Variational Water Wave Modelling: from Continuum to Experiment

Abstract

Variational methods are investigated asymptotically and numerically to model water waves in tanks with wave generators. As a validation, our modelling results using (dis)continuous Galerkin finite element methods will be compared to a soliton splash event resulting after a sluice gate is removed during a finite time in a long water channel with a contraction at its end.

Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

“…(46) can be found in Bokhove and Kalogirou (2016). Rather than using this matrix form directly, we accommodate only the spatial discretization using Firedrake (Rathgeber et al, 2016;Balay et al, 2016Balay et al, , 1997Dalcin et al, 2011;Hendrickson and Leland, 1995), "an automated system for the portable solution of partial differential equations using the finite element method (FEM)".…”

“…To derive the Benney-Luke model, the velocity potential φ is expanded in terms of the sea-bed potential φ(x, y, 0, t) = (x, y, t) and the dispersion parameter µ as in Bokhove and Kalogirou (2016):…”

“…(9) and retaining terms up to second order, Eq. (12) becomes (see Bokhove and Kalogirou, 2016, for details)…”

“…Whilst it has the advantage of conserving both the nonlinear and dispersive properties of the waves (essential to the modelling of a freak wave, for instance), it does not require a mesh moving vertically with the free surface since the model is reduced to the horizontal plane. Pego and Quintero (1999) derived these modified Benney-Luke equations and Bokhove and Kalogirou (2016) recently used them to simulate a soliton splash resulting from a wave running in a restricted channel. Their simulations were in reasonably good agreement with experiments, which confirms that the Benney-Luke approximation is an accurate model of water waves.…”

“…Starting from Luke's variational principle for an inviscid fluid with a free surface (Luke, 1967), a model similar to the one derived by Benney and Luke (1964) for small-amplitude and long waves is obtained. The (numerical) method developed by Bokhove and Kalogirou (2016) is used to derive the relevant variational principle for our Benney-Luke model. This asymptotic model conserves the nonlinear and dispersive properties of the sea waves, which enables comparison with the Kadomtsev-Petviashvili (KP) model for which the modified Miles' theory as expressed in Eqs.…”