Approximate deconvolution model in a bounded domain with vertical regularization

Abstract

We study the global existence issue for a three-dimensional Approximate Deconvolution Model with a vertical filter. We consider this model in a bounded cylindrical domain where we construct a unique global weak solution. The proof is based on a refinement of the energy method given by Berselli in [3].

On the Boussinesq equations with anisotropic filter in a vertical pipe

“…They couple the Navier-Stokes equations for the fluid velocity (whose difficulties are shared) with a transport equation (requiring an additional analysis) for a scalar, usually interpreted as a density or temperature perturbation. An analogous result has been obtained by Ali [1] in a vertical cylindrical domain for the Navier-Stokes equations regularized through the inverse of the differential operator A 3 = I − α 2 ∂ 2 3 , where ∂ 3 is the derivative with respect to the vertical space variable. Here, we further develop ideas coming from [1,2] and we consider the LES model obtained by filtering the Boussinesq equations in a vertical cylindrical domain, i.e., a bounded domain, periodic in the vertical direction, endowed with homogeneous Dirichlet conditions on the lateral boundary, and through the vertical filter given by A −1 3 .…”

“…An analogous result has been obtained by Ali [1] in a vertical cylindrical domain for the Navier-Stokes equations regularized through the inverse of the differential operator A 3 = I − α 2 ∂ 2 3 , where ∂ 3 is the derivative with respect to the vertical space variable. Here, we further develop ideas coming from [1,2] and we consider the LES model obtained by filtering the Boussinesq equations in a vertical cylindrical domain, i.e., a bounded domain, periodic in the vertical direction, endowed with homogeneous Dirichlet conditions on the lateral boundary, and through the vertical filter given by A −1 3 . In particular, we find a suitable setting to handle also the density equation; together with existence and uniqueness of regular weak solutions, we also prove that the energy of the model is exactly preserved.…”

“…In our analysis, we consider a zeroth-order deconvolution modelà la Stolz and Adams (or simplified-Bardina), see [9,18]. We refer to [1,4,7,8] for results concerning Approximate Deconvolution Models (ADM) with general order of deconvolution for the Navier-Stokes equations and the Boussinesq equations. We do not enter the game of considering fractional variants of the filter (anyway a fractional power of the Laplacian larger than one-half will suffice for most results), in order to find the critical spaces and so on, since we are mainly interested to propose an easily applicable method, which can be used and implemented in an efficient way also by practitioners.…”

Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region

“…For a function w, we introduce the horizontal filter (related to the horizontal Helmholtz operator), given by [3,16,17], from the point of view of the numerical simulations, this filter is less memory consuming with respect to the standard one. Further, another interesting feature of this filter is that, even in the case of domains which are not periodic in the vertical direction, there is no need to introduce artificial boundary conditions for the Helmholtz operator (see, e.g., [3,8,9,10,11]).…”

Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering

“…As discussed in [2,15,16], from the point of view of the numerical simulations, this filter is less memory consuming with respect to the standard one. Another significant advantage of this choice is that there is no need to introduce artificial boundary conditions for the Helmholtz operator.…”