Supersonic Flow Simulation with Entropy-Based Artificial Viscosity Stabilization

2016-10-29T00:00:00Z (GMT) by Hessamoddin Abbassi
High-fidelity simulation of supersonic flows is a powerful tool for gaining insight into the complex physics of such flows at a fraction of cost and time of experiments. The availability of supercomputers has made massively parallel numerical simulations a viable option and has inspired a vast amount of research efforts to develop accurate and affordable numerical tools for simulation of real world engineering problems. The accurate simulation of supersonic flows is well suited to higher-order computational fluid dynamics (CFD). Since these cases often involve flow accompanied by strong shock waves, an appropriate shock capturing technique for higher-order methods is necessary. Among the numerous available methods for shock capturing, adding artificial viscosity seems to be the most promising option which has been successfully implemented in classical numerical schemes, e.g. finite volume methods. In this research we embark upon the implementation of an artificial viscosity shock capturing technique in a high-order discontinuous Galerkin (DG) method to accurately simulate shock dominated flows. The artificial viscosity model used in this work is a modified form of the entropy viscosity (EV) method. The artificially added viscosity is proportional to the local size of an entropy production. The entropy satisfies a conservation equation only in the regions where the solution is smooth and satisfies an inequality in shocks. The basic idea used in entropy viscosity method is based on the assumption of a large entropy production at shocks. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions and large in shocks, the dissipation will be virtually added only to shocked regions. However, direct implementation of the entropy viscosity method in our discontinuous spectral element method (DSEM) leads to a non-smooth artificial viscosity, which in turn leads to oscillations and instability of the solution. To smooth the artificial viscosity, the EV method is coupled with a spectral filter and an interface treatment technique. The resulting artificial viscosity is locally large near discontinuities and transitions smoothly to zero in smooth flow regions. The method enables using elements with orders higher than unity while avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Since supersonic flows are, by definition, high speed and naturally turbulent, the shock capturing method should be capable of resolving shocks in presence of turbulence. Consequently, the use of method is extended to compressible turbulence and a special emphasis is placed on distinguishing strong oscillations associated with turbulence from shock waves. A modified formulation incorporating a shock sensor is proposed for turbulent flows and the obtained results confirm the ability of the modified method to capture shocks while preserving the main features of the turbulence structure.