COUPLING DRIFT-FLUX MODELS WITH UNEQUAL SONIC SPEEDS

The well-posedness of a Riemann problem at a junction in a pipeline network is discussed. In addition computational results on the dynamics of the flow of a multi-component gas at such network junctions are presented. The work presented here is a generalisation of [M. K. Banda, M. Herty, and J. M. T. Ngnotchouye, Towards a mathematical analysis of multiphase drift-flux model in networks, SIAM J. Sci. Comput., 31(6): 4633 – 4653, 2010] to models in which the equation of state has different compressibility factors or sonic speeds. KeywordsDrift-flux, multi-component, compressible flow, Riemann Problem, flow networks, coupling conditions, compressibility factor.


INTRODUCTION
We consider pipe networks through which multi-component fluids flow.The pipes in the network intersect at joints or junctions.The idea is to propose coupling conditions for the case in which the individual fluid components have unequal sonic speeds or compressibility factors.The analysis presented in this paper is an extension of the work in [4].In [4] multi-component flow in which sonic speeds of fluid components are equal were considered.An isothermal no-slip drift-flux model for multi-component flows is considered.In this model the closure law, the so called slip condition, has a vanishing slip function.The drift-flux model obtained is presented in Section 2. The same section also discusses a mathematical analysis of the model.These kinds of models have many applications in the chemical, petroleum and nuclear industries.The mathematical analysis of networked flow has recently become an active area of research.Some examples include [2,3,4,8,10].The well-posedness of the coupling of flow at junctions will be presented in Section 3. In Section 4 results of numerical tests of such coupling conditions will be presented.

MODEL FORMULATION AND THE RIEMANN PROBLEM
The conservative isothermal no-slip drift-flux model takes the form where we denote I = ρu, ρ = ρ 1 + ρ 2 .In the above, components of an immiscible mixture of two fluids are denoted by the subscripts i ∈ {1, 2}.Hence the density, volume fraction, velocity and pressure are denoted by ϱ i , α i , u i , p i , respectively, see also [4] and references therein.The density of each component is denoted as , where the positive constants a i are the compressibility factors or sonic speeds of phase i.From the relation α 1 + α 2 = 1, we obtain an equation of state: In the pipe network, at each junction where pipes intersect, we need to solve a Riemann problem [9,11].This consists of solving the model (1) in each pipe with constant data in addition to some coupling conditions at the intesection.We note that the eigenvalues λ 1,2,3 and the eigenvectors r 1,2,3 of the drift-flux model (1) are given by The field 2 is always linearly degenerate while the fields 1 and 3 are genuinely nonlinear [9,11].Moreover since p ≥ 0, ∀ρ 1 , ρ 2 > 0 model ( 1) is hyperbolic and strictly hyperbolic if √ p ̸ = I √ ρ .We now discuss the Lax-curves as a pre-requisite to presenting the solution of the standard Riemann problem.

Shock Curves
The Lax shock curves are derived from the Rankine-Hugoniot jump conditions (3).Let w be a given state and assume that another state w is connected to w by a 1-, 3-shock wave of shock speed s.Then w and w satisfy For the equation of state given in (2), the shock curves are given by with shock speed The forward (respectively, backward) admissible 1-shock curves are obtained using the Lax admissibility conditions [9] denoted as S 1 (ξ; w) given by ( 4) with ξ ≥ 1, (respectively, ξ ≤ 1).Similarly, the forward (respectively, backward) 3-shock curves are given by S 3 (ξ; w) in (4) with ξ ≤ 1, (respectively ξ ≥ 1).

Contact Discontinuity
Let w and w be two given states.Using the linear degeneracy of the 2-field and the Rankine-Hugoniot jump condition (3), we declare that w belongs to the 2-curve emanating from w if w − w = ξr 2 ( w), ξ ∈ R. Eliminating ξ from this system and applying suitable scaling, one obtains the contact discontinuity wave emanating from any state w given by the curve Note that we have continuity of the pressure along the contact discontinuity.

Rarefaction Curves
We first note that rarefaction waves are given as integral curves of the eigenvectors of the flux function The forward (respectively, backward) admissible 1-rarefaction curves for (5) are obtained using the Lax admissibility condition as R 1 (ξ; w) with ξ < 1, (respectively, ξ > 1).Similarly, the forward (respectively, backward) 3-rarefaction curves are given by R 3 (ξ; w) with ξ > 1, (respectively, ξ < 1).In summary the Lax-curves for model (1) with the equation of state ( 2) are given by The shock speeds are given by s

Solution to the Standard Riemann Problem for (2)
Proposition 0.1 [9] We consider the Riemann problem for (1) with initial data For |w + − w − | sufficiently small, there exists a unique weak self-similar solution to this Riemann problem with small total variation.This solution comprises four constant states w 0 = w − , w 1 , w 2 , w 3 = w + .When the i-th characteristic family is genuinely nonlinear w i is joined to w i−1 by either an i-rarefaction wave or an ishock, while when the i-characteristic family is linearly degenerate, w i is joined to w i−1 by an i-contact discontinuity.
Assuming the states w − and w + are given and satisfy the conditions of Proposition 0.1, then the intermediary states, w 1 and w 2 , could be: Denoting the momentum components of w 1 and w 2 as I 1 (ξ 1 ; w − ) and I 3 (ξ 3 ; w + ), respectively.The solution for the Riemann problem is found if we can solve for ξ 1 , ξ 2 and ξ 3 the system ) .
Proposition 0.2 Assume that we have a multi-component fluid described with the equation of state (2) with a 2 1 and a 2 2 given.Let w be a state that satisfies the following condition Then, for w − and w + close to w, the standard Riemann problem with data (w − , w + ) admits a solution.

PIPE-TO-PIPE INTERSECTIONS
We assume two pipes are joined at a junction located at x = 0.The flow in the pipes is defined by (1) and at the junction the flow must also satisfy the coupling conditions: The variables w − and w + denote the flow variables in the left and right pipes, respectively.The notations R + =]0, +∞[ and = [0, +∞[ will also be used.We consider solutions of the Riemann problem at the junction as perturbations of stationary solutions of (1).Consider the subsonic region defined as Further define flow of the density of phase 1 as M (w) = ρ 1 I/ρ, flow of the density of phase 2 as N (w) = ρ 2 I/ρ, flow of the linear momentum as P (w) = I 2 /ρ + p(ρ 1 , ρ 2 ).In general, we are interested in Ψ-solutions, that is, weak solutions depending on the coupling conditions map Ψ.We consider a map ŵ defined as The existence of ŵ+ for a given ŵ− is guaranteed by Lemma 0.1.In a neighborhood of a junction between two pipes, one can integrate the stationary model from (1) to obtain the coupling conditions map [5,6,10] Note that the conservation of mass of each phase and the equality of the dynamic pressure at the junction are preserved.We can prove that when a stationary flow w − is prescribed in the ingoing pipe, we can solve for the flow in the outgoing pipe in general which is also stationary.
Lemma 0.1 Let w ∈ A 0 .Then there exists δ > 0 and a Lipschitz map T : B( w; δ) → A 0 where B( w; δ) is a ball centred at w with radius δ, such that Ψ (w This result is similar to a result presented in [7] in the context of the p-system.Moreover, Lemma 0.1 ensures the "additivity" property [5,9] for the Riemann problem at a junction.For the well-posedness of the Riemann problem at a junction with more than two pipes, we present the following: Proposition 0.3 Let ŵ1 , ŵ2 ∈ A 0 be the data in the ingoing and outgoing pipe connected at x = 0. Assume that the following condition is satisfied ) Then, there exists a constant δ > 0 and for any states w1 and w2 such that | wi − ŵi | < δ, for i = 1, 2, the Riemann problem at the junction with data ( w1 , w2 ) has a unique solution.
The argument in the proof is standard and has also been developed in [2,3,6,8,10] for other models.

NUMERICAL EXPERIMENTS
The numerical schemes used to test the homogeneous no-slip drift-flux multiphase flow model as defined in (1) are the second-order relaxed schemes which were discussed in [1] and references therein.For a comparison with other schemes applied to similar problems refer to [4].Initial conditions are some perturbation of stationary solutions, see Proposition 0.3.Newton's method is used to solve the system in equation ( 7) combined with the Lax curves which gives the boundary conditions at the junctions of the network.For the external (inlet to network or outlet from network) boundary conditions, the transparent boundary conditions are imposed.

A junction connecting two pipes
Here we will firstly verify the qualitative behavior of the coupling conditions.Secondly, we will use it to present the effect of the variation of the sound speed of each phase at the junction.The compressibility factors are taken as a 2 1 = 16.0 and a 2 2 = 1.0 with the Riemann data w − = (3.17123,3.38324, 3.71816), w + = (2.70708,4.0434, 3.5629).The mesh size of N = 400 was employed on a single pipe on which the standard Riemann solver was applied and N = 200 was applied on each coupled pipe.In Figure 2 computational results at t = 0.05 are shown.Qualitatively, there is very good agreement between the two.The results in Figure 3 show the effects of the sonic speeds.Three cases were considered: a 2 1 = a 2 2 = 6.0, 16 = a 2 1 > a 2 2 = 1 and 1 = a 2 1 < a 2 2 = 16.0.We observe that higher values of the  pressure are obtained with smaller values of a 2 1 .

CONCLUSION
We have solved the standard Riemann problem for the multi-phase model (1) with the pressure law (2).For the case of two connected pipes at a junction, the Riemann problem at the junction have been proved to have a unique solution under some conditions.In general we have proven that when the inflow is given and the coupling conditions are defined in a suitable way, one can always solve for the outflow in the outgoing pipe.We have also presented some numerical results that demonstrate the applicability of our model and its possible extension to more general networks.

Figure 1 :
Figure 1: Profiles of the densities, the common velocity and the pressure for the Riemann problem for the drift flux model with different compressibility factors.

Figure 2 :
Figure 2: Profiles of densities, momentum, and the common pressure p for the standard Riemann problem (continuous line) and two coupled pipes (dashed line).

Figure 3 :
Figure 3: Profiles of densities, common velocity and pressure for coupled models with different compressibility factors at time t = 0.06.

Figure 4 : 3 Figure 5 :
Figure 4: Snapshots of the densities for the solution of the Riemann problem at the junction with one ingoing and two outgoing pipes. 1.