The multifield model like its counterpart, the two-fluid model, generally fails to be hyperbolic in its basic formulation. However, interfacial forces such as the interfacial pressure term and the virtual mass force, bringing new differential terms to the system, can change the analysis and make the problem hyperbolic. The aim of this paper is to define how the interfacial pressure default force affects the hyperbolicity of a general multifield model, and consequently, its inherent numerical stability. We characterize the location of the non hyperbolic regions and propose a closure for the interfacial pressure that makes the system hyperbolic in physically relevant regions.
References
1.
HadamardJ., 1923, Lectures on Cauchy's problem in linear partial differential equations, Yale Univ. Press, New Haven, Connecticut.
2.
MizohataS., Some remarks on the Cauchy problem, J.Math. Kyoto Univ., 1(1961), pp 109–127.
3.
LaxP., Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24(1957), pp 627–646.
4.
IvriiV. Ya. and PetkovV. M., Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed, Russian Math. Surveys29 (1974), 170.
5.
Wakabayashi, The Lax-Mizohata theorem for nonlinear Cauchy problems, Comm. in Partial Di. Equ.26(2001), pp 1367–1384.
StuhmillerJ.H., The Influence of Interfacial Pressure Forces on the Character of Two-Phase Flow Model Equations, Int. J. Multiphase Flow, vol.3, pp.551–560, 1977.
8.
ParkJ.-W.DrewD.A.LaheyR.T.Jr, 1999, The analysis of void wave propagation in adiabatic monodispersed bubbly two-phase flows using an ensemble-averaged two-fluid model, Int. J. Multiphase Flow, vol. 24, pp. 1205–1244.
9.
NdjingaM.KumbaroA.De VuystF.Laurent-GengouxP., 2005, “Influence of Interfacial Forces on the Hyperbolicity of the Two-Fluid Model”, 5th Int. Symp. on Multiphase Flow, Xi'an, China.
10.
NdjingaM., “Influence of Interfacial Pressure on the Hyperbolicity of the Two Fluid Model,”Comptes Rendus Mathématique (2007), Vol. 344, Issue 6, 15 March 2007.
11.
LaheyR.T.DrewD.A., 2000, An analysis of two-phase flow and heat transfer using a multidimensional, multi-field, two-fluid computational fluid dynamics model, Japan/US Seminar on Two-Phase Flow Dynamics, California, June 5–8.
12.
RansomV.H., “Numerical Modeling of Two-PhaseFlows”, Numerical Analysis course, Summer School CEA-INRIA-EDF, 12–23 June 1989.
13.
LaheyR. T.JrYinJ. and TiwariP., Analysis of void wave propagation and sonic velocity using a two-fluid model, Multiphase Science and Technology17(4) (2005)293–320.
14.
KumbaroA.NdjingaM., “The Generalized Roe Approximate Riemann Solver for the Simulation of Multifield Two-Phase Flows in the OVAP Code,” to appear.
15.
NdjingaM.KumbaroA.De VuystF.Laurent-GengouxP., Numerical simulation of hyperbolic two-phase flow models using a Roe-type solver, Nuclear Eng. and Design, vol. 238/8, pp. 2075–2083, 2008.
16.
HewittG.DelhayeJ.M.ZuberM., 1987, Numerical benchmark tests in Multiphase Science and Technology, vol. 3, HemisphereWashington DC/New York.
17.
DrewD. A.PassmanS. L., 1999, Theory of Multicomponents Fluids, Springer-Verlag, New York.
18.
YeohG.H.TuJ., 2009, Computational Techniques for Multi-phase Flows, Butterworth-Heinemann, Elsevier Science and Technology.
19.
StewartH.B.WendroffB., Two-Phase Flow: Models and Methods, J. of Comp. Physics, vol. 56, pp. 363–409, 1984.
