Shallow Water Equations for Shear Flows
Open channel flows of ideal incompressible fluid with velocity shear are considered in the long wave approximation. Nonlinear integro-differential models of shallow flow with continuous vertical or horizontal velocity distribution are derived. It is shown that mathematically the models are equivalent and, consequently, the obtained early results for two-dimensional open channel flows with a vertical shear can be applied to the 2D flows with horizontally nonhomogeneous velocity field. Stability of shear flows in terms of hyperbolicity of the governing equations is studied. It is shown that the type of the equations of motion can change during the evolution of the flow, which corresponds to the long wave instability for a certain velocity field. A simple mathematical model describing the nonlinear stage of the Kelvin–Helmholtz instability of shear flows is derived. The problem of the mixing layer interaction with a free surface and its transition into a turbulent surface jet is considered.
KeywordsShear Flow Open Channel Shallow Water Equation Vertical Shear Horizontal Shear
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