The Drift-flux Multiphase Model in Networks of Pipes
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Towards a Mathematical Analysis for Drift-Flux Multiphase Flow Models in Networks
This chapter deals with the dynamics of the multiphasedrift-flux model in a network of pipes. We formulate the model equations from the two-fluid model and obtain a model with a conservation of mass for each of the two phases and a conservation of momentum. The system is closed with an equation of state which gives a formula of the pressure in terms of the densities of the two phases. This chapter focuses on a linear pressure law derived under the assumption that the pressure of each phase is a linear function of the densities. In the next chapter, we will consider a more general pressure law defined as an arbitrary function of the densities. When the model equation for the fluid is adopted, we consider a junction of a network as a set of vectors intersecting at the origin. The vector length represents the pipe and their meeting point is the junction. The dynamic of the flow of the fluid at the junction is stable only if some suitable coupling conditions are prescribed. These are usually derived from the physics of the problem and they play an important role in the proof of the well-posedness of the Riemann problem at the junction. The main results of this chapter are the well-posedness of the Riemann problem at the
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junction and numerical simulations of the dynamics of the flow at the junction for the cases of three and four connected pipes. These results appeared in [10].
3.1 Introduction
We consider anisothermal no-slip drift-flux model for multiphase flows of the form:
∂tρ1+∂x(ρ1u) = 0 (3.1a)
∂tρ2+∂x(ρ2u) = 0 (3.1b)
∂t(u(ρ1+ρ2)) +∂x
(ρ1 +ρ2)(u2+ a2 2 )
= 0 (3.1c)
whereρ1 andρ2are the density of phase 1 and phase 2, respectively,uis the common velocity of the two phases anda is a constant which depends on both phases. This model is derived from the drift-flux model [51] by making the simplifying assumption that the closure law, the so called slip condition, has a vanishingslip function. The slip condition is an algebraic relation that relates the two velocities of the two phases.
The drift-flux model in turn is derived from thetwo-fluid model by summing up the balance laws for the momentum, in canonical form, for each phase. The choice of this model has been motivated by the fact that we would like to concentrate on some basic aspects of the model in order to analyze coupling conditions of pipes at a junction in a network and devise a computational approach for approximating flow at a junction.
The no-slip condition was considered by Evje and Fl˚atten [50] when extending the Weakly Implicit Mixture Flux (WIMF) scheme originally developed for the two-fluid model, to the drift-flux model. In [52] Evje and Karlsen used the same simplification as a basis for proving global existence of weak solutions for the viscous form of the drift-flux model. This model has many applications in the chemical, petroleum and nuclear industries [48, 46]. As a result there has been intense research on such multiphase flows in the recent past. Different models for multiphase flows have been proposed [1, 54, 63, 46, 51] and numerical methods for such models have
been investigated [48, 49, 47, 46]. The mathematical study of the flow of gases in networks is a young field of research and has been under investigation only recently.
The reader is referred to [32, 33, 41, 6, 7], in the context of gas networks. Work has also been undertaken in the context of traffic flow networks, see for example [62].
In this chapter, we investigate the flow of anisothermal no-slip drift-flux model (3.1) in a network of pipes. Firstly, using the properties of Riemann problems for general one dimensional systems of conservation laws, we derive a Riemann solver for the model equation (3.1). Secondly, we consider the flow of (3.1) at the junction of a network of pipes and prove the well-posedness of the resulting Riemann problem at the vertex. Our proof relies on suitable conditions which couple the models from each pipe at the junction. These coupling conditions are motivated by consideration from the physics of the flow. For example, the conservation of mass at the junction forms the cornerstone of such considerations. Similar work has been done for thep- system by Colombo et al. [32, 33] and on the isothermal Euler equations by Banda et al. [6, 7]. Here we consider the case of a multiphase fluid. The constructive proof of our main result allows us to do some numerical simulations of junctions connecting up to four pipes.
This chapter is organized as follows: In Section 3.2, we derive the model equation given in (3.1), study the wave curves in one pipe and define a Riemann solver for the model equation. Section 3.3 is devoted to the modeling of pipe to pipe intersections and the proof of well-posedness of the model at an uncontrolled junction of a network.
Finally, we describe in Section 3.4 a numerical method used to solve the isothermal no-slip drift-flux model on networks. Computational results on some carefully chosen examples are presented and compared with theoretical results.