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.     

The prediction of stratified two-phase flow with a two-equation model of turbulence

A two-equation model is applied to a stratified two-phase flow system to predict turbulent transport mechanisms in both phases.In the present analysis, the effects of interfacial waves on the flow field are formulated in terms of boundary conditions for the gas-liquid interface. For the gas phase, the wavy interface has such flow separation effects as a rough surface in a single-phase flow. While for the liquid phase, the waves generate turbulant energy which is transported progressively toward a lower wall region. The analytical results are in good agreement with available data regarding pressure drop, holdup and velocity profiles.     

Compressibility effects on waves in stratified two-phase flow

Ardron (1980) presented both one-dimensional and two-dimensional analyses of wave propagation in horizontal stratified two-phase flow. He compared the two approaches and concluded that the comparison helped to improve confidence in the use of one-dimensional approximations for the analysis of complex systems such as nuclear reactors.There are several assumptions in Ardron's developments. When alternative assumptions are made the results change. By examining the consequences of several possible assumptions we have learned from this example that considerable care may be necessary in the reduction of a multi-dimensional two-phase flow problem to a simpler form.This paper presents a more complete two-dimensional solution of this problem and discusses the limitations of the approximate solutions.     

Calculations of stratified wavy two-phase flow in pipes

Calculations of fully developed, stratified wavy gas–liquid pipe flow is presented. The wavy interface is represented by an equivalent interfacial roughness obtained from experimental data, which is made non-dimensional following the Charnock formulation. The two-dimensional, steady-state axial momentum equation is solved together with a two-layer turbulence model, which is modified to account for the roughness introduced at the interface. The governing equations are discretized using a finite difference method on a composite, overlapping grid with local grid refinement near the interface and the wall. The immersed interface method is used to make the numerical scheme well-defined across the interface, and a level set function is used to represent the interface. Numerical calculations are found to compare satisfactorily with experimental data.     

Experimental and theoretical investigation on the discharge characteristic of stratified two-phase flow

New measurements of the axial velocity profile of gas/liquid stratified mixture have been carried out. The results demonstrated that there are different types of velocity profiles and different maximum velocities inside the gas and the liquid layers. Simple correlation to determine the average velocity was obtained to characterize the very common mixture in typical industrial pipeline systems. The correlation fits the data well, and it is suggested for engineering purposes.     

A study on the numerical stability of the two-fluid model near ill-posedness

The two-fluid model is widely used in studying gas–liquid flow inside pipelines because it can qualitatively predict the flow field at low computational cost. However, the two-fluid model becomes ill-posed when the slip velocity exceeds a critical value, and computations can be quite unstable before the flow reaches the ill-posed condition. In this work, computational stability of various convection schemes together with the Euler implicit method for the time derivatives in conjunction with the two-fluid model is analyzed. A pressure correction algorithm for the two-fluid model is carefully implemented to minimize its effect on numerical stability. von Neumann stability analysis shows that the central difference scheme is more accurate and more stable than the 1st-order upwind, 2nd-order upwind, and QUICK schemes. The 2nd-order upwind scheme is much more susceptible to instability than the 1st-order upwind scheme and is inaccurate for short waves. Excellent agreement is obtained between the predicted and computed growth rates of harmonic disturbances. The instability associated with the two-fluid model discretized system of equations is related to but quantitatively different from the instability associated with ill-posedness of the two-fluid model. When the computation becomes unstable due to the ill-posedness, the machine roundoff errors from a selected range of short wavelengths, which scale with the grid size, are amplified rapidly to render the computation of any targeted long wavelength variation useless. For the viscous two-fluid model with wall friction and interfacial drag, a small-amplitude long wavelength disturbance grows due to viscous Kelvin–Helmholtz instability without triggering the grid scale short waves when the system remains well posed. Under such a condition, central difference is found to be the most accurate discretization scheme among those investigated.     

Effect of molecular diffusion on stratified shear flow stability

The results of theoretical analysis, which show that molecular heat or salt diffusion can stabilize the flow, are experimentally confirmed with reference to the problem of the motion of a density-stratified fluid towards an opening in a vertical wall. An asymptotic regime in which neither stratification nor diffusion affects the integral characteristics of the flow is established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 35–40, January–February, 1993.The authors are grateful to O. F. Vasil'ev for his valuable contribution to the organization of the research.     

Non-steady two-phase stratified laminar flow of polymeric liquids in pipes

Summary A numerical method of solution is given for the non-steady two-phase stratified. laminar flow of various non-Newtonian fluids in pipes. In particular, the Ellis fluid model is chosen to illustrate inelastic shear thinning effects. It is shown that the method can be applied to the non-steady multi-phase stratified laminar flow of some non-Newtonian fluid models. An Oldroyd six constant model is used to illustrate the fully elastic flow. It is found that the presence of two phases of the same kind of immiscible liquids tends to suppress the typically viscoelastic response to time dependent situations that the same liquids would exhibit as a single phase. Overshoot of flow rates is reduced, if not completely eliminated in both the generation and decay of steady flows. In two-phase pulsatile flows, the flow enhancement is less marked and the time dependence of the individual flow rates can be significantly different. Theoretical results are used to interpret some of the flow instabilities encountered during the capillary flow of polymeric liquids.

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Zusammenfassung Es wird eine numerische Methode für die nichtstationäre geschichtete Zwei-Phasen-Strömung verschiedener nicht-newtonscher Flüssigkeiten durch Rohre angegeben, wobei zur Veranschaulichung unelastischer Scherentzähungseffekte speziell das Ellis-Modell ausgewählt wird. Dabei zeigt sich, daß diese Methode für die Anwendung auf nicht-stationäre Mehrphasenströmungen geschichteter laminarer Strömungen nicht-newtonscher Flüssigkeiten geeignet ist. Zur Veranschaulichung des vollständigen elastischen Fließens wird ein Oldroyd-Modell mit sechs Konstanten gewählt. Es wird gezeigt, daß die Anwesenheit von zwei Phasen nicht-mischbarer Flüssigkeiten die Unterdrückung des typisch viskoelastischen Verhaltens unter zeitabhängigen Bedingungen, wie es beim Vorhandensein einer einzigen Phase beobachtet wird, zur Folge hat. Das Überschwingen der Fließgeschwindigkeit wird sowohl beim Anfahren als auch beim Anhalten einer stationären Strömung verringert, wenn nicht gar völlig verhindert. In pulsierenden Zwei-Phasen-Strömungen ist die Geschwindigkeitserhöhung weniger ausgeprägt, und die Zeitabhängigkeit der beiden Fließgeschwindigkeiten kann wesentlich verschieden sein. Die theoretischen Ergebnisse werden dazu verwendet, einige bei der Durchströmung von Polymerflüssigkeiten durch Kapillaren beobachtete Fließinstabilitäten zu interpretieren.

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With 15 figures     

Interfacial instability of turbulent two-phase stratified flow: Pressure-driven flow and non-Newtonian layers

We derive a flat-interface model to describe the flow of two horizontal, stably stratified fluids, where the bottom layer exhibits non-Newtonian rheology. The model takes into account the yield stress and power-law nature of the bottom fluid. In the light of the large viscosity contrast assumed to exist across the fluid interface, and for large pressure drops in the streamwise direction, the possibility for the upper Newtonian layer to display fully developed turbulence must be considered, and is described in our model. We develop a linear-stability analysis to predict the conditions under which the flat-interface state becomes unstable, and pay particular attention to characterizing the influence of the non-Newtonian rheology on the instability. Increasing the yield stress (up to the point where unyielded regions form in the bottom layer) is destabilizing; increasing the flow index, while bringing a broader spectrum of modes into play, is stabilizing. In addition, a second mode of instability is found, which depends on conditions in the bottom layer. For shear-thinning fluids, this second mode becomes more unstable, and yet more bottom-layer modes can become unstable for a suitable reduction in the flow index. One further difference between the Newtonian and non-Newtonian cases is the development of unyielded regions in the bottom layer, as the linear wave on the interface grows in time. These unyielded regions form in the trough of the wave, and can be observed in the linear analysis for a suitable parameter choice.     

An investigation into the interfacial shear stress contribution in two-phase stratified flow

This paper provides a combined theoretical and experimental investigation into the contribution of interfacial shear stress in certain co and counter-current flows in circular pipes. Based on momentum balance two equations were developed for such flows then two fluid systems of significantly different density ratio were experimentally tested to quantify these equations.     

Modeling of stratified two-phase flow in pipes,pumps and other devices

A model for stratified two-phase flow in pipes. pumps and other devices is presented. Using the assumption of a hydrostatic distribution of pressure over the cross section of a pipe. the effects of stratification are taken into account by means of specific terms in momentum and energy equations. These terms represent the pressure interactions between phases. Four concrete examples (classical earth gravity stratified flow. stratified flow in a regularly curved pipe. annular stratified flow caused by pre- or postrotation. and stratified flow in the impeller of a pump) and a comparison with an experiment are presented.     

Numerical analysis for slugging of steam-water stratified two-phase flow in horizontal duct

Thermalhydraulic transient phenomena of a steam-water two-phase flow was calculated numerically in order to investigate the onset of slugging from a stratified flow in a horizontal duct. Conservation equations were solved by the finite difference method using a two-phase flow analyzer ‘MINCS’. The analysis was performed to investigate the initiation of slugging with and without phase change, or condensation. The present instability criteria for the onset of slugging with no condensation agreed well with that of the Mishima–Ishii relation while it was much lower than that defined by the Kelvin–Helmholtz instability criteria. However, as the temperature difference between phases increased, steam velocity became higher for the onset of slugging condition. The characteristics of flow reversal and water hammering which were the consequences of slugging with condensation, were investigated and described. It is expected that this modeling could be well applied to complicated thermalhydraulic phenomena accompanied by flow reversal and water hammering in power plants.     

A semiempirical model of turbulent stratified flow

A relatively simple mathematical model of the three-dimensional turbulent flow of a stratified fluid is proposed. This makes it possible to calculate the velocity and temperature fields and to estimate the fluctuating characteristics of the flow using three empirical constants. Two problems are solved numerically: stratified flow in an open channel and three-dimensional flow in the Ekibastuz district power station's cooling basin-reservoir. The results are consistent with the experimental data and field measurements.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 29–34, May–June, 1992.     

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.     

Research on the flow stability in a cylindrical particle two-phase boundary layer

IntroductionThecylindricalparticletwo_phaseflowsareofparticularinterestintheprocessingofcompositematerials ,textileindustry ,papermaking ,chemicalengineering ,foodprocessing[1].Thecylindricalparticlesinaflowcanmakethereinforcementofmaterials,thechangeofphysicalpropertyformaterialsandthereductionofdrag .Arranaga[2 ]reportedthatdragreductioneffectsareupto 60 %inpipeflowsbyaddingcylindricalparticlestoflow .Thecylindricalparticleshavealsoeffectsonthemechanismsofflowstability .Theeffectofcylindric…     

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.     

A two-fluid model of non-Newtonian blood flow induced by peristaltic waves

The problem of blood flow induced by peristaltic waves in a uniform small diameter tube has been investigated. Blood has been represented by a two-fluid model consisting of a core region of suspension of all the erythrocytes, assumed to be a Casson fluid, and a peripheral layer of plasma as a Newtonian fluid. The expressions for dimensionless pressure drop and friction force have been obtained. The results obtained in the analysis have been evaluated numerically and discussed briefly. The significance of the present model over the existing models has been pointed out by comparing the results with other theories both analytically and numerically.