects now under way at the David Taylor Model
Basin will lead to broader applications of the theory. This part of the re-
search work, due to its magnitude, must be carried out by large institutions;
the development so far has been hampered by the fact that individuals inter-
ested in the problems have only occasionally found the necessary support.
The present investigation has been initiated and sponsored by the
Director of the Taylor Model Basin, Rear Admiral C.0. Kell, USN, the former
head of the Hydromechanics Laboratory, Captain F.X. Forest, USN, and the Chief
Naval Architect, Dr. F.H. Todd, to whom the author wishes to express his sin-
cere thanks. Dr. Todd's help was essential in completing this ample report.
APPENDIX 1
THE EQUATION OF THE SHIP SURFACE
The axes are chosen as shown in Figure 1. Then the equation of the
hull surface may be written
ay ae w7l(25%4)) 11]
eo nr a 4
eee
Z
Figure 1 - Axes of Reference
81
The double sign appears because the hull consists of two essentially symmetri-
cal halves. In most cases it is sufficient to consider
y = +y(x,z) [1a]
and to double the results.
When dealing with resistance problems it is advantageous to let the
X,Y plane coincide with the free surface and the Y,Z plane with the midship
section. Calculations of buoyancy, however, are commonly performed starting
from the keel. Some differences in definitions and denotations arise because
of the two systems mentioned, but in our present calculations we use only the
coordinates in accordance with Figure 1. The basic elements of the form can
be expressed as follows:
1. Load water line y(x,0) = X[x] [2]
2. Midship section y(0,Z) = Z[z] [3]
3. Longitudinal section y(x,z) = 0 z=k(x] [4]
«[z]
4. Sectional-area curve A[x] = 2 { y(x,z)dz [5]
0
for a rectangular center-plane contour
H
A [x] = 2 J y(x,z)dz [6]
0
Brackets [ ] are used to distinguish operations performed on dimen-
sional coordinates. The volume is given by
+l «[x] +1
v=2 ydx dz = A(x] dx [7]
fa j
=lnO)
_
Dimensionless coordinates are introduced:
g=x/l n=¢ =F [8]
Thus
y = y(x,z) = bn(é, 2) [9]
and the surface is described by its nondimensional shape,
n(&,¢)
a principal dimension and the ratio B/L and H/L. The nondimensional basic
curves are defined as follows:
82
1. Load water line n(&,0) = X(é) = n,(6) [10]
2. Midship section n(0,¢) = Z($) = n,(¢) [11]
3. Longitudinal section n(é,f) = 0 ¢= K(é) [12]
The area coefficients and coefficients of fineness are
code, ve) 8 [J (eae ' [ x(a [13]
B = f 2(eyae [14]
0
6=5(4, + 6p) = BL | nts svaeae + [nts evaear] [15]
The areas of sections are given by
k(é)
ag) =2[{ nlé,cyag
0
However, we define a slightly different dimensionless expression as sectional-
area curve
FE 1 1 k (é)
a’(é) = agAl§) = — | n(édae [16]
0
Throughout the text the symbol n will be used for equations of surfaces as
well as lines. In exceptional cases we add the symbols of independent varia-
bles to avoid ambiguity.
Ship lines and surfaces can be split up, with respect to the midship
section, into a main symmetrical (even) part n,(€) and a secondary asymmetri-
cal (odd) part n(é)- Only n, contributes to the total area or volume of the
curve or surface.
n=, +7, [17]
+1
[ nae = 2 | ngaé : [18]
0
The asymmetrical part alone contributes to the static moment with respect to
the midship section
S = Endé = 2 [ange [19]
83
The position of a centroid x /l = §5 is given by
B gate nas [20]
We define an “elementary ship" as a hull shape described by (1), the equation
m(6,o) = X(€)Z(¢) [21]
and (2) a rectangular center-plane contour. Dimensionless surface ordinates
are obtained as products of the corresponding LWL and midship-section ordi-
nates. Elementary ships are characterized by the following properties men-
tioned earlier:
(a) a*(é) = X(é) [22]
i.e., the dimensionless water line and sectional-area curves are identical;
hence
(b) g=a 6=a8 [23]
(c) All sections are affine to the midship section. It is an advantage of
dimensionless representation that integral properties like area coefficients
of curves, etc., are invariant to affine transformations.
In principle any continuous ship surface can be expressed by a poly-
nomial. This follows from Weierstrass! theorem: A continuous function y(x,z)
within prescribed boundaries can be approximated with any desired degree of
accuracy by a polynomial in x,z. Thus
y= aay
Deeg
are general expressions for the ship surface. Instead of [24] we choose the
[24 ]
special form
i Uh aC Gee [25]*
*The minus sign before 2 has been introduced by analogy with Chapman's parabolas. From a mathe-
matical point of view the plus sign is preferable in all formulas like [25], [27], [28], etc.
84
When the center plane contour is a rectangle even equations with a
small number of terms yield satisfactory results. When dealing with more gen-
eral cases, it may be advantageous to introduce fractional powers. However,
we need not consider such procedures within the scope of the present report.
Derivatives of dimensional and dimensionless values are connected
by
Che 18) Golul
rT [26]
water lines and sectional-area curves are expressed by
n
X(§) = n(§) = 1-Daig [27]
sections by
Z(¢) = n(g) = 1 -S bm [28]
where generally it is assumed that
n(é) = 0 for €= +41
Qos 1
Families with two arbitrary parameters are named basic systems.
These parameters can be expressed by ¢ or @ and Taylor's t-coefficient where
[29]
On
Bs -|23 30
| ae [30]
We use the symbol
(nisms ras) 6)5)) t))) =e | DN ae Te [31]
ny Mens -
with the condition
Don mi
@ may be substituted for ¢.
Further form parameters are
2
= out [32]
OF ve = 0
(3) the dimensionless curvature at the midship section
of the ship line,
(4) Length of parallel middlebody,
(5) Position of the centroid of the fore or aft area
(volume), for instance
85
1
| ngae
fe)
roy [33]
(6) The corresponding moment of inertia coefficients
1
2
Jone ag [34]
(7) Higher degree moments
[ng2as 35]
(8) Position of the point of inflection of the curve. Only condi-
tions (3) and (4) have been used in the present investigation; the latter has
been considered indirectly by varying the degree of the basic form and the
former by using special polynomials.
Various examples of basic families are reproduced in Figures 15 to
17, in addition some curves of $7 and oe are shown in Figure 38. Introducing
explicitly the parameters ¢ and t, the equation of a family can be put into
the form
n= f,(é) + 9f,(é) + tf, (6)
or
(ingryigs Of ©) 2 Allaasep 18 ©) & tellayaseyn Of 1) [36]
f,(é) = (n,njnz; 1; 0) complies with the conditions
on [36a]
[ofi(4)ae = 1
0; 1) with the conditions
PB) = (ian teliae t
=t, =]
ca [36b ]
{este
O€
[jro(elae = @
86
Formulas of the three functions involved are given for various values of
n,n,n, and the graphs drawn in Figure 18; for convenience the function
paar)
(n;nan,; 0.1; 0) = 0.1(n non, ; 1; 0) has been introduced.
8
(Siete
2a
0E a
05 5
he
<1 6
Ch
=
-4
8
+—
(2,4,6;9; 1)
ee |
aha
NEAL
Ae NS
Figure 38a
(2,3,4; 0; 1)
CNS
ING a
es
y
{
Figure 38 - BE
On
Figure 38b
’n
and
0é?
\
Pee
ee eS Ae
fo)
(0)
for Some Ship Lines
87
Finer variations of form such as changes of the curvature may be
performed by polynomials of the type (nyngngn, ; 0; 0) for instance
BA iyo E52) sis qullOnen Leite
(253 40508 0; 0) =& 9° Ur 36 9 § [37]
or
(2,476,850; O)N= G5 = 954" feo = 5e" [38]
complying with the conditions
1
= ba oO =
Jo (dag = 05| 2 I, 0
and
I ) = 0 [39]
&=0
é=1
(Cf. Reference 102). The presence of the square é* is obviously essential
when investigating the curvature of the midship section.
Additional even functions are introduced by using odd powers of abso-
lute values. Especially the cube [é|° is important for various purposes. We
normally omit the symbol | I.
When different equations are used for the forebody ur and afterbody
uN the symmetrical part of the resulting body is given by 1/2(ng + My)» the
asymmetrical by
tory = Bp Ca)
The system of curves representing water lines and sectional area curves dif-
fers from Taylor's system** by:
A. The choice of axes of reference. The origin is located at the mid-
ship section; the fore and afterbodies can be dealt with simultaneously even
if expressed by different equations.
B. A greater variety of forms.
C. The final purpose of obtaining explicit equations of the surface,
not of sets of lines.
D. Parallel middlebodies may be introduced. However, when using equa-
tions of surfaces for resistance calculations it is generally simpler to ap-
proximate a cylindrical part by high powers of &.
88
Surfaces of a more general character than these elementary ships
can be deduced in various ways. For rectangular contours surface equations
can be derived immediately from the expression [25]. However, it seems pref-
erable to begin with simple geometrical concepts.
A. Consider for instance
n = [x(6) - ve)e,(2)] 2(4) [41]
where the "fining function" v(é) complies with the condition
i]
<
(eo)
i]
Oo
v(1) = v(-1)
f, (0) = 0
[41] can be interpreted as an elementary ship minus a layer v(é)f,(¢)Z(¢)
which assumes zero values on the center-line contour. The equation of the
sectional-area curve becomes
at(e) = X(é) - v(é) [42]
with
1
a, = {ztg)e,(e)ae
Putting for instance f(¢) = ¢ we get an inclination of sections at the LWL.
When v(é) > 0, 8, > 0, Equation [42] expresses the fact that the sectionai=
area curve becomes finer than the water line towards the ends. The local sec-
tional-area coefficients are expressed by
a(t) = B- B xe [43]
i.e., they are smaller than 8 when 8, > 0,v(é) > 8.
B. An additional effect is obtained by substituting for Z(¢)
ni
2(.g) = 1 -b, (6)... [44]
By suitably choosing the functions b(é) and exponents n, the fullness and max-
imum curvature of sections towards the ends can be reduced.
When a parallel middlebody is inserted in the forebody of length a)
an appropriate expression for a water line is
89
n
seat “I
Similarly for the afterbody with a parallel length aN
- n
peep srg ps (46
Veen
APPENDIX 2
FORMULAS FOR THE WAVE RESISTANCE
All formulas here collected are valid for uniform motion only.
A. The Wave Resistance of a Source or Sink*
Figure 39 - Source (sink) Beneath the Surface
The flow due to a single source or sink has been studied by Have-
lock.®? Assume the usual axes but with 0Z directed vertically upwards (that
is the direction used in almost all Havelock's papers contrary to Michell's
assumption).
The velocity potential of a source m moving with a constant speed
v in the direction of x at a distance f beneath the surface can be written in
the form
-i+¢ [1]
*Unfortunately there is no easily readable introduction into the theory of wave resistance in any
Western language. Lamb's book stops just when matters become interesting for practice.
90
where
iets fe tates. sel (Alen ie)
)
1
Tesi) me eto (tz, Uw tele
n/r, represents the velocity potential of the source in an un-
bounded fluid,
-n/r, is the potential of a sink of equal strength at the image
point under the same conditions, and
¢?» is the potential due to wave motion.
[2]
For extreme Froude numbers ¢, vanishes; thus the potential is given
by
In the other limiting case, F > 0, the surface acts as a rigid cover;
propriate expression for ¢ is then
62 ee
ry lp
a term, say ¢,, due to waves, disappears again. The expression
m
oho
eH. 2
is legitimate as (1); it has been used by Dickmann.
Using the surface condition
0*¢ Oo Ow.
eum fo) Oa) vibe 7 v
(where w is Rayleigh's friction, Reference 25, K, ==£)
C
and Laplace's equation
Ag = 0
Havelock gives the expression for the velocity potential
+7 J
ey = ae { sec-0d0 { exp [-«(f-z) + ixw] dx
: i | At OT an sec"O + iusec 0
where Re stands for the real part. Dickmann's form yields
[3]
the ap-
[4]
[5]
91
+7 K + ipsec 90) expl[-x(f-z)+ixw
ie | 6 co|| K - Ksec*O + insec@ Ms [9]
+
x35
TE
and leads to a resistance
_ 16mpm*, .-27, peel
R = ee |S l2e,) + K, (27) | = See [10]
where Ko K) are Bessel functions of the third kind.
Tables of wave ordinates in the X,Z plane due to the motion of a
source, have been calculated by Wigley. °*
B. Michell's Integral
Four different methods have been used for calculating the wave re-
sistance of slender ship-like bodies:
1. Michell's original deduction based on the concept of a wedge-like
ship depends upon the computation of pressure changes dp due to the wave
pattern.
as 3 mite oy bs Og OY
Re 2{[apayaz = a[[on SY axaz = 2ov|[ 2? 2 axaz [17]
or
co es ;
ee | ( (ie se) an [12]
Vi= 1
1
where
+e k[z] 2 2
if = | ON gle cos (“ES )axaz
x
25 3
(WS)
iS Oy .-2792/v" hex
ve |) [ sue sin( )axdz
Lyk
z = K[x] is the equation of the longitudinal contour.
Only odd terms of the hull equation contribute to the value of I.
For a symmetrical hull I = 0.
Introducing dimensionless coordinates &é aie and putting
Ways
where ny Seer a ers obtain for a rectangular contour
gril gril H v2
I = HBI‘(y) = HB{ | oS el %! cosvededt [14]
(One)
+1 +1 3 Hv,
J = HBJ’(y) = BB | { = e / % sinyédédé [15]
0 0
15a denote the symmetrical and asymmetrical parts of the hull with respect
to the midship section. The resistance R can be written
ae Se 3 (2)
Ro Segre J feo) wk axa
This form has some slight advantage compared with the original one from the
point of view of computation. For rectangular contours integrations with re-
spect to é and ¢ can be handled independently; the discussion is based on
quadratures of the type:
dy [16]
1
Mm (y) = j é"sin édé
Mn'(y) = [ sPeos édé [17 ]*
0
_() om MESH oe
But?) ee emat eae 7
Especially simple expressions are obtained for the elementary ship: The inte-
gral J‘*(yv) can be written in the form
J (v) = S(y) @(y) [18]
where S(y) depends only on the longitudinal and @(y) only on the vertical
distributions of the displacement. Examples are discussed in the main text.
A rigorous proof of Michell's integral based on potential theory is
due to Sretensky.*%®
Three further methods of calculating wave resistance are mainly due
to Havelock. Explicit results so far have only been obtained in those cases
where a ship form can be represented by images.
Using an approximate connection between the normal velcoity and a
plane surface distribution
2mo = vee [19]
*In the paper by the author JSTG (1930) there is an obvious misprint on page 418 as the double value
of the integral is defined as M, (7). ;
93
Havelock has converted Michell's integral into the form®®’ 7?
m/2
R = 16m K,20{ (P? + Q7)sec? ede [20]
0
Ky - =, where
P= {J cexplix, x secO = K zsec* 0 ]dxdz [21]
iNEn ON)
substituting 9 2 oe a
Havelock's form of Michell's integral®?’"* is obtained
hye
R = PB | (p? + Q?)sec® ode [22]
2
TV 0
gz/v2 sec” 0
P = (2 e cos(=2 seco) dxdz [23]
z 6
Q = (($2 e ae sin(= secé) dxdz {24 ]
Vv
Compared with Michell's formula the direction of the z-axis is reversed (up-
wards). Havelock's form has advantages from the point of view of computation
because of the finite limits of the integral.
Putting sec@ = coshu a third form has been introduced by Havelock.”
The same result can be deduced as follows:
2. Using Lamb's method based on Rayleigh's frictional coefficient u
(frictional force proportional to velocity). With the inclusion of the fric-
tional term in the equations of fluid motion, energy is dissipated at a rate
equal to 2 times the total kinetic energy of the liquid and this must be equal
to the product Rv. This approach is efficient, but highly artificial.2®®
3. Establishing the connection between the wave profile at a great dis-
tance aft of a moving body and the wave resistance of the body. This method
represents an extension of the usual theory of group velocity. The rate of
work done on the fluid by the moving body (otherwise expressed, the power ex-
pended) is equal to the rate of work done across a fixed vertical control
plane minus the rate of flow of total energy across this plane.”
4, By the method of singularities (Lagally's theorem) sketched in the
main text. The resistance is computed from
= Lap {{ o(x,z) u(x,z)dxdz [25]
94
where o(x,z) is the source-sink distribution over the vertical center plane
and u(x,z) the x component of fluid velocity at the point x,z, which must be
calculated from 6¢/Ox.’* Evaluating [25], Michell's integral is again ob-
tained. However, the method of singularities appears to be far superior to
any other known, when determining the forces experienced by singularities in
a given flow; it enables us to calculate the lift and the transverse force as
well as the resistance, and is especially powerful when systems of bodies are
investigated. Havelock's and Dickmann's work furnished beautiful examples of
its application.
We mention finally formulas developed by Kotchine*® which have been
successfully applied by him and by Haskind*** to the calculation of forces on
floating or submerged bodies.
C. Sretensky's Formula for the Wave Resistance in Shallow Water
Sretensky has developed a formula for the wave resistance of "Mi-
chell's ship" in shallow water. The theory is valid under the same assump-
tions as Michell's integral provided the draft/depth ratio is small. The de-
duction is based on methods of potential theory.
_ 8npg eee R OF 20K, aCe ey [26]
Vis 4 yu = ue ~ BE tanh on mh cosh mh
0
where
P = |{ cosh m(z + h) cos) => tanh mh x o(x,z)dxdz [27]
Q = {{ cosh m(z + h) sinV tanh mh x o(x,z)dxdz [28]
Vv
a(x,z) is the source-sink distribution which can generate a ship. Using the
relation between the normal velocity component and source strength:
emo = vey
the integral can be easily rewritten in terms of the surface equation Vixens
The lower limit of the integral mM, is of special interest; it is
given by the equation v2
tanh mh [29]
i]
|
=|
ISP
2
tanh mh =F, m h [29a ]
95
The roots of [29a] can be represented as functions of the parameter
F.:
When the critical velocity Ian = 1 is reached mh becomes zero and
remains zero for ae >1. Thus for the whole supercritical range me OFpeuite
integral (2b) can be put into various shapes when performing actual
computations.
D. The Wave Resistance of Ships in Rectangular Canals
The rather complicated formula for this case has been derived by
Sedov and Keldysh and Sretensky.°®’°°
1G el 2 2 Th 2 2
RS ee es 2h, + Q?)
where
|Jcosh my, (z+h) sin) 2ktanh mh Xo(x,z)dxdz
eateets eae alae DE eee Vee oe ee
k 2
(1 + tar K ) coshem= hy 2 £2
b2m, 2 k Vv
k
ne is a similar expression with the cosine instead of the sin and represents
the influence of odd terms of the surface equation, k is an integer while b is
the width of the canal.
The values of my, are the roots of the equations
ty 2 _ gm ws
2 m 72 tanh mh = 7K
APPENDIX 3
THE VISCOUS PRESSURE RESISTANCE
Observations and some measurements of the velocity distribution at
the stern indicate that with normal models the chief reason for the viscous
pressure resistance is, contrary to earlier opinions, not the separation but
the pressure defect due to increase of boundary layer thickness.
Two methods have been proposed to deal with this problem in a quan-
titative way:
96
‘Figure 40 - Scheme for Computing Viscous Pressure Drag
1. It is assumed that the frictional layer does not disturb seriously
the potential flow outside of it. The static pressure p(A') at the boundary
point A' of the frictional layer is calculated and, using an essential proper-
ty of this layer, the assumption is made that the pressure on the body (Point
A) is equal to p(A'). It is claimed that the viscous resistance so computed
agrees well with experiments.’
2. It is assumed that the body and the frictional wake form a new ef-
fective body, for which the pressure distribution may be calculated from con-
siderations of potential flow. This procedure leads again to a pressure drop
at the stern compared with the calculation applied for the original body.*?
Obviously these two methods can be only applied to infinitely long
cylinders (two-dimensional case) and to bodies of revolution, for which the
boundary layer thickness can be calculated. Thus it seems to be natural that
any research on viscous pressure resistance starts with investigations on
these bodies; a body of revolution is obviously a closer approximation to a
double model than an infinitely long cylinder.
As has been pointed out, experimental data are very scarce. The
following empirical formulas are quoted:
1. Two-dimensional case (symmetrical profiles)
Hoerner's formula for the total viscous drag@*
C d\4
R a ee
a1 + 25+ 70(+) a-thickness [1]
La)
2. Axial symmetry (bodies of revolution)
Hoerner's formula for the total viscous drag@?
Ral Ee ES) :
Bohr e on. (=) d-diameter [2]
—
oT
For double models and bodies of revolution a formula has been given
by Weinig.~”°
It is obviously erroneous to calculate the ratio of viscous pressure
resistance of ship forms to corresponding total viscous drag results valid
for cylinders of the same B/L ratios as can be found even in serious books.
The concept "fairness of lines" which includes some postulates as to
the continuity of their derivatives has been developed only empirically.
The present theories of wave resistance do not yield any answer as
to the order of "smoothness of lines" required since they deal really with
image distributions, not with the actual ship form.
There exist, however, some results on the influence of discontinuity
in curvature on the pressure distribution in the two-dimensional case. At
such points the pressure curve is characterized by a vertical tangent, i.e.,
a sudden change in pressure must be expected. When these critical points lie
in a region of rising pressure, an increasing tendency to separation may be
expected. **
Even a discontinuity in the third derivative of a ship line leads
to an indentation in the pressure curve; it is, however, less probable that
the boundary layer may be adversely affected by it.1?4
Our actual knowledge as to how the ship resistance depends on such
peculiarity of lines is practically nil.
REFERENCES
LIST OF ABBREVIATIONS
TINA Transactions of the Institution of Naval Architects (London)
TSNAME Transactions of the Society of Naval Architects and Marine Engineers
(New York)
J.S.1.G. Jahrbuch der Schiffbautechnischen Gesellschaft (Berlin)
TNECIMES Transactions of the North-East Coast Institution of Marine Engineers
and Shipbuilders (Newcastle)
Proc. Royal Soc. Proceedings of the Royal Society, A, (London)
ZAMM Zeitschrift fir Angewandte Mathematik und Mechanik (Berlin)
=)
FF WNW Pf
1933.
‘oO CO WN WwW
1938.
Wo
V2.
WDe
98
Ackerson, J., TSNAME (New York) Vol. 38, 1930.
Amtsberg, H., J.S.T.G. (Berlin)/Vol. 38, 1937.
Antimonoff, K., Trudy WNITOSS (Leningrad) 4, Vol. 4, 1934.
Baker, G.,,"Ship Design, Resistance, Propulsion," Vol. 1 Liverpool,
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Bragg, E., TSNAME (New York) Vol. 38, 1930.
Chanovitch, I., Theoretical Naval Architecture (Leningrad) 1937.
Childsky, W., "Carene exact" (Brussells)
Dickmann, H., V Int. Congr. Applied Mechanics (Cambridge, Mass. )
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leh
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1925.
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1928.
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Alls
Cr
25)
Graff, W., J.S.T G. (Berlin) Vol. 35, 1934.
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Hogner, E., J.S.T.G. (Berlin) Vol. 33, 1932.
Hogner, E., Proc. Royal Soc. (London) Vol. 155, 1936.
Hoerner, S., Jahrbuch d. Luftfahrtforschung (Berlin) Vol. 5, 1942.
von Karman, Th., IV Int. Congr. Applied Mechanics (Cambridge,
England) 1934.
24,
Lord Kelvin, Scientific Papers, Vol. IV, 193/.
OEE EEE EE EEO re ae
99
25. Lamb, H , Hydrodynamics, Sixth Ed. (New York) 1932.
26. Lamb, H , Proc. Royal Soc. (London) Vol. 111, 1926.
27. Lagally, M., ZAMM (Berlin) Vol. 2, 1922.
28. Lindblad, A., TINA (London), 1948.
29. Lindblad, A., TINA (London), 1933.
30. Loitsiansky, L., Prikladnaja Matematika i Mechanika (Moscow) Vol.
XI, 1947.
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