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Journal ArticleDOI

A fractional step method to compute a class of compressible gas–liquid flows

15 Feb 2012-Computers & Fluids (Pergamon)-Vol. 55, pp 57-69
TL;DR: In this article, the authors present some algorithms dedicated to the computation of numerical approximations of a class of two-fluid two-phase flow models and give the main properties of these models.
About: This article is published in Computers & Fluids.The article was published on 2012-02-15 and is currently open access. It has received 48 citations till now. The article focuses on the topics: Relaxation (approximation) & Uniqueness.

Summary (3 min read)

1 Introduction

  • Two distinct types of models are used in order to compute liquid-gas or watervapour two-phase flows in industrial codes: the homogeneous approach and the two-fluid approach.
  • The two-fluid approach is assumed to be more general, and it is also expected to predict more accurately flows for which the phasic desequilibrium plays a crucial role (see [9, 25, 26]).
  • One must be aware that at least two difficulties are hidden in these sets of PDE.
  • The second section gives emphasis on the approximation of velocity-pressure-temperature relaxation effects.

2.1 The two-fluid model

  • Throughout the paper, indexes l, g refer to the liquid and gas phases; the statistical void fractions of gas and liquid are noted classically αg and αl, which should agree with: αl + αg = 1.
  • The so-called conservative variable W will be defined as: W = (αl, αlρl, αlρlUl, αlEl, αgρg, αgρgUg, αgEg) Moreover, PI(W ) and VI(W ) respectively denote in this paper the interfacial pressure and velocity, and will precised afterwards.
  • The term S1,l will also be introduced later on.
  • External sources might be included but are not considered herein.
  • Standard viscous contributions may be included, which of course comply with the entropy inequality that will be detailed in the next subsection.

2.2 Closure laws for interfacial transfer terms

  • The same holds when tackling three-phase flows, as emphasized in [18].
  • Now, the second requirement (H1) implies that the field associated with λ = VI should be linearly degenerate.
  • As shown in [8, 11], few expressions guarantee this behaviour.

2.4 Main properties of the two-fluid model

  • The authors may now recall in brief the main properties of system (2) using the previous closure laws.
  • Apart from the field associated with the eigenvalue λ =.
  • Otherwise the computation of shock solutions would be meaningless, since multiple shock solutions may be obtained using various -stable- solvers (see for examples [17]) .
  • Obviously, an alternative formulation of jump conditions in genuinely non linear fields that is probably more convenient may be: σ = [ρφUφ]RL/[ρφ].

3.2 Computing the evolution step

  • Many solvers have been proposed in the literature for such a purpose.
  • Approximate solutions in the evolution step may also be obtained using either the non-conservative form of Rusanov scheme, or the non-conservative form of the approximate Godunov scheme VFRoe-ncv [6]; the authors refer to [11, 10] for such a description.
  • Obviously, the ultimate scheme has not been proposed yet.
  • The proof is classical but is briefly recalled.
  • The authors detail afterwards the relaxation step, with special focus on the pressure relaxation step which is rather tricky.

3.3 Computing the velocity relaxation step

  • It may be easily checked that internal energies remain positive through this step.
  • The proof is obvious considering formula (16) and is left to the reader.

3.4 Computing the temperature relaxation step

  • The proof provides a practical way to compute solutions of system (18), that is actually used in the code.
  • The authors emphasize that other -simpler- algorithms may be exhibited ; however, practical computations seem to show that the non-linear temperature relaxation scheme described here provides better results (see [24] and section IV).
  • The authors turn now to the most difficult part which corresponds to the pressure relaxation step.

3.5 Computing the pressure relaxation step

  • The first one is a semi-implicit scheme, that is such that the existence and uniqueness of the discrete solution is ensured, whatever the equations of state would be.
  • The authors focus here on the second one which is totally implicit with respect to the unknown (Pl, Pg, αl).
  • The main lines of the proof are given below, also known as Proof.
  • The implicit scheme (20) is exactly the same as the one introduced for dense granular gas-particle flows in [10], also known as Remark 2.

4 Numerical results

  • In the first subsection, the authors focus on the computation of the evolution step involving all convective terms.
  • The following unities are used : m for distances, kg/m3 for densities, m/s for velocities, Pa for pressures, K for temperatures, s for times and J/s/m2 for heating fluxes.

4.1 Verification of the evolution step

  • The authors use for this test case perfect gas EOS within each phase, setting γg = 1.4 and γl = 1.1.
  • The second test case is another Riemann problem taken from [13], where initial conditions are given in table 2.
  • This Riemann problem also contains a contact wave associated with Ug, and a right-going gas shock wave.

4.2.1 Velocity relaxation substep

  • Two different series of verification test cases have been considered in [24] for the velocity relaxation step.
  • The first one refers to a constant time scale τ3.
  • In that case, the scheme (16) is perfect, since it computes the exact value at time tn+1.
  • In that case, analytic solutions allow computing the true error occuring in the velocity relaxation step.
  • Several examples can be found in [24], which confirm that a first-order rate of convergence is achieved.

4.2.2 Temperature relaxation substep

  • The authors present below some tests corresponding to constant time scales τ4, when computing approximate solutions of (17) with the scheme (18).
  • Φ, unlike the non-linear scheme (18); actually, round-off errors are found for the linear scheme in that case.

4.2.3 Pressure relaxation substep

  • Eventually, the authors provide an example of measured convergence rates in the pressure relaxation step.
  • This one is crucial, and should be handled with great care.
  • Otherwise, both the present model (where the relaxation time scale τ2 is non-zero) and standard two-fluid models (corresponding to τ2 = 0) may be confused, if inadequate “rough” schemes are used to provide approximate solutions of (19).
  • The initial conditions of the test case are given in table 5. Figure 6 and Figure 7 show the behaviour of the scheme (20) and the comparison with another half-implicit scheme introduced in [22] and recalled in [24], focusing on void fractions and pressures within each phase at time t = 10−5.
  • This enables to retrieve the fact that the initial-value problem associated with τ2 = 0 is ill-posed: spurious oscillations arise when the mesh size is sufficiently small, and this is particularly spectacular for void fraction profiles, since the algorithm guarantees bounded variations owing to properties 4 and 5 (see [21] and [17] also for a similar study).

4.3 Two-dimensional numerical results

  • The authors consider now the two-dimensional unsteady computation of a heated wall in an almost square domain, where the wall contains a small cavity in the middle of the lower part .
  • On the contrary the two cells at the exit corners of the cavity do not receive any heat flux.
  • It also seems worth mentionning that almost similar results have been obtained while changing the pressure relaxation time step within the range τ2 ∈ [10−9, 10−6], keeping other relaxation time steps unchanged and using the same mesh.
  • Nonetheless, one must be aware that the 3D counterpart of the present 2D mesh would contain more than 150 millions of cells, which is of course far beyond what one can afford in an industrial situation.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, a HLLC-type scheme is presented and implemented in the context of Arbitrary Lagrangian-Eulerian formulation for solving the five-equation models.

47 citations

Journal ArticleDOI
TL;DR: In this article, an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model is proposed, which is able to cope with arbitrarily small values of the statistical phase fractions.
Abstract: We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions

37 citations

Journal ArticleDOI
TL;DR: In this paper, a class of non-equilibrium models for compressible multi-component fluids in multi-dimensions is investigated taking into account viscosity and heat conduction, subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials.
Abstract: A class of non-equilibrium models for compressible multi-component fluids in multi-dimensions is investigated taking into account viscosity and heat conduction. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production, thermodynamical stability, Galilean invariance and mathematical properties such as hyperbolicity, subcharacteristic property and existence of an entropy–entropy flux pair. For the relaxation of chemical potentials, a two-component and a three-component models for vapor–water and gas–water–vapor, respectively, are considered.

37 citations

Journal ArticleDOI
TL;DR: This is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition.

35 citations


Cites background from "A fractional step method to compute..."

  • ...Actually, comparing with Lax-Friedrichs type schemes is quite significant since for such stiff configurations as vanishing phase cases, these schemes are commonly used in the industrial context because of their known robustness [30]....

    [...]

  • ...This is actually the case when, in addition to the convective system (1), zero-th order source terms are added to the model in order to account for relaxation phenomena that tend to bring the two phases towards thermodynamical (T1 = T2 and u1 = u2) and mechanical (p1 = p2) equilibria (see [12, 21] for the models and [30] for adapted numerical methods)....

    [...]

  • ...However, this work is mainly concerned with the convective effects and these relaxation source terms are not considered here (see [12] for some modeling choices of these terms and [30] for their numerical treatment)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a two-fluid two-phase flow model was validated in some highly unsteady situations involving strong rarefaction waves and shocks in water-vapor flows.

32 citations

References
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01 Jan 1959
TL;DR: In this paper, the authors proposed a method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities.
Abstract: The method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer proposed to use, for the solution of fluid dynamics equations, difference equations into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over several mesh points. Then, it was proposed to proceed with the computations across the shock waves in the ordinary manner. In 1954, Lax published the "triangle'' scheme suitable for computation across the shock" waves. A deficiency of this scheme is that it does not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then transforms any initial data into linear functions. In addition, this scheme smears out contact discontinuities. The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics. Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of characteristics. The agreement of results was fully satisfactory.

1,742 citations

Book
29 Nov 2005
TL;DR: In this article, two-phase field equations based on time average are proposed. But they do not consider the effect of structural materials in a control volume on the two-fluid model.
Abstract: Part I Fundamental of two-phase flow.- Introduction.- Local Instant Formulation.- Part II Two-phase field equations based on time average.- Basic Relations in Time Average.- Time Averaged Balance Equation.- Connection to Other Statistical Averages.- Part III. Three-dimensional model based on time average.- Kinematics of Averaged Fields.- Interfacial Transport.- Two-fluid Model.- Interfacial Area Transport.- Constitutive Modeling of Interfacial Area Transport.- Hydrodynamic Constitutive Relations for Interfacial Transfer.- Drift Flux Model.- Part IV: One-dimensional model based on time average.- One-dimensional Drift-flux Model.- One-dimensional Two-fluid Model.- Two-Fluid Model Considering Structural Materials in a Control Volume.

1,289 citations

Journal ArticleDOI
TL;DR: In this article, a two-phase mixture theory is presented which describes the deflagration-to-detonation transition (DDT) in reactive granular materials, based on the continuum theory of mixtures formulated to include the compressibility of all phases and the compaction behavior of the granular material.

1,155 citations


"A fractional step method to compute..." refers background in this paper

  • ...2 Closure laws for interfacial transfer terms The interfacial transfer contributions have been studied in [4, 5, 17, 10, 28, 29], and a summary can be found in appendix A....

    [...]

  • ...Among other studies, one should at least point out the contributions [4, 28, 5, 27, 32] which are concerned with the modeling aspects....

    [...]

Journal ArticleDOI
TL;DR: A new model and a solution method for two-phase compressible flows is proposed that provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.

906 citations


"A fractional step method to compute..." refers background in this paper

  • ...with “zero time scales”) one can refer to [34] (which deals with instantaneous velocity and pressure relaxation) or [21] (which only deals with instantaneous pressurerelaxation) among others....

    [...]

  • ...Fewer papers tackle the problem of water-gas or water-vapour flows, among which we must quote [33, 34] and more recently the article [12] , that examines a medium of small oscillating bubbles in a liquid medium, and also provides a general formalism in order to derive meaningful governing equations....

    [...]