Circular Shallow Water Profiles

Inspired by pertinent literature in the context of Analogue Gravity, this work analyzes the stream profiles of a fluid film flowing along a radially symmetric, inclined bed with different bottom topographies, in the framework of a 2-D Shallow-water model in radial symmetry. First, steady state profiles are obtained without including turbulent friction, with different slopes and three kinds of bottom topography: flat, stepped and sinusoidal. The flat bottom produces new kinds of orbits with a two Critical Froude Horizons behaviour, because of the interplay between Discharge Conservation and Momentum Balance, which is affected by the presence of slope. The stream profiles obtained for the stepped bottom exhibit instead an energy jump, which allows to interpret discontinuities as orbit jumps. The sinusoidal profile instead can be seen as a sinusoidal perturbation of the flat one: the response of the stream to the amplitude of such perturbation is found to be dependent on an inequality involving Radiant Specific Kinetic Height, and can be amplifying, stabilizing or damping. When turbulent friction (modeled with Chézy parametrization) is included, the stream profiles obtained for the same bottom topographies result to be governed by gravity in case of subcritical regime and by friction in case of supercritical one. In addition, also novel bottom topographies ensuring energy conservation, critical depth and uniform height are obtained. The solutions with Hydraulic jump, represented as a shock discontinuity of zero order, are then obtained and the limits of such representation are analyzed. Finally, a Perturbation Equation of the Shallow-water model is obtained to serve as a future work on Linear Stability analysis.