Conservative Models and Numerical Methods for ...

J Sci Comput DOI 10.1007/s10915-009-9316-y

Conservative Models and Numerical Methods for Compressible Two-Phase Flow Evgeniy Romenski · Dimitris Drikakis · Eleuterio Toro

Received: 10 January 2008 / Revised: 8 October 2008 / Accepted: 7 July 2009 © Springer Science+Business Media, LLC 2009

Abstract The paper presents the computational framework for solving hyperbolic models for compressible two-phase flow by finite volume methods. A hierarchy of two-phase flow systems of conservation-form equations is formulated, including a general model with different phase velocities, pressures and temperatures; a simplified single temperature model with equal phase temperatures; and an isentropic model. The solution of the governing equations is obtained by the MUSCL-Hancock method in conjunction with the GFORCE and GMUSTA fluxes. Numerical results are presented for the water faucet test case, the Riemann problem with a sonic point and the water-air shock tube test case. The effect of the pressure relaxation rate on the numerical results is also investigated. Keywords Hyperbolic conservation laws · Compressible two-phase flow · Finite volume method

1 Introduction Modelling of multiphase compressible flows is one of the most challenging areas of mechanics of continuous media and computational fluid dynamics. Even for the simplest two-phase E. Romenski () Present address: Aerospace Science Department, Fluid Mechanics & Computational Science Group, Cranfield University, Cranfield, MK43 0AL, UK e-mail: [email protected] E. Romenski Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia D. Drikakis Aerospace Science Department, Fluid Mechanics & Computational Science Group, Cranfield University, Cranfield, MK43 0AL, UK e-mail: [email protected] E. Toro Laboratory of Applied Mathematics, Faculty of Engineering, University of Trento, Trento 38050, Italy e-mail: [email protected]

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flow problems there is no a widely accepted formulation for the governing equations, which can form the basis of a complete mathematical theory of initial-boundary value problems and appropriate numerical methods for two- and multiphase flows. Recent developments in multiphase flow modelling are discussed in [2, 4, 11, 16, 23, 24] and references therein. The standard approach to derivation of governing equations for two-phase flow is based on the averaging procedure, which leads to the system of governing equation in the form of mass, momentum and energy balance laws for each phase coupled with phase interaction algebraic and differential source terms. Here we discuss the so-called two-pressure models for two-phase compressible flow, in which pressures of constituents of the mixture are different. The first two-pressure model was proposed in the paper of Baer and Nunziato [3]. Later several modifications of the Baer-Nunziato (B-N) model have been studied in the literature. Note that if the governing equations for specific model of continuum mechanics are hyperbolic and all equations have a conservation-law form, then this model is more advantageous to use than other models. These two features allow one to apply known mathematical means and accurate numerical methods to study different problems of practical interest. The governing equations of the B-N-type models have real eigenvalues and linearly independent set of eigenvectors, therefore they are hyperbolic. But the certain disadvantage of the BN-type model is that not all equations of the system can be written in the conservation-law form. Due to this peculiarity a difficulties arise in the definition of the discontinuous solution and in the development of very high order numerical methods. In this paper another, phenomenological approach, based on the theory of thermodynamically compatible systems [13, 14, 20] is applied to design the governing equations for two-phase compressible flow. In [21] and [22] the isentropic model and the model with different phase temperatures respectively have been proposed for two-phase flow with different pressures. The governing equations for both models are formulated in a conservation-law form and can be transformed to a symmetric hyperbolic form. In [22] it is proved that the equations of conservative models can be transformed to the form of the B-N-type model, which comprises a pairs of interacting phase balance laws for mass, momentum and energy. It turns out that the governing equations for conservative model are similar to the equations of the B-N-type model, but additional differential source terms appear in the momentum equations, which are not included in the B-N-type models [22]. These source terms describe the so-called lift forces which are an essential feature of many kinds of two-phase flow such as particulate flow. The goal of this paper is to formulate a hierarchy of one-dimensional conservative models and select a robust numerical method, which can be applied to various regimes of flow described by the proposed governing equations. The most general model in the hierarchy represents a model with different phase velocities, pressures and temperatures. The governing equations of the model are quite complex and for some kinds of flow, for which an additional assumptions allowing to reduce the set of state parameters can be made, a simplified models are presented. These are a simplified model with equal phase temperatures; and an isentropic model. The three dimensional governing equations for flows with different phase velocities, pressures and temperatures, and for isentropic flow, have been presented in [21, 22]. Here, a new one-dimensional model with equal phase temperatures is proposed, which can be used for flows featuring high rate of phase temperature equalizing. Emphasize again that the main advantage of the presented hyperbolic models is that their governing equations are written in conservation-law form, which is essential for the development of high accuracy numerical methods. The study of discontinuous solutions requires additional careful consideration. For solving the proposed governing equations a finite volume numerical scheme based on the MUSCL-Hancock method with linear reconstruction [25] is developed. The challenge

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here is the solution of the Riemann problem for multiphase media. The development of exact or approximate Riemann solvers for the two-phase models in question is a difficult task due to the high complexity of the governing equations. Only for the simplified isentropic model the Riemann problem can be solved explicitly [21]. The inclusion of more complex models, even for linearized equations, requires the application of numerical linear algebra algorithms. An alternative way to find a solution to the Riemann problem is to apply a centred method for computing the fluxes. Here, the recently proposed GFORCE and GMUSTA fluxes [27] are applied for computing the advective fluxes. These fluxes are implemented in conjunction with the second-order MUSCL-Hancock method. The test problems studied here (some of them are considered in the literature) include the water faucet problem [2, 23], a Riemann problem with a sonic point and a water/air shock tube problem [23]. The numerical results obtained by different methods are compared with known exact solutions. Using the sonic test case, it is shown that the effects of the pressure relaxation rate can become significant. The developed numerical methods admit a straightforward generalization to the multidimensional two-phase problems. The rest of the paper is organized as follows. In Sect. 2 the derivation of conservationform governing equations for two-phase flows with the use of non-equilibrium thermodynamics and thermodynamically compatible system formalism [14, 20], is presented. In Sect. 3 the hierarchy of one-dimensional two-phase governing equations for flows with different phase pressures and temperatures; for flows with different phase pressures and equal temperatures; and for isentropic flow with different phase pressures, are also presented. A brief description of the GFORCE and GMUSTA fluxes is given in Sect. 4. Sections 5 and 6 present the results of the numerical studies and a concluding overview, respectively.

2 Derivation of Two-Phase Conservative Model In this section a method of derivation of governing equations for multiphase compressible flow based on the thermodynamically compatible systems theory is presented. A theory of thermodynamically compatible hyperbolic systems of conservation laws has been developed in [13–15, 19, 20]. This is based on a generalization of the theory of hyperbolic systems of conservation laws with convex extension [9, 10, 12]. The theory provides a mathematical framework to classify systems of conservation laws in mathematical physics and continuum mechanics. Additionally, thermodynamically compatible systems allow to formulate new governing equations for complicated media and processes. Every system of partial differential (conservative-form) equations is written in terms of generating variables and potentials, and can be reduced to a symmetric hyperbolic system. The derivation of this special form is based on the existence of an additional energy conservation law, which represents the first law of thermodynamics for the given system. These properties allow the implementation of known mathematical techniques and advanced high accurate numerical methods for solving initial-boundary value problems of practical interest. There are several key steps in the design of conservative governing equations for a continuous medium: (i) introduce a set of physical variables characterizing the medium; (ii) define a set of thermodynamically compatible conservation laws written in terms of generating potential and variables, which are used to construct the system of governing equations (note that this can be done in different ways); (iii) establish a relationship between the generating potential and variables with the physical variables and thermodynamic potential; (iv) introduce source terms in the system of governing equations. The derivation of the two-phase flow equations is presented in this section.

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2.1 Parameters of State for Two-Phase Flow Two-phase compressible flow is a flow of the mixture in which each phase can be characterized by its own parameters of state. The phase volumes in the mixture are characterized by volume fractions α1 , α2 , which satisfy the saturation constraint α1 + α2 = 1. Phase mass densities ρ1 , ρ2 and velocities uk1 , uk2 (k = 1, 2, 3) can be taken as parameters of state for mass transfer. For the description of heat transfer we choose the so-called hyperbolic heat conduction Cattaneo law [17], from which the classical Fourier heat conduction law can be derived as a relaxation limit. The common formulation of the Cattaneo law for incompressible steady medium reads as ρ

∂E ∂J k + = 0, ∂t ∂xk

τ

∂J k ∂T +κ = −J k , ∂t ∂xk

where T is the temperature, E(T ) is the specific internal energy, J k is the heat flux, κ is the thermal conductivity coefficient, τ is the heat flux relaxation time. The Fourier law J k = −κ ∂T /∂xk can be derived as a zero relaxation limit at τ → 0. We introduce a new variable j k , called the thermal impulse, and the governing equation for this variable is postulated as: ρ

∂T ∂j k ρj k + . =− ∂t ∂xk τ

It is easy to prove that if κ and τ are constant, then equations for j k and J k are equivalent if jk =

τ k J . ρκ

Finally, the thermal impulse j k and specific entropy s connected with the temperature by the first law of thermodynamics dE = T ds can be taken as a parameters of state for the description of heat transfer. In case of two-phase flow we choose phase specific entropies s1 , s2 and phase thermal impulses j1k , j2k as a parameters of state. More details concerning an implementation of thermal impulse in two-phase flow model can be found in [22]. Thus the physical parameters of state for two-phase compressible flow are: αi , ρi , uki , si , jik , where i = 1, 2 is the phase number, αi is the volume fraction of i-th phase (α1 + α2 = 1), ρi is the mass density of i-th phase, uki (k = 1, 2, 3) is the velocity vector of i-th phase, si is the specific entropy of i-th phase, jik (k = 1, 2, 3) is the thermal impulse of i-th phase. In fact, the thermodynamically compatible system can be formulated in terms of mixture parameters of state, which are connected with individual phase parameters by: α = α1 ,

ρ = α1 ρ1 + α2 ρ2 ,

uk = c1 uk1 + c1 uk2 ,

α1 ρ1 , ρ Si = ci si , jik .

c1 = c =

wk = uk1 − uk2 ,

(1)

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In the above equations, ρ is the density of the mixture, ci is the mass fraction of i-th phase, uk is the mixture velocity, wk is the relative velocity, and Si is the partial entropy of i-th phase. Finally, the thermodynamic potential is formulated in the form of specific internal energy, E, which is subsequently a function of α, ρ, c, Si , wk , jkn . 2.2 Thermodynamically Compatible System Generating a Two-Phase Flow Model In this section a derivation of governing equations for two-phase flow based on the thermodynamically compatible systems theory [14, 20] is presented. The theory allows the formulation of a hyperbolic system of conservation laws in terms of generating thermodynamic potential and variables. The key idea is to represent the system of hyperbolic conservation laws ∂U ∂Fk (U) + =0 ∂t ∂xk in the form ∂Lq ∂Fk (q, Lq ) + = 0, ∂t ∂xk where q = (q1 , . . . , qn )T is the vector of generating variables, L(q1 , . . . , qn ) is the generating potential, Lq = (Lq1 , . . . , Lqn )T is the vector of partial derivatives of the generating potential with respect to generating variables. The structure of fluxes Fk (q, Lq ) admits an additional conservation law ∂(qi Lqi − L) ∂k (q, Lq ) + =0 ∂t ∂xk associated with the energy conservation law. Finally, the thermodynamically compatible system can be transformed to the symmetric system L qi qj

∂qj ∂qj + Mijk = 0, ∂t ∂xk

where Mijk forms a symmetric matrix depending on q. The latter system is a symmetric hyperbolic one, if L is a convex function. A generalization of the above scheme can be done in case of the system with differential constraints expressed in the form of steady conservation laws. Many hyperbolic systems of conservation laws, including magnetohydrodynamics equations, nonlinear elasticity equations, superfluid liquid equations, etc., are representative of the class of thermodynamically compatible systems. To obtain equations for the specific medium it is necessary to identify generating variables with physical parameters of state and define the generating potential as a function of these variables. A detailed description of thermodynamically compatible system, including an introduction of dissipative source terms can be found in [14, 20] and references therein. Here we apply the theory to derivation of the governing equations for two-phase flow in the form of hyperbolic system of conservation laws. This derivation follows [22], where the model for the flow with different phase pressures and temperatures is formulated. First, in order to define a structure of fluxes in the governing equations, we formulate the system of conservation laws in which dissipative interphase exchange source terms are not accounted. The set of physical variables for two-phase flow was defined in the end of Sect. 2.1, but for conservative formulation it is more suitable to use the set of conservative variables ρ, ρα, ρui , wk , ρc, ρSk , ρjik .

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The so-called generating variables can be defined as the conjugate variables to the conservative ones with respect to the Legendre transformation of total energy of the mixture as soon as the generating potential L is introduced with the use of thermodynamic identities   ui u i = q1 dρ + q2 d(ρα) + ui d(ρui ) + zk dwk + nd(ρc) + θk d(ρSk ) + vik d(ρjik ) dρ E + 2 = q1 dLq1 + q2 dLq2 + ui dLui + zk dLzk + ndLn + θk dLθk + vik dLvk i

= d(qi Lqi + u

i

Lui + zk Lzk + nLn + θk Lθk + vik Lvk i

− L).

From the above equalities one can find Lq1 = ρ, Lv k = i

Lq2 = ρα,

q1 = E + ρEρ −

ρjik ,

q2 = E α ,

Lui = ρui ,

n = Ec ,

ui ,

ui ui 2

Lzk = w k ,

Ln = ρc,

Lθk = ρSk ,

− αEα − w Ewk − cEc − jik Ej k − θ k Eθ k , k

i

zk = ρEwk ,

θk = ESk ,

vik = Ej k ,

(2)

i

L = ρ 2 Eρ − ρwk Ewk . The thermodynamically compatible system in terms of generating variables and potential, which is used to formulation of a conservative system for a compressible two-phase flow, is chosen as follows: ∂(uk L)qi ∂Lqi + = 0, ∂t ∂xk

i = 1, 2,

∂[(uk L)ui + zk Lzi − δik zα Lzα ] ∂Lui + = 0, ∂t ∂xk ∂Lzk ∂(uα Lzα + n) + = 0, ∂t ∂xk ∂Ln ∂(uk Ln + zk ) + = 0, ∂t ∂xk

(3)

∂((uk L)θl + vkl ) ∂Lθl + = 0, ∂t ∂xk ∂Lvl i

∂t

+

∂((uk L)vl + θ δik ) i

∂xk

= 0,

l = 1, 2, l = 1, 2,

∂Lzk ∂Lzα − = 0. ∂xα ∂xk Here, L is the generating potential, which is assumed to be a convex function of the generating variables q1 , q2 ; u1 , u2 , u3 ; z1 , z2 , z3 ; n; θ1 , θ2 ; v1l , v2l , v3l (l = 1, 2). The last steady equation is added to the system in order to provide the compatibility of (3). Actually, this steady equation is a consequence of the equation for Lzk . One can prove the above, by differentiating the equation for Lzk with respect to xα and subtract the equation for Lzα after its differentiation with respect to xk , thus obtain   ∂Lzα ∂ ∂Lzk − = 0, ∂t ∂xα ∂xk

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and if at t = 0 ∂Lzk /∂xα − ∂Lzα /∂xk = 0, then this equality holds for every time t > 0. The important properties of (3) are the existence of an additional energy conservation law and the possibility to transform the system to a symmetric hyperbolic one. The energy conservation law can be obtained by multiplying (3) with qi , ui , zk , n, θl , vil , i 2u zk and add them up: ∂ (qi Lqi + ui Lui + zk Lzk + nLn + θl Lθl + vil Lvl − L) i ∂t ∂ + (uk (qi Lqi + ui Lui + nLn + θl Lθl + vil Lvl ) + ul zk Lzl + zk n + θl vkl ) = 0. i ∂xk The symmetric hyperbolic system can be obtained from (3) by adding zk ( i

to the left-hand side of the equation for u , and adding side of the equation for zk , thus obtaining the system

∂Lz ui ( ∂xik

−

∂Lzi ∂xk

∂Lzk ∂xi

−

∂Lzi ∂xk

)=0

) = 0 to the left-hand

∂(uk L)qi ∂Lqi + = 0, ∂t ∂xk ∂Lui ∂(uk L)ui ∂zk ∂zα + + Lzi − Lzα = 0, ∂t ∂xk ∂xk ∂xi ∂zk ∂Ln ∂(uk L)n + + = 0, ∂t ∂xk ∂xk ∂(uk L)zl ∂Lzl ∂n ∂uα ∂uk + + Lzα − Lzl + = 0. ∂t ∂xk ∂xl ∂xk ∂xl ∂(uk L)θl ∂Lθl ∂v l + + k = 0, ∂t ∂xk ∂xk ∂Lvl

k

∂t

+

∂(uk L)vl

k

∂xk

+ δik

∂θl = 0. ∂xk

The above is a symmetric system and if L is a convex function then it is symmetric hyperbolic system. Using relationships between generating variables and physical variables (2) one can obtain the following system, which is written in terms of the mixture parameters of state: ∂ρ ∂ρuk + = 0, ∂t ∂x k ∂ρα ∂ραuk + = 0, ∂t ∂x k ∂ρc ∂(ρcuk + Ewk ) + = 0, ∂t ∂x k ∂(ρui uk + ρ 2 Eρ δki + ρwi Ewk ) ∂ρui + = 0, ∂t ∂x k ∂(ul wl + Ec ) ∂wk + = 0, ∂t ∂x k

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∂(ρj1i uk + ES1 δki ) ∂ρj1i + = 0, ∂t ∂x k ∂(ρj2i uk + ES2 δki ) ∂ρj2i + = 0, ∂t ∂x k k ∂ρS1 ∂(ρS1 u + Ej1k ) + = 0, ∂t ∂x k k ∂ρS2 ∂(ρS2 u + Ej2k ) + = 0. ∂t ∂x k

(4)

The equations of this system are conservation laws of the total mass, first phase volume fraction, first phase mass fraction, total momentum, relative velocity, first and second phase thermal impulses, first and second phase entropies. The above system must be supplemented by the last steady equation of the system (3), which in terms of physical parameters reads as follows: ∂wα ∂wk − = 0. ∂xα ∂xk The latter equation expresses a constraint which is a direct consequence of the choice of the structure of fluxes in the generating thermodynamically compatible system (3). It means that the relative velocity is irrotational, and the phase vorticities are identical during rotation of the element of the mixture. Such kind of flow does not exist in reality and introduction of interfacial friction force in the next section leads to the nonzero vorticity of the relative velocity and as a consequence to the appearance of lift forces in the phase momentum equations [21, 22]. The additional energy conservation law can be derived from the equations of the system (4):  ∂ρ E + ∂t

ul ul 2

 +

  ∂ ρuk E +

ul ul 2

+

p ρ

  + wl Ewl + Ec Ewk + Ej k ESi i

∂x k

= 0.

(5)

It is necessary to note that for the study of discontinuous solutions two equations for phase entropies in (4) should be replaced by two relations which describe Rankine-Hugoniot shock conditions correctly. For the single fluid equations the energy conservation law must be used instead of entropy conservation law. In our case we have only one energy conservation law for the mixture (5). Hence, an extra jump condition (for example for one of the phase energies) is needed for correct formulation of Rankine-Hugoniot relations. 2.3 Source Terms The next step in the development of the model is the introduction of source terms to account for various phase exchange and dissipative processes in the two-phase flow. A few interphase exchange processes leading to dissipation are considered. These are phase pressure equalizing, interfacial friction, heat conductivity and phase temperature equalizing. These processes are modelled by adding relaxation source terms to the corresponding conservation-form equations. Although the physical mechanism of relaxation is different for different processes, the meaning of relaxation is as follows: the parameter characterizing the difference of the current state of the flow from the equilibrium state decays exponentially. The explanation of velocity relaxation (interfacial friction) and temperature relaxation

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can be found in [5]. In the case of dispersed particle flow the velocity relaxation represents the Stokes drag force in the phase momentum equations. The temperature relaxation represents the temperature equalizing through the process of heat transfer between phases. The pressure relaxation represents the phase pressure equalizing through the process of pressure wave propagation in dispersed phase and its interactions with the interface boundaries. Finally, the phase heat flux relaxation represents the process of heat waves, which are close to equilibrium, through relative motion coupled with thermal diffusion. Note that here we introduce source terms by a completely phenomenological procedure. In [1] a natural introduction of pressure and velocity relaxations is presented in the framework of discrete model of two-phase flow. The introduction of a source terms in the system cannot be achieved using the existing theories [6], which make use of dissipation potential and relations between thermodynamic parameters and thermodynamic forces. This is due to the fact that individual phase entropies are parameters of state and that the aforementioned dissipative processes lead to entropy production for each phase. Consequently, two nonlinear source terms, which are not thermodynamic forces, appear in the system, posing the following requirements: • • • •

The energy conservation law is satisfied. The total mixture entropy production is non-negative. The partial phase entropy production is non-negative. The Onsager’s principle holds; that is a coefficient matrix of the thermodynamic forces vector is symmetric [6].

Below, we present the dissipative processes with regards to phase pressures equalizing, interfacial friction, heat conductivity and phase temperatures equalizing using the following nomenclature: φ: phase pressure relaxation source term, λ0k , k = 1, 2, 3: interfacial friction source terms, λki , k = 1, 2, 3; i = 1, 2: heat flux relaxation source terms, π1 , π2 : temperature relaxation source terms, 1 , 2 : individual phase entropy production source terms. Taking into account the above, the two-phase conservative governing equations with dissipation are given by: ∂ρ ∂ρuk + = 0, ∂t ∂x k ∂ρα ∂ραuk + = −φ, ∂t ∂x k ∂ρc ∂(ρcuk + Ewk ) + = 0, ∂t ∂x k ∂(ρui uk + pδ ik + ρwi Ewk ) ∂ρui + = 0, ∂t ∂x k ∂(ul wl + Ec ) 1 ∂wk + = −eklj ul ωj − λk0 , k ∂t ∂x ρ ∂(ρj1i uk + ES1 δ ik ) ∂ρj1i + = −λi1 , ∂t ∂x k ∂ρj2i ∂(ρj2i uk + ES2 δ ik ) + = −λi2 , ∂t ∂x k

J Sci Comput k ∂ρS1 ∂(ρS1 u + Ej1k ) + = 1 − π1 , ∂t ∂x k k ∂ρS2 ∂(ρS2 u + Ej2k ) + = 2 − π2 , ∂t ∂x k ∂wi ∂wk − = −ekiα ωα . ∂xi ∂xk

(6)

In the last steady equation of the above system an additional artificial variables ω1 , ω2 , ω3 (multiplied by the unit pseudoscalar ekiα ) are introduced as it is done in [22] in order to save a conservation-law form of the equation for the relative velocity. These variables can be treated as a source terms in the equation for the relative velocity because of the following additional compatibility equation [22]: ∂(ul ωk − uk ωl + λk0 /ρ) ∂ωk + = 0. ∂t ∂xl The source terms in (6) are: φ=

ρ Eα , τ (p)

π1 =

1 E S1 − E S2 , τ (T ) E S1

λk0 = χ00 Ewk + χ01 Ej k + χ02 Ej k , 1

2

π2 =

1 E S2 − E S1 , τ (T ) E S2

λk1 = χ01 Ewk + χ11 Ej k , 1

λk2 = χ02 Ewk + χ22 Ej k , 2

1 (E k λk + c1 Ewk λk0 + c1 Eα φ) (7) 1 = E S1 j 1 1 2  3  3 2 2   1 χ01 c1 χ01 χ02 ρc1 Eα2 + χ E + − − (E k )2 , E χ = k + k 11 00 w j 1 ES1 τ (p) ES1 χ11 E S1 χ11 χ22 k=1 w k=1 1 (E k λk + c2 Ewk λk0 + c2 Eα φ) E S2 j 2 2 2  3  3 2 2   1 χ02 c2 χ01 χ02 ρc2 Eα2 + χ + E + − − (Ewk )2 . = E χ k k 22 00 w j2 ES2 τ (p) ES2 χ E χ χ 22 S 11 22 2 k=1 k=1

2 =

The individual phase entropy productions 1 , 2 are non-negative if the coefficients   χ2 χ2 χ00 − 01 − 02 , χ11 , χ22 , τ (p) χ11 χ22 are non-negative. In [22] the coefficients χij are defined as χ00 = ζ + χ11 =

2 χ01 χ2 + 02 , χ11 χ22

E S1 , c1 κ1

χ22 =

χ01 = −

S1 E S1 , c12 κ1

χ02 =

S2 E S2 , c22 κ2

E S2 , c2 κ2

where ζ is an interfacial friction coefficient (drag coefficient), and κ1 , κ2 are the coefficients of thermal conductivity for the two phases, respectively. The coefficients τ (p) and τ (T ) correspond to the rate of phase pressures and temperatures relaxation, respectively.

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The total entropy production for the mixture is equal to  = 1 + 2 +

1 (ES1 − ES2 )2 ≥ 0. E S1 E S2

τ (T )

The above choice of source terms does not affect the total energy conservation law (5); there is also a symmetry in the source terms coefficients, λki . Thus, all requirements that were set as constraints for the choice of source terms are satisfied. 2.4 Equation of State for the Mixture The last step in the model development is to define an equation of state for the mixture using the known equations of state of the constituents. According to [22], the mixture equation of state (internal energy) is obtained by adding the internal energy e and kinetic energy of relative motion E(ρ, α, c, S1 , S2 , j1k , j2k , wk ) = e(ρ, α, c, S1 , S2 , j1k , j2k ) + c(1 − c)

wk wk , 2

(8)

where e is defined as an average of phase specific internal energies e(ρ, α, c, S1 , S2 , j1k , j2k ) = c1 e1 (ρ1 , s1 , j1k ) + c2 e2 (ρ2 , s2 , j2k ). The above definition of internal energy allows to compute all derivatives of E by means of derivatives of e1 , e2 with respect to their arguments (see [22]): Eρ =

α1 p1 + α2 p2 , ρ2

E S 1 = T1 ,

E S 2 = T2 ,

Eα =

p2 − p1 , ρ

Ej k = c1 1

Ewk = c(1 − c)wk ,

∂e1 , ∂j1k

Ec = μ1 − μ2 + (1 − 2c) Ej k = c2 2

wk wk , 2

∂e2 , ∂j2k

(9)

where μ1 = e1 + p1 /ρ1 − s1 T1 , ∂e1 p1 = ρ12 , ∂ρ1

μ2 = e2 + p2 /ρ2 − s2 T2 ,

∂e2 p2 = ρ22 , ∂ρ2

T1 =

∂e1 , ∂s1

T2 =

∂e2 ∂s2

are phase chemical potentials, pressures and temperatures accordingly. The system of governing equations can be obtained in terms of individual phase parameters of state by using the above formulae together with the relations for mixture parameters and phase parameters of state (1). The one-dimensional counterpart of the 3D system is presented in the next section.

3 One-Dimensional Conservative Equations for Compressible Two-Phase Flow In this section the three one-dimensional models for compressible two-phase flow are presented. These are derived by simplifying the general three-dimensional model described in Sect. 2:

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• A one-dimensional model for two-phase flow with different phase velocities, pressures and temperatures. • A single temperature simplified version of the above, which is obtained using the assumption that the phase temperatures are equal, but velocities and pressures are different. • An isentropic reduced model, in which thermal effects are not taken into account [21]. 3.1 The One-Dimensional System of Balance Laws for the Flow with Different Pressures and Temperatures The one-dimensional system of governing equations presented below is a direct consequence of the general system (6). The flow of two-phase mixture along the x-axis is considered assuming that the mixture is a continuum and the state of each phase denoted by the index, i = 1, 2, is characterized by its volume fraction αi , mass density ρi , velocity ui , specific entropy si and thermal impulse ji . For the volume fractions the saturation constraint α1 + α2 = 1 remains valid. Furthermore, the parameters of state for the mixture are introduced: ρ = α1 ρ1 + α2 ρ2 ,

u = c1 u1 + c2 u2 ,

S = c1 s1 + c2 s2 ,

w = u1 − u2 ,

(10)

where ρ is the mixture density, u is the mixture velocity, S is the mixture entropy, ci = (αi ρi )/ρ is the phase mass fraction (c1 + c2 = 1), and w is the relative velocity of phases. The proposed modelling approach concerns two-phase flows, where the constituents of the mixture have different velocities, pressures and temperatures. Dissipative nonequilibrium processes such as the pressure relaxation, interfacial friction, thermal conductivity and phase temperatures equalizing are included as source terms in the governing equations. The one-dimensional system derived from the general three-dimensional system (6) reads as follows: ∂ρ ∂ρu + = 0, ∂t ∂x ∂ρα1 ∂ρα1 u + = −φ, ∂t ∂x ∂α1 ρ1 ∂α1 ρ1 u1 + = 0, ∂t ∂x ∂(α1 ρ1 u1 + α2 ρ2 u2 ) ∂(α1 ρ1 u21 + α2 ρ2 u22 + α1 p1 + α2 p2 ) + = 0, ∂t ∂x ∂(u1 − u2 ) ∂(u21 /2 − u22 /2 + μ1 − μ2 ) 1 + = − λ0 , ∂t ∂x ρ ∂ρj1 ∂(ρj1 u + T1 ) + = −λ1 , ∂t ∂x ∂ρj2 ∂(ρj2 u + T2 ) + = −λ2 , ∂t ∂x ∂α1 ρ1 s1 ∂(α1 ρ1 s1 u + c1 ∂e1 /∂j1 ) + = 1 − π1 , ∂t ∂x ∂α2 ρ2 s2 ∂(α2 ρ2 s2 u + c2 ∂e2 /∂j2 ) + = 2 − π2 . ∂t ∂x

(11)

J Sci Comput

The above equations are the mixture’s mass conservation law; balance law for the volume fraction of first phase; mass conservation law for the first phase; total mixture momentum conservation law; relative velocity balance law; thermal impulse balance laws for both phases; and entropy balance laws for both phases, respectively. The source terms λ0 , λ1 , λ2 , π1 , π2 , 1 and 2 can be defined by simplifying the general source terms (7). The equations of state (specific internal energy) ei are considered to be known functions of density, entropy and thermal impulse: ei = ei0 (ρi , si ) + Ai

ji2 . 2

(12)

Here, ei0 (ρi , si ) is a known specific internal energy for each phase; Ai is a constant that can be defined for each phase using the thermal conductivity coefficient; and the heat-flux relaxation time τi (see [22] for more details) is given by: Ai = (ρi κi )/(τi Ti ). If the phase equations of state are known, then the phase pressures pi , chemical potentials μi and temperatures Ti are defined as follows: pi = ρi2

∂ei , ∂ρi

μi = ei +

pi − s i Ti , ρi

Ti =

∂ei . ∂si

The source terms in (11) are a consequence of (7) and are defined by the following relations: φ=

1 (p2 − p1 ), τ (p)

λ0 = χ00 c1 c2 (u1 − u2 ) + χ01 c1

∂e1 ∂e2 + χ02 c2 , ∂j1 ∂j2

λ1 = χ01 c1 c2 (u1 − u2 ) + χ11 c1

∂e1 , ∂j1

λ2 = χ02 c1 c2 (u1 − u2 ) + χ22 c2

1 T 1 − T2 1 T 2 − T1 , π2 = (T ) , τ (T ) T1 τ T2 2  χ11 ∂e1 χ01 1 = + c1 c2 (u1 − u2 ) c1 T1 ∂j1 χ11  3 2  c c χ2 χ2 + 1 2 χ00 − 01 − 02 (u1 − u2 )2 + T1 χ11 χ22 2  χ22 ∂e2 χ02 2 = + c1 c2 (u1 − u2 ) c2 T2 ∂j2 χ22 2 3  2 2  c1 c2 χ01 χ02 + − χ00 − (u1 − u2 )2 + T2 χ11 χ22

∂e2 , ∂j2

π1 =

c1 (p1 − p2 )2 , T1 ρτ (p)

c2 (p1 − p2 )2 . T2 ρτ (p)

Here, φ stands for the phase pressures relaxation to equilibrium uniform state; λ0 simulates the interfacial friction; λ1 , λ2 are responsible for the phase dissipative heat fluxes; π1 , π2 stand for the temperature exchange between the phases; and 1 and 2 represent entropy productions for the first and second phases, respectively.

J Sci Comput

The coefficients χ00 , χ01 , χ02 , χ11 , χ22 are defined as χ00 = ζ + χ11 =

2 χ01 χ2 + 02 , χ11 χ22

1 , c1 κˆ 1

χ22 =

χ01 = −

ρs1 , c1 κˆ 1

χ02 =

ρs2 , c2 κˆ 2

1 , c2 κˆ 2

where the interfacial friction coefficient ζ and phase normalized coefficients of thermal conductivity κˆ 1 , κˆ 2 must also be defined to close the system (11). Note that the normalized thermal conductivity coefficient κˆ i is connected to the thermal conductivity coefficient κi by Ti κˆ i = κi . For each medium it is also necessary to define the coefficients of pressure relaxation rate τ (p) and temperature relaxation rate τ (T ) . All these parameters can be functions of the parameters of state. Solutions to (11) satisfy the additional conservation energy law, which is a onedimensional analogue of (5) and reads as:         u2 u2 p1 u21 ∂ ∂ + α1 ρ1 e1 + 1 + α2 ρ2 e2 + 2 + α1 ρ1 u1 e1 + ∂t 2 2 ∂x ρ1 2   2 p2 u 2 ∂e1 ∂e2 + + c2 T2 − c1 c2 (u1 − u2 )(s1 T1 − s2 T2 ) = 0. + α2 ρ2 u2 e2 + + c1 T1 ρ2 2 ∂j1 ∂j2 (13) The one-dimensional system of governing equations for two-phase flow consists of equations that have a conservative form. Although it is difficult to analyse the eigenstructure of the above system, it was shown in Sect. 2 that the three-dimensional thermodynamically compatible system from which the two-phase flow equations are obtained, is hyperbolic if the generating potential is a convex function. Therefore, (5) are also hyperbolic. In some two-phase flows simpler models than the one presented above can be used by neglecting the effects of some interfacial exchange processes, that allows to reduce the set of parameters of state. In the following sections two simplified models are derived, namely a model for equal phase temperatures and an isentropic two-phase model. 3.2 Single Temperature Conservative Equations for Two-Phase Flow In this section the model obtained as a simplification of the general model in Sect. 2 assuming that phase temperatures are equal, is presented. Such approximation is suitable if one is interested in describing the processes with characteristic time much greater than the phase temperature relaxation time in (11), e.g. flow with dispersed phase particles where the relaxation time of phase temperatures to an equilibrium value is proportional to the size of particles [5]. Let us assume that the phase temperatures are equal and there is no heat transfer involved. Then, the system of governing equations can be derived from (11) by neglecting the equations for phase thermal impulses j1 , j2 and assuming that thermal effects can be characterized by the mixture entropy s = c1 s1 + c2 s2 only. Because of the equal phase temperatures, one can derive phase entropies as a function of volume fraction, phase densities and mixture entropy: s1 (α1 , ρ1 , ρ2 , s), s2 (α1 , ρ1 , ρ2 , s), by solving the system of equations T1 =

∂e1 (ρ1 , s1 ) = T, ∂s1

T2 =

∂e2 (ρ2 , s2 ) = T, ∂s2

c1 s1 + c2 s2 = s,

(14)

J Sci Comput

where T is the common phase temperature. The simplified system of governing equations can be obtained from (11) by neglecting equations for jk and replacing a pair of equations for s1 , s2 by the energy conservation law (13). Assuming also T1 = T2 we come to ∂ρα1 ∂ρuα1 + = −φ, ∂t ∂x ∂α1 ρ1 ∂α1 ρ1 u1 + = 0, ∂t ∂x ∂ρ ∂ρu + = 0, ∂t ∂x ∂(α1 ρ1 u1 + α2 ρ2 u2 ) ∂(α1 ρ1 u21 + α2 ρ2 u22 + α1 p1 + α2 p2 ) + = 0, ∂t ∂x   2 ∂ ∂ u1 u22 p1 p2 1 (u1 − u2 ) + − + e1 + − e2 − − (s1 − s2 )T = − λ0 , ∂t ∂x 2 2 ρ1 ρ2 ρ         ∂ u2 u2 p1 u21 ∂ + α1 ρ1 e1 + 1 + α2 ρ2 e2 + 2 + α1 ρ1 u1 e1 + ∂t 2 2 ∂x ρ1 2   2 p2 u 2 + + α2 ρ2 u2 e2 + − ρc1 c2 (u1 − u2 )(s1 − s2 )T = 0, ρ2 2

(15)

where the source terms are φ=

1 (p2 − p1 ), τ (p)

λ0 = ζ c1 c2 (u1 − u2 ),

and τ (p) is the pressure relaxation rate, ζ is the interfacial friction coefficient. Here, the total energy conservation law is included into the complete system of governing equations. Suppose that equations of state for each phase ei are known functions of the density and entropy: ei = ei0 (ρi , si ). The phase pressures pi , chemical potentials μi and temperatures Ti are computed as follows: pi = ρi2

∂ei , ∂ρi

μi = ei +

pi − s i Ti , ρi

Ti =

∂ei . ∂si

The assumption of equal phase temperatures (14) yields specific dependencies for the equations of state: ei = ei0 (ρi , si (α1 , ρ1 , ρ2 , s)). One can prove that solutions to the system (15) satisfy the total mixture entropy balance equation ∂ρs ρ∂su + = , ∂t ∂x where the entropy production Q=ζ is non-negative.

(c1 c2 (u1 − u2 ))2 (p1 − p2 )2 + ≥0 T ρτ (p) T

J Sci Comput

3.3 Isentropic One-Dimensional Conservative Equations for Two-Phase Flow A further simplification of the governing equations can be obtained assuming that the temperature variations in the flow are negligible. This implies that the phase entropy variations are negligible, thereby we may consider the flow as an isentropic one. The simplified governing equations can be obtained from (15) assuming s1 = s2 = 0 and neglecting energy conservation law. The resulting system consists of five equations and is obtained from (15) by neglecting the energy conservation law: ∂ρα ∂ραu + = −φ, ∂t ∂x ∂α1 ρ1 ∂α1 ρ1 u1 + = 0, ∂t ∂x ∂ρ ∂ρu + = 0, ∂t ∂x

(16)

∂(α1 ρ1 u1 + α2 ρ2 u2 ) ∂(α1 ρ1 u21 + α2 ρ2 u22 + α1 p1 + α2 p2 ) + = 0, ∂t ∂x ∂(u1 − u2 ) ∂(u21 /2 − u22 /2 + H1 − H2 ) 1 + = − λ0 . ∂t ∂x ρ To close the system an equation of state, ei (ρi ), for each phase is required. The pressure relaxation φ and interfacial friction λ0 are similar to those in previous sections: φ=

1 (p2 − p1 ), τ

λ0 = ζ c1 c2 (u1 − u2 ).

The system (16) is hyperbolic and its eigenstructure has been analysed in [21]. The availability of this analysis makes possible to solve the Riemann problem for the linearized version of (16). It is interesting to note that the one-dimensional two-phase flow equations (16) can be written in the form of the so-called reduced Baer-Nunziato five-equation model [21]. 3.4 Equation of State (EOS) for Water/Air Mixture In this paper, a few test problems for flows of water-air mixture are studied. The equations of state for both isentropic and non-isentropic cases are presented below. The most common form of equation of state in computational fluid dynamics links pressure, energy and density. Here, the phase density and entropy are used as independent parameters of state and the internal energy is taken as the thermodynamic potential. All the other thermodynamic parameters such as pressure and temperature can be defined as functions of density and entropy [8, 28]. For air the perfect gas equation of state is employed in the form    γ −1 s ρ C2 exp , e(ρ, s) = γ (γ − 1) ρ0 cV

(17)

where ρ0 is the reference density, γ is the adiabatic exponent, C is the velocity of sound at normal atmospheric conditions, cV is the specific heat capacity at constant volume. The

J Sci Comput

pressure and temperature are calculated as p=

ρ0 C 2 γ



ρ ρ0



γ exp

s cV

 T=

,

C2 cV γ (γ − 1)



ρ ρ0



γ −1 exp

s cV

 .

(18)

Denoting p 0 the reference (atmospheric) pressure for water the stiffened gas equation of state is written    γ −1 s C2 ρ0 C 2 − γp 0 ρ , (19) exp e(ρ, s) = + γ (γ − 1) ρ0 cV γρ and the pressure and temperature are given by     s ρ0 C 2 ρ γ ρ0 C 2 − p 0 , exp p= − γ ρ0 cV γ    γ −1 s C2 ρ exp T= . cV γ (γ − 1) ρ0 cV

(20)

For the isentropic case, the energy and pressure are defined as e(ρ) =

C2 γ (γ − 1)



ρ ρ0

γ −1 ,

p=

ρ0 C 2 γ



ρ ρ0

γ (21)

for air, and e(ρ) =

C2 γ (γ − 1)



ρ ρ0

γ −1

+

C 2 ρ0 − γp 0 , γρ

p=

ρ0 C 2 γ



ρ ρ0

γ −

ρ0 C 2 − γp 0 (22) γ

for water.

4 Numerical Method, GFORCE and GMUSTA Fluxes for Systems of Conservation Laws In this sections a finite volume numerical method for solving the proposed one-dimensional systems of conservation laws for two-phase compressible flow is developed. The method can be easily generalized for the multidimensional case. The standard routine procedure in the finite volume method for the system of conservation laws is an inter-cell flux evaluation via solution to the Riemann problem. Due to complexity of governing equations the solution of the Riemann problem for two-phase flow model is a difficult task. Even for the linearized equations of the model it is necessary to use a numerical linear algebra algorithms to solve it. That is why we apply the recently proposed GFORCE and GMUSTA Riemann solvers, which are based on the centred difference scheme for flux evaluation [27]. The comparison of these methods is done for several test problems in order to select the most appropriate solver. A brief description of the GFORCE and GMUSTA Riemann solvers [27] for solving the one-dimensional system of conservation laws ∂U ∂F(U) + = 0, ∂t ∂x

J Sci Comput

is given below. A finite volume discretization of the above system in a control volume [xi−1/2 , xi+1/2 ] × [t n , t n+1 ] of dimensions x = xi+1/2 − xi−1/2 , t = t n+1 − t n gives: Un+1 = Uni − i

t (Fi+1/2 − Fi−1/2 ). x

(23)

Uni is an approximation of U to the cell average and Fi+1/2 = F(Ui+1/2 ) is the numerical flux, which is a function of inter-cell boundary value Ui+1/2 . The latter can be obtained as a solution of the local Riemann problem. Here, the MUSCL-Hancock method with linear reconstruction [25] is used providing second-order of accuracy. The reconstruction is obtained by 1 ULi = Uni − i , 2

1 URi = Uni + i , 2

where for i a minmod limited slope [25] is employed  i =

max(0, min(i−1/2 , i+1/2 )), min(0, max(i−1/2 , i+1/2 )),

i+1/2 ≥ 0, i+1/2 < 0.

The boundary extrapolated values are evolved in time by ˆ Li = ULi − 1 t (F (URi ) − F (ULi )), U 2 x ˆ Ri = URi − 1 t (F (URi ) − F (ULi )). U 2 x Finally, the flux Fi+1/2 in (23) is defined by solving the local Riemann problem with initial ˆ R , UR = U ˆL . data UL = U i i+1 Several methods for flux evaluation have been developed [7, 25] and most of them are based on a complete knowledge of the eigenstructure of the system of conservation laws. For systems of equations of continuous media with complicated properties (like the ones considered here) the eigenstructure analysis is a very difficult task. In this case, it is simpler to employ centred methods, such as Lax-Friedrichs, Lax-Wendroff or FORCE [25], for flux evaluation. Centred fluxes are diffusive as well as dependent on the Courant number coefficient. An improved version of the original FORCE flux, known as Generalized FORCE (GFORCE) flux, has been proposed in [27] and is briefly described below. 4.1 GFORCE Flux The FORCE flux FFi+1/2 obtained as an approximate solution to the Riemann problem with initial data UL , UR is constructed as an arithmetic average of the Lax-Friedrichs FLF i+1/2 and LW Lax-Wendroff Fi+1/2 fluxes [7, 25]: 1 + FLW FFi+1/2 (UL , UR ) = (FLF i+1/2 ), 2 i+1/2 where 1 1 x (UR − UL ) FLF i+1/2 = (F(UL ) + F(UR )) − 2 2 t

(24)

J Sci Comput

and 1 1 t (F(UR ) − F(UL )). ULW = (UL + UR ) − (25) 2 2 x An improvement of the FORCE flux eliminating its diffusivity and dependence on the Courant number coefficient is the GFORCE flux [27] which is a convex average of the Lax-Friedrichs and Lax-Wendroff fluxes: FLW i+1/2 = F(ULW ),

LW LF FGF i+1/2 = ωFi+1/2 + (1 − ω)Fi+1/2 .

(26)

Note that the time step which is used for the evaluation fluxes in (24) and (25) is estimated locally from the local initial data UL , UR as t = tloc = Kx/Smax , where Smax is the speed of the fastest wave in the local initial data and K is the local Courant number (K = 0.9 is usually taken). The weight coefficient ω in (26) is given by ω=

1 , 1+K

0 < K ≤ 1.

As noted in [27] the GFORCE flux is upwind due to the nonlinear dependence of the weight ω on the local wave speed. Furthermore, for the linear advection equation with constant coefficient the GFORCE flux reproduces the Godunov upwind flux. 4.2 GMUSTA Flux The main purpose of the MUSTA (MUlti-STAge) flux is to obtain an upwind numerical flux by evolving in time the initial data in the local Riemann problem [26, 27]. Here, the GMUSTA (Generalized MUSTA) flux is briefly described. For each local Riemann problem with initial data UL , UR let us consider an interval with 2M(−M + 1 ≤ m ≤ M) cells with δx size. Note that δx is not connected with the cell size of the original problem and, in principle, one can take δx = 1. The Riemann problem is solved on this mesh with initial data UL , UR and the interface position located between cells m = 0 and m = 1. Thus, the l−stage (l ≤ M) algorithm can be formulated in steps: (1) 1. Set initial data for k = 1 as U(1) m = UL , m ≤ 0, Um = UR , m > 0 k 2. Compute the GFORCE flux Fm+1/2 using (26) and local time step δtloc for intercell boundaries with number m + 1/2. 3. Open the Riemann fan numerically with the same local time step by

= U(k) U(k+1) m m −

δtloc k (F − Fkm−1/2 ) δx m+1/2

(−M + 1 ≤ m ≤ M). 4. Go to step 2. with initial data U(k+1) m (k) At the final step k = l we have a pair of values U(k) 0 , U1 adjacent to the interface position, which provides an approximation to the original Riemann problem flux: (k) (k) GF (U(k) FGM i+1/2 = F1/2 = F 0 , U1 ).

The cell size δx can be taken arbitrarily due to the self-similar structure of the solution to the Riemann problem. The Courant number, KMUSTA , for time marching is prescribed by the user and the local time step δtloc is calculated from the data Ulm using the conventional formula δtloc = KMUSTA δx/Smax , where Smax is the fastest wave speed.

J Sci Comput

5 Numerical Results In this section test problems for water-air mixture flow governed by the equations of Sect. 3 are presented. The GFORCE and GMUSTA fluxes are used in conjunction with the MUSCL-Hancock method [25]. The GMUSTA flux with K time marching steps is labelled as GMUSTA-K. The test problems are solved for different values of pressure relaxation, τ (p) . The two extreme values of τ (p) are ∞ and 0. The value τ (p) = ∞ corresponds to the case where the characteristic time of the process is small compared to the pressure relaxation time, for example, if the average size of dispersed phase particles is large. The value τ (p) = 0 corresponds to flow in which the average size of dispersed phase is negligibly small and the pressure relaxation is instantaneous. The consequence of instantaneous pressure relaxation is the equality of phase pressures throughout flow. In some cases the pressure relaxation rate may be finite but very high thus leading to numerical stiffness of the differential equations. Here, the backward Euler method with respect to τ (p) ∈ [0, ∞] is implemented. For the sake of simplicity the modification of the numerical method for the isentropic model is described. The finite volume numerical approximation of the equation for volume fraction ∂ρα ∂ραu p1 − p2 + = ∂t ∂x τ (p) reads as (ρα)i(n+1) = (ρα)(n) i −

t t ((ρα)i+1/2 − (ρα)i−1/2 ) + (p) (p1 ((ρ1 )(n+1) ) − p2 ((ρ2 )(n+1) )), i i x τ

where (ρα)i+1/2 , (ρα)i−1/2 are computed through the flux evaluation. Assuming that the valare known from the numerical integration of mass conservation ues (α1 ρ1 )i(n+1) , (α2 ρ2 )(n+1) i laws and taking into account that = (ρ1 )(n+1) i

(α1 ρ1 )(n+1) i (α1 )i(n+1)

,

(ρ2 )i(n+1) =

(α2 ρ2 )i(n+1) 1 − (α1 )(n+1) i

,

the latter (difference) equation is obtained, which is a nonlinear algebraic equation for (α)(n+1) , and can be solved by iterative methods. A similar algorithm can be implemented i for the pressure relaxation in the single temperature model as well as for the relative velocity relaxation. 5.1 Water Faucet Test Problem In this section the so-called water faucet test problem is presented, which is a classical test case in two-phase flow modelling [2, 18, 23]. The case consists of the flow of an initially homogeneous water column in a gas annulus in a tube which accelerates under the effect of gravity and as a result the water column is getting thinner (Fig. 1). In the governing equations the gravitational acceleration term is included as a source term in the right hand side of the mixture momentum equation: f = −ρg, where g = 9.81 m/s2 . We consider a tube of 12 m length and uniform flow at initial time t = 0 with pressure p10 = p20 = 105 Pa and water and air volume fractions α10 = 0.8 and α20 = 0.2, respectively. The water and air velocities are u01 = 10 m/s and u02 = 0 m/s, respectively. As boundary conditions, at the tube inlet the unchanged air volume fraction α2 = 0.2 and water and gas

J Sci Comput Fig. 1 Evolution in time of water-air flow for the Ransom test problem

velocities u1 = 10 m/s, u2 = 0 m/s are posed. At the tube outlet the pressure is set as p = 105 Pa and assumed to remain unchanged. The analytical solution to the problem is derived assuming water as an incompressible medium and neglecting pressure gradients [18]: α2 (t, x) =

⎧ ⎨1 − √ ⎩

1 − α01

α10 u01 (u01 )2 +2gx

if x