6 research outputs found
Stability of shear shallow water flows with free surface
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
We perform the stability analysis for a free surface fluid current modeled as two finite layers
of constant vorticity, under the action of gravity and absence of surface tension. In the same
spirit as Taylor [“Effect of variation in density on the stability of superposed streams of fluid,”
Proc. R. Soc. A 132, 499 (1931)], a geometrical approach to the problem is proposed, which allows
us to present simple analytical criteria under which the flow is stable. A strong destabilizing effect
of stratification in density is perceived when the results are compared with those obtained for the
physical setting where the vorticity interface is also a density interface separating two immiscible
fluids with constant densities. In contrast with the homogenous case, the stratified bilinear shear
current is mostly unstable and can only be stabilized when the background current in the upper
layer is constant
Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current
We study long surface and internal ring waves propagating in a stratified
fluid over a parallel shear flow. The far-field modal and amplitude equations
for the ring waves are presented in dimensional form. We re-derive them from
the formulation for plane waves tangent to the ring wave, which opens a way to
obtaining important characteristics of the ring waves (e.g. group speed) and to
constructing more general `hybrid solutions' consisting of a part of a ring
wave and two tangent plane waves. The modal equations constitute a new spectral
problem, and are analysed for a number of examples of surface ring waves in a
homogeneous fluid and internal ring waves in a stratified fluid. The detailed
analysis is developed for the case of a two-layered fluid with a linear shear
current where we study their wavefronts and two-dimensional modal structure.
Comparisons are made between the modal functions of the surface waves in a
homogeneous and two-layered fluids, as well as the interfacial waves described
exactly and in the rigid-lid approximation. We also analyse the wavefronts of
surface and interfacial waves for a family of power-law upper-layer currents,
which can be used to model wind generated currents, river inflows and exchange
flows in straits. A global and local measure of the deformation of wavefronts
are introduced and evaluated.Comment: 35 pages, 21 figure
Stability Of Shear Shallow Water Flows with Free Surface
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