Preventing numerical errors generated by interface-capturing schemes in compressible multi-material flows

In the present work, errors generated in computations of compressible multi-material flows using shock-capturing schemes are examined, specifically pressure oscillations (when the specific heats ratio is variable), but also temperature spikes and species conservation errors. These numerical errors are generated at material discontinuities due to an inconsistent treatment of the convective terms. Though temperature errors are irrelevant to solutions to the Euler equations, it is shown that they have the potential to lead to problems when physical diffusion is included, i.e., for the Navier-Stokes equations. These errors are studied analytically and numerically by considering the one-dimensional advection of isolated material discontinuities. A methodology preventing such errors for weighted essentially non-oscillatory (WENO) schemes is presented, in which modified WENO weights are used to solve the transport equation for mass fraction in conservative form to prevent temperature and species conservation errors. Pressure errors are prevented by solving an additional transport equation for a given function of the ratio of specific heats. Several multi-dimensional problems with various discontinuities (shocks, material interfaces and contact discontinuities), including the single-mode Richtmyer-Meshkov instability, and turbulence are considered to test the method.

[1]  D. Drikakis,et al.  Assessment of very high order of accuracy in implicit LES models , 2007 .

[2]  Michael L. Corradini,et al.  A thermodynamically consistent and fully conservative treatment of contact discontinuities for compressible multicomponent flows , 2004 .

[3]  Parviz Moin,et al.  Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves , 2010, J. Comput. Phys..

[4]  Smadar Karni,et al.  Multicomponent Flow Calculations by a Consistent Primitive Algorithm , 1994 .

[5]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[6]  Eric Johnsen,et al.  Implementation of WENO schemes in compressible multicomponent flow problems , 2005, J. Comput. Phys..

[7]  Kenneth K. Kuo,et al.  A low-dissipation and time-accurate method for compressible multi-component flow with variable specific heat ratios , 2011, J. Comput. Phys..

[8]  Andrew W. Cook,et al.  Enthalpy Diffusion in Multicomponent Flows , 2008 .

[9]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[10]  Rémi Abgrall,et al.  Computations of compressible multifluids , 2001 .

[11]  R. Abgrall.  How to Prevent Pressure Oscillations in Multicomponent Flow Calculations , 1996 .

[12]  Dimitris Drikakis,et al.  On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes , 2008, J. Comput. Phys..

[13]  S. Osher,et al.  Computing interface motion in compressible gas dynamics , 1992 .

[14]  Rémi Abgrall,et al.  A Simple Method for Compressible Multifluid Flows , 1999, SIAM J. Sci. Comput..

[15]  Smadar Karni,et al.  Hybrid Multifluid Algorithms , 1996, SIAM J. Sci. Comput..

[16]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[17]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[18]  Eric Johnsen,et al.  On the treatment of contact discontinuities using WENO schemes , 2011, J. Comput. Phys..

[19]  M. Brouillette.  THE RICHTMYER-MESHKOV INSTABILITY , 2002 .

[20]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[21]  Keh-Ming Shyue,et al.  An Efficient Shock-Capturing Algorithm for Compressible Multicomponent Problems , 1998 .

[22]  Rémi Abgrall,et al.  An adaptive shock-capturing algorithm for solving unsteady reactive flows , 2003 .