Report 77-0065

DAVID W. TAYLOR NAVAL SHIP
RESEARCH AND DEVELOPMENT CENTER

Bethesda, Md. 20084

DRAG, FLOW TRANSITION, AND LAMINAR SEPARATION
ON NINE BODIES OF REVOLUTION HAVING
DIFFERENT FOREBODY SHAPES

by

John L. Power

| a A paper
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SHIP PERFORMANCE DEPARTMENT
RESEARCH AND DEVELOPMENT REPORT

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‘Report 77-0065

MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS

DTNSRDC
COMMANDER

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TECHNICAL DIRECTOR

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CARDEROCK

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SHIP PERFORMANCE

AVIATION AND

SURFACE EFFECTS
DEPARTMENT ig

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MATERIALS

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CENTRAL
INSTRUMENTATION

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REPORT DOCUMENTATION PAGE | EES a!
1. REPORT NUMBER 2. GOVT ACCESSION NO,| 3. RECIPIENT'S CATALOG NUMBER
DTNSRDC Report 77-0065

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

DRAG, FLOW TRANSITION, AND LAMINAR SEPARATION
ON NINE BODIES OF REVOLUTION HAVING

» AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(a)

John L. Power

. PERFORMING ORGANIZATION NAME AND ADDRESS : PROGRAM ee a Ieee TASK

i AREA & WORK UNIT NUMBERS
David W. Taylor Naval Ship Research
and Development Center SF 434 215 1Z, 1-1520-004 and

Bethesda, Maryland 20084 SSL66007, 1-1552-135

. CONTROLLING OFFICE NAME AND ADDRESS . REPORT DATE
December 1977
” NUMBER OF PAGES
58

. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) - SECURITY CLASS. (of thie report)

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SCHEDULE i

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

. DISTRIBUTION STATEMENT (of this Report)

. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)

. SUPPLEMENTARY NOTES

119. KEY WORDS (Continue on reverse side if necessary and Identify by block number)

(U) Turbulence Stimulation (U) Drag
(U) Transition (U) Submarines

}20. ABSTRACT (Continue on reverse side if necessary and identify by block number)
Resistance has been measured of nine bodies of revolution, having equal volume but varying forebody
| shapes. Forebody shapes ranged from extremely blunt to extremely fine and included two that were flat faced.
The forebodies were altered by changing their length-to-diameter ratios (L/D’s) and prismatic coefficients.
| Drag results indicate that when the prismatic coefficient is fixed and the L/D is decreased, the residual
resistance will increase modestly. Increasing the prismatic coefficient at small L/D’s increases residual resistance,
however, at moderate L/D’s it does not. The results suggest that a flat-faced shape in itself does not increase

resistance. :
(Continued on reverse side)

DD en, 1473 EDITION OF 1 Nov 65 Is OBSOLETE

S/N 0102-LF-014-6601 UNCLASSIFIED

a SS
SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

UNCLASSIFIED

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(Block 20 continued)

‘ In addition to resistance experiments, transition regions on the models were located, using hot film
probes. Calculations predicted laminar separation on five of the model forebodies. The hot film
measurements confirmed that separation did occur at the locations predicted; downstream of the separation
locations, turbulent flow occurred immediately. The remaining four forebodies exhibited well-defined

| natural transition regions. Flow properties in the transition regions, measured by the hot film gages, have
been compared with predicted spatial amplification ratios of disturbances, calculated by linear stability
theory. Results have failed to show a single relationship between measured flow properties and computed
spatial amplification ratios. Correlation of amplification factors with flow regimes varied, both with

forebody shape and Reynolds number.

UNCLASSIFIED

Eee
SECURITY CLASSIFICATION OFTHIS PAGE(When Data Entered)

TABLE OF CONTENTS

Page
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TUREAIN'S HL@ NBT! © CAMO INS eee oss es eee atthe et Nar heat ne, ese eer 15
RESISTANCE RESULTS ....... is SRR RY AERC, SORE Rte OE case Re rtamtede SUA 16
GOIN GIBW STON rer arene etch nanan eee ek est tec cD cies tt Aue Un ENE GRAN AIG a te SER EPR 20
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FUE EVESRUESIN GES etree reey on eo So eis ones ea PETS tre ee SE SR MSS TM em A Cesc, So RUSTIC Se ne Poe 47
LIST OF FIGURES
a Body ots Reviolunty Om seart ceccee ce coro eee Oe eee eee ee OEE a ES RE roan ne eee VD)
D> PHO mles Gir |Bormaloravelas Ort. SSiaVES ll, coosoossdosedeeeeonessbeeceadococecacoosussodbnedcdenlecondonohbteddadeosuse DD)
Si wholes col Honebodies: Of SCLICSi 2 eihesnccns\sececn nace so cers e ere eee scan cease ae ee
4 — Computed Distributions of Pressure Coefficients
OneAxisymmetn Gakorebodieshol Seniesa mecaeeaceee cae ee meee eee en tae 23,
5 — Computed Distributions of Pressure Coefficients
OnyAxisymmeticehonebodiesvOltSemesi essa ss eee eee eee eee eee 24
==" Lokone: Leb cal sg aban enn MONON Scaesecodedasadacooddacoc seouesbencoabaccenoneReneBRORaGABACAeRGcH AG HHSaSaESoBenHecEOCS DS

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Bie
le

Representative Hot Film Signals for

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Idealized Body Frictional Drag Distribution

Rigaiclo ye. OWieel oN KengeXe | [veil o) Pc cabo cddeaaauadasasedacdedsecneaddlteoeseeseaandbeduaaseetede

Measured Flow Regimes on Model 3,

@ComparediwitheAmpliticationt hactones: re. pee eee eee

Measured Flow Regimes on Model 4,

CompareddwithwAmpliticatronpRactom ence ccteesereteceeee eee

Measured Flow Regimes on Model 6,

Compared pwith?Ampliticationthactonmecs. see cte eee eee

Measured Flow Regimes on Model 8,

Compared twithvAmpliftications Factomecse sss soe eeereneeneeneeee eee

Residual Drag Coefficients for Models

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Measured Total Drag Coefficients of

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Variation of Cp with Forebody

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Prototype Total Drag Coefficients of

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LIST OF TABLES

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Computed Incipient Cavitation Speeds

Of NinesROTE BOGIES eer ccske yee rae ae

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Locations of Wire Stimulators and Transition

or Laminan Separation withouteStumulatorsieees eee eee

Correlation between Amplification Factors

and’ MeasuredmiransitioneD ataceaeeeecaeccccttecs aay chee eee eee

Values of Residual Resistance Coehhicients-- 2 .-eeeeeeeeeeee

Effective Horsepower Required for the

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Page

26

26

27

28

HS)

30

31

NOTATION

F Frictional drag coefficient

C Pressure coefficient, Ap/(1/2 p UZ)

Ca Residual drag coefficient

Ca Total drag coefficient

Cp, Prismatic coefficient of forebody

D Maximum diameter of model

D,. Flat-faced diameter or friction drag

DR Residual drag

D,, Total drag

k Turbulence-stimulator height

ky Nondimensional rate of change of curvature on forebody at juncture of flat face
with transition curve

k, Nondimensional rate of change of curvature on forebody at juncture with
parallel middle body

1 Length of model or prototype

IL Length of forebody

L. Length of afterbody

R Reynolds number*

Rg Reynolds number, Rg = Ué/v

I Nondimensional radius of curvature at forward point of body

S Surface area of model

Sp Surface area of forebody

SL-x, Surface area between x = x, and x = L

SR Surface area of afterbody

Sx¢ Surface area between x = 0 and x = xg

*Unless otherwise defined all Reynolds numbers are based on the freestream velocity U, and a length
denoted by the subscript.

XQ-Xo

Subscripts

Surface area between x = x, and x = xq

Arc length along a meridian

Arc length of model along a meridian

Forebody slenderness parameter T = (L,,/D)/Cp,
Potential flow velocity on the model

Speed of model or prototype

Laminar boundary layer velocity at trip height k
Axial distance from model forepoint

Length of laminar flow on model forebody
Forebody radius

Tangent angle of model profile

Stimulator drag coefficient, A D,/C /2 p ug 27 y, k)
Stimulator drag coefficient, AD, /(1/2 p UZ S)
Stimulator drag

Pressure change on model from potential flow
Boundary layer displacement thickness
Nondimensional forebody radius

Boundary layer momentum thickness

Boundary layer momentum thickness at x = xg

6 at x = xg from turbulent flow originating at X = x,

0? d
Pressure gradient parameter, ie

Kinematic viscosity of fluid
Nondimensional axial distance from forward point

Fluid density

Location of stimulator
Laminar flow
Location of virtual origin

Turbulent flow

vi

ABSTRACT

Resistance has been measured of nine bodies of revolution, having equal
volume but varying forebody shapes. Forebody shapes ranged from extremely
blunt to extremely fine and included two that were flat faced. The forebodies
were altered by changing their length-to-diameter ratios (L/D’s) and prismatic
coefficients. Drag results indicate that when the prismatic coefficient is fixed
and the L/D is decreased, the residual resistance will increase modestly.
Increasing the prismatic coefficient at small L/D’s increases residual resistance;
however, at moderate L/D’s it does not. The results suggest that a flat-faced
shape in itself does not increase resistance.

In addition to resistance experiments, transition regions on the models
were located, using hot film probes. Calculations predicted laminar separation
on five of the model forebodies. The hot film measurements confirmed that
separation did occur at the locations predicted; downstream of the separation
locations, turbulent flow occurred immediately. The remaining four fore-
bodies exhibited well-defined natural transition regions. Flow properties in
the transition regions, measured by the hot film gages have been compared
with predicted spatial amplification ratios of disturbances, calculated by
linear stability theory. Results have failed to show a single relationship
between measured flow properties and computed spatial amplification ratios.
Correlation of amplification factors with flow regimes varied, both with
forebody shape and Reynolds number.

ADMINISTRATIVE INFORMATION

This work was funded and authorized by the Naval Sea Systems Command under the
hydrodynamics part of the very high speed submarines program, Task Area SF 434 215 1Z,
Work Unit 1-1520-004, and the improvements in submarine hydrodynamics program, Task
Area SSL66007, Work Unit 1-1552-135.

INTRODUCTION

Experiments have been conducted on nine bodies of revolution having different forebody
shapes. The primary purpose of the experiments was to determine what effect changing
forebody shape would have on resistance. As a means to this end, the roles of transition,
laminar separation, and turbulence stimulation in the analysis of model data have also been
investigated. These phenomena have been examined using hot film probes, located in the
model surfaces - resulting in a new method for analyzing model drag data as reported by
McCarthy, Power, and Huang.! Results of both the resistance and hot film measurements are

presented in this report.

The experiments have been made in two series. In the first, the effect of forebody
length-to-diameter ratio (L/D) was investigated by measuring the resistance and transition
locations of four models with forebodies having different bow entrance L/D’s and a fixed
bow prismatic coefficient of 0.667. The data obtained in this series were also used in
developing the new method of analysis given in Reference 1. For completeness, most of the
results from the Series 1 experiments, reported in Reference |, are repeated.

In Series 2, the effect of very large values of bow prismatic coefficient on resistance was
investigated. Properly shaped blunt forebodies could create strong favorable pressure gradients
in the forward regions of the forebodies where transition could be expected to take place at
full-scale Reynolds numbers. These pressure gradients would extend laminar flow further aft
when compared with flow on more conventional forebodies, possibly producing a more
favorable operating environment for sensors, in the bow region. In Series 2, five blunt fore-

bodies were investigated, including two that were flat faced.

MODELS

To conduct the resistance experiments, nine models were constructed, all bodies of
revolution. Each model consisted of a forebody, parallel middle body, and fixed afterbody:
see Figure 1. Seven of the models had rounded forebodies, and two had flat faces. The
volume and the maximum diameter of ail models were kept constant. In this section, the
method used to generate forebody shapes, the selection of the forebodies, and the construction

of the models will be discussed.

GENERATION OF FOREBODIES
Rounded Forebodies

Six of the rounded forebody shapes were developed using the Granville family of
quadratic polynomials”; the seventh “rounded” forebody was hemispherical in shape. The
nondimensional offsets of the polynomials and the hemispherical shape are given by the

following two equations

McCarthy, J.H. et al., “The Roles of Transition, Laminar Separation, and Turbulence Stimulation in the
Analysis of Axisymmetric Body Drag,’’ Eleventh Symposium on Naval Hydrodynamics, London (1976).

?Granville, P.S., “Geometrical Characteristics of Noses and Tails for Parallel Middle Bodies,’’ NSRDC
Report 3763 (Dec 1972).

tO

Polynomials

n=1R(é) +k, K,(@) + QE) ()

where R(@) = 22Gi1"
K(f) => E- 1)
Q(é) = 1-(§-1)* (4& +1)

Hemisphere
Eee A | (2)

where & =x/L, and n= 2y/D

Dis forebody maximum diameter

L,. is forebody length

x is axial distance from forward point
y is forebody radius

ris radius of curvature at £=0

is rate of change of curvature at & = |

; 3
1 4L, 1 a dn _ 2k dy
Here he nit AEs om Saw 7 aah \ han Amgen y cake eRe
(4) D? (5) dee Oe eer
E=0 X=0

dn? dy?

By selecting proper values of r and k,, various parent shapes can be derived. The values
of L, and D then determine the shape of the particular forebody. Once values of r and k,

have been selected, the prismatic coefficient Cp_ is uniquely determined and is given by
PE

(3)

mic

T
€ a ae
ish mal gyn ae

d :
At x= L.; the slope (22) and the reer a are set at zero, matching
xo Lp

their corresponding values on the parallel middle body. The se Pee shape is fixed

with L, = D/2. At x= L,, the slope matches that of the parallel middle body; however, the

curvature does not.

Flat-Faced Forebodies

The curves connecting the flat face to the parallel middle body of the two flat-faced
shapes were developed, using both an elliptical contour and a contour derived from the
Granville two-parameter cubic polynomial.? The nondimensional elliptical contour offsets are

given by Equation (2), and the Granville cubic polynomial offsets are given by

low EES
n? =— K,(é) +k, K, (&) + Q(é) (4)
ky
and
K, (é) = 6&(é - 1)4
Kw=4ee-1%

O®) = VSESs IY Gee)

2y -D,
where now E sxe and n Day
3
x (4) _(D-D,) (“2)
8L, 3
dn =10 N=0 E dy x=0, y=D,/2

. 3
i, a - 5-5, (2)
3 D-D 3

dé g=1 Bax x=L

where D,. is the diameter of the flat face.

For the Granville shape, the slope and curvature of the connecting curve match the slope and
curvature of the flat face and parallel middle body where the curves join, and the shape of a
parent body is determined by values assigned to k, and a The values of L,. and (D - D,)/2
then determine the contour of the particular forebody. The elliptical transition curve (non-
dimensional) is fixed, and variation in model forebody shape is a function of only L,. and

(D - Dye: The slopes match at the flat face and parallel middle body; however, the

curvatures do not.

Granville, P.S., “Geometrical Characteristics of Flat-Faced Bodies of Revolution,’ NSRDC Report 3710
(Nov 1971).

SELECTION OF SHAPES

For the first series of experiments, four models were selected. Since the purpose of this
series was to investigate the effects of IL //D), only this parameter was varied. All the models
had rounded forebodies with prismatic coefficients equal to 0.667. The first forebody model
selected (4620-3) is designated as the parent model, IL /D = 1.82; the resistance of this model
is then used as a baseline to assess the merits of the other forebodies. The remaining three
forebodies have L,, /D ratios of 0.5, 1.0, and 3.0. The IL /D = 0.5 forebody has a hemispher-
ical shape. The other three have Granville shapes and were derived from a common parent,
ie., r = 0.8333 and k, = 10.0. Figure 2 shows the contours of the four forebody shapes.

Of the five forebodies selected for experiment in Series 2, three had rounded shapes.
They were selected during a parametric investigation of forebody shape, using calculated
pressure distributions and transition locations for guidance. Pressures were computed for
several model shapes, using the Hess-Smith, potential-flow-computer program.* Boundary-
layer stability calculations were made for a few selected shapes at typical full-scale Reynolds
numbers, using the Smith-Gamberoni method.**® Results were then compared with similar
calculations for the parent forebody shape, Model 4620-3. Shapes selected were chosen for
their persistence of laminar flow in the full-scale Reynolds number range. They were derived
from a common parent - r= 3.167, Cp. = 0.850, k, = 5.0 - having L, /D ratios of 0.50, 1.00,
and 1.82.

The first flat-faced forebody selected has a relatively small prismatic coefficient, 0.823,
and a forebody L,, /D of 1.216. The connecting curve between the flat face and the parallel
middle body on this model has an elliptical contour. In contrast, the second flat-faced fore-
body is extremely blunt with a prismatic coefficient equal to 0.933. A Granville two-
parameter cubic polynomial (kK; = 1.0, k, = 3.0) was used for the transition curve on this
model. The contours of the five forebodies of the second series are shown in Figure 3, and
the particulars of the nine models selected are given in Table 1.

Incipient cavitation speeds have been calculated for the nine forebody model shapes; all

are listed in Table 2. The model with the earliest predicted cavitation inception is 4620-7.

4Hess, J.L. and A.M.O. Smith, “Calculation of Potential Flow about Arbitrary Bodies,” Progress in
Aeronautical Sciences, Vol. 8, Pergamon Press, Inc., New York (1966).

>Smith, A.M.O. and N. Gamberoni, “Transition Pressure Gradient and Stability Theory,”’ Douglas Aircraft
Company Report ES-26388 (Aug 1956).

®Wazzan, A.R. et al., “Spatial and Temporal Stability Charts for the Flakner-Skan Boundary-Layer Profiles,”
Douglas Aircraft Company, Report DAC-67086 (Sep 1968).

TABLE 1 — MODEL PARTICULARS

cae Pe D D D D2 D2

SERIES 1

0
0
0
0

D = 0.624 m; L,./D = 4.400; S,,/D? = 40.21; Volume/D® = 6.786.

TABLE 2 — COMPUTED INCIPIENT CAVITATION
SPEEDS OF NINE FOREBODIES

Incipient Cavitation Speeds in Meters per Second
for Water Depths of —

2
3
4
5
6
7
8
9

Note: It is assumed that forebodies have negligible roughness.

At a depth of 15 m this shape had an inception speed of 18.5 m/s. The cavitation calculations
were based on the computed, potential flow, pressure distributions for smooth forebodies.
Rough forebodies would have lower inception speeds. The pressure distributions on the
models of the first series are given in Figure 4 and of the second series in Figure 5. Minimum

values of the pressure distributions are listed in Table 2.

CONSTRUCTION

The models were constructed of molded Fiberglas and consisted of three sections. The
second and third sections were common to all models and consisted of part of the parallel
middle body and the afterbody, respectively. The first section included the forebody and the
remainder of the parallel middle body. For each of the first sections, the length of the
included parallel middle body was adjusted to insure constant volume for all models. The
third section was obtained from an existing model of the research submarine ALBACORE
(AGSS-569) by cutting the model in two parts at the position of maximum diameter.

Special care was taken in the construction of the forebodies of the models. Profile
tolerances were held to + 0.4 mm; all imperfections were removed and meridians were fared.
The Fiberglas was polished to a 0.64-micron, root-mean-square, surface finish. This insured
roughness Reynolds numbers two orders of magnitude lower than the number considered

necessary to trip turbulence prematurely.

EXPERIMENTS
GENERAL

The model drag experiments were conducted in the David W. Taylor Model Basin, using
Carriage 2. This towing basin is 846 m long, 15.5 m wide, and 6.7 m deep. Drag and speed
data were collected for each model in the speed range between 0.51 and 5.15 m/s at intervals
of approximately 1/4 m/s. A waiting time of 5 min between runs was selected because no
effects of residual turbulence from preceding runs could be detected after this time. Flush-
mounted hot films were inserted in the model surface at selected locations to determine the
extent of laminar and transitional flow and the location of laminar separation when it
occurred. Water temperatures were monitored throughout the test period to obtain water

densities and kinematic viscosities.

RESISTANCE MEASUREMENTS

Total drag has been measured using the standard DINSRDC block-gage-type dynamo-
meter. mounted internally, on each model and connected to the carriage with a single stream-
lined strut. This dynamometer has an accuracy of approximately | percent at its maximum
load capability of 55 k. The model centerline-submergence depth was fixed at 2.74 m below
the water surface for all runs. The models were ballasted for a slight amount of negative
buoyancy with zero moment at the towing strut. Previous work has shown that the towing-
strut model, interference-drag coefficient is less than 0.01 - 107? for the size of models
investigated. Since this interference drag is within the experimental accuracy of model drag

data reported here, no corrections have been made for it.

TRANSITION MEASUREMENTS

Table 3 gives the locations of the hot films in the various models. Selection of the hot
film locations was assisted by inspection of pressure distributions, experience from previous
tests, and some trial and error. When laminar separation was predicted, a hot film was
inserted immediately fore and aft of the predicted location. The Curle and Skan modified
Thwaites criterion’ of \ =-0.09 was used to predict the location of laminar separation where

\ is a pressure gradient parameter given by

G2 dU
Ni 5)
Dads ‘
where 6 is boundary layer momentum thickness

v is kinematic viscosity
U is potential flow velocity on the body

s is arc length along a meridian.

The values of 6 were computed using the Granville integral method for bodies of revolution.®

The computed positions of laminar separation are listed in Table 4.

Ke urle, N. and S.W. Skan, “Approximate Methods for Predicting Separation Properties of Laminar Boundary
Layers,” Aeronautical Quarterly, Vol. 8 (1957).

8 Granville, P.S., “The Calculation of the Viscous Drag of Bodies of Revolution,’’ David Taylor Model Basin
Report 849 (1953).

TABLE 3 — LOCATIONS OF HOT FILM GAGES

Nondimensional Axial Distance from Model Forepoint to Hot Film Gages - x/D
Location
4620-1 | 4620-2 | 4620-3 | 4620-4 | 4620-5 | 4620-6 | 4620-7 | 4620-8 | 4620-9

0.13 0.05

—_

Oo ON DOT FW DN

The hot film probes were made by Lingtronics Laboratory. The active elements

(platinum) were approximately 1.6 mm long, fixed to one end of a Pyrex cylinder 12.7 mm in
length and 2.4mm in diameter. The cylinders were carefully mounted in predrilled holes in
the model surfaces and were held in place with rubber cement. Any ridge between the model
surface and the probes was held to less than 0.025 mm, and the holes in the model were
drilled perpendicular to the surface + 0.50 deg. As a further precaution, each probe was
located along a different meridian streamline, to avoid possible spurious signals which might
result from tripping of turbulence by probes, located at successively further forward positions.
Five channels of instrumentation were available and interchangeable between hot films,
allowing the five most significant probes to be activated during a given run. Each channel
consisted of an anemometer, linearizer, oscilloscope, and attentuator. For each speed, 20-s
samples of hot film output were recorded on tape. The output of each channel was kept
close to but less than the 1.4-V overload capacity of the recorder by adjusting the overheat
ratios of the films, the linearizer amplifiers, and the attenuators. Overheat ratios were kept
between 1.035 and 1.050 for most of the tests; however, for some of the low-speed runs,
slightly higher ratios were needed. Since these adjustments had to be made quickly as the
carriage moved down the basin, no record of amplifications or overheat ratios could be kept.
Thus, the information obtained from the hot films was primarily qualitative and was used to
determine the nature of the flow at the hot film locations. Figure 6 shows one channel of

the instrumentation.

TABLE 4 — LOCATIONS OF WIRE STIMULATORS AND TRANSITION
OR LAMINAR SEPARATION WITHOUT STIMULATORS

Locations of
Laminar Separation (LS)
or
Transition (T)

Wire Wire
Diameter Locations
x, /D

LS: x/D =0.472

LS: x/D = 0.894

1.56 < x/D < 2.13,

33.6>R,° 10? 3 Oi.

0.231, 0.405, 0.579
0.405, 0.579 1.26 < x/D < 3.39,
0.579 33.6 > R, x/D 10°° > 3.1.

T: 0.57<x/D< 1.04,
30.6>R lO? 2 20,

T: 0.07 <x/D <0.70,
30.6> Rp -10°° > 3.1
LS: x/D=1 oe

10

As shown in Figure 7, analysis of the hot film, output signals revealed that four different
types of boundary-layer flow could be distinguished:

1. Laminar flow, when hot film signals were of steady amplitude.

2. A smooth wavelike disturbance that had identifiable frequency content, superimposed
on a laminar flow, and no doubt associated with Tollmien-Schlichting waves.? (An early
(Type 2a) and advanced stage (Type 2b) of wavelike flow is illustrated in Figure 7.)

3. An intermittent turbulent bursting flow, reported by Schubauer and Klebanoff,!°
preceded and followed in time by periods of Type (2) flow.

4. A fully turbulent flow of random nature.

Under fixed test conditions, probe outputs indicated that Types (2) or (3) and Types (3) or
(4) could alternately occur on repeat-run, tape-recorded, time samples of the signal at a given
probe location. For a given location of a hot film probe on a forebody, the speed at which

a particular type of flow occurred could be determined to within 1/4 m/s.

TURBULENCE STIMULATORS

Resistance of each model was measured both with and without turbulence stimulators.

In the Series | experiments, Model 4620-3 was used to evaluate stimulators of the following
types: sand strips, wires, and studs. The results of these experiments have been reported in
Reference | and will not be discussed here. The 0.61-mm wire was chosen for all of the
remaining resistance tests because it proved to be an effective trip in the present experimental
arrangement, having a precisely defined geometry consistent from test to test and being easy
to install and remove. More parasitic drag data were also available for the wire than for the
other types of trips.

Table 4 gives the wire sizes used and the locations at which the wires were installed on
the nine models. With the exception of Model 4620-3, only 0.61-mm diameter wires were
used. Whenever possible, experiments were conducted with wires installed at x/L = 0.05, the
traditional location used at the Center for trip wires, where L is the model length. Exceptions
occurred when laminar separation, which acts as a natural trip, or transition took place forward
of this location. When laminar separation occurred forward of x/L= 0.05, a trip was placed a

short distance ahead of the separation location.

” Schlicting, H., “Boundary Layer Theory,” McGraw-Hill, Inc., New York, Chapters 12, 16, and 17 (1955).

10 ¢ chubauer, G.B. and P.S. Klebanoff, “Contributions on the Mechanics of Boundary Layer Transition,”
National Advisory Committee for Aeronautics TN 3849 (1955).

1]

REDUCTION OF RESISTANCE DATA

The value of the residual drag coefficient C= DL, 2 p We S is used to determine the
relative merits of different models, where D, is the residual drag; p, the density of the fluid;
U,, the speed of the model; S, the model surface area. In the present experiments, values of
C, have been determined, using a method developed in Reference 1. In contrast to traditional
methods, the new method takes into account the location of transition on the model in the
analysis. This is important when, as in the present case, the object is to evaluate competing
models with vastly different forebody shapes and Cp values with relatively small differences.
The extent of laminar flow can vary greatly on such models as the forebodies vary from blunt
to fine. This influences the measured model resistance and if not recognized will significantly
alter the relative body drags. An outline of the method is given as follows.

In the analysis it is assumed that two types of flow regimes exist on a model hull:
laminar g and turbulent ,. When a significant length of transitional flow occurrs, the
longitudinal extent xg of the laminar flow regime, having a wetted area Sxo» will be assumed
to terminate where intermittent turbulent bursting first occurs; see Type 3 of Figure 7. When
the wire stimulator is installed, it is assumed that xq will coincide with the wire location,
When laminar separation occurs, it is assumed that Xe coincides with the computed separation
location. The latter two assumptions are in accord with available experimental data.!

The value of C, can be expressed as

Ce CoG oC (6)

where C, is the total drag coefficient D,/1/2 pus S
C,.. is the frictional drag coefficient Dye // 1/2 p We S
C is the wire drag coefficient AD,/1/2 p Wie S

and is the total drag

D,

D,, is the frictional drag
AD,, is the stimulator drag

S is the model surface area

The value of c. is obtained by measurement; however, the value of C and AC, must be
obtained from empirical relations.
Following Reference 1, the drag coefficient AC, of the wire has been taken as 0.75 for

all experimental conditions where AC, is defined as

]
NC. = ANDY.) (6 he 27 Y, " (7)

where k is the wire diameter
u, is the velocity in the laminar boundary layer at height k

Yq is the radius of the model at the wire location

The relationship between AC, and AC, is

2
ur \" 27 yy, k
INC SAC U, ae CSch, (8)

where a is the tangent angle of the model profile at the wire location. Values of u, were
determined from the Pohlhausen one-parameter family of velocity profiles, using laminar
boundary-layer parameters, calculated by the method given in Reference 8.

To calculate C,., a virtual origin x, < xg is defined so that the frictional drag of a

F?
turbulent flow over the wetted-surface area of the model between x = x, and x = L is equal
to the total frictional drag on the model (laminar and turbulent) plus the drag of a stimulator,

if present. This situation is illustrated in Figure 8 and can be expressed by

DE + AD, = De (S,-x,) (9)

Equation (6) can now be written as

Sp - 6)
where
ae | 2
Cr, % Dr, / 5? Wa Shen
By analogy with Equation (9) and by converting to coefficient form, we can also write
Caucs eo op (S ene 11)
RG GO 1d T

This expression provides a means to evaluate x,.

The frictional drag arising from the turbulent flow region S; -~x, on the model is now
considered to be the drag due to the flow over an equivalent flat plate of constant width -
DiVa= Sr -x,L- x,). With this assumption, frictional drag becomes a function of Reynolds

number alone, and Equations (10) and (11) reduce to

13

SL -x,
Cp = G ~ Cp, (Rr -x,)

1 p)

LS Mdig Sx9
Bietes Te a” Can Seq) ae SIGE

where B = (Ryo -x5/Rp)/(Sx¢ -x,/S)- Also, using flat-plate theory, we can write for the flow
over the area SxQ-X,

@3)

(14)
combining Equations (13) and (14) we get

S
so ee
(oneealiamiee ce Sep : + AC, | (15)

The empirical relation used to evaluate the turbulent frictional drag is the Schoenherr
flat-plate formulation!

W/ CE loz REG) 042 (16)

where R, is a Reynolds number based on the length of the plate. Using Equations (16) and
(14) and changing to the present notation, we can write

V Cp, (Rxp-x,.) loki E (Fes 29),| = 0.242

Substitution for Cr, in Equation (14) yields

2
= = 2 (
Ry, = Rxp 34.151 (Reve {Ie (Fess) | (18)

To obtain x Equation (15) is solved iteratively, after initially setting 6 = 1.0 to obtain
Nhs uy

(17)

Then x, can be obtained directly from Equation (18). The value of Cre (SxQ) in

Equation (15) is obtained using laminar boundary-layer theory for axisymmetric bodies.

Mi Schoenherr, K.E., “Resistance of Flat Plates Moving through a Fluid,”’ Transactions of Society of Naval
Architects and Marine Engineers, Vol. 40 (1932).

14

The expression is

&* d /uU\2
= > an U. cos @]| dx (19)
oO

Once the value of x, is determined, Cr, (S; aK) and C, can be obtained, using Equations
(16) and (10). The value of C, obtained should be the same for both tripped and untripped
(AC, = 0) models.

When trips are located near the leading edge, and/or the trip has a high parasitic drag,
the computed value of X, can become negative. When this occurs, the following approxima-
tion is made when computing C, from Equation (1 2)

SL =x L-x

a

S L

The results obtained in Series | experiments were computed using the previously
described equations. However, in Series 2 experiments, whenever x was directly involved in
the calculations, it was replaced by the arc length along the meridian s. Thus the virtual
origin was located at a position s, from the leading edge of the plate of width

day =S;, -x,/(8p - So), Operating at a Reynolds number R = JU (Sia Se) Uamwuitene shanis

SE So
the total arc length of the model. This change was necessary to obtain results on the two
flat-faced models. To compare results, CR values were computed, using both s and x for

Models 5 through 7. There was a difference of approximately 1 percent in the values of (Che

RESULTS
TRANSITION LOCATIONS

On Models 1, 2, 5, 7, and 9, laminar separation was predicted to occur on the forebody
sections of the models at the locations given in Table 4. For the Reynolds number range used
in these experiments, hot films located immediately before the predicted locations indicated
laminar flow. Hot film gages, located immediately aft of the predicted locations, indicated a
disturbed or turbulent flow. It was concluded from these measurements that transition was
occurring between these two gages and that separation was tripping turbulent flow, thus

justifying the use of the predicted laminar separation location as the transition location.

15

Models 3, 4, 6, and 8 exhibited no laminar separation. The flow regimes, measured by
the hot film gages, are given in Figures 9 through 12. Experimental data have been compared
with predicted spatial amplification ratios of disturbance as calculated by linear stability
theory. These ratios were calculated using a computer program developed at the Center,
DABL (disturbance amplification in boundary layers).*!? In Figures 9 through 12, predictions
corresponding to spatial amplification ratios of e,e>,e1/e. el and e!3 are compared with
flow regimes. The results given in these Figures fail to show a single relationship between the
measured flow properties in the transition regions of the four models and the corresponding
computed spatial amplification ratios. Correlations of amplification factors with flow regimes
vary both with forebody shape and Reynolds number. Table 5 gives the principal results
obtained for each model.

It has been assumed in this report that flow transition originates with the start of
turbulent bursting (Type 3 flow of Figure 7). The critical amplification factor correlating
with this flow transition condition varies considerably from model to model and to a lesser
extent, on individual models, with changes in Reynolds number. On Model 3 this variation is
estimated to change from e!* at low values of R, to e® at the highest values of Rp- The
corresponding values on Model 4 are from e!! to e?; on Model 6, from e!? to e!°. The data
available for Model 8 indicate a nearly constant value of e’. However, due to the failure of
a hot film probe, data at the critical location at the higher values of R, are lacking for this

model.

RESISTANCE RESULTS

The C,’s obtained for each model are plotted in Figures 13a through 13i as functions of
Reynolds number R,. The dotted line in each figure represents the faired mean values of Ce
obtained from all of the experiments with a given model, both with and without turbulence
stimulators installed. The solid lines roughly define zones of scatter at Reynolds numbers
greater than R, = 10’. The scatter ranges as much as +0.050 - 107? about the mean value

for some of the models.

*Stability results from this program for Models 3 and 4 (Figures 9 and 10) differ from stability results given
in Reference 1 for the same two models (Figures 6 and 7 of that Reference). Results given in Reference 1
were obtained by hand calculations, using charts from Reference 6. The discrepancy no doubt results from
approximations involved in using interpolations required in determining the amplification factor, using hand
calculations.

12Von Kerezek, C. and N.C. Groves, “Disturbance Amplification in Boundary Layers,” paper in preparation.

Amplification
Pacton Model 3 Model 4 Model 6

TABLE 5S — CORRELATION BETWEEN AMPLIFICATION

FACTORS AND MEASURED TRANSITION DATA

22;< Rp > 1052 < 35)
fully turbulent
5<Rp- 10° < 22,
nearly fully turbulent

ZIEIRy O0> S35,
fully turbulent

IS<IRe On <7)
intermittent

5< Rp + 10° <13,
wavelike

DS Ry 10" <<
intermittent

5< Rp‘ 10°°<20,
wavelike

10°<Riy 105° <20)
wavelike

5<Rp°10°°<10,

laminar

20 < Rp 107° < 35,
fully turbulent

all values of Rp tested

Probably fully turbulent at

20< Rp + 10° < 35,
fully turbulent

26 << Rp *10°° < 35,
fully turbulent

18< Rp - 10° < 26,
intermittent

30< Rp ° 10° < 35,
intermittent
15< Rp + 107° < 30,

wavelike
B< fis ClO? < 16;
laminar

Rp-* 107° = 10, wavelike
Sis Pile? < 10,
laminar
Probably wavelike in range
given by 10< Rp ° 1052 <35

Probably laminar at all
values of Rp tested

TORR Onn 123.

laminar

Probably laminar at all
values of Rp tested

S< ip cle < 1G.

laminar

Probably laminar at all
values of Rp tested

18< Rp 10°? < 32,
between turbulent
and intermittent

5<Rp + 10° <18,
intermittent

9< Rp ° 10° < 32,
intermittent

5<Rp*10°<9,
wavelike

2a AR terme 32)
between intermittent
and wavelike

BGR Rano 124)
wavelike

BS ol? < sy,
wavelike

23EGRe On <2)

probably wavelike

BI<GRewcn|Ons <23)
laminar

Model 8

5<Rp > 10. < 32,
fully turbulent

5<Rp + 10° < 32,
fully turbulent

D< fly O10? ey)
fully turbulent

Close to intermittent
at Rp = 12

S< hip 1 <3,
intermittent

S< kip PO < 11,
wavelike

5 Rp 1002) < 12,
laminar

Probably laminar at all
values of Rp tested

BIR plas)
laminar

Probably laminar at all
values of Rp tested

At Reynolds numbers less than 1.0 * 10’, the spread of the data becomes much larger. This

results primarily from the greater inaccuracy involved in measuring low drags in the low-speed

range. Figures 14a through 14j give the values of the measured C,, both with and without a

turbulence stimulator installed for the nine models.

As stated previously, the analysis method described in Reference | was used to compute

the values of C,. Ideally, Gs obtained from experiments with both tripped and untripped

models should be equal when this method is used. However, an inspection of Figure 13 shows

that a small difference of as much as 0.03 - 107? between the tripped and untripped values

of C, can be discerned for roughly one-third to one-half of the model data.

17

When a discrepancy is evident, the bare-hull data give lower values of C, than do the tripped
data. On Models 4, 6, 7, and 9 for R, > 0.6 ° 107 and on Models 3, 5, and 8 for

Ip 2 2 & 107, no separate trends could be discerned between the two sets of C, data. While
existence of differences may indicate some defects in the analysis method, it is felt that they
are small enough to be ignored when selecting the average values of Cp.

Average values of C, for each model in the absence of significant wavemaking are listed
in Table 6. Included are values of C, as defined previously, based on model surface area,
and values of CR» based on volume to the two-thirds power, Crear = Dp /1/2p U2 (Volume)?/3.
Each value was based on the faired curve - dotted line in Figure 13 - obtained from all the
data on the appropriate model within the R, range 107< Ki S22 10’. For Reynolds
numbers less than R, = 10’, the experimental scatter is too large for the data to be useful.

For Rep=2 2s 10’, wavemaking resistance becomes significant.

TABLE 6 — VALUES OF RESIDUAL RESISTANCE COEFFICIENTS

Table 6 shows that the measured values of CR: based on S, for all forebodies ranged
between 0.00020 and 0.00028. Models 3, 4, and 6 had the minimum Cy, values of 0.00020,
indicating no gain or loss in C, when increasing L, /D from 1.82 to 3.00. For parent Model
3(L,/Di= 1.82), increasing Oe from 0.667 to 0.850 did not change the value of €,: Model
8 is a flat-faced body with a smaller L, /D (1.216) and a larger Cp, than Model 3, these

changes caused only a slight increase in the value C, from 0.00020 to 0.00021.

Decreasing the value of L,./D to 1.00 had the effect of increasing C,. The three fore-
bodies tested with this L,/D (Models 2, 5, and 9) had Cp, values of 0.667, 0.850, and 0.933.
Model 9 with the largest Cp, value has the largest value of C, (0.00026). Model 2
(Cp. = 0.667) had a C, value of 0.00024 and Model 4620-5 (Cp, = 0.850) had a C, value of
0.00023. Decreasing the value of EoD to 0.50 further increased the values of C,. Two
models, | and 7, had L, /D’s of 0.50 and Cp, values of 0.667 and 0.850, respectively. The
measured values of C, were 0.00025 for Model 4620-1 and 0.00028 for Model 4620-5.

The values of C, are given in Figure 15 as a function of a forebody slenderness
parameter T =(L, [D)/Cp... For values of T greater than 2.0 C, assumes a constant minimum
value of 0.00020. As T becomes lower than 2.0 the values of C, rise steadily. Values of T
for each model are given in Table 6.

The values of C, based on model volume to the two-thirds power are also given in
Table 6. The volume C,’s change the relative merits of the models somewhat but not
enough to be significant.

To further illustrate the significance of C, results, prototype values of ins = (Dp
+ ID) (R, ))/C/2 p UL S), and the effective horsepower ehp for nine unappended shapes,
which are geometrically similar to the models, have been calculated.
where Dx is residual drag

D,, is friction drag, at Reynolds number R,

R,_ is Reynolds number, U, L/v

L_ is prototype length

U, is prototype speed

Sis prototype surface area.
Figure 16 gives the values of Crp as a function of R,. The ehp values are given in Table 7
for six representative speeds. Also given in Table 7 are ehp ratios when ehp of Model 3 has
been used as the base. Strictly speaking, this ratio is a function of R,. However, the variation
is not discernable in the three significant figures given. These computations were made using
the Schoenherr friction line, and no roughness allowance was used. The usual assumption
was made of setting C, equal for both the model and prototype scales. A linear ratio of
prototype-to-model of 15.79 was used, resulting in a full-scale length of 110 m for shape 3.

Results indicate a maximum variation in ehp of 4 percent for the nine shapes investigated.

TABLE 7 — EFFECTIVE HORSEPOWER REQUIRED
FOR THE NINE SHAPES

EHP

EHP : 1072] EHP « 1072| EHP » 1072| EHP » 1072| EHP = 107*| EHP - 1077

—_

2
3
4
5
6
7
8
9

Note: Linear ratio = 15.79.

CONCLUSIONS

The following conclusions have been drawn from the results obtained.

1. For the range of conditions covered in these experiments, the occurrence and
location of laminar separation is accurately predicted by the Curle-Skan modified Thwaites
criterion. Separation insures the onset of turbulent flow a short distance after the separation
position.

2. On the four models that did not exhibit laminar separation, the results did not show
a single unique relation between the measured flow properties in the transition regions and
the corresponding computed spatial amplification ratios obtained by linear stability theory.
On Model 3, the range of amplification factors that correlates with the onset of transition
lies between e® and e!?. The corresponding range for Model 4 is from e? to e!! and on
Model 6 from e!® to e!*. The data taken for Model 8 indicate that a nearly constant value
of e’ correlates well with data at the onset of transition.

3. The residual resistance coefficient of the nine models correlates reasonably well with
the forebody slenderness parameter (T = (Ly /D)(Cp,.). The results indicate that for a given
stern and constant total volume, an increase in forebody fullness results in a small increase

in residual resistance coefficient, once the slenderness parameter drops lower than about 2.0.

The maximum spread in values of effective horsepower is about 4 percent. Models 3, 4, 6,

and 8 showed the best resistance characteristics, while Model 7 had the most unfavorable.

ACKNOWLEDGMENTS

The author gratefully acknowledges the assistance of Ms. Laurie M. Higgins and Messrs.
William G. Day, Eugene E. West, Leonard B. Crook, Neil E. Oliver, and Nicholas Santelli for

their assistance in conducting the experiments on which this report is based.

MIDDLEBODY AFTERBODY

Figure 1 — Body of Revolution

L./D
4620-1 0.500 0.667
4620-2 1.000 0.667

4620-3 1.820 0.667
4620-4 3.000 0.667

Figure 2 — Profiles of Forebodies of Series 1

L_/D Cp. D,/D

4620-5 1.000 0.850 0.
4620-6 1.820 0.850 0.
4620-7 0.500 0.850 0.
4620-8 1.216 0.809 0.540
4620-9 1.000 0.933 0.500

Figure 3 — Profiles of Forebodies of Series 2

9)

22

x/D

Figure 4 — Computed Distributions of Pressure Coefficient
on Axisymmetric Forebodies of Series |

23

x/D

Figure 5 — Computed Distributions of Pressure Coefficient
on Axisymmetric Forebodies of Series 2

OSCILLO-

SCOPE
ATTENUATOR

ANEMOMETER

LINEARIZER

TAPE
RECORDER

Figure 6 — Hot Film Instrumentation

25

VOLTAGE OUTPUT OF HOT FILM SHEAR PROBE

1.0V

D_-(x) |

LAMINAR

TYPE 2a

WAVELIKE

TYPE 2b

WAVELIKE

INTERMITTENT

TURBULENT

Figure 7 — Representative Hot Film Signals
for various Flow Regimes

TURBULENT

Figure 8 — Idealized Body Frictional Drag Distribution
with a Turbulence Trip

26

+ D, + AD,

1.0V

1.0 V

LAMINAR (1)
WAVELIKE (2)
INTERMITTENT (3)
TURBULENT

(2) AND (3)

(3) AND (4)

MEASURED
FLOW REGIMES

@oe@eeedeo0d

NEUTRAL
STABILITY

Figure 9 — Measured Flow Regimes on Model 3,
Compared with Amplification Factor

Dil

MEASURED
FLOW REGIMES

AMPLIFICATION FACTOR

e! e?

NEUTRAL
STABILITY

O LAMINAR (1)

© WAVELIKE (2)

@ INTERMITTENT (3)
@ TURBULENT (4)

Figure 10 — Measured Flow Regimes on Model 4,
Compared with Amplification Factor

O LAMINAR (1)
MEASURED ® WAVELIKE (2)
FLOW REGIMES | @ INTERMITTENT (3)
@ TURBULENT (4)

AMPLIFICATION FACTOR
13

11
Cie
e? Seale i 7

NEUTRAL
STABILITY

@
®
@
@
@
)
@
@
@
@
@
@
cc)
i)

Ce G08 8G 868860 OOHhmUMOMUCOUCUCUCOUCUCOWUCUCOUCUCOUCUCO

0.2 0.4 0.6 0.8 1.0
x/D

Figure 11 — Measured Flow Regimes on Model 6,

Compared with Amplification Factor

29

O LAMINAR (1)
MEASURED ® WAVELIKE (2)
FLOW REGIMES } @ INTERMITTENT (3)
@ TURBULENT (4)

AMPLIFICATION FACTOR

®
10)
®

O
NEUTRAL STABILITY

0 0.2 0.4 0.6 0.8 1.0 Ae2.
x/D

Figure 12 — Measured Flow Regimes on Model 8,
Compared with Amplification Factor

30

Figure 13 — Residual Drag Coefficients
for Models 1 through 9

WIRE LOCATIONS
x /L ° 100
@ BARE HULL
O 0.61-mm WIRE
A 0.61-mm WIRE

Cp, = 0.667

L_/D = 0.500

0 1 2 3 4
Mer)
R, ° 10

Figure 13a — Model 1

31

WIRE LOCATIONS
x /L ° 100
@® BARE HULL
O 0.61-mm WIRE 2
A 0.61-mm WIRE 3.5
O 0.61-mm WIRE 5

1 2
2 1n-7
R, ° 10

Figure 13b — Model 2

32

Cp, = 0.667

L_/D = 1.000

WIRE LOCATIONS
x,/L + 100

@ BARE HULL

0 0.61 -mmWIRE
O 0.254-mm WIRE
Y 1.016-mm WIRE
4 0.61 -mm WIRE
O 0.61 -mm WIRE
So 0.254-mm WIRE
CO 1.016-mm WIRE

Cp. = 0.667
L_/D = 1.820

0 1 2 3 4
Saeed
Ry, 10

Figure 13c — Model 3

33

WIRE LOCATIONS
x,/L * 100

@® BARE HULL

O 0.61-mm WIRE 2,3.5&5
A 0.61-mmWIRE 3.5&5
O 0.61-mm WIRE 5

0 1 2 3 4
3 -7
R, * 10

Figure 13d — Model 4

34

WIRE LOCATIONS
x,/L * 100

® BARE HULL
O 0.61-mm WIRE 2.9

Cp. = 0.850

L/D = 1.000

6) 1 2 3 4
5 aReD
Ry. 10

Figure 13e — Model 5

35

WIRE LOCATIONS
x/L ° 100

@® BARE HULL
O 0.61-mm WIRE 5

1 2
5 ~7
R, 10

Figure 13f — Model 6

36

Cp. = 0.850

(fey! 10820

WIRE LOCATIONS
x, /L ° 100
@® BARE HULL
O 0.61-mm WIRE 1.5

Cp, = 0.850

L_/D = 0.500

0 1 2 3 4
-7
Rie WO

Figure 13g — Model 7

i

WIRE LOCATIONS
x, /L * 100

@® BARE HULL
O 0.61-mm WIRE 0.7

Cp. = 0.823

L_/D = 1.216

0 1 2 3 4
5 -7
Rig 10

Figure 13h — Model 8

38

WIRE LOCATIONS
x /L ° 100

® BARE HULL
O 0.61-mm WIRE Ud

Cp, = 0.933

L_/D = 1.000

0 1 2 3 4
Sapa
R. 10

Figure 13i — Model 9

39

Figure 14 — Measured Total Drag Coefficients of Models | through 9

@ BARE HULL
4 0.61-mm WIRE x, /D = 0.107
© 0.61-mm WIRE x,/D = 0.322

Figure 14a — Model 1

BARE HULL

0.61-mm WIRE x,/D = 0.218
0.61-mm WIRE x,/D = 0.382
0.61-mm WIRE x,/D = 0.545

5 10 15 20 25 30 35
- 19-6
Ry. 2 10

Figure 14b — Model 2

40

40

BARE HULL

1.016-mm TRIP WIRE

0.610-mm TRIP WIRE x,/D = 0.223
0.254-mm TRIP WIRE

SCHOENHERR LINE

Figure 14c — Model 3, x, /D = 0.223

BARE HULL

1.016-mm TRIP WIRE

0.610-mm TRIP WIRE x,/D = 0.559
0.254-mm TRIP WIRE

SCHOENHERR LINE

4 -6
R.° 10

Figure 14d — Model 3, x,,/D = 0.559

41

4.2

3.8

3.4

3.0

2.6

2.2

2:2

o 0
OR 2
C)
Figure 14e

BARE HULL

0.61-mm WIRE x,/D =0.231
0.61-mm WIRE x,/D = 0.405
0.61-mm WIRE x,/D = 0.579

ob 0®@®

Y LJ

en: SE

R yr 4 OD Eas
{\ " i)

iS e' )

e

— Model 4

@® BARE HULL
© 0.61-mm WIRE x, /D = 0.311

15 20 25 30
° 19-6
Ri 10
Figure 14f - Model 5

42

35

40

@ BARE HULL
Oo 0.61-mm WIRE x, /D = 0.542

Figure 14g — Model 6

@® BARE HULL
Oo 0.61-mm WIRE x, /D = 0.160

° 19-8
R, * 10

Figure 14h — Model 7

43

@® BARE HULL
© 0.61-mm WIRE x, /D = 0.078

Figure 14i — Model 8

@® BARE HULL
oO 0.61-mm WIRE x, /D = 0.181

22

0 5 10 “15 20 25 30 35 40
. ~6
R, ° 10

Figure 14; — Model 9

44

0.30

0.25

“MODEL NUMBERS

Figure 15 — Variation of Cp with Forebody
Slenderness Parameter

45

2.0

ls7/

3&

9 7* *SHAPE NUMBERS

9!

5
8
4.
&
4 6 8 10 12 14

- 10-8
Ry, 10

Figure 16 — Prototype Total Drag Coefficient of Nine Shapes

46

REFERENCES

1. McCarthy, J.H. et al., “The Roles of Transition, Laminar Separation, and Turbulence
Stimulation in the Analysis of Axisymmetric Body Drag,” Eleventh Symposium on Naval
Hydrodynamics, London (1976).

2. Granville, P.S., “Geometrical Characteristics of Noses and Tails for Parallel Middle
Bodies,’’ NSRDC Report 3763 (Dec 1972).

3. Granville, P.S., “Geometrical Characteristics of Flat-Faced Bodies of Revolution,”’
NSRDC Report 3710 (Nov 1971).

4. Hess, J.L. and A.M.O. Smith, “Calculation of Potential Flow about Arbitrary
Bodies,’ Progress in Aeronautical Sciences, Vol. 8, Pergamon Press, Inc., New York (1966).

5. Smith, A.M.O. and N. Gamberoni, “Transition Pressure Gradient and Stability
Theory,”’ Douglas Aircraft Company Report ES-26388 (Aug 1956).

6. Wazzan, A.R. et al., “Spatial and Temporal Stability Charts for the Flakner-Skan
Boundary-Layer Profiles,’’ Douglas Aircraft Company Report DAC-67086 (Sep 1968).

7. Curle, N. and S.W. Skan, “Approximate Methods for Predicting Separation
Properties of Laminar Boundary Layers,” Aeronautical Quarterly, Vol. 8 (1957).

8. Granville, P.S., “The Calculation of the Viscous Drag of Bodies of Revolution,”’
David Taylor Model Basin Report 849 (Jul 1953).

©, Schlicting, H., “Boundary Layer Theory,’’ McGraw-Hill, Inc., New York, Chapters
loan 7Cligsis)

10. Schubauer, G.B. and P.S. Klebanoff, “Contributions on the Mechanics of Boundary
Layer Transition,” National Advisory Committee for Aeronautics TN 3849 (1955).

11. Schoenherr, K.E., “Resistance of Flat Plates Moving through a Fluid,’’ Transactions
of Society of Naval Architests and Marine Engineers, Vol. 40, (1932).

12. von Kerezek, C. and N.C. Groves, “Disturbance Amplification in Boundary Layers,”’
paper in preparation.

47

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