A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

Abstract

In geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model. In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to approximate the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

“…As it was mentioned in the work of Fernández‐Nieto et al, the nonhydrostatic model () is not completely coherent with . More precisely, the vectorial space of the approximation for u and w , ie, both in , is not coherent with the divergence‐free condition.…”

“…which corresponds to the vertical average of the standard deviation of the compatibility condition (1). In the work of Fernández-Nieto et al, 31 it is shown that (GN) is equivalent, at least for smooth solution, to the classical Peregrine formulation. 2 Let us first recall the main physical properties of (GN).…”

“…More precisely, the vectorial space of the approximation for u and w , ie, both in , is not coherent with the divergence‐free condition. To make the approximation coherent with the divergence‐free condition when u is in , w must be approximated in (see Section 2.2.3 and the work of Fernández‐Nieto et al for more details). It is the case for the Serre‐Green‐Naghdi model () …”

“…The initial condition h (0, x )= h 0 ( x ), , and σ (0, x )= σ 0 ( x ) must satisfy the compatibility conditions and which corresponds to the vertical average of the standard deviation of the compatibility condition . In the work of Fernández‐Nieto et al, it is shown that () is equivalent, at least for smooth solution, to the classical Peregrine formulation . Let us first recall the main physical properties of ().…”

Modeling shallow water waves

“…This is numerically more involved, but the interest of this multi-layer model is that its linear dispersion can approximate the dispersion relation of the full water waves equations with a very good accuracy. We refer to [75] for an analysis of this dispersion relation (the model studied there differs from (52) but only in the nonlinear terms, which do not affect the linear dispersion relation); apart from this, the mathematical analysis (see in particular the open problems mentioned in §3.4) and the numerical implementation of (52) remain to be done. with l 2 = 1 − l 1 and 0 < l 1 < 1.…”