Time-averaged shallow water model: Asymptotic derivation and numerical validation

The objective of this paper is to derive, from the Navier-Stokes equations in a shallow domain, a new bidimensional shallow water model able to filter the high frequency oscillations that are produced, when the Reynolds number is increased, in turbulent flows. With this aim, the non-dimensional Navier-Stokes equations are time-averaged, and then asymptotic analysis techniques have been used as in our previous works (Rodríguez and Taboada-Vázquez, 2005-2012 [8-14]). The small non-dimensional parameter considered, ε, is the quotient between the typical depth of the basin and the typical horizontal length of the domain; and it is studied what happens when ε becomes small. Once the new model has been justified, by the method of asymptotic expansions, we perform some numerical experiments. The results of these experiments confirm that this new model is able to approximate analytical solutions of Navier-Stokes equations with more accuracy than classical shallow water models, when high frequency oscillations appear. To reach a given accuracy, the time step for the new model can be much larger (even four hundred times larger) than the time step required for the classical models.