A Penalization Method for Calculating the Flow Beneath Traveling Water Waves of Large Amplitude

Abstract

A penalization method for a suitable reformulation of the governing equations as a constrained optimization problem provides accurate numerical simulations for large-amplitude travelling water waves in irrotational flows and in flows with constant vorticity.

Extrema of the dynamic pressure in an irrotational regular wave train

“…The approach used in the present paper relies heavily on the irrotational nature of the flow, and cannot be applied to deal with flows having non-zero vorticity. Nevertheless, provided that no flow-reversal occurs, numerical simulations 9,10,15,16,26 and some analytical investigations 14 indicate that we should expect qualitatively somewhat similar results.…”

“…To elucidate the meaning of this terminology, note that the Euler equation, expressed componentwise by the first two equations in (9), is equivalent to the fact that…”

Approximations of steady periodic water waves in flows with constant vorticity

“…We assume that the water is incompressible and inviscid, over a flat bed and acted upon by gravity g. In what follows we make no shallowness or small amplitude approximation. In order to avoid extended revision we refer to the recent works [6,11,12] for the mathematical formulation of the problem. In this section we make only a brief review of the problem, in order to provide sufficient terminology for the fundamental Boundary Value Problem (BVP) (2.10).…”

“…In the last part of the work we illustrate these flow characteristics for several different types of wave-current interactions. Laboratory experiments and numerical simulations for irrotational waves are discussed in [1,2,32], while this type of studies for wave-current interactions in flows of constant vorticity were pursued in [11,16,22,23,29,30]. The results presented below permit a more detailed analysis and numerical simulation.…”

On Irrotational Flows Beneath Periodic Traveling Equatorial Waves

“…Irrotational flows of symmetric periodic traveling water waves over a flat bed are well-studied; from a theoretical point of view [4,10,19], numerically [1,9,11] and experimentally [23].…”