A three-dimensional model for two coupled turbulent fluids: numerical analysis of a finite element approximation

This article deals with the numerical analysis of a coupled two-fluid Reynolds-averaged Navier-Stokes (RANS) turbulence model, such as atmosphere-ocean flow. Each fluid is modeled by the coupled steady Stokes equations with the equation for the turbulent kinetic energy (TKE). In this model, the eddy viscosities for velocity and TKE depend on the TKE, the production (source) term for the TKEs is only in L-1 and the boundary condition for the TKEs on the interface between the two flows depends quadratically on the difference of velocities. To overcome the lack of regularity, we approximate the initial system by a regularized system, in which the eddy viscosities and source terms for the TKEs are regularized by convolution. We perform a full finite element discretization of the regularized model, combined with a decoupled iterative linearization procedure. We prove that the discrete scheme converges to the continuous scheme for large enough eddy viscosities in natural norms. Finally, we present some numerical tests where we study the accuracy of the procedure, and simulate a realistic flow in which an imposed wind in the upper atmosphere generates an upwelling in the oceanic flow.