Nonlinear evolution of sand ripples in a viscous flow
نویسنده
چکیده
We investigate theoretically the nonlinear evolution of sand ripples in a viscous fluid when a granular bed is submitted to a two-dimensional laminar shear flow. On the basis of the hydrodynamic equations and a semi-empirical sand transport law, we derive a non-local and nonlinear equation for the dynamics of the bed profile in the case of infinite flow depth. This equation reveals a coarsening process: the wavelength of the ripple pattern increases indefinitely with time and no final state is selected. The temporal evolution of the amplitude, wavelength and drift velocity of the ripple pattern is carefully analyzed: in particular, the drift speed is found inversely proportional to the ripple amplitude. Finally, we argue that coarsening may interrupt either when the flow is three-dimensional or when the flow depth is finite (i.e., comparable to the amplitude of the ripple pattern).
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