Schemes to Compute Unsteady Flashing Flows

Some ways to compute flashing flows in variable cross section ducts are provided, focusing on the homogeneous relaxation model. The basic numerical method relies on a splitting technique that is consistent with the overall entropy inequality. The cross section is assumed to he continuous, and the finite volume approach is applied to approximate homogeneous equations. Several suitable schemes to account for complex equation of state are discussed, namely, the Rusanov scheme, an approximate form of the Roe scheme, and the volumes finis Roe (VFRoe) scheme with the help of nonconservative variables. To evaluate respective accuracy, the homogeneous Euler equations are computed first, and the L1 error norm of transient solutions of shock tube experiments are plotted. It is shown that the Rusanov scheme is indeed less accurate, which balances its interesting properties, inasmuch as it preserves the positivity of the mean density and the maximum principle for the vapor quality. Computations of real cases are presented, which account for the mass transfer term and the time-space dependent cross sections

[1]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[2]  Jean-Marc Hérard,et al.  Un schma simple pour les quations de Saint-Venant , 1998 .

[3]  J. Hérard,et al.  Finite Volume Algorithm to Compute Dense Compressible Gas-Solid Flows , 1999 .

[4]  V. Rusanov,et al.  The calculation of the interaction of non-stationary shock waves and obstacles , 1962 .

[5]  E. Turkel,et al.  Preconditioned methods for solving the incompressible low speed compressible equations , 1987 .

[6]  J. Hérard,et al.  An Exact Riemann Solver for Multicomponent Turbulent Flow , 2000 .

[7]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[8]  Léon Bolle,et al.  The non-equilibrium relaxation model for one-dimensional flashing liquid flow , 1996 .

[9]  Jean-Marc Hérard,et al.  A sequel to a rough Godunov scheme: application to real gases , 2000 .

[10]  Jean-Marie Seynhaeve,et al.  Experimental and theoretical analysis of flashing water flow through a safety valve , 1996 .

[11]  Thierry Gallouët,et al.  On an Approximate Godunov Scheme , 1999 .

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

[13]  Thierry Gallouët,et al.  On a rough Godunov scheme , 1996 .

[14]  P. Floch.  Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form , 1988 .

[15]  An Approximate Riemann Solver for Second-Moment Closures , 1999 .

[16]  Jean-Marc Hérard,et al.  A Naive Riemann Solver to Compute a Non-conservative Hyperbolic System , 1999 .

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

[18]  Eli Turkel,et al.  Review of preconditioning methods for fluid dynamics , 1993 .

[19]  J. Greenberg,et al.  A well-balanced scheme for the numerical processing of source terms in hyperbolic equations , 1996 .

[20]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[21]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[22]  Jean-Marc Hérard,et al.  Some recent finite volume schemes to compute Euler equations using real gas EOS , 2002 .

[23]  Rémy Baraille,et al.  Une version à pas fractionnaires du schéma de Godunov pour l'hydrodynamique , 1992 .

[24]  Arun In,et al.  Numerical Evaluation of an Energy Relaxation Method for Inviscid Real Fluids , 1999, SIAM J. Sci. Comput..

[25]  Michel Barret,et al.  Computation of Flashing Flows In Variable Cross-Section Ducts , 2000 .

[26]  Jean-Marc Hérard,et al.  An Approximate Roe-Type Riemann Solver for a Class of Realizable Second Order Closures , 2000 .

[27]  Miltiadis Papalexandris,et al.  Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension , 1997 .

[28]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.