On self-similarity properties of isotropic turbulence in numerical simulations of the compressible Euler equations

We present numerical calculations of the parameters $C_{\nu}$, $C_{\epsilon}$ and $C_{\kappa}$ associated with the common closures for turbulence production, dissipation and diffusion. In the case of homogeneous and isotropic turbulence, these parameters are expected to be statistically scale-invariant within the inertial subrange. In order to scrutinise this conjecture, we utilised a generalisation of the Germano filtering formalism, which is applicable to compressible flows as well. The filtering of data obtained from three-dimensional direct numerical simulations of forced isotropic turbulence with Mach numbers in the range $\sim 0.1...1$ then yielded values of the closure parameters associated with different length scales. The results indicate that the closure parameters are nearly universal for subsonic or moderately transonic flows, although the resolution of $432^{3}$ grid cells in our simulations is not quite sufficient to clearly establish scale invariance. In addition, it was found that the customary assumption of a kinetic Prandtl number of about unity for the gradient-diffusion closure is flawed due to the misalignment between turbulent flux and the gradient of the turbulence energy. Nevertheless, sound correlation can be achieved if the flux magnitude rather than the flux vector is locally matched. This conclusion is particularly useful for the family of subgrid scale models based on the turbulence energy equation. Furthermore, the parameter of production $C_{\nu}$ was computed in the fashion of dynamical procedures. Thereby, superior agreement between modelled and explicitly evaluated turbulence stresses in comparison to the eddy-viscosity closure with constant $C_{\nu}$ was verified.