Water waves over a strongly undulating bottom

Abstract

Two-dimensional free-surface potential flows of an ideal fluid over a strongly inhomogeneous bottom are investigated with the help of conformal mappings. Weakly-nonlinear and exact nonlinear equations of motion are derived by the variational method for arbitrary seabed shape parameterized by an analytical function. As applications of this theory, band structure of linear waves over periodic bottoms is calculated and evolution of a strong solitary wave running from a deep region to a shallow region is numerically simulated.

Liquid sloshing in a horizontally forced vessel with bottom topography

“…Thus this approach has clear computational advantages and we use it in this paper to investigate the effect of the vessel topography on the fluid free-surface evolution in a periodically forced vessel. Time-dependent conformal mappings have been used previously to investigate the effect of an inhomogeneous bottom topography on the propagation of periodic water waves (Ruban, 2004(Ruban, , 2005Viotti et al, 2013). These studies present numerical approaches similar to that presented in this paper, however they do not highlight the time-dependence of the conformal modulus, and its significance on the numerical calculation.…”

Water waves over a time-dependent bottom: Exact description for 2D potential flows

“…(23) below and compare with [16]) and substitute proper expressions into Eq. (6), in order to find ψ t .…”

“…Since the bottom motion is assumed to be prescribed, this function has the following structure (compare with [16]):…”

“…Important achievements in this direction were reported in works [9,10,11,12,13,14,15], including analytical and numerical results. A generalization has been recently made to the case when the bottom shape is still static but strongly inhomogeneous in space [16].…”

“…The theory of water waves now is among the most actively developing branches of the hydrodynamic science (see, for instance, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], and many references therein). In particular, two-dimensional (2D) flows of an ideal incompressible fluid with a free surface are very attractive for theoretical study, since they possess many essential features in common with real water waves, and simultaneously are relatively simple for analytical and numerical treatment.…”