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estimation of wave-current interaction effects

K. Takagi

Department of Naval Architecture and Ocean Engineering, Osaka university, 2-1 Yamadaoka Suita, Osaka 565, Japan

The Rankine Source Method is applied to the calculation of wave current interaction effects. Introducting the smaU numerical damping which is equivalent to the artifice due to Rayleigh, a new radiation condition for unsteady flows is developed. The error due to the numerical damping is estimated. The most suitable value for the numerical dam'ping is determined so as to minimize the error. Numerical results are verified with analytical solutions including wave diffraction and radiation by a vertical circular cylinder and a hemisphere. Wave-current interaction effects on wave diffraction and radiation are calculated by means of the above mentioned scheme. Effects on radiation force are small but effects on diffraction force are significant.

Key Words: The Rankine Source Method, wave current interaction, hydrodynamic force.

1. I N T R O D U C T I O N

The Rankine Source Method which has been developed as a numerical method for solving the steady wave making problem is a kind of Boundary Integral Method and uses the simple Rankine source as its Green function. The advantages of this method are:

(1) The programming is easy compared to the ordinary Boundary Integral Method which uses the complicated wave Green function, since the simple Rankine source is used as a Green function.

(2) The C P U time for the calculation under the nonlinear free surface boundary condition is the same as that of the linear case, since the Green function is distributed on the free surface panels.

I n spite of these advantages, the Rankine Source Method is not applied to unsteady free surface flows, since the radiation condition is not known.

Recently, Nakos^ applied it to the unsteady free surface flows. However his radiation condition is restricted to high-speed flows (T = Uo:i/g < 0.25), since he used the condition of the no-propagating waves forward of the body which is equivalent to the condition for the steady wave making problem.

Zhao et al^ and Kashiwagi and Ohkusu^ calculated this problem by making use of a kind of hybrid method in which the near field is calculated by the simple source distribution method and the radiation condition is satis-fied by connecting the inner solution and the outer solution which is expressed by a series of multipoles. Their method is an exact method. However the programing is complicated.

In the case of the steady wave making problem, the region of the progressive waves is restricted behind the

Paper accepted Septeinber t990. Discussion closes September 1992.

© 1991 Elsevier Science Publishers L t d

body and disturbances which are induced by the body propagate backward. Thus simple numerical radiation conditions are developed such as the application of the up-wind finite difference scheme for the free surface boundary condition*, the shift of the collocation point^, and so on. On the contrary, in the case of the unsteady problem, especiaUy a low speed case, some wave systems propagate to the forward direction and this fact prevents the development of the simple numerical radiation con-dition for the unsteady problem.

When we calculate the wave Green function analyti-cally, we avail ourselves of the artifice due to Rayleigh'' which is equivalent to the infinitesimal shift of the pole in the Fourier plane f r o m its real value. Sclavounos and Nakos'' pointed out that the up-wind finite difference scheme and the shift of collocation point causes the shift of the pole in the Fourier plane and the radiation condition is satisfied.

On the analogy of the previous schemes, the new radiation condition for the unsteady free surface flows is developed, that is, the numerical damping which is equivalent to the artifice due to Rayleigh is directly added to the free surface boundary condition and the estimation of the most suitable value for the numerical damping is stated. A detailed discussion about the calculation of a simple free surface flow which is induced by a periodic source beneath the free surface can be found in a previous paper^ and, in this paper, the performance of the present scheme is illustrated in the computation of the hy-drodynamic forces acting on a vertical circular cylinder and a hemisphere in a current and waves.

2. F O R M U L A T I O N F O R U N S T E A D Y F L O W S According to the Rankine Source Method for the steady wave resistance problem, the velocity potential is repre-sented by following;

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An application of the Rankine source method to the estimation of wave-current interaction effects: K. Takagi

<I>(x, y, z) 0-dS + - + -, 1 1

r r dS (1)

r = V ( x - 0^ + (y - nf + (z¬ r' = y ( x - <^)2 + (j; - nf + (z + if

where Sp is the free surface and is the body surface as shown in Fig. 1.

The free surface condition for unsteady free surface flows can be approximated to various forms with the assumption of an ordering of the inflow velocity. The free surface boundary condition, which takes the influences of the steady disturbance up to 0{U^) into account, is represented as follows: 5? + 2 V ^ ^ • V — + V ^ ^ ^ • ^ \ot / ot 1.

ad)

2g [u'-{Vct>,f} dz 8t^ dz = 0 on z (2) V = i dx^ ^' dy

Assuming that the unsteady motion of the fluid is sinusoidal in time and introducing the artifice due to Rayleigh, the free surface condition (2) can be changed as follows: lim £-.0 ffl' 0 $ 1 oz 2g 2e im(l) + (V<?!>B • V 0 az^ on z 0 (3)

Where the steady disturbance potential has to satisfy the free surface condition, however, it is assumed that (f)g can be approximated by the double model flow

Fig. I. Schematic view of a floating body in a current and waves.

potential for the simplification of the numerical calcula-tion.

In order to investigate the radiation condition for unsteady potential (f), the simple free surface condition is used, since the double model flow asymptotically con-verges to the uniform flow at infinity.

lim — + 2e 86 , d^d) dx dx" + 9 dz 1W0 — Ë dx (4)

Let us suppose the periodic source is situated at the point (0, 0, —h). The velocity potential can be divided into two parts.

1

— + (5)

= + y + + h f , '-1 = + y' + iz- hf

Discretising the integral in (1), the regular part of the velocity potential (pi and its derivatives are represented by matrix forms:

{ < / ' i } = [ ^ o ]

dx d ^

dx' (6)

where a is the strength of the source distribution and it is assumed that the strength is constant within a panel. The accuracy and consistency of this scheme is discussed by Sclavounos and Nakos''. Making use of the matrix representation (6), the free surface boundary condition can be changed to a matrix form.

1 U

i - w ' + 2 ( e a > ) [ ^ o ] {a} - 2 - (m + E){AJ {a}

+ ^LA,}{a} + lEUa} = {F] ( 7 )

where [ £ ] is the identity matrix and {F} is an excitation vector.

The eigen values and eigen vectors of the matrix lA^] satisfy the following equation.

( [ ^ ] - i / / c „ [ ü : ] ) { f f } „ = o (8) where the l//c„ is the «-th eigen value and {g}„ is the eigen vector. The components of eigen values and eigen vectors approximately satisfy the following relationships as dis-cussed in the Appendix.

^ /cos 1„3',. sin (n„X; [n : even number)

^' \cos l„y,. cos m„Xi (n : odd number)

I f , + ml = kl (10)

Combining the n-th and n + 1-th eigen vector, the complex eigen vector can be defined.

{G}„ {9]„ + ' { f f l n + l {9}„-i{g}„ + i

{n = 1, 3, 5 . . .) (11)

This complex eigen vector satisfies relationship (8) and, since the matrices \_A{\ and [ ^ 2 ] are obtained by

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differentiating tlie matrix [Ag] with respect to x, these matrices approximately satisfy the following relationships with the complex vector (11).

/)!„

iLAil+{-mYLE:\){G}„^0 (12)

( [ ^ 2 ] + - ^ [ £ ] ) { G } „ ^ 0 (13)

Utilizing the orthogonality of the eigen vector, the strength of the source distribution {a} and the excitation

{F} can be expanded by complex vectors.

W = I « „ { G } „ > { f } = Zi3„{G}„ (14) where the accuracy of equations (12) and (13) depends on the accuracy of equation (9).

Substituting (8), (12), (13) and (14) into (7), the strength of the source distribution is approximately represented by the following form.

KA

^ (co" + 2i£a) + 2i( If {ioj + £)

-gK -gK oK

{G}„

^ + 1 (15) If the element size is infinitesimal and the calculation domain is infinite, the summation in equation (15) would become an integral and equation (15) would be identical to the integral representation of the wave Green function for unsteady free surface flows, that is, the Rankine Source Method converges to the exact solution at the limit of R -> 00 and x -> 0. .

It should be noted that the small variable e is identical to the artifice due to Rayleigh, however, it should be a finite value, since equation (15) is a discrete expression ofthe exact integral representation ofthe unsteady Green function and hereinafter it is referred to as the numerical damping, since it is obvious in the free surface boundary condition (4) that it behaves as a damping.

3. T H E M O S T S U I T A B L E V A L U E F O R T H E N U M E R I C A L D A M P I N G

3.1 Estimation of the error due to the numerical damping

In order to estimate errors due to the numerical damping, a simple two-dimensional flow without current is considered. Let us suppose the periodic source at the origin of the two-dimensional free surface flow field. The strength of the source distribution is given by

A/c

a

„fo K - ipK - k K = CQ-^/g, p = 2£/a)

cos k„X: (16)

where it is assumed that the radius of the calculation domain is infinite and the number of the panel is also infinite. So, the upper bounds of the summation are infinite. The exact integral representation which corre-sponds to the equation (16) is given as the following form.

Fig. 2. Analytically estimated errors due to the numerical damping in a two dimensional unsteady free surface flow.

lim cos kx

K <ik (17)

It is assumed that the n-th eigen value is given by

l/k^^ = l/n-Ak and the wave number K is an integer

multiple of A/c. Neglecting 0(A/c) and 0{p'), the summa-tion in equasumma-tion (16) can be done.

sinh p{Nn — Kxl + in sinh pNn cosh p(Nn • sin Kx: Kx,) cos KX: N (18) sinh pNn

-t- C((KX() cos X x ; -h si{Kxi) sin Kx^

— ip{KXi + s((Xx,.) cos Kxi

- ci{KXi) sin KXj} + 0(A/() + Oip') K/Ak

where si and ci are the sine integral and the cosine integral respectively. Performing the integral in equation (17), the following exact solution is obtained:

ap = {n -\- si(KXi)} sin(Kx,) + {in + cos(JCx,-)} cos(Kx,-)

(19) The error is defined by the following equation:

E = (20)

Errors due to the numerical damping at the point X = l/K for various parameters N are shown in Fig. 2. Errors almost decrease linearly as the numerical damping decreases. However, if the numerical damping is smafler than a certain value, errors increase explosively, because the denominator in equation (16) becomes very small when n is equal to N. I t is defined that the most suitable value for the numerical damping gives the minimum error. I n the case of three-dimensional unsteady flow with a current, the integrand of the wave Green function for this flow has two poles and, since this integrand behaves Hke equation (17) near the each pole and these poles corre-spond to two wave systems, it is considered that the basic nature of errors is the same as in the two-dimensional case. However, the effectiveness of the numerical damping is affected by the wave length, so it will be discussed later.

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An application of the Rankine source method to the estimation of wave-current interaction effects: K. Takagi (k-ki-ipi){k-k2-ilh) + O{£') = 0 ki\ Xo

- 1 h

Fig. 3. Dimensionless numerical damping for k^ and kj wave systems.

The influence of the body for the eigen vector is expected. However, since the body can be regarded as a point disturbance f r o m a distant view, it can be considered that the basic nature of errors is the same as in previous discussions.

In practice, from the complexity of these influences it is difficult to obtain the most suitable value for the numerical damping accurately. So, i t should be decided by means of a numerical experiment which is discussed in detail in section 5.

3.2 Influence of a current

Substituting the velocity potential of the elementary waves into the free surface boundary condition (4), the following equation which must be satisfied is obtained. co" + 2/ccoi7 cos a + C/ "fc" cos 2a - kg

- 2E{iUk cos a + ;co) = 0 (21)

where a is a direction of the propagation of the elementary waves. Neglecting terms of 0(fi"), equation (21) can be factored as follows: 2 cos" a g [1 - 2T COS a + V I - 4T] (22) (23) CO ü /^„= - ( - 1 ) " 2s C O K Q ^ I — 4T COS a X [COQ/C,, COS a + c o g ] (24)

where k^ and /cj are the possible wave numbers of elementary waves and hereafter the system of the elemen-tary waves whose wave number is /c^ (or fcj) is referred to as a kl (or fej) wave system. I t is obvious that and P2 are the substantial numerical damping for k^ and /cj wave systems respectively as is analogized f r o m previous discussions and they should be a certain suitable value. However, since p^ and P2 are the function of the direction of the propagation, usually this condition can not be satisfied.

Figure 3 shows an example of the dimensionless numerical damping defined by the following equation. (The lines for c = 0 correspond to equation (24))

K 2E (25)

I t is obvious from the definition that the dimensionless numerical damping ;c„ must satisfy the condition I K„ I = 1. I n this example, approximately satisfies this condition. On the contrary K^does not. The reason •is that does not work as a numerical damping for k^

wave system. I n order to overcome this difficulty, the following finite difference scheme is introduced.

( - c o " -h 2ieoj)4> - 2U{ia) + E) dx

+ u

-dx hh + g - ^ = 0 (26) oz Ah = 4csU^fg'

It is wefl known that this differential scheme produces an artificial viscosity and it. works as the numerical damping. The lines for c = 1 in Fig. 3 correspond to equation (26) and it can be seen that the change of is not apparent. On the contrary changes drastically.

The constant c in equation (26) can be chosen so that the condition for the dimensionless numerical damping K„ is satisfactory. I t should be noted that the larger value is better than the smaller value, since the smaller damping causes an explosive increase of the error as stated previously. For this reason, c = 1 is chosen i n the present numerical calculations.

4. H Y D R O D Y N A M I C F O R C E S

Considering the time harmonic oscfllation of the body in a current and waves, the following body boundary conditions which are exact up to the order of the amplitude of the motion are obtained.

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8^ dv dv , « = 1 . . . 6 , n = 7 (27)

where (/>„ is a velocity potential due to the «-th motion and 00 is a velocity potential of incident waves whose amplitude is Equation (27) applies on the mean wetted body surface Sg. The v and m components are defined by

(28) V = ( V i , V2, V3) r X V = ( v ^ , V 5 , Vg) m = - (v • VW4>o = i"h, "h> ' " 3 ) - V • V(r X V(/)J = {m^, m^, m^^) r + ix + + kz V - i • ~ + k - ~ dx~^ ^ dy'^ dz

The pressure ofthe fluid due to the ;i-th motion is given by 1 p„= - p ( / f f l + V ( ^ B - V ) 0 „ p - (f2„ • V)(V0B • VCPD) n.. = X r n = 1,2,3 n = 4, 5, 6 ( C i = i , 62 (29) = k) Integrating the pressure on the body surface, hy-drodynamic forces which are proportional to the ampli-tude of the motion are obtained and it is defined that the real part of it is the added mass M„,„ and the imaginary part of it is the damping N,„„.

v„ dS= -[-a'M,„„ + iwN„, (30) The wave exciting force o f t h e n-th and the wave induced drifting force on the restrained obstacle are given by

AR= -piioj + V<^B • V ) ( 0 , + </>o)v„ dS (31)

+

• J _ c 4 ? ico + - ^ C / " - ( V ( / ) J " dz dz dC (32) where the pass of integration C is the water line of the body and * denotes a pair of complex numbers.

5. R E S U L T S A N D D I S C U S S I O N S

5.7. The determination of the value of the numerical

damping

The wave exciting force and the wave drifting force acting on a vertical circular cylinder of infinite draft are calculated by means of the present scheme to illustrate the procedure of the determination of a value of the numerical damping. The ratios of numerical results of the present scheme to exact ones which are obtained by Havelock^ are shown in Fig. 4. The remarkable feature of the ratios for the variation of numerical damping e is

1 5

" \

t o

\

/

\

\

/

F n = 0

Re[F7V

lm[F7-i^

0

0X15

e / o )

Fig. 4. Ratio of the numerical results of wave forces acting on a vertical circular cylinder to exact ones as a function of E/CO.

similar to that of the error due to the numerical damping in the two-dimensional calculation which is stated pre-viously. The radius of curvature of the curves of errors due to the numerical damping shown in Fig. 2 becomes the minimum at a certain point. This point is the most suitable value of the numerical damping. I n the analogy of the two-dimensional flow, it can be considered that the most suftable value of the numerical damping exists in the region of e/co = 0.02 ~ 0.05, since all the hnes i n Fig. 4 have a point where the radius of curvature becomes the minimum in this region.

The real part and the imaginary part of the wave exciting force acting on a vertical circular cylinder of infinite draft are shown in Figs. 5 and 6 respectively and the wave drifting force is shown in Fig. 7. The influence

0.5 Re [F7-1] ja^2' -5.0 1.0 ~ n — g ' 2 • f / a j = 0 . 0 3 A e/cü=O.OA o e/aj=0.05 — Exact Solution

Fig. 5. Real part of the wave exciting force acting on a vertical circular cylinder of infinite draft.

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An application of the Rankine source method to the estimation of wave-current interaction effects: K. Takagi

a

0 0.5 1.0

g 2

Fig. 6. Imaginary part of the wave exciting force acting on a vertical circular cylinder of infinite draft.

3AR-. Pg^aD Q • 1.0 /A /

°

^ — s 0.5

-fo 1 • A O 6 / 0 1 = 0.03 f / 6 L ) = 0 . 0 4 f / ( U ; = 0 . 0 5 Exact Solution 1 0 0.5 1.0 a ; 2 D g 2

Fig. 7. Drifting force acting on a vertical circular cylinder of infinite draft.

of the value of the numerical damping is illustrated by these figures and it should be noted that if the value of the numerical damping is smaller than the most suitable value, the error would be very big. On the contrary, if the value is larger than the most suitable one, the error would not be so big.

Summarizing these results, the best procedure for the determination of the value of the numerical damping for the three dimensional flow may be presented.

(1) Numerical experiments should be done at a certain frequency and Froude number.

(2) A point where the radius of curvature becomes the minimum should be found and the region in which the most suitable value exists should be determined.

(3) The medium value in this region should be chosen as a suitable value.

The weak point of this procedure is that the most suitable value cannot be determined uniquely, however, the error may be limited to less than 10% in the worst choice of a suitable value. 0.5 h 0 . 0 6 4 0 (Rel.10) I I I I 0 0.5 1.0 1.5 9 2

Fig. 8. Added mass for surging hemisphere in a current.

Nil Fn — O — -0.032 1.0 ^ 0 — 0 — 0.064 u 1 n e 1. I U ) j^'^Cf^ D ^^^—-.^ 0.5 0 0.5 1.0 1.5 g ' 2

Fig. 9. Damping coefficient for surging hemisphere in a current.

5.2. Hydrodynamic forces acting on a hemisphere

The added mass and the damping coefficient of a hemisphere in a current are shown in Figs. 8, 9, 10 and 11. The value of the numerical damping (e/co = 0.05) which is used through these results is determined at

g ' 2

(7)

g ' 2

Fig. 11. Damping coefficient for heaving hemisphere in a current. 2.0 ^ - - o o—— Fn 1.0 yó — O — - 0 . 0 3 2 / 0 / —a— 0 . 0 6 4 / 0 (Rel.lO) / i l l 0 0.5 1.0 1.5 T 2

Fig. 12. Horizontal wave exciting force acting on a fixed hemisphere in a current and head waves.

Fn = 0 and co'/g • D/2 = 1.0 by means of the previous

procedure. Froude number is defined by Fn = U/.^/gD and D is the radius of the hemisphere. (The symbol D also denotes the radius of the vertical cylinder.) The exact solutions which are obtained by K u d o ^ ° are also shown in these figures as a reference, and it is found that the numerical results of the present scheme are in good agreement with exact ones.

The wave exciting forces acting on a hemisphere in a

\

2.0 \ — 0 - 0 . 0 3 2 \ 0 \ v — n — 0 . 0 6 4 0 (Rel.10) 1.0 1 1 1 I I I 1 I 0 0,5 1.0 1.5 9 2

Fig. 13. Vertical wave exciting force acting on a fixed hemisphere in a current and head waves.

T " 2

Fig. 14. Horizontal drifting force acting on a fixed hemisphere in a current and head waves.

current are shown in Figs. 12 and 13 and the exact solutions are also shown as a reference. The numerical results of the present scheme are in excellent agreement with exact ones. I t can be concluded f r o m these results that the present scheme has a good performance for the calculation of hydrodynamic forces acting on a body in a current and waves, in spite of the weak point of the present scheme, which was stated previously.

I t is observed in these figures that the influence of the current on the added mass and the damping coefficient is smafl, and it does not have an apparent tendency. On the contrary, the influence of the current on the wave exciting force is significant, and it has an apparent tendency. I t is important from a practical point of view that the added mass and the damping coefficient in a current can be approximated by the values which are obtained by neglecting the influence of the current. However, it should be considered for the calculation of the wave exciting force.

Figure 14 shows the drifting force acting on a hemi-sphere in a current and waves and other numerical results which were obtained by Zhao et al' are also shown as a reference. The numerical results of the present scheme are in good agreement with Zhao's results and both results show that the influence of the current on the drifting force is significant. Zhao et al used the foUowing free surface boundary condition which is exact up to

0{U) and numerical results of the present scheme are

1.0 0.5 0 ( U " ) 0 { U ) Classical Free-surface B.C. F n = 0 . 0 6 4 ^ 1 11 1 g 2

Fig. 15. Nutnerical resuhs of the horizontal drifting force acting on a fixed hemisphere under different free surface boundary conditions.

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All application of the Rankine source method to the estimation of wave-current interaction effects: K. Takagi

6. C O N C L U S I O N S

The Rankine Souree Method is apphed to the calculation of wave-current interaction effects. Introducing the nu-merical damping, a new radiation condition for the unsteady free surface flows which is simple and efficient is developed. The most suitable value of the numerical value for the two-dimensional flow is obtained analyti-cally. I n the analogy of this result the determination procedure of the value for the three dimensional flow with current is developed. The performance o f t h e present scheme is demonstrated by the calculation of hy-drodynamic forces acting on a hemisphere. The numerical results of the present scheme are in very good agreement with other results (kudo's exact solutions and Zhao's numerical resuhs). I t shows the applicability of the present scheme to practical usage.

y-3.46

y=5.20

G = AO S (1^ y ) s in( m ^ x )

O E i g e n v e c t o r

Fig. A-1. An example of an eigen vector of the matrix [A QJ.

also obtained under the following free surface boundary condition.

on z = 0

dz

(33) where the term of the numerical damping is omitted for the simplification.

The comparison of numerical results obtained under different free surface boundary conditions is shown in Fig. 15. Neglecting the steady disturbance, the classical free surface boundary condition is obtained.

-co" licoU

d'x dz 0

on z = 0 (34)

It is apparent that the discrepancy between Zhao's free surface boundary condition and the present one is very small. O n the contrary the discrepancy between the present one and the classical one is significant.

R E F E R E N C E S

1 Nakos, D . E. Free surface panel method for unsteady forward speed flows, Proceedings of 4tli International Wortcsiiop on Water

Waves and Floating Bodies, 1989

2 Zhao, R., Fafdnsen, O., Krokstat, J. R. and Anesfand, V. Wave-current interaction effects on large-volume structures.

Proceedings of tlie Interational Conference on the Behaviour of Offshore Structures, 1988, 623-638

3 Kasiwagi, M . and Ohkusu, M . The effects of forward speed in the radiation problem of a surface-piercing body, Journal of the

Society of Naval Architects of Japan, 1988, 164, 92-104

4 Dowson, C. W. A practical computer method f o r solving ship wave problems. Proceedings of the Second International

Confer-ence on Numerical Ship Hydrodynamics, 1977, 30-38

5 Jensen, G., Soding, H . and M i , Z.-X. Rankine Source Method for numerical solution of the steady wave resistance problem.

Proceedings of the I6th Symposium on Naval Hydrodynamics,

1986,575-582

6 Lamb, H . Hydrodynamics. 6th Edition Dover Publications, New York, 1945

7 Sclavounos, P. D . and Nakos, D . E, Stability analysis of panel method for free-surface flows with forward speed. Proceedings

of the nth Symposium on Naval Hydrodynamics, 1988, 29-48

8 Takagi, K . A n application of Rankine Source Method for unsteady free surface flows. Journal of the Kansai Society of

Naval Architects, Japan, 1990, 213, 1-9 (in Japanese)

9 Havelock, T. H . The pressure of water waves upon a fixed obstaoie. Proceedings of Royal Society, 1940, A.175, 409-421 10 Kudo, K . The D r i f t i n g Force acting on a Three-dimensionai

Body in Waves^ Journal of the Society of Naval Architects of

Japan, 1977, 141, 77-83 (in Japanese)

A P P E N D I X

Since the velocity potential is a harmonic function, i t is expected that eigen vectors of the matrix [AQ\ are given by the following form in the symmetric case.

Qj ^ cos l„yj sin ) » „ X ; or cos I^y^ cos

/«„x,-ll + 'nl = kl

( A l ) (A.2) where the n-th eigen value is defined by l//c„.

In order to investigate the actual relation between eigen values and vectors of the matrix [AQ], the following function is defined:

B„ = Z [0/ - A cos /„y, sin ))!„x,.

- A,, cos l„yi cos m„x,.]" (A.3)

where c/,- is a i-th component of the eigen vector which is obtained numerically. Minimizing the function B under the restriction (A.2), coefficients A^, A,., 1„ and »i„ are

(9)

obtained. An example of the eigen vector and the approximation by (A.l) is shown in Fig. A - 1 and it shows that eigen vectors of the matrix [ / I Q ] are approximately represented by the form (A.l).

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