A new Boussinesq method for fully nonlinear waves from shallow to deep water

作者：Madsen, P A; Bingham, H B; Liu, Hua

来源：Journal of Fluid Mechanics, 2002, 462: 1-30.

DOI：10.1017/S0022112002008467

• Wavenumbers

<jats:p>A new method valid for highly dispersive and highly nonlinear water waves is
presented. It combines a time-stepping of the exact surface boundary conditions with
an approximate series expansion solution to the Laplace equation in the interior
domain. The starting point is an exact solution to the Laplace equation given in
terms of infinite series expansions from an arbitrary <jats:italic>z</jats:italic>-level. We replace the
infinite series operators by finite series (Boussinesq-type) approximations involving up
to fifth-derivative operators. The finite series are manipulated to incorporate Padé
approximants providing the highest possible accuracy for a given number of terms.
As a result, linear and nonlinear wave characteristics become very accurate up to
wavenumbers as high as <jats:italic>kh</jats:italic> = 40, while the vertical variation of the velocity field
becomes applicable for <jats:italic>kh</jats:italic> up to 12. These results represent a major improvement
over existing Boussinesq-type formulations in the literature. A numerical model is
developed in a single horizontal dimension and it is used to study phenomena such
as solitary waves and their impact on vertical walls, modulational instability in deep
water involving recurrence or frequency downshift, and shoaling of regular waves up
to breaking in shallow water.</jats:p>

• 出版日期2002
• 单位上海交通大学