Slowly-varying wave drift forces in short-crested irregular seas

Slowly-varying wave drift forces in short-crested irregular seas M. H. KIM & DICK K. P. YUE Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA A new method for predicting slowly-varying wave drift excitations in multidirectional seas is developed and compared to existing methods based on index and envelope approximations. The present method retains the assumption of narrow bandedness in frequency but treats the wave directional spreading exactly. The method is consequently more applicable for realistic seas but is comparable in computational effort to the other techniques. For the interesting case when storm and swell seas are present from different directions, surprising new results are obtained which indicate that the slowly-varying forces on an axisymmetric body can be greatly amplified when the wave systems are incident from opposing directions. For each of the methods, the slowly-varying force spectra are derived and their statistics confirmed against time-history simulations. For the index and envelope approximations, theoretical probability distributions of the slowly-varying force are reexamined and generalized, and compared satisfactorily to numerically simulated histograms.

1. INTRODUCTION Compliant deep-water structures and moored vessels often have very small restoring forces, and are susceptible to large resonant responses due to higher-order slowly-varying wave drift excitations. There have been many investigations of the second-order slowly-varying phenomena in the past decade j2, but they are mostly limited to unidirectional irregular waves, while studies o f the more realistic short-crested seas are surprisingly rare4,7, t9. A main difficulty o f predicting second-order forces in general is the need to include the contribution of the second-order potential which is computationally difficult to obtain especially for three-dimensional bodies. For slowly-varying excitations, a number of engineering approximations have been proposed which include the index approximation of Newman 2 and the envelope method of Marthinsen 3. These approximations assume that the spectra are narrow banded so that the exact quadratic transfer function (QTF) can be approximated by its monochromatic (mean drift force) value which is given from the first-order potential only t'6. For many applications, the validity o f a narrow-band frequency assumption is confirmed, for example in the numerical work of Faltinsen and Loken ~ for two-dimensional bodies. Several recent experimental and field reports have underscored the importance of wave directional spreading on slowly-varying drift forces and motions. In a series of experiments on the tension-leg platform, Teigen 7 observed considerable reductions of the main

Accepted January 1988. Discussion closes July 1989.

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Applied Ocean Research, 1989, Vol. 11, No. 1

direction drift forces in short-crested waves. Grancini et al. t8 reported severe dynamic responses in the field when their moored tanker encountered storm and swell seas at the same time from different directions. For multidirectional seas which are narrowly spread, the index and envelope approximations can be extended in a straightforward manner 4, although the additional assumption of narrow spreading can often be overly restrictive as pointed out by Marthinsen 4. The fact that the inherent difficulty in solving for the exact QTF is not due to multi-directionality but to multiple frequencies leads us to the present approach, where Newman's narrow frequency band approximation is retained but the directional spreading is treated exactly. This is a useful approximation since in practice wave energies are typically fairly narrow banded and drift response periods very long, while on the other hand wave directional spreadings are often not narrow especially when more than one wave system is present. Thus the present work has a wider range of validity for general short-crested seas, but is otherwise not appreciably different from existing approximations in terms of analytical complexity or computational effort. To provide some understanding of slowly-varying forces in short-crested seas, time-series simulations and spectral analyses of the forces are performed for the present method and for the index and envelope approximations. The statistics obtained from simulations agree well with those from the power spectra. For the probability distribution of the slowly-varying drift force, existing theories for the index and envelope methods are reexamined and generalized. In the index approximation, a remarkably simple closed-form probability density function (PDF) is obtained after taking advantage of the separability of summation expressions,

,- 1989 Computational Mechanics Publications

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue which can be interpreted as a special case of the more general theory of Bedrosian and Rice 13,14. This problem was also investigated in Vinje ~5, which unfortunately contained an important error. For the method of envelope, Langley 16 derived the PDF for long-crested waves, which we extend to multidirectional seas and obtain also the probability distributions of related local variables such as the local amplitude, frequency, wavenumber and direction. All these results are confirmed by histograms obtained from direct numerical simulations of the processes. For illustration, we consider the special case of a uniform vertical circular cylinder in the presence of combined storm and swell seas from different directions. A surprising result is obtained which indicates that the amplitude of the slowly-varying force can be substantially amplified when the wave systems are from opposing directions. This previously unreported phenomenon may be related to the field experience of Grancini et al 8. Definitive experimental investigations are much needed. 2. S L O W L Y - V A R Y I N G DRIFT FORCES

We consider the second-order slowly-varying drift forces on a body in the presence of irregular seas. The linear and second-order hydrodynamic forces on a body due to stationary Gaussian random seas can be in general expressed as a two-term Volterra series 13,14.

FlU) + F2(t) = +

S -co o

h l ( 7 ) ~ ' ( t - r) dr

h2(rl, r2)~'(t - r l ) f ( t - r2) drt dr2

(1) where hi (r) and hz(rl, rE) are respectively the linear and quadratic impulse response functions. For example, h 2 ( t - r ~ , t - r E ) is the second order exciting force at time t due to two unit-amplitude inputs at times rl and r2 respectively, f ( t ) is the ambient wave flee-surface position at some reference point. For unidirectional seas, the surface elevation ~'(t) can be expressed as a sum of frequency components:

~(t) = ~ ai cos(wit + el) = Re ~ Ai e i~'t i

(2)

i

where ai, wi, and ei are the amplitude, frequency, and phase of the i-th wave component, and ei is in general a uniformly-distributed random variable. We can rewrite the second term of (1) in an equivalent form in bifrequency domain: F2(t) = Re )-]~] AiA*Dij e it~°'-~)t i j + Re ~]~-] AiAjSij eit~°~+~j)t (3) i j where ( )* represents complex conjugate of the quantity. Dij - D(wi, wj) and Sij =- D(wi, - wj) are respectively the difference- and sum-frequency quadratic transfer functions (QTF), defined as the double Fourier transform of h2(rl, 7"2):

D(wi, wj) = I ~_~f ~_~ h2(rl, r2) e -i' .... -~'~'-)drl drz

(4)

In this paper, we focus only on the slowly-varying (difference-frequency) part of the second-order force, F(t), represented by the first term of (3): /~(t) = Re ~ AiAffDij e iC~' ,oj)t i j and Dij satisfies the symmetry relation:

(5)

D 0 = 19"

(6)

Note that (6) implies that h2(rl, rz)= h2(r2, rx) which may not be true in general for a quadratic system. However, such a symmetry can always be achieved without loss of generality ~3'14 resulting in a simpler analysis. The time-average of F(t) which gives the mean drift force if, is obtained by setting i = j in (5):

if= ~,, a2Dii = 2 S o S(w)D(w, co) dw i

(7)

0

where S(w) is the (one-sided) wave amplitude spectrum. The exact QTF, Dii, in general depends on quadratic combinations of the first-order potential, and also on the nonlinear potential of the second-order problem. Since the seminal work of Molin 2~ and Lighthil122, a satisfactory treatment of the second-order problem is now available (e.g. Loken 24, Hung & Eatock Taylor23), although the computational effort is still quite substantial and numerical results are limited. For monochromatic incident waves, the differencefrequency component of the second-order force is steady, and the single-frequency QTF, Dii, c a n be obtained in terms of first-order potentials only ~,6. For very low frequency excitations, such as those relevant to the horizontal motions of a moored ship or deep-water compliant platform, this fact can be exploited in a narrow-band approximation z, wherein the bi-frequency QTF, Dii, which depends on the second-order potential is replaced by the single-frequency QTF, Dii, which does not depend on the second-order problem: D0 = D , + 0(wi - w~) so that (5) can be approximated as: / ~ t ) = Re }-]}--] AiA*Dii e i~'°i- ~oj), i

(8)

j

This approximation will be termed "index approximation" hereafter. For narrow-banded wave spectra and/or for slowly-varying excitations due to wave components close to each other in frequency, (8) should be useful provided that the gradient of D o with respect to frequency difference is sufficiently small near the diagonal Dii (Ogilvie ~2). Numerically, the validity in (8) has been supported by Faltinsen & Loken 5 for certain two dimensional bodies, while a detailed validation of three dimensions is still not available. For multidirectional irregular seas, we can write ~'(t) as a double summation with respect to both frequency and incident direction: aik cos(wit+ e i k ) : Re Z Z

~'(t) : Z Z i

k

i

Aik e i~°i' (9)

k

where aik is the amplitude of a wave component of frequency wi and incidence angle/3k, and ei, its uniformly distributed random phase. The difference-frequency drift force in this case is given by: /if(t) = Re Z Z Z Z

Ai*A~Dok, ei~'-~oj)t

(10)

i j k l

Applied Ocean Research, 1989, Vol. 11, No. 1 3

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue where Dokt =--D(wi, wj, 13k,13t) is the bichromaticbidirectional quadratic transfer function, i.e., the (complex) second-order slowly-varying force due to the simultaneous incidence of two unit amplitude regular waves of frequency and direction wi, /3k and wj, /3t respectively. As before, Dijkt satisfies the symmetry relation: Dijkl = D*tk

(1 1)

The mean force with respect to time can be obtained when i = j in (10): i f = Re ~ ] ~ i

aikaitOiikl e i(e~* - e , )

k

(12)

/

where the time average ,ffstill depends on the set of random phases for a specific realization. Upon taking the ensemble average with respect to the phases, we have:

e(;) = ZZ i

oo

.3,

0

If the input spectrum is narrow in directional spreading in addition to narrow-banded in frequency, Newman's index approximation can be extended to the angular spreading and we write: /~(t) = Re Z ~ ] Z Z

A i k A ~ D , k k e i('~' ~')'

2~r

oo

0

0

(16)

i

v S(w)D(w,

~o)

dw

0

An alternative but similar approach to the index approximation is the "envelope method" first suggested by Hsu and Blenkarn 9 who regarded each element of a time series as part of a regular wave so that the slowly-varying drift force could be obtained from successive mean drift calculations within each element. This approach was placed on a somewhat more rigorous basis by Marthinsen 3,4 using the concept of a modulated incident wave. For later reference, the formulation of the envelope method is outlined here. Consider the Hilbert transform pair for the ambient wave:

(cos)

~/(1))

I I 0

R , . --

It(t)

aikOiikk

k 27r

the ratio of (7) and (13):

= ~ ai~sin 3 (wit + el)

(17)

If the input spectrum is narrow-banded, ~'(t) and ~(t) can be rewritten in the form of a slowly-modulated wavetrain:

(14)

i j k l

where Diikk, the QTF for a monochromatic wave with direction Bk, can again be obtained in terms of the firstorder wave-body interaction problem only. Although the assumptions of narrow input frequency band and]or slowly-varying motion responses are usually quite acceptable, the analogous requirement of narrow spreading in incidence direction is often overly restrictive 4, and the approximation clearly fails when one is interested in two or more storms or storm-swell combinations from different directions. A much more reasonable approach is to assume narrow-bandedness for the frequency only but leave the directional spreading arbitrary in (10):

/ ~ t ) -- Re Z Z Z Z

A ik A j t * Oiikl

el(°"- ~j)'

i j k l

05)

This is the basis of our present approximation which has a larger range of validity than (14) for general shortcrested seas, yet the analytical complexity or computational effort required are in fact not appreciably different. This is due to the fact that the major difficulty in calculating the exact QTF in (10) arises from bichromaticity and not the directional spreading. Thus, the monochromatic-bidirectional QTF D.gt can still be evaluated in terms of the first-order potential only. A derivation of D.kl utilizing the far-field approach is given in the Appendix. Dukt can be interpreted as the mean drift force due to an arbitrary combination of two waves of the same frequency from different directions. Equation (13) for the ensemble-averaged mean drift force can be recovered identically from either (14) or (15), and the reduction of the mean drift force in the main direction due to directional spreading is given by

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A p p l i e d Ocean Research, 1989, Vol. 11, No. 1

where a(t) and O(t) are the amplitude and phase of the slowly-varying envelope, and COp the frequency of the carrier wave: aZ(t) = ~'z(t) + rlz(t) (19a,b) O(t) + wpt = tan- ' [~(t)/~(t)] Using the local frequency, WL, defined as the time derivative of the phase: o~L(t) = wp+

d0(t) dt

(20)

the slowly-varying drift force can then be approximated by: l~(t) -- a2(t)D(o~L(t), w,(t))

(21)

Note that since Dii is always positive ~, according to (21), F(t) is also positive definite. If the input directional spectrum is narrow-banded in both frequency and direction, this envelope method idea can be extended directly to multidirectional random seas by considering the Hilbert transform pair: (rt (X, t) =

aik/sin ~ (wit -- k i ' x + elk)

(22)

which can be rewritten in the form: rl (x, t)

I

In this case, the amplitude and phase of the envelope are slowly-varying functions of both time and direction. The local frequency and local wavenumber vector are

Slowly-varying wave drift forces in short-crested irregular seas." M. H. Kim and Dick K. P. Yue defined as: toL(x, t) = top +

00(x, t) 0~

kL(x, t ) = (kL cos /3L, kt. sin /3L)= k p - V 0 ( x , t)

(24) (25)

where the local direction/3L (x, t) is given by: (kp sin/3p - O0/Oy) /3L(x, t) = tan -1 (kp cos ~3p - O0[Ox)

(26)

If we choose x = 0 as the reference point (for the definition of the QTF), the slowly-varying drift force is now given by: l~(t ) = aZ (t )D(coL, COL,/3L,/3L)

(27)

where col and /3L are evaluated at x = 0 from (24) and (26), and F(t) acts instantaneously in the direction/3L. We point out that although in certain applications it may be more convenient to use the local wavenumber kL instead of the local frequency toL in (21) and (27); this cannot be done by direct substitution of the (deterministic) dispersion relation which is no longer valid between the random variables. 3. TIME-SERIES SIMULATION AND SPECTRAL ANALYSIS For a given input amplitude spectrum S(to), a timeseries for the zero-mean Gaussian unidirectional seas can be realized by summing a large number of wave components with random phases*: N

~'(t) = ~ ,2S(toi)Ato cos(tod + ei)

unidirectional seas; or alternatively (10), (14), (15) or (27) for short-crested seas. If direct summation is used (eqs. 5, 8, 10, 14, 15), the QTF is calculated once for all arguments and stored for later times. When the envelope method is employed (eqs. 21,27), however, the QTF need to be calculated at each time instant for the instantaneous local frequency and direction. To avoid possible bias i n / ~ t ) due to particular sets of random phases, several simulations with different sets of random phases are typically made, and their statistics averaged 2o. For the input directional spectrum, we use a PiersonMoskowitz spectrum with cosine-powered directional spreading where the separability with respect to frequency and direction is assumed:

S(to, /3 ) = S(to )S.(/3 ) to5

Cos2n/3

S n ( / 3 ) : Cn

(31)

n ----0, 1,2 .... ;

-~r/2 <~ /3 <<.r/2 Here g is the gravitational acceleration, U the wind speed, and the normalization C, in (31) is chosen so that ISn(/3) d/3 = 1. As n approaches infinity, the unidirectional spectrum S(co) is recovered. For the given directional spreading (31), the spreading reduction factor Rm in (16) can be obtained in closed form for vertically axisymmetric bodies: w/2

Rm = f

Cn

,) -

(28)

i=l

Cos2n+I/3

d/3

7r/2

2

Here N and Ato are the number and interval of frequency division, and es is a random phase uniformly distributed between 0 and 27r. The time series (28) has a periodicity of 2r/Ato, so that a sufficiently small Ato (large N) is necessary for long-time simulations. This can be prohibitive for the direct simulation of b~(t) where the operation count typically increases as N z. Thus, for long simulations, we adopt a modified method1° and write:

g2 e -O'74(g/wU)4

.0081

S(co) -

(32)

(2n!!) z

(2n+ 1)!!(2n- 1)!! where n!! = ( n - 2)!!n and 1!! = 0!! -- 1. The spectrum of the slowly-varying drift force, SF, can be expressed in terms of the wave spectrum for the preceding approximations. We rewrite (5) for F ( t ) in the form: lY(t) = 2 Z Z i j

aiajlDij] cos[ (~i - ~j)t

(i > j )

N

~'(t) = ~ .j2S(cos)Ato cos(colt + cs)

where t0/= cos + 6toi, and 6cos is a random perturbation uniformly distributed between -Ato/2 and Ato/2. Short-crested irregular seas can be simulated in a similar way: N

~'(t) = ~

K

~

,2S(toi,/3k)AtoA/3 COS(to"t + £'ik)

+ ei - ej + ~bii]

(29)

i=l

(30)

i=1 /c=1

where in addition the incident directions are subdivided into K intervals of increment angle At3, and e~, is a random phase uniformly distributed between 0 and 2~r in to - / 3 space. For a given realization of ~'(t), a simulation o f ~ t ) can be obtained from the QTF by evaluating the series (5) or (8), or the expression (21), at each time instant for * As pointed out by Tucker et a # 8, the component amplitude must in general be calculated from the Rayleigh distribution, while (28) relies on the central limit theorem to guarantee the Gaussian property of ~(t) in the limit. In this work, we use the latter for the sake of more direct results such as (34).

(33)

with Dij :

I Dij[ e i¢'j

where the case i = j (i.e. if) is not included here. The autocorrelation function RF(r) is then given by: RF(r) = F ( t ) F ( t + r) =2 ~ a2a2J DijI 2 cos(coi- coj)r i j (i > j)

= 8

S S dtojS(toi)S(wj) [D(toi, toJ)l dtoi

wj

2

o

× cos(toi - ¢oj)r

(34)

After a change of variables (coi-coj=#,coj= co) and using the Wiener-Kinchin relation, we obtain the spectrum of the exact/~(t): SF(#) = 8

S(to)S(to + #) [ D(to, to + lZ)[ 2 dto

(35)

0

A p p l i e d Ocean Research, 1989, Vol. 11, No. 1 5

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue For the index approximation (8), we can follow a similar procedure and obtain the spectrum: SF(II~) ~--- 2

conditions, we write the total spectrum of the mixed seas as a sum of two spectra: S(co,/3) = $1 (co,/3) + S2 (co) a([3 - [30)

S(co)S(co + ~)

(41)

0

× [D(co, co) + D(co + #, co + #)] 2 do)

(36)

In the case o f multMirectional seas, the time-averaged autocorrelation function RF(7) for (10) is still a function o f the phases. Thus if we take ensemble average over the random phases, and using the symmetry relation (11), we obtain the ensemble-mean autocorrelation function: E[ RF(r)] = 2 Z Z Z Z

~.j

where $1 is the spectrum of a short-crested storm sea, and $2 that of a long-crested swell with direction [3o. From (7) and (13), the mean drift force in waves specified by (41) can be obtained by simply superposing each contribution:

a2ka}t l DiJkt I2 cos(co~- coj)r ,~

27r

2rr

0

0

0

× S(co/, [31)1D(coi, coi, & , [3+)12 cos(co~ - coAT (37)

e~

27r

0

0

+2

f

dcoS2(co)D(co, co, [30, [30)

(42)

0

This superposition is, however, no longer valid for the spectrum or the variance of b~(t). Using the spectrum (41) in (38), we obtain the spectrum o f the slowlyvarying force F(t) in the storm-swell irregular seas:

The spectrum of the exact/T(t) is then given by: 0

2~v

2w

0

0

SF(I~) = SF,, (/x) + S ~ ( # ) + SF,~(/z) where

x S(co, [3k)S(co + #, [3t)[D(co, co +/z, fig, [3+)1z (38)

SF.(#) = 8

Corresponding results for the double index approximation (14) as well as our present approximation (15) are respectively: ~o

SF(#) = 2 f

0

2rr

dco f

0

0

d/3lS(co,[3k)S(co+Iz,[31)

0

dco f

0

0

Provided that the wave spectrum is narrow-banded in frequency, the approximation (40) gives reasonable results for small # for all directional spreadings. When one is interested in the response spectra of lightlydamped low-natural-frequency systems, the prediction based on (40) is even more reliable since the transfer function of the system can be expected to filter out the relatively poorly approximated higher-frequency range. 4. APPLICATION TO STORM-SWELL MIXED SEAS

The approximation (15) allows us to study the slowlyvarying forces on a body due to the simultaneous presence of seas from different directions. Grancini et al.8 reported an interesting field observation of the SALS mooring system and a tanker ship installed in the Sicily Channel, where large dynamic roll motions were observed when combined storm and swell seas from different directions were present. To characterize such sea

6

A p p l i e d Ocean Research, 1989, Vol. 11, No. 1

f

d[3tSl(co,/3k)Sl (co + ~, [31) (44a)

dwS2(w)S2(co + #) 0

SF,2(•):8 f 0 dcot"0

(44b)

dfl[Sl(co,[3)S2(w+/x)

X I D(co, co + #, d, [3o)12 + Sl (co + Iz, [3)$2(co) x ] D(co, co + #, rio, [3)] 2] (44c)

d[3lS(co,[3k)S(co+~,[3l)

[ I D(co, co, & , [3t)12 + I D(co + ~, co + #, [3~,&)[ ~ + 2 1D(co, co, & , [3t)l ID(co + ~,co + #,[3. & ) l x cos [ ¢,(co, co, & , fit) + ¢'(co + ~, co + ~, [3. & ) ] i (40)

0

2~r

2-a-

dflk I

d[3k

0

X I D(w, w + #, rio, rio)[ 2

and SF(p.) = 2 f

dco

ex~

SF22(#) = 8

× [D(co, co, [3g, [3k) + D(co +/z, co +/z, [3t ,fit)] 2 (39) 2~r

0

x [ D(co, co +/~, fig, fit)[ 2

27r

d[3kI

(43)

The first and second terms of (43) are respectively the contributions from the storm and swell alone, while the last term represents the additional contribution to the spectrum due to the interaction between the storm and swell. Because of this third term, the variance of A t ) in a storm-swell mixed sea is always greater than that obtained from direct superposition of the individual contributions. If the storm and swell spectra do not overlap and are not close in frequency, (44c) shows that the interaction effect is confined to large # and is therefore relatively unimportant to low-naturalfrequency systems. On the other hand, a change to SF near # = 0 is critical to slowly-varying response. In this sense, usual low-frequency swells are less important than those whose frequency is within the energy band of the storm waves. Confining ourselves to this case, the narrow frequency band approximations of the previous sections can be applied directly. For the double index approximation (39), the interaction term can be written as" oo

SF,2(~) = 2 f

0

2w

rico f

0

d[3 { S l (co + / t , [3)32 (co)

× [D(co,/30) + D(co + #, 13)1 z + Sl (co,/3) Sz(co+lz)[D(co,[3)+ D(co+lz,[3o)] z } (45)

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue

is given by:

while the approximation (40) gives: P

~o

d0

e 27r

I

J0

+ I D(60 + iz, 60 + ~, 3,/3o)1 z + 21D(60, 60, 3o, fl)l × [D(60 + #, 60+ #,/3, 3o)1" cos(¢,(60, 60, 3o, 3) + ~b(60+/z, 60 + ~t, 3, 30))] } (46) Since (46) is not restricted to narrow spreading, it is of interest to investigate the dependence of SF on different incidence directions,/30, of the swell with respect to the storm waves. The results clearly depend on the behaviour of the QTF in /3k-/31 space as well as the shape of the input directional spreading. As will be shown in our numerical results, the interaction effect is sensitive to changes in the direction of the swell, and large amplifications of the slowly-varying force is often possible. This phenomenon has important implications for the operation and safety of moored or dynamicallypositioned vessels in mixed seas.

5. STATISTICS OF S L O W L Y - V A R Y I N G DRIFT FORCES

Index approximation m e t h o d Applying the index approximation, the summations in (8) and (14) become separable and the PDF of b~can be obtained analytically. The following approach was essentially followed in Vinje ~, which unfortunately contains an error in the starting assumption in applying Newman's index approximation to both sum- and difference-frequency terms (his Eqs. 4 & 5) leading to incorrect results. If we define the Hilber transform pairs (x, X ) and

(y, Y):

0

0]

aYZ 0 0 0 o2m

Io

0

(49)

mo~

where S(60) d60

(50)

S ( 6 0 ) D 2 ( 6 0 , 60) do:

(51)

0

and oo

m = E(xy) = E(XY) =

f

S(60)D(60, 60) d60

E(1~) = E ( x y ) + E ( X Y ) = 2m

(53)

and is identical to (7). Noting that x y and X Y are independent random variables whose covariance is zero, the variance o f / ~ i s simply: with

o2

-~-

0"2y = tYxtTy 2 2+ m 2

Z+_ = [X/Ox +-- y / t r y l / 2 ~ +. p )

(55)

where (56)

p = cov(xy)/axOy

is the correlation coefficient of x, y. Defining the nondimensional force f l = xy/axOy, we can express it in terms of z+ and z - : f l = z 2 (p + 1)/2 + z 2_(p - 1)/2

(57)

z 2 / 2 have Gamma distributions whose characteristic functions are given by (1 - iO)- w2. Eq. (57) is a special case of Bedrosian & Rice13's general theory, where the corresponding equation contains an infinite sum of Gamma distribution variables. From the independence of z+ and z - , the characteristic function of f l can be shown to be equal to:

Of,(0) = I[1 - - i ( p + 1)0] [1 - i ( p -

1)0]} -1/2

A similar analysis can also be performed for the random variable f2 - XY/oxay. Using the independence of x y and X Y , we obtain finally the characteristic function of the random variable defined by f = F~oxOy = (xy + X Y ) / O xlTy :

(48)

where x, X, y and Y are zero-mean Gaussian random variables. The covariance matrix of these four variables

-1

(58)

Taking the inverse Fourier transform of (58), we obtain a remarkably simple form for the probability density function p ( f ) which depends on the single parameter p: p(f)=~exp

then (8) can be written in the form:

(54)

This result can also be derived from (36). It is convenient to introduce the normalized Gaussian random variables z+ and z - , which are mutually independent:

Of(O) = [ [ 1 - i(p + 1)0] [ 1 - i(p - DOll

(47)

(52)

0

The mean value of F(t) is then:

o2 = o2+trzr;

In addition to quantities such as mean, variance and frequency spectrum, the probability distribution and in particular the extreme values o f the slowly-varying drift forces are of engineering importance. For general nonlinear Volterra systems, a probability theory was developed in communication theory (e.g., Bedrosian & Ricer3), and was first applied to second-order wave forces by Nea114. In contrast to time-invariant linear systems, the second-order exciting force in a Gaussian sea is in general not a Gaussian process, so that information on the force spectrum alone is of limited usefulness. For the index and envelope approximations, the probability density function (PDF) of F(t) can be obtained in closed form, while for the exact QTF, the PDF must be calculated numerically14.

F(t) = x ( t ) y ( t ) + X ( t ) Y ( t )

m

Cov(x, y, X, Y) = / O

+S,(60, B)$2(60+~)]" [1D(¢o,60,/3o,#)12

IXI= aJI:i°sl(60Jt+es); °'O"I:i°:l

o

- 1

;

f~0

(59)

It is interesting to note that there is a small but nonzero probability of negative f which is confirmed by direct numerical simulation. In the limit of an extremely narrow-banded input spectrum (p ~ 1), (59) is simply

Applied Ocean Research, 1989, Vol. 11, No. 1 7

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue the exponential distribution and f is always positive as is expected for the case of drift force due to a single regular wave1. The parameter p can be obtained from (56), or equivalently from the result of spectral analysis: P = E(F)[20.2 _ Ez (F)]

-1/2

Note that since by definition, [ p [ ~< 1, it follows that the inequality, 0.~ > / E Z ( F ) , is always true. For multidirectional seas, the foregoing analysis can be extended in a straight-forward manner using the Hilbert transform pairs (x, X ) and (y, Y) defined as:

=

aSt~sin~(co_it + es,);

(6o) and identical results are reached upon substituting the following for (50)-(52): oo

27r

0

0

~

27r

0

0

(61)

m = E(xy) = E(XY) 271"

0

0

1

I(~ ~ + ~2)/m,, + c¢~+ ¢)/m~1/2

p(~, r/, ~t, r/t) -- 47rZmomz e (67) Transforming these variables to the set, [a, at, O, Ot}, and integrating with respect to the d u m m y variables a, and 0, we obtain the joint distribution of a and Or: a 2

p (a, Or) -

e- (a2/'"° + a2°~'l"~)12

(68)

mo 2~mz Integration of (68) with respect to Or yields the well known Rayleigh distribution. Upon further transformation of (68) into the variables F and cot, we finally obtain the joint distribution for the slowly-varying drift force and local frequency: (

p(F, cot) = 2.j2~mz moD(coL)t.5 exp [2D- (COL) F × [l/m0 + (col - Ml/Mo)2/m2]]

and

oo

and r/t, the joint distribution can be found easily:

(62)

(69)

An identical formula was also obtained by Langley 16. However, his choice of COp= .~(Mz/Mo), which did not diagonalize (64), led to an incorrect later result for p ( F ) . Integrating (69) with respect to cot (or F ) yields the P D F of F (or cot):

p(F) =

S

p(I~,COL) dcoL

1 p (cot) - 2mo4,m2

(70)

[ l/too + (cot - MllMo)Elm2] -3/2

(71)

Envelope approximation m e t h o d When the envelope approximation is used, the requisite result for the P D F can be obtained using multiple transforms of the local variables. If we redefine the Hilbert transform pair:

I~(x, t)l = a(x, t)lcOSlo(x, t) r/(x, t)

[sin)

(63)

fcos)

= ~i ai~sin~ [ ( w i - cop)t- ( k , - kp)x + el] the covariance matrix of the four slowly-varying Gaussian random variables, ~, r/, ~t and r/t, can be written as: o

cov(~', r/, ~'t, r/~) =

mo

- ml

- ml

m2

0

0

L m~

(64)

The cumulative density function of cot is given by: prob(coL ~< co) = (1 + e~/~,'l + e2)/2;

Co, = 4~mo/m2 (co -- M1/Mo)

kp =

M2/Mog:

m2/

MoO cov(~-, r/, g~x, gr/x) =

(co - cop)"S(co) do:

(65)

The choice of the carrier-wave frequency, coo, is arbitrary at this point, and we can diagonalize the covariance matrix by selecting coo so that the first central moment ml is zero. Thus, we set cop = M~/Mo, where the moment M~ is defined as: t~

w~S(co) do:

(66)

00

Then, from the independence of the variables ~',r/, ~-t

8

Applied Ocean Research, 1989, Vol. 11, No. 1

0 0

0

M, = ~

(72)

From (71) we note that there is a finite probability of negative COL,which is non-physical and for which F -= 0, so that there is an integrable singularity in the probability distribution of F (Eq. 70) at F = 0. Since there is not an explicit relationship between the local frequency and local wavenumber, a direct transformation of (69) or (71) cannot be used to obtain the P D F for kL. If the probability density for kL is desired, it is convenient to start with the variables ~', r/, ~'~ and r/x instead. The covariance matrix can then be diagonalized by selecting

where the n-th central moment m , is defined as:

m, =

where

a 2

=

diag [ M0, Mo, 0"2, 0 2 ]

0 0(73)

where 0.2 =

S

S(co)(co 2 _ co2)2

do:

= M4 -

MZ~/Mo

(74)

o

and deep water is assumed. Note that for the Pierson-Moskowitz spectrum (31), the moment M 4 is unbounded and a suitable spectral cut-off is in general required (see §6). A similar procedure leads to the joint

Slowly-varying wave drift forces in short-crested irregular seas." M. H. Kim and Dick K. P. Yue probability distribution of F a n d kL: P (K k~) :

the variables, F, kL, 13L gives the joint distribution:

g~f 2~2~ moaD(kL) L5

expI~

p(I~ kL, ~L ) =

[l/mo+ g2(kL-M2/gMo)2/a2]]

(75)

g2kLl~ 47rMoolo2D(kL, 13L) I D(kL, 13L)J

f

exp

2D(kL, 13L) [ I/Mo + gZ(kp - kL COS /3L)2/

and the PDF of b=and kL are given respectively by: oz~ + (gkL sin /3L)2/OZ2]] (85)

~o

p ( l ~) = ~ d

p ( l ~ k L ) dkL

(76)

-oo

g

p(kL) = ~

[l/mo + g2(kL - M2lgMo)2l o2] -3/2 (77)

Similar to (70.~, p(b=) in (76) contains an integrable singularity at F = 0. The cumulative density of kL can be obtained from the integration of (77), and is given by (72) with e,~ replaced by c~ defined as: ek = g~Mo ( k - M2/gMo)]o

(78)

For multidirectional seas, the foregoing analysis can be extended by using the Hilbert transform pair:

where F m u s t have the same sign as D(kL, {3z) in (85). The result (85) was also obtained by Vinje 19 via a much more indirect way. Integrating (85) with respect to kL and /3L leads finally to the PDF of F(t): 2~r

P(/~)= I

dkL I 0

d/3Lp(F, kL, f3L)

(86)

0

Note that there is no singularity as F = 0 in the above PDF since there is no finite region in kL -/3L space for which F-= 0. For vertically axisymmetric bodies, the result is simplified:

g2kL

p(ff, kL, ~L) = 47rMoolo2D2(kL)COS 2 [3L ~ (x, t) =

I.sin)

exp[

aig ~sin~ [(ki - kp)" x - (o~i - O~p)t + ei~ ]

2

~

~L~

(87)

(79)

where Q(kL, 13L) is a quadratic polynomial given by:

If we choose the direction and wavenumber of the carrier wave as ~p = 0, and

Q(kL, i3L) = OlO-- a,kL COS t3L + c~2k~ COS213L+ aak z with c~o = MC4,2/q; Ctl = 2gMC.l/q; or2 = g 2 ( M o / q - 1/M4S2); ~3 = g2/MS,2

k . = M c, l / gMo where

and (80) the covariance matrix of the six variables, {~', ~'~, ~'y, r/, fix, fly] can again be diagonalized to yield: (81)

where a z = mC,2 - (MCz~,)2/Mo; o z = MS4,2

(82)

and again deep water and symmetry of the directional spreading are assumed. The joint distribution of these six variables are: 1 -

[(~-2 +

2)/

2340 + g2(~.2 + rtz)/2oz + g2(~.2 + r/yZ)/2oz] }

(83)

Transforming (83) to the variables { a, a~, ay, 0, 0~, 0y 1, and integrating with respect to the variables a~, ay and 0, we have: p ( a , 0~, Oy)

(88)

The joint distribution of k t and /3L can be obtained from (87): p(kL, ~L) = g2kL/TrMooltr2Q 2

cov(~', n,g~x, g~y, gnu, grly) = diag [ Mo, Mo, a 2, 02, cr2 1 , a 2~ 2.1

p = (2r)3MooZto~ exp[

q = MoMC,2 _ (MC.l)2

g2a3 27rMooloz exp{ - 42[ 1/Mo + g202x/O2 + g202/02]/21 =

(89)

Integrating with respect to kL gives the PDF of 13L: p(t3L)--

g2 {2/A + 2 a l A -1'5 COS ~L rMoeltr2

X [7r/2-- tan- l( -- c~l cos ~L/~-A)]I A = 4otoot3+ (4otoot2 -- Ot2)COS2~L

(90)

which is the probability distribution of the direction of the slowly-varying drift forces acting on an axisymmetric body. For numerical integration of (70) and (76), it is convenient to subdivide the domain of integration into three parts, so that for (76) for example, we write: P(,~) =

~+

°+

d k L p ( f , kL)

-=-p , ( F ) + p2(l~) + p3(f') (84)

Integrating (84) with respect to 0~ and Oy yields the same Rayleigh distribution as (68), and transformation into

In the first interval, - ~ ~< kL <<.O, D(kL) is identically zero, and the integral cao be obtained analytically: p l ( F ) = ~olim2-~061 e- ~2Mo~ e r f [ - (M2/aMo) F ~ ]

Applied Ocean Research, 1989, 1Iol. 11, No. 1

9

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue Thus, p~ (b~) behaves like a delta function a t / ~ = 0, and the contribution of p~(F) to the cumulative density of F c a n be obtained from the probability P(kL ~< 0) in (78). In the second interval, 0 <~kL <<.e, D(kc) is typically small and p2 depends on the asymptotic behaviour of D(kL) for small kL. For the uniform vertical cylinder (see Appendix), D(kL) decreases as k ~ f o r kL "~ 1, so that p2(F) has contributions only near F = 0 and decreases exponentially for b~> 0(e 3). The range of p2 can be limited near 0 + by choosing a sufficiently small e, and the cumulative density obtained from P(O <~kL <<.e). The integrand in p3 is regular and the integral is readily obtained by direct quadrature (Romberg quadrature is used in this paper). Similar analyses and numerical procedure are used for (86) and (87), where both the limits kL ~ 0 and /3L~ 7r]2 are treated asymptotically. In this case, negative values of b~are possible when I/3L I >t ~-[2 and there is no singularity at F = 0 since/~is not identically zero in any interval of kL and/3L. 6. NUMERICAL RESULTS A N D DISCUSSION With the preceding formulation, the exact mean and approximate slowly-varying forces and statistics can be obtained for unidirectional and short-crested irregular seas. For simplicity, we consider a vertically axisymmetric body in deep water. Specifically, we choose a uniform vertical cylinder of radius a = 10 m, and wind speed of U = 30 knots in the P i e r s o n - M o s k o w i t z spectrum (31). To ensure the narrow-bandedness of the spectrum 25, the wave energy is assumed to be zero for frequencies co ~< .3 s -~ and co/> 1.3 s -~. In general, the narrow-bandedness can be quantified by the parameter 16 qs2 --- 1 - M~/MoM2, where q2 is equal to

0 for monochromatic seas. For the present truncated spectrum, the value of qs is 0.27, whereas a typical value for a North Sea wave spectrum 16 is qs = 0.3. In this case, the monochromatic bidirectional QTF, Diikt, can be obtained analytically and is presented in the Appendix. Table 1 shows values of Dx(co, co,/3~,/31) for a range of incidence angles /3k and /31, and frequency coZa/g = koa = 0.5. Along the diagonal (/3k =/31), the real part of Dx (or Dy) has cosine (or sine) behaviour, and the imaginary part is zero since the single-wave QTF is real. It is interesting to note that the magnitude of Diikt for different incident angles/3k ~ /31 can be several times greater than that for a narrow directional spreading case. Given two regular waves of the same frequency, the mean drift force on the body is in general a function of the wave amplitude (a~, a2), phases (e~,e2), and incident angles (/31,/32). Fixing the wave frequency at koa = 0.5 and amplitude a~ = a2, we show in Figures la and b the mean drift force in the x and y direction respectively as a function of the difference in phase Ac - e2 - el, for the different incident angles/31 = 0, and/32/7r = 0, 0.25, 0.5, 0.75 and 1. For the main direction steady drift force fix, the maximum amplitude for ~2 = 7r, depending on relative phases, is almost twice as large as that for /~2 = 0. AS expected, the drift force is always positivefor two incident waves in the same direction, whereas Fx is an odd function of Ae for waves in opposing directions. Thus the (phase ensemble-averaged) mean steady force is still largest for t32 = 0. For the transverse drift force, we note another interesting result in that the maxima for any Ae occur when/32 is at an obtuse angle 37r]4 rather than at the normal incidence of 90 ° . These observations are, however, directly dependent on the frequency of the incident waves. This is shown in Figure 2, where the

Table 1. Quadratic transfer function, Diikl = D (w, ~, 13k,~t), f o r the drift f o r c e in the x direction in the presence o f two incident waves, frequency o~2a/g = koa = 0 . 5 , and incidence angles {3k and t3t. The results are normalized by oga. Note that Oiikl = Diitk*. Real(D.k/):

~k/~= 3~]r - 1

- 1

-0.286

-0.75

- 0.75

-0.214 -0.202

- 0.5

- 0.5

0.083 -0.089 0.000

0.25

0.25

0

0.25

0.5

0.000

0.012

0.000

0.012

0.000

0.089

0.083

0.029

0.000

- 0.029

- 0.083

0.202

0.214

0.118

0.029

0.000

- 0.012

0.286

0.214

0.083

0.012

0.000

0.202

0.089

0.000

- 0.012

0.25 0.5

-0.029

0.000

0.75

-0.214

1

0.012

0

0.083

0.75

-0.118

0.286 0.214

- 0.089

- 0.083

-0.202

-0.214

1

0.286

lmag(D.kt):

~/~ ~l]Tr - 1 - 0.75 - 0.5 - 0.25

1 0.000

0.75

0.5

0.25

- 0.032

0.152

0.665

0.962

0.000

- 0.077

0.215

0.665

0.000

- 0.077 0.000

0 0.25 0.5 0.75 1

10

Applied Ocean Research, 1989, Vol. 11, No. 1

0

0.5

0.75

0.665

0.152

0.032

0.000

0.681

0.275

0.000

0.032

0.152

0.275

0.000

0.275

0.152

- 0.032

0.000

- 0.275

- 0.681

- 0.665

0.000

0.25

1

0.032

0.152

- 0.665

0.962

0.000

0.077

- 0.215

- 0.665

0.000

0.077 0.000

-0.152 0.032 0.000

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue ,

....a__.

o

2"

o

7 o

i

1,oo

1.25

1.5o

1,75

;~,0o

AeDr

(b)

-

~

'ooo

~

o'.,~

0'.,o

P," °'"

-7/

o'.,~

,'.~

1'.,~

L~

L,~

~.0.

Aelf

Figure 1. Mean drift f o r c e on a uniform vertical cylinder (radius a) in the presence o f two regular waves, frequency o~2a/ g = koa = 0.5, amplitudes al and az, and phases el and ez. The results are shown f o r (a) xdirection force; and (b) y-direction force; as a function o f the phase difference zl e - el - ez, f o r incident angles 3t = 0 and 3 2 / a - = 0 ( ); 0.25 ( - - - - - - ) ; 0.5

( - - - - - - ) ; 0.75 (-- --); I ( - - - - - ) . •

t~ (.1

long-wave range (koa < - 2 / 3 ) , whereas the opposite is true for shorter waves. These deterministic results have much relevance to the case of multi-directional irregular incident waves as will be discussed later. When the directional spreading is small, the double-index and envelope approximations give reasonable estimates, but fail as the directional spreading increases. Consequently, the interesting dual wave interaction results above are not predicted by these methods. For example, Figure 3 shows the m a x i m u m (over all phase combinations) xdirection drift force as a function of the second incident wave angle 3 2 . Our exact result shows a minimum at 32 ~ 8 0 ° , but a maximum fix of over 1.6 times its value at 32 = 0 when the waves are f r o m opposite directions. As expected, the predictions based on narrow-spreading approximations are poor except for small values of 32. We next consider the time series of the slowly-varying drift force. For these simulations, the input wave spectrum is subdivided into N = K = 25 segments in both the frequency and directional domains. A sampling interval of At = 2 seconds is used which satisfies the Nyquist criterion. First we show the results for unidirectional seas (Figures 4) using the envelope approximation, the index method, and an Inverse Discrete Fast Fourier T r a n s f o r m (IDFFT) method suggested by Oppenheim & Wilson 1°. The method of envelope always gives nonnegative forces, and remains zero whenever the local frequency (or wavenumber) becomes negative. This is a numerical confirmation of t h e int_egrable singularity observed earlier in the P D F of F at F = 0. The index approximation, on the other hand, gives negative values, and although the time history qualitatively resembles that of the envelope method, the amplitudes in general tend to be somewhat smaller. In contrast, the results from I D F F T using the spectrum of F are unacceptable since the second-order force is in fact not a Gaussian process and only the frequency of,Ucan be preserved by i

o

~D.~~' ..~

,

i

i

/

g~

°0.0

0.5

1.0

1.5

%-

2.0

2.5

3.0

Figure 2. M a x i m u m mean drift force (over all possible phase combinations) on a uniform vertical cylinder (radius a) in the presence o f two regular waves, amplitudes al and a2, as a function o f the c o m m o n wavenumber koa. The curves shown are f o r (i) the x-direction f o r c e f o r incidence angles 31 = 0 and 3z = 0 ( ) and 7r ( - - - --); and (ii) the y-direction force f o r 31 0 and 3z = 7r]2 ( - - - - - - ) and 3rc]4 ( - - • --). =

m a x i m u m (over all Ae) of the drift force in the longitudinal and transverse directions respectively for 32 = 0 and ~r, and 3 2 = 71"/2 and 37r/4, are compared over a range of wavenumbers koa. In both cases, the incident waves at obtuse angles have greater m a x i m u m f f i n the

o

•

°0.0

I

0.2

I

t

0.4

0.6

"~...~.,,, I

0.8

,

1.0

P2/,r Figure 3. M a x i m u m x-direction mean drift force (over all possible phase combinations) on a uniform vertical cylinder (radius a) in the presence o f two regular waves, amplitudes al and a2, and wavenumber koa = 0.5, as a function o f incidence angle 3z (3t = 0). Three results obtained using respectively the (i) index approximation ( - - • --); Oi) envelope approximation ( - - - --); and (rio present method ( ) are shown.

Applied Ocean Research, 1989, Vol. 11, No. 1 11

Slowly-varying wave drift forces in short-crested irregular seas." M. H. Kim and Dick K. P. Yue .

.,

,w

.

.

.

.

.

.

(a)

o

.

.

.

.

I

,.

|

i

I

a

I

(b)

"s 7t~ ~

0

I

L'

|

I

,

|

i

I

(c)

"s

(el

o

N

0

•

J

(d)

(d)

"S o X X

0

200

500

800

time (see.)

200

500

8O0

time(sec.

Figure4. Simulated time histories for the case of unidirectional seas incident on a uniform vertical cylinder for (a) the free surface elevation; and slowlyvarying drift force obtained using (b) the envelope approximation; (c) the index approximation; and (d) inverse discrete FFT from power spectrum of the slowlyvarying force.

Figure 5. Simulated time histories for the case of directional seas (cosZ{3 spreading) incident on a uniform vertical cylinder for (a) the free surface elevation; and slowly-varying main direction drift force obtained using (b) the envelope approximation; (c) the double-index approximation; and (d) the present method.

this method ~1. Similar results for short-crested seas with a directional spreading of cos2~ are shown in Figures 5. In this case, the envelope method gives negative values whenever I~LI~ r[2, and as pointed out earlier, has a finite P D F at F = 0. The results from all three approximation methods (envelope, index and the present one) are qualitatively similar, with the present method predicting the smallest amplitudes, which is also indicated in the later spectral analysis results. Using the time history data, the statistics of the slowly-varying drift force can be calculated numerically. This is shown in Table 2 where the results are compared to statistics obtained from the power spectra (Eqs. 36, 39 and 40). Note that since the multidirectional simulations are in general not ergodic 2°, eight simulations with different sets of random phases are made in each case, and the ensemble averages are used for these and later results. From Table 2, we see that the statistics from numerical simulations and theoretical predictions are in

good agreement. Envelope approximation overpredicts both the mean and variance of Fx, whereas the present method for short-crested waves is overestimated by the index method for the main direction mean force, but underestimated for the transverse mean force. The spectrum of the slowly-varying drift force in a multidirectional sea can be obtained in terms of the wave spectrum from (39) and (40) for the index and present approximation respectively. These are plotted in Figure 6 for directional spreadjngs of COS2~and c0s8/3. As the waves become more short-crested, the longitudinal force results deviates more from the unidirectional force spectrum. In all cases, the present approximation predicts lower main direction but higher transverse direction force amplitudes at all slowlyvarying frequencies #. When the directional spreading of the incoming seas is not small, the index or envelope approximations are no longer valid and the present method must be used.

12 Applied Ocean Research, 1989, Vol. 11, No. 1

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue swell, 3o/7r = 0, 0.25, 0.5, 0.75 and 1 are considered. In Figures 7, we plot the part o f the longitudinal and transverse force spectra due to the interaction o f the s t o r m and swell waves (cf. Eq. 43) for the different swell angles. C o m p a r i n g to Figure 6a, we note that the interaction spectra are typically m u c h greater than that due to the storm waves alone. The large amplification o f the spectrum in Figures 7 for certain obtuse values o f 30 especially for small/z is most noteworthy. For example, in the case o f the x-direction slowly-varying force, the increase in the force magnitude near ~ = 0 due to stormswell interaction can be up to 4 and over 5 times larger

(a)

-<..-~

~.0

.

Oil

.

0"2

.

013

.

.

014

.

0.5

.

0.6

0"7

.

0.8

(b)

m

(a)

\ ~

°0.0

:,

--,

O.~t

0.2

~

.

o'.~

o'.~

; ~---v-~-------,0.3

0.4

0.5

0.5

0.7

0.6

/4

Figure 6. P o w e r spectra o f the slowly-varying drift f o r c e as a function o f the slowly-varying frequency #. Two short-crested seas with directional spreading (a) cosZ~; and (b) cosS3 are considered. The curves shown correspond to results f o r x-direction f o r c e f o r (i) unidirectional seas ( ); (ii) present method ( - - - - ---); (iiO index approximation ( - - - m ) ; and f o r ydirection f o r c e f o r (iv)present method ( - - • - - ) ; and (v) index approximation ( ~ 9.

R

~.o "

.

.

.

.

o'., .

o'.,

o'.,

.

o.~ (b)

o~ "~.

To illustrate this, we consider the important case o f the simultaneous presence o f storm and swell seas f r o m different directions. For definiteness, the storm sea is assumed to be given by (31) (mo = 1.55 m 2) with a cos2/3 directional spreading a b o u t x = 0, and the swell is approximated as a long-crested monochromatic waves o f frequency oJo = 0.6 rad/s and amplitude ao = ~'~mo, so that the storm and swell overlap in wave frequency and have the same total energy. Eq. (41) for the swell spect r u m is now simply:

Sz(¢o) = (aoZ/2) 6(¢o - ¢0o)

/

°'.~

°1~.0

O.l

0.2

0.3

0.4

0.5

0.6

0.7

P

Figure 7. Interaction component o f the p o w e r spectrum o f the slowly-varying drift f o r c e due to the presence o f combined storm and swell seas as a function o f the slowly-varying frequency #. The results are obtained using the present method f o r (a) the x-direction force; and (b) the y-direction f o r c e f o r storm wave main direction 31---O, and swell incident angle o f 32/7r = 0 ( ); 0.25 ( - .); 0.5 ( - - - - - ) ; O. 75 ( - - • - - ) ; 1 (-----).

(91)

In this case, our a p p r o x i m a t i o n (46) reduces to a single integral with respect to 3. Five incidence angles o f the

Table 2. Mean, E(ff), and standard deviation, a, of slowly-varying drift forces obtained from time simulations and from theoreticalpower spectra. All values are normalized by pga and given in units of m 2. Resultsfrom power spectra are in brackets ( [... ] ). Unidirectional seas E(ff) envelope approximation index approximation

oF

0.942 0.803 [0.804]

1.167 0.961 [0.932]

cosZ3 directional spread seas E(ff,) envelope approximation index approximation present method

E(ffy)

aFx

ory

0.749

- 0.012

0.902

0.388

0.666 [0.683]

-0.014 [0]

0.776 [0.801]

0.375 [0.369]

0.680 [0.683]

-0.003 [0[

0.696 [0.746]

0.444 [0.422]

Applied Ocean Research, 1989, Vol. 11, No. 1

13

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue for the case when the swell seas are incident at 135 ° and 180 ° to the main direction of the storm waves than when they are arriving from the same direction. For the transverse slowly-varying force, we again observe the interesting result that the interaction contribution is actually larger for a 135 ° swell angle than one at 90 ° to the main storm direction. These observations are also confirmed by direct simulations of (15). We remark that although the variance of the slowly-varying forces due to storm-swell interactions are greater for certain opposing swell angles, the net mean drift forces are always greatest for the case of /30 = 0 and /30 = 7r/2 for the longitudinal and transverse directions respectively. These results have important implications for ocean operations under storm and swell conditions such as those reported by Grancini et al 8. Although the results of Figures 7 are anticipated from our earlier deterministic calculations, we note that existing approximations such as the index method are incapable of making such predictions because of the narrow directional spreading assumptions. Thus a direct calculation of the storm-swell interaction effect based on the double-index approximation (45) leads to qualitatively incorrect results except for small values of /30 (see Figure 8). We now turn to the probability distribution of the slowly-varying forces in unidirectional and short-crested seas. Figure 9 shows the index approximation P D F and CDF for the main direction drift force Fx for three different directional spreadings. For the short-crested waves used, the probability densities are qualitatively similar and tend towards the long-crested result as the spreading is decreased. As expected, the probability of extreme values o f / ~ are higher for smaller directional spreadings. As pointed earlier, there is a small probability for the drift force to be negative. These theoretical P D F ' s are also confirmed by direct numerical simulations of the time-varying drift force. This is shown in Figures 10 where there is good comparison between simulated histograms and the P D F ' s . The results using the envelope method are likewise obtained. For unidirectional seas (Figures 11), there is a

finite probability for negative values of the local frequency WL or wavenumber kL, which results in an integrable singularity in p ( F ) at F = 0. The histograms obtained from simulations are also shown in Figures 11, and the comparisons are satisfactory for all three local

-

o

z~ N

o

o.

~-t.0

0.0

t.0

2.0

3.0

4t.O

5.0

Figure 9. Probability density function and cumulative density function o f the main direction slowly-varying drift force o f the index approximation method. The results are f o r (a) long-crested waves ( ); (b) cos2/3 spread directional seas (-- •--); and (c) cos8/3 spread directional seas (-- - --). oJ

(a)

p~

2

°-t.0

0.0

t.0

2.0

4.0

3.0

~=lpg.

5.0

oa

(b)

_. o L _ j

..-

r I

/

//

°1~ / °0.0

"%.

/

.~" O.l

0.3

0.4

0.5

0.6

I 0.7

P

Figure 8. Interaction component o f the power spectrum o f the x-direction slowly-varying drift force due to ~the presence o f combined storm and swell seas as a function o f the slowly-varying frequency It. The results are obtained by the double-index approximation f o r the xdirection force f o r storm wave main direction of/31 = O, and swell incident angle o f /32/7f = 0 ( ); 0.25 (. . . . ); 0.5 ( - - - - - ) ; O. 75 (-- . . . . ); 1 (-- - - - ) .

14

to

c;

1

~-'-<--_ " ~ - " ~ . ~ _ 0.2

A

=m~x

Applied Ocean Research, 1989, Vol. 11, No. 1

o °-l.

j 0

,,,

0.0

1.0

2.0

3.0

i

4.0

5.0

~xlPg. Figure 10. Comparisons between the theoretical probability density function and that obtained f r o m numerical simulation o f the main direction slowlyvarying drift force using the index approximation method. The results are f o r (a) long-crested waves; and (b) cos2/3 spread directional seas.

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue variables (amplitude, frequency and wavenumber). The PDF and CDF of the slowly-varying force for different directional spreadings are plotted in Figure 12. The probability of extreme values are generally somewhat higher than those predicted by the index approximation. (For example, for cos2/3 seas, the probability P(F~> 4) is 0.012 for the envelope method but only 0.006 for the double-index approximation). For unidirectional waves ~ is always positive, while for short-crested seas,

•

o

"

o

m.

I ,

o

(a) °-i.O

o 0.8

t.6

2.4 a

3,2

4.0

4.8

o

o (b) o

Z. o

o

°.-0.t4

1.0

2.0

'~xlpga

3.0

4.0

5.0

Figure 12. Probability density function and cumulative density function o f the main direction slowly-varying drift force o f the envelope approximation method. The results are f o r (a) long-crested waves ( ); (b) cos2~ spread directional seas ( - - . - - . - - ) ; and (c) cosS~ spread directional seas ( ).

(xl o

°0.0

0.0

-0.07

0.00

01.07

O. 14

0'.21

0.28

%

(c)

the probability of negative force is nonzero corresponding to the situation where the absolute value of the local direction is greater than ~r/2. The PDF for ~ for uni- and multi-directional seas, and for the local direction /3L, i.e. the instantaneous direction of the drift force in short-crested waves, are compared to simulated histograms for the envelope method in Figures 13. The comparisons, including the prediction of negative values in directional seas, are quite reasonable. Although the theoretical methods 14 for unidirectional waves may still be useful, a statistical theory for secondorder forces in general directional seas has yet to be developed and is a subject of current research. In this paper, we show only comparisons of the theoretical PDF's o f / ~ obtained from the envelope and doubleindex approximations which assume narrow directional spreading to the simulated histograms using the present arbitrary-spreading approximation. This is shown in Figure 14 for the case of a cos2~ spreading. It appears that the envelope method overpredicts the probability near the peak at 0, but underestimates the probability of negative values. Overall, the histogram from the present approximation is closer to and compares fairly well to the double-index result. 7. S U M M A R Y

o

~0.4

0,0

0.4

I 0.8

t.2

t.fi

2.0

eL

Figure 11. Comparisons between the theoretical probability density function and that obtained from numerical simulation o f local random variables o f the envelope approximation method in unidirectional seas. The results are f o r (a) local amplitude; (b) local wavenumber; and (c) local frequency.

A new method for the calculation of slowly-varying wave drift forces in short-crested irregular seas is presented and compared with existing theories based upon envelope and index approximations. These methods assume both a narrow band in the frequency of the waves and a narrow spreading in wave directionality. The present method retains Newman's narrowband assumption of the wave frequency, but allows for arbitrary directional spreading which is treated exactly. For typical short-crested storm waves with cos2n/3 spreadings, the present theory predicts respectively lower and higher amplitudes for the main and transverse direction slowly-varying forces. For wide directionally spread waves, such as in the important case of the

Applied Ocean Research, 1989, Vol. 11, No. 1 15

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue o

r

uP

(a)

~'x c o v o

o

o

0.0

° - t .0

t.0

2.0

3.0

4.0

5.0

~1.0

0.0

co

z~x o

o

I t.O

0.0

I 2.0

3.0

41.0

5.0

Figure 14. Probability density function of the main direction slowly-varying drift force in c0s2{3 spread directional seas. The histogram obtained from numerical simulation of the present approximation is compared to the theoretical distributions of (i) the index approximation ( ); and (ii) the envelope approximation (-- • -- • --).

(b)

-t

I 2.0

~xlPS,

~xlPg a

o °-t.0

i t.O

-4.0

3.0

5.0

"FxlPSa g (c)

storm wave direction, the largest amplitude is reached not when the swell is incident at 90 ° but when the swell is coming from an obtuse angle. For the probability distribution of second-order slowly-varying forces in unidirectional and short-crested seas, existing results 15,16,19 for the index and envelope approximations are reexamined and in several cases corrected and generalized. These theoretical probability densities are shown to compare well with numerically simulated histograms. For general wave frequencies and directions, a complete analysis will require not only the exact bifrequency bidirectional quadratic transfer functions (QTF), but also a probabilistic theory for these processes. Research in both these directions are ongoing. ACKN O W L E D G E M E N TS

°-1.5O

-0.75

-0.5O

-0.~

0.00

0.25

0.50

0.75

1.00

flL/It

Figure 13. Comparisons between the theoretical probability density function and that obtained from numerical simulation o f the slowly-varying drift force using the envelope approximation method. The results are for the force magnitude for (a) long-crested waves; (b) cosZ{3 spread directional seas; and (c) the direction o f the force in cos2~ spread directional seas.

This research was supported by the National Science Foundation and the Office of Naval Research• DKPY also acknowledges partial support from the Henry L. Doherty Chair• REFERENCES 1 2

3 4

simultaneous presence of both storm and swell seas from different directions, the existing approximations are invalid and the present approach must be used. For the examples we consider, surprising results are found which indicate that the slowly-varying forces can be several times larger in the main direction when the storm and swell are incident from opposite directions than when they are from the same direction. Similarly, for the slowly-varying drift force transverse to the main

16 Applied Ocean Research, 1989, Vol. 11, No. 1

5 6

7 8

Newman, J. N. The drift force and moment on ships in waves, Journal o f Ship Research, 1967, 11, 51. Newman, J. N. Second order slowly varying forces on vessels in irregular waves, Symp. on Dynamics o f marine vehicles and structures in waves, London, 1974. Marthinsen, T. Calculation of slowly varying drift forces, Applied Ocean Research, 1983, 5, 141. Marthinsen, T. The effect of short crested seas on second order forces and motions, International workshop on ship and platform motions, Berkeley, 1983. Faltinsen, O. M. & Loken, A. E. Slow drift oscillations of a ship in irregular waves, Applied Ocean Research, 1978, 1, 21. Pinkster, J. A. Low frequency second order wave forces on vessels moored at sea, Proc. o f the l l t h Symp. on Naval Hydrodynamics, 1976. Teigen, P. S. The response of a TLP in short crested waves, Offshore Technology Conference, 1983, No. 4642. Grancini, G., Iovenini, L. M. & Pastore, P. Moored tanker behavior in crossed seas: Field experiences and model tests,

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue Symp. on Description and Modeling of Directional Seas, 9

10

11

12

13

14 15

16 17 18

19 20 21 22 23

24

25

Copenhagen, 1984. Hsu, F. H. & Blenkarn, K. A. Analysis of peak mooring forces caused by slow vessel drift oscillations in r a n d o m seas, Offshore Technology Conference, 1970, No. 1159. Oppenheim, B. W. & Wilson, P. A. Continuous digital simulation of the second order slowly varying wave drift force, Journal of Ship Research, 1980, 24, 181. Kaplan, P. C o m m e n t on Oppenheim and Wilson: 'Continuous digital simulation of the second order slowly varying wave drift force', Journal of Ship Research, 1982, 26, 36. Ogilvie, T. F. Second order hydrodynamic effects on ocean platforms, International Workshop on Ship and Platform Motions, Berkeley, 1983. Bedrosian, E. & Rice, S. O. The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs, Proc. oflEEE, 1971, 59, 1688. Neal, E. Second order hydrodynamic forces due to stochastic excitation, Proc. lOth Symp. on Naval Hydrodynamics, 1974. Vinje, T. On the statistical distribution of second order forces and motions, International Shipbuilding Progress, 1983, 30, No. 343. Langley, R. S. The statistics of second order wave forces, Applied Ocean Research, 1984, 6, 182. Gradshteyn, I. S. & Ryzhik, 1. M. Tables of integrals, series and products, Academic Press, 1980. Tucker, M. J., Challenor, P. G. & Carter, D. J. T. Numerical simulation of a random sea: a c o m m o n error and its effect upon wave group statistics, Applied Ocean Research, 1984, 6, 118. Vinje, T. O n the statistical distribution of second order forces, VERITEC report, 1985. Jefferys, E. R. Directional seas should be ergodic, Applied Ocean Research, 1987, 9, 186. Molin, B. Second order diffraction loads upon three dimensional bodies, Applied Ocean Research, 1979, 1, 197. Lighthill, M. J. Waves and hydrodynamic loading, Proc. BOSS "79, London, 1979. Hung, S. M. & Eatock Taylor, R. Second order time harmonic forces on bodies in waves, Second International Workshop on Water Waves and Floating Bodies, Bristol, 1987. Loken, A. E. Three dimensional second order hydrodynamic effects on ocean structures in waves, Norwegian Institute of Technology, report #UR-86-54, Trondheim, Norway, 1986. Longuet-Higgins, M. S. Statistical properties of wave groups in a random sea state, Phil. Trans. Lond., 1984, A312, 219.

APPENDIX. DERIVATION OF THE MONOCHROMATIC-BIDIRECTIONAL

by: =

P

l l a4) 0~* r 2 (90 OO

x

2~ d0rlC°S ~)

a4) a4)* k24)4)*] Or Or

2~

+ fo d0Isin(.cosO] [~r 00-~ + 0--0

z=o

04)

where

(A4)

4) = 4)t+ 4)0, and G(koh) -- tanh(koh) + koh sechZkoh, is a depth factor which goes to unity as koh--* oo. Substituting (A1, A2) into (A4), and using the method of stationary phase for the resulting integral, we obtain the drift force QTF: Dx.,] _ pgkoG(koh) f 2~ Jo Kk(r + O)Kt(Tr + O) Dy,,~ [COS 0~ ipg G(koh)[Kk(Tr + t3t) X (.sin 0) dO---~f c ° s / 3 t l - KT(Tr + 3~)Ic°s 13k)] x (.sin/3t) (.sin /3~

(A5)

which satisfies the symmetry relationship Dkl = D~k. The QTF is related to the mean drift force by: 2

2

Fx,y "= Z Z AkA,*Dx,y,, k=l l=1

(A6)

For vertically axisymmetric body geometries, the Kochin functions Kk need to be calculated only for one incident wave angle, since K~(O) = Kt(O + 13t- 13k), and the computational effort is greatly reduced. In the special case of a uniform vertical cylinder (radius a), the total potential 4)j and hence the QTF can be expressed in closed form:

4)j=_ igAj f(z) ~ eni" 02

QFT

n=O

D(~0, co, fil,, fit). The QTF, Diikl, for a general body in arbitrary water depth for monochromatic bidirectional dual waves is derived using the far-field method t. In the presence of two incident waves, wavenumber ko, and incident angles /3k and/31, the far-field asymptotic forms of the incident (4)1) and diffracted (4)D) potentials can be written as: 4)t - - (ig/w )f(z ) [ Ake ik°r cos(0- &) + Ateikor cos/0- ~,) (A1)

4)D -- - (igloo)f(z)k~/2 7rr X [AkKk(Tr + 0) + AtKt(rc + 0)] e i(k°r+ ~14) (A2) for kor -> 1, and f(z) -- cosh ko(z + h)]cosh koh. Here, Ak, At are the complex amplitudes of the incident waves, and Kk, Kt the Kochin functions defined by: Kj(O) =

dS(-~--nn

4)Oj

f(z)eiko(x cos O+ y sin O)

(A3) j = k, l, where 4)hi is the diffracted potential associated with the jth incident wave alone. Using momentum conservation for the fluid volume, the mean force on the body can be expressed in terms a far-field integral given

x [ J"(k°r)

J'(k°a) H.(kor) ] cos n(O - ~j) H~(koa)

(A7)

where J., 11. are Bessel and Hankel functions of the first kind, primes denote derivatives with respect to argument, and e0 = 1, e. = 2 for n/> 1. Substituting 4)oj in (A7) into (A3), the Kochin function can be evaluated to be: 2i Kj(Tr + 0) = ~ . =0 e. cos n(O-13j)J'(koa)]H'(koa)

(A8) using (A8) in (A5), we have finally:

[D;:,,]_pgaG(kOh)koa tanh k o h , ~=o [ 2 cos n(13k-&, x ( f)C °,S ~/Q T). ' ( ks° a )i + (nc ° s(.sin / f l ~3l T *k ( k ° a ) _

fcos[(n + 1)13k-:~t]lR.(koa ) (.sin [ (n + l)13k -- m J)

_

fcos [(n+ l)fl,-nflk]l R., (ko a) ] (.sin [ (n + l)/3t - n/3k]

(A9)

Applied Ocean Research, 1989, Vol. 11, No. 1 17

Slowly-varying wave drift forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue

where the functions Tn and Rn are defined by: ×[2Real{T.(koa)}-ReallRn(koa)]] T.(koa) = J~(koa)]H'*(koa) R.(koa) = J'+ 1(koa)J~ (koa)/H~+ ~(koa)H" *(koa)

(A10) In the special case of a single incident wave (/3k =/3l =/3), the single frequency and direction QTF, D(w, oJ,/5,/3), reduces to the familiar result:

[

D~,~]_ 2pgaG(koh) Icos/3~ ~] Dyk, j koa tanh koh [.sin 13) .=o

18 Applied Ocean Research, 1989, Iiol. 11, No. 1

(All) which has the asymptotic value of (213)pga{ cos/3, sin/31 in the limit of short waves (koa, koh ~ oo); and the longwave (koa, koh ~ O) asymptote of: Dx**] 57r (COS Dyk~)----8-Oga(k°a)3 ~.sin ~]

(A12)