Fundamental equations of turbulent two-phase flow

A new set of Reynolds equations for predicting turbulent two-phase flows has been developed by means of Reynolds averaging method on the unsteady laminar equations of two-phase flow. These equations involve average terms of products of turbulent fluctuations in some physical parameters in a large degree. The interaction forces between two phases, the pressures for dispersed phase, extra stresses except for pressure and the expressions for energy interchange between two phases have been discussed.     

Empirical equations to predict flow patterns in two-phase inclined flow

Based on experimental data, flow regime maps were drawn for different inclination angles, including horizontal and vertical flow. Different empirical equations for the flow regime transitions are proposed that are functions of inclination angle for both upflow and downflow. In general, the flow regimes and their transitions for upflow were similar to those proposed by Duns and Ros for vertical upflow. For downflow, the flow regimes and their transitions conformed more to the Mandhane et al. type of flow regime map. The stratified flow region in downflow was found to be affected appreciably by the angle of inclination. A detailed comparison of the proposed transition equations with a number of flow regime maps is also presented.     

One-dimensional two-fluid equations for horizontal stratified two-phase flow

Advanced computer codes for water reactor loss-of-coolant analysis are based on the use of the two-fluid model of two-phase flow, in which conservation equations are solved for the gas and liquid phases separately. The standard two-fluid equations, however, sometimes predict the growth of instabilities in the flow, and occasionally become improperly posed. These difficulties have in the past led to the proposal of several different forms for the conservations equations.To help resolve these uncertainties a widely accepted form of the one-dimensional two-fluid equations is used to calculate wave propagation speeds, and stability limits, for the illustrative case of a frictionless horizontal stratified gas-liquid flow. Calculated propagation velocities are shown to agree with the appropriate limit of an exact solution, and the predicted stability limits are found consistent with available observations on the stability of the stratified flow regime.These comparisons help improve confidence in the ability of the two-fluid equations to analyse more complex problems in transient two-phase flow.     

On the various forms of the conservation equations in two-phase flow

The conservation equations for one-dimensional two-phase flow are derived from first principles. The effects of the radial distributions of velocities, enthalpies, and void fraction are taken into account through the use of correlation coefficients. Several simplified separated-flow model formulations that have appeared in the literature are derived from these equations by specializing the values of the correlation coefficients. The equivalence of these formulations under certain assumptions is demonstrated. Finally, new Lagrangian forms of the conservation equations, written in terms of the velocities of the center of mass, momentum, and energy are presented.     

Application of general constitutive principles to the derivation of multidimensional two-phase flow equations

The process of determining appropriate constitutive equations for multidimensional time averaged two-phase flow equations is studied from the point of view of starting from general principles, and proceeding to specific constitutive equations which contain known physical effects. Energetic effects and phase change are not considered. Models are given for the interfacial momentum transfer, the laminar and turbulent (Reynolds) stresses, and the pressure differences between the phases, and between a given phase pressure and the interfacial average pressure.     

Experimental determination of the flow transport coefficients in the coupled equations of two-phase flow in porous media

The transport coefficients in the coupled equations of two-phase flow are defined if the pressure gradient in one of the two flowing fluids is equal to zero. This definition has been used in experiments with oil and water in a sandpack and the four transport coefficients have been measured over wide water saturation ranges. The values of the cross coefficients were found to be significant as they ranged from 10 to 35% of the value of the effective permeability to water and from 5 to 15% of the effective permeability to oil, respectively.     

Separated two-phase flow model: application to critical two-phase flow

A separated flow model has been developed to allow the calculation of critical flow rates for steam-water mixtures. This model considers hydrodynamic as well as thermal non-equilibrium effects which are present due to rapid depressurization. Thus, this model incorporates interphase interaction terms for momentum, energy and mass. The mass transfer, evaporation or condensation rate, is coupled with the heat transfer between the two phases. Certain empiricisms are necessary to be included into this model, e.g. the size and number of nucleation sites at the onset of flashing. Transitions from one flow regime to the other are assumed to occur at certain void fractions. Justification of these assumptions is only possible by comparison with experimental results of different authors which in general shows good agreement.     

The influence of interfacial pressure forces on the character of two-phase flow model equations

Equations for two-phase flow are developed where the interfacial pressure p_i is the closure variable. The assumption that p_i is constant leads to variations of the single pressure model with several aphysical properties. Use of more realistic pressure distributions, for flow about solid particles, produces a model displaying added mass and drag effects and having real characteristic roots.     

Closure of the governing equations for immiscible,two-phase flow: A research comment

The derivation of the governing equations for immiscible, two-phase flow through porous media by Whitaker (Transport in Porous Media 1, 105–125 (1986)) contains an error which is corrected in the present work. The modified equations contain terms not present in the original equations, but their presence does not cause any fundamental changes from the conclusions reached in the original work. However, these extra terms may be important in computations associated with the closure problem.     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

AN IMPLICIT ALGORITHM OF THIN LAYER EQUATIONS IN VISCOUS，TRANSONIC，TWO－PHASE NOZZLE FLOW

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Stability of transonic two-phase flow

The nature of a singular point in the stability of one-dimensional transonic flow of a vapor-drop mixture in a channel of variable cross section is considered within the framework of a two-lquid hydrodynamical model. It is shown that the singular point in the case of any lags of the drops preserves the nature of a saddle inherent to homogeneous gas flow, shifting only towards the divergent part of the channel if the content of condensed phase is not too high. Here the transition of subsonic two-phase flow into supersonic flow is stable and the predominance of drop agglomeration over fragmentation and the positive curvature of the channel profile are stabilizing factors. The saddle nature of the singularity is possible only if the lag of the drops is not too high in the case of flows with a higher content of condensed phase. In the opposite case, the point at which the speed of sound is attained loses the nature of a saddle point.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 163–171, January–February, 1976.     

Energy-separating properties of two-phase flow

A new, substantially nondissipative process of energy separation in two-phase flows has been investigated. Mixtures of air with water, kerosene, and an aqueous solution of diethylene glycol were studied at initial pressures of 3–20 bar. It was found that ice was formed in an air-water mixture issuing from a supersonic nozzle, and for a mixture of air with a nonfreezing diethylene-glycol solution the liquid obtained after nonequilibrium separation had a negative temperature. The possibility of effective freezing out of moisture on an uncooled solid surface exposed to a current of moist air from a supersonic nozzle was demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 49–55, November–December, 1976.     

Transient flow redistribution in annular two-phase flow

A model is described for the prediction of transient flow redistribution in vertical annular two-phase flow. The model is based on an analysis of the local parameters controlling the flow and takes account of the diffusive motion of entrained droplets and the delay time for change in the wave structure on the film. Comparisons are made with experimental results on inlet effects and it is shown that the wall injection experimental results can be described by the model. The jet injection results are not fitted by the model and it is shown that some additional deposition mechanism must be present.     

Effective equations for two-phase flow in porous media: the effect of trapping on the microscale

In this article, we consider a two-phase flow model in a heterogeneous porous column. The medium consists of many homogeneous layers that are perpendicular to the flow direction and have a periodic structure resulting in a one-dimensional flow. Trapping may occur at the interface between a coarse and a fine layer. Assuming that capillary effects caused by the surface tension are in balance with the viscous effects, we apply the homogenization approach to derive an effective (upscaled) model. Numerical experiments show a good agreement between the effective solution and the averaged solution taking into account the detailed microstructure.