A hierarchy of models for two-phase flows - Semantic Scholar

A hierarchy of models for two-phase flows F. Bouchut1, Y. Brenier2, J. Cortes3 and J.-F. Ripoll4

Abstract We derive a hierarchy of models for gas-liquid two-phase flows in the limit of infinite density ratio, when the liquid is assumed to be incompressible. The starting model is a system of nonconservative conservation laws with relaxation. At first order in the density ratio, we get a simplified system with viscosity, while at the limit we obtain a system of two conservation laws, the system of pressureless gases with constraint and undetermined pressure. Formal properties of this constraint model are provided, and sticky blocks solutions are introduced. We propose numerical methods for this last model and the results are compared with the two previous models.

Key-words: two-phase flow – conservation laws with relaxation – conservation laws with constraint – pressureless gas – sticky particles and blocks Mathematics Subject Classification: 76T05, 35L65, 35L85

(1) D´epartement de Math´ematiques et Applications, Ecole Normale Sup´erieure et CNRS, UMR 8553, 45, rue d’Ulm, 75230 Paris cedex 05, France. e-mail: [email protected] (2) Laboratoire d’Analyse Num´erique, UMR 7598, Universit´e P. et M. Curie, B.C. 187, Tour 55-65, 5`eme e´ tage, 4, place Jussieu, 75252 Paris, France. e-mail: [email protected] (3) CMLA UMR-8536, Ecole Normale Sup´erieure de Cachan, D´epartement de Mathe´ matiques, 61, avenue du pr´esident Wilson, 94235 Cachan cedex, France. e-mail: [email protected] (4) Math´ematiques Appliqu´ees de Bordeaux UMR 5466, Universit´e de Bordeaux 1 et CNRS, 351 cours de la lib´eration, 33405 Talence cedex, France. e-mail: [email protected]

1

1 Introduction and governing equations In this paper, we consider two-phase flows of gas and liquid. Our aim is to discuss some simplified models obtained in the situation when the gas is much less dense than the liquid. We start from a general system of nonconservative conservation laws with relaxation from [14], where can be found complete discussions about modeling. The liquid is assumed to be incompressible. By a scaling of the equations, we introduce a small parameter " that represents the ratio of the densities, and derive two models which are obtained as the first-order and zero-order terms in an expansion in ". Then we discuss the mathematical properties of these models. The first-order model is a hyperbolic system of conservation laws with degenerate viscous term. The zero-order model has a less usual structure, it is written as a system of two conservation laws, the system of pressureless gases, to which we impose a constraint on the volume fraction  that must be kept less than 1, and add a Lagrange multiplier associated to this constraint, which takes the form of an undetermined pressure. This system is shown to share most of the good properties of the system of pressureless gases. In particular, special sticky blocks solutions are introduced, generalizing the so called sticky particles that are known for the pressureless gases, see [7]. Our study suggests that the general Cauchy problem for this constraint model is well-posed, and this will be examined in a future work. We propose numerical methods for this last model and the results are compared with the two previous models. The results show the relevance of the asymptotic expansion. The paper is organized as follows. The next subsections are devoted to the asymptotics in ". Then, in Sections 2 and 3 we study the mathematical structure of the models. In Section 4 we introduce numerical methods and sticky blocks dynamics, and in Section 5 we perform numerical experiments.

1.1 The basic two-fluid model Let us briefly present some properties of two-fluid systems that are of relevance for our discussion. We refer to [14] for a complete discussion about modeling. The reduced two-fluid model that we consider consists in two mass equations and two momentum equations for each phase without mass exchanges in the context of bubbly flows. In these circumstances, the governing equations are given as follows, ( @t ((1 )) + @x ((1 )v) = 0; (1) @t (l ) + @x(l u) = 0;

2

and

(

@t ((1 )v) + @x((1 )v ) + (1 )@xp + g = MgD ; @t (l u) + @x (lu ) + @x p + l = MlD ; 2

2

(2)

where  is the liquid volume fraction

0    1;

(3)

and 1  is the gas volume fraction. Moreover,  denotes the gas density, l the liquid density, v the gas velocity, u the liquid velocity and p the common pressure (we refer to [14]). For simplicity, the liquid is assumed to be incompressible

l = constant;

(4)

and the pressure evolution to be governed by an equation of state

p =  ; (5) that will indeed be written as  = (p). Let now U be the unknown vector of conservative variables 0 (1 ) 1 B l CC B@ (1 )v CA : U =B (6) l u The system is closed provided closure relations are given for MgD ; MlD ; g ; l . The right-hand sides MgD and MlD are source terms reflecting inter-phase drag. Here, we choose the standard simplified closure laws

MlD = MgD = (1 )l (v u): (7) The phase pressure fluctuations terms g and l are differential terms which are

mathematically relevant because they affect the well-posed nature of the system. For instance, we can take into account the following pressure correction derived from [15] for bubbly flows,

(

l = Cp l (v u) @x; g = 0: 2

(8)

Similarly we could have considered any alternative pressure correction satisfying

vlim !u k

= 0;

k = g; l: 3

(9)

This physical property is of relevance and a lot of expressions for the pressure fluctuations widely used in industry satisfy property (9). We now introduce the scaling of the two densities. Indeed, a convenient way to describe the behavior of a two-fluid system is to scale the two densities and to consider the ratio of the densities as a small parameter. This approach has been introduced and discussed in [9] and [10]. We have assumed l = constant. The eigen-decomposition of A is obviously complicated by pressure interactions. This is the reason why the density perturbation method has been introduced in [10], where the system (1), (2) is scaled by introducing the two average constant densities l and 0 . This yields the new variables 1 0

~ = = ; 0

Moreover we set

(1 )~ BB  U~ = B @ (1 )~v u

p~(~) = p()= ;

CC CA :

(10)

" =  =l :

(11)

0

0

It can be shown that such an approach is efficient to describe the behavior of the whole system, including source terms effect ([9]). The two-fluid model can now be written in the following condensed form (where for notational simplicity we omit from now on the~symbol)

@t U + @x[F (U )] + A(U; ")@xU = S (U; "): The vector U is given by (10),

0 (1 )v B u F (U ) = B B@ (1 )v u

2

2

1 CC CA ;

(12)

0 1 0 BB CC 0 B C S (U; ") = B B@  (1 )(u v) CCA ; " (1 )(v u)

and

0 0 0 BB 0 0 A(U; ") = B B@ (1 ) @U1 p (1 ) @U2 p " @U1 p " @U2 p + Cp (v u) 0

0 1

1

0

1

0

1

2

0 0 0 0

In the following, the system (12)-(14) will be referred to as Model I.

4

0 0 0 0

1 CC CC : A

(13)

(14)

1.2 First-order model derived from Model I by Chapman-Enskog expansion The source term S (U; ") contains a stiff relaxation term

 (1 )(u v): "

(15)

Several authors ([2], [8], [18]) have shown that the long time behavior of hyperbolic systems with stiff relaxation terms is governed by local equilibrium systems. When the relaxation parameter " is much smaller than the sound wave velocity, one might expect that the solution tends to the solution of the local equilibrium system, derived from Model I by setting " to zero. We perform now the Chapman-Enskog expansion for Model I. We seek solutions to Model I of the form v = u + "w: (16) We rewrite the third equation of Model I in the following way, (1 )w = @t ((1 )u)+ @x((1 )u2)+(1 ) 0 1@xp + O("): (17) Moreover, the first and fourth equations of Model I give

@t ((1 )) + @x ((1 )u) = O(");

(18)

@t (u) + @x (u ) = O("):

(19)

and

2

Now, using (18) and (19), a simple computation gives (1 )w = (1 ) 0 1@x p + O("): Therefore, introducing (20) in (12), we finally get the first-order model

@t V + @x(G (V ) + "G (V )) = "@x(D(V )@xV ); 1 0 (1 ) V =B @  CA ; u 0 (1 )u 1 CA ; u G (V ) + "G (V ) = B @

u + " p 1 0 @V1 p @V2 p 0 1  1 D(V ) =    B @ 0 0 0 CA : 0 0 0 0

with

0

1

1

2

0

0

(21)

(22)

(23)

1

1

In the following, the system (21)-(24) will be referred to as Model II. 5

(20)

(24)

1.3 Zero-order model

Let us now try to let " ! 0 in Model II. If we neglect the term in " in (23), we get from the two last equations of (21)

(

@t  + @x (u) = 0; @t (u) + @x(u ) = 0; 2

(25)

the system of pressureless gases. However, it is known ([3], [7], [5]) that this system gives Dirac distributions on  in finite time, even for smooth initial data. But since here   1, this limit is obviously wrong. Here we have to underline that Models I and II do keep this inequality satisfied, as numerical simulations show, even if we are not able to justify it rigorously. Thus since the pressure term cannot be neglected, it must become singular in the limit, and we make a more realistic assumption

" ! 0;

(26)

@t  + @x(u) = 0; @t (u) + @x (u + ) = 0:

(27)

" p !  0

which yields at the limit

(

1

as

2

We also assume that in the regions where  remains less that 1, the limit in (26) in indeed 0. Therefore, we complement (27) with the constraints

0    1;

(1 ) = 0:

(28)

Since in (21)  = (p), we see that according to (26), we must have  ! 1 where  6= 0. However, we can conjecture that (1 ) !  as " ! 0, where   0 satisfies @t  + @x (u) = R; (29)

where R is the limit of the possibly singular diffusion term in the right-hand side of (21). Equation (29) describes the convection of the gas density, while (27)-(28) describe the evolution of the liquid. But since the system (27)-(28) is closed, we shall drop (29), which can be solved afterwards, for example using the theory of [4]. The constraint model (27)-(28) will be denoted by Model III. Its mathematical properties are discussed in Section 3. It can be interpreted as a coupling of two systems in the respective domains where  < 1 (liquid-gas mixture) and where  = 1 (pure liquid), in which we solve respectively the pressureless system (25) and the incompressible gases equation

@t u + @x = 0; 6

@xu = 0;

(30)

where  is the Lagrange incompressible pressure. This equation is very simple in one dimension. However, the transmission conditions at the interface are not given explicitly, they are hidden in the whole system (27)-(28). We can remark that the pressureless system has a sound speed that vanishes identically, while the incompressible system is elliptic and has infinite sound speed. We can interpret the fact that (27)-(28) is closed by saying that since the gas is infinitely light, it cannot influence the evolution of the incompressible liquid, that can occupy all the volume, ( can reach the value 1). Equation (29) says that the gas will then follow the liquid. The situation is different in Model II where the gas is always heavy enough to repulse the liquid by its pressure, so that  cannot reach the value 1.

2 Wave structure for Models I and II 2.1 Wave structure for Model I Model I is written in nonconservative form because it contains some nonconservative products A(U; ")@x U . Such terms must be carefully handled in the presence of discontinuities but we do not discuss on that point in this paper and refer to [11]. The wave structure for model I has been discussed in [10]. The Jacobian matrix AI of Model I is given by

AI (U; ") = @U F (U ) + A(U; "): (31) It can be shown that, for small ", such matrix admits four distinct eigenvalues provided that u 6= v . Indeed we have the decomposition AI (U; ") = PDP (U; "); 1

with

0 v pc 0 0 BB 0 g u pcl 0 D=B @ 0 0 u + pcl 0

where

0

0

u 6= v; 0 0 0p v + cg

1 CC CA + O(");

(32)

(33)

cg =  @ p > 0 and cl = Cp(v u) > 0: (34) Unfortunately, when u = v we see that cl = 0. Moreover, it is shown in [10] that 0

1

2

we get a Jordan block related to the degeneracy of the second and third eigenvectors. This structure is clearly related to the physical assumption (9). The Jacobian matrix AI now writes

AI (U; ") = RJR (U; "); 1

7

u = v;

(35)

with

0 u pc + O(") 0 0 g BB 0 u 1 J =B @ 0 0 u

1 0 CC 0 CA : 0 0 0 u + pcg + O(")

0

(36)

Finally, we see that Model I is a conditionally hyperbolic system written in nonconservative form.

2.2 Model II: a strictly-hyperbolic viscous system in conservative form We see obviously that Model II is written in conservative form. Let us study the convection part of Model II. The Jacobian matrix writes

AII = @V (G + "G )(V ); 0

(37)

1

where G0 + "G1 is given by (23). A straightforward computations gives

 1  u  u   C 0 0 1 A;

" @V1 p " @V2 p u 2u and we can easily compute the three eigenvalues of AII , s s c g u " (1 ) ; u; u + " (1cg ) ;

0 AII = B @

1

1

0

1

0

1

(38)

2

(39)

with cg given in (34). Remember that  is now dimensionless. We see that these eigenvalues are distinct, hence Model II is a strictly hyperbolic system. If we now look at the diffusion term "@x (D (V )@x V ) in the right-hand side of (21), we easily get that D (V ) has 3 eigenvalues given by

1  c ; 0; 0:  g

(40)

These eigenvalues are nonnegative. Hence, we can finally conclude that Model II is a strictly-hyperbolic viscous model in conservative form.

8

3 Structure of the constraint model This section is devoted to the (mainly formal) mathematical properties of Model III ( @t  + @x(u) = 0; (41) @t (u) + @x (u2 + ) = 0; with the constraints 0    1; (1 ) = 0; (42) and with initial conditions

(0; x) =  (x); (0; x)u(0; x) =  (x)u (x): (43) The initial data  and u must satisfy 0    1,  must not be identically 1 at infinity, u must be bounded, and locally constant in the regions where   1. It is obvious from the formulation that the total mass and momentum do not depend on time. Of course,  remains between 0 and 1, but there is also an estimate on u which is quite hidden, the maximum principle inf u  u(t; x)  sup u : (44) 0

0

0

0

0

0

0

0

0

0

0

It is not easy to justify it, but as we explain in Section 4 we can build numerical methods that satisfy this property, and even more, the velocity u is total variation diminishing. This is also the case for pressureless gases ([3], [7], [5]). However, here, the situation is slightly different due to the Lagrange incompressible pressure which prevents concentrations. We have to put also entropy conditions to select the right solution, in practice we ask that   0 and that the characteristics associated to the single eigenvalue u enter the generalized shocks, i.e. that @x u  0 in shocks. In particular, this needs that the intermediate values of u are between the left and right values (maximum principle). The Oleinik entropy condition @x u  1=t should hold indeed, see Section 4.3. In fact, we can even get entropy inequalities, obtained as follows. Let S be a convex function. Then, subtracting the first equation of (41) multiplied by u to the second, we get (for smooth solutions)

(@t u + u@xu) + @x = 0: (45) Then, we multiply (45) by S 0 (u) and add the result to the first equation of (41) multiplied by S (u). We obtain @t [S (u)] + @x [uS (u) + S 0 (u)] = S 00(u)@xu  0: (46) This generalizes entropy inequalities that hold for pressureless gases. Existence of weak solutions for (41)-(43) will be examined in a future work. The volume fraction  is continuous with respect to time, but an unusual property 9

here is that the momentum density u is not, because  can contain Dirac functions in time, see Section 4.3. However, the same problem with viscosity has been solved in [17] in multidimension. Other hyperbolic problems with constraints have been studied in [12], [1], [16]. Let us now study the resolution of the generalized Riemann problem for the system (41)-(42). We look for a generalized shock solution (t; x), u(t; x) and (t; x) of the following form,

8 > < (; u; ) = (l ; ul; 0) (; u; ) = (1; u (t); a(t)x + b(t)) > : (; u; ) = (r ; uc r ; 0)

if x < x1 (t); if x1 (t) < x < x2 (t); if x > x2 (t):

(47)

In the initial value problem, we are given l , ul , r , ur , and x01  x02 (and also uc(0) if x01 < x02 ). For simplicity we are going to consider only the self-similar case where x01 = x02 . We also assume that ul > ur , in order to get a shock. Otherwise, the solution is just the same as in a pressureless gas,   0 and the left and right states evolve independently with vacuum in the middle. Theorem 3.1 For any initial data l , ul , r , ur , x0 satisfying 0 < l ; r < 1 and ul > ur , there exists a unique entropy solution to (41), (42) of the form (47) for some smooth functions x1 (t) < x2 (t), uc (t), a(t), b(t) satisfying x1 (0) = x2 (0) = x0 . Moreover, x1 (t) and x2 (t) are linear functions oft, u and  are constant between x1 (t) and x2 (t), uc (t) = u , a(t) = 0, b(t) =  , x_ 1 = (u l ul )=(1 l ), x_ 2 = (u r ur )=(1 r ), where the intermediate values are defined by

q

q

(1 r )l (1 l )r q q ul + q ur ; u = q (1 r )l + (1 l )r (1 r )l + (1 l )r

l r (ul ur )  = q q 
(1 r )l + (1 l )r

(48)

2

2

(49)

Proof. Along the two discontinuity lines x = x1 (t) and x = x2 (t), a weak solution (47) to (41) must satisfy the Rankine Hugoniot jump conditions for the mass conservation uc(t) l ul = x_ 1 (t)(1 l ); (50) uc(t) r ur = x_ 2 (t)(1 r ); (51) and for the momemtum conservation

a(t)x (t) + b(t) + uc(t) 1

2

l ul = x_ (t)(uc(t) l ul); 2

10

1

(52)

u_ c(t) + a(t) = 0; a(t)x (t) + b(t) + uc(t) r ur = x_ (t)(uc(t) r ur ): 2

2

2

2

(53) (54)

By subtracting (50) to (51), we obtain

x_ (t)(1 l ) x_ (t)(1 r ) = r ur l ul: 1

(55)

2

Similarly, subtracting (52) to (54) yields

u_ c(t)(x (t) x (t)) = l ul r ur + x_ (t)(uc(t) l ul ) x_ (t)(uc(t) r ur ): (56) Since by (50) and (51), x_ (t) and x_ (t) can be written as functions of uc (t), (56) can be written as u_ c(t)(x (t) x (t)) = G(uc(t)); (57) 2

2

1

2

1

1

2

2

with

2

1

G(uc) = (uc1 l ul) l

(uc r ur ) +  u l l 1 r 2

2

r ur :

2

2

(58)

Now we define

Z (t) = (1 l )x (t) (1 r )x (t); W (t) = (uc(t) l ul)x (t) (uc(t) r ur )x (t): 1

(59) (60)

2

1

2

We have according to (55) and (56)

Z_ = r ur l ul;

W_ = r ur l ul ; 2

2

(61)

therefore we obtain

Z (t) = (r ur l ul )t + Z (0); W (t) = (r ur l ul )t + W (0): From equations (59) and (60), we can obtain x (t) and x (t) x (t) = D(u1 (t)) ((r ur uc(t))Z (t) + (1 r )W (t)); c 1 x (t) = D(u (t)) ((l ul uc(t))Z (t) + (1 l )W (t)); 2

2

1

(62)

2

1

2

(63)

c

with

D(uc) = (1 r )(uc l ul ) (1 l )(uc r ur ); (64) which does not vanish in [ur ; ul ] since it is affine and D (ul ) = r (1 l )(ul ur ) < 0, D(ur ) = l (1 r )(ul ur ) < 0. Thus we get D(uc(t))(x (t) x (t)) = (r l )W (t) (r ur l ul )Z (t) (65) = (r l )W (0) (r ur l ul)Z (0) l r (ur ul) t: 2

1

2

11

Then, D (uc(t)) is bounded when t ! 0 because uc (t) 2 [ur ; ul ], and since x2 (t) x1 (t) ! 0, we must have (r l )W (0) (r ur l ul)Z (0) = 0. Therefore,

x (t) x (t) = l r (ur ul ) t=D(uc(t)): 2

2

1

(66)

Now from (57) we deduce

(uc(t)) = D(uc(t))G(uc(t)) : u_ c(t) = x G (t) x (t) l r (ur ul ) t 2

2

1

(67)

The function G in (58) is polynomial of degree at most two. One can check that G(ul) = (ur ul )2r =(1 r ) < 0, G(ur ) = (ur ul )2l =(1 l ) > 0, therefore G has exactly one zero in [ur ; ul ], and it is indeed given by u in (48). Thus G > 0 in [ur ; u[ and G < 0 in ]u ; ul ]. Since D < 0 in [ur ; ul ], we deduce that the only solution uc(t) to (67) which remains in [ur ; ul ] as t ! 0 is the constant uc (t) = u . We finally deduce Z (0), W (0) in terms of initial data from (59), (60), x1 (t), x2 (t) from (63), and a(t) = 0, b(t) =   from (53), (54), (51). Thus uniqueness is proved. Conversely, it is obvious that if uc (t) = u , (57) is satisfied, and defining x_ 1 , x_ 2 by (50), (51), a(t) = 0, b(t) from (54), we get (50)-(54), thus we obtain a weak solution to (41).

4 Numerical schemes 4.1 Upwind schemes for Models I and II Our main concern for models I and II is to provide numerical results for comparison with the ones obtained for Model III. Hence we will not provide details. We have seen in Section 2.2 that Model II is a strictly hyperbolic system written in conservative form. It is well known that there are a lot of efficient upwind schemes available for such systems. We refer to [13] and all the references therein. Here, we use the Lax-Friedrichs scheme. It is well known that this scheme can be very diffusive, and we use mesh refinement to limit this drawback. Moreover, we do not compute the diffusion part of Model II for the simulations. As previously mentioned, our aim is not to discuss the relevance of this scheme. Similarly, a simple numerical scheme is used to solve Model I. This scheme is detailed in [9]. It simply consists in a splitting scheme between convection and relaxation. We use Roe’s scheme for the convection step ([20]), and we explicitly solve the ordinary differential equation related to the stiff source term. The drawbacks of splitting schemes in the context of relaxation are fully discussed in [19]. We refer to [21] for more details about approximate Riemann solvers for two-phase flows. We now concentrate on the derivation of numerical schemes for Model III. 12

4.2 Transport-projection approach for Model III In this part, we build a finite difference scheme for the constraint model

(

@t  + @x(u) = 0; @t (u) + @x (u + ) = 0;

(68)

2

with

0    1;

(1 ) = 0:

(69)

The main difficulty is to impose the bound on the mass fraction , that could produce a Dirac function otherwise, as in the system of pressureless gases (68) with   0, see [3]. Our method consists in solving the problem in two steps, similarly to [6]. In the first step, we just solve the pressureless system. Then, we perform some kind of projection onto the space of constraint states. We have to notice that in the pure incompressible liquid region where  = 1, the system becomes elliptic, and has infinite speed of propagation. Thus the scheme needs information from the boundary of such a region, and cannot be written as a local scheme globally. It is only possible in the mixture regions, where the speed of propagation, which is u there, is bounded. In our approach we detect the pure liquid and mixture regions, and we treat them differently. However, an important feature is that we must have the global conservation of mass and momentum. A first guess to solve (68) is to write a finite volume formula, without any pressure,

(

in = in ((u)ni = (u)ni = ); in uni = inuni ((u )ni = (u )ni +1

+1

+1 2

+1

1 2

2

2

+1 2

=

1 2

);

(70)

where  =
t=
x. In order to simplify the presentation, let us only consider states with nonnegative velocities u. This property will indeed be preserved during the resolution. Now let us choose numerical fluxes with a special form

(u)ni

=

+1 2

= inuni

=

+1 2

(u )ni

;

2

=

+1 2

= inuniuni

=

;

(71)

=

);

(72)

+1 2

for some values uni+1=2 to be chosen. Then (70) becomes

(

in = in (inuni = in uni = ); in uni = inuni (inuniuni = in uni uni +1

+1

+1

+1 2

1

1 2

+1 2

1

1

1 2

or equivalently

(

in = (1 uni = )in + uni = in ; in uni = (1 uni = )inuni + uni = in uni : +1

+1

+1

+1 2

1 2

+1 2

13

1

1 2

1

1

(73)

We also obtain an incremental form by subtracting uni times the first equation of (72) to the second,

in (uni +1

uni) = uni

+1

n = i

1 2

1

(uni

uni):

1

(74)

The advantage of these formulas is that under the CFL condition

0  uni

=

+1 2

 1;

(75)

the scheme preserves the nonnegativity of  and u, satisfies the maximum principle for u, and moreover, u is total variation diminishing. Since uni +1 is a convex combination of uni and uni 1 , it also satisfies discrete entropy inequalities for any convex function S

in S (uni )  inS (uni) (inS (uni)uni +1

+1

=

+1 2

in S (uni )uni 1

1

=

1 2

):

(76)

Let us now examine the constraint. We start from values at time tn satisfying 0  in  1, 0  uni  umax, with umax  1. Then the necessary and sufficient condition for in  1 (for a fixed i) is obviously n n 1  i n ui =  n + i n uni = ; (77) i i +1

1

+1 2

or equivalently

uni

=

1 2

1 2

 1ni + ni uni = : i i n

1

n

1

+1 2

(78)

This condition can be interpreted as a recursive constraint on the values uni+1=2 . Let us now explain how we proceed. First, we compute some predicted values ipred , upred i , by the formulas (72) where we take the simple upwind flux uni+1=2 = uni. these predicted values can have mass fraction greater than 1, indeed this corresponds to the resolution of the pressureless gas system. we compute the predicted region of pure liquid I = fi s.t ipred  1g = S IThen, j j , where Ij are the ”connected components” of I . The number of components is not a priori determined. For example, we can have two components if we take an initial velocity with two decreasing discontinuities. The next step is to define the real values uni+1=2 that will be used to compute n+1=2 , un+1=2 . Outside I we simply take the upwind value intermediate values i i n n ui+1=2 = ui , while for each component Ij = fim;


; iM g we compute the values uni+1=2, i 2 Jj  fim 1;


; iM g, by the following procedure. We select an index ic 2 Jj (approximately the middle of Jj ), and we use a predicted value uj that will 14

be chosen later on. We define unic +1=2 = uj , and deduce the fluxes by the recursive relations ! n n 1  i 1 n i n n i = ic + 1;


; iM ui+1=2 = max ui ; n + n ui 1=2 ; i i ! (79) n n  1  i i n n n i = im ;


; ic: ui 1=2 = min ui 1; n + n ui+1=2 ; i 1 i 1 n+1=2 , un+1=2 with (72) and the above defined values Then we compute values i i n+1=2  1, except eventually for the cells immediof uni+1=2 . Obviously we have i ately around the boundary of I , but we do not care about this. Now, the problem that remains to be faced is that the previous recursive procedure to compute the fluxes in I does not necessarily give a consistent result. In particular, u should be constant in the pure liquid regions. Thus we take in+1 = in+1=2 , but for u weS perform a projection in the new pure liquid region I n+1 = fi s.t in+1  1g = j Ijn+1, by  X n+1=2 n+1=2  X n+1=2  uni +1 = k uk = k ; i 2 Ijn+1; (80) k2Ijn+1 k2Ijn+1 and uni +1

= uin

= if i 2 = I n+1. Since

+1 2

X

i2Ijn+1

in uni = +1

+1

X i2Ijn+1

in

= un+1=2 ; i

+1 2

(81)

there exist some unique values in+1=2 , i 2 Jjn+1 , such that

= un+1=2 + ( n n n+1 i i+1=2 i 1=2 ) = 0; i 2 Ij ; (82) in+1=2 = 0; i = min Jjn+1 or max Jjn+1: Therefore, by defining in+1=2 = 0 if i 2 = [j Jjn+1, we obtain the discrete form of (68) in+1 in + (inuni+1=2 in 1uni 1=2 ) = 0; n +1 n+1 n i ui i uni + (inuniuni+1=2 + in+1=2 in 1uni 1uni 1=2 in 1=2 ) = 0: (83) Note that with these formulas it is not obvious to see if in+1=2  0 everywhere. It only remains to choose the uj used in the flux construction. For best consistency with (80), we use the following formula  X pred pred  X pred uj = (84) i ui = i : i2Ij i2Ij

in uni +1

+1

in

+1 2

15

We can also derive entropy inequalities, for any convex function S , as follows. Define

Sin = +1

 X

k2I +1 n j

kn

=

+1 2

n+1=2 ) if and Sin+1 = S (ui Si+1=2 such that

in Sin +1

+1

S (ukn

i 2= I n

in

=

+1 2

=

+1 2

+1

S (uin

 X

)=

k2I +1 n j

kn

=

+1 2

 ;

i 2 Ijn ; +1

(85)

. Then as above, there exist entropy fluxes

=

+1 2



) +  Si

Si

=

+1 2

=



1 2

= 0;

(86)

and satisfying Si+1=2 = 0 for i 2 = [j Jjn+1 (and even for i in the boundary of n+1=2 ), and since obviously each Jjn+1 ). Adding this to (76) which involves S (ui S (uni +1)  Sin+1, we get

in S (uni ) inS (uni) + inS (uni)uni = + Si +1

+1

+1 2

=

+1 2

in S (uni )uni 1

1

=

1 2

Si

=

1 2



 0:

(87)

Numerical results, for this scheme, are performed in Section 5. We have to mention that for roundoff error reasons, it is necessary in practice to use a testing value 1  with   1 instead of 1 in the definitions of I and I n+1.

4.3 Sticky blocks In this section, we introduce a particular class of exact solutions to the system (41)-(43) consisting of sticky blocks, that generalize the sticky particles solutions involved in the pressureless gas system ([7]). This induces obviously a particle method for the numerical resolution of Model III. Let us consider a volume fraction (t; x), that can only take the values 0 and 1, of the form n X (t; x) = 1Iai (t)