Interaction of weak shocks in drift-flux model of compressible two-phase flows
Introduction
In the recent past, study of hyperbolic systems of partial differential equations (PDEs) have been the subject of great interest both from mathematical and physical point of view due to its applications in variety of fields such as magnetogasdynamics, astrophysics, engineering physics, multi-phase flows, aerodynamics and plasma physics etc. The Riemann problem was introduced by Riemann in [1] for system of hyperbolic conservation laws describing gas dynamics [2]. Riemann [1] initiated the concept of weak solutions and the method of phase plane analysis; Riemann’s solutions reveal that the elementary waves of isentropic flows: shock waves and simple waves. His result was extended to adiabatic flows by Courant and Friedrichs [3], and a new kind of elementary wave, contact discontinuity (slip line), was added. In Lax’s comprehensive discussion of such systems [4], it was proved that for strictly hyperbolic systems (i.e., the eigenvalues of the Jacobian matrix are distinct), there is a unique solution of the Riemann problem provided the initial data given by two constant states Ul and Ur are sufficiently close (in a precise sense). Liu [5] proposed the entropy condition and solved the Riemann problem for general 2 × 2 conservation laws. Since then, a lot of interesting work has been contributed to the one dimensional Riemann problem for various hyperbolic systems of conservation laws (see, [6], [7], [8], [9], [10]); indeed, the Riemann problem has been playing an important role in all three areas of theory, applications and computation. Smith [11] investigated existence and uniqueness of solution of the Riemann problem, with arbitrary initial data Ul and Ur, for the equations of compressible fluid flow in one space variable.
The solution of the Riemann problem involves shock waves, centered simple waves and contact discontinuities which are called elementary waves. Another interesting feature of nonlinear systems is that interaction of elementary waves. Based on the solution of Riemann problem, the interaction of one dimensional elementary waves in gas dynamics has been studied by Courant and Friedrichs [3], Smoller [2], Chang and Hsiao [12]. Solution of Riemann problem and wave interactions for various physical problems have been discussed by Raja Sekhar et al. (see, [13], [14], [15]).
It has been demonstrated that considerable research has been devoted to the Riemann problem for two-phase flows with multiple components. Two-phase flows can be described by means of different models: mixture, drift (homogeneous or not), two-fluid or even multi-fluid models are currently used in industrial thermo-hydraulic codes. A general transient two phase problem is formulated by Goda et al. [16] using a two-fluid model or a drift-flux model and this depends on the degree of coupling between the phases. The basic concept of the drift-flux model is the consideration of two separate phases as a mixture phase. Therefore the fluid properties are represented by mixture properties making the drift-flux formulation simpler than the two-fluid formulation. The general expression of the drift-flux model, proposed by Zuber and Findlay [17], accounts for the effects of non-uniform flow and void distribution across the flow channel, and local relative velocity between phases. The drift-flux model on the other hand is formulated by Ishii and Hibiki [18] considering the mixture as a whole rather than two phases separately. Numerical solution of one-dimensional drift-flux two-phase flow model have been studied in the literatures [19], [20], [21].
In this article, we consider the Riemann problem for an isothermal no-slip drift-flux model of multi-phase flows [17]. We establish the one-parameter families of curves for simple waves, shock waves and contact discontinuities. The main motivation of the present work is to study the existence and uniqueness of simple waves, shock waves and contact discontinuities. Lax’s paper [4] leaves no doubt that such a solution exists, but it seems to us that there may be interest in a brief and explicit proof favorable to numerical computations. A necessary and sufficient condition is derived for arbitrary initial data for the existence of solution either in terms of shocks or simple waves or both. Further, we discuss the interaction of two weak shocks. This interaction has been treated by von Neumann [22].
The structure of this paper is as follows. In Section 2, we recall the drift-flux model. In Section 3, we derive elementary waves; simple waves, shock waves and contact discontinuities of the Riemann problem. In Section 4, we construct the one-parameter families of Riemann solution. In Section 5, we prove a result on global existence and uniqueness of solution of the Riemann problem and identify the possibilities for shocks or simple waves to occur in a 1-family and 3-family. Section 6 deals with the interaction of two weak shocks of same family and finally, in Section 7 we close the paper with the conclusions.
Section snippets
The drift-flux model
The basic concept of the drift-flux model is the consideration of two separate phases as a mixture phase. Therefore the fluid properties are represented by mixture properties making the drift-flux formulation simpler than the two-fluid formulation.
Elementary waves of the Riemann problem
From (2.6) to (2.7) it is easy to see that the characteristic fields λ1 and λ3 are genuinely nonlinear and the characteristic field λ2 is linearly degenerate. Thus the elementary wave solutions of the system (2.4) consists of contact discontinuities, shocks and centered simple waves. Let U denote the right unknown state for either a simple wave or a shock wave of 1-family and the left unknown state for either a simple wave or a shock wave of 3-family.
One-parameter families of curves
To construct the elementary waves of the Riemann problem (2.4) and (2.5) somewhat more explicit, we introduce a new parameter y, as Hence, from (3.5) to (3.7) for a shock wave and a simple waves of 1-family, respectively, we have So, from the Theorems 3.1 to 3.2, we can conclude that y < 0 for shock waves and y ≥ 0 for simple waves, respectively. Similarly, for 3-family, we consider Hence, from (3.6) to (3.8) for a shock wave and a simple waves of
Solution strategy
It is useful to express the above one-parameter families in general notation, thus let We consider transformations where the left state and the right state of j-shock wave or j-simple wave are connected through and for the contact discontinuity it is connected through . Using (4.1)–(4.3), it is defined by Now, we are going to
Interaction of weak shock waves
Now we define the initial function with two jump discontinuities, at x1 and x2, as follows: with an appropriate choice of Vm and Vr in terms of Vl and arbitrary x1 and . Hence, we have two Riemann problems locally with the above initial data. An elementary wave of first Riemann problem may interact with an elementary wave of second Riemann problem and at the time of interaction a new Riemann problem is formed. Here we consider the interaction of two
Results and conclusion
The Riemann problem for an isothermal compressible gas-liquid drift-flux model of two phase flows is considered. One parameter family of elementary wave curves of solution of the Riemann problem are derived explicitly for different compressibility factors. The liquid density and velocity profiles for 1-family, and gas density and velocity profile for 2-family are shown in Fig. 1. It is noticed that the velocity profile of 1-shock wave is concave and increasing straight line for 1-simple wave of
Acknowledgments
We would like to thank the anonymous referees for their valuable comments and suggestions to improve presentation of the manuscript. Research support from Science and Engineering Research Board, Department of Science and Technology, Government of India (Ref No: SB/FTP/MS-047/2013) is gratefully acknowledged by second author (TRS).
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