Moon-Sun Chung

Numerical solution of hyperbolic two-fluid two-phase flow model with non-reflecting boundary conditions

Abstract Flux vector splitting method is applied to the two-fluid six-equation model of two-phase flow, which takes account of surface tension effect via the interfacial pressure jump terms in the momentum equations. The latter terms using the concept of finite-thickness interface are derived as a simple function of fluid bulk moduli. We proved that the governing equation system is hyperbolic with real eigenvalues in the bubbly, slug, and annular flow regimes. The governing equations do not need any conventional artificial stabilizing terms like the virtual mass terms. Sonic speeds obtained through characteristic analysis show excellent agreement with the existing experimental data. Edwards pipe problem is solved as a benchmark test of the present two-phase equation model.

Wave propagation in two-phase flow based on a new hyperbolic two-fluid model

A high-resolution up wind scheme based on the flux vector splitting method is developed for the two-fluid six-equation model to solve the wave propagation problems of two-phase flow. The interfacial pressure jump terms make the governing equations hyperbolic without any conventional source terms in the momentum equations. Real eigenvalues are obtained for all the bubbly, slug, and annular flow regimes. Calculated speeds of sound have shown excellent agreement with the existing experimental data. Solutions to wave propagation problems withinitial pressure and void distribution are presented. The Edwards pipe problem accompanied by sudden depressurization and flashing also is solved as a benchmark test.A high-resolution up wind scheme based on the flux vector splitting method is developed for the two-fluid six-equation model to solve the wave propagation problems of two-phase flow. The interfacial pressure jump terms make the governing equations hyperbolic without any conventional source terms in the momentum equations. Real eigenvalues are obtained for all the bubbly, slug, and annular flow regimes. Calculated speeds of sound have shown excellent agreement with the existing experimental data. Solutions to wave propagation problems withinitial pressure and void distribution are presented. The Edwards pipe problem accompanied by sudden depressurization and flashing also is solved as a benchmark test.

A NUMERICAL STUDY OF TWO-PHASE FLOW USING A TWO-DIMENSIONAL TWO-FLUID MODEL

Two-phase flow is studied numerically in two dimensions using a two-fluid equation system. The development is based on the surface-tension force terms incorporated in the momentum equations. The governing equations become hyperbolic type for which an upwind method such as flux vector splitting (FVS) avails. As a benchmark problem, a two-phase shock tube is first used to study the wave propagation characteristics and interaction between the gas and liquid phases. Fluid sedimentation due to the density difference and cavity growth in a duct with a bend are showed next. Advantages and capabilities of the present formulation are discussed in some detail.

AN INTERFACIAL PRESSURE JUMP MODEL FOR TWO-PHASE BUBBLY FLOW

Interfacial pressure jump terms based on the physics of phasic interface and bubble dynamics are introduced into the momentum equations of the two-fluid model for the bubbly flow. The pressure discontinuity across the phasic interface due to the surface tension force is expressed as a function of the fluid bulk moduli and the bubble radius. The consequence is that we obtain from the system of equations the real eigenvalues representing the void-fraction propagation speed and pressure wave speed in terms of the bubble diameter. Inversely, we can obtain an analytic closure relation for the radius of bubbles in the bubbly flow by using the kinematic wave speed given empirically in the literature. It is remarkable to see that the present mechanistic model can indeed represent both the bubble dynamics and the two-phase wave propagation in bubbly flow. Finally, it has been shown that the numerical stability is improved significantly if the interfacial pressure jump terms are used in lieu of the virtual mass terms.Interfacial pressure jump terms based on the physics of phasic interface and bubble dynamics are introduced into the momentum equations of the two-fluid model for the bubbly flow. The pressure discontinuity across the phasic interface due to the surface tension force is expressed as a function of the fluid bulk moduli and the bubble radius. The consequence is that we obtain from the system of equations the real eigenvalues representing the void-fraction propagation speed and pressure wave speed in terms of the bubble diameter. Inversely, we can obtain an analytic closure relation for the radius of bubbles in the bubbly flow by using the kinematic wave speed given empirically in the literature. It is remarkable to see that the present mechanistic model can indeed represent both the bubble dynamics and the two-phase wave propagation in bubbly flow. Finally, it has been shown that the numerical stability is improved significantly if the interfacial pressure jump terms are used in lieu of the virtual mass terms.

On the wave dispersion relevant to the virtual mass terms in the two-phase two-fluid model

Interfacial pressure jump terms are introduced in the momentum equations of the two-fluid two-phase flow while at the same time we keep the conventional virtual mass force as a nonobjective formulation. The pressure discontinuity across a thin interface due to the surface tension is compactly represented by a function of the fluid bulk moduli. The governing equations with the interfacial pressure jump terms produced a hyperbolic equation system having real eigenvalues for the bubbly flow, regardless of whether the virtual mass terms are added or not. The mixture sound speed for the two-phase flow evaluated using the combined interfacial pressure jump and conventional virtual mass terms has shown increasing dispersion of the small-amplitude waves when the virtual mass coefficient is larger. When the virtual mass terms are added to account for the accelerating flow, care should be exercised therefore not to introduce nonphysical wave dispersion.

Numerical Calculation of Two-Phase Flow Based on a Two-Fluid Model With Flow Regime Transitions

Numerical test and eigenvalue analysis for a two-phase channel flows for energy conversion systems like fuel cells or water electrolysers with flow regime transitions are performed by using the well-posed system of equation that takes into account the pressure jump at the phasic interface. The interfacial pressure jump terms derived from the definition of surface tension which is based on the surface physics make the conventional two-fluid model hyperbolic without any additive terms, i.e., virtual mass or artificial viscosity terms. The four-equation system has three sets of eigenvalues; each of them has an analytical form of real eigenvalues relevant to the sonic speeds with phasic velocities of three typical flow regimes such as dispersed, slug, and separated flows. Further, the eigenvalues for the flow transition regions can also be obtained numerically for smooth calculation of flow regime transitions. The sonic speeds agree well not only with the earlier experimental data but also with those of an analytical model. Owing to the hyperbolicity of this model, we can adopt an upwind method, which is one of the well-known Godunov type upwind methods. A typical example of two-phase flows shows that the present model can simulate the phase separation caused by density difference of two-phase fluids.© 2012 ASME

Upwind Solutions for Two-Phase Flow Problems Using a Hyperbolic Two-Fluid Model

This study discusses on the implementation of an upwind method for a new 2-dimensional 2-fluid model including the surface tension effect in the momentum equations. This model consists of a complete set of 8 equations including 2-mass, 4-momentum, and 2-internal energy conservation equations having all real eigenvalues. Based on this equation system with upwind numerical method, the present authors first make a pilot 2-dimensional code and then solve some benchmark problems to verify whether this model and numerical method is able to properly solve some fundamental one-dimensional two-phase flow problems or not.Copyright

Two-Phase Critical Flow Simulations by Using a New Non-Equilibrium Model and a Modified Equilibrium Model

An accurate prediction of a critical flow discharged from a pressurized pipe system is of most importance in such a safety analysis of nuclear power plants, since it provides the transient boundary conditions during the depressurization transients initiated by a pipe break in primary or secondary systems and during the over-pressurization transients resulting in a relief of coolant through valves. Mass and energy discharge through the opening of pressure boundary affects the system thermal hydraulic responses, that is, phase changes and flow distribution in the system, and the mass inventory remaining in the system necessary to remove core decay heat of a nuclear reactor. Therefore, the safety significance relating to the critical flow led to a development of various empirical and mechanistic critical flow models. However, the accuracies of these models are still in question especially during two-phase critical flow condition. A good example of that is a homogeneous equilibrium model (HEM). The HEM is the basis of several system codes, such as early versions of RELAP, for nuclear loss-of-coolant accident (LOCA). The major non-equilibrium phenomena that are ignored in the HEM are vapor bubble nucleation and interface heat, mass, and momentum transfer. Henry-Fauske empirically handled non-equilibrium vapor generation by introducing a non-equilibrium parameter that allows only a fraction of the equilibrium vapor generation to occur. This approach boils down in essence to a correlation of the deviation between the measured flow rate and the prediction from the HEM: The details of the flow path do not have to be worked out and only needs to know the upstream conditions. However, if we treat non-equilibrium phenomena with this model, it requires an empirical database of the non-equilibrium parameters or their correlations that are so far unknown. Further, because the coefficients are not applied separately to the subcooled liquid and two-phase mixture, we have not been able to treat the non-equilibrium phenomena with the phase change properly. For this reason, we propose the non-equilibrium parameters for subcooled liquid and two-phase mixture, respectively, and then we adopt their combinations according to the flow conditions through the phase change process using the RELAP5/MOD3 code. In addition, we discuss the assessment results of Marviken LBLOCA tests using these non-equilibrium parameter sets with those from the non-equilibrium model by Trapp-Ransom and Chung et al.Copyright