We consider a 3D Approximate Deconvolution Model (ADM) which belongs to the class of Large Eddy Simulation (LES) models. We work with periodic boundary conditions and the filter that is used to average the fluid equations is the Helmholtz one. We prove existence and uniqueness of what we call a "regular weak" solution (w N , q N ) to the model, for any fixed order N ∈ N of deconvolution. Then, we prove that the sequence {(w N , q N )} N ∈N converges -in some sense-to a solution of the filtered Navier-Stokes equations, as N goes to infinity. This rigorously shows that the class of ADM models we consider have the most meaningful approximation property for averages of solutions of the Navier-Stokes equations.This section is devoted first to the definition of the function spaces that we use, next to the definition of the filter through the Helmholtz equation, and finally to what we call the "deconvolution operator." There is nothing new here that is not already proved in former
In averaging the Navier-Stokes equations the problem of closure arises. Scale similarity models address closure by (roughly speaking) extrapolation from the (known) resolved scales to the (unknown) unresolved scales. In a posteriori tests scale similarity models are often the most accurate but can prove to be unstable when used in a numerical simulation. In this report we consider the scale similarity model given by ∇ • w = 0 and w t + ∇ • (w w) − ν∆w + ∇p = f. We prove it is stable (solutions satisfy an energy inequality) and deduce from that existence of weak solutions of the model.
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