Virtual mass and slender body theory for bodies in waves

ARCh

STEVENS INSTITUTE OF TECHNOLOGY

Davidson Laboratory _i

Hoboken, New Jersey

Lab. y. Scheepowue

Technische Hoescoo

DeHt

VIRTUAL MASS AND SLENDER-BODY THEORY FOR BODIES IN WAVES

by

Dr. Paul Kaplan

and

Dr. Purig Nien Hu

Submitted to the Midwestern Conference

on Fluid and Solid Mechanics, University

àf Texas September 1959.

June 1959 _{Note 543}

Errata for "Virtual Mass and Slender-Body Theory fôr Bodies in Waves"

Second terni in Eq. [41] should be -Im(n)

Eq. [46) should Iave term in

f

}as

-lb-R1

(b-R)3 R1'2

14 R R2 (b-R)2(b+R) (.. + sin 4 b

(_.+1+-.)+

8b3 R 1/2 14 R R2)

(b-R)2(R)4

(T 1 b-R i òc

_L

(3

E

+ + 8b3 . + sin Eq. [47] should be

r___

(b-R)3 R 1/2 14 R R2 (b-R)2(b+R)4 (li

bR1

+

s1n1

A42=

L

(-3..

+ 1. +_..) 8b3 Eq. [48) should be dL

= -

D i

L'

Sentence following Eq. [48) should have the word "negative"

before "rate of change".

Abstract

The hydrodynamic forces and moments actiron slender submerged bodies and body-appendage combinations which move oblique to the crests of regular waves are found by application of slender-body theory, utilizing two-dimensional cross-flow techniques. A simple

interpretation of the results leads to identificätionof two components,

one due to change of fluid momentum and the other a convergence effect due to a pressure gradient in spatially nonuniform flow (an extension

of a result due to G.I. Taylor). Corrections are made to dome. previous erroneous results for this problem. Particular examples illustrate the

interference between a body and its appendages in regard to the wave

forces.

N-543

-1-Nomenclature

A(x) cross-sectional area of submerged body

A virtual moment Qf inertia

42 ¡

a wave amplitude

b fin semi-span

c wave propagation speed D

total time derivative

g gravitational acceleration

h depth of centerline of submerged body below calm free

surface . imaginary unit 2ír g k=--

--

wave number L rolling moment M pitching moment

m mass of displaced fluid

virtual mass

N yawing moment

p pressure

R radius of circular cross-section r radial polar coordinate

t time V

V forward speed

v,w lateral and vertical orbital velocities, respectively

y ,w values of v,w taken at center of cross-section o o

X surging force

x,y,z rectangular coordinates V Y lateral force

Z vertical fórce

angle between x-axis and normal to wave crests

= y+iz complex coordinate

angular polar coordinate X wave length

p water density

N-543

-3-total velocity potential

(2) total velocity potential in reference (2) velocity potential of waves

øw value of Ø taken at center of cross-section

.øvo velocity ooential due to forward motion Of body

Introduction

In a previous

paper(l)1, one of the authors applied the

slender-body theory to evaluate the forces and moment acting on slender

sub-merged bodies aid surface ships moving normal to the crests of regular

waves.

The same method was used in (2) for slender bodies and

body-appendage combinations, in oblique waves.

The well-known slender-body

theory, which was originally dealing with either steady or unsteady

potential flows representing a body pláced in a spatially uniform

stream, has thus been extended to this particular flow problem which

is both time-dependent and spatially nonuniform.

Since the technique

is two-dimensional, a great simplification is therefore accomplished

as compared with the three-dimensional treatments by Havelock

(3) and Cummins (4).

Also, the influence of appendages can be easily

determined by this method.

However, due to an erroneous boundary

condi-tion and an incomplete expression for the pressure2 (which, fortunately,

does not effect the results for a body of revolution), a simple physical

interpretation could not be given to the results obtained in (2) for the

forces acting on finned bodies.

In the present paper, the corrected

boundary condït'ion will be used to investiate the same problem, and the

nonlinear terms in the pressure equation will be taken into account.

In addition, the forces and rolling moment acting ori

.n asymmetrically

finned body will also be evaluated.

It will be shown that the use of the

correct boundary condition and the nonlinear terms in the pressure equation

leads to a simple physical interpretation of the results.

'Nurnibers

in parentheses refer to the Bibliography at the end of the paper.

2The

authors are indebted to Dr. John R.

preiter for his discussion about

these points.

N-543

4-A cartesian coordinate system is chosen with the axes fixed in the body, which is constrained to translate horizontally to the right. The x-axis is positive in the cirection of motion, the y-axis is positive to port, and the z-axis is positive upward. The origin is placed on the

axis of the body, located at a depth h below the free surface..

The free surface i disturbed by waves which propagaté with speed

c in a drection oblique to the forward motior of the submerged body. The angle betweèn the x-axis and the normal to the crests is denoted by

, which lies in the range

-T1/2rT/2

as shown in Fig. 1.

/

N'

Velocity Potential of Waves Near the Body

Fig. i Relation of body to waves

The wave propagation speed c is > O for following seas and c<.O for head seas.

The velocity potent ial of the waves with wavelength X , amplitud , with wave propagation speed c is given by

N-543

-kh k[z+i[xcos + sinn

+(v

cos -c)t)}

=ace

e

where k = = , = is the imaginary unit and only the real part of the velocity potential is to be taken.

In the plane of the cross-section of the body, i.e., the planes parallel to the y-z plane, the vertical and lateral components cf the orbital velocity of waves are

w=-

W

k

ace_ek

{+tx

cos + sin

+(v

cos-

c)tJ} [2]

or

fil

where the spatial nonuniformity of the flow ìs evidenced by the dependence

of the velocities on the coordinates y and z.

Assuming that the linear dimensions of the. cross-section are small

relative to the wavelength, the velocity potential of the waves may then be expanded near the body intö a power series around the center, y=z=O,

of the crosssection. Neglecting the higher order terms beyond the linear,, this yields

ac e [l+k(z+j y sin)

ek cos+(vcos-c)tJ

-

V Y-W Z

o o

where

₀

,

y and w are the values of

w o o

o

taken at the center of the cross-section.

y, and w, respectively, [5]

quatïon [5J shows that the velocity potential of waves near the body may be recognized to be contributed by three components: a uniform

stream y in the y-direction, another uniform stream w0 in the z-direction,

and a spatially independent (in the plane of the cross secion) velocity

N-543

-6-a rid

potential term. Consequently, the potential due to the presence of the body in waves, known as the wave-body interaction potential, may be

determined by the usual, two-dimensional treatment for unifOrm potential flows.

Forces and Moments on a Submerged Body of Revolution

FOr a body of revolution, the cross-section is a circle with its

center located at the point y=z=O . The wave-body interaction potential,

denoted by 0wb' is therefore found to be

2

--(vcos9+wsin9)

r o o

where r and = tan - _{are the polar coordinates, arid}

R=R (x) is the local radius of the body.

Due to the effect of thé forward motion of the body, tiEnormal velocity on the boundary of the cross-section is no longer zero as in the case of pure two-dimensional streaming flow past the section. For a body of revolution, this normal velocity yields a boundary condition for the

total velocity potential , which is

() =V

_r

r=R dx

Since, the velocity potential terms previously considered (2) satisfy the boundary candit ion

-s _[8]

r=R

the additional term due to forward speèd, _{, also satisfies Eq. [7] and}

it is fòund to be

dR

= VR .- log r

Which represents a to-dimensional point source located at the center o

the circle.

The total cross flow potential ' near the body is then

[7]

[g]

N-543

=

=0w0wb'V

in consideration of the order of magnitude of the .nqnlinear terms, where

Therefore, the pressure on the body contour, r=R , is given by

p = -ipage_<'[l + 2(kR - i _{-)(sin9 +. isin cose))}

'.

eb05c0stpy2dj21ogR

-2pa gk e (cos Q. - i sin sin2Q - sin sin Q).

i2k[x.co+(Vco -c)tJ

i .2 (dR2

-p

Integrating the pressure along the body contour, the elemental

lateral and vertical forces are given by

[14)

N-543

-8-2 2

0-v0(r+) cos9

_- (r + _.)siri9 + VR log r [10)

A substitution of the values of _Ø'

, V

and w gives

W O O

o.

= ace

h[11(

_{)(siriO + sin}

_os9)3

eiXC5 + (V cos-c)ti,.

VR log r

This form of the potential obviously satisfies the _{same boundary condition}

[7) , and will be used to determine the pressure

on the boundary of the cylinder.

Lighthill (5) and Spreiter (6) pointed out that the pressure equation for the slender body should be written as

2 2]

[U

.

and Xb and N xs = 2îr sine Rd9 o

= i2page_lth(A(x) -

..L)e1k[XCOS

+ (Vcos -c)t]

2cdx

where A(x) = hR2 is the cross-section area.

Integrating the elemental forces over the length of the body,

where x and _Xb denote the stern and bow x-c.00rdinates respectively, the total lateral force Y , for the body o revolution, is given by

Y = -2pagk sin

eh(l_

cos)

A(x)e.X. cosp+(vcosc)tJdX

[17]

dY

X dx

Where the fact that

A(x)

= A(xb) = O has»bè.en used.

Similarly, the total vertica1fòrceis equal to

Z =

pagkeh(1_

_cos)

A(x)eWOS0_t]

[16]

['9]

[20]

N-543

-9-The moments about the y- and z-axes, denoted as M and N respectively, are given by

M = - ( x j dx Jx s.

- = -

dY

_{J p cos@}

Rd dx o -kh( V

dA e[0

₊ (Vcos -c)t],

Substituting [15] and [16] leads to M =

2pageh

_-

ik(1-cos)x1

A(x)ei XC0S

C0S-)tJ

_dx Xb xcos+(vcos_c)t)dX

f 22 J

The longitudinal surging force due to Waves may also be determined

from the present analysis. Following theprocedure used in (2) with the present pressure expression, the final value for the surging force is found

tobe

.kh

X -pagk coso e ,

A(x)etXCO5C9_t3dx

X

s

where only linear terms in kR have been retained (it is assumed throughout. dR

the analysis that the quantities kR and are of the same order of magnitude).

The results given by [17], [18], [21], [221 and [23] are exactly the same as those of Cummins (4), who evaluated the forces and moments on a

lender body of revolution due to waves by application of his extension of Lagallyts theorem to unsteady flows (7), Le. by use of a three-dimensional theory. They are also the same as the resultS of (2) for the _{case of a body} of revolution, therefore demonstrating that the additional _{source potential}

has no influence on the values of the forces and moments _{for this class}

:f bodies.

[23] N-543 lo

-[i

L

+k(i-

f

co5)x]

A(x)e

N = -2pg sinn e_kh

f

X

s

[21]

Fig. 2 _{Transverse cross-section of a symmetrical'finned body}

The finned circle contour of Fig. 2 is transformed into _{the upper and lower} side of the flat plate strip c'Ç _{, where}

= b + R2

f24)

by means of the transformation

N-543

Symmetrically Finned Bodies

One of the major advantages of the present technique is the fact that conformal mapping methods may be utilized because of the

two-dimensional nature of the slender-body treatment. Slender bodies which are not bodies of revolution, and hence do not have circular cross-sections, may be treated by proper mpping of representative transverse contours

into either ¿ cIrcle or any other simple shape that may be easily solved by

ùse of the methods described above. _{Some particular configurations will )5e}

treaedherein to illustrate the application of these methods.

A particular body that cannot be treated easily or properly by any

of the previous techniques applied to the study of forces _{in wavés is a} slender body of revolution with tail fins terminating at the stern end

of the body. Thi _{restriction is imposed in order to avoid any complications}

due to the trailing vortex sheets behind the fins.

Consider a body with a pair of horizontal tail fins. _{A transverse} cross-section through the tail region is shown in Fig. 2.

AZ

[27] N-543 12 -R2 rl = + -z-. _[25]

The velocity potential. for the finned circle of Fig. 2 in.the .-plane

when it is placed in a horizontal stream. y

_'O,.

and, a vertical stream W

, o

will be the safne as that of the flat plate'.in the r'plane placed in the same streams. _{This potential may be written as}

- vRe(rl) w Re(c2... rl2)i/2

[26]

where the upper sign hold _{for the 'upper"side of the body (and a similar} statement for the lower sign), and Re _{epresents the réai part of the} complex quantities. _{The. wave-bödy interactió'ñ pQtential will be the} potential .given by [26] after subtractin thepotential due to the uniform horizontal and vertical streams y and w . Together with

- O ' O

the potential of the waves and the potentiaÏ duetO the forward motion

of the body, the total potential _{near the-contour is}

=Ø

- vRe()

w0Re(2_)1/2+

VR log r -,

Substituting thevalues and expressing the _{coordinates in polar form in the}

-plane results in

ace<e

[xcos+(Vcos13_c)t

_{siñ(r+ )cösQ}

r 4 _{-., 4}

_11/2

+k Re [b2+ - _- (r2+ .)cos2Q - i(r2- ..)sin2Qj

-

b r r

dR

+ VR log r

, _[28]

where this form of the potential satisfies the boundary _condition

=V

at r=R.

dx

The pressure near the body is iven by [12 j. Substituting [ 28 j and setting r = R , _{the pressure on the circle is (neglecting the second}

p

pgi e11XC0

0tj

2ksin Rcos9

C

- _d

(b2 - 2R2cos2e)1/2_i_i i sink os9

cdx

b2 C dx

ik(b+ _{- 2R2cos2e)1Í21} - çV2 q::: log R

- d2R2 _{log r -} R2 fdR\2

2 'dx1 r

The lateral force on the body is not influenced by the presence of the fins and hence the same value is obtained äs in[15]. _{The vertical elemental} forces on the circular body and the fins are calcu1ted separately _in

'accordance with the dif.ferent pressure expressions, and for the _circle

the vertical force is given by '

4pagehe1k[05C0_t]

L

+ (b2+R2) sin 2Rb b2+R2 2h 4Rb2 + ikR I b2-R2 (b2+R2)2

sin1

2Rb L 2b 4Rb2 b2+R2 1 2 dR 2 v ₍ [29]

Siniilarly, setting 9 = O , the pressure on the fin becomes (neglecting

[30]

N-543

-13-the second harmonic terms with larger depth

-kh ik[xcos(Vcos-c)t] Pf Page e

-Vd

2 R4 R4)1/2 - - (b +

-cdx

2 2 r attenuation) R2

sin(r

+ { V R dR (r-R4 2 R4

2

R4 -1/2

--

)(b + ---r b r -

-1

r -. 2. R4 2 R4 1/2 V 3R _{R3 dR} s in (_. + ik (b +.

-b.

r - _-) r i r

n

[31]

When

b=

R , this value reduces to that obtained for a circle alone, which serves as a check' on the answer.

For the fins, the vertical force per unit length is

b

1dZ\

'dx'f

JR

(P»

dr

,

[32]

where (Pf) arid (P» are the pressures on

the

lower and upper sides of

£ u

the fin respectively0 Substituting [30 1 leads to

= a e

ik[xcos+(vcosp-c)tjívf d (bR\

dx g e

cI2

dx '; b '

+ sin

b2-R2

d

(b+R)

IT dR

(b2-R2)2

2 dR

22 dx

b 2 dx R 2 dx - b dR

(b2+R2)2

-1

b2R2i

.

[

b2-R

ir

b-R2

2 d -sin. 2 2 ' > Rb

b+RJ

b2+R2 2 -1

b2-R2

'

Combining the forces on the cylinder and the fins, the

total elemental force

on the cross sect-ion may 'be expressed as

dZ

-kh[

2 R4 .

V d

'2

R4

=

ìîrpage

Lk(b

)-i -.

(b +

etxc05_c0s

)t]

[34]

Since the virtual mass in the vertical direction of the configuration in

Fig. 2 is

- 4

& =

rrp(b2+ ì-

-

R2)

b.

(see Kuerti, McFadden and

Shanks (8)), Eq.

[343 may then be expressed as Dw dZ ₌ D

i,

o

tm W)+ m

)] [333 N-543 14 -2 2

b +R

N-543

where th = pirR is the mass of displaced fluid.

The above result demonstrates that the, force on the body cross-section is cdntributed by two components. The first term n the right hand side may be identified. as the time rate of change of fluid momentum associated

with the particular section, and the second term s the force arising dUe

to the spatial nonuniformity of the flow (the action of a pressure gradient

across the. body). This latter term is expressed as the product of the mass

of the displaced fluid with the orbital wave acceleration at the center of

the section. When the body is purely two-dimensional, then %- and

the result given in Eq. [36] reduces to that obtained by Taylor () for a two-dimensional body placed in a spatially nonuniform flow. Theréfore, [36].. may be considered as an extension of Taylor's result to the case of slender bodies of varying' cross-section.

For a body with a pair of vertical tail fins of semi-span b, a simi.-lar'résult will be obtained, atìd the valué 6f the'hòrizontal eiethentalforce on:the cross-section is

-ïpag sin

ek(b2+

4)_i

(+

-

R)

ik[xcos+ (Vcos-c)t_J

f37]

which may again be expressed in the same manner, since the virtual mass

in this case is also given by [34].

In each of the above cases, there is no rolling moment because of the symmetry of the configuration.

Asymmetrical Finned Body

Consider now the case of a body of revolution with a single vertical

tail fin terminating at the stern end. A transverse cross-section through

the tail region is shown in Fig. 3..

liz

"y

Fig0 3 _{Transverse cross-section of an asymmetrically finned body}

The finned circle contour of Fig. 3 is transformed _{into the right arid léÍt}

The total velocity potential due to the waves, the wave-body interaction

and the f or.ward motion of the body is then bound to be

= 0W wRe(,u vRe(/42+c2)1/2+ \ÏR L _{log r}

[413

where the upper sign holds for the right side _{holds for the right side of}

the. body (and a similar meaning for the lower sign). _{A substitution of}

the proper values leads to

N-543 16

-side of the flat plate strip

-j/k

_{i, where}

by means of two successive transformations

1 = _- j;-and /.4=T)-[38] [39] [40)

dY -kh = - ilpag sinn e _k

i-L4

b+R)4_2R2]

the above result _{agrees. exactly with that in [36].}

The elemental rolling

moment about -the x-axis is only contributed by

the force on the fin, and _{may be expressed as}

-

JL]

r dr. [45]

where _{(Pf) and (Pe)}

are the pressure on the left _{and right side of} r

the f-in respectively..

Substituting the pressure yields

N-543 17

-= ace hik[ xcos+(Vcos-c)t

j. + k(r+ )sirie

p

+ 1k -sin Re

[(2+

-)cos28 - 2R2+ (b2+R2)

+ (r+

R2)(bR)2.9

+

i(2-

)sin29 -

i(r-R2)(bR2)Q]h/2}

- R)4_3R2]} 1k[xcos+(Vcos-)tJ

. [43]

Since the virtual mass in this _{case (8) is}

=

p [(b+R)4_12R2b2] ,

[44J + VR log r

[42 J

The pressure on the body may be calculated from _{{ 12}

J, but the result is

too complicated to be written down here. _{However, integrating the pressure} over the contour, the vertical elemental force _{on the contour is the same} as obtained for a body of revolution, and the horizontal _{elemental force.} is

= k Í2

()l/2 (3b+2Rb43R)

4b2 b+R)3 (b-R)2 (- b-R)] - . V d

f

2(b-R)3 Rl'23b2+2Rb+3R2

1cdxl

b b l2b ) + _.(b+R)3(b-R)2(- + sin

eXCO5CO5_tj

[46]

Since the virtual moment of inertia about the x-axis of the finned circle due to a horizontal motion can be shôwn to be

- 12(b-R)3 i/ 3b2+2Rb+3R2)

b l2b

.+

;.

(b+R)3(b-R)2(-+sin )] [47]

where A42 is the appropriate tensor element in the general expression of the virtual inertia tensor (see Landweber nd YÌh (lo)), Eq. [47] may be

written as

-dL D r

;z=t-

LA42v0. [48]

which shows that the moment on the bòdy is'equal to the rate of change of the ¿ngular momentum of the fluid. In this particular case., the spatial nonuniformity does not hsve any effect on the' moment because of the fact that 'the moment is contribütéd by the fòrce on the fih alone, where the convergence effect along the y-direction is zero.

N-543 18

-Conclusions

The results obtained above have demontrated the _successful application of slender-body theory _{to the evaluation of the forces}

and

moments acting on a submerged body moving under waves. _{The extension} of slender-body theory to take account of the spatial _{non-uniformity} of the wave orbital

velocities leads to results' for a body' of revolution

that òheck with those of

other investigations, which were obtained by

different and more.jnvolved methods _{Complicated cross-sections of}

interest,

such as body-appendage _{combinations, can also be treated}

by the same technique and three cases are evaluated. _{All of the results}

can be inter-preted in terms of virtual _{masses and inertias, and reduce}

to Taylor's results in pure two-dimensional _flows.

Although it has not yet been

demonstrated that this simple

interpretation is also true' in the general

case of an arbitrary body, the téchnique _{offers a great simplification} over the existing methods because of

the two-dimensional nature of the slender-body theory,

N-543 19