Second-order hydrodynamic effects on ocean platforms

Transformation of Coordinates Boundaru-Value Problem

Force and Moment Equations of Motion

LARGE DRIFT MCTIONS Two Time Scales

Transformation of Coordinates Boundary-Value Problem

Force and Moment Equations of Motion

Slowly Varying Motions Comments

NEGATIVE DRIFT FORCE ACKNOWLEDGMENTS REFERENCES

LOW-FREQUENCY FORCES ON A FLOATING STRUCTURE

1 October 1983

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SECOND-ORDER HYDRODYNAMIC EFFECTS

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ON OCEAN PLATFORMS T. Francis Ogilvie by

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Prepared for INTERNATIONAL WORKSHOP ON SHIP AND PLATFORM MOTIONS University of California, Berkeley

October 26-28, 1983 CONTENTS

INTRODUCTION

SECOND-ORDER ANALYSIS OF FREE WAVES

Sinusoidal Unidirectional Progressive Waves in Deep Water Two Sinusoidal Waves (Same Frequency) Moving in Opposite

Directions

Two or More Sinusoidal Waves of Different Frequencies Moving in the Same Direction

Free Waves in Water of Finite Depth STEADY DRIFT FORCE AND MOMENT

SECOND-ORDER HYDRODYNAMIC EFFECTS ON OCEAN PLATFORMS

by

T. Francis Ogilvie

Department of Ocean Engineering Massachusetts Institute of Technology

Cambridge, MA 02139

Professor John V. Wehausen has educated a generation of students of hydrodynamics. He taught them not only concepts and methodologies but also how to be scholars. By his own example, he persuaded them of the value of precise thinking although muddling through often seems to reach the answer more quickly. As one of his early students, I take great pleasure in dedicating this paper to Professor Wehausen.

Abstract

This paper is a survey of the current state of the art of predicting steady and low-frequency forces on ships and other marine structures. Little methodology is available for predicting effects related to either viscosity or large-amplitude waves, and so most of the paper is devoted to the development of perturbation theory based on the ideal-fluid concept. This, unfortunately, represents the state of the art. Emphasis is placed on the concepts and methods of analysis, :7ather than

on results.

INTRODUCTION

Until about ten years ago, problems of time-dependent interactions between waves and marine structures fell generally into two categories:

Naval architects were concerned with the motions of a ship in a seaway and with the structural loads imposed on a ship by the waves and the motions. Remarkable success was achieved in predicting motions through the use of classical linearized water-wave theory, based on the ideal-fluid, irrotational-flow model. Only the excitation and damping of roll motion could not be treated adequately by this approach, but even in that case the basic framework could be used after being fixed up in a relatively simple way.

Civil engineers were concerned with wave loads on fixed structures, which could be analyzed approximately by the use of Morison's formula

(see Morison et a/. (1950)), sometimes modified according to the results of Keulegan and Carpenter (1958), in which the drag coefficient is con-sidered to be a function of amplitude of motion. The marine structures designed by civil engineers are frequently built on pilings, which typically have transverse dimensions that are very small compared with the wavelengths of incident waves. Under these conditions, Morison's assumption is quite reasonable, namely, that the loads can be expressed entirely in terms of the velocity field that would exist in the absence

---t(11

of the structures. Furthermore, Morison included the loads associated with both viscosity and inertia of the water, although in a phenomena- 1

logical way. His formula does not allow for any diffraction effects.* With the rise of interest

in

offshore platforms in recent years,

cially platforms for use in deep water, both of these approaches have been used for the prediction of forces and motions. Neither is ade-quate in all cases. In many Such problems, diffraction of incident waves must be considered in making accurate predictions. Viscosity effects, manifest primarily in flow separation, may strongly affect

the forces too. As Huse (1977) has shown convincingly, crossflow drag (caused by viscosity) can cause a platform to drift toward the advanc-ing waves, which is opposite to what is predicted by ideal-fluid .

water-wave theory. The effects of viscosity and of wave diffraction can well be of the same order, and predictions based on the neglect of either may be seriously in error.

There is no adequate theory that encompasses both

types

of phenomena. In fact, outside of Morison's formula, there does not appear to be any generally useful method of analyzing viscosity effects, and Morison's formula is really very primitive. So, in this survey, I shall concen-trate on the state of the art of predicting wave load S and motions on, the basis of ideal-fluid theory. I emphasize that this is sometimes a severe restriction.

For a discussion of the Meaning and validity Of Morison's equation, see Lighthill (1979). There is a large literature on the use of Mori-son's equation; for example, Pijfers and Brink (1977) apply it to the

case of a structure subject to both waves and current.

There are still other limitations that ate

sometimes

severe, An impor-tant one arises when the waves are very steep; our Analysis is based entirely on perturbation theory, which, although it may be Able to predict nonlinear phenomena, is quite invalid if wave amplitude and wave slope are not fairly small. Our perturbation theory also fails if the side walls of the structure do not meet the undisturbed free surface at right angles, since the linear theory then shows singulari-ties at these junctures, and one can expect further that the second, order theory may contain nonintegrable singularities there.

In classical ship-motions problems, since linearized theory gives

generally excellent predictions, little is to be gained from an attempt to extend perturbation analyses to second order. In platfonnproblems, on the contrary, there are some fundamentally second-order phenomena that are of great importance. Linear theory does not give even a

first Approximation for predicting these phenomena. The simplest such phenomenon is the steady drift force. It had earlier received some

attention in the Ship-hydrodynamics literature with respect to two problems: (i) A moored ship takes a particular equilibrium attitude with respect, to the direction of the incident waves (if wind effects can be neglected), this attitude depending on second-order forces and moments. (ii) Ships advancing through waves experience an added resistance above their resistance in calm water; this is also not predictable from a strictly linear theory.

*Morison's formula does allow for the fact that the presence of the body modifies the incoming flow. In a sense, this is the infinite-fluid equivalent of diffraction. But there is no way to include true wave diffraction in the Morison formula.

espe-v.)

0.5

1.0

to in rad

sec-I

Figure 2. Sway spectrum (from Verhagen and Sluijs (1970))

1.5

0.5

1.0 1.5

w in rad sec-I

Figure 1. Wave spectrum for sway experiments (from Verhagen and Sluijs (1970))

Suyehiro (1924) attempted to explain the steady drift force

on a

ship

in

terms of the effects of wave reflection (i.e., diffraction) by the

ship. Later, Watanabe (1938) postulated a different mechanism,

namely, interaction between the incident waves and the motions of the ship. Havelock (1942, 1940, respectively) provided more mathematical analyses of both concepts. Maruo (1960) was the first to treat both in a single coherent analysis. His work has been extended to more general cases by Newman (1967), Lee and Newman (1971), Lin and Reed

(1976), and others. If the linearized theory is appropriately for-mulated so that Platforms can be treated as well as ships, the same general approach can also be used to predict steady forces and moments on Platforms, as shown, for exa.awle, by Faltinsen and Michelsen (1974) and Karpoinen (1979). Some of this work will be discussed in detail presently.

The subject of added resistance will not be considered in this paper, inasmuch as I am limiting my attention to the case of zero or very low speed of advance. This decision was made only to keep the scone of this survey to a reasonable size; it does not imply that the

added-resistance problem is either unimportant or uninteresting.

An unrestrained ship or platform has no natural frequencies in its horizontal-plane motions (surge, sway, yaw). But a floating offshore

platform must be controlled with respect to its horizontal position, and so some kind of restraint system is always provided, most often by mooring lines fixed to anchors. A moored-platform system does have some natural frequencies of oscillation in the horizontal plane, and these frequencies are usually much smaller than the frequencies of the incident waves that cause the familiar heave, pitch, and roll motions. This situation leads to the occurrence of another phenomenon, reso-nance of the system with

low-frequency

second-order excitations. This phenomenon appeared clearly in experiments reported by Verhagen and Sluijs (1970), from which Figures 1 and 2 are taken. A floating

structure was exposed to a spectrum of incident waves with the distri-bution shown in Figure 1. The largest part of the sway response

occurred in a frequency band well below the lowest frequency at which there was any significant energy in the incident waves. Verhagen

and

Sluijs offered an explanation: Because of the nonlinearity of the free-surface conditions, the existence of two waves of different fre-quencies always implies the existence of waves at the sum and differ-ence (beat) frequencies. The latter may occur near the resonance fre-quency of the platform in sway, surge, or yaw. If the incident wave system consists of a continuous spectrum of waves, one is assuredthat

there is always some disturbance at any very low frecuency, and, if the damping is small (as it usually is in such motions), a highly amplified resonant motion must be exoected. it is important in prac-tice to predict the magnitude of the low-frequency horizontal excur-sions of a platform and to ensure that they are kept within acceptable bounds. This is one of the most important hydrodynamics problems that

must be solved in the designing of ocean platforms. It will be a major topic in this paper.

Many investigators have attacked this problem, and more-or-less ade-quate solutions exist. In a series of papers, Pinkster has provided one of the more thorough analyses. To a large extent, I shall follow his approach, as presented in Pinkster (1976, 1980), Pinkster and van

Oortmerssen (1977), and Pinkster and Huijsmans (1982). It is a straightforward perturbation approach.

Triantafyllou uses the method of multiple-scale perturbation theory. He assumed that a platform undergoes two kinds of motion that can be treated more or less separately: (i) The usual ship-motion response

to the incident waves occurs at frequencies that are fairly high; the amplitude, velocity, and acceleration are small, considered to be

0(e) , where e is the usual perturbation parameter. (ii) The low-frequency motion has velocity that is 0(e) , whereas its amplitude is

0(1) . This implies that the frequency of this motion is 0(e)

. If t is the time variable that is normally used, he uses a new time variable,

t =

Et , for analyzing the slow-drift motion. The motions at wave-spectrum frequencies are treated as rapid oscillations about a slowly changing position of the structure. The boundary-value problem separates into slowly and rapidly varying parts, both of which are linear, and a procedure is developed for solving for the slow-drift motions. Triantafyllou's approach has not yet been validated through

a comparison of predictions and measurements, but I believe that it may hold great promise. It will be discussed in some detail.

Low-frequency motions in the vertical plane are sometimes important too. In particular, large-volume structures are sometimes designed with small waterplane areas for the specific purpose of keeping the frequencies of heave, pitch, and roll resonance outside of the range of expected ambient waves. Natural periods of 30 to 60 seconds are not uncommon. Such structures can be expected to undergo second-order vertical-plane resonances just like those described in the horizontal plane. Unfortunately, the Newman (1974) argument, mentioned already,

is not so likely to be valid for these vertical-plane motions, since the corresponding resonance frequencies are considerably higher than in the horizontal-plane motions. The state of the art is not so good for such cases.

It should also be noted that, whenever nonlinear effects lead to the occurrence of beat-frequency phenomena, one should also expect sum-frequency phenomena. These may be very important in certain applica-tions. For example, ships sometimes experience a phenomenon known as

"springing," in which the ship resonates as a vibrating beam in response to Periodic wave excitation. Since the lowest natural fre-quency of a hull girder generally corresponds to ambient waves of very short wavelength, which can hardly be expected to excite overall hull vibration, springing may very well be a response to the harmonics and sum-frequency components that appear in the second-order

analysis

(see Troesch (1982)). A similar mechanism may also be important in

analyz-ing the vertical-plane motions of a tension-leg platform; such motions are not likely to be important of themselves, but they may

lead to fatigue problems in the tension legs. The analysis of such problems has not progressed far, and they will not be considered

further.

The paper ends with a short discussion of the possibility of develop-ing a simple system for canceldevelop-ing the second-order drift and low-fre-quency force on a structure. Viscosity-induced crossflow drag may contribute to this development. In addition, the careful use of passive lifting surfaces deserves attention, as shown by Isshiki

(1982a, 1982b). An apparently practical system has been reported by Terao (1982).

Almost all of the extant papers on this subject are based on some kind of assumption that eliminates the need actually to solve for the

second-order potential. A particularly simple explanation and justi-fication for the most commonly used simplifying assumption has been provided by Newman (1974). A more physical argument was given by Hsu

and Blenkarn (1970). The assumption allows one to express the low-frequency force (which in principle does depend on the second-order potential) in terms of the mean drift force (which does not depend on the second-order potential). To prove the validity of this assump-tion, the most obvious approach is to solve the second-order problem. But this is a formidable task. Papanikolaou, in a series of papers from 1977 through 1982, has solved several parts of the second-order problem in the two-dimensional case, but he has still not gone far enough to draw a general conclusion about the common assumption. Fal-tinsen and Lpiken (1978a, 1979a, 1979b) managed to avoid having to solve the second-order problem: They formulated the problem precisely to second-order, expressed the drift force in terms of first- and second-order potentials, and then used Green's theorem to eliminate the

explicit dependence on the second-order potentials. They concluded that Newman's simplification is a reasonable one for practical use, although it does not give precisely correct results. (They say that the neglect of other phenomena, for example, viscosity, causes greater errors.) Faltinsen and LOken treated only the two-dimensional

prob-lem, but there seers to be little reason to doubt that the same con-clusion would be reached in three dimensions.

There is one case in which the second-order potential may be very important. If one considers only free waves (no body present) of at least two frequencies, low-frequency disturbances arise in two ways: (i) There are beat-frequency motions that can be found directly from the first-order potentials. (ii) The second-order potential contains components at the beat frequencies. It is the latter that are

neglected according to the conventional assumption. In fact, it is found in general that the amplitude of the fluid velocity associated with the second-order potential approaches zero as the beat frequency approaches zero. This result (which can be proved explicitly only for

free waves) suggests that the second-order potential may really be unimportant. But it sometimes happens that the wavelength associated with these beat-frequency waves is very, very long, so that one must

treat them as shallow-water waves. In this case, the velocities asso-ciated with the second-order beat-frequency potential do not vanish as the beat frequency approaches zero. Note that the first-order waves in the incident-wave spectrum may very well still be deep-water waves, but the second-order waves may be shallow-water waves, because of

their very low frequency. Bowers (1976) recognized this phenomenon and accounted for it, at least partially. This phenomenon does not contradict the analysis of Faltinsen and LOken, because they consid-ered only the case of infinitely deep water.

One of the limitations of the usual perturbation analysis can be over-come through a reformulation of the problem. The horizontal excur-sions of a platform may be very large in comparison to the lateral dimensions of the platform, a situation that invalidates the basic assumptions of the perturbation theory. Triantafyllou (1982)observed however that the platform velocity may still be quite small (in the sere sense that fluid particle velocities in the waves are small), and he took advantage of this fact to develop a mathematical model that involves only linear hydrodynamics problems, even while it permits large excursions of the platform in the horizontal plane.

SECOND-ORDER ANALYSIS OF FREE WAVES

Before we consider the interactions between waves and bodies, it is

useful first to consider the fluid motion in waves that are unimpeded

by solid boundaries (other than possibly the sea bottom).

This

sec-tion, which is based entirely on material presented in Wehausen and

Laitone (1960), concerns some aspects of the second-order analysis of

such wave motions, which are frequently called "free waves."

The

ideas are fundamental and rather well known.

But some implications

for the analysis and design of ocean platforms have not been widely

recognized.

In this section, we consider only fluid motions that

can be described in terms of a velocity potential, the gradient of

which gives the fluid velocity.

There are two free-surface boundary conditions to be satisfied:

lr 2

2 21

0 = gC

¢t

¢y±

c:)zi

on

z = C(x,y,t)

0 ₌

_q'tt

_{2[4)x(Sxt+cPyq'yt+Cbz4)zt]}

2 2 2

4)x4)xx÷ y9yy+ (Pz9zz

2`Px4)17cPxy÷ 2q5174zq'yz+ 4zg5xcPzx

on

z = C(x,y,t)

where

_¢(x,y,z,t)

_{is the velocity potential,}

_C(x,y,t)

_{gives the}

elevation of the free surface, and

g

is the gravitational

accelera-tion.

The origin of coordinates lies in the undisturbed free surface

with the

z

axis pointing upwards.

The first condition states that

pressure is a constant (zero) on the free surface.

The second

condi-tion is obtained by writing the Bernoulli equacondi-tion and taking its

substantial derivative on the free surface.

These conditions are

valid for all of the problems that we consider in which viscosity and

surface-tension effects are negligible.

In order to carry out a perturbation analysis, we assume that there

is a small parameter

s

that provides a basis for ordering all

quan-tities that arise.

We can think of this parameter as the maximum

wave slope, for example, although its precise definition does not

really matter.

Eventually we absorb

E

into other quantities that

are precisely defined.

We assume further that Quantities such as

¢

and

C

can be expressed as power series in

C:

cp(x,y,z,t)

Ei(x,y,z,t)

+

0(eN+1)

₍₃₎

j=1

N

C(x,y,t)

X

E3Ci(x,y1t)

0(6N+1)

(4)

j=1

Since

¢

satisfies the Laplace equation, which is linear, each

_¢i

also satisfies the Laplace equation.

We substitute these expansions

into the two free-surface conditions, and we further assume that all

quantities that are supposed to be evaluated on

z = C

can be

eval-uated alternatively by extensions with respect to

z = 0

.

In this

way, we obtain the following pairs of free-surface boundary

condi-tions for the first- and second-order problems:

g.z

=

(1)

In addition, we generally need a condition on the sea floor (or an asymptotic condition at great depth), a radiation condition, and, if other solid surfaces are present, a kinematic condition on such

sur-faces.

Sinusoidal Unidirectional Progressive Waves in Deep Water

Let the first term in the potential expansion, ( 3), be written: alw kz

01(x,z,t)

= - e sin (wt-kx+ di)

where al is the wave amplitude (assumed positive) and di is an arbitrary phase angle. When (9 ) is substituted into the free-sur-face condition, (5 ), the dispersion relation is obtained:

co2

k = .

This potential represents sinusoidal waves traveling in the +x

direction. (Since the motion is purely two-dimensional, we have omit-ted y in the arguments of the potential function.) The

correspond-ing wave elevation is obtained from (6): ci(x,t) = al cos (wt - kx+ di) .

For future reference, we write out the velocity components:

4)1x = al (11 ekz cos (wt-kx+ 61)

(Piz = - ai w ekz sin (wt- kx+1) .

The second-order solution is obtained by substituting the first-order solution into the right-hand side of (7). But the right-hand side of (7) vanishes when we carry this out, and so 02 satisfies the

same free-surface condition as

01 . Thus it must have the same

form:

(x,z,t) = a2w ekzsin

_(wt-kx+2)

. (12)

One can, of course, simply absorb this into the linear solution, and so the statement is frequently made that the usual linear solution,

(9), is valid through second order.

The same statement is not valid with respect to the free-surface shape given by (11). From (8), we find that

a r 2 2g (15y-4-4)1z} +licoit?;-{4)1tt+g4) 1 + (P2 + 4)2 lx ly lz lz z=0 on

z =

0 (7) (2) gb2tt÷gcP2z

2 = {2t

4)

g t 'Pitt g4)1. 0 on z = 0 , (5) = (6) z=0 -(10) (11)

.

1 1 g)1t 1 2 =

+

2(x,t) = a2

cos (wt - kx+

62) + 4a21

k cos 2

(wt - kx + 61) ₍₁₃₎ The second term here, when combined with , has the effect of sharpening the wave crests and making the troughs more shallow.

Consideration of fluid particle velocities and trajectories will _give us considerable insight into some aspects of the analysis of platform forces and motions. _{Of course, the gradient of the velocity} poten-tial gives us fluid velocity, but it tells us the velocity at a par-ticular point in space, regardless of what fluid particle happens to occupy that space at any instant. What we want to find now is the motion of a particular fluid particle. _{Consider a particle that has} coordinates X0 and Zo _{in the absence of any fluid motion, and let}

the coordinates of this particle in general be X(t) and Z(t)

Clearly, we can set:

k(t) = (1)x(x,z,t) (14a)

' (t) = qbz(x,z,t) . (14b)

Since the right-hand sides here are expanded in power series in E we should be able to do the same for the left-hand sides, and _{so we} write:

X(t)

xo +

Ex1(t) + E2X2(t) + 0(c3)

Z(t)

Zo +

EZ1(t) + _{E2Z2(t) + 0(c3)}

We substitute these expansions into the previous equations, noting that x = X(t) and z = Z(t) _{on the right-hand sides. Reorganizing} quantities with respect to E , we obtain for X(t)

X(t) = EX + E23(2

+ 0(E3)

1

ealw exp k (20+EZi+...) cos [wt - k

(X0+EXi+...)

+ 61] + c2a2w exp_k (Z0+...) cos [wt - k(X0+...) + 62] + 0 (e3) = ,z{ai w e0 cos (wt- kX0 + 61) /

+ E2{a2 w ekZ0 cos (wt - kX0

_{+ 62)}

+

alw k

ekZ0 [Zi

cos

(wt - kX0 + 61) +X1

sin

_{(wt - kX0 + 61)}

+ 0(c3) - (15)

Similarly for Z(t) we obtain: (t) =

+ E22

+ 0

(c3)

= - Efai ekZ0 sin (Wt- kX0 + 51)

c2{a2

sin (wt - kX0 + 62)}

+ aiw

k

ekZO[Zi sin (wt- kX0 + 61) - Xi cos (wt kX0 + (31) /

+ 0(c3) ₍₁₆₎

Identifying coefficients of E _{, we get a pair of equations for}

X1 (t) and z1 (t) , which are readily solved:

XI (t) = a1ekZ0 sin (wt- kXo + 61)

(17)

-kZ0 _{oit - k.X0+ 61)} Z1(t) = ale cos (

These show the well-known fact that, to first order, a fluid particle with equilibrium coordinates X0 and Zo moves in a clockwise

Zo circle about the equilibrium position. The circle has radius afk Note that we could'add constants to these expressions in general. But, for particles given by Zo = 0 , the above expression for Z1 (t) must be identical to the expression giving the free-surface shape,

(11), and so the constant must be zero. Any additive constant on

X1(t) can simply be absorbed into X0 .

We now continue this process for X2(t) and Z2(t) . In (15) and (16), we identify coefficients of E2 to obtain a pair of differen-tial equations for X 2(t) and Z2(t) , into which we must substitute the results from (17) and (18). These equations are also easily solved, yielding:

kZ, . 2

2kZot

X2 (t) = a2e sin (wt - kX0 + 52) + a iwke

Z2 (t) = a2e1Z0 cos (wt- kX0 + 62) + b2

where _b2 is an additive real constant that should not be omitted, fora reason to be discussed presently.

The second term on the right-hand side of (19) is the interesting

one: It indicates that the fluid Particles have a steady horizontal

velocity component of magnitude E2a.f.wkexp 2kZ0 . This is generally

small compared with the fluid particle velocity associated with orbi-tal motion, but it may be by no means negligible. For example, for a wave of 100 in wavelength and amplitude 3 m, the transport velocity at

Z = 0 is about 20% of the orbital velocity at the same place. What

0

really makes the steady-transport velocity important, however, is the fact that it always has the same sign.

The transport-velocity term never shows up in the velocity potential explicitly (see (9) and (12)). But it really is present in the linear potential. We can see this by integrating 4), to obtain the volume rate of flow across a vertical plane. To second order, we obtain:

eci+...

N1(t) = 4),(x,z,t) dz -co

saiw

1.1 + Ekai cos (wt - kx+ 51) + ...} cos (wt- kx+ 61)

c2a2w

cos (wt - kx + 2) + 0(c3) .

The time average of this quantity is

122

M = aiw

If we calculate the same quantity from the transport-velocity term in (19), we obtain the same result, showing that _p1 does indeed include the steady transport motion.

(18)

.

t

- (20)

There is no contradiction between this result and the fact that 4)1

predicts circular orbits. The latter are only approximate, as can be seen from the derivation of (17) and (18). In fact, we can compute the exact trajectories predicted by (14) with the right-hand side approximated by

Oix and 01, .

An example is shown in Figure 3 for the case of a wave of wavelength 100 m and amplitude 2 m. The trajectory is shown for a fluid particle with an equilibrium depth slightly over three times the wave amplitude. The steady drift is clearly evident. If the trajectory is computed from Equations

(17)-(20), the curve is indistinguishable from that shown in the figure. It is interesting to note that the periodicity of the vertical motion of the particle is not equal to the Periodicity of the wave itself. At the start of the trajectory shown, the fluid particle is directly under a wave crest. After one period, t = 27/w , the particle has moved to the right a small distance, and the next wave crest has not yet arrived at this horizontal location. This crest will catch up with the particle only after yet a small time interval, at which time the particle will have reached its highest point again. In this sense, the periodicity of the particle motion is a function of depth. The solution obtained for Z2 , (20), when we set Zo = 0 , appears to be quite different from the expression for 2 , (13). But we must recall again that x and X0 are different. If we set X0 = x- EX1 in (18) and substitute for X1 from (17), we find that

Z1 = a1 cos (wt - kx + (Si)

r 1 2 1 2

+ t aik cos 2 (wt - kx+ 1) - a1c + 2

The first term on the right-hand side gives us back Ci , as in (11), and the first term on the second line gives us the extra term in 2

(see (13)). But we have one term left over here. If we choose the constant in (20):

11U_AIPIF

a

= 2m

X = 100m

Z

= -6.41m

X

Figure 3. Trajectory of fluid particle in sinusoidal wave, computed from first-order potential

-1 2

b2 = 2- a1k

this extra term is canceled. And so we have agreement between our results for eC + e2 + . and Z (t) , the latter being calculated

for particles for which Zo = 0 . The constant b2 cannot be

obtained from the kinematic analysis based on (14), since the location of the free surface is determined from the dynamic boundary condition.

Two Sinusoidal Waves (Same Frequency) Moving in Opposite Directions When we have waves moving in more than one direction, even if they have the same frequency, an interesting additional phenomenon occurs: There is a second-order oscillatory pressure field that is uniform in space. This phenomenon was mentioned by Miche (1944) and analyzed in detail by Longuet-Higgins (1950), who suggested that such pressure fields in the ocean are the cause of microseisms.

Suppose we have two wave systems moving in opposite directions, the linear potential being given by

kz

cpi(x,z,t) - e sin (wt - kx + (Si)

kz

e sin (wt.+ kx

+ Ei)

The second-order potential satisfying (7) is

4)2(x,z,t)

= aibiw sin (2wt + 61+ el) .

(Now, and from here on, we omit the second-order terms that can be absorbed into the first-order solution.) The second-order potential leads to a contribution to the fluid pressure that is uniform through-out the fluid:

- 2e2pw2a1b1 cos (2wt + di +

Such a pressure field might have a significant effect on the total vertical commonent of force on a floating structure, if loads at high frequency are of any concern.

From an analysis of the particle motions, one finds that the steady mass-transport velocity is

2 ,2, 2kZo E2cok(ai - pi)e

Comparison with the result for a single wave train shows that the mass-transport velocities associated with the two wave trains are algebraically additive. Such a result is hardly obvious, since mass transport is a nonlinear phenomenon.

This gives an interesting result when we consider the case of waves incident on a body. Let the amplitude of the incident waves be al and the amplitude of the reflected waves b1 . Then, if the amplitude

of the transmitted waves is c1 , we must have

2 2 2

c1 = al - b1

in order that the mass transport rate be the same on both sides of the body. But (21) is also the condition that energy flux be balanced if

the body does no net work on the fluid.

(21)

alw

-.

-Two or More Sinusoidal Waves of Different Frequencies Moving in the Same Direction

Now we consider two wave trains, both moving in the

+x

direction,

but with different frequencies.

Let the first-order potential be

(x,z,t)

- -

ari

eak2iwz9

sikn2zsn

k2

(wi.it_(wk::: 51(2x1)

(22)

The second-order potential can be found as before:

(1)2 (x,z,t) =

w1aia2e(k1-k2)z sin [(w1-w2)t - (k1-k2)x+ 61 - 62]

.

(23)

Here it has been assumed that

which implies also that

2 2

ki = wi/g > k2 = w2/g .

The wave elevation is

(x,t)

=

dal cos (wit - kix+ 61) + a2 cos (w2t-k2x+ 52)1

+ e2{1k2

cos 2 (wit - kix+ 61) +

ifc2ai cos 2 (w2t - k2x + 52)

1

-

(k1-k2)ai a2 cos [ (wi-w2)t - (k1-k2) X+ (61-62)

1

+

(ki+k2)ala2 cos [(w1+w2)t - (k1+k2)x+ (51+52) )1 + 0(e3)

.

Note that there is a sum-frequency wave, although

(>2

has no such

terms.

The fluid particle trajectories are given by

X(t)

=

X0 + sia1ek1Z0 sin (wit - kiXo+ 61)

+ s2ek2ZOsin (w2t-k2X0 +62)1

[+

e2 {a2iwikle2k1z0 + ,226.12k2e2k2Zo}t

+ a a

{w1k1+u)2k2 e(kl+k2)Z0

wl(kk2)

e(ki-k2)Zol

1 2 wi -up,

w -w2

x sin 1 (w1-w2)t - (k1-k2)X0 ± (61-62) i

_{4" 0(63)}

Z (t)

=

Zo + ela1ek1Z0 cos (wit - kiX0 + 51)

+ a7ek2Z0 cos (w2t - k2X0 + 52)1

÷ e

( wl2

ala2{w1kliww2k2 e(k1+k2)Z0

2 w1

(k1-k2)

_{e(k1_k2)z0}}

wl-w2

> cos [ (w1--,;,..)t - (k1-k2)X0 + (51-62)]

+ 0(e3)

.

In (24), there are steady transport terms just like those for the

individual wave trains, as given in (19) .

In addition, there are two

terms indicating oscillations at the difference frequency (but not at

the sum frequency).

For clarity of discussion, we write out the

expression f

the diiference-frequency horizontal velocity component:

)Zo

e2a1a: (w1k1+w2k2)e

(kl+k2) Z 0 -

[w1(k1-k2) ]e (kl-k2l

x cos [ (wi-w2)t - (k1-k2) X0 + (61-62)]

.

(25)

(24)

-+

-2

-.

The first term comes from the direct interaction between the two first-order waves; its rapid decay with depth is comparable to that of the steady-transport terms. The second term in (25) comes from

cp2 ; note that its depth dependence is appropriate for a wave of

wavenumber (k1-k2) . The most important observation to be made from

(25) is that the second term vanishes as wi-w2 0 , whereas the amplitude of the first term approaches a finite limit, 2w1k1 . In fact, the displacement amplitude of the latter approaches infinity in this limit, as is evident from (24); this must happen, of course, since the velocity amplitude remains finite while the period goes to infinity.

In many problems of platform dynamics, one is most concerned with forces and motions at very low frequencies. The above results show explicitly that a platform in mixed waves is subject to both steady and low-frequency flows. The steady incident flow is fully described by the first-order potential, just as in the case of asingle-frequency wave motion. Furthermore, the important part of the low-frequency incident flow is also included in the first-order potential, since the amplitude of velocity associated with the second-order potential goes to zero as the frequency difference goes to zero. This suggests (but does not prove) that the second-order potential will be unimportant in most platform problems. An exception to this statement will be pointed out presently.

In problems of practical interest, one must consider a superposition of many waves or, most generally, a continuous spectrum. For our pur-poses, it will be sufficient to consider the situation in which the first-order potential is given as a sum of terms like those in (22), with index j going from 1 to N . The mean horizontal velocity component is

7 2

x

E2 _{L a.3w.k.e2kiZo} (26)

3

j=1

and the low-frequency horizontal velocity component is N i-1 X(t) E2 1 a.a.3f(wik133.k.)e(ki+kj)Z0. i=2 j=1 - wi(ki-kj)e(ki-kj)Zol

xoes ((wi-wi)t- (k1-ki)X0 + (6i-6j)] + O(E3)

(26')

By analogy with an analysis by Newman (1974), we can go one step fur-ther in simplifying these. If we are concerned only with very small difference frequencies, we can write

wk i + wiki =

_+0(cui-i)3,

ki + kj = 2ki [1 + 0(wi-toi)].

In (26'), we note that the second term, containing the factor (ki-k)

is 0(wi-w) , and so it will be included only in the remainder term.

If we look on aiaj as a matrix, (26) is a summation over the diago-nal elements, and (6') is a summation over the triangular matrix elements on one side of the diagonal. By treating the matrix in a symmetrical way,- we can combine (26) and (26') into one expression:

11-'

N N

X + X(t) = e2 1 aiajwikie2kiZ° i=1 j=1

x cos ( ) t - (ki-ki ) Xo+ (di-ój ) ] + 0 (wi-wj ) . N

=

-In the slowly varying part, the first term comes from the interaction between the first-order motions, and the second comes from ¢2 . Both

approach non-zero limits as w1-w2 0 , and so they are both gener-ally of comparable importance. If (k1-k2)h » 1 , the second term becomes very small.

In the cases considered by Bowers, the typical wave period in the spectrum of incident waves was about 5 sec. This corresponds to a wavelength of about 40 m, which, for the water depth of 23 m, can be

considered as a deep-water wave. The natural frequency of the moor-ing system was 0.1 rad/sec or less, correspondmoor-ing to wave lengths of 250 m or more (wavelength = 27/(k1-k2) ). So the difference-frequency motion is definitely a shallow-water motion. In such a situation, both terms in (29) must be included. This was further evident in Bowers' computed results for surge movement, which he showed both with and without the effects of the second-order potential.

Triantafyllou (1982) points out the second-order potential in (28) is, in a sense, not second order at all. If we multiply the denominator by (w1-w2) and set (i) Q = (w1-w2) and (ii) K = (k1-k2) , the denominator becomes KsinhKh-gQ2 cosh Kh . If this quantity is set

equal to zero, it gives the usual dispersion relationship for waves in water of depth h . Since ¢2 satisfies the nonhomogeneous

free-sur-face condition (7), the dispersion relationship is not satisfied by 2 and K . But there is one exception to this: If the water is very

shallow, all waves travel at the same speed (in the linearized theory), and then the denominator in (29) approaches zero. In fact, for very small Kh , the denominator is 0(2) = 0(w1-w2) . If this frequency

difference is considered to be 0(e) (as it is in Triantafyllou's analysis), we have E2¢2 = 0(c2/E) = 0(E) .

Pinkster and Huijsmans (1982), in a very thorough study of the low-frequency forces on a semisubmersible in waves, find that their com-puted forces are consistently 30 to 40 per cent lower than their

measured forces. Since their analysis of second-order forces is based entirely on first-order potentials, and since the wavelength of their low-frequency waves was about 30 times the water depth, the question arises whether the neglect of the second-order potential might be the cause of the discrepancies between their computed and measured results. From Bowers' work, it seems that a difference of about 30 per cent is

quite likely in

such a case. On the other hand, it must be recognized

that there are major differences between the cases that they studied: (i) Bowers considered a ship in head waves only, and he neglected dif-fraction effects. (ii) Pinkster and Huijsmans were concerned with a semisubmersible, and they carefully considered diffraction effects, at least within the scone of the first-order potential.

Bowers makes another point that should be recognized by anyone under-taking experiments of this kind: A laboratory wavemaker (in his case a paddle) generates second-order waves other than those considered here. This was discussed in detail by Madsen (1981).

-Again it should be emphasized: Everything in this expression comes from the first-order potential functions. Also, the corresponding expression for the vertical second-order velocity component

is

0 (wi-wi ) .

We have worked here with X(t) instead of with X(t) for the purpose of avoiding the term in (24) with amplitude going to infinity as

wi-w2 0 . This may seem artificial, but one should note that the steady-transport term in (24) causes a similar problem: Its magnitude depends on an arbitrary choice of time for t = 0 , and it can take

any value from -0. to +co .

Free Waves in Water of Finite Depth

Bowers (1976) presented experimental data and a theoretical analysis for a problem in which it is apparently important to include the second-order potential. He studied the longitudinal drift force on a moored ship in head waves. The frequencies in the spectrum of inci-ent waves were such that the wave motion could be treated as if the water were infinitely deep, but the second-order difference-frequency waves were effectively shallow-water waves.

First consider a wave train in water of finite depth. The first-order potential can be written

(;)1(x,z,t)

ga cosh k (z+h)

sin (wt-kx)

where h is the water depth at equilibrium. The corresponding free-surface elevation is still given by (11). In contrast to the deep-water case, there is now a nontrivial second-order potential:

(02-

(x,z,t) 3wa2 cosh 2k (z+h)_2 sin 2 (wt -kx) .

2[ cosh 2kh -11

One can readily check that this vanishes as kh . The steady

transport velocity is

akok cosh 2k (Zn+h) X =

(27)

cosh 2kh -1

Now consider a superposition of two wave trains, each having the form in (27). Bowers gives the full second-order potential, but it will be sufficient here to simplify the situation: We assume that kih >> 1 for i = 1,2 but that (k1-k,)h is not large. Then we find the following second-order potential:

2a1a2w1w9 cosh (k1-k2)(z+h)

4)2(x'z't) (w1+w2) sinh (k1-k2)h- (w1-w2) cosh (k1-k2)h

xsin [(w1-w2)t- (k1-k2)x] . (28)

The sum of the steady and low-frequency particle velocities is ar + x(t) = e-la2iwikle2k1Z0 4w2k2e2k2Zol + e2ala2 (wiki+w2k2)e(kl+k2)Z0 [2w1w2(w1+w2)/g) cosh (k1-k2) (Zo+h) ) (u,-w,) (wl+w2) sinh (k1-k2)h - (w1-w2) cosh (k1-k2)h x cos [(w1-w2)t - (k1-k2)X0] + 0(e3) . (29) -.

-STEADY DRIFT FORCE AND MOMENT

In order to calculate the steady drift force and moment on a ship or platform in waves, it is not necessary to solve the second-order boun-dary-value Problem even though the force and moment are second-order quantities. Furthermore, Procedures have been developed for computing at least some components of the force and moment without the necessity

of

integrating

_{the pressure over the surface of the body. These are}

both tremendous simplifications in practical problems. _{It should be} recognized that, in the cases to be discussed here, these simplifica-tions do not involve assumpsimplifica-tions beyond those already made, namely, irrotational flow (no viscosity effects), negligible surface-tension effects, and perturbation theory. The presentation here follows closely that of Newman (1967).

Let the body surface, SB , be given by an equation of the form

SB(x,y,z,t) = 0 , ₍₃₀₎

and let

n = (n1,n2,n3)

be a unit vector normal to the body surface, pointing outwards _from the fluid, thus into the body. _{As before, we assure that the fluid} motion can be described in terms of a velocity potential _¢(x,y,z,t) which is a solution of the Laplace equation. The axes Oxyz are the

same as in the previous section, i.e., fixed in space. The potential satisfies (i) the free-surface condition (2), (ii) a body condition

= 11-v) = Un on SB(x,y,z,t) = 0 , (31)

Dn

where Un _{is the normal component of velocity of the body itself,} (iii) an outgoing-wave radiation condition, and (iv) a bottom

condi-tion. If the water is very deep, the last condition is replaced by

Vc}t 0 as z -= . (32)

The steady drift force and moment will be derived from _{an expression} for the rate of change of momentum of the fluid. _{Let S. be a closed} surface, which can be defined in an arbitrary way (not necessarily related to the fluid motion). Let the momentum of the fluid inside S be given by

M(t) =

Iff

p V ch ,

where V = Vc} = fluid velocity and p is the fluid density. Then a fundamental formula of calculus gives the rate of change of M

dm(

av f

at = P

ff.)

dT + v Un dc5 ,

where Un is the normal component of velocity of the surface S itself. The procedure is then (see Newman (1967)) to substitute for 3V/at from the Euler equation of motion and use Gauss' theorem to change the volume integral into a surface integral. _{The result is:}

dM

If

[n(p+pgz) + pV(vn - Un) do

dt

-(33)

where

vn

= = n-7(1) (34)

From the intermediate step (not written out here), it is clear that the term containing gz contributes only to the 'z _{component of} dM/dt .

The force on the body is given by

F(t) = If Ilp(x,y,z,t) do

SB

where SB is the [exact] wetted surface of the body, and p(x,y,z,t) is the fluid pressure. On SE , (31) and (34) show that

following definitions:

(n1,n2,n3)

= n =

unit normal vector on S outward from

fluid;

(n4,n5,n6) = X x n where

X = (x,y,z) = (x1,x2,x3) (38)

denotes a position vector (the notations Oxyz and _0x1x2x3 being

The Proof described above for i = 1,2,3 can be modified to show that this equation is valid also for i = 4,5,6 . It starts with the

defi-nition of fluid angular momentum

K(t) = III P (X x V)dr

and proceeds in directly parallel steps (see Newman (1967)).

(35) Rearranging terms, we

F =dM

_{- a-- Pg}

Vn -have now n z do -if SB Un = 0 . r

in(

) V ( fl p pgz p Vn-Un) }

JS-SB

. (36)

We can generalize (36) to include the moment on the body. We use the

used interchangeably for the fixed axes);

(VI,V2,V3)

= V =

fluid velocity; (39a)

(V4'Vs'V6) = X x V

; (39b)

(1411M2,M3)

= M =

momentum of fluid inside S (40a)

(M4,M5,M)

= K

= angular momentum of fluid inside S ; (40b)

(F1,F2,F3)

= F

= fluid force on body; (41a)

(F4 ,F5 ,F,)0

= G

= fluid moment on body. (41b)

Now we rewrite (36) with these notations: dMi

Fi pg f i ni z do - f I ini (p + pgz) + pVi (Vn-Un) }du . (42)

dt

Next we take the time averages of all terms to obtain expressions for the steady force and moment components. The average value of dMi/dt over all time must be zero; otherwise there would eventually be an unbounded amount of momentum or angular momentum inside S . Denoting time averages by a superbar, we have:

Fi = - pg If niz do - if Ini(t)±pgz) + pV1 (Vn - Un) }do . (43)

SE S-SE

Note that it would be incorrect at this point simply to use time aver-ages of the integrands, since the surfaces SE and S-SE generally change in time.

We consider the cases i = 1,2,6 (surge and sway force, yaw moment) in more detail. As already noted, the terms containing z make no net contribution in these modes, and so we drop those terms. We choose the control surfaces that make up

S-SB

SR : a fixed vertical circular cylinder of large radius, Ro

SF : the part of the free surface bounded by SB and SR , and

SD :

a horizontal plane at great depth.

Note that

on SR ,

U, = 0

on SF , a) p = 0 and b) V, = ;

on SD , a) nl = n2 = n6 = 0 and b) Vi = 0 .

After dropping the terms containing z , we have

Fi = - {nip+ pViVR} da , i = 1,2,6 , f (44) SR where VR =

1L

= radial component of fluid velocity 3R

on SR .

Equation (44) was derived by Maruo (1960) for the cases i = 1,2 Newman (1967) extended it to the important case i = 6 .

The above formula is exact under the assumption of irrotational flow. In order to .carry out computations, however, one has to make some approximations. We assume that the pressure p(x,y,z,t) can be expanded with respect to e . This expansion can be found by

substi-tuting the 4) expansion from (2) into the Bernoulli equation: = Plgz ct 3 17cH₂

1 ;,2 2 2 1

= - pgz - pe¢lt - pe2(4) +-f-19ix+

cl)iy+¢izsj

0(e3) . (45)

Let the surface SR in (44) be divided into two parts: (i) SR0 the surface up to z = 0 , and (ii) the small remaining part of

'SR

between z = 0 and z = . Then we can write the Pressure integral

in (44):

.

ifnip da = - p ni[gz + O1tE214)2t A-4-[4)1x+$21y+(i)21z]l da

SR _SR0

-

pg 0 dZ ni

f

iz -

Ecifdz 0(E3)

'CR

the average value of which is

1 2r,2 1 r 2

11 nip da = - p

ff

ni(gz + -ye 1.9ix+q) ₁ )da + -le2 pg 0 d2,

1

SR SR0 CR

The single integrals are taken along the contour CR , which is the intersection of SR and the plane z = 0 . The boundary condition

(6) has been used to approximate g51t. near z = 0 . Now we have for Fi

Fi = - tnip+ pViVRIda -r 1 2pg 2 di + 0(e3)

SR0 CR _{i = 1,2,6}

with

= E2P{21x+Cbiy+

(The hydrostatic pressure integrates to zero over

SR0 )

Maruo (1960) discussed an important and elegant special case of (46). Consider a two-dimensional body in incident sinusoidal beam waves that come from x = -00 , the potential for which can be written

c/), (x,z, t) = - ekz sin (wt - kx + 6)

with a representing the amplitude of these waves. There will also be outgoing waves at the left and the right, given by the potentials

bw

4)R(x,z,t)

= - ek z sin (wt+kx+e)

cw _kz

= - sin (wt - kx+ -y)

respectively. In (46), the surface SR0 is replaced by two vertical planes at x = ±0, , and the line integral over CR is replaced simply

by

2 Cl

From (46) thus modified, the steady force F1 is found to be

F1 =

/pg

[a2+b2-c21

. (48)

The condition that the net average energy flux through S be zero yields the condition

a2 =. b2 c2 ₍₄₉₎

which, when substituted into (48), gives

F1 = pgb2 . (50)

If the body is fixed, b is the amplitude of the reflected waves and

c the amplitude of the transmitted waves. In this case, F1 is

f

proportional to the square of the amplitude of the reflected waves. If the body is buoyant and free to move with the waves, b _includes

both the reflected waves and the motion-generated waves moving to the

left. In either case, the average force depends only on the square of

the amplitude of the waves that travel in the direction against that of the incident waves.

One additional assumption was introduced above, namely, that no net work is done (or absorbed) by the body. So (49) and (50) are not valid if the body is forced to move by some external force, if it acts as a wave-energy absorber, or if there is significant loss of energy because of, say, viscosity effects.*

Since (49) is identical to (21), which was derived from considerations of mass flux, the Question arises whether the validity of (50) is

really dependent on the existence of energy-flux equilibrium. The answer is clearly affirmative. Equation (21) is not valid, for example, if the body is forced to oscillate in otherwise calm water, since in such a case a = 0 but neither b nor c is generally equal to zero. In this situation, in fact, there are only outgoing waves on both left and right sides, and so there is a net outward mass transport in both directions.

An important asymptotic limit can be found from (50): In the two-dimensional problem of a surface-piercing body, if the incident waves are very short compared with the dimensions of the body,t the ampli-tude of the reflected waves equals the ampliampli-tude of the incident waves. Thus

F1

1-ba2

, as X 0 , (51) 2 '

where, as before, a is the amplitude of the incident waves. This is true whether the body is restrained or free. (Even if it is free, it has no motion in-the limit as w , and so its freedom to respond is irrelevant.)

In general, we still have to solve the first-order boundary-value problem in order to compute Fi . It is not my purpose here to

dis-cuss methods for doing this except to comment on which methods appear to be most useful for obtaining the drift force. Various approxima-tions and simplifying assumpapproxima-tions have been tried, since it is gener-ally either impossible or very difficult to find the velocity poten-tial exactly, even for the linearized problem.

In order to Predict the steady drift force and moment on a ship, Newman (1967) used a version of slender-body theory based on an

assumption that the wavelength is comparable to or much greater than ship length. Unfortunately, this assumption does not lead to

*The body was further assumed not to move in response to the steady drift force, which means that a steady force must be applied to the body by some external means. But a steady force acting on a body that only moves sinusoidally does no net work on the average, and so the existence of such a force does not invalidate (49) and (50).

tStrictly, we should assume that X is small compared with the radius of curvature of the body at its intersection with the free surface. We should also recall that the body surface and the undisturbed water surface are required to intersect at 90°.

acceptable predictions of ship motions under many conditions, and so it does not lead to good predictions of the steady force and moment either. Faltinsen and LOken (1978a) used Newman's general formula,

(46), but they calculated the potential by strip theory (slender-body theory with X comparable with ship beam). They called this the

"Newman-Helmholtz method" because the diffraction potential satisfies the Helmholtz equation in the near field (reducing to the two-dimen-sional Laplace equation for the beam-seas case).

Faltinsen and Michelsen (1974) used (46) to compute the steady force and moment on a large platform structure. To obtain the potential

function, they solved the three-dimensional boundary-value problem numerically, distributing sources on the body surface and solving an integral equation to find the strengths of the sources. This has since become a common method of solving such problems, but it was a major advance in 1974. It is the nearest thing to an exact solution that exists.

Faltinsen and Lleken (1976b) calculated drift force and moment by a variety of methods and compared the results with each other and also with the sparse experimental data that existed then in the open literature. Figure 4 is one of their figures: A loaded tanker in beam seas is considered to be either fixed or free, and predictions of the drift force are shown as predicted by the methods of Maruo

(1960), Newman-Helmholtz (described in Faltinsen and LOken (1978a)), and Faltinsen and Michelsen (1974). Maruo's method is based on the same assumptions as the Newman-Helmholtz method; discrepancies between them occur only for the unrestrained ship at wavelengths near roll resonance ( X/L = 0.8 , where L is ship length), where there is a

difference in the way roll damping is treated (a viscosity effect).

0.3

0.2

0.1

,ASYMPTOT1C

MARUO'S BEAM SEA FORMULA

,/f

SOLUTION

Y (FREE MOTIONS)

().5

₀

MARUO'S BEAM SEA FORMULA

(SHIP RESTRAINED) 0.4

..

FALTINSEN AND MICHELSEN FORMULA

0

ti-,..-..

\

v _{7.-s ::}

(SHIP RESTRAINED)

-.."*;,. . FALTINSEN AND MICHELSEN FORMULA

N.s.,...a (FREE MOTIONS) N91g'-... NEWMAN-HELMHOLTZ FORMULA (FREE MOTIONS) NEWMAN-HELMHOLTZ FORMULA

A

(SHIP RESTRAINED) '0

Figure 4. Drift force on loaded tanker in beam seas (:.rom Faltinsen and LOken (1978b))

0.5 1.0 1.5 2.0

-O...

J. WI

0.2

0.1

A

ASYMPTOTIC

SOLUTION

FY p

NEWMAN'S LQNG WAVE LENGTH p FORMULA (CFA = 01.6)

(FREE MOTIONS)

NEWMAN-HELMHOLTZ FORMULA (FREE MOTIONS) (CB = 0.6) OGAWA'S EXPERIMENTAL VALUES

(SHIP RESTRAINED) (CB = 0.7) NEWMAN-HELMHOLTZ FORMULA

(SHIP RESTRAINED) (C/ = 0-6)

Figure 5,. Drift forte on a Series 60 ship, CB = 0.6, heading

angle 60° (from Faltinsen and LOken (1978b))

For the most part, the ,Faltinsen-Michelsen method agrees with the others. Actually, it is capable of yielding the greatest accuracy, since it is a fully three-dimensional theory. For small wavelengths, however, a very fine grid of source panels must be distributed over the hull surface; this requirement sometimes limits its usefulness. For conditions other than beam seas, there is less agreement between the Faltinsen-Michelsen and Newman-Helmholtz methods, probably because of limitations inherent in a strip theory. The latter breaks down completely for head seas,

as

shown by Troesch (1976).

Figure 5 is adapted from Faltinaen and LOken (1978b). For a Series-60' ship with block. coefficient CB = 0.6 at a heading angle of 60°, it presents two informative comparisons with respect to lateral drift

force: <i) For the unrestrained ship, predictions by Newman's (1967) original method (based on the long-wave assumption) are compared with predictions from the Newman-Helmholtz method. There are large dis-crepancies for small A/L and also near roll resonance. Both methods are based on (46); differences are caused by the inadequacy of the

long-wave assumption. (ii) For the restrained ship, the Newman-Helm, holtz predictions are compared with experiments by Ogawa (1967), show-ing generally good agreementw,,

In both Figures 4 and 5, the high-frequenty limit for the steady drift force is indicated. In Figure 4, which relates to a beam-sea condi-tion, this limit value comes directly from (51); Figure 5 relates to am oblique-sea

condition,

for which (51) has to be modified. In any

case, as, A/L 4- 0 the force approaches the appropriate limit

15

_X/L

0.3-,

regardless of whether the ship is free or restrained. At lower fre-quencies (larger

A/L ),

there is a.very large difference between the drift force on the restrained ship and on the free ship. As wave-length becomes larger and larger, a free ship moves more and more as if it were a fluid particle, and so the reflected-wave amplitude becomes smaller and smaller.

From consideration of the short- and long-wave behavior, we see that wave-diffraction effects are dominant for the short waves, and

body-motion effects nearly cancel diffraction effects for long waves. In general, the neglect of either diffraction or motions

effects may lead to invalid predictions of drift force.

An exception may arise in the case of a ship in head waves. In Sal-vesen's (1974) theory for steady force, diffraction waves are

neglected. Faltinsen and LOken (1978h) showed that this gives poor results for beam and oblique wave headings, but Salvesen's predictions of longitudinal drift force in head seas agree well with predictions made by a computer program at Det norske Veritas that includes dif-fraction effects. Bowers (1976) also neglected diffraction effects in computing the drift force on a ship in head waves, and he nevertheless obtained good agreement with experimental data.

The fact that Salvesen and Bowers neglected diffraction waves and still obtained reasonable predictions must be treated with caution. The diffraction of head waves by a ship is very complicated, as is evident from Faltinsen (1972), Maruo and Sasaki (1974), and Ursell

(1975). The idea that diffraction effects are small simply because of

the small frontal area that a fine ship presents to the waves is quite incorrect; the incident waves are diffracted all along the length of the ship, with a gradual but cumulative effect from bow to stern. Furthermore, the two-dimensional results typified by (50) suggest that even for a ship free to oscillate the dominant cause of drift force is wave diffraction, the motions of the ship serving mostly to reduce the diffraction drift force. In the beam- and oblique-sea cases shown in Figure 4 and 5, this is evident. The situation is not so clear for the head-sea condition.

Lee and Newman (1971) developed an analysis similar to that described above for predicting the mean vertical force and pitch moment on a fully submerged body. Equation (43) is valid for these cases, but the terms containing z must be retained, with the result that the inte-gral over SF cannot be eliminated directly. For the submerged body, we can choose SF as either the actual free surface or the mean free

surface.* We must then evaluate the integral -Ifni(p+ogz)da over SF as part of (41). If SF is the actual free surface, p = 0 but

z = c

in this integral. On the other hand, if SF is the plane

z = 0 , we are left with the integral of nip . Either way, we must

compute an integral over SF .

Lee and Newman (1971) expressed the potential in terms of the Kochin function, which allowed them to transform the free-surface integral into a line integral far away at the intersection of the free surface *For a surface-piercing body, this is not an arbitrary choice. If one Chooses the plane z = 0 as SF , the surface SB is not the exact body surface. And so (35) does not give the fluid force on the body correctly.

and the cylindrical control surface. Thus they were able to express the steady vertical force and pitch moment in terms of the asymptotic

solution at great distance.

This procedure is not valid for a surface-piercing body. In that case, the integral over SF can be transformed in a similar way, but two line integrals remain, one at the control surface far away and the other at the intersection of the body and the free surface. It is not evident that the latter can be eliminated.

Steady force and moment have been extensively studied by many investi-gators for the reasons noted at the beginning of this section: (i) One does not have to solve a second-order boundary-value problem, and

(ii) there are attractive general alternatives (in some cases) to integrating pressure over the body. Neither of these situations pre-vails when we compute the low-frequency force and moment caused by nonlinear interactions between first-order phenomena at multiple fre-quencies. Much effort has been devoted to finding procedures to accomplish similar simplifications, and these efforts have been

par-tially but not entirely successful.

Therefore it is desirable to formulate the second-order problem in a more detailed manner. We do that next.

THE SECOND-ORDER PROBLEM

When the simple procedure of the preceding section is not valid, one must in general find the second-order force and moment by a direct

integration

of pressure over the hull. So we now set up the problem

statement complete to second order.

Transformation of Coordinates We define two sets of axes:

Oxyz = 0x1x2x3 : inertial (space-fixed) axes; O'x'y'z = 01x1:qx'3 : body-fixed axes.

The Oxyz axes have their

origin in

the plane of the undisturbed free surface with the z axis pointing upwards. The two sets of axes coincide when the body is at rest. Let the position of 0' with respect to 0 be denoted by the vector

=

(1,Z2,3)

r (52) and let the position vector to a point in space be denoted by

X = (x,y,z) = (x11x2,x3) X' = (x',y',z') = (lei,x12,x'3)

respectively, in the two coordinate systems. The position vectors are related by a linear transformation

=

D(X-)

(53a)

X =

+ bx'

, (53b)

where D is a matrix and 13 is its transpose. For such matrices, we

note that the product DD is the unit matrix.

,

In setting up the first-order problem, it is convenient and conven-tional to represent the rotations approximately in terms of a vector. In (52), the components denote, respectively, surge, sway, and heave displacement. We extend this scheme by defining the infinitesimal-rotation vector

a = = 0(e) , (54)

where the components denote, respectively, the angles of roll, pitch, and yaw. With this notation, we can write

=

X-

-axX+ 0(e2)

, (55) where we are now assuming that the displacements, (52), are also small. Equation (55) is equivalent to (53a) if we set

D =

(-6

1

[ 1

5 1

and keep only those terms that are linear in the .

The representation of the transformation in terms of a vector opera-tion, (55), is valid only in the linearized problem. In the nonlinear problem, we must use Euler angles (or an equivalent) to specify the

instantaneous orientation of the body, and the resulting expression for D depends on the order in which the three rotations are taken. The order is, of course, arbitrary, since any displacement of a rigid body can be described as the sum of a translation and a single rota-tion about some axis. But that axis is constantly changing in time, and so we must use a systematic method of describing the kinematics of the body. My choice is to take roll, pitch, and yaw, in that order. These are not the Euler angles described in textbooks, but they are more useful for our present problem.

For the moment, neglect the translational motion and consider only rotations (thus 0 and 0' coincide). Define a new coordinate system ORSil that is identical to the Oxyz system except for a positive rotation about the x axis. Thus c = x . The

trans-formation from

X

to

X

is

simple (see Figure 6a):

yl

(a)

Ron (b) Pitch

5

(c) Yaw

Figure 6. Transformation of coordinates

If Ax I cos E6 C I E6 0 E6±E4' 5

1-

WI++ E62r) cos Es 0 sin EsMO

CoS Et+ sin E,6

+ sin E4 sin Es cos E6

cos E4 cos E6 - sin ELF.

sin'Es

sin Es

sin.ELF cos Es

E.5 and. E6 are stall, this can

1 2 2

2

-sin 0

cos E6 0

4 1

Then we make a second rotation, this time through an angle E5 about the y- axis (see Figure 6b) .

Let the new axes be denoted by 0.

The transformation is given by

X = BX

where

sin Es

0

cos

Es

Finally, the third rotation, through the angle E about the 2

axis (see Figure 6c), brings the axes into coincidence with the Ox' y' z' axes:

x' ,

where

The ^complete transformation is obtained by applying these in, order, according to the usual rules of matrix multiplication:

= CEA

cos E5 cos E,6

- cos E5 sin. Es

sin Es

sin sin E6 cos E4 sin Es cos E

sin Ei+ cos E6

+ cos

ELF sin Es sin E6

cos

E4 cos Es be approximated: (57) El-p+ 1 2 2 1- -TE,i+ + Es/A

we see that this agrees with (56)

+ 0(z3) (57t

If we had taken the rotations in a differen4:- prder, i(:7") would differ.

only in the locations of the quadratic quantities in the. off-diagonal elements.

where

f 1 0 0

A

=

I Q

COS L4 sin ELF

0 Ei+ cos E,4,

If we keep only linear quantities, - ,E6 E5 = - s in L D = , 1 - + . 0 B = 1

-The above forms of D can now be used in (53a). The reverse trans-formation will be needed several times. It can, of course, be

obtained by writing the transpose of (57) or (57'). But the follow-form will be especially useful:

X = X' 4-

[E-i-axxI]

+

c2HX' + 0(E3)

(58)

where

or

n =

n'

+ axfl' + 62Hn' + 0(63) . (61') It will be convenient to introduce a generalization of the unit normal vector for use in problems of rotation (Cf. Equation (37b)):

X xn = (n4,n5,n6) (62)

=

[X' + (+ co<X1)

+ c2Hx11 x

[nu

+axn' +

E2fin + 0(63)

= x ni + x

n' + ax (x'xrit

+ 4

x.

(axn') + c2H(x'xn') +

0(e3) . (62')

The first term on the right-hand side of (62') can be identified as

(n4,n15,4)

.

An orthogonal transformation, such as (53a) and (53b), always allows two interpretations: It can simply transform the representation of a vector in one reference frame into the representation in the second reference frame, or it can change one vector into another vector, both being expressed in the same reference frame. We shall need both

interpretations. As already noted, the vectors n and

n'

both represent the same vector. But we can also interpret

n'

as the unit normal vector to the body in its equilibrium position, in which case

(61) and (61') give the time varying unit normal vector n in terms of the constant vector

n'

on the body at rest.

The angular velocity can, of course, always be expressed as a vector. It can be obtained directly from the rotations in Figure 6. In terms

0 0

E H2- = 1

- 2

_{(EL++ C6)}2 2 0

2 (59)

2 2

-2

-2 5C6

₍₁₄

Equation (58) shows explicitly the correction that must be made to (55) when second-order quantities are considered.

If

x'

gives the position of a point fixed in the body, we can use (58) to obtain the velocity of that point, expressed in the inertial reference frame:

U = x = + + 0(c3) , (60)

where H denotes the H matrix with all elements differentiated with

respect to time.

Let n be a unit vector normal to the body, directed into the body. In the O'x'y'z' axes, the same vector is denoted by

n'

. The two vectors differ only in their representations with respect to the

respective sets of axes. The translation represented by is irrel-evant, and so

fl = (n11n,n3) =

at'

= b(111,n12,n) (61)

Z4

Z65

+ 0(c3) (63a ' )

6 514 ,

the last being valid if Ei = 0(c) for i = 4,5,6 . The same vector

expressed in the body-fixed reference frame is

IL, cos E5 cos E6 + Z5 sin E6

w' = E5 cos E6 - E4 cos E5 sin 6 (63b)

E6 + sin 'Es

e

'456

41+6, + 0(c3) . (63b')

46

Equation (63a) can be obtained in another way, using a principle that will be useful several tires later. If X' is a point fixed in the body, its velocity as observed in the inertial reference frame is

= d + -DX11

dt

= + =

+ bp(x-E)

=

t

+

Lox (x-

,

the last following from the definition of angular velocity. It can be checked by.a lenqthy calculation, based on (57) and (63a), that the operator b1, is equivalent to the operator 43 x that is, for any vector y

bpy = X y . (64a)

If

y' varies with time, we also have

y

= Dy + (.4 x y (64h)

It is generally necessary to obtain time derivatives in the inertial reference frame. But once this has been done, the derivative can be transformed into the body-fixed reference frame.

Boundary-Value Problem

The free-surface condition on the velocity Potential has been given already in Equation (2). When the potential is expanded in a pertur-bation series, as in (3), the first two terms satisfy (5) and (7), respectively, on the plane z = 0 . These conditions are applied on that part of this plane lying outside of the vessel in its mean position.

In the first-order problem, the radiation

condition

is conventional: At infinity, must represent (i) the specified incoming waves and

of the three angles of rotation defined above, it is:

+ Z6 sin E5

03 = E5 cos ' 14. - E6 sin EL, cos E5 (63a)

E5 sin EL, + E6 cos EL, cos Es,

=

-E,)

,

w