An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model

Laura Gastaldo; Raphaèle Herbin; Jean-Claude Latché

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 2, page 251-287
  • ISSN: 0764-583X

Abstract

top
We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative. Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases. To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step. The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model. Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability.

How to cite

top

Gastaldo, Laura, Herbin, Raphaèle, and Latché, Jean-Claude. "An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model." ESAIM: Mathematical Modelling and Numerical Analysis 44.2 (2010): 251-287. <http://eudml.org/doc/250805>.

@article{Gastaldo2010,
abstract = { We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative. Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases. To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step. The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model. Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability. },
author = {Gastaldo, Laura, Herbin, Raphaèle, Latché, Jean-Claude},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Drift-flux model; pressure correction schemes; finite volumes; finite elements; drift-flux model},
language = {eng},
month = {3},
number = {2},
pages = {251-287},
publisher = {EDP Sciences},
title = {An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model},
url = {http://eudml.org/doc/250805},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Gastaldo, Laura
AU - Herbin, Raphaèle
AU - Latché, Jean-Claude
TI - An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 2
SP - 251
EP - 287
AB - We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative. Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases. To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step. The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model. Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability.
LA - eng
KW - Drift-flux model; pressure correction schemes; finite volumes; finite elements; drift-flux model
UR - http://eudml.org/doc/250805
ER -

References

top
  1. G. Ansanay-Alex, F. Babik, J.-C. Latché and D. Vola, An L2–stable approximation of the Navier–Stokes advection operator for low-order non-conforming finite elements. IJNMF (to appear).  Zbl1321.76035
  2. M. Baudin, Ch. Berthon, F. Coquel, R. Masson and Q.H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law. Numer. Math.99 (2005) 411–440.  Zbl1204.76025
  3. M. Baudin, F. Coquel and Q.-H. Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline. SIAM J. Sci. Comput.27 (2005) 914–936 (electronic).  Zbl1130.76384
  4. S. Becker, A. Sokolichin and G. Eigenberger, Gas-liquid flow in bubble columns and loop reactors: Part II. Comparison of detailed experiments and flow simulations. Chem. Eng. Sci.49 (1994) 5747–5762.  
  5. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991).  Zbl0788.73002
  6. G. Chanteperdrix, Modélisation et simulation numérique d'écoulements diphasiques à interface libre. Application à l'étude des mouvements de liquides dans les réservoirs de véhicules spatiaux. Energétique et dynamique des fluides, École Nationale Supérieure de l'Aéronautique et de l'Espace, France (2004).  
  7. P.G. Ciarlet, Finite elements methods – Basic error estimates for elliptic problems, in Handbook of Numerical AnalysisII, P. Ciarlet and J.L. Lions Eds., North Holland (1991) 17–351.  Zbl0875.65086
  8. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Revue Française d'Automatique, Informatique et Recherche Opérationnelle (R.A.I.R.O.)3 (1973) 33–75.  Zbl0302.65087
  9. K. Deimling, Nonlinear Functional Analysis. Springer, New York, USA (1980).  Zbl1257.47059
  10. S. Evje and K.K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys.175 (2002) 674–701.  Zbl1197.76132
  11. S. Evje and K.K. Fjelde, On a rough AUSM scheme for a one-dimensional two-phase model. Comput. Fluids32 (2003) 1497–1530.  Zbl1128.76337
  12. R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal.18 (1998) 563–594.  Zbl0973.65078
  13. R. Eymard, T Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical AnalysisI, P. Ciarlet and J.L. Lions Eds., North Holland (2000) 713–1020.  Zbl0981.65095
  14. T. Flåtten and S.T. Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model. ESAIM: M2AN40 (2006) 735–764.  
  15. T. Gallouet, J.-M. Hérard and N. Seguin, A hybrid scheme to compute contact discontinuities in one dimensional Euler systems. ESAIM: M2AN36 (2003) 1133–1159.  
  16. T. Gallouët, L. Gastaldo, R. Herbin and J.-C. Latché, An unconditionally stable pressure correction scheme for compressible barotropic Navier-Stokes equations. ESAIM: M2AN42 (2008) 303–331.  Zbl1132.35433
  17. T. Gallouët, R. Herbin and J.-C. Latché, A convergent finite-element volume scheme for the compressible Stokes problem. Part I: The isothermal case. Math. Comp.78 (2009) 1333–1352.  Zbl1223.76041
  18. L. Gastaldo, R. Herbin and J.-C. Latché, A pressure correction scheme for the homogeneous two-phase flow model with two barotropic phases, in Finite Volumes for Complex Applications V – Problems and Perspectives – Aussois, France (2008) 447–454.  
  19. L. Gastaldo, R. Herbin and J.-C. Latché, A discretization of the phase mass balance in fractional step algorithms for the drift-flux model. IMA J. Numer. Anal. (2009) doi:.  Zbl05853329DOI10.1093/imanum/drp006
  20. J.-L. Guermond and L. Quartapelle, A projection FEM for variable density incompressible flows. J. Comput. Phys.165 (2000) 167–188.  Zbl0994.76051
  21. J.L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows. Comput. Meth. Appl. Mech. Eng.195 (2006) 6011–6045.  Zbl1122.76072
  22. H. Guillard and F. Duval, A Darcy law for the drift velocity in a two-phase flow model. J. Comput. Phys.224 (2007) 288–313.  Zbl1119.76067
  23. F.H. Harlow and A.A. Amsden, A numerical fluid dynamics calculation method for all flow speeds. J. Comput. Phys.8 (1971) 197–213.  Zbl0221.76011
  24. D. Kuzmin and S. Turek, Numerical simulation of turbulent bubbly flows, in 3rd International Symposium on Two-Phase Flow Modelling and Experimentation, Pisa, 22–24 September (2004).  
  25. B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys.95 (1991) 59–84.  Zbl0725.76090
  26. M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in Handbook of Numerical AnalysisVI, P. Ciarlet and J.L. Lions Eds., North Holland (1998).  Zbl0921.76040
  27. J.-M. Masella, I. Faille and T. Gallouët, On an approximate Godunov scheme. Int. J. Comput. Fluid Dyn.12 (1999) 133–149.  Zbl0944.76041
  28. F. Moukalled, M. Darwish and B. Sekar, A pressure-based algorithm for multi-phase flow at all speeds. J. Comput. Phys.190 (2003) 550–571.  Zbl1076.76074
  29. R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Methods Part. Differ. Equ.8 (1992) 97–111.  Zbl0742.76051
  30. J.E. Romate, An approximate Riemann solver for a two-phase flow model with numerically given slip relation. Comput. Fluids27 (1998) 455–477.  Zbl0968.76052
  31. A. Sokolichin and G. Eigenberger, Applicability of the standard k-ε turbulence model to the dynamic simulation of bubble columns: Part I. Detailed numerical simulations. Chem. Eng. Sci.54 (1999) 2273–2284.  
  32. A. Sokolichin, G. Eigenberger and A. Lapin, Simulation of buoyancy driven bubbly flow: Established simplifications and open questions. AIChE J.50 (2004) 24–45.  
  33. B. Spalding, Numerical computation of multi-phase fluid flow and heat transfer, in Recent Advances in Numerical Methods in Fluids1, Swansea, Pineridge Press (1980) 139–168.  Zbl0467.76094
  34. P. Wesseling, Principles of computational fluid dynamics, Springer Series in Computational Mathematics29. Springer (2001).  Zbl1185.76005

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.