We consider a general Euler-type two-phase flow model with relaxation towards phase equilibrium. We provide a complete description of the transition between the wave dynamics of the homogeneous relaxation system and that of the local equilibrium approximation. In particular, we present generally valid analytical expressions for the amplifications and velocities of each Fourier component. This transitional wave dynamics is fully determined by only two dimensionless parameters; a stiffness parameter and the ratio of the sound velocities in the stiff and non-stiff limits. A direct calculation verifies that the stability criterion is precisely the subcharacteristic condition. We further prove a maximum principle in the transitional regime, similar in spirit to the subcharacteristic condition; the transitional wave speeds can never exceed the largest wave speed of the homogeneous relaxation system. Finally, we identify the existence of a critical region of wave numbers where the sonic waves completely disappear from the system. This region corresponds to the casus irreducibilis of the describing cubic polynomial.
DOI : 10.1051/m2an/2014048
Keywords: Relaxation, subcharacteristic condition, phase transfer
Solem, Susanne 1 ; Aursand, Peder 1 ; Flåtten, Tore 2
@article{M2AN_2015__49_2_601_0,
author = {Solem, Susanne and Aursand, Peder and Fl\r{a}tten, Tore},
title = {The dispersive wave dynamics of a two-phase flow relaxation model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {601--619},
year = {2015},
publisher = {EDP Sciences},
volume = {49},
number = {2},
doi = {10.1051/m2an/2014048},
mrnumber = {3342220},
zbl = {1316.35193},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2014048/}
}
TY - JOUR AU - Solem, Susanne AU - Aursand, Peder AU - Flåtten, Tore TI - The dispersive wave dynamics of a two-phase flow relaxation model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 601 EP - 619 VL - 49 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2014048/ DO - 10.1051/m2an/2014048 LA - en ID - M2AN_2015__49_2_601_0 ER -
%0 Journal Article %A Solem, Susanne %A Aursand, Peder %A Flåtten, Tore %T The dispersive wave dynamics of a two-phase flow relaxation model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 601-619 %V 49 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2014048/ %R 10.1051/m2an/2014048 %G en %F M2AN_2015__49_2_601_0
Solem, Susanne; Aursand, Peder; Flåtten, Tore. The dispersive wave dynamics of a two-phase flow relaxation model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 601-619. doi: 10.1051/m2an/2014048
, An extension of Roe’s upwind scheme to algebraic equilibrium real gas models. Comput. Fluids 19 (1991) 171–182. | Zbl | DOI
and , Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361–396. | MR | Zbl | DOI
, and , A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577–616. | MR | Zbl | DOI
and , On the dispersive wave-dynamics of relaxation systems. J. Hyperbolic Diff. Eq. 9 (2012) 641–659. | MR | Zbl | DOI
and , A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12 (1986) 861–889. | Zbl | DOI
, A reduced stability condition for nonlinear relaxation to conservation laws. J. Hyperbolic Diff. Eq. 1 (2004) 149–170. | MR | Zbl | DOI
, , and , A simple finite-volume method for compressible isothermal two-phase flows simulation. Int. J. Finite 3 (2006). | MR | Zbl
, and , Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787–830. | MR | Zbl | DOI
, Relaxation schemes for the multicomponent Euler system. ESAIM: M2AN 37 (2003) 909–936. | MR | Zbl | Numdam | DOI
, and , Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium. ESAIM: M2AN 46 (2012) 1029–1054. | MR | Zbl | Numdam | DOI
and , Relaxation two-phase flow models and the subcharacteristic condition. Math. Models Methods Appl. Sci. 21 (2011) 2379–2407. | MR | Zbl | DOI
, and , Wave propagation in multicomponent flow models. SIAM J. Appl. Math. 70 (2010) 2861–2882. | MR | Zbl | DOI
, and , On solutions to equilibrium problems for systems of stiffened gases. SIAM J. Appl. Math. 71 (2011) 41–67. | MR | Zbl | DOI
and , Relaxation models of phase transition flows. ESAIM: M2AN 40 (2006) 331–352. | MR | Zbl | Numdam | DOI
H.-O. Kreiss and J. Lorenz, Initial-boundary value problems and the Navier-Stokes equations. Academic Press (1989). | MR | Zbl
, Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108 (1987) 153–175. | MR | Zbl | DOI
, A hierarchy of relaxation models for two-phase flow. SIAM J. Appl. Math. 72 (2012) 1713–1741. | MR | Zbl | DOI
and , Two-phase flow of with phase transfer. Energy Procedia 23 (2012) 246–255. | DOI
A. Morin, P.K. Aursand, T. Flåtten and S.T. Munkejord, Numerical resolution of CO transport dynamics. In Proc. of SIAM Conference on Mathematics for Industry: Challenges and Frontiers (MI09). SIAM, Philadelphia (2009) 108–119.
A. Morin and T. Flåtten, A two-fluid four-equation model with instantaneous thermodynamical equilibrium. Submitted. | MR | Numdam
and , A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202 (2005) 664–698. | MR | Zbl | DOI
, Convergence to equilibrium for the relaxation approximation of conservation laws. Commun. Pure Appl. Math. 49 (1996) 795–823. | MR | Zbl | DOI
R. Natalini, Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws. In vol. 99 of Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL (1999) 128–198. | MR | Zbl
and , A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves. J. Comput. Phys. 259 (2014) 331–357. | MR | Zbl | DOI
and , A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467. | MR | Zbl | DOI
, and , Modelling phase transition in metastable liquids: application to cavitating and flashing flows. J. Fluid Mech. 607 (2008) 313–350. | MR | Zbl | DOI
, and , Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures. J. Comput. Phys. 228 (2009) 1678–1712. | MR | Zbl | DOI
and , Expression for predicting liquid evaporation flux: Statistical rate theory approach. Phys. Rev. E 59 (1999) 429–440. | DOI
W.-A. Yong, Basic aspects of hyperbolic relaxation systems, in Advances in the Theory of Shock Waves. Vol. 47 of Progr. Nonlin. Differ. Eq. Appl. Birkhäuser Boston, Boston (2001) 259–305. | MR | Zbl
, and , Modeling phase transition for compressible two-phase flows applied to metastable liquids. J. Comput. Phys. 229 (2010) 2964–2998. | MR | Zbl | DOI
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