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Two-Dimensional Wave-Making Problem C.R. Chou, R.S. Shih, J.Z. Yim Department of Harbor and River Engineering, National Taiwan O

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Two-Dimensional Wave-Making Problem C.R. Chou, R.S. Shih, J.Z. Yim Department of Harbor and River Engineering, National Taiwan O Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X Two-dimensional wave-making problem C.R. Chou, R.S. Shih, J.Z. Yim Department of Harbor and River Engineering, National Taiwan Ocean University, Bee-Ning Road 2, Keelung 202, Taiwan Abstract In this study, generation of two dimensional nonlinear waves is simulated numerically using boundary clement method. The present scheme is based on a Lagrangian description together with finite differencing of the time step. An algorithm to generate waves with any prescribed form is also implantec in the scheme. The numerical model was first verified by studying the case of finite amplitude waves impinging against a vertical wall. Time histories of evolution of a soliton running up on a sloping beach, as well as over a submerged obstacle are then presented. 1 Introduction Problems associated with generation, propagation and deformatio.i have been studied numerically by many researchers. Based on a mixed Eulerian- Lagrangian method, solitary wave interacting with a gentle slope, and wave breaking on water of gradually varying depth are simulated numerically in two- dimensional fluid region by Sujjino and Tosaka[l], Ouyama[2] explored soli- tary wave set up on a slope by boundary element method. Pedersen & Gjevik[3 ] developed a numerical model to study run-up of long waves governed by a set of Boussinesq equations. Lagrangian coordinate description was used to sim- plify the numerical treatment of a moving shoreline. Based on the Green';* formular, Nakayama[4] applied a new boundary element technique to analyze nonlinear water wave problems. Faltinsen[5] presented a numerical method fo: studying the sloshing in a wave lank with a two-dimensional flow. In this study, problems of generation and propagation of nonlinear waves are considered nu- merically by boundary element method. The algorithm is based on a Largrange Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X 650 Boundary Elements description and finite differencing of time steps to solve the problem of twc- dimensional wave deformation. In the present numrical model, a scheme to generate waves with any prescribed form was implanted. 2 Theoretical analysis Figure 1: Defintion sketch As shown in Fig. 1, a Cartesian coordinate system is employed. The origin of which is located on the still water level with the z-axis pointed positively upwards. The flow field is bounded by a pseudo-wave-generating boundary FI, a free water surface l\ and an impermeable sea bed Fg. Assuming that th^ pseudo-wave-generating boundary FI is sufficiently far away from the coastal zone, wave scattering induced by undersea topography or obstacle can be ne- glected. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. Fluid motion has velocity potential $(%, z; t) satisfying ilie fol- lowing Laplace equation: 2.1 Boundary conditions (1) Boundary condition on pseudo wave making boundary FI The boundary, F% is assumed to represent a wave-generating device. In this study, a piston wave generator is assumed for simplicity, although it is clear that paddle of any desire type can be simulated. Requiring that the horizontal velocities of the pseudo wave-paddle U(t) and fluid flow be continuous, we obtain: Wave with prescribed form can be simulated by selecting suitable U(t) as input. For finite amplitude wave, U(t) can be expressed as: = -acrsmcrZ (3) ^ sinhkh • coshkh -f kh . Where a (=2vr/T) is the angular frequency, (o is the amplitude, k is the v/avti number, and T the period of the wave to be generated. Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 651 For the case of a solitary wave, U(t) can be expressed as: (Z--Zc)] (5) Zc = 7T/U (8) Where XQ denotes the semi-stroke of wave paddle and (o is the wave height of the solitary wave to be generated. (2) Boundary condition on free water surface Atmospheric pressure is assumed to be constant, boundary condition on the free water surface can be obtain from the kinematic and dynamic condition as: where D is the Lagrange differentiation, g is the gravitational acceleration, and ( is the surface elevation. (3) Boundary condition on the impermeable sea bed The water particle velocity is null in the normal direction on impermeable seabed: ^T=0 (12: 0Z/ where i/ is the unit normal vector. 2.2 Integral equation According to Green's second identity law, velocity potential $(£,;:; t) ai: any point within the region can be expressed using velocity potential on the. boundary and its normal derivative as: where r = [(( - 2)% 4- (77 - z)^]L When the inner point (x,z) approaches the boundary point (£',77'), due to its. singularity, the velocity potential $(f , 77; t) can be expressed as: where R= £-? Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X 652 Boundary Elements To proceed with numerical calculation, the boundaries FI through F?, are divided into N\ to TVs discrete segments with linear elements. Eq.(14)can be expressed in matrix form as: W = [0]ffi (15) where [$] and [$] are, respectively, the potential function and its normal deriva- tive on the boundaries, [0] is a matrix related to the geometrical shape of th: boundary. The numerical scheme ihas been discussed in detail by Chou[6]. To facilitate substituting the boundary conditions into each boundary, w^ rewrite Eq.(15) as follows: [*.-] = [O*] fc] , U = l~3 (16) 2.3 A system of equations 2.3.1 Initial conditions The initial conditions on each boundary are summarized as follows (1).Pseudo-wave-generating boundary, FI Requirement of continuity between horizontal velocity of the pseudo-wave paddle U(t) and the fluid motion, we obtain: (2).Free water surface F] Assume that water surface is initially at rest (t=0), the velocity potential is therefore null ,i.e. #0 = 0 (18) (3). Impermeable sea bed F] The flow is null in the normal direction of an impermeable seabed, that is: *> «*§_« where superscript "0" denotes the first time step of simulation. 2.3.2 Finite Difference of related terms The tangential derivative on the free water surface, (3$2/<9s);, can be ap- proximated through: ~&s^' (20) 5' = As;+i + A/;,- , 5" = As; • As;+i Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 653 On the free surface, we have 302 302 . -— - — sm/3- —cos/) OX OV OS (21) —30 -2 = —90 -cos/34-2 . ^$— -sm/2 • ,) where (3 denotes the angle between the free water surface and the x-axis. At k-th time step, the profile of free water surface is expressed as (c;\ z^), from Eq.9 and Eq.10 we can then evaluate (x^\ z^ at the k + l-th time step through: /3(T>^ (22) (23) where Af is the discrete time differencing interval. From Eq.ll and Eq.21, velocity potential of the free surface at the k+l-th time step, 0^+*, can be approximated through: (241 Substituting Eq.2, Eq. 12 and the above equation into Eq. 16, we can obtain the following simultaneous equations. k+l k+l I -On 0 On 0 0,3 $2 0 -022 0 021 -/ 023 $2 (25. $3 0 -032 / 031 0 033 $3 2.3.3 The iterative scheme 1. Substituting Eqs.l7~19 into Eq.16. Initial values for the normal deriva- tive of velocity potentials on the water surface, d^/dv, the velocity potentials on the pseudo-wave-generator, $^, as well as the velocity po- tentials on sea bed, $°, can be obtained. 2. Tangential derivative of velocity potentials on water surface, d^jdv, is then calculated using Eq.20. 3. Surface elevations, (x', z'), for the next time step can be obtained from Eq.22andEq.23. 4. The velocity potentials on free water surface for next time step are given by Eq.24. Transactions on Modelling and Simulation vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-355X 654 Boundary Elements 5. At the next time step, t=At, recalculate the coefficients of the matrix [0\ in Eq.16, using profiles of the water surface obtained in procedure 3 and new position of the pseudo wave-paddle. 6. With the velocity potentials on the new water surface given by procedure 4, the horzintal velocity, U(t), of the pseudo wave paddle given by Eq.2, and the boundary condition on sea bed given by Eq.12 substituted into Eq.25, to obtain the normal derivative of velocity potentials, d$\j'dv on the water surface, the velocity potentials, 0} on the pseudo wave paddle and the velocity potentials on sea bed, $3, for the time step t = At. 7. Repeating procedure 2 through 6, the time history for the generation propagation and deformation of waves can be simulated. 3 The numerical results and discussions 3.1 Finite amplitude wave To varify the present numerical scheme, the case of finite amplitude waves h considered. All incident waves are assumed to have an unique amplitude of (o = 0.05/i, but with various dimensionless angular frequencies of a^h/g = 0.25—1.2.5. The pseudo wave paddle is placed at a distance of 5 times the wave length away from the origin of coordinates. For the case of vertical wall, each wave length are divided into 32 elements, i.e. As = Z//32, and a discrete time interval of At = T/160 was taken, whereas for the case of an inclined slope As = L/40 and At = 7/400 are used.
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