Brevig (1981), Baker, Meiron and Orszag (1981) and Roberts. (1983), using a number of different integral formulations to model the motion of two-dimensional ...
AN EFFICIENT BOUNDARY-INTEGRAL METHOD FOR STEEP UNSTEADY WATER WAVES J.W. Dold and D.H. Peregrine (School of Mathematics, University of Bristol)
1.
INTRODUCTION
A new method for computing the unsteady motion of a water surface, including the overturning of water waves as they break, has been developed. It is based on a Cauchy theorem boundary integral for the evaluation of multiple time-derivatives of the surface motion. The integrals are cast into the form of full matrices which can be solved iteratively for the instantaneous motion of discrete surface particles. The size of each timestep is determined to maintain a specified order of accuracy by using a truncated Taylor-series to perform explicit timestepping. The numerical implementation of the method is efficient and accurate. For sufficiently large time-steps a growing "sawtooth" instability appears in the region of greatest point density, but the method is found to be stable for small enough time steps. A deliberate reduction in time step size to restore stability without smoothing normally only becomes necessary with the formation of the jet of a breaking wave where surface particles tend to become strongly concentrated. Boundary-integral methods can be used to reduce the calculation of the inviscid incompressible irrotational motion of a body of fluid to the evaluation of the motion of its surface alone. In a numerical scheme it is possible therefore to solve for the fluid motion using only a point discretisation of the surface, thereby substantially reducing the number of unknown variables in the problem. This approach has been implemented by Longuet-Higgins and Cokelet (1976), Vinje and Brevig (1981), Baker, Meiron and Orszag (1981) and Roberts (1983), using a number of different integral formulations to model the motion of two-dimensional gravity waves on water. In order to obtain accurate solutions for complicated surface· motions, ranging from breaking waves to instabilities in
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672
a train of travelling waves it is, nevertheless, necessary to use fairly large numbers of surface p oints. (For example , a steepening wave in shallow water, part of which is shown in Fig. 1 was computed using up to 200 points). This involves running times per time step which vary as the cube of the number of points for algorithms which solve the integral equation using matrix inversion or factorisation methods, or as the square, using iterative techniques. It is therefore important to ensure computational efficiency particularly for more complicated wave phenomena.
y=1
' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Fig. 1
2.
Nonlinear steepening of an initially gently sloping surface between uniform levels, y = 1 and y = 2.25. The dotted line marks the path of a surface particle. Only about 20% of the computed region is shown.
MODEL AND ALGORITHM:
With the fluid velocity determined by the gradient of a velocity potential ~' Laplace's equation is satisfied in the body of the fluid, 2
'V qi
= o.
( 2. 1)
With the values of~ known on the surface,£=~( ~ ), equation (2.1) can be solved to give the gradient of ~on the surface. Hence the kinematic boundary condition, DR -
Dt
= u
-
=
'V q>
and the dynamic boundary condition, given b y Bernoulli's equation,
( 2. 2)
BOUNDARY INTEGRAL METHOD FOR WATER WAVES
u
673
2
D -= 2 Dt
p
(- + p
gy)
( 2. 3)
(in which the pressure P may be taken to be constant on the surface) provide the basic information for time-stepping the surface motion. Equation (2.1) is not only satisfied by , but also by derivatives of , e.g.
