On the Boussinesq equations with anisotropic filter in a vertical pipe

We propose a new Large Eddy Simulation (LES) model for the Boussinesq equations. We consider the motion in a three-dimensional domain with solid walls, and in a particular geometrical situation we look for solutions which are periodic in the vertical direction and satisfy homogeneous Dirichlet conditions on the lateral boundary. We are thus modeling a vertical pipe and one main difficulty is that of considering regularizations of the equation which are well behaved also in presence of a boundary. The LES model we consider is then obtained by introducing a vertical filter, which is the natural one for the setting that we are considering. The related interior closure problem is treated in a standard way with a simplified-Bardina deconvolution model. The most technical analytical point is related to the fact that anisotropic filters provide less regularity than the isotropic ones and, in principle, the density term appearing in the Boussinesq equations may behave very differently from the velocity. We are able to define an appropriate notion of regular weak solution, for which we prove existence, uniqueness, and we also show that the energy associated to the model is exactly preserved.