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Existence and features of similarity solutions for non-Boussinesq gravity currents. (English) Zbl 1176.76023

Summary: The flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions. The current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of much lower density. Unlike earlier investigators, we assume that the ambient fluid exerts no significant resisting action on the current, and therefore the flow depth is expected to drop to zero at the front in the absence of friction. In this context, the shallow-water equations are highly appropriate for computing the mean velocity and flow depth of the current. The boundary condition imposed at the front leads to technical mathematical difficulties. Indeed, unlike in the Boussinesq case, no regular solution to the shallow-water equations satisfies the downstream condition, but when the flow is supercritical at the channel inlet, it is possible to construct a piecewise solution by patching a regular solution to an exceptional solution, which represents the head behavior. To better understand this result and make sure that the result is physically relevant, we consider the Navier–Stokes equations within the high-Reynolds-number limit. Approximate similarity solutions can be worked out, which support our earlier analysis on the shallow-water equations. While the flow body is self-similar and weakly rotational, the head is not self-similar, but tends toward a self-similarity shape at long times. It is characterized by a strong vorticity, a straight free surface, and a nonuniform velocity profile, which becomes flatter and flatter with time.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
35Q31 Euler equations
35Q86 PDEs in connection with geophysics
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