Importance of Spectral Properties of Viscous Flux Discretization and Gradient-Based Reconstruction Approach

Abstract

In this talk, I present the importance of viscous flux discretization for a wide variety of flows and the novel gradient-based reconstruction approach.First, I will present our observations regarding viscous flux discretization. While investigating nonlinear instability encountered in a high-resolution viscous shock-tube simulation, we have discovered that using an appropriate viscous scheme is important in preventing spurious oscillations around shocks. The existing viscous scheme of Shen and Zha and α-damping scheme are used for the simulations. Even though both schemes are sixth-order accurate, the α-damping approach gave superior results and is free of oscillations. The main difference between both schemes is in the spectral properties. It was concluded that having good spectral properties (high-frequency damping) is essential. We extended this idea further to turbulent shock-free flows, where we studied six different methods, divided into two classes, with poor and better representation of spectral properties at high wavenumbers. Both theoretical and numerical results have revealed that the method with better properties at high wavenumbers, denoted as α-damping type discretization, produced superior solutions compared to the other methods for the viscous Taylor-Green vortex test case.Second, I will present the Gradient-Based Reconstruction (GBR) approach for the convective fluxes. The original idea was to compute the gradients once and re-use them in both viscous and inviscid fluxes, and the importance of viscous fluxes was an observation. GBR is a novel approach for computing the cell interface values using gradients. Problem-independent shock-capturing techniques are considered, and the results for the benchmark cases are presented. It is observed that the proposed methods produced significantly better results than the existing methods. Furthermore, the GBR approach is efficient as the gradients are shared between several subroutines. Even for convective fluxes, the spectral properties might be more important than the order of accuracy alone.Finally, I will briefly present various applications where these methods are used.