Abstract
In this article, we present the current status of the derivation of a viscous Serre–Green–Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary layer) is described by Navier–Stokes equations. We consider a particular regime binding the Reynolds number and the shallowness parameter. The computations presented in this article are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prandtl-like equation that we claim to be irreducible. Further approximations are necessary to obtain a tractable model.
Highlights
The water wave theory has been essentially developed in the framework of the inviscid and, very often, irrotational Euler equations
A straightforward energy balance asymptotic analysis shows that the main dissipation occurs at the bottom boundary layer [1] [Section §2]
The corresponding long wave and small amplitude Boussinesq-type systems have been derived taking into account the boundary layer effects [4]
Summary
The water wave theory has been essentially developed in the framework of the inviscid and, very often, irrotational Euler equations. A straightforward energy balance asymptotic analysis shows that the main dissipation occurs at the bottom boundary layer [1] [Section §2] (or at the lateral walls if they are present [2,3]) In this way, the corresponding long wave and small amplitude Boussinesq-type systems have been derived taking into account the boundary layer effects [4]. This solitonic behavior was rediscovered in a seminal work by Zabusky and Kruskal (1965), which opened the whole research direction in infinite-dimensional integrable systems It is precisely a numerical simulation of this reduced equation that suggests that there exist solutions that do not vanish and even remain invariant (solitary waves). A full derivation appears very unlikely since it would require being able to reduce the Prandtl’s equation found in the boundary layer
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