Second order wave diffraction around two-dimensional bodies by time-domain method

MICHAEL ISAACSON and KWOK FAI CHEUNG

Departnwnt of Civil Engineering, University of British Cohtmbia, Vancouver, B.C., V6T 1Z4, Canada

A time-domain second order method is developed to study the nonlinear wave forces and runup on a surface piercing body of arbitrary shape in two dimensions. The free surface boundary conditions and the radiation condition are satisfied to second order by a numerical integration in time and the field solution at each time step is obtained by an integral equation method based on Green's theorem. The solution is separated into a known incident potential and a scattered potential. The initial condition corresponds to a Stokes second order wave field in the domain, and the scattered potential is allowed to develop in time and space. The stability and numerical accuracy of the proposed solution and the treatment of the radiation condition to second order are discussed. Comparisons of wave forces are made with previous theoretical and experimental results for the case of a semi-circular cylinder with axis at the still water level and a favourable agreement is indicated.

Key Words: Nonlinear diffraction, runup, second order theory, waves, wave forces.

1. I N T R O D U C T I O N

Numerical modelling of nonlinear wave diffraction around large offshore structures has been the subject of investigation for a number of years. The motivation of such studies arises primarily because of the need to obtain more accurate wave force and runup predictions than those of linear diffraction theory which is based on the assumption of infinitesimal wave heights. In general, two categories of methods have been developed to solve the nonlinear wave diffraction problem. One is a second order solution obtained by a perturbation method while the other is a full nonlinear solution obtained by a time-stepping procedure.

In the perturbation methods, a second order correction term is developed and applied to the linear solution in the frequency domain. The fundamental case of wave diffraction around a fixed vertical circular cylinder has received particular attention (e.g. Molin ~, Hunt and Baddour 2, Chen and Hudspeth 3, Eatock Taylor and Hung 4, and Abul-Azm and Williams 5) while the problem involving axisymmetric bodies has been treated by Kim and Yue 6. The more general case of a structure of arbitrary shape in three dimensions has been treated by Garrison 7 and the two-dimensional problem with structures of arbitrary shape in infinite water depth has been treated by Kyozuka 8, Vada 9 and Miao and Liu 1~ In general, these methods have been found to be algebraically complicated and are restricted for applications to regular or bichromatic wave diffraction.

A second approach has been to develop a full nonlinear solution for the boundary value problem numerically by

Paper accepted June 1990. Discussion closes February 1992.

�9 1991 Elsevier Science Publishers Ltd

a time-stepping procedure. Since the free surface boundary conditions are applied at the time-dependent water surface, a new system of simultaneous equations must be generated and solved at each time step. The method was introduced by Longuet-Higgins and Cokelet 1 on the basis of a complex potential and considerable progress has been made on the accurate calculation of two-dimensional flows, including breaking waves in the presence of a fixed obstacle in the flow ~2, and breaking waves generated by a wave-maker z 3. The three-dimensional problem involving fixed or floating bodies of arbitrary shape was treated by Isaacson ~4 on the basis of an integral equation method based on Green's theorem. This approach requires a substantial computational effort which limits its application, and possible numerical instabilities or errors and the effects of waves reflected from control surfaces surrounding the fluid domain of interest impose further limitatons to the length of simulation time.

The present paper presents an alternative approach to the nonlinear wave diffraction problem which modifies the time-stepping procedure by incorporating a perturba- tion expansion and a radiation condition. A variant of the present method with a different expansion procedure and a different initial condition has been given by Isaacson and Cheung ~ 5. However, their method does not account for the radiation condition at second order in an entirely satisfactory manner, the treatment of surface-piercing bodies led to difficulties and the duration of simulation is somehow restricted. The alternative developed here appears to overcome these drawbacks. Although the method is developed in the context of the two-dimensional problem with a surface piercing body, the extension to three dimensions can readily be made on the same basis.

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Second order wave diffraction around two-dimensional bodies by thne-domahl method." M. Isaacson attd K. F. Cheung

2. THEORETICAL FORMULATION

Full nonlinear problem The full nonlinear boundary value problem defining

the fluid motion is first considered here. With reference to Fig. I, the two-dimensional problem is defined with a right-handed Cartesian coordinate system (x, z), in which x is measured horizontally in the direction of incident wave propagation and z is measured vertically upward from the still water level. Let t denote time and tl the free surface elevation above the still water level. The seabed is assumed horizontal along the plane z = - d . With the fluid assumed incompressible and inviscid, and the flow irrotational, the fluid motion can be described by a velocity potential 4, which satisfies the Laplace equation within the fluid domain:

V24, = 0 (1)

and which is subject to the following boundary conditions:

04, = 0 at z = - d (2) Oz

O_~ = 0 on S,,, (3) On

04, 0,1 04, D,I = 0 on S I (4)

0z 0t ax 0x

t- O'l + ~ I V4,12 = 0 on S.r (5) Ot

Here 9 is the acceleration due to gravity, n denotes distance i n the direction of the unit normal vector n directed outward from the fluid region, S,,. is the instantaneous wetted body surface, S s is the free surface z = q and n_. is the direction cosine of n with respect to the z direction.

Equations (2) and (3) correspond to the kinematic boundary conditions on the seabed and body surface respectively, while equations (4) and (5) correspond to t h e kinematic and dynamic free surface boundary conditions respectively. In addition suitable radiation conditions are to be applied on the control surfaces S~1 and S~2 some distance from the body as indicated in Fig. 1. The major difficulty associated with this problem arises primarily from the poorly defined radiation condition involving transient waves and from the two

,..----z . . . . .

i !

i Scli

z t ....

-- Sw Sc2

d

/ ] V / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Z G

Fig. 1. Definition sketch

nonlinear free surface boundary conditions applied on the time-dependent free surface which itself is unknown a priori.

Second order expansion For moderate wave heights it is possible to reduce the

two nonlinear free surface boundary conditions to con- ditions evaluated at the still water level. As in the development of Stokes second order wave theory, the nonlinear free surface boundary conditions are expanded about the still water level with a Taylor series expansion:

0(~ z O, l 04,0,1 , (9 (04, O, l 04, 0,1~ Ot Ox -~x ) + q ~z fizz Ot Ox Ox ]

+ . . . . 0 on So (6)

1 V4,12) 1 0 (~t+g,,+_ ~ (~--~- + e,/+ ~ I v4, I z) + ,, ~ I

+ . . . . 0 on So (7)

The problem may now be a cast in a time-independent geometry which includes the still water surface, denoted as S o, in place ofS I. It is now convenient to separate the first order and second order quantities in the formation by introducing perturbation expansions for 4, and q:

4,' = ~4,1 + ~24,2 + " " (8)

I l = FAll + 132112 + ' ' " (9)

where e is a perturbation which is small. The first and second order potentials and free surface elevations are further separated into incident and scattered components,

4, -- ~(4,~' + 4,]) + ~(4,~' + 4,9 + " " (IO)

t 1 = r + 11]) + t;2(ll~ ~ + 1/Z ) + " " (1 I)

The superscript s indicates the scattered components, while the superscript w indicates the incident components which are known from a Stokes second order wave theory.

Upon substituting equations (I0) and (11) into equations (1), (2), (3), (6) and (7), and collecting terms at first and second order, the Laplace equation and boundary conditions at first and second order are obtained for the scattered potential.

First and second order boundary conditions In the first order problem, the seabed and body surface

boundary conditions, the kinematic and dynamic free surface boundary conditions are given respectively by

04)] = 0 at z = --d (12) Oz

04,] 04,'~ an an on S b (13)

04,] o,1] Oz Ot = 0 on S O (14)

04,] O--i-- + all] = 0 on S O (15)

where S b is the body surface below the still water level. At second order, the seabed boundary condition, the body surface boundary condition and the two free surface

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Second order wave diffraction around two-dimensional bodies by thne-domaht method: M. lsaacson and K. F. Cheung

boundary conditions are given respectively as

34,[ = 0 az

a4, ' On On

at z = - d

. o n S ~

(16)

(17)

1 a.,a+, az aS-= \ az at/+oxo---x ,h

on So (18)

at + a ' t 2 ) - 2 Iv4'tl -'h azat on So (19)

Each term on the right-hand side is either known from the first order solution or from the incident wave components.

With the problem now defined in terms of the scattered components, a radiation condition applied to the scat- tered potential can now be introduced. The Sommerfeld radiation condition, originally derived on the basis of spatially and temporally periodic conditions of a poten- tial function, has been extended to account for unsteady wave motions by Orlanski ~6. In the time domain, the radiation condition for the scattered potential at k-th order is given by

344, 34, , a--~ "+ c ~ = 0 on S~l and Sc2 (20)

where c is the time-dependent celerity of the scattered waves on the control surfaces. Literally, the radiation condition is applied locally and momentarily on the control surfaces with the celerity calculated near the control surface every time step.

Integral equation The problems at first and second order may now be

defined for a time-independent domain such that q~] and q5" in turn satisfy the Laplace equation within the domain �9 2

and are subject to the boundary conditions on its boundary. Since q~] and q~ both satisfy the Laplace equation within this time-independent domain, Green's theorem may now be applied on the time-independent domain to provide integral equations for ~] and ~ .

The potential at k-th order at a point x on the surface of the domain is given by the following integral equation

~bi(x) = re1 G(x, ~) ~ (~) -- ~b~,(~) On ~) dS

S'

(21)

Here ~ represents a point (~, () on the surface S' over which the integration is performed, G is a Green's function and n is measured from the point ~. In the present context the surface S' would Comprise of the body surface Sb below the still water level, the still water surface So, the control surfaces S~x and S~2 and the seabed, as

indicated in Fig. 1. However, because of the assumption of a horizontal seabed, it is more efficient to exclude the seabed from S' and to choose a Green's function that accounts for the symmetry about the seabed. This is

G = l n r + I n r' (22)

where r is the distance between the points x and r and r' is the distance between x and the point ~ '=(~, - ( ( + 2d)), which is the reflection of the point ~ in the seabed.

An alternative integral equation for the potential at second order, which contains additional second order terms due to the expansion of Green's theorem about the still water level, has been given by Isaacson and Cheung ~5, but equation (21) is consistent with Stokes second order wave theory and is preferred for more general applications. With either the potential or its normal derivative on each portion of the boundary S' evaluated from the corresponding boundary conditions, the solution to the integral equation (21) can be obtained by a numerical procedure from which the remaining unknowns can be evaluated.

3. NUMERICAL PROCEDURE

Field solutions at first and second order The integral equation (21) is solved by a numerical

procedure in which the boundaries Sb, S~1, S~2 and So are discretized into finite numbers of facets. The integral equation involving the scattered potential 4'~, at k-th order may then be rewritten without approximation as

[,; ~b],(x 3 = - ~ G(x~, r ~ (~)dS j = l

As~

l f g a ~ ( , j ) ~ ( x , , r . . . . . N)

as~ (23)

in which N is the total number of facets in the formulation and AS i is the area ofthej-th facet. To develop a numerical solution, the values o f ~ and 0q~/&z are taken as constant over each facet and applied at the centre. Equation (23) is reduced to the following set of simultaneous equations:

a $ $

y. Y. " \ a,, Jj - =tZ 1 = 1 j f N b + l

N

-- ~ Ao(a~i (i= l . . . . ; N) (24) ] = N b + l

in which Nb is the number of facets on Sb. It is understood that the surfaces are treated in the order of Sb, Set, Sc2 and S O as the index j increases from 1 to N.

The coefficients B o and A u correspond respectively to integrals of the Green's function and its normal derivative over area of thej-th facet as in equation (23). When i =j , the coefficients are evaluated by a closed form integration and when i # j , the integration is performed by a Gaussian numerical integration. The coefficients A o and B u are

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Second order u'are diffraction around two-dimensional bodies by time-domaht method: M. Isaacson and K. F. Cheung

given by:

ASs n'( -x) n'-(~,-x) (ir 2re , . = 1 r m

Ais = (25) ASi .u n"(~m--X)

l - - - ~ - ~ Wm ,2 ( i=J) m = 1 r m

in terms of the solutions obtained at previous time-steps:

qS~,(x, t + At) = 1 -- cAt~An l -+ cAt/An ~b~,(x, t - At)

2cAt~An Jr 1 + cAt/An qS~,(x- nan, t) (28)

AS ,~t . , , ~ ,~ ,,=El w,,(ln r,, + in r,,; (i # j)

Bq = (26) AS i [- [ASi" ~ -] AS i .-. T [ l n ~ , - 5 - ) - l J +2-n--n ,,~='l w,,, In r~, ( i= j )

where m is the index for the Gauss points, M is the number of Gauss points used in the numerical integration, w,, is the weight function at the Gauss point m and n ' = (nr -nr is the normal vector at the image point ~'.

It is important to note that the coefficients in the system of simultaneous equations (equation 24) are invariant in time. Thus, the solution to the system of equations is required only once rather than at each time step, compared to a more conventional nonlinear time-domain solution in which a new system of simultaneous equations must be generated and solved at each time step. Furthermore, the coefficient matrices are functions of geometry only and consequently the same set of equations can be applied to different incident wave conditions with modifications only made to the right-hand side input vector.

The input vector on the right-hand side consists of the normal derivative of the scattered potential on the body surface which is known from the incident potential, and the potential on the control surfaces and still water surface which is to be evaluated by an explicit time-integration procedure applied to the corresponding boundary conditions.

Time-hztegration o f radiation condition In order to be able to simulate a sufficiently long

duration in a reasonably sized domain, the application of a radiation condition on the control surface so as to avoid reflected waves requires particular consideration. In the present method, based on Ref. 16, the Sommerfeld radiation condition with a t ime-dependent celerity is integra!ed numerically in time to obtain the required scattered potentials at first and second order on the control surfaces.

Considering a point x o n the control surface, the radiation condition for the scattered potential at k-th order at time t is expanded by a central difference in time and a leap-frog difference in space:

(a~k(X' t + At) - (a~(x' t - At) § c {~ 2At ~ [~b[(x, t + At)

At)] -- ~b~,(x - nan, t)} = 0 (27) + 4,1(x, t

where At is a time-step size and An is a small distance of order a facet length. From equation (27), the potential on the control surface at time (t + At) can be expressed

Literally, the boundary value is extrapolated from the values of ~b~, near the boundary inside the domain at previous time-steps through the knowledge of the celerity c on the control surface.

For diffraction of regular waves at first order, a constant value of c corresponding to the linear celerity of the incident wave train may be used in equation (28) to calculate the scattered potential on the control surfaces. For diffraction of regular waves at second order, the scattered potential is made up of second order free waves at twice the incident wave frequency as well as second order forced waves associated with the inhomogeneous terms in the second order free surface boundary conditions. Therefore, the radiation condition cannot be satisfied with a unique celerity and is evaluated numerically near the control surfaces as

a~llat c = (29)

O ~ l / O n

As a physical requirement of the radiation condition in the time domain, information must be allowed to flow from interior points in the fluid domain to the boundary. The celerity on the control surface at time t is therefore approximated by that calculated at (t - At) and (x - nan) instead. From equation (27), the numerical value of c is given by

A,l 4,~(x,, t) " " -- q~k(xl, t -- 2At) C ~ - -

At ~,(xl, t) + ~b~,(x 1, t - 2At) - 2~b~(x2, t - At)

(30)

in which x I = x -- nan and x 2 = x - 2nAn. Unfortunately, even at first order there is a numerical difficulty in using equation (30) in that at a zero-crossing of the free surface and the still water surface both O~b]/Ot = 0 and 0~]/On = 0 and equation (30) becomes undefined. Even at locations near zero-crossing points the numerical values of c calculated by equation (30) are inaccurate. At second order, the numerical procedure is further complicated by the presence of both the forced and free waves. The value of 0qS~/a), is not necessarily zero when O~/On = 0, and the resulting celerity calculated by equation (30) may be very large.

In order to circumvent these numerical difficulties, the following numerical scheme is carried out. When both the absolute values of Odp[/gt and g~bW0n are less than a certain prescribed value, the celerity is not calculated from equation (30) and the scattered potential ~b~(x, t + At) in equation (28) is simply given by q~(x, t), since a ~ / 0 t is small and ~, changes very little in one time step. As the Sommerfeld radiation condition requires the scattered waves at large distance from the body be outgoing, only the outward celerity (i.e.c.~ 0) is used in describing the boundary condition. Finally, for stability reasons the maximum value of the celerity calculated by equation (30) is limited to have a maximum value of An/At. To

178 Applied Ocean Research, 1991, Vol. 13, No. 4

Second order wave diffraction around two-dinlensional bodies by thne-donlain nlethod: M. Isaacson and K. F. Cheung

summarize, the numerical values of c is given by:

O for c'~<0 c = c' for 0 < c' < An~At (31)

An~At for c' >~ An~At

where c' is the initially calculated value of the celerity from equation (30).

According to Lee and Leonard ~7, the control surfaces have to be positioned at least three times the water depth away from the body in order to exclude the effects from evanescent modes or local disturbances. This criteria has been confirmed by numerical testing and has been adopted in the computations of the results presented.

Time-hltegration of free surface boundary conditions The free surface boundary conditions are used in an

iterative time-stepping procedure to provide the free surface elevation and the velocity potential on the still water surface at each time step. Initially, the free surface elevation at a new time (t + At) is first evaluated in terms of the known solution up to time t by the first order Adam-Bashforth equation as

'l~(t + At)='l~(t)+A---~ [3(c911~ _(&li'~ 1 (32) L \ Ot/, \ Ot/,iaeJ

The dynamic free surface boundary condition, equations (15) and (19), can now be used to provide c3~b~/0t at (t + At). In turn the potential ~bl may then be obtained by the first order Adam-Moulton equation:

</>l(t + At)= </>l(t)+-7- Lk at ), +\ at ,/,+A,J

With the normal derivative of the velocity potential on Sb known from the incident wave potential and the velocity potential on Sol, Sc2 and S O evaluated from the time-stepping equations, the flow in the domain at the advanced time (t +At) can now be solved from the discretized boundary integral equation (24). The output vector provides cOqSl/dn on So at (t + At). The kinematic free surface boundary condition, equations (14) and (18), can now be used to provide the corresponding values of aql/Ot at the advanced time (t + At).

This completes an initial calculation of all the boundary parameters at time (t + At) and the computation could then proceed to the next time step. However, an improvement to these values may be obtained by applying an iterative procedure in which improved values of the free surface elevation at (t + At) are obtained by using the first order Adam-Moulton equation in place of equation (32):

, At F/a,ll~ fa,llk 1 +tw),., /

The improved value of ill can then be used to obtain in turn Ocbl/Ot from equation (15) or (19), ~b I from equation (33), O~bl/0n from equation (24) and Oql/at from equation (14) or (18).

Wave forces In the present application, the pressure distribution

over the body surface may be determined by the unsteady

Bernoulli equation,

p = - p ~ - f - ~ plVqS, I z -pgz (35)

where p is the fluid density. The values of a~/at and Vq51 can be obtained by applying a central difference approximation to q5 in time and space respectively.

Once the potentials at first and second order have been obtained, the wave forces acting on the body can be determined by carrying out an integration of the pressure over the instantaneous wetted body surface S.,. The force vector is then given as

t" F = I pn dS (36)

S.

To a second order approximation, equation (36) can be simplified by assuming the value of dg~/Ot within the region bounded by the free surface and still water level be given by that evaluated at z = 0. Equation (36) is then simplified to

f ' f F= pn dS + ~ p# ,l~n dl (37)

Sb I

where l is the waterline contour and it is assumed that the body intersects the still water surface perpendicularly. The second integral in equation. (37) accounts for the wave force due to the fluctuation of the free surface around the body and represents a resultant between the hydrostatic and dynamic pressures. In two-dimensional problems, it is reduced to two terms evaluated at the two free surface-body intersection points.

For regular wave diffraction to second order, the total force acting on the body may be expressed in the usual way as a sum of three component forces:

F = F, + F / + F 2 (38)

where Fx, F o and F 2 are respectively the linear wave force at the incident wave frequency, the second order steady drift force and the second order oscillatory force at twice the wave frequency. The expression for each force component is given as:

F~=-pf~-~ndS s~

(39)

l p<flV~bll2n dS>+~pg<ftl2n dl> (40) F o = - ~

s, t

F 2 = --p ndS - -~p IV~b112n dS

S~ s~

, f +2 pv ,t~n d l - F o (41) I

where ( ) denotes a time average. Each of the above

three force components is further made up of horizontal and vertical components. In addition, the preceding

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Second order wave diffraction around two-dimensional bodies by thne-domahz method: M. Isaacson and K. F. Cheung

formulation can readily be extended to include also three components to the moment acting on a three-dimensional body.

4. COMPUTATIONAL CONSIDERATIONS

To illustrate and examine the present method, and to compare with known theol:etical and experimental results, the special case of deep water regular wave diffraction around a submerged semi-circular cylinder with axis at the still water level is considered in this paper. The computation was performed on an IBM 3090/150S computer at the University of British Columbia and double precision was used throughout. The matrix coefficients in equations (25) and (26) are evaluated by a 4-point Gaussian integration.

Initial conditions aml development of scattered potential The initial condition corresponds to a regular Stokes

second order wave train in the domain. The development of the scattered potential in time and space is attained by the imposition of the body surface boundary condition (equations (13) and (17)). In the simulation of full nonlinear wave diffraction with the same type of initial condition, Stansby and Slaouti 18 and Isaacson and Zuo 19 applied the body surface boundary condition instantly at the first time step corresponding to an impulsive- started condition. The transient effects associated with the abrupt initial condition were found to decay rapidly in time and the established flow is reached after relatively few time steps.

On the other hand, by modulating the body surface boundary condition in time it is possible to avoid such an abrupt initial condition and instead allow a gradual development of the scattered potential. Before applying in equation (24), the right-hand sides of equations (13) and (17), which correspond to the body surface boundary condition are multiplied by the following modulation function,

Fro= 2 l - c o s for t < T , ,

1 for t >/T,, (42)

where T,, is a modulation time. Consequently, a smooth transition between the fully developed initial condition and the subsequently developed scattered potential is obtained by the gradual imposition of the body surface boundary condition in the computation.

In order to illustrate the influence of T,,, Fig. 2 shows comparisons of free surface profiles obtained at time tiT = 5 using different modulation times, T J T = 0, 0.5 and 1, where T is the wave period. The incident wave conditions correspond to ka =0.6, H/L = 0.1 and deep water, where a, k, H, L are respectively the radius of the cylinder, the wave number, height and wavelength of the incident wave train. In the figure, A = HI2 is the wave amplitude, and the cylinder axis is located at x/L = 2. When the modulation time T,, = 0, some high frequency dispersive components associated with the impulsive- started condition are just noticeable in the flow at first order. However, their effects are considerably amplified at second order. On the other hand, the same phenomenon is not observed in the profiles corresponding to T,,/T = 1 and 0.5. Since the amplitude of these high

q

I T m / T = 0 . 0 A .h t,

o ~ ! T [ 1 I i i

0.0 1.0 2.0 3.0 4.0 o.

,.<o ~ o ~ o ,-4 f ~ !

0.0 1.0 2.0 .3.0 4.0 o

:<q - - - - ~ ~o

o

0.0 I.O 2.0 3.0 4.0 x,/t_

Fig. 2. Comparisons of fi'ee smface profiles obtained by different modulation thnes T m for ka = 0.6, H/L = 0.1, t i t = 5 and deep water. - - - , sohttion to first order; - - , sohttion to second order

frequency components is generally small, their effects are mainly confined to the region near the free surface. The influence on the resultant force acting on the body is usually small except for relatively small bodies. However, the steadiness of the free surface elevation is affected to a greater extent.

Through extensive numerical testing, it has been found that a smooth transition between the initial condition and the fully developed flow is desirable for a stable and steady solution at second order. For ka less than about 0.3, it is difficult to obtain a steady solution at second order without the use of a non-zero modulation time. The impulsive-started condition also affects the perform- ance of the radiation condition and eventually the duration of simulation. In the results subsequently reported in this paper a modulation time T, J T = I is adopted and generally a steady state solution is obtained immediately after the duration of the modulation.

Despite the use of the modulation, the present procedure is still more efficient and reliable in comparison to Isaacson and Cheung's 15 earlier approach in which the initial condition corresponds to still water everywhere in the domain and an incident wave train is generated on the control surface. In the earlier method, it takes a few cyclesbefore a steady state condition can be developed, since the incident wave train modifies itself before reaching the body. In the simulation of irregular wave diffraction, the present procedure has the advantage of providing a firm control on the incident wave condition at the test section.

Stability of thne-steppin9 procedure Similar to other problems which incorporate time

stepping, the stability of the proposed solution requires that the time step size At is sufficiently small in relation to a characteristic facet size AS. This condition corresponds to a minimum value of T/At, for any given L/AS. For the purpose of illustration in this section, a constant facet size is used on the entire boundary while in practice smaller facets are used on the body surface and on the nearby free surface.

For incident wave conditions correspond to ka = 1 and

180 Applied Ocean Research, 1991, Vol. 13, No. 4

Second order wave diffraction around two-dimensional bodies by time-domain method: M. lsaacson and K. F. Cheung

Table 1. Maxhman T/At as a fimction o f L / AS for a stable sohttion o f regular wave diffraction for ka = 1 and deep water

Table 2. Ware rzmup components as fimctions o f Nbfor ka = 0.6 and deep water

No iteration, (T/At)min 1 iteration, (T/At)rain Upwave Runup Downwave Runup

L/AS 1st order 2nd order 1st and 2nd order Nh RI/A Roa/A 2 R2"t/A 2 RI/A Ro:t/A 2 RI:dA 2

10 1.8366 0.4~99 0.6058 0.4246 (}.1)336 0.2346 I0 13 15 7

15 1.8419 0.4942 0.7859 0.4202 0.0365 11.2554

20 19 21 9 2o i.8474 0.5044 0.9295 0.4208 0.0392 0.260(i

30 28 30 12 25 1.8494 ( I .5067 1 . 0 0 2 4 0.4207 0.0400 0.2634

30 i.8498 0.5092 1 . 0 3 5 2 0.4204 0.0412 0.2683 40 31 33 13 35 1.8520 0.5147 1.0976 0.4209 0.0418 0.2661

50 34 36 15 40 1.8512 0.5148 1.0890 0.4206 0.0424 0.2734

deep water, Table 1 shows the minimum values of T/At for various values of L/AS which gave rise to stable solutions at first and second order. For smaller values of T/At, the solution exhibits an oscillatory instability, while for larger values the solution remains quite stable. In the present application, the time-stepping procedure converges rapidly and the solution appears not to be affected by increasing the value of T]At above the minimum requirement or by using the iterative approach involving equation (34). However, when the time-stepping procedure with one iteration is used, the minimum value of T/At required to maintain stable solutions at first and second order is found to be less than half that when iteration is not used. It is interesting to observe that for L/AS = I0, a stable solution to second order can be obtained with T/At as low as 7.

In the subsequent applications, smaller facets are used on the body surface and on the nearby free surface in order to render more accurate calculations of the wave force and runup without increasing the number of facets on the entire boundary. The smallest facet in the discretization is then used to estimate the required time step size to maintain stable solutions at first and second order. Furthermore, the iteration procedure involving equations (32), (33) and (34) is only applied for the first few time steps to account for the transient effects associated with the initial condition, and the subsequent development of the flow is obtained by the applications of equations (32) and (33) without the use of iteration.

Accuracy of mmlerical solution Quite apart from the question of stability, the

numerical accuracy of the solution must also be assessed in terms of the facet size chosen. The computed wave forces and runup are found to depend on the number of facets used to represent the body surface, and in general, the convergence of the computed wave force and runup with increasing number of facets depends on a number of factors such as the relative size and water depth. To illustrate the effects, the special case corresponding to ka = 0.6 and deep water is considered here.

The runup components at first and second order in their dimensionless forms are shown in Table 2 as functions of N~, the number of facets on the body surface. The first order runup Ra is found to converge rapidly with the number of facets and an accurate result can be obtained with Nb less than 15. Since the second order steady runup R o can be calculated from the first order wave components, its convergence is slower due to

accumulation of numerical errors, but generally follows that of the first order solution. In the deep water case considered here, the wavelengths of the second order free and forced waves are respectively one-quarter and one-half of the first order wavelength, and more facets per first order wavelength are required to obtain a satisfactory resolution of the wave profile at second order. Therefore, the convergence of the second order residual runup R 2 is much slower compared to other components and a smaller facet size is required to achieve accurate results.

The convergence of the wave force components at first and second order in their dimensionless forms are shown in Tables 3 and 4. The first order force components in the x and z directions (denoted by Fix and FI : respectively) are found to converge rapidly with the number of facets representing the cylinder and very accurate results can be obtained with N b as low as 10. The convergence of the second order steady force components (denoted by Fox and F0:) and the second order oscillatory force component due to ~b~ (denoted by U1} and Ft21: ~) is more rapid for the z direction than for 2x the x direction. This is because the force components in the x direction contain an additional contribution from the free surface fluctuation as well as the velocity squared term in the Bernoulli equations and hence more numerical errors are involved. The convergence of the second order oscillatory force components, F~2.~ and F~]), due to ~b2, is slower compared to that of F ~ and F~), due to ~b~, while numerical errors are carried over to the resultant second order oscillatory force components, denoted by F2. ~ and F2: .

Table 3. Wave force components hz x direction as fimctions o f Nbfor ka = 0.6 and deep water

(1) (2) Nb Flx/f)gaA Fox/pgA 2 F2x/pgA 2 F2x/OgA 2 F2x/pgA 2

10 1.1631 0.3716 1.3482 1.1040 0.3721

15 1.1764 0.3740 1.3574 1.1701 0.3918

20 1.1804 0.3814 1.3612 1.1897 0.4316

25 1.1826 0.3860 1.3606 1.2010 �9 0.4500

30 1.1839 0.3889 1.3587 1.2046 0.4667

35 1.1844 0.3928 1.3593 1.2096 0.4783

40 1.1852 0.3934 1.357i 1.2060 0.4763

Applied Ocean Research, 1991, Vol. 13, No. 4 181

Second order wave diffraction around two-dhnensional bodies by tbne-domaht method: M. lsaacson and K. F. Cheung

Table 4. IVavc force components ht z direction as fimctions of Nbfor ka = 0.6 and deep water

0) - 2 (2) NI, Fi,/pg:tA Fo,/pgA 2 F2~lpg~ F2,/pgA 2 F2,/pgA 2

10 1.0441 -0.3622 0.3433 1 .1356 0.8022 15 1.0433 -0 .3612 0.3423 1 .1776 0.8444 2(1 1.0430 -0.3602 0 .3411 1 .2025 0.8689 25 !.0430 -0.3599 0.34(]6 1 .2127 0.8790 30 1.0429 -0.3597 0.3404 1~135 0.8797 35 1.0429 -0 .3595 11.3401 1 .2150 0.8810 40 1.0429 -0 .3595 0 .3401 1 .2163 0.8820

Effectiveness of radiation condition at second order The radiation condition at first order is satisfied by the

application of a constant celerity corresponding to the first order incident waves. At second order, the radiation condition is satisfied by a time-integration involving a time-dependent celerity, and its performance depends on the accuracy of the numerical scheme in which the celerity is evaluated.

In order to examine the effectiveness of the radiation condition, the development with time of the second order oscillatory force due to q52 predicted by two models of different lengths, L J L = 2 and 4 is compared in Fig. 3. In both cases the cylinder axis is located at the still water level and half way along the domain's length, and the cylinders a r e subjected to identical incident wave conditions corresponding to ka = 0.6, H]L = 0.1 and deep water. The figure shows results for the smaller domain with a rigid wall condition applied at the radiation boundaries, and with both a constant celerity and a time-dependent celerity used in the radiation condition. These are compared to results for the longer domain for which reflected waves are absent.

It is evident from the figure that the solution obtained using the tme-dependent celerity gives the best agreement with the solution obtained with the large domain.

Although a slight effect of reflection is observed in the wave force predictions, the use of the time-dependent celerity is expected to give even better results in three-dimensional problems in which the scattered wave amplitude decays with distance away from the body. Results using a constant celerity are somewhat worse, on account of the reasons already given, and results using a rigid wall condition indicate significant reflected waves as expected.

In frequency domain methods, the validity of a radiation condition has immediate effects on the accuracy of the solution and considerable research has been directed towards the development of a proper radiation condition at second order. In the present method, the solution is independent of the radiation condition applied for a sufficiently long duration of simulation and the adverse effects of the radiation condition on the solution can confidently be removed regardless of the application of the radiation condition or not.

6. RESULTS AND DISCUSSION

Free surface profiles and rmmp Figure 4 shows the development of the free surface

profiles with time to first and second order for ka = 0.6, H/L = 0.1 and deep water. In the figure, the cylinder axis is located at x /L = 2. The flow immediately adjacent to the body appears to repeat itself after the first cycle, while the flow further from the body takes somewhat longer to reach a steady state. The scattered waves at first and second order propagate steadily away from the body at their corresponding group velocities. In the upwave region, the first order scattered and incident waves propagate in the opposite direction forming a system of partial standing waves and giving rise to a strong second order forcing in the generation of second order forced and free waves. In the downwave region, the first order scattered and incident waves are out of phase and propagate in the same direction. The resultant wave amplitude is small

q

~t-~ ~ [ , , , ~ ~; ,~ L, ~, ~, ~ t' ~, , ' '~, I,' ~ ' ; i:, ,, I

0 . 0 2 .0 4 .0 6.0 8 .0 10.0 o

Lt- O.

7 0 . 0 2 .0 4 . 0 5.0 8 . 0 10.0

VT

Fig. 3. Development with time o f horizontal and vertical wave force components due to qb2for ka = 0.6, H]L = 0.1 and deep w a t e r . - - , L D / L = 4 ; . . . . , L D / L = 2 w#h time-dependent celerity; ...... , L D / L = 2 with constant celerity; - - - - , L J L = 2 without radiation condition

o

t~ / , , t ' /

0.0 1.0 2.0 3.0 4.0 o. r | -- j ~ ,

0.0 1.0 2.0 5.0 4.0 o

. =

0.0 tO 2.0 3.0 4.0 o

, 1 "~ , , ~ " - " , , 0.0 l.O 2.0 3.0 4.0 o

~ q

0.0 1.0 2.0 3.0 4.0

~/L

Fig. 4. Development o f f ree surface profiles with time for ka = 0.6, H/L = 0.1 and deep water. - - - , sohttion to first order; - - , sohttion to second order

182 Applied Ocean Research, 1991, Vol. 13, No. 4

Second order wave diffraction aroumt two-dimensional bodies by t#ne-doma#l method: M. Isaacson and K. F. Cheung,

q i .~ ~ i ' I i i '

o ,r

I

0.0 1.0 2.0 3.0 4.0 5.0 o ,.5 i i ' i i

" b .

q

0.0 1.0 2.0 ,.'3.0 4.0 5.0

Fig. 5. Development of nmup with time for ka = 0.6, H/L=0.1 amt deep water. (a) upwave, (b) downwave, - - - , solution to first order; - - . , sohaion to second order

and the second order forcing is weak. Figure 5 shows the corresponding variation of the runup indicated by the maxima of the curves in dimensionless form to first and second order with time on the two sides of the body. Despite a short duration of transient effects associated with the initial condition, the runup approaches its steady state after only one wave cycle. It is observed that the second order effects contribute significantly to the total runup on the upwave body surface and that a steady component of runup is also present.

The various components of the runup on the upwave and downwave body surfaces are plotted in dimensionless form as R/A against the relative body size ka in Fig. 6. In the figure, the steady component of runup associated with the increase in mean water level near the body is due to the second order interactions between the first order incident and scattered waves. As ka~O, each component of the runup is expected to approach the value corresponding to regular incident waves, as if the body is not present. Thus the first order component of R/A should approach unity, the second order steady component should approach zero, the second order residual component should approach the second order wave amplitude c~; and the total runup should approach (1 + c 0. The second order interactions between the body and the surrounding wave field are observed to be most significant for small values of ka and the second order residual runup is relatively large having a peak near ka = 0.1 on both the upwave and downwave body surfaces. The total runup in this frequency range is dominated by second order effects. For large values of ka, each component of the runup including the total runup, approaches a constant value corresponding to the total reflection of incident waves. For the cases considered here, the runup at second order is important and accounts for about one-fifth to one-half of the total runup on the upwave body surface depending on the frequency range of interest.

Wave force Figure 7 shows the development with time of the

horizontal and vertical wave force components (indicated by Fx and F_. respectively) to first and second order for ka = 0.6, H/L = 0.1 and deep water. At t = 0, the wave forces correspond to those obtained by the Froude-

o

,4

o e~

o

o c5

(a )

i i i

i

0.0 0.4 0.8 1.2

ka

I D ,,..t

d -

o

d

o.o 0.4 o.a ~.z

(b) ka

Fig. 6. Rmmp components as fimctions of ka for H/L=O.1 and deep water. (a) upwave, (b) dowmrave. 1--t, first order rumtp; x , second order stead), rtmup; A, second order residual rumtp ; l , total rumtp to second order

Krylov assumption in which the body is assumed to have no effects on the incident flow. A steady state solution is developed gradually in one wave cycle and the results indicate the presence of strong second order effects which include a horizontal drift force in the positive x direction and a vertical drift force in the negative z direction.

o N

o N

I

0

N t J _

q

0.0 1.0 2.0 3.0 4.0 5.0

i i I i

0.0 1.0 2.0 3.0 4.0 5.0

VT Fig. 7. Development of horizontal and vertical wave force components with time for ka = 0.6, H/L = 0.1 and deep water. - - - , solution to first order; , solution to second order

Applied Ocean Research, 1991, Vol. 13, No. 4 183

Second order wave diffraction around two-dimensional bodies by time-domaht method: M. Isaacson and K. F. Cheung

Comparisons of the oscillatory force components at first and second order have been made with those obtained by two previous theoretical studies 8J~ and also with the experimental results of Kyozuka 8, and are presented in Figs 8 and 9. At first order, the present results give excellent agreement with the theoretical and experimental results of Kyozuka (Fig. 8). At second order, t~o the present results give good agreement with Kyozuka's ,~ theoretical results, whereas the oscillatory variation with " ka predicted by Miao and Liu and indicated in Fig. 9a t n

is not reproduced. Kyozuka's experimental results exhibit d some scatter but have the same trend as other theoretical results.

Composition of wave forces at second order As indicated in equation (40), the second order steady

drift force is made up of two components, one due to the free surface fluctuation on the body and the other due to the velocity squared term in the Bernoulli equation, both of which can be computed on the basis of the first order potential. Figure 10 shows the drift forces in the x and z directions and their component terms. Since higher surface elevations occur on the upwave side of the body, the horizontal force due to runup is higher on the upwave body surface than on the downwave body surface and

123

x L L

d

# 0 0"<'~--o o o o

- / " ~ - . . . o O o o ~ oO

O d , f , '

0.0 0.5 1.0 (a)

' I

1.5 2 . 0 2 . 5

ka

0 . 0 (a)

0 u5

I 1 i i

0 0

0

/ " ~ . .

0

�9 . : o . / / ~ ~ - - - . . . ~ o I . I

o \ .

0.5 1.0 1.5 2 .0

ka

I i

0

o / �9

2.5

~ y o o o o

o.o o.5 ~.o ts z o z s

(b) ka

Fig. 9. Comparisons of computed wave force amplitude at second order with Kyozuka"s o theoretical and experi- mental restdts and with Miao and Lht's ~~ theoretical restdts. (a) Horizontal component, (b) vertical component. - - , Kyozuka 'S theory; - . - , Miao and Lht 'S theory; O, Kyozuka "S experhnents ; 0 , present study

' i i �9 I �9 i

~ �9

L ~

0 d w * ~ t v I

o.o o.5 ;.o ;.s z o z s

( b ) k a

Fig. 8. Co)nparisons of computed wave force amplitude at first order with gyozuka 'Ss theoretical and experhnental results. (a) Horizontal component, (b) vertical component.

, Kyozuka's theory; O, Kyozuka's experhnent; O, present study

0

c~

?

�9 �9 �9 -_ -_ �9

o c ~ f l o o

q

0.0 0 . 5 1.0 1.5 2 . 0 2 . 5

ka

Fig. 10. Composition of second order steady drift forces hi x and z directions. V], horizontal drift force due to velocity squared term; A, horizontal drift force due to free surface fluctuation ; I , resultant horizontal drift force; O, resultant vertical drift force

184 Applied Ocean Research, 1991, Vol. 13, No. 4

Second order wave diffraction around two-dimensional bodies by time-domahz method: M. Isaacson and K. F. Cheung

therefore the resultant drift force component always acts in the direction of wave propagation. On the other hand, the higher surface elevations in the upwave region are associated with higher fluid velocities and hence lower pressure there, and consequently the horizontal drift force component due to the velocity squared term always acts opposite to the direction of wave propagation: the two components ofthe horizontal drift force are counteracting each other. Due to the balance of fluid momentum, the resultant horizontal drift force always acts in the direction of wave propagation and for large values ofka approaches a constant value corresponding to a total reflection of incident waves by the body. On the other hand, the free surface fluctuation has no effects on the drift force in the vertical direction, which itself always acts in the negative z direction due to the negative pressure induced by the velocity squared term in the Bernoulli equation.

The second order oscillatory force in the x direction consists of a component from the potential at second order as well as two components from the potential at first order as described above. Figures 1 la and 1 Ib show the amplitudes of the various components of the second order oscillatory force in the horizontal and vertical directions respectively. In Fig. 1 la, the two components due to q~ in the x direction are of the same phase, while

o

o

Cn O.

h. o

o d

o

0.0 0.5 tO 1.5 2.0

ka

2.5

Q.

o

o

0.0 o.s to t5 zo 2.s ka

Fig. 11. Compos#ion of second order oscillatory wave forces. (a) Horizontal component (b) vertical component. x , component due to velocity squared term; [~, component due to free sttrface flttctttation; A, component due to ~l ; O, component due to ~2; 0 , restdtant force amplitude

the component due to q~2 has the opposite phase over most of the frequency range, and consequently gives rise to a smaller resultant force. Similar observations are also made in Fig. 1 lb for the second order oscillatory force in the z direction, but the force component due to q~2 is found to dominate. It is interesting to note that for large values of ka both the horizontal and vertical force components due to ~2 increase steadily with ka, while the components due to qSt remain fairly constant and do not depend on the size of the body. In terms of the contributions toward the total wave force, the potential at second order plays an important role especially for large values of ka therefore should be accounted for accurately.

7. CONCLUSIONS

A time-domain second order diffraction method has been developed to treat the nonlinear interactions of ocean waves with large two-dimensional structures of arbitrary shape. By integrating the free surface boundary conditions and the radiation condition to second order in time, the present method does not require the complicated algebra as in the conventional perturbation method, and avoids the need to solve a system of simultaneous equations every time-step as in other nonlinear time-stepping solutions.

The initial condition corresponds to a second order incident wave train in the domain, and a scattered potential is allowed to develop in time and space. Through the gradual imposition of the body surface boundary condition, steady state solutions in the vicinity of the body are obtained after one wave cycle. A number of computational considerations inherent in the method has been discussed. The present procedure is more efficient and reliable in comparison to those which require the incident waves to propagate into initially still water and provides a firm control of the incident wave conditions at the test section in simulating irregular wave diffraction.

The present method is applied to a study of the nonlinear diffraction of deep water regular waves around a surface piercing semi-circular cylinder in two dimensions. The runup and wave forces at first and second order for various relative cylinder sizes are presented and compari- sons with previous theoretical and experimental results indicate a favourable agreement. The importance of the second order potential in the evaluation of the total runup and wave forces is also highlighted. The application of the present method in three dimensions with regular and irregular incident waves is currently under way.

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Applied Ocean Research, 1991, Vol. 13, No. 4 185

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