A numerical study on parasitic capillary waves using unsteady conformal mapping

This paper describes fully nonlinear computation of unsteady motion of parasitic capillary waves that appear on the front face of steep gravity waves progressing on water of infinite depth, within the framework of irrotational plane flow. As an alternative to the widely-used boundary integral method with mixed-Eulerian–Lagrangian (MEL) time updating, we focus on a numerical method based on unsteady conformal mapping, which will be hereafter referred to as the unsteady hodograph transformation (UHT) method. In this method, we solve the nonlinear evolution equations to find an unsteady conformal map in a complex plane with which the flow domain is mapped onto the unit disk while the free surface is fixed on the unit circle. The aim of this work is to compare the UHT method with the MEL method and find a more efficient method to compute parasitic capillary waves. From linear stability analysis, it is found that a critical difference between these two methods arises from the kernel of cotangent function in singular integrals, and the UHT method can avoid some numerical instability due to it. Numerical examples demonstrate that the UHT method is more suitable than the MEL method for not only parasitic capillary waves, but also capillary dominated waves. In particular, the UHT method requires no artificial techniques, such as filtering, to control numerical errors, in these examples. In addition, another major difference between the two methods is observed in terms of the clustering property of sample points on the free surface, depending on the restoring force of waves (gravity or surface tension).