The deformation of steep surface waves on water - I. A numerical method of computation

Abstract

Plunging breakers are beyond the reach of all known analytical approximations. Previous numerical computations have succeeded only in integrating the equations of motion up to the instant when the surface becomes vertical. In this paper we present a new method for following the time-history of space-periodic irrotational surface waves. The only independent variables are the coordinates and velocity potential of marked particles at the free surface. At each time-step an integral equation is solved for the new normal component of velocity. The method is faster and more accurate than previous methods based on a two dimensional grid. It has also the advantage that the marked particles become concentrated near regions of sharp curvature. Viscosity and surface tension are both neglected. The method is tested on a free, steady wave of finite amplitude, and is found to give excellent agreement with independent calculations based on Stokes’s series. It is then applied to unsteady waves, produced by initially applying an asymmetric distribution of pressure to a symmetric, progres­sive wave. The freely running wave then steepens and overturns. It is demonstrated that the surface remains rounded till well after the over­turning takes place.

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

“…The numerical method gives rise to "spurious oscillations" similar to those reported in other BIE methods, 30,[38][39][40] and these errors may grow in time as the interface evolves. 39 Aitchison and Howison 39 note the introduction of small wavelength instabilities into the solution of the numerical problem is a result of rounding errors and approximate solution techniques and shows that the frequency of the error oscillations scale with N. The difficulty being the more mesh points that are included (i.e., increasing N) shorter wavelength errors are permitted, and it is these which grow fastest in time.…”

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

“…We now present the time evolution of the layered Boussinesq inviscid fluid with initial zero vorticity and temperature given by (36)- (38). Although for these particular conditions, the flow has four-fold symmetry, we do not use this property to achieve higher resolution but instead compute the solution in the whole domain, [0, 1] × [0, 1].…”

“…To reduce the dispersive error inherent in centered differences, we filter θ and ω separately in ξ and η every time step using the following fourth-order filter [38]: u j ← 1 16 (−u j−2 + 4u j−1 + 10u j + 4u j+1 − u j+2 ). This filter can effectively eliminate the small amplitude mesh-scale oscillations without affecting the accuracy of the physical solution.…”

A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

“…Early numerical studies on the non-linear wave propagation appeared in 1976 when LonguetHigguin and Cokelet [17] presented their bi-dimensional (2D) approach to simulate the transient surface waves. Their approach was based on the potential flow theory and used the Boundary Element Method (BEM) to solve the Laplace's equation in conjunction with a Mixed EulerianLagrange (MEL) technique to update the free surface.…”