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Title: A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations

Abstract

A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel,more » A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures.« less

Authors:
 [1];  [2];  [2];  [2]
  1. Polytech'Marseille, University Institute of France, Universite de Provence and SMASH Project UMR CNRS 6595 - IUSTI-INRIA, 5 rue E. Fermi, 13453 Marseille Cedex 13 (France). E-mail: Richard.Saurel@polytech.univ-mrs.fr
  2. Polytech'Marseille, University Institute of France, Universite de Provence and SMASH Project UMR CNRS 6595 - IUSTI-INRIA, 5 rue E. Fermi, 13453 Marseille Cedex 13 (France)
Publication Date:
OSTI Identifier:
20991578
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 223; Journal Issue: 2; Other Information: DOI: 10.1016/j.jcp.2006.10.004; PII: S0021-9991(06)00476-1; Copyright (c) 2006 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPRESSIBLE FLOW; EFFICIENCY; EQUATIONS OF STATE; EXACT SOLUTIONS; MIXTURES; MULTIPHASE FLOW; OSCILLATIONS; RELAXATION; THERMAL EQUILIBRIUM

Citation Formats

Saurel, Richard, Franquet, Erwin, Daniel, Eric, and Le Metayer, Olivier. A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations. United States: N. p., 2007. Web. doi:10.1016/j.jcp.2006.10.004.
Saurel, Richard, Franquet, Erwin, Daniel, Eric, & Le Metayer, Olivier. A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations. United States. doi:10.1016/j.jcp.2006.10.004.
Saurel, Richard, Franquet, Erwin, Daniel, Eric, and Le Metayer, Olivier. Tue . "A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations". United States. doi:10.1016/j.jcp.2006.10.004.
@article{osti_20991578,
title = {A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations},
author = {Saurel, Richard and Franquet, Erwin and Daniel, Eric and Le Metayer, Olivier},
abstractNote = {A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures.},
doi = {10.1016/j.jcp.2006.10.004},
journal = {Journal of Computational Physics},
number = 2,
volume = 223,
place = {United States},
year = {Tue May 01 00:00:00 EDT 2007},
month = {Tue May 01 00:00:00 EDT 2007}
}
  • The relaxation-projection method developed in Saurel et al. [R. Saurel, E. Franquet, E. Daniel, O. Le Metayer, A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations, J. Comput. Phys. (2007) 822-845] is extended to the non-conservative hyperbolic multiphase flow model of Kapila et al. [A.K. Kapila, Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration to detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002-3024]. This model has the ability to treat multi-temperatures mixtures evolving with a single pressure and velocity and is particularly interesting formore » the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures. The non-conservative character of this model poses however computational challenges in the presence of shocks. The first issue is related to the Riemann problem resolution that necessitates shock jump conditions. Thanks to the Rankine-Hugoniot relations proposed and validated in Saurel et al. [R. Saurel, O. Le Metayer, J. Massoni, S. Gavrilyuk, Shock jump conditions for multiphase mixtures with stiff mechanical relaxation, Shock Waves 16 (3) (2007) 209-232] exact and approximate 2-shocks Riemann solvers are derived. However, the Riemann solver is only a part of a numerical scheme and non-conservative variables pose extra difficulties for the projection or cell average of the solution. It is shown that conventional Godunov schemes are unable to converge to the exact solution for strong multiphase shocks. This is due to the incorrect partition of the energies or entropies in the cell averaged mixture. To circumvent this difficulty a specific Lagrangian scheme is developed. The correct partition of the energies is achieved by using an artificial heat exchange in the shock layer. With the help of an asymptotic analysis this heat exchange takes a similar form as the 'pseudoviscosity' introduced by Von Neumann and Richtmyer [J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21 (1950) 232-237]. The present Lagrangian numerical scheme thus combines Riemann solvers and artificial heat exchanges. An Eulerian variant is then obtained by using the relaxation-projection method developed earlier by the authors for the Euler equations. The method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the Euler equations in the context of interface problems. The method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks.« less
  • We investigate the performance of the multiresolution method. The method was recently proposed as a way to speed up computations of compressible flows. The method is based on truncating wavelet coefficients. The method showed good performance on one-dimensional test problems. We implement the method in two space dimensions for a more complex compressible flow computation, intended to simulate conditions under which many production CFD codes are running. Our conclusion is that we can in many cases reduce the CPU time, but that the gain in efficiency is not as large as for the one-dimensional problems. We furthermore observe that itmore » is essential to use the adaptive multiresolution method, which mixes centered differences with TVD methods. 5 refs., 14 figs., 1 tab.« less
  • A numerical study is undertaken comparing a fifth-order version of the weighted essentially non-oscillatory numerical (WENO5) method to a modern piecewise-linear, second-order, version of Godunov's (PLMDE) method for the compressible Euler Equations. A series of one-dimensional test problems are examined beginning with classical linear problems and ending with complex shock interactions. The problems considered are: (1) linear advection of a Gaussian pulse in density, (2) Sod's shock tube problem, (3) the ''peak'' shock tube problem, (4) a version of the Shu and Osher shock entropy wave interaction and (5) the Woodward and Colella interacting shock wave problem. For each problemmore » and method, run times, density error norms and convergence rates are reported for each method as produced from a common code test-bed. The linear problem exhibits the advertised convergence rate for both methods as well as the expected large disparity in overall error levels; WENO5 has the smaller errors and an enormous advantage in overall efficiency (in accuracy per unit CPU time). For the nonlinear problems with discontinuities, however, we generally see both first-order self-convergence of error as compared to an exact solution, or when an analytic solution is not available, a converged solution generated on an extremely fine grid. The overall comparison of error levels shows some variation from problem to problem. For Sod's shock tube, PLMDE has nearly half the error, while on the peak problem the errors are nearly the same. For the interacting blast wave problem the two methods again produce a similar level of error with a slight edge for the PLMDE. On the other hand, for the Shu-Osher problem the errors are similar on the coarser grids, but favors WENO by a factor of nearly 1.5 on the finer grids used. In all cases holding mesh resolution constant though, PLMDE is less costly in terms of CPU time by approximately a factor six. If the CPU cost is taken as fixed, that is run times are equal for both numerical methods, then PLMDE uniformly produces lower errors than WENO for the fixed computation cost on the test problems considered here.« less
  • A new algorithm based on spectral element discretizations and flux-corrected transport concepts is developed for the solution of the Euler equations of inviscid compressible fluid flow. A conservative formulation is proposed based on one- and two-dimensional cell-averaging and reconstruction procedures, which employ a staggered mesh of Gauss-Chebyshev and Gauss-Lobatto-Chebyshev collocation points. Particular emphasis is placed on the construction of robust boundary and interfacial conditions in one- and two-dimensions. It is demonstrated through shock-tube problems and two-dimensional simulations that the proposed algorithm leads to stable, non-oscillatory solutions of high accuracy. Of particular importance is the fact that dispersion errors are minimal,more » as show through experiments. From the operational point of view, casting the method in a spectral element formulation provides flexibility in the discretization, since a variable number of macro-elements or collocation points per element can be employed to accomodate both accuracy and geometric requirements.« less
  • Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here the authors present a method for calculating time-dependent incompressible inviscid flow which combines a projection method with a Cartesian grid approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volume-of-fluid representation. A redistribution procedure is used to eliminate time-step restrictions due to small cells where themore » boundary intersects the mesh. The projection step uses an approximately projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and for the time-dependent algorithm in two dimensions. The method is also demonstrated on flow past a half-cylinder with vortex shedding.« less