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WEST-EAST HIGH SPEED FLOW FIELD CONFERENCE 1 INRIA, 2004 Route des Lucioles, 06902 Sophia-Antipolis, France. 2 Institut de Math´ematiques de Bordeaux, Universit´e de Bordeaux1, Talence, France. Vincent.Perrier@math.u-bordeaux1.fr Key words: Hyperbolic systems, Second-order perturbations, Rankine-Hugoniot relations, Trav-eling waves, shock waves, Two-phase flows .
A traveling wave analysis of a two phase isothermal Euler model is performed in this work. This analysis allows to exhibit the inner structure of shock waves in two-phase flows.
In the model under investigation, the dissipative regularizing term is not of viscous type butinstead comes from relaxation phenomena toward equilibrium between the phases. This givesan unusual structure to the diffusion tensor where dissipative terms appear only in the massconservation equations. We show that this implies that the mass fractions are not constant insidethe shock although the Rankine-Hugoniot relations give a zero jump of the mass fraction throughthe discontinuities. We also show that there exists a critical speed for the traveling waves abovewhich no C 1 solutions exist. Neverthless for this case, it is possible to construct traveling solutionsinvolving single phase shocks.
The correct definition of shock solutions in two-phase models is an open question.
Actually, many two-phase models are in non-conservative form and in addition tothe classical relations expressing the conservation of mass, momentum and energycontain additional equations in non conservative form. In principle, traveling wavesanalysis1,5 provide a satisfactory way to describe the inner structure of a shock andconsequently should allow a rigorous definition of shock solutions for these systems.
However, the practical realization of a travelling wave analysis requires to iden-tify the precise shape of the dissipative tensor. Actually, the regularizing effect ofthis tensor precisely dictate the amplitude of the jump relations connecting the twostates of the discontinuity. In the framework of two-phase system, this implies thatthe dissipative tensor cannot be arbitrary and must in some sense encode the rightphysic of the inner structure of a two-phase shock.
Many works on the regularisation of non-conservative hyperbolic systems considerviscous effects as the leading mechanism allowing the definition of shock waves.
H. GUILLARD et al./Shock Structure in Two-Phase models However, in two phase systems, we would like to emphasize that other effects thanviscous regularization can exist. We will present here another kind of these possibleregularizing effects based on the existence of relaxation phenomena in two-phasesystems that drive the two phases toward mechanical and thermodynamical equilib-rium. The existence of these relaxation mechanisms gives a particular and unusualstructure to the dissipative tensor. We will see that a traveling wave analysis ofthe dissipative system shows some unusual consequences on the structure of shockwaves in two-phase flows.
The model we consider in this work has been introduced in ref2 as a model forisothermal dispersed bubbly flows. We refer to this work for its derivation. Numer-ical simulations performed with this model has shown that despite its simplicity, itis able to reproduce two-phase computations usually performed with more complexmodels. In one dimension, this system can be written This system describes a two-phase medium composed of two immiscible fluids k =1, 2 where the pressures in the phases 1 and 2 are equal and given by barotropicstate. To be more specific, ρ here denote the total density of the flow, u its velocitywhile Y is a mass fraction expressing the relative proportion of the mass of one ofthe two fluid over the total mass. For definitiveness, we will assume that this massfraction is relative to the fluid 2 : Y = Y2. Finally, the pressures pk, k = 1, 2 in thetwo phase are given by barotropic state laws pk = pk(ρk). The phase densities aswell as the pressure p are then found by solving the system of equations expressingthe equality of the pressures in the two phases as well as the saturation constraintα1 + α2 = 1 with αk = ρYk/ρk giving : The partial mass conservation equation (1.2) contains a diffusive term that expressesthe fact that the velocities of the two phases are not exactly equal. They differsfrom the centre of mass velocity u by a relative velocity ur. An asymptotic analysis2provides the following estimate for this term H. GUILLARD et al./Shock Structure in Two-Phase models where α is the volume fraction. The second-order perturbation of this model presentssome unusual features. In contrast with many dissipative systems, this model doesnot contain any regularizing term in the momentum equations but only containa second-order perturbation in the partial mass equation. It can be shown thatthis second-order perturbation comes from the existence of relaxation phenomenabetween the two phases that drive the system toward mechanical equilibrium andthat system (1) results from a rigorous Chapman-Enskog asymptotic analysis ofnon-equilibrium two phase models.2 In this section, we will use this model to study the possible regularizing effect intwo-phase models of second-order perturbations of the form displayed in equation(1.2). Actually, since the model (1) does not contain a viscous regularisation in themomentum equation, one may wonder if the diffusive term in (1.2) is sufficient fora regularizing effect to occur. This is the main motivation of this work. Therefore,we now focus on the possible existence of a certain class of solutions of this system,namely the traveling waves solutions defined by U(t, x) =t (ρ, ρY, ρu) is a traveling wave solution of (1) if 1. There exists a real s and a one-parameter function ˆ U(ξ) such that U(t, x) = 2. There exist two state vectors UL and UR such that ξ→+∞ U (ξ) = 0 If such solutions exist, they are characterized by the differential system of degree 2 : Since system (1) is in conservative form, it can integrated once to yield a first-ordersystem. Algebraic manipulations of the resulting first-order system of odes gives thefollowing result3 : Lemma 0.1 The differential system (4) associated with the right boundary condi-tions limξ→+∞ U(ξ) = URlimξ→+∞ U (ξ) = 0 can be reduced to the following first order ode : Y (1 − Y )(α − Y )z H. GUILLARD et al./Shock Structure in Two-Phase models where the scalar variable z = p − pR denote the pressure while pR is the pressureof the downwind side of the travelling wave and where Y and α are functions of zdefined by : τ (z)(pR + z) with M the mass flux and τ = 1. The existence of traveling wave solutionsthus reduces to the sudy of this ode. It is then easily seen that z = 0 andz = zL = M2τR − pR are two equilibrium points. The linearization of (5) in thevicinity of these two points shows easily that z = 0 is a stable equilibrium whilez = zL is unstable. To end this study, we then have to check if (5) can becomesingular and thus we are lead to study the function Y (z)(1 − Y (z))(α(z) − Y (z))for z ∈ [0, zL]. This study3 then reveals the existence of a critical mass flux Mcritsplitting the set of solutions into two classes as follows : Weak shock case If the mass flux verifies M2 < M 2 the ode (5) is never singular and therefore, it exists a unique C1 solution connecting the two equilibrium z = 0and z = zL = M2τR − pR and in consequence a viscous profile connecting the twostates UL and UR.
These continuous profiles are displayed in figure 1 from ref3 where they have beencomputed by a numerical integration of (5) as well as by a finite volume approxi-mation of the original PDE system (1).
Figure 1: Profiles in the shock for the weak shock case, upper left : pressure, upper right : velocity,lower left : mass fraction H. GUILLARD et al./Shock Structure in Two-Phase models The second set of solutions is given by : Strong shock case If the mass flux verifies M2 > M2 , there is no C1 viscous profile connecting the states UL and UR. The ode (5) is singular in z∗ and z∗ and there exist an infinite number of orbits connecting the two equilibria z = 0 andz = zL. These orbits are composed of - a C1 two-phase solution connecting the equilibrium points z = 0 and z = z∗ or - a discontinuous one-phase shock connecting the two states U∗ and U∗ - a C1 two-phase solution connecting z = z∗ and the equilibrium z = z These discontinuous solutions again from3 are displayed in figure 2 Figure 2: Profiles in the shock for the strong shock case obtained from numerical integration of(5) upper left : pressure, upper right : velocity, lower left : mass fraction In this work, we have investigated in a simple model of two phase flow, a particular structure of dissipative tensor that differs from the usual viscous tensors constructed Traveling wave analysis for this model has been performed. This analysis has re- vealed that despite the zero mass fraction jump implied by the Rankine-Hugoniot relations, the mass fraction is not constant in the shock region. Moreover, this anal- ysis has also shown the existence of a critical speed of the waves above which noC 1 solutions exist. However above this critical speed, we have shown the possibility to construct traveling solutions involving single phase shocks. This analysis thus reveals interesting features that cannot a priori be deduced from the non-dissipative H. GUILLARD et al./Shock Structure in Two-Phase models system. The same type of analysis performed with more complex two-phase modelsis currently under investigation and show a similar behavior.4 [1] G. Dal Maso, P. G. Lefloch, and F. Murat. Definition and weak stability of nonconservative products.
J-Math-Pures-Appl, 74(6):483–548, 1995.
[2] H. Guillard and F. Duval. A darcy law for the drift velocity in a two-phase model. J. Comput. Phys., [3] Herve Guillard and Vincent Perrier. Shock structure in a two-phase isothermal Euler model. Research [4] Mathieu Labois and Herve Guillard. A five equation two-phase dissipative model l. Research report, [5] L. Sainsaulieu. Traveling-wave solutions of convection-diffusion systems in nonconservation form. SIAM J. Math. Anal, 27:1286–1310, 1996.

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