Finite element treatments for free surface waves caused by a body in uniform flow

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r.

L

THEORETICAL

AND

APPLIED

MECHANICS

Volume 2.6

Proceedings of the 26th Japan National Congress for Applied Mechanics, 1976

Edited by Japan National Committee forTheoretical and Applied Mechanics Science Council of Japan

UNIVERSITY OF TOKYO PRESS

Technische Hogeschool

Dem

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Body in Uniform Flow

Yoshiyuki YAMAMOTO and Hiroshi KAGEMOTO

Department of Naval Architecture, University of Tokyo, Tokyo

The method of superposition of analytical and finite element solutions is applied to solving the linearized theory of free surface waves caused by a body in uniform flow. In the case of a floating body, a difficulty due to the so-called line integral can easily be resolved by proper treatment using the finite element method, which can be formulated on the basis of a Galerkin-type equation.

I. INTRODUCTION

The linearized theory for free surface waves caused by a body in uniform flow encount-ers difficulty in treatment of the free surface condition. In the case of a floating body, a sin-gularity occurs along the intersection line of the body surface with the original free surface, and it is called the line integral.1-3) In the present paper, the line integral is investigated on the basis of variational principles. For the linearized theory, a Galerkin-type equation is derived from the variational principle and can be used for finite element formulation. The method of superposition of analytical and finite-element solutions,4) which has been applied for wave problems caused by a body in harmonic motion, can be used for solving the pre-sent problem numerically; numerical results for a two-dimensional problem are prepre-sented.

II. VARIATIONAL PRINCIPLES5-7>

Water motions caused by a floating or immersed body in uniform flow with velocityU

will be discussed herein. Water is assumed to be inviscid and incompressible as usual, and the depth h of the water field is uniform. This problem is closely related to water motions caus-ed by a moving body. The x- and y-axes are taken on the original free surface SÇ, the sense of the x-axis being opposite to the direction of flow, and the z-axis is upward normal to the original free surface (cf Fig. 1). The free surfzce S1 is expressed by

z = (x, y)

(1)

where is the surface elevation. The water motion is described by the velocity_potential and the surface elevation .

In water fields far from the body, the velocity potential Jia and the free surface elevation

¿ _{can be determined analytically within a certain number of unknown parameters. Such}

water fields are bounded by a fictitious surface Sr chosen properly around the body surface Sh. The surfaces Sr and S are assumed to intersect the x, y-plane along the closed curves Cr and Ch, respectively. If h is finite, the bottom of water field is denoted by _{S. The water}

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Fig. 1. Domain of analysis.

field D surrounded by Sr, S, Sn and S can be assumed to be finite without lossof genera-lity. In cases where the water depth is not uniform, such parts areconsidered to be parts of

Sn.

The general variational principle is given in the following form: Theorem 1. The solutions 'Í and 4 satisfy the problem

= -

VP Vdxdydz -

I

- dS +

¡7(Ç)

_1f

D

i5,. a

..f(gz_

U2) dxdy

= stationary (2)

where g is gravitational acceleration, j7 is the gradient operator, n is the outward unit normal vector, and b and satisfy the conditions

=

on S,.

=

on Cr

(5)

By virtue of Eq. (3), the variation vanishes on C. Then the variation of ¡7 is given by

=

5D P ô(Pdxdydz

-

f

a

ô(P dS

+f

SJnaDL2

[1v.v+ (g_u2)oC

2

It then follows that

in D

This shows the validity of the theorem.

In the following, it will be assumed that the surface elevation is small and the velocity potentials (P and (Pa can be defined beyond Sf by analytical continuation. Correspondingly, Sr can be defined beyond S. The field surrounded by Sr, s), Sa and Sb will be denoted by

D°. For the sake of convenience, the following nomenclature will be introduced:

= SrflaD°

and S = sanaD°

The functional ¡7 can be rewritten as II((P, )

=

ifD. v(P

V(Pdxdydz +

--

L[f r(P

Vcbdz + (g2 - U2)]dxdy

-

ç a(Pa / Caa(Pa

(PdS

(f dz}dC

an

_5Cr'

a2

where fl2 is the two-dimensional outward normal vector on the boundary of S. Since is

small, (P and (Pa and their derivatives with respect to x and y can be considered to be inde-pendent of z near S, and (P and (Pa are expressed in the form

(P=Ux+çb and (Pa_Ux+6a

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where ç and Ø are the velocity potentials corresponding to the disturbed water motion. Then an approximate expression fl2(, ) for H((P, ) can easily be obtained.

Theorem 2. In case of small surface elevation, the solutions Ø and satisfy the problem

H2(ø,O=.f(vvø_2U+U2)dxdydz

dxdy (4) 6n

=

o on a(Pa S1, S and Sb on Sr

on S

(5) an (U2

_{- v(P}

p7(P)

(6)

+

1

J,

a

2

2V2-2U + I L' dz+g

ax) o azi

raa

( Ux +

) dS

f L2 ( Ux +

Ça an

= stationary (8)

where V2 is the two-dimensional gradient operator, and and satisfy the conditions

3a

on

Sfl

r on

_c7Ì

on C, an2

aa

_Uan2 on an2 ' a

36

az2 V2[ÇV2(ç Ux)I = O on S1°

=

-gax

2g This theorem is equivalent to:

on S (11)

Corollary 2.1. fi2 in Theorem 2 can be replaced by

=

2fD,

dxdydz

f U ndS

-f

dS_f

adc

n ,. n

This validity of this corollary can be confirmed by the following relation:

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(12)

The natural condition of Eq. (8) is given by

VP=0

=0

aa

an in on on D° S, Sb r (10)

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1121(çS,O = H2(ç,

-

U2dxdydz

r

acpa ,..

ai

-

an

UxdS

-

_iCr an2 UxÇadC

The last three of Eqs. (10) are the conditions of continuity of the flow flux above Sf. The flow flux overCh or Cr will be denoted by w:

an2

- Un2x

= W

on C

and Cr (14)

The flow flux is treated as a singular distribution of velocity along C and Cr for analysis in D°.

(a)

Sç

(b) Linearized theory

Fig. 2. Flow flux over C.

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III. LINEARIZATION AND METHOD OF SUPERPOSITION

In cases where disturbances caused by the body is considered to be very small, çb and can be regarded as small quantities, and therefore, terms of higher order with respect to

Ø and can be disregarded. Omitting third-order terms in 1721 leads to

=

25D. V vçbdxdydz -f

sJ\

(u

ax

-

g2)

dxdy

-

a

_{5 dS + f Uan2dC}

-

f Un4SdS.

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or

= UÇn2x Ofl C (18)

where Ç is the surface elevation along Ch and may be an unknown quantity. This condition corresponds to the so-called line integral for the theory of wave making resistance of ships. Since Eq. (17) is obtained by introducing the second condition of Eq. (9), it then follows that Theorem 3. Solutions of the linearized problem satisfy the Galerkin-type equation given by

+

iäØ dC = 0 (19)

where satisfies Eq. (9).

Eq. (19) gives the following natural conditions together with Eqs. (17) and (18):

= O in D° (20) (21)

-

on S, (22)

- an

= Un on S, (23) on

S/

(24) ax The variational problem

173 = stationary (16)

yields an inconsistent natural condition

UÇn2X=0 on

Ch

which corresponds, in the framework of the linearized theory, to the fifth condition of Eq. (10). By linearization, the flow flux over Cr, defined by Eq. (14), becomes

= - UÇn

= - UÇan2

on Cr (17)

and on Ch is also given in a similar form. In reality, açb/an2 in Eq. (14) may be finite in view of Eq. (11) and cannot be disregarded; it gives an additional flow flux (cf. Fig. 2) which will be denoted by ii". Now the fifth condition of Eq. (10) or the condition that the flow flux vanishes on Ch is given by

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r-"

gx

Introducing Eq. (25) into l7(Ø, ) leads to the functional 1731(ç1);

=

ifD. vc

vc dxdydz

-

--f1

()2XY

-

ç dS

-J' Unç6dS

+ f

Utn2çbdC (26)

By introducing the flow flux over Ch and Cr, it follows that Corollary 3.1. Solutions of the linearized problem satisfy the equation

U9çS

ö1731

+ C

ii"ôç3dC + ,Li + - - fl2x I ô?S dC = 0 (27)

Ch

gax

where satisfies the first condition of Eq. (9).

The natural conditions corresponding to the second of Eqs. (9) and Eq. (24) become as

y

Un2,

(

=---fl2}

U2 aa

_gax

aç U2a2

5z

gax2

These are the continuity conditions for flow flux above S.

When a problem is described by linear equations, the method of superposition4) can easily be applied, and therefore the present problem can be solved numerically in the follow-ing sequence:

Let ° be a numerical solution of Eqs. (18), (20), (21), (23), (24), (25) and (29), and the conditions given by

on Sr°

(30)

yj=O

on Cr

The analytical solution ça which is effective in the far field beyond S,' can be obtained by the use of Green's function of the problem or its asymptotic expansions, and is given in the form N = mk (31) /c=1 on S (25) On Cr (28) on S (29)

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where

where i,u is the flow flux over C caused by Ø.

The solution of the original problem is determined within a number of unknown con-stants mk's in the following form:

N

=

_{° +}

_m1ç3 in D° (33)

k1

The constants mk's are determined by the condition of continuity on S,' given by the first of Eqs. (9):

N N

m0+° +

mk3=>mkø

on S

(34)

Ic-i k=1

The constant m0 is added since the velocity potential is determined within a constant. Eq-uation (34) can be solved with the aid of the collocation method by evaluating both sides at a number of points chosen properly on S;.

As for numerical methods for determining çì° and çSZ's, the finite element method8) will be used in the present paper for the sake of convenience of formulation. For the same purpose, the finite difference method and the integral equation method can also be used

effe-ctively.

The numerical procedure described above is based on Corollary 3.1. A similar proce-dure can also be developed on the basis of Theorem 3.

IV. NUMERICAL EXAMPLES FOR TWO-DIMENSIONAL

PROBLEMS

Consider the two-dimensional water field of depth h shown in Fig. 3. Assume that S; is given by the planes x = Xi and x2. From physical considerations, the upstream analytical solution can be taken as a null function, and the downstream analytical solution can easily be obtained from the field equation and the conditions on S; and Sb if U[s/ìï < 1:

O

xx1-m1 çi + mz3a2 x X2

(35)

are unknown constants to be determined. Let ç6

(k = 1,

.

of Eqs. (20), (21) and (29), and the conditions given by

e o on an Y'a on an e U2a9s Wk

-

flax on . . , S1

S;

C N) be numerical solutions (32)

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=

cos K(x - Xk) cosh ,c(z ± h)/cosh ,ch k

=

1, 2 (36) X

=

tanh xh for U

=

o an

aai

an

-

_2x

g ax

=

o

Fig. 3. Domain of analysis.

Since çl's should be independent of each other, it is assumed that sin ,c(xi - X2) does not vanish. Introduce the velocity potential

a1 _X1 _X _X2 ₍₃₈₎

Then aøa/ax corresponding to çl given by Eqs.(35) and (38) is continuous on x

=

x1 and x2.

For the sake of convenience of numerical treatments, it is assumed that is given in the form

= 3a

₊

₍₃₉₎

where çbni is a solution of Eqs. (20), (21) and (29) and the equations given by

on x=x1andx2

On S

at C, at Cr (37) (40)

where *i and yjr1 are caused byç5r1.

The velocity potential obtained in D° is given by

Øo

+ m1(i +

in D° (41)

Since velocity potentials are determined within a constant, the condition on x

=

x1 cor-responding to Eq. (34) is equivalent to

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+ mj

a(ai + øri)

=

az

The constant m1 in Eq. (41) is determined by evaluating this condition at a point on x = x1 near S. Similarly, m2 can be determined by the condition of continutiy of açs/az on x = X2. Consider a thin plate supported vertically at x = O in water as shown in Fig. 4. The floa-ting plate named Fis piercing the free surface, and its draft with respect to Sì is d(=h/8). The immersed plates I and J with depth h/8 locate so that the z-coordinates of their upper edges are h/24 and O, respectively. With the use of the mesh subdivision shown in Fig. 5,

Fig. 5. Mesh subdivision by triangles.

on x=x1

(42) z (b) PLATE I z X ::

INUNR

ádUI IN

UI'...

1//lì

X

(a) PLATE F _{(c) PLATE J}

Fig. 4. Floating and immersed plates.

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finite element calculations are performed for D° with x1

_{= - x2 =}

6h18. If K6h/8 irand/

or sin ic6h/8 0, x and X2 should be modified.

Surface elevations obtained are shown in Figs. 6, 7 and 8. The surface elevation for the plate F is very large in comparison with the other cases and the water depth; it may show inadequacy of linearization for piercing models of a floating body in the two-dimensional

case.

Results obtained on the basis of Theorem 3 are also shown in Fig. 6, and they are prac-tically the same as those obtained above.

10.0 8.0 6.0 4.0 2.0 o -6.0 4.0 -2.0-

j

q q -6.0--8.OE -10.0- -120-4.0 2.0 o -2.0

Fig. 7. Surface elevations (immersed plate I).

In cases where U//h

1, sinusoidal waves disappear in downstream, and the analy-tical solutions in the far field are given by

Fn0.998

- Fn=0.825(Corollary 3.1)

---Fn0.586.

_u

(Fn_=)

Fig. 6. Surface elevations (floating plate F).

-4.0 j 2.0 2.0 4.0 6.0 Fn0.825 (Theorem 3) Bottom

If ii! /

/ I/I111/f/ / If

X/d 4.0 6.0 = m1 Øa1 m2 3a2 X X1 X X2 (43)

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4.0 20

20

Fig. 8. Surface elevations (immersed plate J).

where

= exp [(- 1)k ,c(x - Xk)] cos ic(z + h)/cos ich k

=

1,2

and ic is the positive lowest root of the equation

K=tanKh

for

l.

-Numerical solutions are obtained according to the general procedure described in the pre-vious section.

If KX and icX2 are large enough, both a1 and qa2 can be regarded as a null function, and çt° itself is the final solution çl.

V. DISCUSSION AND CONCLUSION

A linearized theory for free surface waves caused by a body in uniform flow is formulat-ed on the basis of the variational principle, and the method of superposition of analytical and finite-element solutions is used for numerical calculations.

In the case of a floating body, the flow flux above the original free surface is taken into consideration, and it gives an additional condition along the intersection of the body surface with the original free surface, which corresponds to the line intergral in the theory of wave-making resistance of ships. In the case of a thin body like a Michell-type ship, this additional condition may be disregarded because the nonlinear term in ii' becomes small. In the case of a floating bluff body, it becomes essentially important, and it causes large surface eleva-tion, which is n contradiction to the assumption of small surface elevaeleva-tion, though this the-ory is applicable to a deeply immersed bluff body.

It is notable that any numerical method like the integral-equation method or the finite difference method can be used in place of the finite element method for numerical calcula-tions in the present paper.

The authors would like to express their sincere thanks to Professor T. Inui, Professor H. Maruo, Professor T. Bessho, Professor H. Kajitani, Dr. Isshiki, and Dr. K. J. Bai for their instructive suggestions and comments.

REFERENCES

Q-

X/d

2O 40 60

1) Brard, R., The representation of a given ship by singularity distributions when the boundary condition on the free surface is linearized. J. of Ship Research, Vol.16,(1972), pp. 79-92.

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Kusaka, Y., On the contribution of line integral to the wave resistance of surface ships. International Se-minar on Wave Resistance, Soc. of Naval Architects of Japan, 1976, pp. 249-254.

Bessho, M., Line integral: Uniqueness and diffraction of waves in the linearized theory. ibid. pp. 45-55. Seto, H. and Yamamoto, Y., Finite element analysis of surface wave problems by a method of

superposi-tion. Proceedings First International Conf on Numer. Ship Hydrodynamics, N.S.R.D.C., 1975, pp. 49-70. Bessho, M., Variational approaches to steady ship wave problems. 8th Symposium on Naval Hydrodyna-mics, O.N.R., ARC-179, 1970, pp. 547-572.

Ikegawa, M. and Washizu, K., Finite element method applied to analysis of flow over a spiliway crest. mt. J.for Numer. Meth. in Engng., (1973), pp. 179-189.

Luke, J. C., A variational principle for a fluid with a free surface. J. Fluid Mec/i., Vol. 27, Part 2, (1967), pp. 395-397.

Bai, K. J., A localized finite element method for steady, two-dimensional free surface flow problems. Proc. First litt. Conf on Numer. Ship Hydrodynamics, N. S. R. D. C., 1975, pp. 209-229.

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