Modeling error of \(\alpha\)-models of turbulence on a two-dimensional torus
The article focuses on different turbulence models in a two-dimensional periodic domain: the Leray-\(\alpha\) model, the simplified Bardina model, and the modified Leray-\(\alpha\) model, which are obtained from the Navier--Stokes equations by certain regularizations of the nonlinear term. Denoting by \(u_\alpha\) the solution to the respective \(\alpha\)-regularized model and by \(u\) the solution to the Navier--Stokes equations, the authors derive rates of convergence of \(u_\alpha\) to \(u\) as \(\alpha\to0\) in the norm of both \(L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H^1(\Omega))\) and \(H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^2(\Omega))\), which improve known convergence rates for these models. Convergence rates for the corresponding pressure fields are also deduced, and the authors explain how to transfer their results to the more technical case of a bounded domain with zero Dirichlet conditions. The paper concludes with additional remarks on the difficulties that occur if one may derive analogous estimates in the case of three dimensions or the Euler equations.
scientific article; zbMATH DE number 7360946
scientific article