Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition

doi:10.1016/j.compfluid.2014.02.008

Computers & Fluids

Volume 94, 1 May 2014, Pages 112-138

https://doi.org/10.1016/j.compfluid.2014.02.008 Get rights and content

Highlights

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The ALE formulation for the computation of the five-equation models is presented.
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A second-order time-accurate geometrically conservative HLLC-type solver is proposed.
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Assessment on 1-D and 2-D applications including moving boundaries is performed.
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The heat and mass transfer modeling is computed using a fractional step approach.
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Good agreement with experiments is obtained for flashing/condensation phenomena.

Abstract

Computation of compressible two-phase flows with single-pressure single-velocity two-phase models in conjunction with the moving grid approach is discussed 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. In addition, the extension to multicomponent cases is also examined. The method is first assessed on a variety of Riemann problems including both fixed and moving grids applications showing its simplicity and robustness. The method is also tested on 2-D moving mesh applications including fluid–structure interactions. The heat and mass transfer modeling is finally examined for two-phase mixtures. Computations using a fractional step approach of water hammer and fast depressurization with flashing are performed. Good agreement is obtained with available experimental data. All computations are performed with the Europlexus fast transient dynamics software.

Introduction

Two-phase compressible flow phenomena involving moving geometries undergoing deformations or displacements induced by the pressure forces can be found in a variety of engineering applications in industrial processes including water hammer flows or pipe break in circuits of power plants. In addition, the numerical simulation of those kinds of phenomena is a challenging research topic. Thus, development in efficient numerical algorithms adapted to such problems is of interest for both researchers and engineers. Two major classes of model are used to represent two-phase flows: the two-fluid class and the homogeneous one. The two-fluid approach is the most complete as it represents the dynamics of the quantities mass, momentum, energy and statistical void fraction with the consideration of pressure–velocity-temperature-Gibbs free energy relaxation. One of the best known two-fluid model is the one proposed by Baer and Nunziato [1]. The computation of the two-fluid model is also known to be a real challenge [2], [3], [4], [5], [6], [7], [8], [9] in particular due to the corresponding large number of waves. As an alternative way, simpler reduced models have been proposed and applied with success ranging from the three-equation models to the five-equation models [10]. An important class of reduced models is composed by the five-equation models [11], [12], [13], in which pressure equilibrium and velocity equilibrium are assumed. In this approach the two-phase fluid is described as a mixture or as a two-fluid medium and thus, can be treated as a single fluid. The five-equation models contain four balance equations for conservative quantities: two for mass, one for momentum and one for energy. The last equation describes the evolution of a non-conservative quantity: the statistical void fraction in [11], [13] for mixture problems or the so-called “color function” in [12] for interface problems.

In order to deal with fluid dynamics problems with moving meshes and boundaries, several approaches have been proposed. The Arbitrary Lagrangian–Eulerian (ALE) formulation introduced by Hirt et al. [14] is based on the description of the flow field on a moving frame of reference which is attached to the moving surface(s). The ALE scheme is actively used in the Finite-Volume community for compressible aerodynamic applications, see for instance [15], [16], [17], [18] and the references therein. For suitable computation on moving/deforming meshes, the time integration should preserve the stability and accuracy of its fixed mesh counterpart. For example, the mesh motion should not deteriorate the preservation of uniform flow. The influence of the Discrete Geometric Conservation Law (D-GCL) has been identified in terms of accuracy and stability [15], [16], [17], [18]. In the ALE Finite-Volume approach, the mesh velocity appears in the conservative convective flux term of the formulation. Thus, the ALE extension of classical approximate Godunov-type methods can be done easily for conservative equations as the compressible Euler system [19]. The ALE extension of the compressible single-pressure, single-velocity two-phase flow models characterized by the presence of non-conservative terms as in the case of the Baer-Nunziato model [20] is now examined in details.

The first Section is devoted to the brief presentation of the compressible single-pressure single-velocity two-phase flow models. These models are also extended to the multicomponent case. The second Section concerns the ALE formulation of the five-equation models. The spatial discretisation is based on a HLLC-type solver which is detailed on unstructured grids and in the multidimensional framework. As in the monophasic context [17], the time integration is performed in order to respect the D-GCL. This method is then applied in the third Section on several applications ranging from 1-D Riemann problems to 2-D moving mesh applications including fluid–structure interactions. The fourth Section is devoted to the heat and mass transfer modeling in conjunction to the “mixture” five-equation model. Two cases are finally examined: the Simpson’s valve closure induced water hammer and the Canon fast depressurization experiments. Phenomena like flashing, pressure wave propagation in a two-phase mixture are present in the two tests. Comparisons between computational results and experimental data are performed.

Section snippets

Reduced five-equation model for interface problems

We consider the reduced five-equation two-phase flow model of Allaire et al. [12], [21] designed for interface problems: $\{\begin{matrix} \partial_{t} (ρ_{1} z_{1}) + \nabla \cdot (ρ_{1} z_{1} u) = 0 \\ \partial_{t} (ρ_{2} z_{2}) + \nabla \cdot (ρ_{2} z_{2} u) = 0 \\ \partial_{t} (ρ u) + \nabla \cdot (ρ u \otimes u) + \nabla p = 0 \\ \partial_{t} (ρ e) + \nabla \cdot [(ρ e + p) u] = 0 \\ \partial_{t} z_{1} + u \cdot \nabla z_{1} = 0 \end{matrix}$ where $z_{k}$ is the “color function” of the phase k, i.e. an abstract parameter that takes the value 1 in fluid k and 0 otherwise. Thus, the following constraint has to be considered $z_{1} = 1 - z_{2}$ . $ρ_{k}$ is the density of the phase k. $ρ, u, p, e$ are respectively the density, the velocity vector, the

Integral form with moving grids

The Eq. (6) for a moving control volume can be expressed in an integral form as: $\frac{d}{d t} \int_{C (t)} U d V + \int_{\partial C (t)} F^{ALE} (U, v) n d S + \int_{C (t)} B (U) \nabla \cdot u d V = 0$ with the ALE flux-vector defined as $F^{ALE} (U, v) \equiv F (U) - U \otimes v$ where $C (t)$ is the moving control volume, $\partial C (t)$ its boundary whose the mesh velocity is denoted $v$ and the unit outward normal vector $n$ .

When moving/deforming grids are considered, to ensure the conservative property, the following geometric conservation law (GCL) has to be satisfied [17]: $\frac{d}{d t} (| C (t) |) - \int_{\partial C (t)} v \cdot n d S = 0$ This

Numerical tests

Here we assess the robustness and the accuracy of the present HLLC-type Riemann solver. Both first- and second-order approximations are checked on several applications ranging from 1-D shock tube problems to 2-D explosion using both fix and moving grids. 1-D applications are computed by using the first-order accurate method based on the present HLLC-type approximation whereas 2-D computations are performed with the second-order scheme.

Heat and mass transfer modeling

The heat and mass transfer modeling is now examined on the two-phase mixture five-equation model. In this purpose, the single-pressure single-velocity two-phase flow model of Kapila et al. [11], [52] is considered. The hyperbolic part of this PDE system corresponds to Model 2 described in Eq. (3). In conjunction to this model, a modified form of the stiffened gas EOS [53] is used here. The temperature $T_{k}$ and the specific entropy $s_{k}$ of phase k are defined as: $\{\begin{matrix} T_{k} = \frac{p_{k} + π_{k}}{C_{v_{k}} ρ_{k} (γ_{k} - 1)} \\ s_{k} = C_{v_{k}} \ln (\frac{T_{k}^{γ_{k}}}{(p_{k} + π_{k})}) \end{matrix}$

Conclusion

A HLLC-type scheme is presented and implemented in the ALE framework for the computation of the compressible single-pressure single-velocity two-phase models on moving grids. The extension of these five-equation models to multicomponent cases is also examined. Focus is given on the respect of the D-GCL in order to ensure the free-stream preservation as in the monophasic case of the compressible Euler equations. The solver is proposed on unstructured grids in the multidimensional framework. The

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¹: Address: LIMSI-CNRS, Université Paris-Sud, Orsay, France.

²: Address: SOCOTEC, Montigny-le-Bretonneux, France.

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Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition

Highlights

Abstract

Introduction

Section snippets

Reduced five-equation model for interface problems

Integral form with moving grids

Numerical tests

Heat and mass transfer modeling

Conclusion

Int J Multiphase Flow

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J Comput Phys

J Comput Phys

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J Comput Phys

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Comput Methods Appl Mech Eng

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Comput Methods Appl Mech Eng

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Numerical modeling of two-phase flows using the two-fluid two-pressure approach

Math Models Methods Appl Sci