Simulation of directional waves in a numerical basin by a desingularized integral equation approach

Ocean Engineering 27 (2000) 603–624www.elsevier.com/locate/oceaneng

Simulation of directional waves in a numericalbasin by a desingularized integral equation

approach

A.N. Williams*, W.W. CrullDepartment of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4791,

USA

Received 5 October 1998; accepted 6 January 1999

Abstract

A numerical solution is developed to investigate the generation and propagation of small-amplitude water waves in a semi-infinite rectangular wave basin. The three-dimensional wavefield is produced by the prescribed “snake-like” motion of an array of segmented wave gener-ators located along the wall at one end of the tank. The solution technique is based on theboundary element approach and uses an appropriate three-dimensional Green function whichexplicitly satisfies the tank-wall boundary conditions. The Green function and its derivativeswhich appear in the integral equation formulation can be shown to be slowly convergent whenthe source and field points are in close proximity. Therefore, when computing the velocitypotentials on the wave generators, the source points are chosen outside the fluid domain,thereby ensuring the rapid convergence of these functions and rendering the integral equationsnon-singular. Numerical results are shown which illustrate the influence of the various wav-emaker and basin parameters on the generated wave field. Finally, the complete wave fieldproduced by the diffraction of oblique waves by a vertical circular cylinder in a basin ispresented. 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Directional waves; Laboratory waves; Wave diffraction; Wave basin; Wave generation

* Corresponding author. Tel.:1 1-713-743-4269; fax:1 1-713-743-4260; e-mail: [email protected]

0029-8018/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0029 -8018(99 )00016-5

604 A.N. Williams, W.W. Crull /Ocean Engineering 27 (2000) 603–624

1. Introduction

It is well known that, in certain applications, wave directionality plays animportant role in the hydrodynamic loading experienced by coastal and ocean struc-tures. Many laboratory wavetank facilities now consist of a relatively wide rectangu-lar basin in which scale models can be tested in short-crested waves. Directionalseas are usually produced in this type of facility by the prescribed action of an arrayof segmented wave generators located along one or more walls of the basin. Thegenerator motions required to produce the specified wave field are obtained on thebasis of the so-called “snake principle”. This is based on the concept that the spatiallysinusiodal motion of an infinitely long segmented wavemaker produces regular wavespropagating obliquely to the generator plane (see, for example, Dean and Dalrymple,1984). However, in reality, the basin is of finite width and so the characteristics ofthe generated wave field are influenced by sidewall reflections. Therefore, the properinterpretation of the results of model tests in this type of facility requires knowledgeof the characteristics of the wave field produced in the basin, including the influenceof the sidewalls.

A methodology outlining the extension of the snake principle to include the effectof finite tank width and reflecting sidewalls has been described by Funke and Miles(1987). By modifying the wave generator motions near the sidewalls they show howthe dimensions of the reflection-free test area in the tank can be significantlyincreased over that obtained by conventional directional wavemaker theory. Dalrym-ple (1989) applied the mild-slope equation to predict the wave field generated bysnake-like wavemaker motions in a finite basin with fully reflecting sidewalls.Numerical results were presented for the instantaneous surface elevations in basinsboth of constant depth and also with sloping bottoms. Isaacson (1995) has presenteda boundary element solution for the directional wave field in a finite basin producedby one or more segmented wave generators oscillating according to the snake prin-ciple. Both partial and full sidewall reflections were considered. The wave field wasdecomposed into propagating and evanescent components, numerical results werepresented for the propagating components only, and the region of applicability ofthe solution was discussed. Finally, the surface elevations resulting from locating avertical cylinder in the test area of the basin were presented.

In recent years, there have been several related studies involving the influence ofsidewalls on the hydrodynamics of bodies situated in narrow channels. Boundaryelement techniques utilizing Green functions, which explicitly satisfy the boundaryconditions on the sidewalls of the tank, have been developed by Thomas (1991),Kashiwagi (1991), Butler and Thomas (1993), Williams and Vazquez (1993), Chen(1994) and Vazquez and Williams (1994, 1995), while rapidly convergentexpressions for Green functions for channel problems have been developed by New-man (1992) and Linton (1993). Analytical studies of wave interactions with bodiesin channels employing a variety of techniques have been carried out by Yeung andSphaier (1989), Calisal and Sabuncu (1989), Linton and Evans (1992), McIver(1993), McIver and Bennett (1993) and McIver and Linton (1994).

In the present paper, the three-dimensional Green function technique developed

605A.N. Williams, W.W. Crull /Ocean Engineering 27 (2000) 603–624

by Vazquez and Williams (1994, 1995) is employed to investigate the generationand propagation of small-amplitude water waves in a semi-infinite rectangular wavebasin. The three-dimensional wave field is produced by the prescribed “snake-like”motion of an array of segmented wave generators located along the wall at one endof the tank. Particular attention is focussed on ensuring the rapid convergence of theGreen’s function and its derivatives which appear in the integral equation formu-lation. These functions can be shown to be slowly convergent when the source andfield points are in close proximity. Therefore, when computing the velocity potentialson the wave generators, the source points are chosen outside the fluid domain, ther-eby ensuring the rapid convergence of these functions and rendering the integralequations non-singular. This approach has been used in related three-dimensional,potential-flow problems by Cao et al. (1991) and by Scorpio and Beck (1998). Thenumerical results presented herein clearly illustrate the influence of the various wav-emaker and basin parameters on the generated wave field. Finally, the complete wavefield produced by the diffraction of oblique waves by a vertical circular cylinder ina basin is presented. It is found that the tank geometry may have a significant influ-ence on the wave field experienced by a structure located in the basin.

2. Theoretical development

A segmented wave generator occupies one wall of a semi-infinite rectangular basinof width 2b (Fig. 1). The water depth in the tank is denoted byh. Cartesian coordi-nates (x,y,z) are employed with thex- andy-axes in the horizontal plane and thez-axis directed vertically upwards from an origin at the tank centerline, on the tankbottom. The tank walls are located aty 5 6 b. The wave generators are assumedto undergo prescribed small-amplitude oscillations of frequencyv so that the associa-ted fluid motion may be described by linear (Airy) wave theory. Therefore, alldynamic quantities may be expressed as the real part of!e−ivt where!(x,y,z) is thecomplex-valued amplitude of the quantity under consideration.

Under the assumption of an inviscid, incompressible fluid undergoing irrotationalmotion, the wave motion may be described in terms of a velocity potentialf(x,y,z,t)5 Re{F(x,y,z)e−ivt}, such that the fluid velocity vectorq 5 =f. The governingequation for the fluid motion may be written as

=2F 5 0 in the fluid domainV (1)

subject to the following boundary conditions

∂F

∂z5 0 on the tank bottomSB (2)

2 v2F 1 g∂F

∂z5 0 on the quiescent free-surfaceSF (3)

∂F

∂y5 0 on the tank wallsSw (4)


Fig. 1. Definition sketch.

∂F

∂n5 Vn on the mean wave generator surfacesSo (5)

whereg is the acceleration due to gravity.The horizontal wavemaker displacement about the mean positionx 5 0 is given by

X(y,z,t) 5 Re{A(y)H(y,z)ei(loy 2 vt)} (6)

in which H(y,z) is the generator shape function which describes the vertical depen-dency of the wave generator motion andA(y) is the maximum amplitude of the wavegenerator strokes atz 5 h, at a given location along the tank. In Eq. (6),lo is they-component of the wavenumber of the desired waves,lo 5 k sin u whereu is theangle the waves make with thex-axis, and the wavenumberk satisfies the lineardispersion relationv2 5 gk tanhkh. In Eq. (5),Vn 5 dX/dt, and so the wave generatorboundary condition can be written

∂F

∂n;

∂F

∂x5 2 ivA(y)H(y,z)eiloy on the mean wave generator surfacesSo (7)


Finally, at large distances from the wave generators, the potential must satisfy asuitable radiation condition. This condition ensures that, asymptotically, the wavecomponents are propagating away from the wave generating surface, or are vanishin-gly small.

3. Integral equation solution

The solution to the above boundary-value problem will be obtained by an integralequation technique utilizing an appropriate Green functionG(r ; r 9), r 5 (x,y,z), r 95 (x9,y9,z9) which is required to satisfy Eq. (1) except asr→r 9 and Eqs. (2)–(4)together with an appropriate radiation condition. A Green function may be obtainedby an eigenfunction expansion technique outlined by Morse and Feshbach (1953),and is given by Vazquez and Williams (1994, 1995) as

G(r ; r 9) 54p

b O`m 5 0

emGom(x,z; x9,z9)cosSmpyb DcosSmpy9

b D1

8p

b O`m 5 1

Hom(x,z; x9,z9)sinF(2m 2 1)py2b GsinF(2m 2 1)py9

2b G1

4pib OP

m 5 0

emGpm(x,z; x9,z9)cosSmpyb DcosSmpy9

b D (8)

18p

b O`m 5 P 1 1

Gem(x,z; x9,z9)cosSmpyb DcosSmpy9

b D1

8pib OQ

m 5 1

Hpm(x,z; x9,z9)sinF(2m 2 1)py2b GsinF(2m 2 1)py9

2b G1

8p

b O`m 5 Q 1 1

Hem(x,z; x9,z9)sinF(2m 2 1)py2b GsinF(2m 2 1)py9

2b Gin which eo 5 1 andem 5 2 for m > 0, and

Gom 5 O`n 5 1

e−mmnux 2 x9u

(2knh 1 sin 2knh)kn

mmn

cosknzcosknz9 (9a)

Gpm 5eimmo

ux 2 x9u

(2kh 1 sinh 2kh)k

mmo

coshkzcoshkz9 (9b)

Gem 5e−m9

moux 2 x9u

(2kh 1 sinh 2kh)k

m9mo

coshkzcoshkz9 (9c)


Hom 5 O`n 5 1

e−lmnux 2 x9u

(2knh 1 sin 2knh)kn

lmn

cosknzcosknz9 (10a)

Hpm 5eilmo

ux 2 x9u

(2kh 1 sinh 2kh)k

lmo

coshkzcoshkz9 (10b)

Hem 5e−l9

moux 2 x9u

(2kh 1 sinh 2kh)k

l9mo

coshkzcoshkz9 (10c)

where

m2mo 5 k2 2 Smp

b D2

5 2 m92mo (11a)

m2mn 5 Smp

b D2

1 kn2 (11b)

l2mo 5 k2 2 S2m 2 1

2bpD2

5 2 l92mo (12a)

l2mn 5 S2m 2 1

2bpD2

1 kn2 (12b)

In Eqs. (9)–(12), thek*ns, n 5 1,2,... are the positive real roots ofv2 1

gkn tanknh 5 0. In Eq. (8),P and Q are defined as the greatest integers such thatm2

mo > 0 andl2mo > 0, respectively. For the case wherek , p/2b, i.e. l2

1o , 0, thefifth summation term in Eq. (8) is not executed andQ 5 0 should be used whenevaluating the last summation term in that equation.

It is noted from Eqs. (8)–(10) that the rate of convergence of the Green functiondepends critically on the value of |x 2 x9|. As x→x9 the series becomes slowlyconvergent and analytical techniques involving the subtraction and separate analyti-cal integration of the series aymptotics must be used to achieve convergence(Williams and Vazquez, 1993). This issue is especially important for the presentproblem since all wave generator nodes lie in the planex 5 0. Therefore, whencomputing the velocity potentials on the wave generators, the source points arechosen outside the fluid domain, thereby rendering the Green function series rapidlyconvergent and the integral equations non-singular. In particular, a set of pointsr 95 ( 2 x*,y*,z*) is defined, wherex* > 0, 2 b , y* , b, and 0, z* , h. Sub-sequently, this set of points will be chosen such that the computed solution will beindependent of their locations.

Applying Green’s second identity toF(r ) andG(r ; r 9) over V,

ES

HF(r )∂G∂n

(r ; r 9) 2 G(r ; r 9)∂F

∂n(r )JdS 5 0 (13)

for r 9 exterior to the surfaceS consisting ofSo, SF, SB, Sw and S̀ , andn is the unit


normal toS directed intoV. Imposing the tank bottom, free-surface, radiation andsidewall boundary conditions leads to the following integral equation:

ESo

F(r )∂G∂n

(r ; r 9)dS 5 ESo

Vn(r )G(r ; r 9)dS (14)

wherer 9 is located exterior to the surfaceS. The wave generator surfaceSo is nowdiscretized intoM quadrilateral panels,Sm, with central nodesrm 5 (0,ym,zm) andthe potential within each panel is assumed constant. Then, letting the pointr 9 varyasr 9

m 5 ( 2 x*,ym,zm), for x* > 0, Eq. (14) may be rewritten in terms of these nodalvalues as

OMm 5 1

F(r m)ESm

∂G∂n

(r ; r 9i)dS 5 OM

m 5 1

Vn(rm)ESm

G(r ; r 9i)dS (15)

for i 5 1,2,...,M. The integration of the Green function and its derivative over anelement is accomplished using Gaussian quadrature. Eq. (15) may thus be used todetermine the velocity potentialF at the nodal pointsrm on So.

Once the velocity potentials on the wave generator surfaces have been determined,the potential on the free-surface (and hence the wave elevation) may be determinedby re-applying Green’s second identity, only this time withr 9 5 r 9

o 5 (x,y,h) onSF. Therefore,

F(r 9o) 5

12p

OMm 5 1

HF(rm)ESm

∂G∂n

(r ; r 9o)dS2 Vn(rm)E

Sm

G(r ; r 9o)dSJ (16)

The water surface elevationsh(r 9o,t) 5 RehG(r 9

o)e−ivtj may then be obtained fromthe dynamic free-surface boundary condition, namely

G(r 9o) 5

ivg

F(r 9o) (17)

4. Numerical results and discussion

First, the convergence of the numerical technique with respect to the number anddistribution of panels on the wave generator surface and the location of the fieldpoints (specifically the value of the parameterx*) were investigated. The wave gener-ating surface was taken to be square, i.e.h 5 2b, with b/L 5 1.6, x*/L 5 0.5 andu 5 30°, whereL is the incident wavelength defined byL 5 2p/k. Flap-type wavegenerators hinged at the tank bottom were considered, i.e.H(y,z) 5 z/h andA(y) 5Ao. Fig. 2 presents the instantaneous free-surface elevationh(r 9

o,0)/Ao along the linesy 5 0, 6 b/2, 0 # x/L # 10 for two different levels of discretiztion on the wave


Fig. 2. Comparison of instantaneous surface elevationh(r 9o,0)/Ao along the lines: (a)y 5 2 b/2; (b) y

5 0; and (c)y 5 1 b/2 for two different discretizations of the wave generator surface. The results fordx/b 5 dz/h 5 0.1 are denoted by the symbols, while those fordx/b 5 dz/h 5 0.05 are denoted by thelines. Additional parametric details are given in the text.



5 0; and (c)y 5 1 b/2 for two different values of the parameterx*. The results forx*/L 5 0.2 aredenoted by the symbols, while those forx*/L 5 0.1 are denoted by the lines. Additional parametric detailsare given in the text.


generator surface, namelydx/b 5 dz/h 5 0.1 and 0.05, wheredx and dz are thepanel dimensions in thex- and z-directions, respectively. It can be seen from thefigures that convergence has essentially been achieved for this wavelength–tankwidthcombination. For each source–field point combination, the numbers of terms usedin the series for the Green’s function and its derivative were determined adaptivelyaccording to specified convergence criteria. Generally speaking, more terms arerequired in the vicinity of a tank resonant frequency, particularly when the sourceand field points are in relatively close proximity. Fig. 3 shows the instantaneousfree-surface elevationh(r 9

o,0)/Ao along the linesy 5 0, 6 b/2, 0 # x/L # 10 fortwo different values of the parameterx*/L 5 0.1 and 0.2, for a panel discretizationon the wave generator surface ofdx/b 5 dz/h 5 0.1. Since the convergence of theGreen’s function series is quicker for larger values ofx*/L, based on the results inFig. 3, a value ofx*/L 5 0.2 was used in all subsequent calculations.

Fig. 4. Comparison of the free surface elevationh(r 9o,0)/Ao obtained from the present program withu

5 0° (symbols) and the analytical solution of Dean and Dalrymple (lines) for (a) piston- and (b) flap-type wave generators. Additional parametric details are given in the text.


Results for theu 5 0° case may be compared with the analytical, two-dimensionalsolutions for planar wavemakers presented by Dean and Dalrymple (1984). Fig. 4presents a comparison between the instantaneous free-surface elevationsh(r 9

o,0)/Ao

for 0 # x/L # 10 for both piston- and flap-type wave generators forh/L 5 0.25.For the flap-type wave generatorH(y,z) 5 z/h while for the piston wave generator,H(y,z) 5 1. The discretization adopted wasdx/b 5 dz/h 5 0.1 and the parameterx*/L 5 0.2. Excellent agreement is observed between the two sets of results.

Fig. 5. Contour plot of instantaneous surface elevationh(r 9o,0)/Ao for b/L 5 1.67,h/L 5 0.67, andu 5

15° (top), u 5 30° (middle), andu 5 45° (bottom).


Fig. 6. Contour plot of instantaneous surface elevationh(r 9o,0)/Ao for b/L 5 3.33,h/L 5 0.67, andu 5

15° (top), u 5 30° (middle), andu 5 45° (bottom).

615A

.N.

Willia

ms,

W.W

.C

rull/O

cea

nE

ng

ine

erin

g2

7(2

00

0)

60

3–

62

4

Fig. 7. Contour plot of instantaneous surface elevationh(r 9o,0)/Ao for b/L 5 5.0, h/L 5 0.67, andu 5 15° (left), u 5 30° (middle), andu 5 45° (right).



5 0; and (c)y 5 1 b/2 for two different discretizations of the wave generator and cylinder surfaces.The results fords/rcp 5 0.125 anddz/h 5 0.2 are denoted by symbols, while those fords/rcp 5 0.0625and dz/h 5 0.1 are denoted by lines. Additional parametric details are given in the text.


A parametric study is now performed for a variety of basin widths and angles ofwave generation. The parameters chosen wereh/L 5 0.67,u 5 15, 30 and 45° andb/L 5 1.67, 3.33 and 5.0. The results of this parametric study are presented as contourplots of the instantaneous free surface elevationsh(r 9

o,0)/Ao in Figs. 5–7. The effec-tiveness of a particular tank configuration for a given directional wave conditionmay be evaluated by considering the effective test area of the wave basin. Thisregion is roughly triangular in shape. The location of the triangle apex is a functionof the wave angle and width of the wave basin. One leg of this triangular area isbounded by the reflected waves off one wall, while the other side of the triangle isbounded by the beginning of the diffracted wave pattern. However, the regiondirectly in front of the wave generator is also not considered part of the effectivetest area due to local disturbances associated with the generation of the wave field.Depending on the basin geometry and the angle of wave generation, repeated zonesof reflected waves may extend down the tank with (possibly) intermittent areas popu-

Fig. 9. Contour plot of instantaneous surface elevationh(r 9o,0)/Ao (top) and amplitude of surface elev-

ation |G(r o)9|/Ao (bottom) forb/L 5 3.33,h/L 5 0.67, andu 5 0°.


lated only by diffracted waves. This phenomenon can clearly be seen in the figures,however, for some of the tank configurations, the surface profiles are not graphed asufficient distance along thex-axis to satisfactorily display the repeated patterns. Forcertain basin geometries an extremely acute angle of wave generation will not resultin repeated diffracted wave zones in a finite tank. Also, the special case,u 5 0° isexcluded from this discussion since, for this angle of wave propagation, no reflectedwaves are generated.

It can be seen that for the narrow wavetank geometry,b/L 5 1.67, shown in Fig.5, wave reflection from the sidewalls occurs almost immediately and quickly disruptsthe desired wave patterns. In the worst case shown,b/L 5 1.67,u 5 45°, the heightof the triangle of the effective test area isx/L 5 1.2, and so a physical wavetankpossessing this geometry would be of little use in studying directional waves of thegiven frequency (wavelength) at this angle of propagation. The results shown in thisfigure may be compared with those shown in Fig. 7 forb/L 5 5.0,u 5 45°, where the


ation |G(r 9o)|/Ao (bottom) forb/L 5 3.33,h/L 5 0.67, andu 5 15°.


height of the triangle of the effective test area isx/L 5 4.8. One possible extension ofthe work presented in this paper is the deliberate use of sidewall reflection to selec-tively increase the effective test area in the wave basin for a given design wave (see,for example, Funke and Miles, 1987). However, depending on the physical wavebasin under consideration, actual implementation of this concept may not be possiblesince the basin may not possess wave generators capable of the required action.

The effect on the wave field of locating a vertical circular cylinder in the testarea of the wave basin is now investigated. The cylinder, of radiusrc, is positionedsymmetrically in the wavetank with geometric center at (xc,0). One example of theconvergence of the numerical results with respect to the number and distribution ofpanels on the cylinder surface is presented in Fig. 8. In this case, the wave basincharacteristics areb/L 5 3.33,h/L 5 0.67 andu 5 30° while the cylinder size andlocation are such thatrc/b 5 0.125 andxc/L 5 3.33. The results shown in Fig. 8are typical. It has been found that, in general, convergence may be achieved with acylinder surface discretization similar to that adopted for the wave generator surface.




Figs. 9–12 present a series of contour plots of instantaneous free surface elevation,h(r 9

o,0)/Ao and amplitude of the water surface, |G(r 9o)|/Ao, for u 5 0, 15, 30 and 45°,

and b/L 5 3.33, h/L 5 0.67, rc/b 5 0.125 andxc/L 5 3.33. As the angle of wavegeneration increases, the (fixed) cylinder is located closer to, or beyond, the apexof the triangular region describing the effective test area for the basin. Therefore, asthe angle of wave generation increases, the cylinder’s exposure to reflected wavesalso increases. The exception to this is the case where the generated waves pass infront of the cylinder and the waves reflected from the sidewall pass behind the cylin-der. Theu 5 0° case, shown in Fig. 9, shows the largest reduction in wave amplitudebehind the cylinder (due to the absence of reflected waves for this angle ofincidence), while for theu 5 45° case, shown in Fig. 12, there is minimal reductionin wave surface amplitude behind the cylinder due to wave reflections from thebasin sidewalls.

The influence of the cylinder on the wave field inside the basin is further shown




in Figs. 13 and 14 which present the amplitude of the water surface, |G(r 9o)|/Ao, along

the linesy 5 0, 6 b/2 both with and without a cylinder present forb/L 5 3.33 andu 5 0 and 30°. The other basin and cylinder parameters areh/L 5 0.67, rc/b 50.125,xc/L 5 3.33, and the wave generators are piston-type. It can be seen that thepresence of the cylinder results in a significant modification in the amplitude of thewater surface elevation, wave amplitudes in front of the cylinder are generallyincreased due to reflection, while in the immediate lee of the cylinder amplitudesare decreased due to shielding.

5. Conclusions

A numerical solution has been developed to investigate the generation and propa-gation of small-amplitude water waves in a semi-infinite rectangular wave basin. The

Fig. 13. Comparison of water surface amplitudes |G(r 9o)|/Ao for u 5 0° along the lines: (a)y 5 6 b/2;

(b) y 5 0, for a basin with (———) and without (- - -) a cylinder present. Additional parametric detailsare given in the text.


Fig. 14. Comparison of water surface amplitudes |G(r 9o)|/Ao for u 5 30° along the lines: (a)y 5 2 b/2;

(b) y 5 0; and (c)y 5 1 b/2, for a basin with (———) and without (- - -) a cylinder present. Additionalparametric details are given in the text.


three-dimensional wave field is produced by the prescribed “snake-like” motion ofan array of segmented wave generators located along the wall at one end of thetank. The solution technique is based on the boundary element approach and usesan appropriate three-dimensional Green function which explicitly satisfies the tank-wall boundary conditions. Particular attention has been focussed on ensuring therapid convergence of the Green function and its derivatives which appear in theintegral equation formulation. When computing the velocity potentials on the wavegenerators, the source points are chosen outside the fluid domain, thereby renderingthe integral equations non-singular. Numerical results have been shown which illus-trate the influence of the various wavemaker and basin parameters on the generatedwave field, and the complete wave field produced by the diffraction of oblique wavesby a vertical circular cylinder in a basin has been presented.

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Simulation of directional waves in a numerical basin by a desingularized integral equation approach