CABARET scheme implementation for free shear layer modeling

In present paper we reexamine the properties of CABARET numerical scheme formulated for a weakly compressible fluid flow basing the results of free shear layer modeling. Kelvin–Helmholtz instability andsuccessive generation of two-dimensional turbulence provide a wide field for a scheme analysis including temporal evolution of the integral energy and enstrophy curves, the vorticity patterns and energy spectra, as well as the dispersion relation for the instability increment. The most part of calculations is performed for Reynolds number $\textrm{Re}=4\times10^5$ for square grids sequentially refined in the range of 1282–20482 nodes. An attention is paid to the problem of under resolved layers generating a spurious vortex during the vorticity layers roll-up. This phenomenon takes place only on a coarse grid with 1282 nodes, while the fully regularized evolution pattern of vorticity appears only when approaching 10242-node grid. We also discuss the vorticity resolution properties of grids used with respect to dimensional estimates for the eddies at the borders of the inertial interval, showing that the available range of grids appears to be sufficient for a good resolution of small-scale vorticity patches.Nevertheless, we claim for the convergence achieved for the domains occupied by large-scale structures.
The generated turbulence evolution is consistent with theoretical concepts imposing the emergence of large vortices, which collect all the kinetic energy of motion, and solitary small-scale eddies. The latter resemble the coherent structures surviving in the filamentation process and almost noninteracting with other scales. Thedissipative characteristics of numerical method employed are discussed in terms of kinetic energy dissipation rate calculated directly and basing theoretical laws for incompressible (via enstrophy curves) and compressible(with respect to the strain rate tensor and dilatation) fluid models. The asymptotic behavior of the kinetic energy and enstrophy cascades comply with two-dimensional turbulence laws $E(k)\propto k^{-3}$, $\omega^{2}(k)\propto k^{-1}$. Considering the instability increment as a function of dimensionless wave number shows a good agreement with other papers, however, commonly used method of instability growth rate calculation is not always accurate, so some modification is proposed. Thus, the implemented CABARET scheme possessing remarkably small numerical dissipation and good vorticity resolution is quite competitive approach compared to other high-order accuracy methods