Slowly-varying wave drift forces in short crested irregular seas

r

La

ì.

Scheephouwkun

Technische Hogeschoo

Deth'

Slowly-varying wave drift forces

in short-crested

irregular seas

_U4GEKOMEN

M. H. KIM & DICK K. P. YUE

65 JUNI 989

Department of Ocean Engineering, Massachusetts institute of Technology. Beantw. Çambridge, Massachusetts, USA

A new method for predicting slowly-varying wave drift excitations in multidirectional seas is

developed and compared to existing methods based on i?dex 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 theother 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

con-lirmed against ime-history simulations. For the index and envelope approximations. theoret-kai probability distributions of the slowly-varyingforce are reexamined and generalized, and compared satisfactorily to numerically simulated histograms.

I. INTRODUCTION

Compliant deep-water structures and moored vessels

often have very small restoring forces, and are suscep-tible to large resonant responses due to higher-ordçr slowly-varying wave drift excitations. There have been

many investigations of thesecond-order slowly-varying phenomena in the past decade'2, but they are mostly limited to unidirectional irregular waves, while studies

of the more realistic short-crested seas are surprisingly

rare47'9.

A main difficulty Of predicting second-order forces in

general is the need to include the contribution of the second-order potential which is computationally dif-ficult 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 Newman2 and the envelope method of Marthinsen3. 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'.6. For many applications, the validity of a narrow-band frequency assumption is confirmed, for example in the numerical work of Faltinsen and Loken5 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,

Teigen7 observed considerable reductions of the main

Accepted January 1988. Discussion closes July 1989.

direction drift forces in short-crested waves. Gräncini et al. 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 manner4, although the additional assumption of narrow spreading can often be overly

restrictive as pointed out by Marthinsen4.

The fact that the inherent difficulty in solvingfor 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 approxima-(ion is retained but the directional spreading istreated 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 validityfor general short-crested seas, but is otherwise not

appreci-ably different from existing approximations in termsof 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 approx imations. The statistics obtained from simulations agree

well with those from the power spectra. For the pro-bability distribution of the slowly-varying drift force,

existing theories for the index and envelope methods are

reexamined and generalized. In the index approxima-lion, a remarkably simple closed-form probability density function (PDF) is obtained after taking advan-tage of the separability of summation expressions.

which can be interpreted as a special case of the more general theory of Bedrosian and Rice'34. This problem

was also investigated in Vinje'5, which unfortunately

contained an important error. For the method of

envelope, Langley ' 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 con-lirmed 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 swéll seas from different direc-tions. 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 previpusly unreported

phenomenon may be related to the field experience of Grancini et al8. Definitive experimental investigations

ar' much needed.

2. SLOWLY-VARYING DRIFT FORCES

We consider the second-order slowly-varying drift

torces 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''4:

F(r)+ F2(t)=

h1(r)(tr)dr

+ r /12(71, rz)T(1 - - T2) dri dr2

(I)

where /i1(r) and h2(ri, r2)are respectively the linear and

quadratic rmpulse response functions. FOr example, /72(1 - Ti, t - T) is the second order exciting force at time t due to two unit-amplitude inputs at times Ti and

72 respectively. (t) is the ambient wave free-surface

osition at some reference point.

For unidirectional seas, the surface elevation (1) can e expressed as a sum of frequency components:

(1) E a, cos(w( + e,) = Re E A, e'" (2)

where a, w,, and e are the amplitude, frequency, and phase of the i-tb wave component, and e is in general

a uniformly-distributed random variable. We can

rewrite the second term of (I) in an equivalent form in hifrequency domain:

F2(i) = Re >iE AjAj*D,j etu1

ii

+ Re EE AAjS,,, e"'t'

ii

(3)

where ( ) represents complex conjugate of the quantity. D(w,, Wj) and Sj D(w,, -w,,)are respectively the difference- and sum-frequency quadratic transfer

func-t ions (QTF), defined as func-the double Fourier func-transfOrm of h2(ru, 72):

Ill (Ti, r) e -j(.,fl - dr1 di2 (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):

Êt) = Re EE A,Aj*DiJ eit (5)

¡J

and D satisfies the symmetry relatiOn:

D,J=DJ (6)

Note that (6) implies that h2(ri, rl) = h2(r2, r) which

may not be true in general for a quadratic system.: However, such a symmetry can always be achieved

without loss of generality'3"4 resulting in a simpler

analysis. The time-avèrage of F(t) which gives the mean

drift force F, is obtained by setting i =j in (5):

F=

aDii=2Ç

S(w)D(w,w) dw (7)

where"S(w) is the (one-sided) wave amplitude spectrum.

The exact QTF, D, 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 Mohn2' and Lighthill22, a satisfactory treatment of the second-order problem is

now available (e.g. Loken24, Hung & Eatock Taylor23),

although the computational effort is still quite

substan-tial and numerical results are limited. For

monochromatic incident waves, the

difference-frequency component of the second-order force is

steady, and the single-frequency QTF, D,, can be

ob-tained in terms of first-order potentials only .6 Forvery 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 approximation2, wherein the bi-frequency QTF, Thj, which depends on the second-order potential is replaced by the single-frequency QTF, D,, which does not depend on the second-order problem:

D0 = D,, + O(w, - w)

so that (5) can be approximated as:

Fit) = Re EE AA,,*D,j e1'

-ii

(8)

Applied Ocean Research, 1989, Vol. 11, No. 1 3 S/a wlv- varvii.g n'aie clrij'iJòrces in s/i on-crested irregular seas:M. H. Ni,,: and Dick K. P. Yac

This approximation will be termed "index approxima-tion" hereafter. For narrow-banded wave spectra andfor for slowly-varying excitations due to wave

com-ponents close to each other in frequency, (8) should be

useful provided that the gradient of D with respect to diagonal D, (Ogilvie Il) Numerically, the validity in (8) has been supported by Faltinsen & Loken5 for certain

frequency difference is sufficiently small near the

two dimensional bodies, while a detailed validation of three dimensions is still not available.

For m Itidirectional irregular seas, we can write (1)

as a double summation with respect to both frequency and incident direction:

(t) = alk COS(w,t + tik) = Re EE Alk e'' (9)

¡k

ik

where ak is the amplitude of a wave component of

fre-quency wj and incidence angle 13k, and Cik its uniformly

distributed random phase. The difference-frequency drift force in this case is given by:

where DlJk, D(wl,wJ,ßk,ßl) is the bichromatic-bidirectional quadratic transfer function, i.e., the

(com-plex) second-order slowly-varying force due to the

simultaneoUs incidence of two unit amplitude regular

waves of frequency and direction w, ß

and Wj,

respectively. As before, Dk, satisfies the symmetry

relation:

(Il)

The mean force with respect to time can be obtained

when i=j in (10):

F= Re

_EEE

akaj,Djuc! ettA - (12)

1k!

where the time average PstilI depends on the set of ran-dom phases for a specific realization Upon taking the ensemble average with respect to the phases, we have:

E() = >I UkDlIkk

ik

ç

=2 ç21 ÇS(w,ß)D(w,w,13,ß)dwdß

(13)

1f the input spectrum is narrow in directional spreading in addition to narrow-banded in frequency, Nlewman's

index approximation can be extended to the angular

spreading and we write:

f(i)=Re

AikAjDukk e''

'''

jfk!

where Djkk, the QTF for a monochromatic wave with

direction ¡3k, can again be obtained in terms of the

first-order 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 narrowspreading in incidence direction is often overly restrictive4 and the approximation clearly fails when one is interested in two or more storms or storm-swell

combinations from different directions. A muc.h more reasonable approach is to assume narrow-bandedness

for

the frequency only but leave

the directional

spreading arbitrary in (10):

F(t)

Re

jfk!

AjkAjDjjk! e''1'

(15) This is the basis of our present approximation which has

a larger range of validity than (14) for

general short-crested seas, yet the analytical complexity or

computa-tional effort required are in fact not appreciably

dif-fèrent. This is di.ie 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 Duki can still be evaluated in terms of the first-order potential on-ly. A derivation of Djk, utilizing the far-field approach is given in the Appendix. Dik, can be interpreted as the mean drift force due to an arbitrary combination of two waves of the same frequency from difTerentdirections. 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 directioñ due to directional spreading is given by

4 Applied Ocean Research, 1989, Vol. 11, No. ¡

the ratio of (7) and (13):

(21T . S(w, f3)D(w, w, ¡3, ¡3) dw d13 n 0 JO "In = f S(w)D(w,w)dw JO

An alternative but similar approach to the index

approximatiòn is

the "envelope method"

first

sug-gested by Hsu and Blenkarn9 who regarded each ele-ment 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 rigorou.s

basis by Marthinsen 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:

=

_aj{'?}wii

+

e)

(17)

If the input spectrum is narrow-banded, (i) and nU)

can be rewritten in the form of a slowly-modulated

wavetrain:.

=

{}

a '

(14)

a(i) e'"'"

O(i))

where a(i) and O(i) are the amplitude and phaseof the

slowly-varying envelope, and w1, the frequency of the

carrier wave:

a2(1)

.2()

+

(l9a,b) O(i) 1- Wpt =tan

_{' [n(t)I(1)]}

Using the local frequency, w!, defined as the time

derivative of the phase

(20)

the slowly-varying drift force can then be approximated

by:

a2(t)D(wi. (1),WL (t)) (21)

Note that since D is always positive', according to

(21), F(t) is also positive definite.

1f the. input directional spectrum is narrow-banded in

both frequencyanddirection, this envelope method idea

can be extended directly to multidirectional random seas

by considering the Hilbert transform pair:

{: }

=

ai,c{C?}

_{(w,t - k-x + (k)}

which can be rewritten in the fOrm: n(x,1)) = (1m)

fReÌ(()

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

(18) 5/o t'I%'-vcIrvi1:g wave drift fones in sFiori -crested irregular seas:M. H. Kin: and Dick K. P. Yac

Slot'lv-varying wave drift forces in short-crested irregular seas: M. H. Ku,, and Dick K. P. Yue

unidirectional seas; or alternatively (IO), (14),

(l5j or

(27) for short-crested seas. If direct summation is used (24) (e4s. 5, 8, lO, 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 ïo be calculated at each time instant for the instantaneous local frequency and direction. To avbid possible bias in F(t) due to particular sets of randóm phases, several simulations with different sets of ran-dom phases are typically made, and their statistics

averaged20

Forthe input directional spectrum, we use a

Pierson-Moskowitz spectrum with cosine-powered directional spreading where the separability with respect to

fre-quency and direction is assumed: (26)

As pointed oui by Tucker e: al", the componeñt amplitude must in general be calculated from the Rayleigh distribution, while (28) relies on (he centra! limit theorem to guarantee the Gaussian property of (1) in the limit. In this ssork. we use the latter for the sake of more drci results such as (34).

AppliEd Ocean Research, 1989, Vol. 11, No. ¡ 5 S(w,3) = S(w)S,,(fi)

$(w) g2 (31)

S,,(f3) = C,, cos2"ß n = 0, 1,2, ...; - 7r/2

_fi

irf2

Here g is the gravitational acceleration, U the wind

speed, and the normalization C,, in (31) is chosen so that

S(ß) dß = I. As n approaches infinity, the

unidirec-tional spectrum S(w) is recovered.

For the given directional spreading (31), the spreading

reduction factor R,,, in (16) can be obtained in closed form for vertically axisymmetric bodies:

R,,,= C,,cos2"'f3d13

J -r/2

2 (2n!!)2

(32)

r (2n+ l)!!(2n I)!!

where n!!=(n-2)!!n and 1!!=O!!

1.

The spectrum of the slowly-varying drift force, Sp,

can be expressed in terms of the wave spectrum for the preceding approximatiOns. We rewrite (5) for F(t) inthe

form

EE a,a1ID,jlcos[(wjwj)I

('>1)

+CiCj+I,ij]

(33) with

D,j= ID,I e'"

where the case ij (i.e. P) is flot included here. The autocorrelation function Rp(r) is then given by:

RF(T) = F(t)F(t+ r)

=2 EE aaJIDI2 cos(w,w)r

= 8 L dw L dwS(w)S(w1) I D(w:, w)J2

X cos(w - w1)r (34)

After a change of variabi,es (WjWj=IL,Wj=W) and using the WienerKinchin relation, we obtain the spec-trum of the exact F(i):

sF(L)8s(W)s(W+L)ID(W,W+1L)I2 do,

(35) defined as:

WL(X, t) = ci',, + 30(x, 1)

k,. (x, t) = (k1 cos (3L. k,. sin 13L) = k,, -VO(x, t) (25)

where the local direction j3L(x t) is given by: ¡3,(x,t) tan

(k,, cos f3,, - âO/âx)

(k,, sin fi,, - ao/ay)

If we choose x = O as the reference point(for the

defini-tion of the QTF), the slowly-varying drift forceis now

given by:

E(i) = a2 (l)D(WL, fiL, fiL) (27) where and fi,. arc evaluated at x = O from (24) and

(26), and F(t) acts instantaneously in the direction fiL. We point out that although in certain applications it

may be more convenient to use the local wavenumber kL

instead of the local frequency w:. in (21) and (27); this cannot be donc by direct substitution of the

(deter-'iinislic) dispersion relation which is no longer valid

.2tween the random variables.

3. TIME-SERlES SIMULATION AND SPECTRAL

A N A LYS IS

For a given input amplitude spectrum S(w), a

time-.crics for the ¿ero-mean Gaussian unidirectional seas can be realized by summing a large number of wave components with random phases:

= E 2S(w)w

cos(wt + e,) (28)

Here N and A, are the number and intérval of fre-quency division, and e is a random phase uniformly

distributed between O and 27r. The time series (28) has

a periodicity of 2tr/,, so that a sufficiently small w

(large N) is necessary for long-time simulations. This can be prohibitive for the direct simulation of F(i) where the operation count typically increases as N2.

Thus,

for long simulations, we adopt a modified

method'° and write:

(t) = E 12S (Wi )Aw cos(wI ¡ + e) (29)

where üj/ = w + ôw, and ôwj is a random perturbation uniformly distributed between - ¿w/2 and ¿w/2.

Short-crested irregular seas can be simulated in a similar way:

. K

(t) = E

2S(w,,ßk)w iß cos(o,f I + c,) (30)

i=l 11=1

wherein addition the incident directions are subdivided into K intervals of increment angle f3, and (1k isa

ran-dom phase uniformly distributed between O and 2ir in

wf3space.

-For a given realization of (I), a simulation of F(t) can

be obtained from the QTF by evaluating the series (5) or (8), or the expression (21), at each time instant for

!l'iJF)

ing tatc di if! forces in short crested irregular

For the index approximation (8), we can follow a similar procedure and obtain the spectrum:

SF(tL)2Ç S(w)S(w+)

x(D(w,o,)+D(w+,w+/L)i2dw

(36)

In the case of ,nultidireciional seas, the time-averaged

autocorrelation function Rb(T) for (10) is still a function

of the phases. Thus if we take ensemble average over the

random phases, and using the symmetry relation (Il),

we Obtain the ensemble-mean autocorrelation function:

E(Rb(r)j

2 EEE a.&aj,IDuk,j2 cos(w,w)r

i j k I (I>J) ,oe poe p21 8 dw dw d/3k d/3,S(, (3k) WI O O JO x S(1, (3,) I D(w,, O)j, ¡3k, /3,) 2 cos(o,

-The spectrum of the exact i) is then given by:

SF(S)= 8 d dßk dß,

O

.xS(wj3k)S(o,+,L,ß,)ID(o.,w+1L,f3k,í3i)I2

Corresponding results for the double index approxima-tion (14) as well as our present approximaapproxima-tion (15) are

respectively: poe r2T

ri

SF(i)z2

dc dßk\

dß,S(,ßk)S(±1L,ßI)

o o O xLD(o,,w,ßk,/3k)±D(w+,,W+IL,f3i,i3l)]2 (39) and poe r21T pIT SF(c) 2 do, dj3k d/3,S(o,,ßk)S( + ¡,ß,) Jo Jo Jo

+ 2ID(w,w,ßk,ß,)IlD( + i,w +

x cosh&(o,,w,ßk,ß,)+ i,&(w + c,w + L,f3I,ßk)] I (40)

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 lightly-damped 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 slowly-varying forces on a body due to the simultaneous presence of seas from different directions. Grancini ei 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 dif-ferent directions were present. To characterize such sea

6 Applied Ocean Research, 1989, Vol. ¡1, No. ¡

seas: M. H. Ku,, and Dick K. P. Yue

conditions, we write the total spectrum of the mixed

seas as a sum of two spectra:

S(,ß)=S1(,ß)+Sz(o)ô(ßßo)

(41)

where S1 is the spectrum of a short-crested storm sea,

and S2 that of a longcrested swell with direction ¡3e.

From (7) and (13), the mean drift force in

vaves specified by (41) can be obtained by simply superposing each contribution:

poe pZr

E(P)=2

dßS1(,,/3)D(,,/3,ß)

Jo Jo

do,S2(w)D(w, w,f3o, /3o) (42)

This superposition is, however, no longer valid for the spéctrum or the variance of F(i). Using the spectrum (41) in (38), we obtain the spectrum of the slowly-varying force F(i) in the storm-swell irregular seas:

Sb(;L) = S,,(tc) + S(i) + S,u:(it) (43) where poe pl pIT

St,(t)=8

dw d/3k _{dß,S,(w,(3k)St(w+,ß,)} Jo Jo Jo

x ID(w,+,(3k,ß,)I2

(44a)

S,()=

8Ç dwS2wS2( + ti) X ID(o,w+ii,/3o,ßo)I2 (44b) poe 2w S,..(1z)=8 dw

_{dß[Si(w,ß)Si(+ji)}

Jo Jo

X ID(w,w+It,(3,ßo)I2+SI(+ti,ß)S2(w)

X D(,w+ii,i3o,ß)I2]

(44c)

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 F(t) in

a storm-swell mixed sea is always greater than that obtained from direct superposition of the individual contributions. 1f the storm and swell spectra do not

overlap and are not close in frequency, (44c) shows that

the interaction effect is confined to large

i and is

therefore relatively uñi'mportant to lownatural-frequency systems. On the other hand, a change to S,

near i = O 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:

poe

Sp,.(ti)2

d/3Si(w+ti,ß)S2(w)

Jo Jo

x[D(w,ßo)+D(w+ti,/3)]2+Si(w,ß)

S2(w +ti)[D(ca,,f3) + D(co + tzßo)]21 (45)

Sb i'/y- va,yin g wave drift forces in s/ion-crested irregular seas: M. H. Ku,, and Dick K. P. Yac

while the approximatiOn (40) gives:

-

.

r2

-S.()=2

dw _{d131IS,(o+,L,ß)S2(w)} .io Jo +S,(w,ß)S2(w+)j EID(w,c,ßo,13)I2 + I D(c + ¡L,(.) + (3, ßo)l +21D(w, wj3o,ß)I x'ID(w +tL,w+ ,,,i3u)I.cosW'(w,w,13o,ß) (46) Since (46) is not restricted to narrow spreading, it is of interest to investigate the dependence of S,- ondifferent incidence directions, ¡3o, of the swell with respect to the

storm waves, The results clearly depend on the behaviour of the QTF in. 13k - /3, 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 slowlyvarying force is often

possible. This phenomenon has important implications for the operation and safety of moored or dynamically-positioned vessels in mixed seas.

'5. STATISTICS OF SLOWLY-VARYING DRIFT FORCES

In addition to quantities such as mean, variance and

Ire-quency spectrum, the probability distribution and in

particular the extreme values of 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 &

Rice'3), and was first applied to second-order wave forces by Neal

. 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

infor-mation 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 numerically'4.

Index approximation method

Applying the index approximation, the summations

in (8) and (14) become separable and the PDF of Fcan

'essentially followed in Vinje'5, which unfortunately be obtained analytically. The following approach was

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):

{} =

{.} -=

aiDu{ } (ct + ei) then (8) can be .ritten in the form:

(48)

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

where x, X, y and Y are zero-mean Gaussian random

variables. The covariance matrix of these four variables

is given by: ci ni 0 0 'n O O Cov(x, y, X, Y)_{= ni} O

O m

where and

,nE(xy)=E(XY)

S(w)D(w,w) dw (52)

The mean value of f(t) is then:

E(f) = E(xy) + E(XY)

= 2iii (53)

and is identical to (7). Noting that xy and XY are

in-dependent random variables whose covariance is zero, the variance -of F is simply:

a=

+ ay;

with a,.

cdy=.aa-i- in2

(54)

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

2_ 2_

av a y --D

a=a1

S(w)dw

L

(49) ;(50) LS(cz,)D2(w,w) d (SI) z± = ExIa ±

y/av]/.j(I ± p)

where p = cov(xy)/axay (56)

is the correlation coefficient of x, y. Defining the non-dimensional force f, xy/aa,, we can express it in

terms of z+ and z-:

f,=z+(p+

l)/2+z_(p l)/2

(57) z/2 have Gamma distributions whòse characteristic functions are given by (I - i0 "s. Eq. (57) is a special case of Bedrosian & Rice'-3's general theory, where the

corresponding equation contains an infinite sum of Gamma distribution variables. From the independence ofz+ and z-, the characteristic function of f, can be shown to be equal to:

0(0)= ([I i(p+ 1)0] [1 i(p l)0]J_l2

A similar analysis can also be performed fOr the random

variable f2 XYfa1ay. Using the independence of xj' and XY, we obtain finally the characteristic function of the random variable defined byf F7axav=(xy+XY)/

ej(e)= lEI i(p+ l)0J[l i(p-- l)O]r'

(58) Taking the inverse Fourier transform of (58), we obtain

a remarkably simple form for the probability density

(47) function p(f) which depends on the single parameter p:

p(f)!expl

₂

_l+p)

4Ì; jo

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

(55)

(59)

the exponential distribution and fis always positive as is expected for the case of drift force due to a single regular wave'. The paraméter p can be obtained from

(56), or equivalently from the result of spectral analysis:

Note that since by definition, l 1, it follows that the

inequality, c .E2(F), is always true.

For multidirectional seas, the foregoing analysis can be extended in a straight-forward manner using the Hubert transform pairs (x, X) and (y, Y) defined as:

(X

jx

(wit + e1,);

{

= akDukk{.j (wet +

and identicäl results are reached upOn substituting the

following for (50)(52): oo 2i

a=d= Ç

_dwf dßS(w,ß); Jo .lo and

,n= E(xy) = E(XY)

,.cø

hr

= dw dßS(w,ß)D(u,wJ3,ß) Jo Jo

Envelope approximation method

When the envelope approximation is used, the

requi-site result for the PDF can be obtained using multiple transforms of the local variables. If we redefine the Hilbert transform pair:

{-(xt)1

= a(x,l)ÍC0SÌO(x,t)

,i(x,t»

sIn) ₍₆₃₎

cos

=> ai1.J[(wi_WP)i_(ki_kP)X+ti1

the covariance matrix of the four slowly-varying Gaus-sian random variables, , i, , and ,, can be written as:

In0 O O liii'

O i?i

-ini

O

0

lili

1fl2 O

Ini O O ifl

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

(64)

Inn

=

L

- w)"S(w) dw (65)

The choice of the carrier-wave frequency, we,, is

arbit-rary at this point, and we can diagonalize the covariance matrix by selecting w1, so that the first central moment ,n is zero. Thus, we set Wp M,/Mo, where the moment

M,, is defined as:

M,, -w"S(w)dw (66)

Then, from the independence of the variables

, i,

,

8. Applied Ocean Research, 1989, VoI. ¡1, No. ¡

and ,,, the joint distribution can be found easily:

P( ) 2 e

-I (C + l7)frmn +

4r momfl7

(67)

Transforming these variables to the set, a, a,,O,O,I, and integrating with respect to the dummy vari,ables a,

and O, we obtaiñ the joint distribution of a and O,:

-. (68)

p(a, O,) = e -(crfn'i,,

uno,J27rm2

Integration of (68) with respect to O, yields the well known Rayleigh distribution Upon further transforma '60' tion of (68) into the variables Fand wi, we finally obtain

'

_{/ the joint distribution for the slowly-varying drift force}

añd local frequency:

-Ir

cr.

p(F,WL)= - expt

-2J2rm2 ,hoD(w,)'5 (2D(WL)

x [1/uno +-(WL M1/M0)2/Inh)} (69)

Au identical formula was also obtained by Langley

6

However, his choice of Wp = ,(M2/Mo), which did not dia&onalize (64), led to an incorrect later result for p(F). Integrating (69) with respect to (or F) yields the PDF of F (or w): (62) p(F) =

p(Fw,) dw,.

(70) p (wL) I [1/177(1 + (w,. - M ¡Mro)2/,,, j -.I2 2mo.mz (71)

The cumulative density function of wg. is given by:

prob(WL

w) = (1 e,/Ï +)/2; where

- ei,, = no/rn2 (w -M,/M0) (72)

From (71) we note that there is a finite probability of

negative wL, which is non-physical and for Which F O,

so that there is an integrable singularity in the

probabI-ity distributión of F(Eq. 70) at F= O. Since there is not

an explicit relationship between the local frequency and

local wavenumber, a direct transförmation of (69) or (71) cannot be used to obtain the PDF fòrk,. lithe probability density for kL is desired, it is convenient to start with the variables

, i,

and , instead. The covariance matrix can then be diagonalized by selecting

ki = M2/Mog: 1Mo

Io

L cov(r,i, gx, g'7x) = 2 =diag[Mo, O a O Mo,a2,a2] O O O a2 (73) Where =

L

S(w)(w2 - w)2 dw = M4 - Mi/Mo (74)

and deep water is assumed. Note that for the

PiersonMoskowitz spectrum (31), the moment M4 is unbounded and a suitable spectral cutoff is in general

required (see §6). A similar procedure leads to the joint

Sb wly-varying wave

dnf,

forces in s/iou-crested irreg u/ar seas: M. H. Kim and Dick K. P. Yue

000

Mi)

00

= = _dwf dflS(w,ß)D2(w,c',flrß)

(61)

probability distribution of F and k,.:

gF

22,r ,noaD(kd'5

1/lilo+ g2(k,. M2/gMo)u/a2J} (75)

and the PDF of rand k,. are given respective y by:

p(f)=

p(rk,.) dk,.

(76)

p(kL)

_=.;;;

[1/lilo + g2(k,. - M/gMo)2/o21 -3/2

(77)

Similar to (70,, p(P') in (76) contains an integrable singularity at F= 0. The cumulative density of k,. canbe

obtained from the integration of (77), and is given by (72) with e, replaced by ¿'k defined as:

¿'k = g.,Mo (k M2/gMo)/a (78)

P=

2 2 expl

-(2,r) Moo Ici

2Mo + g2(2 + )/2a + g2( +

)/2U

Transforming (83) to the variables [a, a, a,., O, O..,., O),

and integrating with respect to the variables a., a,. and

O, we have: 9273 (83)

i

n n.

°

pia, x,

y3 27rMooIa2.

exp[ -

a2

_{[1/Mo + g2O/cf + g2O/a] /2}

(84)

Integrating (84) with respect to O and O, yieldsthe same

Rayleigh distribution as (68), and transformation into

the variables, Fk,., 13,. gives the joint distribution:

g2k,J

p(F, ,.,ß)

_{4MocD(k 13,)! b(k,.,13,.)I}

exP{-2D(/,'f3,) E 1/Mo + g2(k,, k,. cosß,)2/

of + (gk,. sin 13L)2/of

i]

.85)

where Pmus have the same sign asD(k,.,f3,.) in (85). The result (85) was also obtáined by Vinje'9 via a much

more indirect way. Integrating (85) with respect tó k,.

and (3,. leads finally to the PDF of F(t):

,. ,2

p(F)=

dk:.\ dß,.p(rk,.,,3,.) (86) tJot that there is no inguIarity as

r=

O in the above

PDFsiñce there is no finite region in k,. ßL space for

which

FEO.

For vertically axisymmetric bodies, the result is

the slowly-varying drift forces acting on an

axisym-metric body.

FOr nümerical integration of (70) and (76), it is

conve-nient to subdivide the domain of integration into three parts, so that for (76) for example, we write:

(.0

j.0 j.,O'

p(F)=j

-

+ + I dk,.p(F,k

o

t)

E p (F) + P2 (F) + p (F)

In the first interval, - oe kL 0, D(kL) is identically

zero, and the integral ca be obtained analytically:

Pi(F) = 11m

2M06 erf[ - (M2/aM0)F7öJ

Applied Ocean Research, ¡989, Vol. Ii. No. ¡ 9

I ç (AAÇ\ 'JI

(M))

2ir I I i

= i

d,

.df3S(c,j3)w J(i JO ( J sinß (80) and. q = M0M.2 -

(M.1)2

(88)

the covariance matrix of the six vàriables, , , ,, The joint distribution of k,. and ¡3'. can be obtained

can again be diagonalized to yield: from (87):.

cov(,

g, g)

p(kL, ¡3L) =g2kL/7rMooIa2Q2 (89)

= diag[ Mo, M0, cf,aL cf, aJ (81) Integrating with respect to k,. gives the PDF of f3,:

where

Of= M2 (MÇ1 )2/Mo;

ci

= M2

(82) g

_+2a

cosß,. p(13L)=

[2/a

7rMooia2

x [ir/2

tan(a, cos ß,./&)] I

and again deep water and symmetry of the directional

spreading are assumed. The joint distribution of these 4a0a +(4croa2 - af)cos2i3L (90)

six variables are: which is the probability distribution of the direction of

For ,nziltidirectional seas, the foregoing analysis can simplified: be extended by using

xt)Ì

=a(x,t)1CO51O(X,t)

the Hilbert transform pair:

r

p( ,k,.,f3L)

g2kLF

4irMociia2D2(k,.)cos2 a,.

(7(xt))

sIn) exp- rQ(k,.,ß,.) ₍₈₇₎ l 2D(k,)cos 13,. Qik{c. } i k

[(ke - k)'s -

(ai, p)t + Ciic]

(79) where Q(kL,13,.) is a quadratic polynomial given by: If we choose the direction and wavenumberof the car- Q(kL,13L)= a

crjk,. cos (3,. + a2kj cos2ß,. + a3kì

rier wave as f3,., = 0, and with

kp =

Mi/gMo

cro = M2/q; ai =

2gMi/q;

cr2= g2

(Mo/q - l/M.2);

Cr)=

where

Sia%t'lv-t'a, Ping (t'avedrijiforcesinshori-c'resled irregular seas: M. H. Kim andDick K. P. Yac

Thus, p (P behaves like a delta function at fl= 0, and the contributión of

pi(F)

to the cumulative density of Fean be obtained from the probability P(k 0) in

(78). In the second interval, O kL t, D(k:.) is

typically small and P2 depends on the asymptotic

behaviour of D(kL) for small kL. For the uniform

ver-tical cylinder (see A2pendix), D(kL) decreases as

klfor

k1. 1,so thatp2('F) has contributions only near F=0

and decreases exponentially for F> O(e3). The range of P2 can be limited near 0+ by choosing a sufficiently

small e, and the cumulative density obtained from

P(O kt e). The integrand in p3 is regular and the integral is readily obtained by direct quadrature

(Romberg quadrature is Ì.ised in this paper).

Similar analyses and numerical procedure are used

for (86) and (87), where both the limits kL -.O and

13L -, 42 are treated asymptotically. In this case

negative values of F are possible when I I 42 and

there is no singularity at F= O since Fis not identically

zero in any interval of k,, and fiL.

6. NUMERICAL RESULTS AND I)ISCUSSION

With the preceding formulation, the exact meanand

ap-proximate slowly-varying forces and statistics can be

obtained for unidirectional and short-crested irregular seas. For simplicity. we consider a vertically

axisyni-metric body in deep water. Specifically, we choose a uniform vertical cylinder of radius a = IO m, and wind

speed of U= 30 knots in the Pierson-Moskowitz spec-trum (3l) To ensure the narrow-bandedness of the

spectrum25, the wave energy is assumed to be zero for frequencies w .3 s' and w 1.3 s1. In general, the

narrow-bandedness can be quantified by the

pararneterU'» q

I - M/MoM2, where q

is equal to

Table I. Quadthzic transfer function. = D(w, , ß, e,).for the drift force in the x.direciion in the presence of tieo incident waves. frequen:v w2 og = k1a = 0.5. and incidence angles ß and ß,. Theresults are normalized by pga. Noie

O for monochromatic seas. For the present truncated spectrum, the value of q5 is 0.27, whôreas a typical value

for a North Sea wave spectrum(6 is qs 0.3.

In this casé, the monochromatic bidirectional QTF,

DIk,, can be obtáincd analytically and is presented in the Appendix. Table I shows values of D.r(w, w, 13h, 13i ) for

a range of incideñce angles fl and flu, and frequency 2a/g = koa = 0.5. Along the diagonal (ß = ), the real part of D.,. (or D,.) 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 Dk: fòr different incident angles 13k fl can be several

times greater than that

for a narrow directional

spreading case.

Given twô regular waves of the same frequency, the mean drift force on the body is in general a function of

the wave amplitude (ai, a2), phases (Ci,Ci), and incident angles (131,132). 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 ydirection respectively as

a function of the difference in phase c

c - ti, for

the different incident angles fl = O, and flz/ir = 0, 0.25, 0.5, 0.75 and 1. For the màin direction steady drift

fòrce F.,, the maximum amplitude for ¡3 = ir, depending

on relative phases, is almost twice as large as that for

132 = O. As expected, the drift force is always positive_for two incident waves in the sanie direction, whereas F,. is

an odd function of ¿c for waves inopposing directions. Thus the (phase ensemble-averaged) mean steady force

is still largest for ¡3 = O. For the transverse drift force,

we note another interesting result in that the maxima for any c occur when 132 is at an obtuseangle 344 rather than at the normal incidence of900. These observations

are, however, directly dependent on the frequency of

the incident waves. This is shown in Figure 2, where the

IO Applied Ocean Research, 1989, Vol. ¡1, No ¡

II

-I

-0.75 -0.5 -0.25 0 0.25 0.5 0.75 - i -0.286 -0.214 -0.083 -0.012 0.000 -0.012 -0.083 -0.214 -0.286 - 0.75 -0.202 -0.089 0.000 0.012 0.000 -0.029 -0.118 -0.214 - 0.5 0.000 0.089 0.083 0.029 0.000 - 0.029 - 0.083 ((.25 0.202. 0.214 0.3 1$ 0.029 0.000 -0.0(2 I) 0.1(16 0.234 0.083 0.0(2 0.000 ((.25 0.202 0.0(19 0.000 -0.0(2 0.5 0.75 -0.202 -0.2(4 0.000 -0.089 -0.083 -0.286 lnhtg(D,,A,): thfr = -0.75 -0.5 -0.25 0 0.25 0.5 0.75 - I 0.000 -0.032 0.352 0.665 0.962 0.665 0.352 -0.032 0.0(8) -0.75 0.000 -0.077 0.213 0.663 0.68 I 0.275 0.000 0.032 0.5 0.000 -0.077 0.152. 0.275 0.0(8) -0.275 -0.152 0.25 0.000 -0.032 0.000 -0.275 -0.681 -0.665 (I 0.000 0.032 -0.152 -0.665 -0.962 0.25 0.000 0.077 -0.215 -0.665 0.5 0.000 0.077 -0.152 075 0.000 0.032 0.000

'000 0.23 0.50 0.70 I

(b)

0.00 0.23 0.50 0.73 7.00 7.23 1.00 7.75

Figure I. Mean drift force on a uniform

cylinder (radius a) in the presence of two regular waves,

freque,icy ,2a/g = koc = 0.5, amplitudes ai and a2,and

phases c and C2 The results are shown for (a) » direction force; and (b) y-direction force; as a function

of the phase difference C

- f2, for

incident angles

j3 = O and 132/7r = O (

);

0.25 (- - - -); 0.5

(---1; 0.75 (-

.); J

Slot'l.v-vaiying wave clrïftforces in short-crested irregular seas: M. H. Kini and Dick K. P. Yue

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. Consequent

ly, the interesting dual wave interaction results above

are not predicted by these methods. For example, Figure 3 Shows the maximum (over all phase combinations) x-direction drift force as a function of the second incident

wave angle ¡32. Our exact result shows a minimum 'at

132 800, but a maximum F of over 1.6 times its value at ¡32 = O when the waves are from opposite directions. As expected, the predictions based on narrow-spreading approximations are poor except for small válùes of 132. We next consider the time series of the slowly-varying

drift f»rce. For these simulations, the input wave spec-trum 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

Transform (IDFFT) method suggested by Oppenheim &

Wilson'0. The method of envelope always gives non-negative forces, and remains zero whenever the. local

frequency (Or wavenumber) becomes negative. This is a

numerical confirmation of the integrable singularity observed earlier in the PDF of Fat F= 0. The index ap-proximatión, 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 EDFFT using the spectrum of Fare unacceptable since the second-order force is in fact not a Gaussian

process and only the frequency of Fcan be preserved by

7.25 7.50 1.75 2 00

2 00

vertical

Figure 2. Maximum mean drift force (Over all possible phase combinations) on a uniform vertical, cylinder

(radius, a) in the presence of two regular waves, amplitudes ai and a2, as a function of the common

wavenumber koa. The curves shown are for (i) the

»direction force for incidence angles ¡3, = O and (32 = O

(

) and ir (- - -); and (ii) the y-direction

force

for fi, =Oand132='ir/2 (---) and3ir/4(--).

maximum (over all

¿xc) of the drift force in the

longitudinal and transverse directions respectively for

ßz = O and ir, and ¡12 = ir/2 and 3 ir!4, are compared over a range of wavenumbers koa. In both cases, the incident waves at obtuse angles have greater maximum F in the

Applied Ocean Reseqrch, 1989, Vol. 1/, No. ¡ 11

0.2 0.4. 0.6 0.6 1.0

Figure 3. Maximum x-direction mean drift force (over all possible phase combinations) on a uniform vertical

cylinder (radius a) in the presence of two regular waves,

amplitudes a1 and Qz, and wavenumber koa = 0.5, as a.function of incidence angle ßz (iIi = O). Three results obtained using respectively ¡he (i) index approximation (_. . -); ('iO envelope approximation (- - -); and ciii) present method ( ' ) are shown.

0.5 i.0 1.5 2.0 2.5 3.0

Slow/i-varying wave drift forces in short-crestedirregular seas: M. H. Ki,iz and Dick K. P. Yue e o (d)

frXï,\

(c) 200 00 800 time(sec.)

Figure 4. Simulated lime histories for the. case of unidirectional seas incident on a uniform vertical cylinder for (a) the free surface elevation; and slowly-varying drift force obtained using (b) the envelope

approximation; Çc,) the index approximation; and (d) in-verse discrete FFT from power spectrumof the slowly-varying force.

this method Similar results for short-crested seas with a directional spreading of cos2ß are shown in Figures 5. In this case, the envelope method gives negative values whenever I 13L .1 irJ2, and as pointed out earlier, has a

finite PDF at F= O: The results from all three

approx-imation methods (envelope, index and the present one)

are qualitatively similar, with the present method predicting the smallest amplitudes, which is also

in-dicated in the later spectral analysis results.

Using the time history data, the statistics of the

slowlyvarying 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 simular are in general notergodic20, 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

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

(a)

(b)

(d)

200 500 800

Figure 5. Simulated time histories for the case of direc-tional seas (cos2f3 spreading) incident on a uniform

ver-tical 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.

good agreement. Envelope approximation overpredicts both the mean and variance of F, 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

pre-sent approximation respectively. These are plotted in

Figure 6 for directional spreadings of cos2í3 and cos8i3.

As the waves become more short-crested, the

longitudinal force results deviates more from the

unidirectional force spectrum. In all cases, the prçsent approximation predicts lower main direction but higher

transverse direction force amplitudes at all slowly varying ftequencies ¡.

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.

Sb tvly-t'an'in.' ;tave dnft forces in short-crested irregular seas: M. H. Kim and Dick K. P. Yue

swell, 130/ir = 0, 0.25, 0.5, 0.75 and I are considered. In

Figures 7, we plot the part of the longitudinal and transverse force spectra due to the interaction of the

storm and swell waves (cf. Eq. 43) for the different swell

angles. Comparing to Figure 6a, we note that the in-teraction spectra are typically much greater than that

due to the storm waves alone. The large amplification of the spectrum in Figures 7 for certain obtuse values of ßo especially fòr small is most noteworthy. For example,

in the case of the x-direction slowly-varying force, the

increase in the force magnitude near ¡ O due to stormy swell interaction can be up to 4 and over 5 times larger

0 0.1 0.2 0,3 6.4 0.0 0.6 0.7 0 I p

Figure 6. Power spectra of the slowly-varying drift force as a function of the slowly-varying frequency ¡. Two short-crested seas with directional spreading (a)

(os2ß and (b) cos8I3 are considered. The curves shown

correspond to

results for xdirection force for (i)

unidirectional seas ₍ ); (ii) present method

( - ); (iii) index approximation ( );

and for

y-direction force for iv present method (-- ); and (y)

index approximation ( - ).

To illustrate this, we consider the important case of the simultaneous presence of storm and swell seas from

dif-ferent directions. For definiteness, the storm sea is

assumed to be given by (31) (m0 = 1.55 m2) with a cos2f3

directional spreading about x = 0, and the swell is

ap-proximated as a Icng-crested monochromatic waves of frequency o = 0.6 rad/s and amplitude Co = 2mo, so

that the storm and swell overlap in wave frequency and have the same total energy. Eq. (41) for the swell

spec-rum is now simply:

S2(w)=(a/2) ô(wc,o)

(91)

In this case, our approximation (46) reduces to a single integral with respect to f3. Five incidence angles of the

(b)

Figure 7. Interaction component of the Po wer spectrum

of

the slowly-varying drift force due to the presence

of

-combined storm and swell seas as a function of the slowly-varying frequency . The results are obtained

using the present method for (a) the x-directïon force; and (b) the ydirectionfòrceforstormn wave main

direc-tion ß = 0,

and swell incident angle of f3z/ir O

(

); 0.25 (----); 0.5 (---); 0.75 ( S-);

Table 2. Mean. E(f), and standard deviation, o. of slowly-varying drift forces obtained from time simulations and

fron, theoretical power spectra. A lt vatues are normalized by pga and given in units of in . Results from power spectra

are in, brackets ((...J).

Applied Ocean Research, ¡989, Vol. ¡1, No. ¡ 13 Unidirece'ional seas

OF

envelope approximation 0.942 1.167

index approximatión 0.803 10.804 J 0.961 (0.9321

cos2I3 directiónal spread seds

E()

OFi envelope approximation irdex approximation present method 0.749 0.666 10.6831 0.680 (O.683J 0.012 0.014 101 0.003 [0[ 0.902 0.776 (0.8011 0.696 10.7461 0.388 0.375 10.3691 0.444 10.422 J 0.7 01 - 0.2 03 0.4 p 0.5 0.6 .0

S/u ly-varviiìg ta'e drift forces in short-crested irregular

for the case when the swlI seas are incident at 135° and

1800 _{w 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

con firmed by direct simulations of (15). We rematk that

although the variance of the slowly-varying forces due

to storm-swell interactions are greater for certain

oppos-ing swell angles, the net mean drift forces are always greatest for the case of ßo=0 and 13o=.ir/2 for the longitudinal and transverse directions respectively.

These results have important implications for ocean operations under storm and swell conditions sùch as

those reported by Grancini et al8.

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 stormswell interaction effect based

on the

doUble-index approximation (45) leads to qualitatively incorrect results except for small values of

j3 (sec Figure 8).

We now turn to the probability distribution of the

slowly-varying forces in uñidirectional and short-crested seas. Figure 9 shows the index approximation PDF and

CDF for the main direction drift force F 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 of F are higher for smaller directional spreading. As pointed earlier, there is a small prob-ability for the drift force to be negative. These

theoretical PDF's are also confirmed by direct

numerical simulatións of the time-varying drift force. This is shown in Figures 10 where there is good

com-parison between simulated histograms and the PDF's.

The results using the envelope method are likewise

obtained. For unidirectional seas (Figures 11), there is a

,'

,

----.

,-Kb.c 0.1 0.2 .

\.

-.-.

s..' N

-.- -.--.. ,.----.

-.-

.-.-0.3 0.4 0.5 0.6 0 7 p

Figure 8. Interaction component of the po wer spectrum

of the x-direction slowly-varying drift force due to the

presence of conibined storm and swell seas as a function

of the slowly-varying frequency ¡z. The results are

ob-tained by the double-index approximation for the

x-direction force for storm wave main x-direction ofß 0,

and swell incident angle of

132fr =0 ( ); 0.25

(----);0.5(---);0.75(-- --);1(---).

14 _{Applied Ocean Research, 1989, Vol. 11, No. 1}

seas: M. H. Ku,, and Dick K. P. Yue

finite probability for negative values of the local fre-quency WL or wavenumbeL 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

K , e ca .0 e 1.0

Figure 9. Probability density function and cumulative density function of the main direction sb w/y- varying

drift force of ¡he index approximation method. The

results are for(a) long-crested waves ( ); (b) cos2ß

spread directional seas (--); and (c) cos8f3 spread

directional seas (- - -J.

(a)

-i.0 0.0 1.0 2.0 3.0 4.0 5.0

F,lpg.

Figure 10. Comparisons between the theoretical prob-ability

density function and that

obtained from

numerical simulation of the main directiön

slOwly-varying drift force usiÑg the index approximation

method. The results are for (a) long-crested waves; and (b) casi13spread directional seas.

0.0 1.0 2.0 3.0 4.0 5.0 Fipga 50 4.0 0.0 1.0 2.0 3.0 'n'pg.

variables (amplitude, frequency ànd wavenumber). The

PDF and CDF of the slowly-varying force for different directional spreádings are plotted in Figure 12. The probability f extreme values are generally somewhat higher than those predicted by the index approximation

(For example, for cos2ß 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 F,. is always positive, while for short-crested seas,

a 0.

.S7ow/'- tari'illg it'm'e c/rift forces in s/ion-crested irregular secs: M. H. Kii; and Dick K. P. Yue

(b)

0.21 0.20

Figure II. Comparisons between the theoretical prob-ability

density function and (hai

obtained from

nu,nerical simulation

of

local random variables of the envelope approximation method in unidirectional seas.

The results are for (a,J

loca! amplitude; ('b) local

wavenumber; and (c) local frequency.

Figure ¡2. Probability density function and cumulative density function

of

the main direction sb wly varying drift force

of

the envelope approximation method. The

results are for (a) long-crested waves ( ); (b) còs2I3

spread directional seas (

. ); and (c) cosj3

spread directional seas ( -

).

the probability of negative force is nonzero correspon-ding to the situatiöñ where the absolute value of the local direction is greater than r/2. The PDF for f for uni- and multi-directional seas, and for the local direc tion ßL, i.e. the instantaneous direction of the drift

force in short-crested waves, are compared to simulated

histograms for the envelope method in Figures l3 The

comparisons, including the prediction of negative values in directional seas, are quite reasonable.

Although the theoretical methods'4 for unidirectiónal waves may still be useful, a statistical theory for

second-order 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 of F obtained from the envelope and

double-index 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 cos2j3 spreading. lt 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. SUMMARY

A new method for the calculation of slowly-varying wave drift forces in short-crested irregular seás 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 direction ality. The present method retains Newman's

narrow-band assumption of the wave frequency, but allows for arbitrary directional spreading which is treated exactly.

For typical short-crested storm waves with cos2"f3 spreadings, the present theory predicts respectively

lower and higher amplitudes for the main and transverse

diiection slowly-varying forces. For wide directionally spread waves, such as in the important case of the

5/o 111v- i'ar'ilg llave drift forces in short-crested irregular seas: M. H. Ki,iz and Dick K. P. Yzie

(C)

PLI'

Figure 13. Comparisons between the theoretical prob-ability

density function and that

obtained from numerical simulatioñ of the slowly-varying drift force using the envelope approximation method. The results

are for the force magnitude for (a) long-crested waves; (b) cos 2ß spread directional seas; and (c) the direction of the force in cos2í3 spread diréctional seas.

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 maih

16 Applied Ocean Research, ¡989, Vol. ¡1, No. ¡

Figure ¡4. Probability density function of the main dfrection slow!yvúrying drift force in cos2ß spread

dfrecuional 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 approx' imation (

- -).

storm wave direction, the largest amplitude is reached

not when the swell is incident at 9Ø0 but when the swéll

is coming from an obtuse angle.

For the probability distribution of second-order

slowly-varying forces in unidirectional and short-crested

seas, existing results'5'69 for the index and envelope

approximations are teexamined and in several cases

cor-rected and generalized. These theoretical probability densities are shown to compare well with numerically

simulated histograms.

For general wave frequencies and directions, 'a

corn-p!eteanalysis will require not only the exact bifrequenëy

bidirectional quadratic transfer functions (QTF), but

also a probabilistic theory for these processes. Research

in both these directions are ongoing

ACKNOWLEDGEMENTS

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

I Newman. J. N. The drift force and moment on ships in waves. Journal of Ship Research, I967 Il, SI.

2 Newman, J. N. Second Order slowly varying forces on vessels in

irregular waves, Synp on Dynamics of marine vehicles and

strUctures iñ waves, London, 1974.

3 Marthinsen, T. Calculation of slowly varying drift forces,

Applied Ocean Research, 1983, 5, 141.

4 Marthinsen, T. The effect of short crested seas on second order

forces and motions International workshop on ship ánd plat. forni motions, Berk eky, 1983.

5 Falcinsen O M & Loken A E Slow drift oscillations of a ship

¡n irregular waves. Applied Ocean Research, 1978, 1, '21.

6 Pinkster, J. A. Low frequency second order wave forces on

vessels moored at sa, Proc. of the 11th Symp. on Naval

Hydrodynamics. 1976.

7 Teigen, P.S. The response of a TLP in short crested waves, Off-shore Technology Conference, 1983, No. 4642.

8 Grancini, G., lovenitti, L. M. & Pastore, P. Moored tanker

behavior in crossed seas: Field experiences and model tesis.

.0 0.0 1.0 2.0 3.0 4.0 5.0 I-

_{- e}

SIoitIr-wiini.' lI(1%C (Ii[( /)IC'S ¡11 sIIori'-cresle(/ iI7(''IlIUrSCüS ¡'4. l-I. Ku,, and Dick K. I'. Yue S,,,i1,. (Ill ¡)crIi,,1iè)1r (liii! ,t!(,(IeIi,,g uf ¡)i,e1 I Seu.ç.

(:tpcuI!I:Ic!1. I94.

9 II sti , F. I I. & I I ki k i i u , K . A . A I1aysis of pca k m1mri ii Forces

I.. ucctI by slowvL&%d cli ill uscii!itiOii in u tndoiii st.i' OJlsho'c

Tcc/noIugr Conlere,,re. 1970, No. I 159.

IC) O1pciiIiciiii, U. \V. & \Vikuii, I'. A. Cci,Iniiotis IigiIa

simula-I Oil Or simula-I me sceoi,d ortkr slowly v;iryiimg w:ive drm 1tiri:,. J nul

. ()j 5jj,, Re.wu,c/,, I 980. 24, I 8 I.

II K aptan . P . Conimimemi t oil Oiij,cnheimmm amid V'i sou : 'Coni immnous

digital sitimiilatioim of the secottd order slowly varyitíg wave drift force', Jourizal of S/tip Re.warclt, 1982, 26, 36.

12 Ogiivic, T. F. Second order hydrodyiiatnk ellecis oti ocean

plat-forms, I,tlernaliwta/ I FosAs/top on Ship(111(1 l'luiforin lt lotions,

Berkeley, 1983.

13 Iiedrosiati. E. & Rice. S. O. 1Imc output properties Of Volterra

systems (non lt mica r svsi ti ç w tilt tncniory) dt is Ctl b) liti titOliIL

and Gaussian inputs, Proç. ofIEEE, 1971, 59, 1688.

14 Neal, E. Second oidcr liycirodyìiatnic forces due to stoiiastic

excita t ion. ¡'roc. 1016 Symp. on Naval Iimdrodv,,w,,ics, 1974.

IS Vinje, r. On time siatisiicál distribution or sccöncl order forces

and tnoiions, l,,ter,u,tiona/ Shipbuilding I'rogress, 1983, 30, No.

343.

16 Langley, R. S. Time statistics of second order was-c forces,

Applied Oeca,, Research, 1984; 6, 182.

17 Gradshtcyn, !. S. & Ryzliik. I. M. 'rabies of integrals, series and

products, Aadcmic l'rcss, 1980.

18 Tucker, M. J., Clialkitor. I'. G. & Carier, I).J. r. Niitucrical

siimtl;ttioii nl ¡t r:ttitlsiiii se;t: a cimitmutott cirmiramid ils effect iipi'ii

Wavi' group StuliStiCs, /1/.iplii'(l Oct'afl Restsiic/t, 198'l, (p. 118.

19 Vitije, T. On time statistical distribittioti or second order forces, VERITEC report, 1985.

20 .Icllerys, E. R - Directioital seas should be etgodic, ,lpplicd Ocean Research. l9H7, 9, 186.

-21 Mohn, I3. Second order diflration basis upon three diitictmsiotmtl

bodies, Applied Ocean Rereurcl,. 1979, I, 197.

22 LighihibI, M. J. Waves amid Imydrodynamic luadiiig, l'sue. IIOSS 79, Lotidoim, 1979.

23 Hung, S. M. & Eatock Taylor. R. Sceotid order little Itariiiotii forces on bodies itt waves, Second h,,U'rnaiiu,,aF iI'ork.cIto, u:

Water lI'ave.r rmd Floating Bodies, l3rislol, 1987.

24 Loketi, A. E. Three dimetisional secotid order hydrodynantic effects oit ocean smritctures in waves, Norwegian Itislitute of Technology, rettori U R-86-54, Trondltciiti, Norway, 1986.

25 LoiigueiI liggitis. M. S. Statistical properties of wave groups in ¡t ratidotim sea state, P/ti!. Trusts. Load., 1984, A3 12. 219.

API'ENDIX. DERIVATION OF THE

MONOCIIROMATIC-IIIIMRECTIONAL QFT D(w, w, ¡3k, ¡3,).

The QTF, DjM,, for a general body ii arbitrary water epth for nionochromatic bidirectional dual waves ¡s Icrived using the far-field method'. In the presence of

two iñcident waves, wavenumber k0, and incident angles ¡3k and ¡3,, the far-field asymptotic forms of the incident

(b,) and diffracted

(v)

potentials can be written as: (ig/w )f(z)f A A etkus os(0 - I)+ A,e" cos(U - 13,)

(AI)

d'!, _{(ig/o.i)f(z).jko/2irr}

X E kKk(ar + O) + A,K,(ir+ 0)1 e11" '! (A2)

for k0r

I, and 1(z)

cosli kn(z + Iz)jcosh k01,. Here,

A, A, are the complex amplitudes of the incident

waves, and K&, K, the Kochin functions defined by: A'1(0)_{= ,Ç Ç w}

)fze0

coi + t' sin 0)

body

(A3)

j = k, I, where Qj is the diffracted potential associated

with the jth incident wave alône. Using momentum

con-servation for the fluid volume, the mean force on the

body can be expressed in terms a far-field integral given

by: =Gtk01z) .2,r dOrIL?S f,.) 8k0 u O f I _a a

2

_*1 X J -f f

dOI' O'1±

'

.l,

(cos o) ,ar ac

äO

_{¡fr j=tt}

whet-c (A4)

=

, f

,,, and G(k11/i) tanli(ko/z) + ku/i sech 2k1/i,

is

a depth factor which goes to unity as k0!,

Substituting (Al, A2) into (A4), and using the tnethdd of stationary phase for the resulting integral, we obtain the drift force QTF:

ÇD.s.M1

-

pgkuG(ktth) 8r e ti Kk(lr+O)K,(7r+O) icos O'

oj

dO a(kO/1)[Kk( + (3,) [sin ¡3J

K,

13»t'

ai]

(cos j3,) * Çços 13k (AS)

which salisfies (lie symmetry relationship Dk, = D. The QTF is related IO the mean drift force by:

2 2

= > /1*Aitr.r1, (A6)

k=l ¡=1

For vertically axisytnmetric

body geometries,

the

Koch in functions X need to be calculated only for one incident wave angle, since K(0) = K,(0 + 13, ¡3k), and the computational effort is greatly reduced.

In the special case of a uniform vertical cylinder

(radius a), the total potential Ç'j and hence the QTF can be expressed in closed form:

cki= ijA11(z)

nO

X [Jit(kor) Hn(kor)]cos

n(O ¡3j)

(A7)

where J,,, H,, arc Bessel and Hankel functions ol the

first

kind, primes denote derivatives with respect to

argument, and e0 = I, e,, = 2 for n I. Substituting in (A7) into (A3), the Kochin function can be evaluated to be:

Kj(7r+O)=.L >

e cos n(Oß)J,(k1pa)/H,(koa)

k11 ,,,

(A8)

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

(D,151 pgaG(koh) . fe,, I

I -

i cos n(13 13i)

(l)s.,,J kc1a tanli k0!, =0 12

x

(1c?5 13kÌ

_{T(koa) +}

1c:)s (3,' T1'(koa)) sin (3k) sin ¡3,)

-

fcos[ (n + 1)13k _{- 'l13,fl} ,Sll [(n + ¡ )ßA- - "fi,] JR,, (k0 a) fcos (n + 1)13,

nß I)

sin [(n + 1)13, _iiß

j

I?:(kua)] (A9)

.pp!ied Ocean Research, ¡989, VoI. 11, No. 1 17

X

Sia It'!)'- t'aiying wave c/rift forces in short-crested irregular seas A1 H. Kim and Dick K.. P. l'ue

where the functions T,1 and R are defined by: T,(ko.a) =Jh(koa)IH,

*(ka)

R,,(koa)= j,1,+,(kua)J,(kua)/Jl,hl+i(kO(.1)1I,*(ko(1)

(A IO)

In tue special case

of a

singic incident wave

(13k 13, = fi) the single frequency and direction QTF,

D(w, w,fi, fi), reduces to the familiar result:

ÇDx 2pgaG(koh) Çcos

_Ê

1D,J

k0a tnh koIi

siì ßj ,,o

18 AppliedOcean Research,

¡989, Vol. ! I, No. ¡

x ReaIIT,,(coa)l - RcallRi(koa)l]

(All)

which has the asyïnptotic value of (2f3)pgalcos fi, sin ßj

in the liìnit of short waves (koa, kJ: -. ); and the

long-wave (koa, k11/z -' Ø) asymptOte of:

(D.,.1) 511.2 _{(cos fi}

t ----pga(koa);.

8 sin fi