Analytical approximation and numerical simulations for periodic travelling water waves

Abstract

We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity.This article is part of the theme issue 'Nonlinear water waves'.

On rotational flows with discontinuous vorticity beneath steady water waves near stagnation

“…For the case of constant vorticity, it is known that when the vorticity is below a critical value , the stagnation point first appears at the bottom and at that stage numerical continuation breaks down (Ko & Strauss 2008 b ). However, Amann & Kalimeris (2018) and Kalimeris (2018) found that when but is very close to , the branch of the bifurcation diagram ( plot) has another part which is bounded below and above by two values of the bifurcation parameter . These bounds correspond to two different waves in flows with stagnation point at the bottom and at the surface.…”

“…Amann & Kalimeris (2018) expanded upon the method of Ko & Strauss (2008 b ), by allowing non-uniform grid points and enabling the continuation with vorticity as a bifurcation parameter. Using the latter scheme they successfully discovered a new part of the bifurcation curve for some critical constant vorticity values (see also Kalimeris 2018). Furthermore, Constantin, Kalimeris & Scherzer (2015 b ) proposed a penalization method to solve for large-amplitude waves.…”

Some quasi-analytical solutions for propagative waves in free surface Euler equations

“…Some other explicit solutions have been presented in the literature, for example by Boulanger et al [2] and Daboussy et al [6] for the steady state. Following the methodology of Constantin and Strauss [3], Kalimeris [9] proposes an asymptotic expansion of the Euler system reducing the problem resolution to a cascade of ODEs. On the one hand, the result of Kalimeris is not reduced to flows without vorticity, on the other hand the proposed solutions -also exhibiting singularities of the free surface -are not analytical because obtained through an iterative numerical process.…”

Capturing the flow beneath water waves

“…The flow beneath water waves can be captured by different numerical strategies, as outlined above. The following articles [7][8][9][10][11][12][13][14][15][16] are of interest to the problems discussed here. As will be presented herein, we adopt a boundary integral formulation (through Green's third identity) as well as conformal mappings, in order to compute our features of interest with great precision.…”