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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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01 Aug 2016
Abstract: Despite the fact that solid-liquid mixing plays a key role in the production of a wide variety of consumer goods such as pastes, paints, cosmetics, propellants, pharmaceuticals, and food products as well as in the operation of chemical reactors with solid catalysts, it still faces considerable challenges. Most research on solid-liquid mixing has focused on the fully turbulent regime of operation even though many industrial operations take place in the laminar or transitional regime. In particular, it is unclear how the rheology of a suspension, particle interactions, and a complex rotating geometry impact flow patterns and particles distribution and dispersion in these regimes. Although more is known about the turbulent regime of operation, most research on this type of regime has been devoted to the prediction of the just-suspended speed (Njs ), which is the impeller speed at which all particles are suspended in the liquid phase. However, numerous mixing operations require a different state of operation. For these processes, operating at Njs can lead to energy overconsumption, product fouling, or inhomogeneous reactions due to the presence of dead zones. Consequently, more information on the velocity patterns and distribution of particles in agitated vessels is required. To shed light on issues related to solid-liquid mixing, numerical and experimental investigations are essential. However, due to the opacity of most viscous suspensions, local measurements of the flow field using optical techniques are highly problematic. Consequently, almost all experimental measurements have been limited to determining the global characteristics of the mixing flow such as the fraction of suspended particles or the torque acting on the impeller. However, CFD simulations of these systems do not suffer from these drawbacks. A variety of models have been developed to simulate solid-liquid flows. These include the classic Eulerian-Eulerian (or two-fluid) model and the combination of the Discrete Element Method (DEM) for the particles and CFD method for the liquid phase (CFD-DEM). Although it possesses enormous potential due to its formulation, notably as regards to its natural capacity to reproduce the maximal packing fraction of solid particles, the ability of the CFD-DEM approach to accurately model solidliquid flows in complex geometries has not yet been proved. In addition, the method has not been validated experimentally for solid-liquid flows. However, this type of model could theoretically allow for a quantitative assessment of flow patterns, particle distributions, and the fraction of suspended particles. In this thesis, a CFD-DEM model is developed to model the suspension of particles in a stirred tank, from start-up to steady state and in all regimes of operation. The model is

9 citations


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

  • ...of analytical Riemann solvers and wave ordering, as for example in [124], these solutions are limited to the context of inviscid compressible flows....

    [...]

Dissertation
07 Oct 2016
TL;DR: In this paper, a new schema of convection, robuste et efficace, capable of gerer toute equation d'etat, is proposed. But it is not suitable for the simulation of transitoires of eau-vapeur.
Abstract: Cette these traite de la modelisation et de la simulation des ecoulements diphasiques transitoires eau-vapeur. Dans de nombreuses installations industrielles, des ecoulements monophasiques d'eau liquide sont susceptibles de devenir diphasiques lors de certaines situations transitoires. La modelisation de ces ecoulements peut s'averer delicate car deux phenomenes physiques interagissant fortement entre eux, le changement de phase et la propagation d'ondes de pression, sont alors a prendre en compte. Une approche bifluide statistique, ne supposant aucun equilibre entre les phases, est utilisee afin de modeliser de tels ecoulements. Les modeles obtenus sont de type convection-source et s'apparentent au modele de Baer-Nunziato. Differentes lois de fermeture pour ces modeles sont comparees sur des cas experimentaux de transitoires eau-vapeur tels que les coups de belier et la depressurisation d'une tuyauterie d'eau liquide suite a une rupture. La simulation numerique des differents modeles est effectuee grâce a une methode a pas fractionnaires. Un nouveau schema de convection, robuste et efficace, capable de gerer toute equation d'etat est utilise dans la premiere etape de cette methode. La seconde etape est dediee au traitement des termes sources et requiert differents schemas implicites. Une grande attention est accordee a la verification de tous les schemas numeriques utilises grâce a des etudes de convergence. Une nouvelle modelisation du transfert de masse est egalement proposee, sur la base de travaux dedies a la brusque depressurisation d'eau liquide en approche homogene. La validation du modele est effectuee grâce de nombreuses comparaisons calcul-experience.

7 citations


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

  • ...Based on previous work [25], a fractional step method has been used to approximate solutions of the three models....

    [...]

  • ...As in [25], the relaxation time step is divided into different substeps to treat the different relaxation phenomena....

    [...]

  • ...Enfin, des approches par relaxation ont été appliquées au système barotrope [7, 25] puis étendues au système complet par Coquel et Saleh [26, 101]....

    [...]

  • ...Grid convergence studies have been carried out for most of steps of the method in [25] and similar work is done here for the remaining steps of the different models....

    [...]

  • ...The verification of such a scheme can be found in [25]....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a three-phase flow model is presented to represent complex flows involving liquid metal droplets, liquid water, and its vapour, and a fractional step method that computes successively the convective part and the relaxation effects is used to obtain approximate solutions.
Abstract: We focus here on a three-phase flow model in order to represent complex flows involving liquid metal droplets, liquid water, and its vapour. The governing equations and its main properties are given, and focus is given on the pressure–velocity relaxation process on the one hand, and on the structure of solutions of the one-dimensional Riemann problem associated with pure convective effects. A fractional step method that computes successively the convective part and the relaxation effects is used to obtain approximate solutions on unstructured meshes. Details of algorithms are provided, and it is shown that the numerical method preserves positive values of statistical fractions and partial masses. Verification and validation test cases are presented, and some perspectives are eventually drawn.

7 citations

01 Dec 2016
TL;DR: In this article, a comparison of numerical methods for the convective part of a two-phase flow model is performed using Riemann problems or smooth solutions, and the main criteria for comparing the methods are: the accuracy for a fixed size of mesh, the convergence, the robustness and the CPU time cost necessary to reach a fixed error.
Abstract: The present paper is devoted to a comparison of numerical methods for the convective part of a two-phase flow model. Four explicit schemes are tested using Riemann problems or smooth solutions. The main criteria for comparing the methods are: the accuracy for a fixed size of mesh, the convergence, the robustness and the CPU time cost necessary to reach a fixed error. The conclusions of this study are the following. The classical Rusanov scheme is not competitive, as expected. The relaxation scheme seems to be more efficient than the VFROE-ncv method, especially regarding robustness. The staggered scheme, recently investigated for this model, is sometimes less accurate for a fixed size of mesh than the VFROE-ncv and relaxation schemes but is the most acurate for a fixed run time.

6 citations


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

  • ...• A more common way of proceeding is to use a splitting method for the source terms (see [17] for the model considered and [18] for the general framework): for each time iteration, convection and source terms are resolved successively....

    [...]

  • ...This kind of method is very convenient: it only requires a solver for the convective part, for which there are a lot of schemes ([22], [10], [7], [17] which will be tested in this paper, but also [9], [24], [2], [25] this list is not exhaustive), and source terms solvers....

    [...]

  • ...The first one is the classical Rusanov scheme (see [21] for the historical approach, then [17] for the adaptation to the BN model)....

    [...]

Journal ArticleDOI
TL;DR: Some of the main difficulties arising in the numerical simulation of solutions of these complex and highly non-linear systems of PDEs are discussed, and some numerical results when tackling two-phase flows with mass transfer are shown.
Abstract: We give in this paper a short review of some recent achievements within the framework of multiphase flow modeling. We focus first on a class of compressible two-phase flow models, detailing closure laws and their main properties. Next we briefly summarize some attempts to model two-phase flows in a porous region, and also a class of compressible three-phase flow models. Some of the main difficulties arising in the numerical simulation of solutions of these complex and highly non-linear systems of PDEs are then discussed, and we eventually show some numerical results when tackling two-phase flows with mass transfer.

6 citations

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"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....

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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....

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  • ...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....

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