Shock Regularization for the Burgers Equation

Abstract

As is well-known, the inviscid Burgers equation is prone to shock formation. A class of regularized Burgers equations is proposed that do not have an explicit artificial viscosity term. The steepening eect of the Burgers nonlinear convective term is mitigated by spatial averaging or low pass filtering of the convective velocity. Numerical simulations are performed to investigate the eect of this regularization. Specifically, the regularized Burgers equation is numerically shown to compare favorably with the viscous Burgers equation and to avoid spurious oscillations (unlike other conservative regularizations, such as the KdV equation). The numerics indicates that the width of the filter or the diameter of the spatial averaging domain controls the thickness of the regularized shock. While the governing equations of fluid flows (the Euler and Navier-Stokes equations) are well-known, their computation are still a challenge. There are two particular challenges associated with the computation and the understanding of solutions of these equations: one is turbulence modeling and the other is the formation of shocks and their regularization. Both challenges are attributed to the nonlinear convective term in the governing equations. In this article we will show that both diuculties can potentially be fixed by a particular class of averaging or low pass filtering. While the general case of the Euler or Navier-Stokes equations are not investigated here, we will show evidence of shock regularization in the Burgers equations. Lagrangian averaging is a technique for modeling the mean flow of incompressible turbulent flows. 8‐10 The Lagrangian Averaged Euler (LAE- ) equations for incompressible flow and their viscous counterpart are interesting from both the analytical and the numerical points of view. In these equations, is a spatial scale below which rapid fluctuations are smoothed by linear and nonlinear dispersion. The distinctive feature of the Lagrangian averaging approach is that averaging is carried out at the level of the variational principle and not at the level of the Euler or Navier-Stokes equations, which is the traditional averaging or filtering approach used for both the Reynolds averaged Navier-Stokes (RANS) and the large eddy simulation (LES) models. Therefore, the resulted LAE- equations possess conservation laws for energy and momentum, as well as Kelvin circulation theorem. The behavior of the LAE- solution approximates the behavior of Euler equations well to spatial scales of order , while truncating the energy spectrum for scales smaller than . This averaging or filtering is done without adding viscosity, but by a nonlinear dispersion from the large scales. The numerical simulations of the Lagrangian Averaged Navier-Stokes- (LANS- ) equations performed by Chen et al 4 and Mohseni et al 16 for isotropic homogenous turbulence demonstrated the good features of this model in reproducing large scales of turbulence. Recently, these techniques have been extended to wall bounded flows 18 and compressible barotropic flows. 2