Some issues related to the modeling of interfacial areas in gas–liquid flows I. The conceptual issues

Two-phase flow modeling has been under constant development for the past forty years. Actually there exists a hierarchy of models which extends from the homogeneous model valid for two-phase flows where the phases are strongly coupled to the two-fluid model valid for two-phase flows where the phases are a priori weakly coupled. However the latter model has been used extensively in computer codes because of its potential in handling many different physical situations.The two-fluid model is based on the balance equations for mass, momentum and energy, averaged in a certain sense and expressed for each phase and for the interface between the phases. The difficulty in using the two-fluid model stems from the closure relations needed to arrive at a complete set of partial differential equations describing the flow. These closure relations should supply the information lost during the averaging of the balance equations and should specify in particular the interactions of mass, momentum and energy between the phases. Another requirement for the interaction terms is that they should satisfy the interfacial balance equations. Some of these terms such as the added mass term or the lift force term do not depend on the interfacial area but some others do, such as the mass transfer term, the drag term or the heat flux term. It is then necessary to model the interfacial area in order to evaluate the corresponding fluxes. Another benefit resulting from the modeling of the interfacial area would be to replace the usual static flow pattern maps which specify the flow configuration by a dynamic follow-up of the flow pattern. All these reasons explain why so much effort has been put during the past twenty years on the modeling and measurement of the interfacial area in two-phase flows.This article contains two parts. The first one deals with the conceptual issues and has the following objectives:

1.
to give precise definitions of the interfacial area concentrations;
2.
to explain the origin of the interfacial area concentration transport equation suggested by M. Ishii in 1975;
3.
to explain some paradoxical behaviors encountered when calculating the interfacial area concentration transport velocity.

     

Direct numerical simulation of a freely decaying turbulent interfacial flow

Whereas Large Eddy Simulation (LES) of single-phase flows is already widely used in the CFD world, even for industrial applications, LES of two-phase interfacial flows, i.e. two-phase flows where an interface separates liquid and gas phases, still remains a challenging task. The main issue is the development of subgrid scale models well suited for two-phase interfacial flows. The aim of this work is to generate a detailed data base from direct numerical simulation (DNS) of two-phase interfacial flows in order to clearly understand interactions between small turbulent scales and the interface separating the two phases. This work is a first contribution in the study of the interface/turbulence interaction in the configuration where the interface is widely deformed and where both phases are resolved by DNS. To do this, the interaction between an initially plane interface and a freely decaying homogeneous isotropic turbulence (HIT) is studied. The densities and viscosities are the same for both phases in order to focus on the effect of the surface tension coefficient. Comparisons with existing theories built on wall-bounded or free-surface turbulence are carried out. To understand energy transfers between the interfacial energy and the turbulent one, PDFs of the droplet sizes distribution are calculated. An energy budget is carried out and turbulent statistics are performed including the distance to the interface as a parameter. A spectral analysis is achieved to highlight the energy transfer between turbulent scales of different sizes. The originality of this work is the study of the interface/turbulence interactions in the case of a widely deformed interface evolving in a turbulent flow.     

Experimental study of interfacial area transport in air–water two phase flow in a scaled 8 × 8 BWR rod bundle

In order to accurately predict nuclear reactor behavior, the ability to predict the transfer of mass, momentum and energy between the phases in two-phase flows, whether in the Reactor Pressure Vessel (RPV) or steam generator, is essential. A significant component of this prediction is the area available for transfer per unit volume, called the interfacial area concentration. Current thermal-hydraulic system analysis code predictions use empirical models to predict the interfacial area concentration; however accuracy and reliability can be improved through the use of an Interfacial Area Transport Equation (IATE). The IATE requires rigorously developed models for sources and sinks due to bubble interactions or phase change and an extensive database to validate those models. To provide this database, experiments using electrical conductivity probes to measure the interfacial area concentration at several axial positions have been performed in an 8 × 8 rod bundle which was carefully scaled from an actual BWR rod bundle.     

Local instant formulation of two-phase flow

The local instant formulation of mass, momentum and energy conservations of two-phase flow has been developed. Distribution, an extended notion of a function, has been introduced for this purpose because physical parameters of two-phase flow media change discontinuously at the interface and the Lebesgue measure of an interface is zero. Using a characteristic function of each phase, the physical parameters of two-phase flow have been defined as field quantities. In addition to this, the source terms at the interface are defined in terms of the local instant interfacial area concentration. Based on these field quantities, the local instant field equations of mass, momentum and total energy conservations of two-phase flow have been derived. Modification of these field equations gives the single field representation of the local instant field equations of two-phase flow. Neglecting the interfacial force and energy, this formulation coincides with the field equations of single-phase flow, except in the definition of differentiation. The local instant two-fluid formulation of two-phase flow has also been derived. This formulation consists of six local instant field equations of mass, momentum and total energy conservations of both phases. Interfacial mass, momentum and energy transfer terms appear in these equations, which are expressed in terms of the local instant interfacial area concentration.     

A model for droplet entrainment in heated annular flow

The ability to accurately predict droplet entrainment in annular two-phase flow is required to effectively calculate the interfacial mass, momentum, and energy transfer, which characterizes nuclear reactor safety, system design, analysis, and performance. Most annular flow entrainment models in the open literature are formulated in terms of dimensionless groups, which do not directly account for interfacial instabilities. However, many researchers agree that there is a clear presence of interfacial instability phenomena having a direct impact on droplet entrainment. The present study proposes a model for droplet entrainment, based on the underlying physics of droplet entrainment from upward co-current annular film flow that is characteristic to light water reactor safety analysis. The model is developed based on a force balance and stability analysis that can be implemented into a transient three-field (continuous liquid, droplet, and vapor) two-phase heat transfer and fluid flow systems analysis computer code.     

Nucleation and flashing in nozzles—2. Comparison with experiments using a five-equation model for vapor void development

A quasi-one-dimensional, five-equation, homogeneous, nonequilibrium model has been developed and utilized on a microcomputer to calculate the behavior of flowing, initially subcooled, flashing water systems. Equations for mixture and vapor mass conservation, mixture momentum conservation, liquid energy conservation and bubble transport were discretized and linearized semi-implicitly, and solved using a successive iteration Newton method. Closure was obtained through simple constitutive equations for friction and spherical bubble growth, and a new nucleation model for wall nucleation in small nozzles combined with an existing model for bulk nucleation in large geometries to obtain the thermal nonequilibrium between phases. The model described was applied to choked nozzle flow with subcooled water inlets based on specified inlet conditions of pressure and temperature, and vanishing inlet void fraction and bubble number density. Good qualitative and quantitative agreement with the experiment confirms the adequacy of the nucleation models in determining both the initial size and number density of nuclei, and indicates that mechanical nonequilibrium between phases is not an important factor in these flows. It is shown that bulk nucleation becomes important as the volume-to-surface ratio of the geometry is increased.     

Non equilibrium gas-liquid flows in variable cross section ducts

The behaviour of two-phase high velocity flows in variable cross section ducts was investigated using a one-dimensional numerical model developed for the study of the annular flow configuration. Heat, mass, and momentum transfer between the phases during the flow were considered. The validation of the calculation procedure was made with some experimental data for the air-water couple, while the main application concerned the evaluation of momentum transfer from an expanding gas to an entrained liquid stream in droplet form. A liquid metal-gas flow was considered to simulate the process taking place in a plant where electrical power is generated by a liquid metal flowing in a magnetic field (MHD). The effectiveness of energy and momentum transfer between the liquid and the gas phase during the expansion was evaluated and the influence of nozzles with different convergence angles was investigated.     

One-group interfacial area transport equation and its sink and source terms in narrow rectangular channel

The characteristics of two-phase flow in a narrow rectangular channel are expected to be different from those in other channel geometries, because of the significant restriction of the bubble shape which, consequently, may affect the heat removal by boiling under various operating conditions. The objective of this study is to develop an interfacial area transport equation with the sink and source terms being properly modeled for the gas–liquid two-phase flow in a narrow rectangular channel. By taking into account the crushed characteristics of the bubbles a new one-group interfacial area transport equation was derived for the two-phase flow in a narrow rectangular channel. The random collisions between bubbles and the impacts of turbulent eddies with bubbles were modeled for the bubble coalescence and breakup respectively in the two-phase flow in a narrow rectangular channel. The newly-developed one-group interfacial area transport equation with the derived sink and source terms was evaluated by using the area-averaged flow parameters of vertical upwardly-moving adiabatic air–water two-phase flows measured in a narrow rectangular channel with the gap of 0.993 mm and the width of 40.0 mm. The flow conditions of the data set covered spherical bubbly, crushed pancake bubbly, crushed cap-bubbly and crushed slug flow regimes and their superficial liquid velocity and the void fraction ranged from 0.214 m/s to 2.08 m/s and from 3.92% to 42.6%, respectively. Good agreement with the average relative deviation of 9.98% was obtained between the predicted and measured interfacial area concentrations in this study.     

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.     

Effect of drag force on backward-facing step gas-particle turbulent flows

An improved drag force coefficient of gas-particle interaction based on the traditional Wen’s 1966 model is proposed. In this model, a two-stage continuous function is used to correct the discontinuous switch when porosity less than 0.2. Using this proposed correlation and the Wen’s 1966 model, a gas-particle kinetic energy and particle temperature model is developed to predict the hydrodynamic characteristics in backward-facing step gas-particle two-phase turbulent flows. Numerically results showed that they are in good agreement with experiment measurements and presented model are better due to a improvement of momentum transport between gas and particle phases. Particle dispersions take on the distinctively anisotropic behaviors at every directions and gas phase fluctuation velocity are about twice larger than particle phases. Particle phase has a unique transportation mechanism and completely different from the gas phase due to different density. Furthermore, the correlation values of axial–axial gas-particle are always greater than the radial–radial values at fully flow regions. The gas-particle two-phase interactions will make influence on two-phase turbulent flow behaviors.     

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.     

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.     

Interfacial measurements in stratified types of flow. Part I: New optical measurement technique and dry angle measurements

The development of a new non-intrusive computerized image analysis and optical observation method for accurately detecting the vapor–liquid interface in stratified two-phase flows is presented. This technique is applied to a round horizontal sight glass tube using a monochromatic laser sheet for observing two-phase flow patterns and for measuring cross-sectional dry angles and void fractions in these types of flow. The cross-sectional image observed externally is distorted by refraction and is thus reconstructed by computer. From this image, the shape of the vapor–liquid interface is detected and the dry angle and void fraction are accurately determinable over a wide range of conditions for a glass tube of 13.6 mm diameter. Results for dry angle are reported here while the test facility and void fraction measurements are presented in Part II. R-22 and R-410A are the test fluids. Dry angles were quite close to values predicted for stratified flow and much larger than comparable values for air–water flows.     

Study of two-phase flows in reduced gravity using ground based experiments

Experimental studies have been carried out to support the development of a framework of the two-fluid model along with an interfacial area transport equation applicable to reduced gravity two-phase flows. The experimental study simulates the reduced gravity condition in ground based facilities by using two immiscible liquids of similar density namely, water as the continuous phase and Therminol 59^® as the dispersed phase. We have acquired a total of eleven data sets in the bubbly flow and bubbly to slug flow transition regimes. These flow conditions have area-averaged void (volume) fractions ranging from 3 to 30% and channel Reynolds number for the continuous phase between 2,900 and 8,800. Flow visualization has been performed and a flow regime map developed which is compared with relevant bubbly to slug flow regime transition criteria. The comparison shows that the transition boundary is well predicted by the criterion based on critical void fraction. The value of the critical void fraction at transition was experimentally determined to be approximately 25%. In addition, important two-phase flow local parameters, including the void fraction, interfacial area concentration, droplet number frequency and droplet velocity, have been acquired at two axial locations using state-of-the-art multi-sensor conductivity probe. The radial profiles and axial development of the two-phase flow parameters show that the coalescence mechanism is enhanced by either increasing the continuous or dispersed phase Reynolds number. Evidence of turbulence induced particle interaction mechanism is highlighted. The data presented in this paper clearly show the marked differences in terms of bubble (droplet) size, phase distribution and phase interaction in two-phase flow between normal and reduced gravity conditions.     

Benchmarking of the one-dimensional one-group interfacial area transport equation for reduced-gravity bubbly flows

In order to properly design and safely operate two-phase flow systems, especially those deployed on future space missions, it is necessary to have accurate predictive capabilities. The application of a novel predictive method, the interfacial area transport equation (IATE), to dynamically predict the change of interfacial area concentration for reduced-gravity two-phase flows is described in this paper. Fluid particle interaction mechanisms such as coalescence and breakup that are present in reduced-gravity two-phase flows have been studied experimentally as reported in a previous paper by the current authors [Vasavada et al., 2007]. These mechanisms represent the source and sink terms in the IATE and their mechanistic models are benchmarked using experimental data obtained in a 25 mm inner diameter ground-based test section wherein reduced-gravity conditions were simulated. The comparison of the predictions from the model against experimental data shows good agreement. It has been found that, in contrast to the hypothesis extended in the literature, the wake entrainment based coalescence mechanism is present in reduced-gravity two-phase flows and in some cases is more important than coalescence due to random collision. Physics based arguments are extended to support this conclusion.     

Three-dimensional simulations of air–water bubbly flows

Interfacial area transport equation (IATE) is considered promising to evaluate dynamic changes of the interfacial area concentration in gas–liquid two-phase flows, which is of significance in characterizing the interfacial structure of the flows. Efforts were made by the authors in the past on the implementation of the IATE into computational fluid dynamics codes, such as Fluent. However, it remained unclear whether the IATE model coefficients derived from one-dimensional IATE model calibrations can be applied to three-dimensional simulations. The current study aimed to examine, primarily by investigating the lateral profiles of phase distributions, the applicability of the coefficients obtained from the one-dimensional IATE model calibration to a three-dimensional simulation of bubbly flow in a pipe. In addition, effects of the lift force on the lateral phase distributions were studied. A new set of the IATE model coefficients was suggested for a three-dimensional bubbly flow simulation. Good agreement was obtained with the updated coefficients between the predicted and measured flow parameters.     

Effect of wall friction and vortex generation on the radial distribution of different phases

Arguments are presented to prove the existence of rolling vortices in single-phase and two-phase flow. In the liquid phase, they appear in a boundary layer near a wall while in the continuous vapor phase they occur near the interface with a liquid film. The intensity and size of these vortices depend on the local velocity gradients normal to the wall. The interaction between the rotational field associated with such vortices and bubbles in liquid flow or droplets in vapor flow is discussed. This interaction may be called the wall-vortex effect. It appears that several, apparently unrelated, phenomena observed in two-phase flow systems may be interpreted in terms of this mechanism. Among these are: (i) radial void peaking near the walls (ii) vapor velocities less than liquid velocity observed also in vertical upward flow (iii) reduced droplet diffusion near the liquid film and (iv) reduced vapor mixing between subchannels at low steam qualities. The cause of secondary flows in non-circular channels may also be explained in terms of rolling vortices near the walls. Finally, a comparison is made with the well known Magnus effect.     

Modeling and computation of cavitation in vortical flow

The paper presents an investigation of Euler–Lagrangian methods for cavitating two-phase flows. The Euler–Euler methods, widely used for simulations of cavitating flows in ship technology, perform well in regions of moderate flow changes but fail in zones of strong, vortical flow. Reasons are the strong approximations of cavitation models in the Euler concept. Alternatively, Euler–Lagrangian concepts enable more detailed formulations for transport, dynamics and acoustic of discrete vapor bubbles. Test calculations are performed to study the influence of different parameters in the equations of motion and in the Rayleigh–Plesset equation for bubble dynamics. Results confirm that only Lagrangian models are able to describe correctly the bubble behavior in vortices, while Eulerian results deviate strongly. Lagrangian formulations enable additionally the determination of acoustic pressure of cavitation noise. Two-way coupling between the phases is required for large regions of the vapor phase. A new coupling concept between continuous fluid flow and discrete bubble phase is developed and demonstrated for flow through a nozzle. However, the iterative coupling between the phases via volume fractions is computationally expensive and should therefore be applied only in regions where Eulerian treatment fails. A corresponding local concept for combination with an Euler–Euler method is outlined and is in progress.     

Two-fluid modeling in analyzing the interfacial stability of liquid film flows

The area-averaged two-fluid model formulation of a separated two-phase flow system is used to investigate interfacial stability of liquid film flows. The analysis takes into account the effects of phase change at the interface as well as the dynamic effects of the adjacent vapor flow on the interfacial stability. Wave formation and instability criteria are established in terms of the generalized fluid and flow parameters. The criteria are applied to investigate the stability of laminar liquid film flow with interfacial shear and phase change. The influence of various dimensionless parameters characterizing film thickness, gravity, phase change and interfacial shear are studied with respect to the neutral stability, temporal growth factor and the wave propagation velocity. The results of the present study indicate that the interfacial stability analysis developed within the frame of the two-fluid model formulation proves to be quite accurate as judged by comparing its results with the available experimental data and with the results of much longer and more complex analytical investigations which are valid only for the liquid film free of interfacial shear.