KdV and BBM models in open cylindrical channel under precession

Abstract

The Korteweg–de Vries partial differential equation that has nonlinear-dispersion balance was derived under potential conditions to match the case of a single Kelvin mode that was noticed revolving on the outer periphery of an open cylindrical channel under precession conditions, which is assumed the solitary wave case in the channel. This led to a new version of the equation with a forcing term that includes the tilt effect with coefficients include the rotation effect. It was solved numerically using Fourier transformation methods for space discretization and the fourth order Runge–Kutta method for time discretization; the results were in a good match with the experiment. The rotational case led to a new Benjamin–Bona–Mahony equation that has variable coefficients with time and space mainly coming from the Coriolis effect in the axial direction of motion, with a forcing term comes from the gravity force. It was also solved numerically using a simple implicit finite difference scheme. This equation has two versions, one in terms of the velocity and one in terms of the amplitude. The first was compared with the bore velocity signal, which reflected the cnoidal type of waves, and the results were in a satisfactory match with the extracted signals; the second one was tracked with time to see the role Coriolis and gravity forces play on the single Kelvin wave form.