Stability Of Shear Shallow Water Flows with Free Surface

Abstract

Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is shown that all shear flows having monotonic convex velocity profiles are stable. The hydrodynamic approximations of the model corresponding to the classes of flows with piecewise linear continuous and discontinuous velocity profiles are derived and studied. It is shown that these approximations possess Hamiltonian structure and a complete system of Riemann invariants, which are found in an explicit form. Sufficient conditions for hyperbolicity of the governing equations for such multilayer flows are formulated. The generalization of the above results to the case of stratified fluid is less obvious, however, it is established that vorticity has a stabilizing effect.

Effect of variation in density on the stability of bilinear shear currents with a free surface

“…The extent of the consequences of doing so with many more vorticity layers (as in Ref. [33]) is currently not known, as only sufficient conditions are known for the stability of the flow in the longwave limit [22], and warrants further study. Comparison with the setting where the vorticity interface becomes also a density interface, separating two immiscible liquids with constant densities, reveals that stratification in density has a strong destabilizing effect on the stability of the flow.…”

“…Here, we investigate the conditions under which bilinear shear currents in a homogeneous fluid are stable. The case when the shear current is modelled by N constant vorticity layers was recently addressed by Chesnokov, El, Gavrilyuk, and Pavlov [22], and sufficient conditions for the long wave limit stability were proposed (see Lemma 3.2 in [22]). Here, dispersive effects of Euler equations are considered, but the analysis is limited to bilinear shear currents (N = 2).…”

Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography

“…Due to this fact the terms ε 2b and ε 2ḃ can be neglected during the derivation of system (5). The first two equations in (5) correspond to the balance of mass and momentum in the non-hydrostatic and almost potential lower layer.…”

“…Due to this fact the terms ε 2b and ε 2ḃ can be neglected during the derivation of system (5). The first two equations in (5) correspond to the balance of mass and momentum in the non-hydrostatic and almost potential lower layer. They are obtained by averaging the incompressibility and horizontal momentum equations in (1) over the depth taking into account the boundary conditions (3) and (4).…”

“…They are obtained by averaging the incompressibility and horizontal momentum equations in (1) over the depth taking into account the boundary conditions (3) and (4). We also assume that u| z=b+h = U, since this is the only case the first two equations in (5) are compatible with the energy equation for the lower layer (see [10] for details)…”