A non-linear residual distribution scheme for real-gas computations

Abstract This paper deals with a high-order accurate Residual Distribution scheme for the numerical solution of dense gas flows on unstructured grids. Dense gas-dynamics studies the flow of gases in the thermodynamic region above the upper saturation curve, close to the liquid–vapor critical point. In such conditions, some fluids may exhibit negative values of the fundamental derivative of gas-dynamics, leading to non-classical gas-dynamic behaviors, such as rarefaction shock waves, mixed shock/fan waves, and shock splitting. Due to the complexity in performing reliable experimental studies for non-classical gas-dynamics, accurate numerical simulations of dense gas flows are of paramount importance. In this work, advantages in using high-order methods are highlighted, in terms of number of degrees of freedom and computational time used, for computing the numerical solution with a greater accuracy compared to lower-order methods, even for shocked flows. Several numerical experiments are also performed to assess the influence of the thermodynamic models on the problem solution.

[1]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[2]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[3]  Calin Zamfirescu,et al.  Admissibility region for rarefaction shock waves in dense gases , 2008, Journal of Fluid Mechanics.

[4]  Moulay D. Tidriri,et al.  Preconditioning Techniques for the Newton-Krylov Solution of Compressible Flows , 1997 .

[5]  Alfred Kluwick,et al.  On the propagation of waves exhibiting both positive and negative nonlinearity , 1984, Journal of Fluid Mechanics.

[6]  Alberto Guardone,et al.  Multiparameter equations of state for selected siloxanes , 2006 .

[7]  Chi-Wang Shu,et al.  High order residual distribution conservative finite difference WENO schemes for convection-diffusion steady state problems on non-smooth meshes , 2007, J. Comput. Phys..

[8]  Layne T. Watson,et al.  Supersonic flows of dense gases in cascade configurations , 1997, Journal of Fluid Mechanics.

[9]  Rémi Abgrall,et al.  High Order Fluctuation Schemes on Triangular Meshes , 2003, J. Sci. Comput..

[10]  Yu-Chun Hou,et al.  Development of an equation of state for gases , 1955 .

[11]  Paolo Silva,et al.  Dense Gas Thermodynamic Properties of Single and Multicomponent Fluids for Fluid Dynamics Simulations , 2003 .

[12]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[13]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[14]  Pietro Marco Congedo,et al.  Inviscid and viscous aerodynamics of dense gases , 2007, Journal of Fluid Mechanics.

[15]  P. A. Thompson,et al.  A Fundamental Derivative in Gasdynamics , 1971 .

[16]  Rémi Abgrall,et al.  High order residual distribution scheme for Navier-Stokes equations. , 2011 .

[17]  Stefano Rebay,et al.  Numerical simulation of dense gas flows on unstructured grids with an implicit high resolution upwind Euler solver , 2004 .

[18]  Homer F. Walker,et al.  Choosing the Forcing Terms in an Inexact Newton Method , 1996, SIAM J. Sci. Comput..

[19]  Brian Argrow,et al.  Construction and Validation of a Dense Gas Shock Tube , 2001 .

[20]  Roland Span,et al.  Equations of State for Technical Applications. I. Simultaneously Optimized Functional Forms for Nonpolar and Polar Fluids , 2003 .

[21]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[22]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[23]  Pietro Marco Congedo,et al.  Numerical investigation of dense-gas effects in turbomachinery , 2011 .

[24]  Chi-Wang Shu,et al.  High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes , 2006, J. Comput. Phys..

[25]  Brian Argrow,et al.  Application of Bethe -Zel'dovich-Thompson Fluids in Organic Rankine Cycle Engines , 2000 .

[26]  Rémi Abgrall,et al.  Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes , 2011, J. Comput. Phys..

[27]  P. Cinnella.  Transonic flows of dense gases over finite wings , 2008 .

[28]  Pietro Marco Congedo,et al.  Numerical Solver for Dense Gas Flows , 2005 .

[29]  Homer F. Walker,et al.  NITSOL: A Newton Iterative Solver for Nonlinear Systems , 1998, SIAM J. Sci. Comput..

[30]  R. Stryjek,et al.  PRSV: An improved peng—Robinson equation of state for pure compounds and mixtures , 1986 .

[31]  Alberto Guardone,et al.  Siloxanes : A new class of candidate Bethe-Zel’dovich-Thompson fluids , 2007 .

[32]  Antony Jameson,et al.  Lower-upper implicit schemes with multiple grids for the Euler equations , 1987 .

[33]  Pietro Marco Congedo,et al.  Sensitivity analysis of dense gas flow simulations to thermodynamic uncertainties , 2011 .

[34]  M. Cramer,et al.  Nonclassical Dynamics of Classical Gases , 1991 .

[35]  Alberto Guardone,et al.  Three-dimensional shock tube flows for dense gases , 2007, Journal of Fluid Mechanics.

[36]  Pietro Marco Congedo,et al.  Optimal airfoil shapes for viscous transonic flows of Bethe-Zel'dovich-Thompson fluids , 2008 .

[37]  Barry Koren,et al.  Euler and Navier-Stokes solvers using multi-dimensional upwind schemes and multigrid acceleration : results of the BRITE/EURAM projects AERO-CT89-0003 and AER2-CT92-00040, 1989-1995 , 1997 .

[38]  Rémi Abgrall,et al.  Residual distribution schemes: Current status and future trends , 2006 .

[39]  T. Chan,et al.  Nonlinearly Preconditioned Krylov Subspace Methods for Discrete Newton Algorithms , 1984 .

[40]  Stefano Rebay,et al.  Influence of Thermodynamic Models in Two-Dimensional Flow Simulations of Turboexpanders , 2010 .

[41]  Thomas J. R. Hughes,et al.  Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations†‡ , 1987 .