One-pressure Model Research Articles

A modified Rusanov scheme for shallow water equations with topography and two phase flows

In this work, we introduce a finite volume method for numerical simulation of shallow water equations with source terms in one and two space dimensions, and one-pressure model of two-phase flows in one space dimension. The proposed method is composed of two steps. The first, called predictor step, depends on a local parameter allowing to control the numerical diffusion. A strategy based on limiters theory enables to control this parameter. The second step recovers the conservation equation. The scheme can thus be turned to order 1 in the regions where the flow has a strong variation, and order 2 in the regions where the flow is regular. The numerical scheme is applied to several test cases in one and two space dimensions. This scheme demonstrates its well-balanced property, and that it is an efficient and accurate approach for solving shallow water equations with and without source terms, and water faucet problem.

Hyperbolicity Criteria for System of Field Equations of One-Dimensional Two-Phase Flow with Compressible Components

As part of an effort to improve the stability of the RELAP5-3D computer code, a characteristic analysis of the governing differential equations for a compressible, one-dimensional, two-fluid, nonhomogeneous nonequilibrium model is presented. The study is limited to the case when small timescale relaxation terms can be neglected, and therefore, a two-pressure model can be reduced to an equivalent volume-average, one-pressure model. The primary focus of the work is to consider flow with compressible components and to compare hyperbolicity criteria with the results of commonly used limitations of flow with incompressible phases. Based on a review of current achievements in this area, a generic form of momentum conservation equations that are invariant from the definition of differential interfacial terms is suggested. New analytical criteria of strict hyperbolicity of the governing system for the compressible two-phase-flow model are developed and supported by numerical calculations and comparisons. Furthermore, overrestriction of results of eigenvalue analysis based on an incompressible components model is demonstrated.The derived criteria are applied to RELAP5-3D in the form of modifications to momentum equations. Upon implementing the developed criteria, the simulation results show marked improvement in stability without otherwise affecting the calculations. Theimportance of well-posedness of the initial value problem for numerical solution stability is demonstrated.

An advancement in iterative solution schemes for three-dimensional, two-fluid modeling of two-phase flow in PWR fuel bundles

This paper outlines a fully three-dimensional two-fluid one-pressure model with a semi-implicit finite difference scheme coupled with heat conduction which can be applicable to thermal non-equilibrium two-phase flow field in subchannel geometry of Pressurized Water Reactors (PWR). The system of equations was linearized using the Newton–Raphson method and was collapsed into the pressure equations forming a system of the Poisson type. Then, two-phase flow modeling was combined with Krylov methods as advanced computing techniques to investigate the feasibility of implementing preconditioned Krylov subspace solvers as the numerical scheme to solve pressure equations. Six popular Krylov subspace solvers were considered: GMRES, FGMRES, DQGMRES, CGNR, BCG, and TFQMR combined with the block incomplete LU factorization with a dual truncation strategy (BILUT) preconditioner. These proposed iterative solvers were applied to the constructed linear pressure equations in the inner iteration in combination with the outer-Raphson iteration loop. Evaluation was performed in two stages. First, two-fluid numerical scheme capability was evaluated against OECD/NRC NUPEC PWR Bundle tests (PSBT Benchmark). The results for steady-state (PSBT) bundle show that an overall agreement can be found. At the second stage, convergency, stability, and accuracy of the proposed schemes were studied based on PSBT steady-state data through a comparison of utilized Krylov solvers and the direct inversion method as the pressure solution modeling options. It was found that the Krylov solvers do not introduce deviations to the two-phase flow results. Moreover, the convergence history of candidate Krylov solvers demonstrated that GMRES, FGMRES, and DQGMRES have a more efficient and stable convergence performance.

ON THE MODELLING OF TWO-PHASE FLOW IN HORIZONTAL LEGS OF A PWR

This paper aims at presenting the state of the art, the recent progress, and the perspective for the future, in the modelling of two-phase flow in the horizontal legs of a PWR. All phenomena relevant for safety analysis are listed first. The selection of the modelling approach for system codes is then discussed, including the number of fluids or fields, the space and time resolution, and the use of flow regime maps. The classical two-fluid six-equation one-pressure model as it is implemented in the CATHARE code is then presented and its properties are described. It is shown that the axial effects of gravity forces may be correctly taken into account even in the case of change of the cross section area or of the pipe orientation. It is also shown that it can predict both fluvial and torrential flow with a possible hydraulic jump. Since phase stratification plays a dominant role, the Kelvin-Helmholtz instability and the stability of bubbly flow regime are discussed. A transition criterion based on a stability analysis of shallow water waves may be used to predict the Kelvin-Helmholtz instability. Recent experimental data obtained in the METERO test facility are analysed to model the transition from a bubbly to stratified flow regime. Finally, perspectives for further improvement of the modelling are drawn including dynamic modelling of turbulence and interfacial area and multi-field models.

Multi-Dimensional Two-Phase Flow Modeling Applied to Interior Ballistics

Complex phenomena occur in a combustion chamber during a ballistic cycle. From the ignition of the black powder in the primer to the exit of the projectile through the muzzle, two-phase gas-powder mix undertakes various transfers in different forms. A detailed comprehension of these effects is fundamental to predict the behavior of the whole system, considering performances and safety. Although the ignition of the powder bed is three-dimensional due to the primer’s geometry, simulations generally only deal with one- or two-dimensional problem. In this study, we propose a method to simulate the two-phase flows in 1, 2 or 3 dimensions with the same system of partial differential equations. A one-pressure, conditionally hyperbolic model [1] was used and solved by a nonconservative finite volume scheme associated to a fractional step method, where each step is hyperbolic. We extend our study to a two-pressure, unconditionally hyperbolic model [2] in which a relaxation technique was applied in order to recover the one-pressure model by using the granular stress. The second goal of this study is also to propose an improved ignition model of the powder grains, by taking into account simplified chemical kinetics for decomposition reactions in the two phases. Here we consider a 0th-order solid decomposition and an unimolecular, 2nd-order gas reaction. Validation of the algorithm on several test cases is presented.

格子-粒子ハイブリッド法の二流体一圧力モデル化と計算安定性および解析能 力に関する考察

格子-粒子ハイブリッド法の二流体一圧力モデル化と計算安定性および解析能 力に関する考察

A well-balanced scheme for a one-pressure model of two-phase flows

We consider a model of two-phase flows in which one phase is compressible and the other is incompressible. The system is a closed system with four governing equations for four unknowns. We construct a well-balanced scheme for this system. We then supplement the scheme with a practical modification so that it is always well defined. Our scheme is shown to be capable of maintaining equilibrium states.

Numerical Methodology of Sodium-Water Reaction with Multiphase Flow Analysis

A numerical methodology of sodium-water reaction (SWR) and a coupling method of SWR and multiphase flow analysis are proposed. Two SWR models are considered. One is a surface reaction model, which assumes that water vapor reacts with liquid sodium at the gas-liquid interface. The surface reaction is likely to be dominant in the initial phase of SWR. The analogy between mass and heat transfers is assumed to evaluate the diffusion-controlled reaction rate. The other is a gas-phase reaction model. If chemical reaction heating due to the surface reaction is large enough to vaporize the liquid sodium, it turns over in the gas-phase reaction. In the gas-phase reaction, water vapor reacts with sodium gas. The reaction mechanisms in the gas-phase reaction are investigated using an ab initio molecular orbital method. The reaction rate of the gas-phase reaction described by the Arrhenius law is obtained from the transition-state theory or the capture theory. The reaction models are employed in a compressible multifluid and one-pressure model using the Highly Simplified Marker and Cell method for multiphase flow analysis. As numerical examples, surface reaction with multiphase flow analysis and simplified gas-phase reaction analyses are carried out. It is confirmed that the present method is practically applicable to the coupling phenomena of SWR and multiphase flow.

Numerical Approach to the Safety Evaluation of Sodium—Water Reaction

A numerical simulation method of multi-dimensional and multi-phase reacting flow (SERAPHIM code) has been developed to evaluate the sodium-water reaction (SWR) phenomena in a steam generator of liquid metal fast reactor (LMFR). A compressible multi-fluid and one-pressure model is adopted and pressure and velocity fields are updated simultaneously by the HSMAC method. Two types of reaction models are considered; one is a surface reaction and the other is a gas-phase reaction. The surface reaction model assumes that water vapor reacts with the liquid sodium at the gas-liquid interface. If chemical reaction heating is large enough, liquid sodium is vaporized resulting in a gas-phase reaction. In the surface reaction, the reaction rate is assumed to be infinitely large. Several overall reaction equations are taken into account in the gas-phase reaction and the reaction rates are described in the form of the Arrhenius law. In the present study, adequacy of the analytical procedures for compressible multi-phase flow is validated by a benchmark calculation of the Edwards pipe blowdown problem. As a numerical example, two- and three-dimension analyses of the single-tube geometry and the two-dimension analyses of the 43-tubes geometry are carried out. It is concluded that the numerical quantification of the SWR accident by the SERAPHIM code is practicable and further use of the SERAPHIM code is useful to resolve safety issues immanent in the SWR.