A localized artificial diffusivity approach inspired by TVD schemes and its consistent application to compressible flows

Abstract

We present a novel computational model for simulation of compressible flows involving shocks and contact discontinuities using a localized artificial diffusivity (LAD) approach. Owing to their non-dissipative nature and secondary conservation properties, second order central operators have been shown to be very accurate for simulation of turbulent flows and acoustics. Committing to second order central spatial operations, we control their large dispersion errors by augmenting the mass, momentum and total energy equations with artificial mass diffusivity, artificial shear/bulk viscosities and artificial thermal conductivity, respectively. Utilizing traditional LAD models, we observe that problem-dependent tuning of the nondimensional user-defined constants is required for acceptable levels of accuracy. Moreover, there are no guarantees of robustness for such LAD schemes. As an alternative approach, inspired by analysis of second-order total variation diminishing (TVD) flux limiters, we present a novel approach for localizing the artificial diffusivity terms that in contrast to previous phenomenological LAD schemes, achieves high accuracy using CD without problem-dependent tuning or de-aliasing of the solutions via filtering. We derive the model in a one-dimensional (1D) setting and assess its performance using canonical 1D tests. Finally, we present an approach for extending the proposed model to multi-dimensional settings for collocated Cartesian grids.