Theoretical - experimental study of the effect of viscosity on wave resistance

DAVIDSON

LABORATORY

Report 1236

THEORETICAL-EXPERIMENTAL STUDY OF THE EFFECT OF VISCOSITY ON WAVE RESISTANCE

by King S. Eng and

John P. BreslIn

October 1967 R- 1236

Report 1236

October 1967

A THEORETICAL-EXPERIMENTAL

STUDY OF THE EFFECT OF

VISCOSITY ON WAVE RESISTANCE

by

King S. Eng

and

John P. Breslin

Prepared for the Office of Naval Research

Department of the Navy under

Contract Nonr 263(65) (DL Project 2957/085)

Dlstributon of this document is unlimited.

Approved

1236

ABSTRACT

The problem of assessing the effect of viscosity on wave resistance

Is considered. The approach Is to regard the decisive effect of viscosity as stemming from the difference between the measured pressure distribution on the stern and that predicted by. invisc id-f low theory. Results based on

the linearized theory show thatthe pressure deficiency at the stern of the body has little or no effect in the determination of wave resistance. 1ndications are that the pressure distribution around the mid-portion of the body makes the Important contribution to wave resistance.

KEYWORDS

Wave Resistance Viscosity

FIGURES (1 through 8)

TABLE OF CONTENTS

Abstract Iii

INTRODUCTION .

1

RELATIONSHIP BETWEEN PRESSURE DISTRIBUTION

AND SOURCE DENSITY 3

WAVE RESISTANCE 6

SELECTION OF MODEL 9

DESCRIPTION OF EXPERIMENTS 12

RESULTS AND DISCUSSION 13

CONCLUSIONS 17

APPENDIX A (Determination of the Constant of

the Homogeneous Solution). . . 19

APPENDIX B (Evaluation of the Integral) . . . 21

REFERENCES 25

R-l236

INTRODUCTION

Theorists and experimentalists In the field of ship resistance have for many years recognized the fact that thin-ship theory is inadequate in certain respects. The Consensus is that the apparent deficiencies may account for the lack of close agreement between the wave-resistance coeffi-cient computed in accordance with the theory and the residuary resistance derived from model tests;. it seems possible to break down these

deficien-cies Into the following three categories:

No representation of the effect of viscosity in changing the pressure distribution in the stern region of the hull No correction for the influence of the free surface on

the relationship between source-strength and hull-geometry The Inability of distributions on the longitudinal

center-line plane to provide a sufficiently close fit to an arbitrarily shaped hull

Work has been done on each deficiency. Havelock,1 for example, long ago proposed that the stern lines be modified by adding the displacement thickness of the boundary layer, in an effort to account for the "smoothing" effect of viscosity which is strongly suspected as the cause of much more weakness in the stern wave system than is predicted by theory. Martin2

actually attempted such modification, without much success. Inui3 proposed an empirical method of correcting for this effect of viscosity, but his curves for the correction parameters do not seem to have general validity.

lnu13 was the first to calculate streamlines based on a thin-ship source distribution for the condition corresponding to zero Froude number; and he showed that the streamline pattern does not fit the lines for which

the sources are calculated. Moreover, he later showed that experiments with forms corresponding to the streamlines give results which are

con-siderably closer to the theoretical wave drag than are the results of experiments with forms built to fit ship lines.

a SerIes 60 (CB= 0.60) hull, using the hull-surface scurce distribution

5

arrived at by Hess and Smith, which applies to the cae of zero Froude number and guarantees that the streamlines and the sources are correct for the hujl lInes Used. When the results of these calculations were compared

with the residuary resistance recorded in model tests of this hull, the discrepancy was found to be greater than that shown in comparisons with results based on thin-ship theory. It was believed that the increase in

error stems from neglect of the influence, of viscosity and the Influence of the free surface, on the source-density/hull-geometry relationship.

it therefore seems reasonable to pursue the required corrections separately, to assess their relative Importance to the problem of predicting _ship

wave reslstahce.

The present study is concerned with an assessment of the effect of viscosity on wave resistance,. _{The approach taken here Is to regard the} decisive effect of viscosity as stemming from the difference _{between the} measured pressure distribution on the stern and that predicted _{by inviscid} flow theory. _{One way of looking at the wave train generated by a ship Is}

to consider it as generated by a moving pressure distribution which, to the first order, _{is that which would be developed by the same form reflected}

in the free surface. Thus, In theory, large waves originate from the

stag-nation-pressure region in the bow, and equally large waves originate from the stagnation-pressure region in the stern. But in real fluid, s'tagnaflon pressure is not developed at the stern, and hence the stern wave system will not be developed to anything like the extent predicted by the theory. This requires construction of a more realistic theoretical modelwhich does not merely alter the stern .1 ines, by adding the displacement thIckness to provide a stagnation region a bit further aft, but provides the correct distribution of pressures.

In this repOrt, therefore, the analytical work is concerned with finding a velocity potential to fit an experimental pressure distributIon, and with investigating the effect which such' a velocity potential has on' wave resistance. Specifically, this procedure is applied to a strut-like

RELATIONSHIP BETWEEN

PRESSURE DISTRLBUTION

AND SOURCE DENSITY

Within the implications of thin-ship theory, we retain the center-plane distribution of sources as the disturbañcê potential, and its

attend-ing pressure distribution as independent of Froude number. Thus, the velocity potential with the coordinate system as shown in Sketch 1 is

SKETCH 1 C p C p R- 1236

I

I

m(,C) dC

d .1

%J(x)2+

y2 +

(z)2

-L -h

where m is source density per unit velocity.

Neglecting the square of the perturbation velocities, the linearized pressure can be wrtten as

P-Pa,

___

= 1 2

-

U

- UX

pU

,.2 h 211 -L -h 3

(x-) m(,) dC d

.+ y2 +

(z-p(x,y,z) U

where. x = x..L

Equation (1) _{is a pressure and source-density relationship based on potential} theory. _{In the present analysis, the same relationship is used to find a}

source distribution to fit an experimental pressure distribution.

The use of a source density which corresponds to a real-fluid pressure distribution is i,n a sense a strategy to represent the effects of _{a viscouS} flow by a potential flow model. _{To be entirely conslstent it wbuld be}

necessary to Include an external singularity distribution to represent the effects of the vIscouS wake. _{This was omitted here for the sake of maktng} the analysis as tractable as possible, In the hOpe. that thewake effects would be negligible.

The general

C

p

Then for a strUt-like ship with large draft and Small beam-length ratio, the pressure coefficient can be closely related to the source density by a singular Fredholm equation Of the first kind.

hivers ion of Equation (1) is

-1

,-2

c(.1)

Tri_xi2

V1'i

1_

( l-x1

where m0 _{is an arbitrary constant of the homogeneous solution. A number}

of different consraInts have been Investigated for the determination of m0 _{The nature of these constraInts and other aspects of Equation (1) are} discussed in Appendix A, where the only constraint which _{seems physically} meaningful is the closure condition. That is,

Jm(xi)

dxi

(1)

1236

m from the linearized pressure distribution gives the same body shape as the direct determination of m from the familiar relation m = 21.1

it may be objected that the use ôfthe condition

£m(x) dx = 0

Implies a zero drag force on the form as It would for the case of a body in an infinite fluid moving with uniform stream, whereas the real-fluid pressure distribution certainly gives a non-zero viscous pressure drag. The closure condition may still be Invoked without Implying zero drag when the presence of an external disturbance (such as a wake simulated by

potential-f low singularities) provides a non-uniform longitudinal flow

u(x,y). Then by Lagally's theorem,

£

drag

= - p J

u(x,0)

-L

5

It is clear that the Integral j'2m(x) dx = 0 Implies zero drag only when

u(x0) is a constant.

where

_P+

_iQ=

WAVE RESISTANCE

The wave-resistance express ion for _{a Source distribution6 is given by}

where p1 + = g/U2

=k2

0

V sec e h1 ' h/L L 2L Normalizing _R by

fpU212,

TT/2

J

₀

(P2+

e'1 dx1

Substituting (2) into (La) and interchanging the order of inte9ratlon, we obtain -th1 sec2 9 ivx e

)2

9 d9 (L) (4a) (3) (3 a) R = l6Trp k02 J:2 + Q12 sec3 9 9 .ivx1 k0z sec29 e e _{dx1 dz} 2 1

Denoting I =

ç

I

= j(v)

0

I

cos v' 1 236

The definite integrals are evaluated by

J

cosV-cosV'

siny'

cos iv

sin IV'

= TI

0

which Is valid for all Integer values of J . The wave-resistance

ex-press ion can be written as

l)

I2rn

'2n R2m,2n +

2m-1

2n-1 R-,2-i

m=1 n=

e'1

dx1

F

2

and letting x1 = cos v and = cos , one may express

e'

COS V In a Series Involving Bessel functions:

COS V

=

j

(v) + 2 (-1)

J2E(V)

cos 2nV + 21

(_l)1

j2..1(v) cos (2n-l)i

0 n= n=l n .TI I cos 2nV

,dy

I)

COS v -

cos v

0

IT )

1..

cos(2n-l)V

j Cos v -

cos 0 (6) V cos n +1 + 21 n=

J2n_1 (

where cj = -

2J

_{C(s) sin (j}

_cos1

) d ₍₇₎ TT/2 =J.. j,k 2TT

J

Sec 8)

Ik(K

sec 8) (1 e-th1 sec28"

1cos8d8

0

(8)

The term Rik as given by Equation (8)

_{can be calculated with a good}

degree of accuracy, as indicated in Appendix B; and if _C, is prescribed, can be easily determined by Equation (7). _{in the Froude-number range} for which _{CR was computed,}

RJ,k decreases with increasing values of

J and k. Therefore _CR may be computed efficiently because (6) is an alternating series with converging elements.

R-

1236

SELECTION OF MODEL

A symmetrical generalized Joukowskl section was selected to demon-strate this procedure (see Fig. 1 at end of report). This section was selected becaUse the actual pressure distribution had been measured (Fig. 2

at end of report) In a wind tunnel by Fage .et The profile Is obtained by transforming a circle

(g -

kL)2

+ 2

from the circle plane

(C-plane)

Into

the

physical plane (z-plane), by the transformatiofl function shown below (Sketch 2 Illustrates the relationship).

(z-nf\ \z+nLJ ₌

The geometrical interpretation of this transformation is

A'P'

=

(EV

BP

\BP/

To compute the theoretical _{pressure for the foil, first consider the}

potential of the outer circle in the circle plane

p2 W(C)

- (C - k2)

+ (C-k)

U

-The complex velocity In the physical plane is

U - iv = - = - !±\ (. \dzJ _{\dC/ \dz}

2+

2 Since C is defined as C = 1 U , C _{can be written as} p _p U2 P C _{sin2 (1 + cos} ) + k2) - (I - k2)

[(4

where A = - 4 (A + ICE) i(

l-k2

cos

_{- -j-}

+ sin2 =

tank

;in1

LCOS

_l+kJ

The normalized coordinates in the physical _{plane where these pressures} occur are

and

cos nQ + 2 cos 2na!]

xl

1236

[(k

"1

-

.T)J(A2" -1)

Ag" +l-2A"

cos (nc)

2[l A" sin (ncx)

- A2" + 1

-

2A" cos (nO)

For the particular section used here, k = 0.1 and

n = L95.

This gives

a thickness ratio of about 0.15. FIgure 1 shows the section and elevation

of the experimental set-up, and Figure 2 dIsplays both the experimental and the theoretical pressure distribution. It is to be noted that the exact theoretical pressure distribution agrees closely with that obtained from measurements (with wall correction) except in the vicinity of the trailing edge; that is, for locations aft of 80 percent of the chord. For

compari-son purposes, the linearized pressure distribution is also exhibited in Figure

2.

A5 may be expected, there are significant differences in the

linearized and exact theoretical pressure distributions.

11

k

DESCRIPTION OF EXPERIMENTS

The model used in the experiment is a strut whIch measures 60 by 30 inches and has a thickness that is 15 percent of the length (see Figs. I

and 3; Fig. 1 shows the setup of the experiment).

Since the wave resistance of the lower part of the strut is small _{compared with viscous drag and}

form drag, the upper part of the strut (which _{has a depth of 7.5 inches)} is detached and independently supported by _{the balance so that its total} resistance can be measured separately In the presence of the rest of the strut. _{This reduces error when wave resistance is to be separated from} measured total resistance. _{A theoretical estimate showed that the lower} part of the strut contributes at most only about _{15 percent of the total} wave resistance.

The strut was towed in Tank 3 at Davidson _{Laboratory, at Froude} numbers from 0.1 to 0.34. _{The gap between the upper and lower pieces of} the strut was kept to about 1/8 inch throughout _{the experiment.}

Figure 4 shows the total drag coefficient _{obtained in the tank test.} It is based on the wetted _{area of the upper part of the strut, excluding} the bottom area. _{Also given in Figure 4 are the total drag}

coefficient from the wind-tunnel test with wall correction, _{the Schoenherr friction} curve, and the estimated gap resistance. _{(The estimated drag from the} gap, _{CDgap 0.34 X} 1O , is based on the wetted area used for the total

drag coefficient.) _{The wave resistance is obtained by subtracting the} sum of the total drag obtained in the wind-tunnel test and the estimated gap resistance from the total drag obtained in the tank _{test. In principle,} this should be the value closest to the actual wave resistance of the body, if one ignores wave-boundary layer interaction.

CR=E

_nl

(-n=

where

1236

RESULTS AND DISCUSSION

Figure 5 shows that the wave resistance based on the linearized pressure distribution corresponds more closely to the data obtained from tank testing. The remaining curve in Figure 5 is the theoretical wave resistance based on the measured pressure distribution from wind-tunnel

tests. To determine the importance of the influence of the pressure at the stern,* calculations of wave resistance based on the exact theoretical pressure of the Joukowski section were also performed. The results are not significantly different from the results based on the viscous pressure distribution in the Froude-number range considered (i.e., 0.22 0.140). This finding clearly demonstrates that the pressure distribution at the

stern has no effect on the wave resistance as predicted by linearized

(Mitchell) theory. The reason for this lack of sensitivity to the disturb-ing pressures at the stern is that the structure of the weightdisturb-ing function provided by the theory completely suppresses the contribution of the pres-sures in the imediate vicinity of the bow and stern for the Froude-number

range investigated. This can be seen by examining the expressions involved in the wave-resistance calculation as given by Equations(6), (7), and (8) below: c

= -

2

j

C(x1)

sin (J cos' x1) dx1 -1 0'2m-1

'2n-1

R2m-1

,2n-1 +

(6)

(7)

may be recalled that the exact theoretical pressure distribution agrees very closely with the pressure distribution from measurement - except in

the extreme after-body, as pointed out on page 11 and as is shown in Figure 2.

TT/2

J J(t sec

)

_{Jk(Sec e)}

_thisec28)2

cos

e dO

e

0

Let x1 = cos V in Equation (6). Then _{can be written as}

= -

2

J

C(y) [sin y sin (jy)] dy

The leading and trailing edges correspond _{to V = 0 and V = IT}

,

respec-tively. _{Equation (7-1) shows that is not strongly dependent}

on C,

at the end points, because it _{is weighted by the factor sin V sin (jy)} which goes strongly to zero at each end. It also shows that is the

Fourier sine coefficient of the _{quantity [-2 sin V C(cos V)]. Thus, if} [-2 sin V C(cos V)] _{is a regular function in the interval 0}

then _{decreases with increasing value of}

J . This is also true for

k as given by Equation (7), which also decreases with increasing j

and k . Hence, the wave-resistance coefficient

CR given by Equation (8) converges very rapidly. _{In the cases which are treated here (i.e., the} ogive and the Joukowskl sections), the

CR curve can be determined by taking no more than eight terms of (i.e., c to

An ogive with thickness-length ratio equal _{to 0.1 is selected for} the purpose of clarifying the above discussion. _{Figure 6 shows three} different pressure distributions, _Cpexact Cp11 , and

Cpmod , of the

same body, where

Cp is determined from the exact potential theory..

C1,11 _{is determined from the linearized potential}

theory.

C _{is the modification of C near the stern of the}

mod Pii

body to simulate the real-fluid flow _{condition (the} modification is arbitrary but realistic),

R- 1236

illustrate the effect of the weighting factor of Equation (7-1). ;t is clear from this figure that the integrand for is insignificantly changed in spite of the large difference between

C11

and

Cp,d

in

the extreme after-body.

To determine the wave resistance of the ogive, the coefficients

(fs) which are defined by Equation

(7-1), are needed. The following

table shows two sets of j'S based on _Cpexact and

When _J is even, all the vanish because of the symmetry of the theoretical pressure distribution.

The as based on

Cp,d

are prac-ticaily the same as those based on

Cp1. The even terms of

j'S (e.g.,

2' a'4 etc.) are too small to cause any

significant. change in the wave

resistance, In the range of Froude number considered (i.e., 0.22

0.1+0).

in the wave-resistance calcUlationS there is no appreciable change between the values based on C and C .. This example clearly shows that

Plin Pmcjd

the pressure distributions at the ends of the body do not contribute to the wave resistance, In spite of the large difference between Cnrljn and C

in the extreme after-body. However, the difference !n the resistance

derived from C and that derived from .0 is considerable (Fig. 8).

Pexact Plin

This difference can be explained by the difference in c . In the

prac-tical range of Froude numbers (i.e., 0.22 0.32), the wave resistance derived from C averages about 62-percent larger than that derived

Pexict

from Cp11.

Noting that the resistance varies as the square of a_i

*

The authors are well aware that the use of the exact theoretical pressure distribution in these calculations is inconsistent with the use of the linearized theory; use of the exact pressures is made to illustrate the

Importance of the difference in the distributions through the central por-tion of the secpor-tion.

15 J 3 a' Based on C j Pexact 0.691+8765 0.3707759 a' Based on C J Plin 0.5333333 0.3200000 5 -0.0595277 -0.0761905 7 0.0258622 0.0355556 9 -0.011+2138 -0.0207792 11 0.0089009 0.0136752

and taking

2

c _(exact)

Qi2(linearized) - 1 = 0.69

we see that the difference in

CR can be best accounted for by the differ-ence in o _{since the first few 0J'S are shown to be dominant In} deter-mining _CR

The term or _{can also be written In terms of}

source density

m

by substituting Equation (1) _{into (7) and interchanging the order of} integration. That is,

Qj =

25

m(x1) cos (j cos1 x1) dx1 -1

is significant that the _{source density is weighted by}

a factor which goes linearly to zero at the ends, whereas _{the representation in}

terms of (7-2)

The transformation used earlier (viz., _{V = cos x1) gives Equation} (7-2) the form

=

2J

_{m(y) Fsin y cos (Jy)] dy}

(7-3)

Equation (7-3) shows that is not strongly dependent _{on m and hence} on waterline slope at the end points, since _{sin V-.' 0 at 0 and}

ii

Again the small values of

J are the important terms at practical Froude

numbers (0.22 _{0.32), and therefore the values of m}

at the ends are not emphasized; quite to the contrary, they are weighted by the factor _zero at each end, regardless of the value of _{J , and hence the values of}

m In the neighborhood 0f the ends do not contribute significantly _{to the value} of the Integral.

123 6

CONCLUSIONS

The intuitively conceived proposition which led to the Initiation of

thIs study Is not supported by results of the study. Calculations based on linearized wave-resistance theory show that the pressure deficiency at the stern of the body has little or no effect in the determination of wave

resistance. t Is now clear that the pressure

distribution in the

mid-port ion of the body makes the important contribution to wave resistance as estimated from linear theory.

In an effort to seek a rational explanation, for the present result, a consistent second-order theory has been formulated. Some preliminary

numerical results revealed that the finite Froude-number effect In the correction of the kinematic boundary and the pressure correction on the

free surface are the dominant factors in reducing the strong stern wave system which was predicted by the first-order theory. The detail of this study will be presented in the near future.

APPENDIX A

DETERMINATION OF THE CONSTANT OF

THE HOMOGENEOUS SOLUTION

The various side conditions considered in determining the constant of the homogeneous solution of Equation (1) are stated below.

Case 1: The source density at the stern vanishes

[i.e., m(-1) = 0]; then -i J1

iiEi

c(1)

d1

m _TI -1

Case:

The thin-body approximation relates the source

density and the

slope of the offset by the equation

y(x)

m(x1) - - 2 or

y(x1)

₂ TX: dx1

Suppose y(-l) is prescribed; then

rn = y(-1)

(A-2)

where y(-1) can be stipulated as the boundary-layer displacement

thickness

at the stern of the body.

The constant, m0 , for the first two cases Is positive.

This means

that the body is open because there is an excess of sources.

These

selec-tions also give higher values of wave resistance than does the case where

(11)

APPENDIX B

Evaluation of the Integral

11/2

=

J Jt

sec sec 0) cos e dO

0

When we change the variable 0 to x = . sec e

, takes the form

2f

dx

=

J(x) J(x)

x2_t2

Noting that the integrand converges as does

+

rLV

I.L,v where and 2 (0) r1 - 21T 2

-

211 .(]) - 211 R- 1236 ii J1(x) J(x)

J(x)

J(x)

, we might write (B -2) 2 dx 1

[J2(

1

J

'L

(x)

;.

=.

2(+1)

+

(2_1)

k2

Jk2N]

when

>1

(B-3a) 21 (B-la) (B-lb) (B-2a) (B-2b)

Except for the case when ii. = = 1 ,

r0

has a closed form solution.9 /

JA/

_212.2

_X3

(0) 1 = 2Tr(+v+2) Reference

I

(K)

_(v-IL-k)

i

(t)j

(it) IL-i +

_{(v_p) [2_(v_2)21}

(v_IL+k) j

(t)J

(K) J

(K)J

₍

LL-2

V-i

V

+

(v-IL) 1v2_(IL_2)21

+

v2-(IL-2)2

j

K2 + (+v_1)2_1 (K)+' 2

()Jv(I1

IJ2

J (K)J

(t)+J

IL-i IL-i + K EJ + v(K)] + J

(pt) j

i.'

v-

ILl

when IL and

IIL ± vf

2

(B-3b)

r°

K2 +

2>2 k2(k) -

_i (IL2+3IL2)

J2 (K)

l6rr

IL(IL+1)(I.i+2) IL

-o

k=1 - IL IL-i

(K) +

IL

(IL+l)

2 (K) IL+2

+9

J()

JJ (pt)

- j

_(K)I

J

(K) j (K)

IL _IL+2 TI I&+2 _p. when IL (B-3c) 9 demonstrates a very efficient and accurate method of computing the Bessel function of arbitrary orders _{and arguments, so that computing} r0)v is no problem. _{The remaining integral r0)v can be written as}

=

JlL(x) J(x) (X2

X2K2

the upper limit V was investigated for values V = 5t, lOt, and 2O. The computed result shows that r

the three values of Y .

R- 1236

23

*

REFERENCES

*

HAVELOCK, T. H., "Calculation illustrating the Effect of Boundary Layer on Wave Resistance." Trans. Inst. Naval Arch., March 1948. MARTIN, M., "Analysis of Ship Forms to Minimize Wave-Making Resistance."

Report

_845,

Davidson laboratory (DL), Stevens institute of Technology, May 1961.

INW, T., "Study on Wave Making Resistance of Ships." Soc. Naval Arch. Japan, 60th Anniversary Series, Vol. 2, 1957.

1+. BRESLIN, J. P. and ENG, K., "Calcülat ion of the Wave Resistance of a

Ship Represented by Sources Distributed Over the Hull." Paper

presented at the International Seminar on Theoretical Wave Resistance, University of Michigan, Ann Arbor, Michigan, August 1963 (DL Report 972, July 1963).

HESS, J. 1. and SMITH, A. H. 0., "Calculation of Non-Lifting Potential Flow About Arbitrary Three-Dimensional Bodies." Douglas Aircraft Company Report E.S. 40622, March 1962.

LUNDE, J. K., "On the Theory of Wave Resistance and Wave Profile." Skipsmodelltanken Meddelelse NR. 10, April 1952.

FAGE, A., FALKNER, V. M., and WALKER, W. S., "Experiments on a Series of Symmetrical Joukowski Sections." Aeronautical Res. Comm. R & M No. 121+1, 1929.

TULIN, M. D., "Steady Two-Dimensional Cavity Flows About Slender Bodies." Report 831+, David Taylàr Model Basin (DTMB), May 1953.

RANDELS, J. B. and REEVES, R. F., "Note on Empirical Bounds for Genera-ting Bessel Functions." Communications of the Association for

Computing Machinery, Vol. 1, No. 5, May 1958.

LUKE, V. L., "Integrals of Bessel Functions." McGraw-Hill Book Co., 1962.

1236

UNCITED REFERENCES

GOLDSTEiN, S., "Modern Developments In Fluid DynariCS."

Vol. 11,

Oxford at the Clarendon Press, 1950.

WU, T. V. T., "IntcractiOfl Between Ship Waves and

BoundarY LaYe!." International Seminar on Theoretical Wave Resistance, Vol. III,

August1963.

3 WIGLEY, C , "Effects of Viscosity on

Wave_Resistance " International

Seminar on Theoretical Wave_Resistance, Vol.

III, August 1963.

L WIGLEY, C , "The Effect of Fluid

Viscosity on Wave Resistance Using a Modification of Laurent ieff's Method - A More Accurate Calculation "

C Wigley, Flat 103, 6-9 CharterhoUse Square,

London Ed, England, May 1966.

KARP, S ,

KOTK, J

and LURVE, J

, "On the Problem of Minimum Wave

Resistance for Struts and Strut-Like Dipole Distributions

" Proc

3rd Symposium on Naval Hydiodyflamics, .5 ningen, Holland,

1960.

WL

71/211

-c15

----1.01

O.

06

₀₄

₀₂

_04

-0.6

-I.

FIGURE

2.

PRESSURE DISTRIBUTION OF A GENERALIZED JOUKOWSKI

_SECTION

PPa

I/2PV

0.5

0

WIND-TUNNEL MEASUREMENT WITh WALL

CORRECTION

2U

EXACT THEORECTICAL Cp

-z

I'J 2.4

20

0.8

0

C.,

4

0

0.4 MODEL CHARACTERISTICS 2.09' 0 0.08 0.12

L-5'

DISPLACEMENT 1.556 FT3 WETTED AREA =6.403 FT2

(EXCLUDED THE BOTTOM AREA)

0.16

BEAM

LENGTW - 0.15

BEAM

DRAFT - 1.2

TOTAL DRAG COEFFICIENT FROM WIND-TUNNEL _TEST (WITH WALL CORRECTION)

TOTAL DRAG COEFFICIENT

FROM TANK TEST

FIGURE 4.

_{RESISTANCE COEFFICIENT OF A}

_GENERALIZED

_{JOUKOWSKI SECTION}

SCHOENHERR FRICTION CURVE _{ESTIMATED GAP RESISTANCE}

0.20 Q24 0.28 FROUDE NUMBER,

4

Ui 3.6

0

I-Ui

z

0

o

4 Ui U)

4

N

0

'

I-z

Ui C., Ui

0

0.

Ui U

z

I2

'U.

>

4

05 0.4 0 OL8 THEORECTICAL RESULT

BASED ON LiNEARIZED PRESSURE, Cp2j BASED ON WIND TUNNEL Cp

O DIFFERENCE IN DRAG COEFFICIENT BETWEEN

TESTING IN THE TOWING TANK AND THE

WIND TUNNEL

/

I

/

/0

1

0

I

/00

J o9

I I I . . 0.12 0.16

020

024

0.28 0.32 0.36 0.40

FROUDE NUMBER,k

0

0

I

/

/

/

0

0.2

... S..

.1

CPEXACT(PRESSURE COEFFICIENT BASED ON EXACT POTENTIAL THEORY)

P

LIN (PRESSURE COEFFICIENT BASED ON LINEARIZED POTENTIAL THEORY)

-CpM0D

(CpLIN

MODIFIED NEAR THE STERN TO SIMULATE REAL-FLUID FLOW CONDITION)

I

I I I. I _J

0

-02

-0.4

-0.6

-0.8

-1.0

FIGURE 6.

PRESSURE

_{COEFFICIENT OF A 10% THICKNESS-LENGTH RATIO}

_,

_{OGIVE SECTION}

I

0.8 0.6 Q4

Cp 02

.

Is.) a . .AI

a'

I _I I I

I.

I _I

0.4

I I _I 1.0 0.8 0.6 04

02

2.8 2.4

x

C,

z 2.0

LU

z

0

m 1.6

x

I-z

LU C, La. Ia. LU

0

C, LU

z

Cl) LU O.8 0.4 0

0.2

1236

BASED ON EXACT PRESSURE

BASED ON LINEARIZED PRESSURE

0.16 _{0.20 0.24}

FROUDE NUMBER, _.!_

FtGURE 8. WAVE RESISTANCE COEFFICIENT

_{OF A 10%}

THICKNESS-To-LENGTH OGIVE

36

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UNCLASSIFIED

DD

I NOV 55 I

FORM 1473

S/N 0101-807-6811

(PAGE 1) IJNCLASS IFIED

Security Classilicatton _{A- 3* 4 05}

S,turiI v Cl.Ibi Fjction

DOCUMENT CONTROL DATA. R & D

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tI')fied)

i ORIGINS lUNG AC TIVI iv (Corpotate author)

-Davidson Laboratory

Stevens Institute of Technology

20. REPORT SECUR!TV CLASSIFICA lION

Unclassified

2b. GROUP

REPORT ,I,tr

-THEORETICAL-EXPERIMENTAL STUDY OF THE EFFECT OF VISCOSITY ON WAVE RESISTANCE

4. DESCRIPTIVE NOtES (Type Of repor)afld.iflCtUaiVe dates)

Final

. Au THOR(S) (irs1 name, middle initial, last name)

King S. Eng and John P. Breslin 6 REPORT DATE

October 1967 .

7S. TOTAL -NO. OF PAGES

36

,b. NO. OF REFS 10 + 5 Uncited

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_263(65)

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II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

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13. ABSTRACT

The problem of assessing the. effect of viscosity on wave resistance is

considered. The approach is to regard the decisive effect of viscosity as stem-ming from the difference between the measured pressure distribution on the stern and that predicted by inviscid-flow theory. Results based on the linearized theory show that the pressure deficiency at the stern of the body has little or no effect

in the determination of wave resistance. Indications are that the pressure dis-tribution around the mid-portion of the body makes the important contribution to wave resistance.

II.

KtV WORDS

Wave Resistance

Viscosity

LINK A NO L t NT LINK N LWKC ROLE [ NT flOL NT