Stable, entropy-consistent, and localized artificial-diffusivity method for capturing discontinuities

Abstract

In this work, a localized artificial-viscosity/diffusivity method is proposed for accurately capturing discontinuities in compressible flows. There have been numerous efforts to improve the artificial diffusivity formulation in the last two decades, through appropriate localization of the artificial bulk viscosity for capturing shocks. However, for capturing contact discontinuities, either a density or internal energy variable is used as a detector. An issue with this sensor is that it not only detects contact discontinuities, but also falsely detects the regions of shocks and vortical motions. Using this detector to add artificial mass/thermal diffusivity for capturing contact discontinuities is hence unnecessarily dissipative. To overcome this issue, we propose a sensor similar to the Ducros sensor (for shocks) to detect contact discontinuities, and further localize artificial mass/thermal diffusivity for capturing contact discontinuities. The proposed method contains coefficients that are less sensitive to the choice of the flow problem. This is achieved by improved localization of the artificial diffusivity in the present method. A discretely consistent dissipative flux formulation is presented and is coupled with a robust low-dissipative scheme, which eliminates the need for filtering the solution variables. The proposed method also does not require filtering for the discontinuity detector/sensor functions, which is typically done to smear out the artificial fluid properties and obtain stable solutions. Hence, the challenges associated with extending the filtering procedure for unstructured grids is eliminated, thereby, making the proposed method easily applicable for unstructured grids. Finally, a straightforward extension of the proposed method to two-phase flows is also presented.