An extended charnock estimate of interfacial stress in stratified two-phase flows

The existing theory of interfacial shear and roughness in fully-developed flow is generalized with the aid of the Colebrook-White formula to encompass smooth, transitional and rough interfaces, and an empirical correction improves the agreement with experiment near regime transition.     

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.     

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.     

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow. The Project Supported by the National Natural Science Foundation of China.     

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.     

A charnock-based estimate of interfacial resistance and roughness for internal,fully-developed,stratified, two-phase horizontal flow

Charnock's (1955) relation between interfacial roughness and drag is utilized in order to determine these two quantities for fully-developed flows in tubes of arbitrary cross-section.     

An investigation of pressure transients in pipelines with two-phase bubbly flow

This paper presents a two-dimensional model for the analysis of the pressure transient of a two-phase homogeneous bubbly mixture flowing in a pipeline and the numerical integration using the centre implicit method (CIM). Experiments were conducted to confirm the proposed sonic speed equation of an air–water mixture for an air concentration of less than 1%. The 2D CIM model is compared with the method of characteristics (MoC) for a two-phase bubbly flow in a pipeline. The comparisons show that the proposed 2D CIM model generally gives good agreement with the method of characteristics.     

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     

Heat transfer and interfacial drag in countercurrent steam-water stratified flow

Local condensation heat transfer coefficients and interfacial shear stresses have been measured for countercurrent stratified flow of steam and subcooled water in rectangular channels over a wide range of inclination angles (4–87°) at two aspect ratios. Dimensionless correlations for the interfacial friction factor have been developed that show that it is a function of the liquid Reynolds number only. Empirical correlations of the heat transfer coefficient, based upon the bulk flow properties, have also been set up for the whole body of data encompassing the different inclination angles and aspect ratios. These indicate that the Froude number as a dimensionless gas velocity is a better correlating parameter than the gas Reynolds number. As an alternative approach, a simple dimensionless relationship for the beat transfer coefficient was obtained by analogy between heat and momentum transfer through the interface. Finally, a turbulence-centered model has been modified by using measured interfacial parameters for the turbulent velocity and length scales, resulting in good agreement with the data.     

Electro-magnetic-field-induced flow and interfacial instabilities in confined stratified liquid layers

The electro-magneto-hydrodynamic (EMHD) flow and instabilities engendered by the Lorenz force arising from interaction between externally applied perpendicular electric and magnetic fields are investigated in layers of two immiscible liquids in a channel. A new finite wave-number EMHD instability mode is uncovered by the Orr–Sommerfeld analysis, in addition to the interfacial and shear modes which also arise in the pressure-driven flows. Thus, EMHD can be controlled for micro-channel transport, heat and mass transfer, mixing, micro-emulsion generation, etc.     

On the interfacial roughness scale in turbulent stratified two-phase flow: 3D lattice Boltzmann numerical simulations with forced turbulence and surfactant

Numerical 3D simulations of turbulent, stratified two-phase shear flow with a surfactant laden interface were used to test and develop a phenomenological interfacial roughness scale model where the energy required to deform the interface (buoyancy, interfacial tension, and viscous work) is proportional to the turbulent kinetic energy adjacent to the interface.The turbulence was forced in the upper and lower liquids in the simulations, to emulate the interfacial dynamics without requiring (prohibitively) large simulation domains and Reynolds numbers. The addition of surfactant lead to an increased roughness scale (for the same turbulent kinetic energy) due to the introduction of interfacial dilatational elasticity that suppressed horizontal motion parallel to the interface, and enhanced the vertical motion.The phenomenological roughness scale model was not fully developed for dilatational elasticity in this work, but we proposed a source term that represents surfactant induced pressure fluctuations near the interface. This source term should be developed further to account for the relation between surfactant density fluctuations and turbulence adjacent to the interface. We foresee that the roughness scale model can be used as a basis for more general interfacial closure relations in Reynolds averaged turbulence models, where also mobile surfactant is accounted for.     

Asymptotic investigation of two-phase couette flow

We consider plane and cylindrical Couette flow for a two-phase medium. The motion of the medium is described by the equations obtained in [1]. Collisions between the particles are disregarded, and their motion, in addition to the inertial forces, is determined by the pressure gradient of the carrying phase and the forces of viscous interaction between the carrying phase and the particles. We obtain simple asymptotic solutions of the indicated problems for small and large values of the dimensionless determining parameters. In a number of cases the solution has the nature of a boundary layer on solid walls.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 67–73, July–August, 1978.     

Numerical investigation of two-phase rotating flow

Finite difference solutions of the two-fluid equations of motion for a particle (droplet)-fluid mixture in a rotating finite axisymmetric cylinder are presented. The numerical method, which can be regarded as an extension of the Harlow & Amsden approach, employs forward time and centred space discretization and treats implicitly the pressure, Coriolis and volume flux terms. The computed flow fields are examined via a detailed comparison to previous analytic approximations, which illuminates both the physical and numerical aspects and the validity of these approximations.     

Characteristics of an interfacial wave in a horizontal air-water stratified flow

Interfacial wave parameters, in this case the frequency, height, velocity, and slope, were investigated experimentally in a horizontal air-water stratified flow. Experiments were conducted with a parallel wire conductance sensor and PIV visualization in a rectangular channel, of which the width and height are 40 mm and 50 mm, respectively. In the experiments, the flow condition covered the liquid Reynolds number Re_l range of 450 to 3540 and the gas Reynolds number Re_g range of 14,000 to 70,000. The results revealed that the observed wave types according to the flow conditions in the rectangular channel are similar to those in a horizontal pipe. The frequency, height, and slope of the interfacial wave show complicated tendencies according to the combination of Re_g and Re_l, which affects the coalescence and breakup of the wave. Specifically, the wave height and wave slope have opposite tendencies regarding the criterion of Re_g = 34,000. For cases in which Re_g  ≥  34,000, the interfacial drag force significantly affects the height and slope of the disturbance wave. In contrast, for Re_g < 34,000, the growth of the wave has an important effect on the wave parameters. Finally, new empirical correlations for the frequency, height, and slope of the interfacial wave were proposed for application to the development of a droplet entrainment model in a horizontal stratified flow.     

Structural and interfacial stability of multiple solutions for stratified flow

Prediction of the liquid level in stratified two-phase upwards flow shows that one may have multiple solutions. In this case it is necessary to determine which solutions will actually occur and whether hysteresis is possible, namely whether it is possible to have two or more solutions for the same operating conditions. In this work the stability of the solutions for stratified flow is considered using two types of stability analyses: (1) structural stability analysis; and (2) interfacial stability analysis (Kelvin—Helmholtz, K—H). For the K—H stability analysis we used two methods: an approximate simplified method suggested by Taitel & Dukler; and a more rigorous method suggested by Barnea, which is based on a combination of the viscous K—H and inviscid K—H analyses. The results show that whenever three solutions exist only the first, i.e. the solution with the thinnest liquid level, is stable. The middle solution is always structurally unstable (linearly), whereas the third solution is structurally unstable to large disturbances (non-linear stability). The third solution is usually also unstable to the K—H type of instability. As a result it is concluded that hysteresis is not possible and that only the thinnest solution will be observed practically.     

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.     

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.