Estimates of the Modeling Error of the $\alpha$-Models of Turbulence in Two and Three Space Dimensions

Abstract

This report investigates the convergence rate of the weak solutions $\mathbf{w}^{\alpha }$ of the Leray-$\alpha$, modified Leray-$\alpha$, Navier–Stokes-$\alpha$ and the zeroth ADM turbulence models to a weak solution $\mathbf{u}$ of the Navier–Stokes equations. It is assumed that this weak solution $\mathbf{u}$ of the NSE belongs to the space $L^4(0, T; H^1)$. It is shown that under this regularity condition the error $\mathbf{u}-\mathbf{w}^{\alpha }$ is $\mathcal {O}(\alpha )$ in the norms $L^2(0, T; H^1)$ and $L^{\infty }(0, T; L^2)$, thus improving related known results. It is also shown that the averaged error $\overline{\mathbf{u}}-\overline{\mathbf{w}^{\alpha }}$ is higher order, $\mathcal {O}(\alpha ^{1.5})$, in the same norms, therefore the $\alpha$-regularizations considered herein approximate better filtered flow structures than the exact (unfiltered) flow velocities.