The large-time asymptotic expansion of the impulse response function for a floating body

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The large-time asymptotic expansion of the impulse

response function for a floating body

F. T. K O R S M E Y E R A N D P. D. S C L A V O U N O S

Department of Ocean Engineering Massachusetts Institute o f Technology, Room 5-326 Cambridge, Massachusetts 02139, USA

Irregular-frequency effects are present in the time-domain, as well as in the frequency-domain analysis of the first-order, free-surface wave radiation problem. In the time domain, the effect is an oscillatory behavior in the impulse-response function at large time. The nature of this behavior is investigated relative to the temporal and spatial discretization of the integral representation of the problem. In an effort to remove these effects in both domains, a method for obtaining the large-time asymptotic expansion of the impulse-response function for a floating body is presented. This method is applicable to arbitrary bodies in any mode of motion. The procedure is to Fourier transform the low-frequency expansion of the complex impedance function. This expansion is available to any order. The use of the large-time expansion of the impulse-response function in eliminating irregular-frequency effects is demonstrated.

Key words: time domain, large-time asymptotics, low-frequency asymptotics, impulse-response function, added-mass, damping

INTRODUCTION

Recently there has been increased interest in studying the linearized wave-body interaction problem in the time-domain. 1'2'3 The motivation for time-domain analysis in preference to the traditional frequency domain approach includes: interest in transient responses in zero-speed problems; the ease of handling forward speed in the ship motion problem; the possibility of computational efficiencies for very complicated bodies; interest in quasi-nonlinear approaches, where the free- surface boundary condition is linearized but the body boundary condition is not; and the use of linear outer solutions to close fully nonlinear inner analyses by matching.

All of the above cited authors point out that problems associated with irregular frequencies encountered in the frequency domain are also present in the time domain. Adachi and Ohmatsu 4 state that in the time domain the effect of irregular frequencies is manifested as an oscillation of the impulse-response function at large time. If the impulse-response function is Fourier transformed to obtain the added mass and damping coefficient curves in the frequency domain, these large- time oscillations cause inaccuracy in the vicinity of the irregular frequencies. In the frequency-domain potential formulation of the radiation problem, solutions may be computed to a desired accuracy by panel methods at frequencies which are close to an irregular frequency by increasing sufficiently the accuracy of the body discretization. The analogous situation in the time Accepted May 1988. Discussion closes September 1989.

domain is that the amplitude of the oscillations in the impulse-response function may be reduced and the time at which the oscillations become apparent may be delayed by increasing the accuracy of the body discretization; and to a certain extent, by increasing the accuracy of the temporal integration. Since in principal we require an infinite record of the body response, Newman ~ suggests that the impulse-response function be constructed by matching a suitably accurate numerical solution to a large-time asymptotic representation.

The large-time asymptotic expansion of the impulse- response function may be obtained from the low- frequency asymptotic form of either the added-mass or damping coefficient functions through the Fourier transform. One or two terms of the low-frequency expansion for the damping coefficient function are readily available from the Haskind relations applied to well-known, one- or two-term approximations of the exciting force. Simon and Hulme 5 however, show that the frequency-domain velocity potential, and therefore the hydrodynamic coefficients, may be expanded to any order in the wavenumber through the direct expansion of the Green formulation of the frequency-domain radiation problem.

This is the method employed in the present paper to produce the required frequency domain results. The frequency-domain Green formulation for the velocity potential in the radiation problem is expanded directly for low frequencies by using an ascending series representation for the Green function. This provides a hierarchy of problems for the coefficients of the velocity potential expansion. From these problems, some of the

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The large-time asymptotic expansion." F. T. Korsmeyer and P. D. Sclavounos coefficients in the added-mass and damping expansions

m a y be found in closed form, and the first two terms o f the damping expansion are shown to agree with values derived independently by using a low-frequency exciting force expansion in the Haskind relations. The low-frequency expansion o f the complex impedance function for the heaving hemisphere is Fourier t r a n s f o r m e d and used in the time domain to remove the oscillatory behavior in the impulse-response function. On transfor- mation back to the frequency domain, we find that the effects of the irregular frequencies are removed. This technique is applicable to simple, three-dimensional bodies in any mode of motion.

THE LINEAR T R A N S I E N T R A D I A T I O N PROBLEM

We are interested in the fluid motion due to an impulsively accelerated body which pierces or is near the free surface. As shown in Fig. l, a Cartesian coordinate system is fixed in space with the z-axis positive up, and the x- and y-axes in the undisturbed plane of the free surface o f a semi-infinite fluid. This fluid is incompressi- ble and inviscid, and the flow is irrotational. The fluid density, the acceleration due to gravity, and a representative b o d y length have been set equal to one. The fluid and the body are at rest for time t < 0, and then the body velocity takes a constant, non-zero value at t = 0 ÷ . The exact problem is linearized on the assumption of small wave slope, and so the b o u n d a r y conditions are applied on t h e initial b o u n d a r y positions. For a body with velocity U and inward directed unit normal vector ~, the potential must satisfy the following equations:

V2~(~, t ) = 0 in the fluid domain, (1) the kinematic and dynamic free-surface conditions m a y

Sr ~ y So /

s. ___.._~/ /

/ /

Figure 1. The coordinate system for the transient radi- ation problem. Surface identifications apply to the initial positions o f the surfaces

be combined to yield

~tt -q- ~Oz = 0 on the free surface, SF, (2)

Cn = U . ~, t > 0 on the b o d y surface, Ss, (3) ¢, V¢, Ct are uniformly

bounded on So, for t finite. (4) As in any transient problem, we require initial conditions. In the exact, nonlinear, radiation problem these are that we must specify the position and velocity of every fluid particle at t = 0 +. Since ¢ ( ~ , t) is a harmonic function, such specification on the boundaries alone is sufficient. In a radiation problem, surfaces Sn and S= are specified, and the potential or its normal derivative, on those surfaces, is specified as well. On SF, it can be shown by integrating the exact free-surface conditions f r o m t = 0 to t = r, and letting r ~ 0, that the exact potential and the exact free surface elevation are zero at t = 0 ÷ . 6 Application o f the free-surface conditions then yields:

~ ( x , y , 0, 0 + ) = 0, (5)

and

~t(x, y, 0, 0 + ) = 0. (6)

Following Wehausen, 7 we employ Green's second identity with function ¢~(Y,r) and Green function G(Y, ~, t - r) to reformulate the problem as an integral equation to be solved on the body surface,

SB

S~

$8

where the Green function G(Y; ~', t) satisfies the same b o u n d a r y conditions as ~(Y, t) except on Sn, and has been derived by Wehausen and Laitone.8 This function m a y be written: 1 1 + 2 I ~ G ( ~ , ~ , t ) R R' o x [1 - cos(kl/Zt)]e-kYJo(kX) dk, (8) where R = [ ( x - ~)2 d- ( y - ,~,)2 -k- (Z -at- ~*)2] 1/2, Y = - ( z + ~'), and X 2 = ( x - ~ ) 2 q.. ( y _ ~,)2.

Equation (7) m a y be referred to as the 'potential formulation' to distinguish it f r o m the integral equation obtained by distributing sources over the b o d y surface, the 'source f o r m u l a t i o n ' . This p r o b l e m also m a y be formulated for a body with disl~lacement, not velocity, which is a Heaviside function.

We construct a discrete a p p r o x i m a t i o n to equation (7) by panelling the body with N p l a n e , quadrilateral panels

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The large-time asymptotic expansion: F. T. K o r s m e y e r and P. D. Sclavounos upon which ~o(2, t) is assumed to have a constant value,

and performing the convolution by the trapezoid rule. A linear system results from the enforcement o f the integral equation at a discrete set o f collocation points at the panel centroids. If we specialize equation (7) to the cases o f unit sway or heave, (the two modes for which results are presented), the discrete form which must be solved at each time-step, M, up to the last time-step M r , is: N s~ M-1 j=IN f i _ (F _" = - Z ' A t Z ~Oj.m d ~ G , , t { x , ~,tM-m) it/=0 & N

+

I

j = l d d sj i = 1 , 2 . . . N M = O, l . . . M r , (9) where n is the component of the normal vector appro- priate to the particular mode, the prime on the temporal summation indicates that a weight o f one-half is applied when m = 0, and G (°) and G (F) a r e the Rankine and

wave parts o f G(Y; ~, t), respectively. Two important aspects o f equation (9) are that: one, the Fredholm kernel is independent o f time so that the left-hand side matrix is reduced once and for all time at t = 0+; and two, G(r)(Y; ~',0)= 0 so that the current value o f the unknown ~o(Y,t) does not appear inside the discrete Volterra integral. The time-dependent potential

resulting from the solution o f (9) may be integrated over the body to determine the transient added-mass

M(t)= I I

d~(~,t)~.(~),

(I0)

SB

which can be differentiated in time to provide the impulse-response function

L(t)= I I

d~¢t(~,t)¢.(~).

(II)

Sa

The impulse-response function is related to the added-mass and damping coefficients o f the linear, frequency-domain, radiation problem through the Fourier cosine and sine transforms

a(o~) = (~(oo) + d t L ( t ) c o s cot, (12)

0

and,

b(___~) _ d t L ( t ) s i n wt. (13)

CO 0

A computer code has been written which solves equation (9) for general, three-dimensional bodies in any mode o f motion. Here we present and discuss results for the hemisphere in the heave and sway modes. In Figs 2 and 3, the impulse-response functions in these modes are presented. For simple bodies, we expect the impulse-response function to monotonically approach zero at large time. However we find that at large time, the impulse-response function begins to oscillate around

o o w z " ° C:) I - - L.) Z b _ o lad c,r) Z 0 o tJJ o r r " t ~ ( / ) . . J : [ o w ~ o I I I I JO. O0 10. O0 20. O0 30. O0 40. O0

Figure 2. Heaving hemisphere; 64 panels, A t = 0 . 2 . _ _ I

50. O0

TIME. t

I I J I

60.00 70,00 BO.O0 90,00 tO0.O0

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0 ,z ._f o

4

o

~.0o

A I I I I I I I I tO.O0 20.00 30.00 40.00 50.00 80.00 70.00 80.00 90.00 tO0.00 TIME. t

The large-time asymptotic expansion: F. T. Korsmeyer and P. D. Sclavounos

Figure 3. Swaying hemisphere; 64 panels, At = 0 . 2 _ _

what we regard to be the correct solution. This is the phenomenon of irregular frequencies manifested in the time domain. As several authors 1'2 have noted. the period o f the most prominent oscillation co- responds to the lowest irregular frequency for the particular body. Other oscillations superposed on this most prominent oscillation are the successively higher irregular frequencies. Fourier analysis reveals this fact, which may be seen in Figs 4 and 5, which present frequency-domain results calculated by equations (12) and (13), from the time-domain results o f Figs 2 and 3. The irregular behavior in the time and frequency domains is introduced by the integral formulation of the problems. The severity o f this behavior is affected by the manner in which the continuous equations are discretized. In Fig. 6, it is clear that the time-domain solution is like its frequency-domain counterpart in that the effect of irregular frequencies may be mitigated by improving the accuracy of the spatial discretization of the body. Empirical evidence suggests that for hydrodynamic coefficients computed from the frequency-domain formulation for the potential, the bandwidth of the inaccurate region centered on an irregular frequency may be narrowed by a more accurate body model. In the time domain, the amplitude of the oscillatory behavior may be reduced by a more accurate body model. Extremely fine discretizations of the body may render the oscillations undetectable over a time history like that shown in Fig. 6, but they will show up eventually, as seen in Fig. 9 where there are results for a 4608-panel hemisphere.

The effect of reducing the time-step is less clear. In Fig. 7, it is evident that the effect of reducing the time- step from A t = 0 . 5 to A t = 0.3, is a reduction in the amplitude of the oscillations, and some change in the results in the earlier-time, non-oscillatory portion of the record. However, the reduction to a time-step of At = 0.05 has increased the oscillatory amplitude and shifted the location of the extrema on the time axis. As discussed below, however, the accuracy over the entire time-history o f the computation with / x t = 0 . 0 5 , is superior to those computations with larger time-steps.

Quantification of the accuracy of solutions at various levels of discretization has been done by comparison to a very finely discretized model: N = 4608, At = 0.05. In Fig. 8, the mean-square error relative to this fine model, for a matrix of different spatial and temporal discretizations presented. Although we cannot make a general statement about the reduction in the amplitude of the oscillatory behavior as the time-step is reduced, the mean-square error for any particular body model decreases approximately as a function of (At) z, as may be expected for trapezoid rule convolution. Consider- ation of Figs 6 and 8 together, reveals that the reduction in the mean-square error, as panel size is reduced, is largely a consequence of the reduction in the amplitude of the irregular-frequency effect.

Because the Green function oscillates with increasing frequency in an exponentially thin region at the free surface as time increases, we may expect that panels near the free surface have a greater effect on the solution than panels at greater depth, at least at large times. Con-

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The large-time asymptotic expansion: F. T. Korsmeyer and P. D. Sclavounos o

l

la. h t J t I l ~ r t-~ o

°1

~ o t J ~.

, ~ °

o . I g ,*0 .o ~.o , 2 o , 3 o , 4 . 0 , 5 . 0 , 8 . 0 WAVE NUMBER. k

Figure 4. Heaving hemisphere; 64 panels, A t = 0.2, truncated at t = 2 0 _ _ ; 64 panels, A t = 0 . 2 , truncated at t = 5 0 . . . . ; 256 panels, A t = O . 1 , truncated at t = 2 0 _ _ _ _ . Coefficients are non- dimensionalized by the displaced f l u i d mass

sequently the idea of cosine spacing the vertical distri- bution of panels as the free surface is approached may be justified. The mean-square errors for several such body models are also plotted in Fig. 8; and indeed, for exactly the same numerical effort as with a regular distribution of the same number of panels, the cosine distribution slightly improves the results. Later we will make use of the impulse-response function for a cosine distributed body model, which is shown in Fig. 10, and it may be seen that this panel distribution has the effect of shifting the oscillations on the time-axis relative to the regularly spaced model results of Figs 2 or 6. In general, changing panel aspect ratios, without changing the number of panels, affects results unless the discretizations are very fine. This fact makes it im- perative that panel aspect ratios be preserved when con- vergence studies are undertaken. In summary, Fig. 8 suggests that the maximum accuracy for any particular amount o f computational effort may be had by a cosine distribution of the least required number of panels. It is inefficient to obtain this accuracy by a larger number of panels combined with larger time-steps because the computational effort is quadratic in the number of unknowns and linear in the number of time-steps.

Z • U'/ I g IAI o o 0 0 Z I,.I_ b_ I,~ 0 Z I 0 0 0 0 I I I I i ~ i i ! , 0 0 WAVE NUMBER. k

Figure 5. Swaying hemisphere; 64 panels, A t = 0.2, truncated at t = 2 0 _ _ ; 64 panels, A t = 0 . 2 , truncated at t = 5 0 . . . Coefficients are non- dimensionalized by the displaced fluid mass

In practice, we numerically transform a truncated, rather than infinite, impulse-response function, but the choice of truncation point affects the accuracy of the frequency-domain result• We might be inclined to transform as long a record as we can realistically com- pute, in order to correctly define low-frequency behavior. However, the inclusion of an increasingly long sample of an oscillating record, leads to a magnification of the irregularities in the corresponding transform. In Figs 4 and 5, the different Fourier transforms of the impulse-response functions shown in Figs 2 and 3 show the effect of truncation points at t = 20 and t = 50. In the former case, there is irregular behavior in a broad frequency band centered on the first and second irregular frequencies for each mode. In the latter case, the longer record has concentrated the ir- regular behavior in a narrower frequency band around the irregular frequencies and increased its magnitude.

Consideration of the radiation problem in the fre- quency domain can be of use in the transient problem. In particular, the low-frequency asymptotic form of the

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T h e l a r g e - t i m e a s y m p t o t i c e x p a n s i o n : F . T. K o r s m e y e r a n d P . D . S c l a v o u n o s o ! .i g - o Z i,,..1 p - r ~ t+, (,-j Z 0 S.Ll ...I g o

-o L

1 o ' 0 + 0 0 2 . 0 0 4 . 0 0 | . 0 0 | . 0 0 F i g u r e 6. H e a v i n g h e m i s p h e r e ; A t = 0 . 1 ; 6 4 p a n e l s _ _ __ tO.O0 t2.00 TIME. t , 1 4 4 p a n e l s _ _ _ _ t4.O0 l l . O0 t|.OO 20.00 , 2 5 6 p a n e l s w .a g * o Z @,4 t~ D Z t ~ ~ o LI.J O'J o +0.90 ~. O0 4. O0 6. O0 g. O0 l O . O0 12. O0 T I M E , t F i g u r e 7. H e a v i n g h e m i s p h e r e ; 1 4 4 p a n e l s ; A t = 0 . 5 _ _ _ _ , A t = 0 . 3 . . . . 1 4 . 0 0 1 6 . 0 0 1 8 . 0 0 2 0 . 0 0 • A t = 0 . 0 5 80 A p p l i e d O c e a n R e s e a r c h • 1989, Vol. 11, N o . 2

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The large-time asymptotic expansion: F. T. Korsmeyer and P. D. Sclavounos t . t . o lad t . f e e- aS X 64 panels u 144 panel8

256

panel6 I N -- i g

~.oo

o~os

olto

o~ts

olzo

o ~

olno

o~35

o~4o

o ~

o.so

Time Step

Figure 8. Heaving hemisphere. Mean square error in the impulse response function (compared to a 4608-panel with A t = 0.05). Solid lines are splines through the marked data-points f o r regularly distributed panel models. Cosine distributed panel models are marked: 64 panels + , 100 panels A, 144 panels []

damping or added-mass functions, available for arbitrary bodies, can be used to construct the large-time asymptotic expansion o f the impulse-response function. Since the form of this time-domain expansion can be found for an arbitrary body, and the effort required to calculate the coefficients is small compared to convolution for large time, it is reasonable to match this expansion to the calculated impulse-response function at some time which allows the Fourier transform o f a function without oscillatory behavior.

THE LINEAR FREQUENCY-DOMAIN

RADIATION PROBLEM

We consider the same problem as discussed above, but now with harmonic body velocity which exists f o r all time. This is the familiar frequency-domain radiation problem with implied time dependence o f ~(.~, t) = ~(.~, k )e i~t, with only the real part understood to have physical significance. Recall that the problem is normalized by setting the acceleration due to gravity, the fluid density, and a representative body length equal to one. The heave mode, alone, will be studied in the re- mainder o f the paper, although the analysis is valid for any mode. If the body heave velocity is e i~t and n is the vertical component o f the body unit normal vector, then

the problem formulation for

~ ( ~ , k) is

V2~(~, k) = 0 in the fluid domain (14)

- - r..d2~ "1 - ~ z = 0 o n SF

(15)

~, = n on Sa (16)

1 e_i(l~_~t ) /~ --* ao, /~2 = X2+ y2"

~ e i°Jt o l - - ~

(17) In the usual manner, Green's second identity with ~(.~, k) and G(.~; ~, k) leads to

So

S a

(18)

where the Green function is defined by Wehausen and Laitone s to be:

G(.~; ~', k) = ~ + 1 _{--R7 + 2k}1 I ~° _-1 _{J o ( K X ) e -KY dK} o K k

(19) with R , R ' , X and Y defined as in the time-domain Green function, following equation (8), and k = ¢,,.0 2. The

contour over the real k-axis is indented above the pole at K = k in order to enforce the radiation condition (17). This Green function has been shown by Hulme 9 to possess the convergent ascending series expansion,

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The large-time asymptotic expansion." F. T. Korsmeyer and P. D. Sclavounos which through

O(k 2)

is:

1 1

c(~; ~ , k ) = ~ + ~ - 2 k log k - 2

× [ [log(R' + Y ) - l o g 2 + 3 ' ] +Tri]k + 2 Y k 2 log k

+ 2 [ [ l o g ( R ' + Y ) - l o g 2 + 3 ' ]

× Y - R ' + 7rYi]k2+ O(k s log k). (20) where 3' is Euler's constant. Insertion of this series in equation (18) suggests a similar series expansion for so(Y,k). In a method demonstrated by Simon and Hulme, 5 (a reference we were not aware of when we developed this approach) a series representation for ~(Y, k) is assumed and both sides of equation (18) are expanded to produce a hierarchy of problems for the unknown coefficients of the series for ~o(£,k). This series, unlike that for G(Y; ~, k), will contain powers of log k. Using the convenient nomenclature, where for in- stance, G appears as

G(.~; ~,k) = g00 + gllk log k + glok

+ g21k 21og k + gzok 2 + . . . , (21) ~0(£, k) has the expansion

M(n) log"'

~o = ~,, k" Z Pro,, k, (22)

• q = O m = O

where M(n) = (n + 1)/2, when n is odd, M(n) = n/2, when n is even, and M = 0 when n = 0. This pattern for the introduction of logarithmic terms in sa(,~,k) is a result of the fact that the normal derivative of G(.~; ~,k) has no term at O(k log k). The resulting problems for the complex coefficients of the radiation potential are: O 27rpoo+ I i d ~ P ° ° ~n ( g ° ° ' = f l dgn3goo (23) SB S8 0 2 7 r p l l '~- I f d~Pll ~---~ (goo) = i f d ~ n 3 g l l (24' SB 5,'~ 0 27rplo + f l d~plO -~n (gOo)

S~ 0 (gl0)) (25) $8 0 2 r p 2 ' + I f d ~ p 2 1 ~ n ( g O o ) : 1 3 " d g $8 $8

(

o

)

x nagzl - P o o On ( g E t ) - Pll ~ (glO) (26) 0 SB Sa

(

o

)

x n3g2o - Poo -~n (g2o) - Pro On (glo) (27)

a3 (co) = Then,

b r i m ~ - - -

The infinite set of equations represented by (23) through (27) may be solved to obtain the coefficients of the expansion for ~o(~, k) to any order. Note that the kernel is the same at all orders, but the right-hand sides are increasingly more complicated combinations of lower-order solutions and coefficients from the expansion of G ( £ ; ~ ' , k ) . The coefficients of the expansions for the added-mass and damping functions are found by integration over the body surface of the potential coefficients. We can use the hierarchy of integral equations to find, in closed form, the first few coefficients of the expansions for the heave damping and added-mass a3 (w) and b3 (w); and in particular, we can recover the first two terms of the heave damping expansion derived from the low-frequency approximation for the heave exciting force

)(3(o:) - Aw + iwb3 - w2(a3(O) + ¥), (28) where v is the body displacement and Aw is the waterplane area. When this expression is inserted into the Haskind relations, the low-frequency expansion for the heave damping function is found to be

b3 (~o) A2w

k - Aw(a3(O) + V)k:. (29)

w 2

The hierarchy of integral equations to order k2 recovers this result. For the added-mass, we can find in closed form, terms at orders 1, k log k, and k 2 log k. Consider the following definitions

b 3 ( w )

_-

~ kn M~ n)

_b,,.

log'" k (30) O.~ r t = 0 n l = O M(n) l o g ' " ~] k" E an,,, k. (31) n=O m=O and, amn =

f I d~p,,,,ns,

SB

(32)

f d~Rp,,,,ns. (33) $8

Looking order by order at equations (23) through (27):

O(1)

This equation is real, so ,~poo = 0 and hence boo = 0. The equation for poo is

2"a'Poo + I f d ~ * ~ (g°°)p°°= I f dgn3goo. (34, Sn SB This is 0 1 $8 S~

the equation for the rigid-lid potential, ~'Rt [i.e.

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The large-time asymptotic expansion: F. T. Korsmeyer and P. D. Sclavounos

(a/an)~bm

= 0 o n S F ] . Therefore, aoo is the infinite-

fluid, dilating-body added-mass,

art.

O(k log k)

This equation is also real, so ~p,~ = 0 and b~t = 0. The equation for p ~ is

0

27rp~, + I f d~'P~ ~n (g°°)= I f

d~n3g~l.

(36)

Sa S~

Since the right-hand side is a constant, we have a constant solution; and since

Sa it follows that A~ (38) P l l - - 2~" ' and A~w a~, = - - - (39) 27r"

O(k)

This equation is complex. The real part has not yielded to analysis, hence the exact coefficient for as (co) is not available at this order, although some coefficients at higher orders may still be found in closed form. Tak- ing the imaginary part of the equation we obtain

27r~plo + I f d ~ ' a

an

(g°°)~Px° = I f d~'n3~g~o

SB SB

=

- 27rAw.

(40)

Since the right-hand side is a constant, we have a constant solution

-Aw

(41) ~ p l o - 2 ' thus

A~.,

bto = - - . (42) 2 O ( k 2 log k)

This is another real equation, s o b21 = 0. For P21 we have

O

SB Sa

[

0

0 ]

x nsg21-POOon(g21)-Pll~(glo)

• (43)

We have found that poo =

t~Rt,

and p , , =

-A~[27r,

so we have

a

Sa S~

x [2ns(Y+ ~bRt)-A--Ew fl~- l°g(Y+ R " ]

an

(44)

Letting p21 = / ~ +/~, w e separate the integral equation (44) into the following components

sf

°

I!

27r/~ + d~/~ ~nn (g°°) = d ~ 2 ( - r + ~bRt)ns So

(45)

and

Sf

°

2r/~ + d~/~ Onn (goo) - SB X f l d ~ n l o g ( Y + R ' ) - 2 A w z . S s (46)

where the definition Y= - ( z + ~') has been used. The equation for 15 has a right-hand side which is a constant function over the body surface, equal to 2(as(0)+ V). Since its kernel is non-singular, it must possess a unique solution. Here, we seek it to be a constant function.

It is easy to verify that

1

/~ = ~ (as(0) + V) (46)

solves the integral equation (45), thus is its unique solution. To solve the equation for /~, we first use the identity

St~ S..

- 1 1 S S d ~ ( l + ~ _ ) (48,

xd~=~

Sw

where

Sw

is the body waterplane surface. Identity (48) follows by conserving the mass flux across the closed surface

SB U Sw

due to the flow generated by the harmonic function log(Y + R ' ) which is singular at a point above the free surface. Applying next Green's second identity with the potentials ~, =

lJR + llR',

and ~2 = in the domain interior to the body surface, we obtain

S,,. Sa

x + n 3 - f ~ n n R + + r Z (49) where the term r z on the right-hand side is the contribution from the hemispherical indentation around the location of the Rankine source 1/R on the body surface Ss. Utilizing (48) and (49) in the right-hand side of the integral equation (46) we obtain

1 Aw

S s SB

The right-hand side of equation (50) consists of three terms. The first term is proportional to the right-hand side of the rigid-lid integral equation. The corresponding contribution to the velocity potential# is

Aw]27r~bnt.

The last two terms can be obtained if we substitute

-z(A~[2r)

f o r / ) on the left-hand side. Therefore, this

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The large-time asymptotic expansion." F. T. Korsmeyer and P. D. Sclavounos is the second c o m p o n e n t of the solution for the velocity

potential/~. Thus, the solution o f the integral equation (46) is

A~

t0 = ~ ( ~ R L -- Z ) . ( 5 1 )

Combining (47) and (51) gives

1 A ~

Pzl = ~ (a3(0) + ¥) + ~ (ffRL -- Z). (52) Integrating for the term a2~ in the expansion for the added mass coefficient function, we obtain

A~

a21 = - - ( a 3 ( 0 ) + V ) . ( 5 3 ) 71"

O ( k 2)

This equation is complex. The real part has not been solved in closed form. Taking its imaginary part

÷ I I

= f l

Sn Sn

[

0 o

]

x n3~g2o- poo~ On (g2o) - ~pJo On (glo) • (54) We have found that poo = ~RL, and ,~Plo = -A~[2, so we have

Se St~

[

o

]

x 2~rn3(Y+ ~gL) -- Aw-~n l o g ( Y + R ' ) . (55) However, the right-hand side of (55) is just that of (44) with the multiplicative factor o f ~r. Therefore

~ P 2 0 = 7rp21, (56) and so

b2o = -Aw(a3(O) + ¥) (57)

recovering the second term in equation (29), which was derived f r o m the low-frequency expansion o f the exciting force. Summarizing, in closed f o r m we have found the first two terms in the low-frequency expansion for the heave added-mass coefficient function

A~

a3(w) = a R L - ~ k log k + O(k), (58) and the first two terms in the low-frequency expansion for the heave damping coefficient function

b3(w) _ AZw

k - Aw(a3(O) + ¥ ) k z + O ( k a log k).

w 2

(59)

T H E F O U R I E R T R A N S F O R M

The impulse response function m a y be determined f r o m the inverse forms o f either equation (12) or (13); or m o r e generally, we m a y write:

L ( t ) = - I 9~ ei~t[ W(w) - a ( ~ ) l dw, (60)

7I" 0

where W(w) = a(w) - ib(w)/w is the complex impedance

function. Using low-frequency expansions for a(w) and b(w)/w we consider two methods for the large-time asymptotic evaluation of equation (60): in the complex w-plane by contour integration, and on the positive real w-axis by Fourier theory involving generalized functions.

Contour integration

For contour integration o f W ( w ) - a(oo) we require knowledge o f that function in the complex w-plane, and particularly on the imaginary w-axis. The ascending series representation for G(Y; ~, k) is exact and m a y be continued for complex k. When this is done, the hierarchy of integral equations is continued into the complex k-plane and hence so is the associated complex impedance functions, W(w). O f course we no longer expect that the real and imaginary parts o f W(w) repre- sent the added-mass and damping functions o f the original wave-body problem. When the expansion for G(Y; ~, k) is moved into the complex k-plane to just above the cut on the negative real k-axis, that is k = - I k I + ic as c --, 0, the expansion may be rewritten with coefficients ~,,,, related to the original g .... as follows:

I n e v e n

g,,o = +-- (~g,o + i2~g,O) ( n odd

(61)

= +- gnl I'n even

(

n odd '

and at least through O ( k 5) the new integral equations have simple relations to those equations on the positive real k-axis. In fact to all orders, the kernel is not changed, although the right-hand sides are. The W(w) corresponding to the solution of these equations with this particular complex k will define the complex impedance function on the positive, imaginary w-axis, where w = i f ( f a positive real number). This W(f~) m a y be written in terms o f the coefficients o f W(w), for real w; and the imaginary part which we will need in the contour integration is:

, ~ W ( f ) - 2 [blof/z - b 2 0 f 4 + 2b3z~ 6 log ~ + b30~ 6 - 2 b 4 1 f 8 log f ~ - baof8 + 4bszfl°log 2 f

+ 2 b 5 1 f 10 log f + (b5o - r 2 b 5 2 ) f 10] (62) Alternatively, substitution o f w = in into W(w), as defined on the real w-axis, results in ,~W(i2):

~ W ( ~ ) - ( b l o - " a - a l l ) f i E - (bE0 -- 7 r a 2 1 ) f l 4 + (b31 - 4"a-a32)Q 6 log f + (b3o - 7 r a 3 1 ) ~ 6

- (2b41 - 47ra42)~] 8 log fl - ( b 4 o - a-a41)f 8 + (4b52 - 12~ra53)f a° log 2 f

+ (2b51 - 47rasE)f a°log f~

+ (b50 - 7r2b52 + 71"3a53 -- 7 r a s 1 ) f 10. (63) If we equate (62) and (63), we recover the relations between the coefficients o f the expansions for a(w) and b(w)[w, a concept suggested by the Kramers-Kronig relations, and discussed by Greenhow. lo

By contour integration around the first quadrant of the complex w-plane, we can set equation (60) equal to a Laplace integral plus some u n k n o w n set o f residues f r o m poles which m a y exist in this quadrant. As t ~ oo

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The large-time asymptotic expansion: F. T. Korsmeyer and P. D. Sclavounos their contribution decays exponentially so that asymp-

totically L(t) may be determined from:

L(t) - - 1 e-at~W([2) d[2. (64)

~" 0

Olver 11 presents an asymptotic analysis of this type of integral if the function W(9) can be represented by a power series which may include powers of logarithms. Results required for the evaluation of terms through O(k 5) are:

that function k" log 2 k

inverse sine transform

87r ( - l)"(2n)! [log 2 t - 2¢(2n) log t [ 2n 1 7rZ]]t_(2n+l) + (¢(2n)) 2 - ~ j2 j = !

]2JJ

(73) function [2" [2" log [2 9 " log 2 [2 Laplace contribution - 1 n!t_(n+ l ) 7r - 1 n! [¢(n) - log t ] t -("+ i) 7r -Trl n! [log 2 t - 2¢(n)log t

(

:

+ CZ(n) + --6-- ~] t - j = l where ¢ ( N ) = ~a - - - - 1 ,)/. n = l n (65) (66) ¢. + 1) (67) Fourier integration

For this analysis it is more convenient to work with the inverse form of either equation (12) or (13) than with the general form (60). We choose the inverse sine transform of the damping function:

L(t) =-2 ~= b(co) sin cot dco. (68)

71" " 0 CO

Evaluation of this transform is facilitated by the theory of generalized functions detailed in Lighthill. 12 Table I of this reference can be used to obtain the following results:

function inverse sine transform

k" 2 ( _ 1).(2n)!t_~Zn+ 1) (69)

71"

k" log k 4 ( _ 1)"(2n)! [¢(2n) - log t] t -(2n+ l)

7r

(70) The term of order k 5 log z k requires additional effort. Using the definition

log[ x[ = lim (1 - I x l - 9 / c , (71)

e ~ 0

we set

l o g 2 1 x l = l i m 1 e ~ o - ~ [IX[e + I X I - e - - 2 ] , (72) where the transforms for the right-hand side of (72) are available in the above cited table. Since xmf(x) transforms to ( - 2 r d ) - " ( d m ) / ( d y " ) g ( y ) , it follows

T H E I M P U L S E - R E S P O N S E F U N C T I O N

Simon and Hulme 5 present the following expansion for the damping coefficient function of a heaving hemisphere. This was derived from a subset of the hierarchy o f integral equations, with the help of relations between the added-mass, damping and exciting-force functions. Their expression includes all terms to order kS: b--23- 27r---~2 [ 3 k - 1.830951k 2 + 34k3 log k co 3 + 2.542917k 3 - 2.746427k 4 log k -2.374082k4 + ~6 k5 log 2 k + 5.646816k 5 log k - 0.442638k 5 }. (74) Using either equation (62) and the Laplace integral results in equations (65) through (67); or equation (30) with Fourier integral results in equations (69), (70) and (73) gives the same expression for the asymptotic form of the impulse-response function. Equations (67) and (73) appear to imply that the Laplace and Fourier analysis methods produce different time-domain results for terms at order with squares of logarithms. However this apparent difference in the time domain, here specifically at O(t-l~), is offset by the difference in the coefficients of the impedance functions at O(k 5) on the real co-axis and at 0([2 ~0) on the imaginary co-axis. We find that: L(t) _ __2 [2!blot_ 3 _ 4!b2ot_5 L - 2(6!)b3xt -7 log t + 6[(b3o + 2¢(6)b31)t -7 + 2(8!)b4x t -9 log t - 8!(b4o + 2¢(8)b41)t-9 + 4(lO!)b52t T M log 2 t - 2(10!)(4¢(10)b52 + b51)t -11 log t j = l ₁ 7r2 _{~ ] b52} + 2¢(10)b51 + b50] t-11].

It is interesting to note that equations (62) and (63) suggest that the coefficients a,o are superfluous to this analysis. This is supported by the Fourier integration as well. If the inverse cosine transform of the added-mass expansion is used, the a,o'S contribute nth order derivatives of 6(t) (where b(t) is Dirac's delta function) which only affect L (0).

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The large-time X . . d U Z u, taJ 0 ta t O bO t r . . J n

Figure 9. Heaving hemisphere; 64panels, A t = 0.2, × ; asymptotic results (labeled) to O(t-11)

asymptotic expansion." F. T. Korsmeyer and P. D. Sclavounos ¢ J g o ? 5 ?

\

? ? ? ? / o _. _,. _, _,. _, _,. _, _, _,. JO.O0 0 02 0.04 0 06 0 . 0 8 0 tO 0 . 1 2 0 . 1 4 0 16 INVERSE TIME. i / t ; 1 0 2 4 p a n e l s , A t = 0 . 0 7 5 , ~ _ ~ I~L~ ~Ltl ~LT ~Lb ~L3 i I 0 . 1 0 0 . 2 0 ; 4608 panels, A t = O. 05, c } .6J o (:~ o-4 F - c.J t l d t l ] Z C } tad o U J t . q . J o ~0.00 2.00 4.00 6.00 8.00 lO.O0 12.00 14.00 16.00 10.00 20.00 TIME. t

Figure 10. Heaving hemisphere; superposition o f numerical results f o r 100 cosine distributed panels, A t = 0.2 and the ZL9 asymptotic result

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The large-time asymptotic expansion: F. T. Korsmeyer and P. D. Sclavounos

APPLICATION OF THE LARGE-TIME

A S Y M P T O T I C S

Equation (75), when combined with the low-frequency asymptotic coefficients for the damping function o f the heaving hemisphere provided in equation (74), results in the specific asymptotic expression for the impulse- response function o f the heaving hemisphere at large time.

Let 1,,, have a time-domain definition which is analogous to that of b,m in equation (30), and consider the additional definitions

L , ( t ) ₌t-" M~,) _{l,,, log m t,} ₍₇₆₎

m = 0

where M ' ( n ) = ( n - 1 ) ] 4 - 1 / 2 when ( n - 1)/2 is odd, ( n - 1 ) / 4 - 1 when ( n - 1)/2 is even, and M ' = 0 when n = 3, 5; and,

N

ZLN(t) = ~] Ln(t). (77)

n = 0

L , ( t ) is the contribution to L(t) at O(t-") including any logarithmic terms, and

ZLN

is the complete contribution to L(t) up to O(t-N). Figure 9, is a plot of the ZLN for N = 3, 5, 7, 9, 11, along with numerical results found by using 1024 panels with a time step o f 0.075 and 4608 panels with a time step of 0.05. These numerical results have converged to at least four decimals. [Note that the oscillatory behavior which is present for t > 12 is hardly reduced in spite o f the increased effort in computation.] There are two important features o f the asymptotic results. The first is that ZLI~ appears to be inconsistent with the trend in the asymptotic results up to ZL9. This may be the first diverging partial sum in the asymptotic series, but it would be necessary to com- pute at least ZL~3 to confirm this. It is more likely that in the time range before the asymptotic results have sufficiently converged, this trend is misleading. A similar behavior is evident in the frequency-domain damping function (equation (74)) from which these results were transformed [cf. Fig. 2, Reference 5]. The second is that these results also share with their frequency-domain counterparts a rather slow convergence; although the most accurate numerical results and the ZL9 asymptotic result differ in their prediction of L(t) by only 2 x 10 -3 beyond t = 8.0. For computations for any simple body, there will be a time beyond which the asymptotic representation will be a better estimate o f the correct L(t) than the numerical result. The most efficient method for obtaining an accurate impulse-response function, then, is to perform the computation with sufficiently fine spatial and temporal discretization to reduce the oscillatory behavior in the time history until the time when the asymptotic representation may be adopted.

In Fig. 10, this situation is demonstrated. Figure 8 was used as a guide in the selection o f a model which produces reasonable accuracy for moderate computational effort. This model for the heaving hemisphere consists o f 100 panels with a cosine distribution as the free surface is approached, and a time-step o f At = 0.2. In Fig. 10, the numerical results for this model are shown with the asymptotic result to O(t-9). In this case, it is apparent that the numerical effort could have been terminated just beyond t = 8.

To obtain frequency-domain results for this model, the records were patched at t = 8.2 by simply disconti- nuing the numerical results and picking up the asymptotic approximation. The combined record was then Fourier transformed with a truncation point at t = 30, where the asymptotic result, [ ZL9[ < 3 x 10 -3. The results o f the transform, added-mass and damping coefficient functions, are presented in Fig. 11. In this figure, they are plotted for comparison with the numerical results without asymptotic correction, and spherical harmonics results of Hulme. 9 The frequency-domain results from the patched impulse-response function contain no visible irregular behavior, and the overall accuracy o f this representation o f L (t) is sufficient to produce a maximum error in the frequency domain of less than 1.5 percent at the peak o f the damping curve. In Fig. 12, the frequency domain results from a less accurate model (64 regularly spaced panels and At = 0.2) are presented. In this case the irregular behavior is also removed, but the overall accuracy of the patched L(t) is less, leading to less accuracy at the maxima o f the hydrodynamic coefficient curves.

LL U'J ¢.,q ~E 0 0 t~ a o ,4

8 .

Q. ]E Q o o o

~.0

Zo 60 WAVE NUMBER. k

Figure 11. Heaving hemisphere. Transform o f the pat- ched functions o f Fig. 10 (numerical portion f o r 100 cosine distributed panels, At = 0.2) truncated at t = 30, ; the same model without patching truncated at t = 30, . . . . Results from Hulme 9 x . Coefficients are non-dimensionalized by the displaced fluid mass

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The large-time a s y m p t o t i c expansion: F. T. K o r s m e y e r a n d P. D. S c l a v o u n o s o u - b_ to 0 ¢~ o o Z Z o

o

' O . O

~ y

xx WAVE NUMBER. k 6 . 0

Figure 12. H e a v i n g hemisphere. T r a n s f o r m o f the pat- c h e d f u n c t i o n s : n u m e r i c a l p o r t i o n f o r 64 panels, A t = 0.2 t r u n c a t e d at t = 30, _ _ ; the s a m e m o d e l w i t h o u t p a t c h i n g t r u n c a t e d at t = 30, R e s u l t s f r o m H u l m e 9 × . Coefficients are n o n - d i m e n s i o n a l i z e d

b y the displaced f l u i d mass

C O N C L U S I O N S

T h e c h o i c e o f the heave m o d e f o r d e m o n s t r a t i o n o f this m e t h o d was d i c t a t e d b y t h e fact t h a t the r e q u i r e d coefficients were a v a i l a b l e . 5 It is e x p e c t e d t h a t g e n e r a t i o n o f coefficients b y t h e s o l u t i o n o f t h e h i e r a r c h y o f i n t e g r a l e q u a t i o n s will be m o r e u s e f u l in h o r i z o n t a l m o d e s o f m o t i o n . T h e r e a r e t w o r e a s o n s f o r this: o n e , t h e r e is o n l y a l e a d i n g o r d e r t e r m a v a i l a b l e a n a l y t i c a l l y for the d a m p i n g f u n c t i o n in these m o d e s ; a n d t w o , i r r e g u l a r f r e q u e n c y effects are m o r e severe in h o r i z o n t a l m o d e s . T o find the coefficients, t h e c o m p u t e r c o d e n e c e s s a r y for the s o l u t i o n o f t h e r i g i d - l i d p r o b l e m with its r i g h t - h a n d - side f o r c i n g as s h o w n in e q u a t i o n s (23) t h r o u g h (27) c a n be easily a d a p t e d f r o m t y p i c a l f r e q u e n c y - d o m a i n , p a n e l - m e t h o d codes.

It is i m p o r t a n t to n o t e t h a t the s o l u t i o n s so g e n e r a t e d a r e for the e x p a n s i o n o f the p o t e n t i a l itself, a n d it is p o s s i b l e t o t r a n s f o r m t h e p o t e n t i a l , on a p o i n t w i s e basis, to o b t a i n a n e x p a n s i o n f o r the t i m e d o m a i n p o t e n t i a l at large t i m e . This m a y be o f v a l u e in the s e c o n d - o r d e r t r a n s i e n t p r o b l e m where a c c u r a t e k n o w l e d g e o f the f i r s t - o r d e r p o t e n t i a l is n e c e s s a r y f o r a c c u r a t e s e c o n d - o r d e r results. A l i m i t a t i o n o f the m e t h o d is t h a t it a p p a r e n t l y m a y n o t be u s e d to a s y m p t o t i c a l l y c o r r e c t the i m p u l s e - r e s p o n s e f u n c t i o n o f a c o m p l e x s t r u c t u r e . [By ' c o m - p l e x ' we m e a n a s t r u c t u r e w h i c h is c o m p o s e d o f m u l t i p l e large v o l u m e e l e m e n t s . ] I n this case, t h e r e a r e p h y s i c a l r e s o n a n c e s at an infinite discrete set o f f r e q u e n c i e s . In the t i m e d o m a i n , this m e a n s t h a t the i m p u l s e - r e s p o n s e f u n c t i o n will not a p p r o a c h z e r o m o n o t o n i c a l l y at large t i m e . Since the a s y m p t o t i c r e p r e s e n t a t i o n o f the i m p u l s e - r e s p o n s e f u n c t i o n is m o n o t o n i c f o r a n y b o d y , p a t c h i n g o f a n a s y m p t o t i c e x p a n s i o n to a n u m e r i c a l result f o r a c o m p l e x b o d y will r e m o v e b o t h the i r r e g u l a r - f r e q u e n c y effects a n d , to s o m e extent, the p h y s i c a l l y r e l e v a n t r e s o n a n c e s . F o r t u n a t e l y , n u m e r i c a l evidence i n d i c a t e s t h a t i r r e g u l a r - f r e q u e n c y effects are w e a k for these c o m p l e x s t r u c t u r e s . 13

This w o r k suggests t h a t a p r a c t i c a l m e t h o d o l o g y for d e v e l o p i n g the h i g h - f r e q u e n c y e x p a n s i o n s o f f r e q u e n c y - d o m a i n q u a n t i t i e s m a y be to F o u r i e r t r a n s f o r m t h e s m a l l - t i m e e x p a n s i o n o f the i m p u l s e - r e s p o n s e f u n c t i o n . T h i s w o u l d b e o b t a i n e d in a n a n a l o g o u s w a y to t h e a d d e d - m a s s o r d a m p i n g e x p a n s i o n s at low f r e q u e n c y , b y direct e x p a n s i o n o f the G r e e n f o r m u l a t i o n in the t i m e d o m a i n , using a s m a l l - t i m e e x p a n s i o n f o r t h e G r e e n f u n c t i o n .

T h e a u t h o r s wish to a c k n o w l e d g e the financial s u p - p o r t p r o v i d e d b y the N a t i o n a l Science F o u n d a t i o n ( C o n t r a c t M E A 8210640) a n d the Office o f N a v a l R e s e a r c h ( C o n t r a c t N0014-8222-k-0198).

R E F E R E N C E S

1 Newman, J. N. Transient axisymmetric motion of a floating cylinder. J. Fluid Mech. 1985, 157, 17-33

2 Beck, R. F. and Liapis, S. Transient motions of floating bodies at zero speed. J. Ship Research 1987, 31 (3) 164-176

3 Korsmeyer, F. T. On the solution of the radiation problem in the time domain. The First International Workshop on Water Waves and Floating Bodies. 1986, 61-66

4 Adachi, H. and Ohmatsu, S. On the influence of irregular frequencies in the integral equation solutions to the time-dependent free surface problems. Jr. Soc. Nay. Arch. Japan 1979, 146, 127-135

5 Simon, M. J. and Hulme, A. The radiation and scattering of long water-waves. Symposium on Hydrodynamics of Ocean Wave- Energy Utilization of the International Union of Theoretical and Applied Mechanics. 1985, 419-432

6 Korsmeyer, F. T. The first- and second-order transient free- surface wave radiation problems• PhD Thesis, Massachusetts Institute of Technology, Cambridge 1988

7 Wehausen, J. V. The motion of floating bodies• Ann. Rev. Fluid Mech. 1971, 3, 237-268

8 Wehausen, J. V. and Laitone, E. V. Surface waves. Handbuch der Physik. 1960, 9, 446-778

9 Hulme, A. The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. J. Fluid Mech. 1982, 121, 443-463

10 Greenhow, M. High- and low-frequency asymptotic conse- quences of the Kramers-Kronig relations. J. Eng. Math. 1986, 20, 293-306.

11 Olver, F• W. J. Asymptotics and Special Functions• Academic Press, New York, 1974

12 Lighthill, M. J. Fourier Analysis and Generalised Functions. Cambridge University Press, Cambridge, 1958

13 Korsmeyer, F. T., Lee, C.-H., Newman, J. N. and Sclavounos, P. D. The analysis of wave effects on tension-leg platforms. OMAE Conference, Houston 1988