Second-order, slowly-varying Forces on Vessels in Irregular Waves

SYNOPSIS

An analysis is made of the slowly varying second-order wave force which results from the nonlinear interactions between adjacent portions of the wave spectrum. This force is of practical importance in a variety of problems including added resistance and course variations of ships. horizontal oscillations of moored vessels. and vertical oscillations of vessels with small _waterplane areas. Approximate results are derived which depend only on the steady time-average force in _{regular waves.} and obviate the need to determine the _second-order hydrodynamic force on the vessel in the_{presence of two} simultaneous regular waves. The validity of:the

approxi-mate results is

_{illustrated with a simple numerical} example.

1. INTRODUCTION

THE HYDRODYNAMIC FORCE _{and moment acting upon a}

marine vehicle or structure in regular waves will generally include a second-order nonlinear component propor-tional to the square of thewave amplitude. This second-order force or moment will include, in_{turn. asteady-state} component independent of time. plus a second-harmonic

oscillatory term. In spite of its small

_second-order

magnitude. the steady-state component is of practical importance if the corresponding static restoring force or moment of the vehicle is small or zero. For surface vessels this is true in the horizontal plane. so that

second-order forces affect the added resistance _and course-keeping ability of ships in waves, as well as the drifting motion of unpropelled vessels. Similarly, _{if the static} restoring force is non-zero but small. as in the horizontal oscillations of moored vessels, or the vertical oscillations ofsmall-waterplane-area vessels, a highly tuned resonance

will generally occur at very low _{natural frequencies.} in an irregular wave _{spectrum this resonance will be} excited by the slowly varying second-order_wave excita-tion which corresponds to the _{steady-state force in a} regular monochromatic wavesystem.

There have been numerous studies of the steady second-order hydrodynamic force in regular _{waves, including} works on added resistance. drifting motions, and on the vertical force affecting a submerged vessel._{Asynthesis of} the time-averaged second-order _{wave resistance in a} spectrum of irregular waves has also beeñ _developed. e.g. by Gerritsma, van der Bosch) and Beukelmann

_(I).

But in irregular waves the slowly_{varying forces, which} occur at the differen frequencies between _all com-ponents of the spectrum, must also be _{examined to} complete our understanding of the problem, particularly in those cases where a low resonant frequency exists. Our knowledge of the slowly varyïng _second-order

The MS of this paper was accepted for publication on 4th January 1974 f Universities of New South Wales and Adelaide, Australia.

References are given in Appendix I 182

Paper 19. Second-order,

_{SowIy-varying Forces}

_on

Vesse!s n IrreuIar Waves

J. N. Newmant

hydrodynamic forces is not well devéloped, being restricted to the theoretical study be Lee (2) _{of forced} oscillations for two-dimensional cylinders. A very generai experimental approach based on bi-spectral_{analysis has} been outlined by Hasselmann (3) for ship_{problems, but} this synthesis does flot exploit the slowly-varying nature of the important nonlinear forces.

In this brief note I shall outline _{a relatively simple} approach to the slowly varying wave forces. in _an Irre-gular wave system composed of a discrete spectrum of regular waves. Attention will be focused on the contr-bution from difference-frequency terms of very low frequency relative to the fundamental, and _asymptotic approximations will be derived for the slowly _varying force and moment which depend only on the steady. state hydrodynamic transfer functions. The results _of this analysis will be illustrated by computing the_second-rn order pressure in an irregular wave system.

1.1 Notation

Am complex wave amplitude . -.

Fm first order transfer function

Fm,, second order transfer function

f

force

g gravitational acceleration

km wave number

(,n. n) _{indices denoting harmonic components}

p pressure t time x horizontal coordinate z _{vertical coordinate} free-surface elevation p fluid density

Z THE SLOWLY-VARYING

_FORCE

We shall assume that the incident wave system can be approximated by a discrete spectrum. and hence the

(linearized) wave elevation_{can be expressed as a finite} series.

(t) = ReAmeamt

(1)

Here Re denotes the realpart. '(t) is theinstantaneous free surface elevation at a prescribed point in space at time t. A,,, is a complex amplitude with random phase. and (0mthe radian frequency of the nth component of the

spectrum. It will be assumed hereafter that the index'n has been ascribed in ascending order. and that m+ t >

e-0m for all values of in.

If the wave system (1) acts _{upon a floating or} sub-merged body. the linearized first-order hydrodynamic force (or moment) acting on the body can be expressed

§ A preliminary approach ro the slow/v varying force in irregular waicS. which is closely related to the present study, has been outlined by Remery and Hermans (5).

in the form

= ReAmFVeid1r

In

where F,,P F '(w) is the first-order transfer function. or response amplitude operator. The analogous expres-sion for the second-order force is

f

= Re AmAn F»

±

In n

where an asterisk (*) denotes the complex conjugate.

'4)

_{F2(Wm.Wr is the transfer function for the}

second-order 'sum frequency' force and F» Ft2 (w. w,,) is the transfer function for the difference-fre-quency force.

We shall focus our attention here on the slowly varying second-order force. associated with the

difference-frequency terms in (3). and thus we shall delete the super-script designation hereafter and write, simply

f(z) = Re

Fmn e1'"_"

m n

First it may be noted that the time-average of this

force is

J= Re>AmA,F,,,m.

Since the imaginary part of has no significance in these equations. it will be assumed hereafter that Fa,,,, is reaL and can be interpreted physically as the second-order steady force. acting on the vessel in regular waves of unit amplitude and frequency W;,1. When (5) is extended to a continuous spectrum it is essentially the synthesis used by Gerritsma and Beukelmann (1) to evaluate the time-averaged added résistance of a ship in irregular

waves.

Turning now to the off-diagonal elementsin the double summation (4). a degree of ambiguity will exist in the matrix of coefficients in this summation. Physically. Fm,, represents the amplitude and phase of the second-order (difference-frequency) force due to the presence of

two simultaneous waves with frequencies W and W,,

But sin the designation of these.two indices, to represent the two waves, is arbitrary. the symmetry of the matrix Fm,, needs to be prescribed We could. for example. impose the restriction thatWm ? w,,.so that the matrix would be

triangular, with Fm,, = O for nz < n. This is computation-ally convenient, to avoid summing over redundant terms. but in the analytical development here it is preferable to assume a square matrix, ascribing to the two opposite off-diagonal elements an equal contribution to the total force, and this criterion will be satisfied if

p _p*

_{n',, nm}

Generalizing the concept of the 'pure' time-average(S). let us now consider the 'slowly varying' force which will

be denQted by f(t). The contribution to this slowly varying force will be associated with those off-diagonal elements in (4) for which w,,, w,, or. more precisely. where the differences frequency is very small compared to the average:

SECOND-ORDER. SLOWLY-VARYING FORCES ON VESSELS IN IRREGULAR WAVES

(4)

The slowly varying force will be associated then with those terms in the summation of (4) which are very close to the principal diagonal. In fact. if the difference fre-queñcy is sufficiently small, and the force coefficients Fm,, are regular functions of.the two frequencies. it follows that

F,,,,, = Fm,n + O(w,,, - w,,) . .. .(8)

With this approximation (4) takes the form

= Re AmAn* F,,,m e'"'' + O(w,,, - w,.) ...(9) m n

Equation (9) does not appear very much simpler than the exact result (4). but its virtue lies in the fact that it gives an asymptotic approximation to the slowly varying second-orderforce which depends only on the regular-wave, second-order transfer function F,,,,, As noted in the

Introduction, much more is known aboút the latter coefficients, from a hydmodynamic standpoint, by com-parison with the off-diagonal elements Fm,,. so that in fact (9) can be used in practice to evaluate the slowly varying second-order forces in a wide range of problems where (4) is unusable.

On the other hand. the direct summation of (9) still requires the large computational effort associated with a double-summation. and it is not obvious that this process will be convergent. Theseproblems can be circumvented noting that

(Re Am .11 F,,,,, I eict)2 = 4Re AmAn jI I

x

em")l + Re

m ,

x ,,jIFmmFnnIec)m_)(

(10)

and, once again,

,JI F mm1'n,,I = Fmm + O(Wm - w,)

(11)

Thus, the slowly varying part of the square of a suitably chosen single series is identical to the double series (9) or. symbolically,

î(=íL]2_L')]2+O(Wm_W,,) (12)

where

L (t) = Re Am .J( ± 2Fmn') ekmt ...(13) and these sums are tO be carried out only for those terms

where the argument of the square root is positive in each case. (For a positive-definite second-order Fmm> 0. L = O and only the series involving L need be con-sidered).

Ifa suitable low-pass filtering can be imposed. (12) is a coñvenient and relatively simple representation of the slowly varying second-order force. Moreover this filtering process can be avoided if the slowly varying, force is not to be. regarded as an end of itself, but instead is to be used in an integrated sense. fOr example to find the speed loss in waves or, after twofold integration, to

find the trájectory of the vessel in response to the slowly varying force. In that case the large inertial forces asso-ciated with the virtual mass of the vessel will effectively

filter out the .higher harmonies of (L'

- L2) or.

alternatively, will regard these as negligible compared to the oscillatory first-order forces. Thus the potential 183

/ Wm max j

- MARiNE VEHICLES

difficulties associated with the averaging of(12) can be avoided in most cases of practical importàñce.

3. AN ILLUSTRATIVE EXAMPLE

As a simple example, for which the exact and approximate slowly varying foroes may be compared. let us consider the second-order pressure field associated

with an

undisturbed incident wave system. in infinitely deep water. This pressure can be used as an approximation to the vertical exciting force. analogous to the

Froude-Kriloff approximation in the first-order linear theory. The first-order velocity potential corresponding to (1) is

= Re ig (A,,.jw,,) exp [(Wmt k,,,X) _{+ ka,:]} . ..(14)

Here g is the gravitational acceleration. km = w,/g is the

wave number. and long-crested waves moving in the + x

direction are assumed. with z the vertical coordinate. directed positive upwards. and z = O the plane of the undisturbed free surface. The second-order velocity potential 2)

can be derived, after noting that this must satisfy the second-order boundary condition

4 +

g/

= 2VV'1

on: = O

...(15)

(cf. Wehausen and Laitone (4), equation (10.12). An additional contribution to the right side of(l5) vanishes. since is of the form (14) and satisfies the linear free-surface condition exactly at all depths z).

The i-ight side of(15) may be evaluated from (14). and it follows that the solution of Laplace's equation satis-fying the boundary condition (15) is

ç21=Rei>AmA

x max

( 1)m'exp[lk

- kz + i(w.

\

W,,J

i(kmk,)x]

...(16) where -wm form> n co forn > ni

O forn=m

The second-order pressure (2) is. from Bernoulli's equa-tion. or p(2) =

(2)

₊

Vm.v1))

_...(17) p(2)

= pRe>AAp,,(z)

x exp [1((Om - Wr)t - (km

k)x]

.. .(18) where i _{1k -k} I-p,,.,,,(z) = (O.m - w,.) max _{ie "'}

-o,we"

.119)

The second-order. slowly varying pressure (18) at the position x = O will now be regarded as an illustrative example of the second-order slowly-varying force (4). and thus

= p(z)

. . .120)

0 250 _s500 750

Fig. _{la. First-order free-surface elevation, for 30-knot} Pierson-M oskowitz spect runt

J, -2000 I, g. -000 -6000 -8000 o Fig. lb. 2000 .ic

0k1(y1

e-2000-I _i f 'J) g.-'000-' -6000 -8000 o

Fig. ic. Second-order pressure using equation (9). with second-order transfer functions approximated by their

values on the principal diagonaL

2000 O -2000 -4000 -6000 8000 u, D o (r) o Q. 2000 o 250

t,s

500

Second-order pressure at depth = 0'Ol m.

250 500 750 750 1000 icoc o 250 500 750 loco

t,s

Fig. id. Second-order pressure using equations (12-13). in which the double summation is replaced by thesquare

- of a single senes.

For numerical computations we consider a discrete Pierson-Moskowitz spectrum for a fully developed

30-knot wind, with 100 frequenciesWm equally spread in the interval (02. 1.2) radiáns per second. or wave periods between 5.2 and 31.4 seconds. The Pierson-Moskowit.z energy spectrum is used to determine the magnitude of each wave Am. and the phases are determined by. a random-number generator. The resulting first-order wave height _{(t). computed from (1). is shown in Fig la. as a}

function of time with O < r

< 1000 seconds; The probability-density function of is plotted in Fig. 2a-and compared with a Gaussian distribution. Figures Ib and 2b show the corresponding second-order slowly

varying force fit) and probability-density functI1.

and z = 0.01 m. in this case corresponding to the second-order pressure very near the free surface. Fiures Ic and 2c are the corresponding results. but with J(t) computed from (9). so that the transfer functions Fm are replaced by their values on the principal diagonal. Figures id and 2d show the corresponding results using equations (12-13) to compute J(t). Finally. Figs. 3 and 4 show the analogous results for a depth z = 10 m.

04

0025

SECOND-ORDER- SLOWLY-VARYING FORCES ON VESSELS IN IRREGULAR WAVES

00 -36 -2L -12 0025 00

'i'

--3000 -2000 -1000 O 12. 2 36

Fig. 2a Probability-density functiôn (xl00) for the

first-order free-surface elevation shown in Fig. la (Continuous

curve is a Gaussian function; pmean and =standard

deviation).

00

-3000 -2000 -1000 0 1000 2000 3000

Fig. 2b. Probability-density function (x l0Ó) for the

second-order pressure shown in Fig. lb.

-715 ci 1268

idoo 2000 3000

Fig. 2c. Probability-density function (xl 00) for the

second-order pressure shown in Fig. ic.

002S

00

-3000 -2d00 -1X0 o idoO 2d00 30.00

Fig. 2d. Probability-density function (xl0O) f rthe

second-order pressure shown in Fig. id.

4. DISCUSSION OF RESULTS

The results shown in Figs. l-4. for the secoñd-order

pressure at depths of 0.01 m and 10 m. serve to illustrate

the analysis we have carried out. and may be used to judge the validity of our assumptions. In this connection

O 250 500

Fig. 3a. First-order free-surface elevation, for 30-knot

PiersonMoskowitz-spectrum. 2000 O 8 -2000-8. -L000--6000 -8000 o 2000 O In 8 -2000 'n -0D0- -6000--8000 o 2000 o.. u, -o -2000-'J -L000--6000 Q. -8000 o .250 500 t, s

Fig. 3b. Second-order pressure at depth = 10 m.

20 _t,s 7° 750 1000 1000 1000

Fig. 3d. Second-order pressure using equations (12-13), ifl which the double summation is replaced by the square

of a single series.

it should be noted that the second-order transfer func-tion (19) depends significantly on depth. in so far as the principal diagonal terms p,,

= -

exp (2kmZ) are

attenuated rapidly with increasing depth. Thus our

basic approximation (8). in which we replace the

second-order transfer function by its value on the principal diagonal, may be expected in the present illustrative case to be validonly very near the free surface = 0. Thi is confirmed by Figs. 1 and 2. where the exact (b) and approximate (c) results are virtually indistinguish-able, whereas in Figs. 3 and 4 for the depth of 10 m.

noticeable differences exist.

The additional approximation, in which the double summation is replaced by the square of a single series. yields the results shown in Figs (ld-4d). The results here are noticeably distinct, in particulàr the time-history

185

70 1000 500

250 _{t, s}

Fig. 3c. Second-order pressure using equation (9). with

second-order transfer functións approximated py their

values on the principal diagonaL

/.L r-719 u 1261

04 0025 J_L 2L0 O =469 0025 0-0 - -3000 0-025 186 0-0 - i i i i i -3-6 -2-4 -12 0 12 24 3-6

Fig. 4a. Probability-density function (x 100) for tge

first-order free-surface etevation shown in Fig. 3a.

00

-3000 -2000 -1000 0 1000 2000 3000

Fig. 4b. Probability-density function (xl 00) for the

second-order pressure shown in Fig. 3b.

J_L 0

= 1-231

-241 0. 333

-3000 -20à0 -iobo o iobo 2d00 3000

Fig. 4d. Probability-density function (xlOO) forthe

second-order pressure shown in Fig. 3d.

MARINE VEHICLES

shown in Figs id and 3d now includes the sum-frequency components. and there is no oscillation in the sign of the

pressure. which is in this approximation always less than or equal to zero.

The probability-density functions (Figs. 2d. 4d) are noticeably more peaked. but it should be noted that the

standard deviations a are

still well approximated. especially at the shallow depth.

It follows from equation (5) that the mean value n will depend only on the principal-diagonal elements. Thus the value of t in Figs. (2b-d) and (4b-d) are consistent

and do not depend on the differences of those respective treatments of the second-order pressure.

In conclusion, we have shown that the slowly varying second-order wave forces acting on marine vehicles can be approximated from a knowledge of the mean second-order forces in regular waves. The quantitative accuracy of these approximations may vary from one case to another, and cannot be rigorously established without some knowledge of the off-diagonal second-order forces. resulting from the simultaneous presence of two discrete wave systems. However, this knowledge is lacking in most cases of engineering interest, and thus for practical purposes the present approximate approach offers the only possibility for analysis of the slowly varying second-order forces.

5. ACKNOWLEDGMENTS

This research was initiated at M.I.T. under Office of Naval Research Contract N00014-67-A-0204-0023 and National Science Foundation Grant OK-10846. lt has been completed during a visit to the University of New South Wales. with support from the Australian-American Education Foundation and The John Simon Guggen-heim Memorial Foundation. The numerical results shown

in Figs. 1-4 were obtained by Mr W. K. Soh. of the

University of New South Wales.

APPENDIX i

REFERENCES

GERRITSMA, J., VAN DEN BoSCH, J. J. and BEUKF.LMANN. W..

Propulsion in Regular and Irregular Waves',mt. Sliipbldg. Prog.

1961,8, No. 82.

LEE, C. M., 'The Second-Order Theory for Nonsinusoidal Oscilla-tions of a Cylinder in a Free Surface,Proc. Eighth Sy,np. Nacal Hydrodynamics, 1960. 905-951.

HASSELMAN. K.. 'On Nonlinear Ship Motions in Irregular

Waves'.J. of Ship Research 1966, 10. 64-68.

WEHAUSEN,J. V. andLATTONE, E. V.,'Surface Waves'. Handbuch

der Physik 1960. D((Springer Verlag).

REMERY. G. F. M.andHERMANS. A. J.'The Slow Drift Oscillations

of a Moored Object in Random Seas', Soc. of Petroleum Engineers Journal1972. 191-198.

-2000 -1000 0 1000 2000 3000

Fig. 4c. Probability-density function (xl 00) for the

second-order pressure shown in Fig. 3c.

/1 =-240 0' 406