KdV Equation Model in Open Cylindrical Channel under Precession

TL;DR

This paper derives a modified KdV equation for Kelvin waves in a rotating, tilted cylindrical open channel, incorporating a forcing term and validating results with numerical simulations and experiments.

Contribution

It introduces a new non-integrable KdV model with a forcing term for Kelvin waves in a rotating cylindrical system, validated by numerical and experimental data.

Findings

The model captures the formation of Kelvin solitary waves under tilt and rotation.

Numerical solutions agree well with experimental observations.

The forced KdV equation is non-integrable and requires numerical methods.

Abstract

A new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope {\tau} from the inertial vertical z, in uniform rate {\Omega}_1={\tau}{\Omega}, and the whole tank is elevated over other table rotating at rate {\Omega}. Under these conditions, a set of Kelvin waves is formed on the free surface depending on the angle of tilt, characterized by the slope {\tau}, volume of water, and rotation rate. The resonant mode in the system appears in the form of a single Kelvin solitary wave, whose amplitude satisfies the Korteweg-de Vries equation with forced term. The equation was derived following classical perturbation methods, the additional term made the equation a non-integrable one, that cannot be solved without the help of numerical methods. Invoking the simple finite difference scheme method, it was found that the numerical results are in a good agreement with the experiment.