Entropy stable artificial dissipation based on Brenner regularization of the Navier-Stokes equations

Abstract

In contrast to conventional shock-capturing methods that introduce dissipation based on smoothness of a discrete solution, we propose to regularize the compressible Navier-Stokes equations by adding an artificial dissipation operator introduced by Howard Brenner. This regularization satisfies the first and second law of thermodynamics, ensures positivity of thermodynamic variables, and preserves the translational and rotational invariance at the continuous level. In this paper, we present a new class of artificial dissipation spectral collocation operators of arbitrary order of accuracy, that mimic some key properties of the continuous Brenner-Navier-Stokes diffusion operator at the discrete level. The new artificial dissipation operator preserves superconvergence of the corresponding baseline spectral collocation scheme, satisfies the summation-by-parts property and discrete entropy inequality, thus facilitating a nonlinear L2-stability proof for the symmetric form of the regularized Navier-Stokes equations. Numerical results demonstrating accuracy and non-oscillatory properties of the new schemes are presented for both continuous and discontinuous flows.