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.     

On the momentum equations in dispersed two-phase systems

For so-called dispersed two-phase flow systems the equations of motion are derived for each phase separately. The equations are developed using a minimum of mathematics by applying mass and momentum balances respectively over a small cubic volume element. Special attention is paid to those particles which are cut by the boundaries of this control volume element. In fact these particles are treated by means of two methods, both methods giving the same final result.     

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.     

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.     

On the surface equations in two-phase flows and reacting single-phase flows

This paper summarizes the mathematical surface equations which are useful in two-phase flows and single-phase reacting flows. The connection between the interfacial area concentration transport equation for two-phase flows and the flame surface density transport equation for turbulent reacting flows is established. Several analytical examples are given to clarify the physical significance of the different quantities involved in the different transport equations. An introduction to the mathematical treatment of anisotropic interfaces is also given. This theory is illustrated on two different numerical examples: a single inclusion in a simple shear and a single inclusion in an uni-axial elongation.     

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.     

On the characteristic equations of elasto-plastic flow

Three dimensional characteristic surfaces (slip surfaces) of elasto-plastic Navier's equations and the criteria for their existence are discussed, and the solutions are also applied to two dimensional cases. By making use of isotropic yield function, the following results are proved. If, and only if the plastic/elastic moduli ratio is zero and Det$\text{(φ}{}_{\mathrm{ij}}\text{)=0,(φ}{}_{\mathrm{ij}}\text{=}\text{?φ}\text{?σ}{}_{\mathrm{ij}}$, φ: yield function,σ_ij: stress tensor), characteristic surfaces exist. There are two and only two characteristic surface elements at each point, and they are identical with the surfaces of maximum shearing stress.     

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.     

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.     

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.     

Multicomponent fluid flow analysis using a new set of conservation equations

In this work hydrodynamics of multicomponent ideal gas mixtures have been studied. Starting from the kinetic equations, the Eulerian approach is used to derive a new set of conservation equations for the multicomponent system where each component may have different velocity and kinetic temperature. The equations are based on the Grad's method of moment derived from the kinetic model in a relaxation time approximation (RTA). Based on this model which contains separate equation sets for each component of the system, a computer code has been developed for numerical computation of compressible flows of binary gas mixture in generalized curvilinear boundary conforming coordinates. Since these equations are similar to the Navier-Stokes equations for the single fluid systems, the same numerical methods are applied to these new equations. The Roe's numerical scheme is used to discretize the convective terms of governing fluid flow equations. The prepared algorithm and the computer code are capable of computing and presenting flow fields of each component of the system separately as well as the average flow field of the multicomponent gas system as a whole. Comparison of the present code results with those of a more common algorithm based on the mixture theory in a supersonic converging-diverging nozzle provides the validation of the present formulation. Afterwards, a more involved nozzle cooling problem with a binary ideal gas (helium-xenon) is chosen to compare the present results with those of the ordinary mixture theory. The present model provides the details of the flow fields of each component separately which is not available otherwise. It is also shown that the separate fluids treatment, such as the present study, is crucial when considering time scales on the order of (or shorter than) the intercollisions relaxation times.     

On the suitability of first-order differential models for two-phase flow prediction

The stability features of a general elass of one-dimensional two-phase flow models are examined. This class of models is characterized by the presence of first-order derivatives and algebraic functions of the flow variables, higher-order differential terms being absent, and can accommodate a variety of physical effects such as added mass and unequal phase pressures in some formulations. By taking a general standpoint, a number of results are obtained applicable to the entire class of models considered. In particular, it is found that, despite the presence of algebraic terms in the equations (describing, e.g. drag effects) the stability criteria are independent of the wavenumber of the perturbation. As a consequence, reality of characteristics is necessary, although not sufficient, for stability. To illustrate the theory, three specific models are considered in detail.     

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

ＡＮＩＭＰＬＩＣＴＡＬＧＯＲＩＴＨＭＯＦＴＨＩＮＬＡＹＥＲＥＱＵＡＴＩＯＮＳｉＮＶＩＳＣＯＵＳ,ＴＲＡＮＳＯＮＩＣ,ＴＷＯ－ＰＨＡＳＥＮＯＺＺＬＥＦＬＯＷＨｅＨｏｎｇ－ｑｉｎｇ（何洪庆）ＨｏｕＸｉａｏ（侯晓）ＣａｉＴｉ－ｍｉｎ（蔡体敏）ＷｕＸｉｎｇ－...