Regularizing isotropic turbulence

Abstract

Since most turbulent flows cannot be computed directly from the (incompressible) Navier-Stokes equations, a dynamically less complex mathematical formulation is sought. In the quest for such a formulation, we consider nonlinear regularizations of the Navier-Stokes equations that can be analysed within the mathematical framework devised by Foias et al. If the regularization preserves symmetry and conservation properties, an inertial subrange exists. It can rigorously be proven, that the inertial subrange of the regularized system follows the usual –5/3 law of Kolmogorov. Compared to Navier-Stokes, the inertial subrange is shortened yielding a more amenable problem to solve numerically. To analyse the primary properties of the regularization method both direct and regularized numerical simulations of homogeneous, isotropic forced turbulence at a Reynolds number of about 500 (based on the Taylor scale) are performed. With the help of the numerical results the regularization mechanism can be clarified. Therefore triad interactions (nonlinear interactions in Fourier space) are computed around the end of the inertial subrange of the regularized system. By comparing the regularized interactions with the nonregularized, we characterize the regularization in terms of modified local and nonlocal triads. Finally, the restrained production of small vortices is measured directly from the simulations with and without the regularization.