Raquel Taboada-Vázquez (Universidade da Coruña).
The objective of this talk is to present some new shallow water models derived from the Navier–Stokes equations considering a shallow domain and using asymptotic analysis.
We will focus on two different cases, in the first place we study the Navier–Stokes equations with anisotropic viscosity. The shallow water model obtained (see [1]) considers the possibility of a non-constant bottom and calculates the depth-averaged horizontal velocity, but also the three components of the velocity for all z (when the vorticity is not zero). New diffusion terms are revealed in this way. The numerical computations performed confirm that the new model is able to approximate the solutions of Navier–Stokes equations with dependence on z (reobtaining the same velocities profile), whereas the classical model only computes the average velocity.
In the second case we introduce a new shallow water model able to filter the high frequency oscillations that are produced, when the Reynolds number is increased, in turbulent flows (see [2]). The results of the numerical 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 than the time step required for the classical models.
[1] J.M. Rodríguez, R. Taboada-Vázquez: Derivation of a new asymptotic viscous shallow water model with dependence on depth, Appl. Math. Comput. 219 (2012) 3292–3307.
[2] J.M. Rodríguez, R. Taboada-Vázquez: Time-averaged shallow water model: Asymptotic derivation and numerical validation, J. Math. Anal. Appl. 428 (2015) 930–950.