Numerical continuation method for large-amplitude steady water waves on depth-varying currents in flows with fixed mean water depth

Abstract

A numerical method is proposed for computation of large amplitude water waves interacting with depth varying currents. The proposed method is developed for fixed mean-depth of water for which a theoretical formulation has recently been introduced in the literature Henry (2013b, 2013c) but no numerical studies have been performed yet. The proposed method relies on a numerical continuation approach for computing large amplitude waves bifurcating from the steady laminar flows. This study hence is on the numerical continuation method for computing two-dimensional (2D) large-amplitude steady water waves on rotational flow with an arbitrary current profile, based on an emerging formulation of the problem. For given mean water depth, current profile, and wave length, the method is able to generate a group of waves from the laminar flow solution to the limiting wave of the largest wave height. Due to the normalization effect in the formulation, the computational domain is fixed regardless of the wave length and water depth, allowing for efficient computation even when dealing with long waves in deep water. The method is first applied to study non-linear wave-current interactions in ocean engineering demonstrating its potential in capturing the wave period changes with increasing wave amplitude in the presence of current. Furthermore, wave characteristics are investigated for cases of constant and discontinuous vorticity, revealing some interesting features. In particular, for waves on linear shear flows, the surface profiles of the waves with the largest amplitude for a range of the vorticity values tend to pass through two fixed points in the 2D plane, and this characteristic seems to be invariant of the water depth.