Waves over Curved Bottom: The Method of Composite Conformal Mapping

TL;DR

This paper introduces a compact numerical method using conformal mapping to analyze ideal fluid flows over curved bottoms, extending previous models to include rotating frames and complex geometries, with applications to tsunami modeling, solitons, and centrifuge dynamics.

Contribution

The paper generalizes the conformal mapping method to potential flows in rotating frames and complex geometries, providing new numerical results for free boundary dynamics in such systems.

Findings

Modeling of tsunami waves over nonuniform bottoms

Dynamics of Bragg solitons over periodic profiles

Numerical results on free boundary in rotating centrifuges

Abstract

A compact and efficient numerical method is described for studying plane flows of an ideal fluid with a smooth free boundary over a curved and nonuniformly moving bottom. Exact equations of motion in terms of the so-called conformal variables are used. In addition to the previously known applications for shear flows with constant (including zero) vorticity, here a generalization is made to the case of potential flows in uniformly rotating coordinate systems, where centrifugal and Coriolis forces are added to the gravity force. A brief review is given of previous results obtained by this method in a number of physically interesting problems such as modeling of tsunami waves caused by the movement of nonuniform bottom, the dynamics of Bragg (gap) solitons over a spatially periodic bottom profile, the Fermi-Pasta-Ulam (FPU) recurrence phenomenon for waves in a finite pool, the formation of anomalous waves in an opposing nonuniform current, and the propagation of a solitary wave in a shear current and its runup on a depth difference. In addition, a number of new numerical results are presented concerning the nonlinear dynamics of a free boundary in closed rotating containers partially filled with a fluid -- centrifuges of complex shape. In this case, the equations of motion differ in some essential details from those of $x$-periodic systems.