Application of Sle 00 Garr

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LIBRARYTECHNICAL REPORT SECTIONNAVAL POSTGRADUATE SCHOClMONTEREY, CALIFORNIA 93240

NPS-69GM77051

NAVAL POSTGRADUATE 3C Monterey, California

OL

APPLICATION OF SLENDER BODY THEORY IN SHIPHYDRODYNAMICS AT HIGH-FROUDE NUMBER

C.J. GarrisonMay 1977

Approved for public release; distribution unlimited

FEDDOCSD 208.14/2:NPS-69GM77051


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NAVAL POSTGRADUATE SCHOOLMonterey, California

Rear Admiral Isham LinderSuperintendent Jack R.BorstingProvost

The work reported herein was supported by the Naval Postgraduate School,Monterey, California.

Reproduction of all or part of this report is authorized.

This report was prepared by:


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1. REPORT NUM9ERNPS-69GM77051

2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

4. TITLE (and Subtitle)Application of Slender Body Theory in ShipHydrodynamics at High-Froude Number

5. TYPE OF REPORT A PERIOD COVERED

6. PERFORMING ORG. REPORT NUMBER7. AUTHORf*;

C.J. Garrison

8. CONTRACT OR GRANT NUMBERfaJ

9. PERFORMING ORGANIZATION NAME AND ADDRESSNaval Postgraduate SchoolMonterey, California 93940

10. PROGRAM ELEMENT, PROJECT, TASKAREA 4 WORK UNIT NUMBERS

11. CONTROLLING OFFICE NAME AND ADDRESSNaval Postgraduate SchoolMonterey, California 93940

12. REPORT DATEMay 1977

13. NUMBER OF PAGES18

U. MONITORING AGENCY NAME 4 ADDRESS^// different from Controllint Office) 15. SECURITY CLASS, (of this report)unclassified

15a. DECLASSIFICATION/ DOWNGRADINGSCHEDULE16. DISTRIBUTION ST AT EMEN T (of thla Report)

Approved for public release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different from Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS fConlfnue on reverse tide if necettary and taentlly by block number)

Ship HydrodynamicsSlender Body Theory

20. ABSTRACT (Continue) on reverse tide If necestory and Identify by block number)The slender body theory utilizing the method of inner and outerexpansions is applied to evaluate the yawing force and moment on a slendership hull at high-Froude number. The theory is developed for hulls ofarbitrary cross-section, and a Green's function method is outlined forsolving the inner problem for hulls of arbitrary section. The theory isapplied to a slender cylindrical hull, and the potential is represented inthe inner region by a multipole expansion. Closed form results are obtainedfor the yawing force and moment for the cylindrical hull.

DD ) jan 73 1473 EDITION OF 1 NOV 65 IS OBSOLETES/N 0102-014- 6601 I UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Dats Sr.tered)


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h InSrodacU on.Slender body theory has enjoyed great success in aerodynamics, and

in recent years there has been considerable interest in applications innaval hydrodynamics. Ship hulls are long and slender, and this is alogical feature to exploit in simplification of the governing equations.Michell's classical theory of wave-making resistance published in 1898considered the ship to be a thin body and represented the hull by use aof distribution of Havelock sources over a vertical mid-plane. How-ever, ships tend to be slender as opposed to thin because the beamand draft tend to be of the same order of magnitude and both tend tobe small in comparison to the length. It was hoped, therefore, thatslender body theory might represent a significantly better alternativeto Michell's theory in the area of wave resistance and at the same timeyield useful results for the hydrodynamic forces in the transversedirection both in steady and unsteady motion. Newman (1970) has givena review of the application of slender body theory to ship hulls whichdiscusses various kinds of hull motion and wave resistance.

Major advances in the application of slender body theory in shiphydrodynamics have been made largely in the past decade or soprincipally by Tuck (1963,1964), Newman (1964,1965), Ogilvie (1969),Ursell(1968) and Faltinsen(1971) . The method of approach involvesthe use of matched asymptotic expansions which is a powerful methodfor dealing with singular perturbation problems. Potential flowpast a slender body is recognized as a singular perturbation problemand in such cases an inner and outer expansion is developed. Theinner expansion is valid in the near-field, i.e., near the body andthe outer expansion is valid in the far-field. The unknowns re-maining in either of the expansion are determined by a matchingprocedure. The general method of matched asymptotic expansionsis discussed by Van Dyke (1964).

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In the present report the application of the slender body theoryto a ship hull in steady motion at high-Froude number is considered.While most displacement ships operate in the zero to intermediate Froudenumber range, there is a new generation of high speed ships generally ofthe catamaran configuration which utilize rather thin or slender hullsand operate at rather high Froude number. The surface effects ship hullsrepresent the principal example of this type of hull even though the flowabout an SES hull is somewhat more complex than the one considered herebecause of the pressure difference between the two sides of the hull.In the following a single slender hull operating a high-Froude numberis considered from the view point of slender body theory.

2. Boundary-value Problem For Ship Hydrodynamics

We assume the fluid to be ideal (inviscid, incompressible andhomogeneous) and the motion to be irrotational. If (x,y,z) representsCartesian coordinates fixed in space with the gravitational forceacting in the negative y-direction and t denotes time, a velocitypotential


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where C(t) may be, in general, a function of time, p denotes the massdensity of the fluid, and g, the acceleration of gravity The functionC(t) is at most a function of time, and can be determined by evaluationof P+p(j) 4%>V ,V(J)+pgy at some reference point. This point is generallytaken at the free surface at infinity where all of the values are known,

The surface of the hull is described by the equation s(x,y,z, t)=0.Since this surface is considered to be impermeable the kinematic boun-dary condition, which imposes the condition that the relative fluid velo-city in a direction normal to the hull must vanish, is given by

?7 = T? + V.Vs=0 on s=0 (4)Dt o tOn the free surface there are two boundary conditions which must

be applied. If the elevation of the free surface is described byy=n(x,z,t) the kinematic boundary condition obtained in a fashionsimilar to (4) results from setting the substantial deriative of thesurface n-y=0 to zero,

(n-y)=n +$ n +4> n - = on y=n (5)Dt t x x z z y

In addition to (5) a dynamic boundary condition must also be imposed.This boundary condition results from simply applying P=0 on the freesurface and is given by

+ JsV


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In addition to the boundary conditions (4-6) the potential functionmust satisfy certain radiation conditions at x2+y 2 -> appropriate tothe particular problem under consideration. Also, on the bottom thekinematic boundary condition

= on y=-hy (7)

is applied if the water depth is finite and denoted by h.

3. Perturbation Procedure For a Slender BodyAt this juncture let us consider a particular problem: a slender

hull is positioned with its long axis coincident with the x-axis andheld fixed in space as indicated in Figure 1. A uniform stream of

Figure 1 Definitions

magnitude U makes an angle a with respect, to the x-axis so that theresulting x,y, and z components are -U cos a, 0, -U sin a, respectively.The coordinate system is placed at the mean free surface so that they=0 plane represents the mean free surface, and the water depth is takenas infinite. The motion will be restricted to steady motion, and the

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hull is described by s(x,y,z;e) = where the parameter e = beam ordraft-to-length ratio.

Outer Problem

The problem which is here defined is recognized to be dependenton two dimensionless parameters, the beam/length ratio, e, and theincidence angle, a. Thus, as an initial step we may expect the solu-tion to be expandable in a power series in both e and a. (At leastthe solution should behave in a regular manner with the exception ofthe region near the body.) Introducing the body potential, , thetotal potential may be expressed as

$ = C.x,y,z;e,a)-Ux cos a-Uz sin a (8)

where the body potential $ is expressed by

$=e^+0LO:+O( 2 )+0{a. 2 ) (9)

in the outer region. The first term of the Taylor series, 4> (0,0)= , isof course, zero because if both e=0 and a=0 the uniform stream isdirected parallel to a line of zero thickness and the body potentialvanishes. Thus, expanding sina and cosa about a=0 in (8) and substi-tuting into (9) gives the total potential as

= e j + a(4> -Uz)-Ux+0(e 2 )-K)(a 2 ) (10)

and substituting this form of the potential into (2) shows that

V2 i=v2^ =0 (11)since a and e are independent parameters.

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In similar fashion,we may express the free surface elevation inthe form of (9),

n=en^Wj[+0(e2 )+ 0(a 2 )40(cie) (12)

Substituting (8) , (9) and (12) into (5) and collecting coefficients oflike powers of e and a gives:

$? + Un? = on y=0 (13)

and4>? + Un? =0 on y=0 (14)ly lx

The dynamic free surface boundary condition is obtained by use of(6) by evaluating C(t)=^U2 at x= on y=0 and substituting in the formof and n as given in (12) and (8). Here again two separate boundaryconditions result because of the independence of the two small parameters,e and a. The first of these is obtained by setting a=0 as

nf+ 2- f =0 on y=0 (15)1 g lxand the second by setting e=0,

T\,+ -JL.$, = on y=0 (16)1 g lx

Eqs. (13) and (14) together with (15) and (16) yield the linear freesurface boundary conditions,

*L =0 (17)ly g Ixxand

*y e ixx

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Eqs. (17) and (18) represent familiar linear boundary conditions forthe free surface. If we now let the Froude number U2 /gL (where L isconsidered to be 0(1)) become large, (17) and (18) become simply

d> 1 = on y=0 (25)

The boundary condition on the hull is in this case to be replaced by amatching condition.

Inner Problem

We now consider the inner potential which is valid near the slenderbody. The solution to the slender body problem is singular in the innerregion but the degree of singularness at the outset is unknown. We there-

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fore begin by writing the body potential as

= A (e) (x,Y > Z;a)+A 1 (e)$ 1 (x,Y,Z,;a)+ ... (26)

where the Y and Z coordinates are stretched by the parameter e,

Y=y/e (27)Z=z/e (28)

and A (e),AT (e), etc. are called gage functions as defined by Van Dyke(1964). These gage functions will eventually be determined by use ofthe principle of least degeneracy also discussed by Van Dyke.

The body potential is expressed in (26) as an inner perturbationseries in e, the beam-length ratio. Here the incidence angle, a, isallowed to remain finite and is not considered in any limiting process.We shall return to this point later.

If the inner representation of the body potential given by (26)is substituted into (8) and the result substituted into the Laplacian,(2) , we find,

,2 ^ ^ -L- Ai(e)L +n +^ + , U 2^ +


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s(x,y,z;e) = S(x,Y,Z;e) = then the hull kinematic boundary condition,which is obtained by use of (26) and (8) with (4) > gives

(i2 *ox

s*+*oYWz)+ vfy' (2,Jxy%Wz) (3i)e 2US. eUaS7 =A (e) A (e)o o

Now if a were to be considered to be fixed at 0(1) then the principle ofleast degeneracy would require that A (e) = e, and the hull kinematicocondition would read

*oYSY+*oZS Z=UaS Z n S=0 C32)

This boundary condition simply indicates that in any transverse planeperpendicular to the x-axis the body potential must just cancel thecross-flow velocity, Ua. It is noted that in such a case the cross-flow is an order of magnitude larger than the perturbation associatedwith the thickness distribution of the body and thickness effect is,therefore, lost.

Alternately, if the angle of attack were taken to be zero then theprinciple of least degeneracy leads one to A (e)= 2 and (31) gives,

$ S +$ S =US on S=0 (33)OY Y 0Z Z X

a familiar result in slender body theory.

A third possibility is to take a to be of the same order of magni-

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tude as e and retain both of the terms on the right hand side of (32)and (33). This gives A (e)=e 2 and

SV$og S Z - U(SX + f Sz ) on S=0 (34)An alternate way of writing (34) is

-< +M)where N=n/e and n represents distance measured normal to the curve in theY-Z plane described by the equation S(x,Y,Z;e) = with x held fixed.

The free surface condition to be applied to the inner solution isobtained by eliminating ri between (5) and (6). The boundary conditionobtained in this way is

4> (VVcf> )+(f> (V-V )+g =0 on Y=ri/e (36)x x z z ySubstitution of (26) and (8) into (36) gives

( 2$0x-u) [ 2$0xx ( e 2^-U)+s^ ^o xY+^o xZ (e *0z-Ua)+ (c $oZ-Ua) [e *Oxz(e2*ox-U)+e *oY oYZ+*oZZ (e*oZ-Ua)+ge G^0 on Y=0 C3Z)

Now, if e-K) with a=fixed, (37) gives

*oZZ= n Y= (38)

or from this we can deduce that$ =0 on Y=0 (39)o

is the appropriate boundary condition in the inner region. The problemfor the first term of the inner expansion for the high-Froude number

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slender body theory is, in summary,

$ + =ooYY oZZ (40a)

3$ TT/S x ^ 3N,3N U(^ +7 3Z ) n S= (40b)$o =0 on Y=0 (40c)

There is no radiation condition specified in (40) ; this condition isreplaced by matching with the outer solution.

4. Solution for the Inner Problem

The inner problem specified in (40) is essentially a two-dimensionalproblem as indicated in Figure 2. Such problems can be solved for rather

$ n =

C(Y,Z)=0

$n =

a function which iso = specified on the hull

surface, C=0.

Figure 2 Inner Problem

arbitrary shapes by use of a Green's function of the form

G(Z,Y;,n)=ln>/(Z-C) 2+(Y-n) 2 -ln\/( z-^ 2 + (Y+rl>


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The first term in (41) represents a source, and the second term representsa negative image with respect to the Z axis. The potential $ may berepresented by a surface distribution over the hull of these two-dimen-sional singularities.

In particular, the potential may be represented by

o 2tt I F(x;n,?)G(Z,Y;C,n)dC (42)where C denotes the curve in the Y-Z plane representing the cross-section of the hull and dC denotes an arc length along that curve.Using (41) for G, (40a) and (40c) are automatically satisfied. Thehull boundary condition (40b) then yields the integral equation forthe source strength distribution, F(x;Y,Z) as

'(x;Y,Z)+^ /F(x;ti,?) || (Z,Y;C,n) dC(43)

This equation can be solved numerically by subdiving C into a finitenumber (say N) of segments. Eq. (43) can then be reduced to a set ofN linear equations in N unknown values of F. over each of the segments.

Cylindrical HullFor the case of a semi-immersed circular cylinder, a multipole

series may be used to represent the solution. Consider the surfacedescribed in cylindrical coordinates by

S(R,9,x)=R-RH (x) (44)

where R=r/e denotes the radial coordinate in inner variables and R^

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denotes the radius of the hull in the inner variable (R^ =r /e) andis, in general, a function of x. The kinematic boundary condition onthe hull, (40b), takes the form,

ff = u(\ + | cose > on R=RH W)where the polar coordinate system is related to the cartesian system by

Z=R cos 6 (46a)Y=-R sin 9 (46b)

The solution for $ satisfying both (40a) and (40b) is given byoo

$o=L an sin(n8)n=l Rn (47)Application of the hull boundary condition (45) now gives

oo a n t2 (n+1) sin n^-URj T U cos9 (48)Rn=l HBy use of the results

J sin(n6)de=y Cos2(n6)de=7r/2 (49)o ~oTT

sin(n9)sin(m9)=/ cos (n6) cos (m6) d9=0 if m^n (50)sin (n0 ) sin (m6 ) =/IT/ sin(n9)cos6 d9= -4-., n^l (51)n -1

/oir l-cos(mr) l-(-l) nsin(n9)d9= ^-*- = v ' (52)n n v

we find

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(n+1) 9 /=_L i^H i_/ cosQ sin(n9)de, n=2,4,6, (53)n e n Hor, carrying out the integration,

andan= . i*& ^nlj n.1>3>5 ,... (55)

The inner potential function is, therefore, represented by the series

Wit / \n^-l ) -n sin(n6)(56)

n. n+1 .sin(n9)

n= 1 , 3 , . . '

The dynamic pressure on the hull can be obtained from Bernoulli'sequation, disregarding terms of OCe*4 ) and o(e a 2 ),0 1U$o -(r>^o 9sine+ (7> U2sin2 9 (57)The normal (or yaw) force per unit length along the hull canbe determined through integration of the pressure over the immersedsurface. That is,

dFz=r/ p Rh

cose deldx (58 ^

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After substitution of (56) into (58 ) and some lengthy manipulationsand integrations we get, for the normal force on the segment dx,

1

/2+y^ [ l-(-l) n _ n(l+(-l) nn=2 \

dx (59)

The net normal force is obtained by integration from the bow whereRjj=0 to the stern where Rh=Rit

F - -a 1 oU2 ZL r 2 I V^ A-(-D n n(l+(-l) )F z- -a 2 pU ^^^ 2+^ I ^n=2 \The dimensionless normal (or yaw) force coefficient based on the sternarea, therefore, becomes

f -^V -M**^ (^=^- - n( i+ ;- i)n > ) ; a *sPU^tt r2- HS I n=2

where r is defined as R^ =rIJ /e and rH represents the hull radius in outerHs Hs Hgvariables. For boat-tailed hulls we get the familiar result F z=0 becauseRhs=o.

The summation in (61) can be somewhat rearranged to read

I n=3,5,7 /

The series converges rapidly to approximately 0.128 so

f =0.75a (63)2 \The value of f =0.75a for the cylindrical hull at high Froude number canzbe compared with the classical value of f =2ct for the same hull operatingz

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with the low-speed free surface boundary condition, 8$ /8Y=0 on Y=0.oIt is reasonable to expect a somewhat smaller value for fz for thehigh-Froude number case because the high -Froude number free surfaceboundary condition makes the Y=0 plane act like a zero pressure sur-face and, therefore, right at the free surface there can be no pres-sure difference across the hull. The low Froude number (actually lowto moderate) boundary condition makes the free surface act like a rigidwall and a pressure difference across the hull can exist at the Y=0level.

The moment about the Y-axis is given byb/M - -/ xdF < 64)y / z

x=-a

where a and b denote the distance from the origin to the stern and bow,respectively, so that the hull length is L=a+b. Thus, using (59) with(64) gives

My* rf PU2V ' n=2,3).. /'where

As=area of the stern section below the mean waterlineV=hull displacementL=hull length

When the series is evaluated we get(66)my -o.75(l- 4-^)

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and for boat-tailed (or closed) bodies the moment coefficient becomesmy =0. 75a. This value may be contrasted with the low Froude numberresult of 2.0a which is often referred to as Munk's moment.

5. Conclusions

Slender body theory has been applied to a slender ship hull opera-ting at high-Froude number. For the case of a semi-circular hull,closed-form results were obtained for the yaw force and moment. These valuesare less than half the values of the equivalent coefficients associatedwith the rigid-wall free surface boundary condition.

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REFERENCES

Faltinsen, O.M. (1971) Wave Forces on a Restrained Ship in Head-SeaWaves , Ph.D. thesis, The University of Michigan (Also: A RationalStrip Theory of Ship Motions: Part II, Department No. 113, Depart-ment of Naval Architecture & Marine Engineering The University ofMichigan, Ann Arbor, Michigan.)

Newman, J.N., (1970) Applications of Slender-Body Theory in ShipHydrodynamics , Annual Reviews of Fluid Mechanics, Vol.2.

Newman, J. N. , (1965) David Taylor Model Basin Report. No. 2127

Newman, J.N. , (1964) A Slender-body Theory for Ship Oscillation inWaves , Journal of Fluid Mechanics, Vol.18 pp 602-608.

Newman, J.N. and Tuck, E. 0. ,( 1964) Current Progress in the Slender-BodyTheory for Ship Motion, Procurement Proceedings Fifth Symposium onNaval Hydrodynamics, Bergen, Norway , (1964) (U. S. Government PrintingOffice, Washington, D.C.) pp 129-162.

Ogilvie,T.F. and Tuck,E.O. , ( 1969) A Rational Strip Theory of Ship Motions ,University of Michigan, Department of Naval Architecture and MarineEngineering Dept. No. 013.

Tuck,E.O. , (1964) A Systematic Asumptotic Expansion Procedure for SlenderShips , Journal of Ship Research , Vol.8, No.l, pp 15-23.

Tuck, E. 0. , (1964), On Line Distributions of Kelvin Sources Journal of ShipResearch , Vol.8 No. 2, pp 45-52.

Tuck,E.0. , (1963) Steady Motion of a Slender Ship , (Doctoral Thesis, Universityof Cambridge.)

Ursell,F. (i968), 0n Head Seas Travelling along a Horizontal Cylinder , J,Inst. Math. Applications, Vol. 4 (414-427)

Van Dyke, M. (1964) /'Perturbation Methods in Fluid Mechanics (Academic Press.New York and London, 229 p.p.)

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Distribution List

Defense Documentation Center 2Cameron StationAlexandria, Virginia 22314Library 2Naval Postgraduate SchoolMonterey, California 93940Office of Research Administration 1Naval Postgraduate SchoolMonterey, California 93940Chairman, Department of Mechanical Engineering 1Naval Postgraduate SchoolMonterey, California 93940Professor Allen E. Fuhs 10Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940Allen Ford 1SES Division (Code 163) , ASEDDTNSRDCBethesda, Md. 20084Dr. Nan King 1SESPODTNSRDCBethesda, Md. 20084Arnold Anderson 1SESPODTNSRDCBethesda, Md 20084Dr. CM. Lee 1DTNSRDCBethesda, Md. 20084Dr. Odd Faltinsen 1University of TrondheimNorges Tekniske HogskoleInst, for SkipshydrodynamikkN-7034 Trondheim-NTHNORWAY


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