A Subgrid-Scale Turbulence Model for Simulating Compressible Astrophysical Flows

My research team in the Laboratory for Computational Science & Engineering (LCSE) at the University of Minnesota has been simulating compressible, turbulent flows in stellar astrophysics for many years using our PPM gas dynamics codes. These codes solve the Euler equations of inviscid flow. They capture turbulent phenomena automatically, since for very high Reynolds number flows the development of turbulence and the nature of the turbulent cascade at longer wavelengths is not affected by viscosity. However, just as one finds from solving the Navier-Stokes equations for such flows, energy tends to accumulate in the region of the velocity power spectrum just above the very short wavelength region where viscous dissipation, either real or numerical in origin, sets in. In this “near dissipation” range, the velocity power spectrum tends to have a k−1 rather than a k−5/3 behavior. For our PPM codes, this near dissipation range is from wavelengths of about 8 to 30 grid cell widths, and for underresolved Navier- Stokes simulations (i.e. for the smallest viscous coefficients that yield stable behavior) it extends to somewhat longer wavelengths. This pile up of energy just before the dissipation range is a real physical effect, but it is undesirable when we are trying to get the best possible approximation to the infinite Reynolds number limit of viscous flows. Over the last few years, David Porter and I have been using very highly resolved PPM Euler simulation data for turbulent flows in the place of experimental data. We are able to filter these flows to yield high quality information on the various terms involving sub-filter-scale information that appear in the fluid equations for the filtered variables. We have compared this data with subgrid-scale turbulence modeling ideas, and we have developed a model that when added to the PPM Euler scheme eliminates the accumulation of energy in the near dissipation range of the spectrum. This model includes terms that mimic the form of some of the numerical truncation error terms, but in turbulent flow regions these terms are about an order of magnitude larger than their numerical error counterparts. This new model will be presented and the data that supports it discussed. Although the model was developed in the PPM family of codes, it should work equally well with any modern, dissipative numerical scheme for compressible flow by means of a readjustment of a single, dimensionless free parameter.