Interfacial Wave Motion of Very Large Amplitude: Formulation in Three Dimensions and Numerical Experiments

Abstract

Interfacial gravity driven motion of a two-fluid system bounded above and below by rigid lids, is studied. The interfacial motion is three-dimensional, fully nonlinear and fully dispersive. By the method of successive approximations, various approximations of the method are derived, where the truncated versions are computationally fast, still being highly accurate. The ac- curacy is tested out by numerical calculations and comparisons to accurate interfacial solitary waves in two dimensions wherein the reference computations, the full nonlinearity and dispersive effects are kept. The calculations show that the methods expressed by its quadratic and cubic approximations provide very accurate representation of the waves. Error estimates obtained by Euclidian norm tend to zero when the amplitude goes to zero. The cubic approximation of the normal velocity along the interface has an error of less than 0.02 percent on the side of the lower, deep fluid layer and 0.24 percent on the side of the upper, shallow fluid, respectively, when the wave amplitude is equal to the upper layer depth. The cubic approximation is still very good for solitary waves of very large amplitude; even as large as close to the conjugate flow limit, which in the present computations is 9.66... times the upper layer depth (depth ratio of 20.4, density ratio of 0.986). A downward shift of the reference level improves the calculation of the normal velocity in the thin layer.