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A spectral cascade-transport model has been developed and applied to turbulent channel flows ( Re τ = 550, 950, and 2000 based on friction velocity, u τ ; or Re δ = 8500; 14,800 and 31,000, based on the mean velocity and channel half-width). This model is an extension of a spectral model previously developed for homogeneous single and two-phase decay of isotropic turbulence and uniform shear flows; and a spectral turbulence model for wall-bounded flows without resolving the boundary layer. Data from direct numerical simulation (DNS) of turbulent channel flow was used to help develop this model and to assess its performance in the 1D direction across the channel width. The resultant spectral model is capable of predicting the mean velocity, turbulent kinetic energy and energy spectrum distributions for single-phase wall-bounded flows all the way to the wall, where the model source terms have been developed to account for the wall influence. The model has been implemented into the 3D multiphase CFD code NPHASE-CMFD and the latest results are within reasonable error of the 1D predictions.

This paper is primarily concerned with the development of three dimensional (3-D) multiphase computational fluid dynamics models for use in Pressurized Water Nuclear Reactor (PWR) design and analysis. These models were developed during the last 40 years, that is, during the time since the first NURETH meeting in 1980. The major topics in this paper include: the development of a 3-D two-fluid model for the MCFD prediction of phase distribution in turbulent adiabatic and diabatic bubbly flows. The mechanistic prediction of departure from nucleate boiling (DNB), and the direct numerical simulation (DNS) of bubbly flows. While significant progress has been made in the modeling and prediction of bubbly flows, specific recommendations are made in this paper for further improvements and for the extension of this type of MCFD model to other flow regimes.

Methodology for performing large scale parallel numerical simulations of two-phase flow is presented that uses a three-dimensional (3-D) finite element method (FEM) solver, PHASTA, to perform the simulation and a parallel mesh adaptation code, phParAdapt, to iteratively refine and coarsen regions of the solution domain. This version of PHASTA uses a stabilized FEM to discretize the 3-D Incompressible Navier–Stokes (INS) equations plus a level set method to capture the two-phase interface. The phParAdapt code takes the current parallel mesh along with the PHASTA flow solution and, using various user-specified correction indicators as input, performs a parallel mesh refinement/coarsening procedure. Utilizing local mesh modification operators, element sizes are adjusted to equally distribute the errors across the distributed mesh during the mesh refinement/coarsening procedure. Numerical simulations ranging from simple canonical test problems up to an experimental annular steam/water flow condition were performed to illustrate this methodology. The annular flow simulation captured the instabilities that generate both the ripple and roll wave like structures on the steam/water interface and the droplet entrainment mechanisms expected in annular two-phase flows.

The quantitative prediction of bubbly flow around a maneuvering surface ship is critical in determining its hydrodynamic performance and its acoustic and optical signatures. It is a challenging multiscale problem that relies heavily on subgrid models of turbulence and air entrainment. In this manuscript we analyze this problem using a phenomenological air entrainment model that predicts the location and rate of air entrainment around a surface ship. This subgrid model was coupled with a two-fluid Reynolds averaged Navier Stokes (RaNS) bubbly flow model and used to evaluate the flow field around a naval surface ship in straight ahead and turning motions. For straight ahead motion the predicted void fraction distributions aft of the stern were compared with experiments at three different ship speeds and good agreement was found. The qualitative differences in the location of the air entrainment and the resulting bubbly flow between straight ahead and steady turning motions were discussed and compared with experimental observations at sea. To our knowledge this study presents the first quantitative numerical prediction of void fraction distributions around a full-scale surface ship, well matching the experimental measurements.

Both RaNS (Reynolds-averaged Navier–Stokes) and DES (Detached Eddy Simulation) type turbulence models were used in conjunction with a two-fluid model of bubbly flow and a new subgrid air entrainment model to predict air entrainment and transport in a hydraulic jump. It was found that the void fraction profiles predicted by both methods are in agreement with the experimental data in the lower shear layer region, which contains the air bubbles entrained at the so-called toe of the hydraulic jump. In contrast, in the upper roller region behind the toe, the averaged results of the DES turbulence model gives accurate predictions while a RaNS turbulence model does not. This is because the DES turbulence model successfully captures the strong fluctuations on the free surface which allows it to entrain air near the top of the roller region. In contrast, RaNS type turbulence model results in a steady, smooth interface which fails to capture the wave-induced bubble sources in that region. To our knowledge, this study is the first successful quantitative numerical simulation of the overall void fraction profiles in a hydraulic jump.

DNS simulations of two-phase turbulent bubbly channel flow at Re τ = 180 (Reynolds number based on friction velocity and channel half-width) were performed using a stabilized finite element method (FEM) and a level set approach to track the air/water interfaces. Fully developed turbulent single-phase solutions obtained previously using the same stabilized FEM code were used as the initial flow field, and an appropriate level-set distance field was introduced to represent the air bubbles. Surface tension and gravity forces were used in the simulations to physically represent the behavior of a bubbly air/water two-phase flow having a liquid/gas density ratio of 858.3. The simulation results were averaged to obtain the liquid and gas mean velocity distributions, the local void fractions as well as the local turbulent kinetic energy and dissipation rate of the liquid phase. The liquid phase parameters were compared with the corresponding single-phase turbulent channel flow to determine the bubbles’ influence on the turbulence field.

The simulation of free surface bubbly flows using a two-fluid model remains a challenging problem in part due to the lack of a comprehensive air entrainment model that can predict the rate and location of air entrainment for a wide range of flows. In this study we derive a sub-grid model and implement it in a computational multiphase fluid dynamics (CMFD) framework to solve the Reynolds-averaged two-fluid equations. We assess the performance of our model in simulating bubbly flows underneath a plunging liquid jet and a hydraulic jump while varying the characteristic velocity. We compare the void fraction predictions with their experimental counterparts and conclude that the air entrainment model and the two-fluid modeling approach yield accurate results everywhere for the plunging jet and in the turbulent shear layer for the hydraulic jump. The inability of the proposed approach to recover the high void fraction in the roller region of the hydraulic jump is attributed to the failure of RaNS model to resolve the large coherent vortices observed in this region.

This paper presents an analysis of the interaction of pressure-drop oscillations (PDO) and density-wave oscillations (DWO) for a typical NASA type phase change system. A transient lumped parameter model is developed for use in the analysis of the dynamics of this type of system. A compressible volume (e.g., an accumulator vessel) dynamics model was also developed and PDO/DWO interactions are investigated.

The development and assessment of spectral turbulence models requires knowledge of the spectral turbulent kinetic energy distribution as well as an understanding of the terms which determine the energy distribution in physical and wave number space. Direct numerical simulation (DNS) of turbulent channel flow yields numerical “data” that can be, and was, analyzed using a spatial Fast Fourier Transform (FFT) to obtain the various spectral turbulent kinetic energy balance terms, including the production, dissipation, diffusion, and the non-linear convective transfer terms.

This paper is concerned with three-dimensional numerical simulation of a plunging liquid jet. The transient processes of forming an air cavity around the jet, capturing an initially large air bubble, and the break-up of this large toroidal-shaped bubble into smaller bubbles were analyzed. A stabilized finite element method (FEM) was employed under parallel numerical simulations based on adaptive, unstructured grid and coupled with a level-set method to track the interface between air and liquid. These simulations show that the inertia of the liquid jet initially depresses the pool's surface, forming an annular air cavity which surrounds the liquid jet. A toroidal liquid eddy which is subsequently formed in the liquid pool results in air cavity collapse, and in turn entrains air into the liquid pool from the unstable annular air gap region around the liquid jet.

The accurate prediction of air entrainment is critical in simulating various important multiphase (air/water) flows. In this paper, we present a sub-grid air entrainment model that quantitatively predicts the rate of air entrainment and subsequent disperse bubbly flow for a plunging jet. The derivation of this model is based on the two-stage (i.e., low and high liquid jet velocity) air entrainment mechanisms suggested by Sene [Sene KJ. Air entrainment by plunging jets. Chem Eng Sci 1988;43(10):2615–23]. This model was validated against extensive experimental data for water jets in air over a wide range of liquid velocities (from around 1 to 10 m/s) for the total rate of air entrainment. It was then implemented into an Eulerian/Eulerian two-fluid computational multiphase fluid dynamics (CMFD) model, wherein the liquid and the bubbles are modeled as two distinct continua. This multiphase model, supplemented by the new sub-grid air entrainment model, was used to predict the void fraction distribution underneath plunging water jets at different depths and water jet velocities. It was found that this approach yields results that match the experimental observations very well.

This paper presents the direct numerical simulation (DNS) of various incompressible and compressible single and two-phase flows. It is shown that very detailed information on the flow structure can be obtained using a suitable adaptive grid 3D solver. In addition, it is demonstrated how to use these detailed numerical “data” to obtain interfacial closure models for use in two-fluid, computational multiphase fluid dynamic (CMFD) formulations. These advanced generation 3D CMFD models can, in-turn, be used for nuclear reactor design and analysis.

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.

The analysis of turbulent two-phase flows requires closure models in order to perform reliable computational multiphase fluid dynamics (CMFD) analyses. A spectral turbulence cascade-transport model, which tracks the evolution of the turbulent kinetic energy from large to small liquid eddies, has been developed for the analysis of the homogeneous decay of isotropic single and bubbly two-phase turbulence. This model has been validated for the decay of homogeneous, isotropic single and two-phase bubbly flow turbulence for data having a 5 mm mean bubble diameter. The Reynolds number of the data based on bubble diameter and relative velocity is approximately 1400.

Direct numerical simulations (DNS) of air entrainment by the breaking bow waves of naval surface ships are outside of the computational reach of the most powerful computers in the foreseeable future. This creates a need for physically-based models of air entrainment for applications in numerical simulations for ship design. Due to the non-linear dependence of the terminal bubble velocity with diameter most air entrainment flows have a high void fraction region immediately below the free surface. We present a model that locates this region employing the local liquid velocity and the distance to the interface. Using this model and the bubble size distributions measured by Deane and Stokes [Deane BD, Stokes MD. Scale dependence of bubble creation mechanisms in breaking waves. Nature 2002;418:839–44] we have simulated the air entrainment in the breaking wave experiments of Wanieski et al. [Waniewski TA, Brennen CE, Raichlen F. Measurement of air entrainment by bow waves. J Fluids Eng 2001;123:57–63]. Comparison against these experimental data is good. We then apply this model to simulate the flow around naval combatant DTMB 5415 and the research vessel Athena. The model predicts air entrainment in regions where it was actually observed at sea, namely the breaking bow wave, along the water/air/hull contact line and in the near-wake. To the best of our knowledge this is the first model of air entrainment that compares favorably with data at laboratory scale and also presents the right trends at full-scale.

Advanced generation three-dimensional (3-D) two-fluid models and computational multiphase fluid dynamic (CMFD) solvers give good predictions of dispersed flows on earth (i.e., at 1 g), where the lift, wall, and turbulent dispersion forces determine the lateral void fraction distribution. However, for microgravity (μ-g) conditions these buoyancy-related forces become quite small and two-fluid model predictions are generally inadequate. This implies that some of the physics lost during ensemble averaging was not included back into the two-fluid model with the closure laws that were used. Recent modeling advances that include the effect of the phasic velocity fluctuations on the phase distribution are presented. The resultant two-fluid model was evaluated and compared with various data sets for steady, fully developed turbulent conditions using a novel one-dimensional (1-D) CMFD solver that is numerically very efficient. It was found that with the addition of these new physical mechanisms to the closure laws...

The hydrodynamics of the implosion and rebound of a small (10@mm diameter) air bubble in water was studied using a three-dimensional direct numerical simulation (DNS). To study this problem, we developed a novel stabilized finite element method (FEM) employing a combination of ghost fluid and level set approaches. This formulation treats both the air and water as compressible fluids. Using this method, a transient three-dimensional (3-D) solution was obtained for the implosion (i.e., collapse) and rebound of an air bubble. These simulation results obtained were qualitatively similar to those observed/predicted in previous experimental/numerical studies. The 3-D simulations show that the conditions within the bubble are nearly uniform until the converging pressure wave is strong enough to create very large temperatures and pressures near the center of the bubble. These dynamics occur on very small spatial (0.1-0.7@mm), and time (ns) scales. The motion of the air/water interface during the initial stages of the implosion was found to be consistent with predictions using a Rayleigh-Plesset model. However, the simulations showed that during the final stage of energetic implosions, the bubble can become asymmetric, which is contrary to the spherical symmetry assumed in many previous numerical studies of bubble dynamics. The direct numerical simulations predicted two different instabilities, namely Rayleigh-Taylor type interfacial/surface and shape instabilities. During the violent collapse stage, the bubble deviates from spherical symmetry and deforms into an ellipsoidal-shaped bubble. A linear stability analysis based on spherical harmonics also indicates that an ellipsoidal bubble shape could be expected. Moreover, interfacial instabilities also appear during the later stage of the implosion process. Distinguishing these phenomena with the help of numerical simulations opens new opportunities to understand many features of recent experiments on sonoluminescence and sonofusion.

In a bubbly flow, the presence or absence of the dispersed phase (gas) at a given instant and a given point in space depends on two factors, namely: (a) the forces acting on the bubble (or, more generally the dispersed particle) as a whole and responsible for its movement, and (b) the shape (geometry) of the bubble, which contributes to define if the spatial point in question lays inside or outside the bubble at that instant. Based on that point of view, we propose herein a two-fluid model that involves a particle-center-averaging procedure for the disperse phase, and a re-interpretation and post-processing of the results obtained. This center-averaged approach averages the disperse phase (bubbles) based on a particle center indicator function, while retaining standard phase indicator averaging for the continuous phase (liquid). The solution fields obtained are then post-processed to introduce the geometry of the bubbles in order to recover the values that are representative of the measured fields. The key idea here is to separate the geometric aspect from the dynamic aspect of the problem into two independent, successive steps. Tracking the bubble centers makes it possible to model the wall-induced forces more accurately. We take advantage of this fact to propose models for the wall normal force for very small Weber numbers and for Weber numbers of order unity. The additional tangential drag induced by the wall is also examined in light of recent experimental evidence. The resultant two-fluid model may be easily incorporated into existing two-fluid model codes. Results obtained with the new model showing agreement with experimental data are also presented.

A model for the wallinduced forces on bubbles for application in two-fluid model simulations of bubbly flows has been derived. The wall-induced force is obtained as the integral of an excess pressure calculated from the solution of the lubrication equation. Our numerical results are compared with experiments and it is shown that the wall-induced force is a function of the buoyancy and added mass forces, modulated by a smoothed Heaviside function that accounts for the proximity to the wall. This model explicitly accounts for the Reynolds, Eotvos and Weber numbers of the bubble and is valid in a larger range of these numbers than previous models.

We present analytical results for stable stratified wavy two-phase flow and functional forms for the various interfacial force densities in a two-fluid model. In particular, we have derived analytically the components of the non-drag interfacial force density [Drew, D.A., Passman, S.L., 1998. Theory of Multicomponent Fluids. Springer-Verlag, New York; Nigmatulin, T.R., Drew, D.A., Lahey, R.T., Jr., 2000. An analysis of wavy annular flow. In: International Conference on Multiphase Systems, ICMS’2000, Ufa, Russia, June 15–17], Reynolds stress tensor, and the term, (p˜cli−p¯cl)∇αcl, where p˜cli is interfacial average pressure, p¯cl the average pressure, and αcl is the volume fraction of the continuous liquid phase. These functional forms should be useful for assessing two-fluid closure relations and Computational Multiphase Fluid Dynamics (CMFD) numerical models for stratified wavy flows. Moreover, it appears that this approach can be generalized to other flow regimes (e.g., annular flows).

Two-fluid turbulent dispersion models have been compared with direct numerical simulations (DNS) of a decaying turbulence bubbly flow in the low Stokes number limit, St≈10−3. Because of the absence of empiricism, DNS results represent an excellent means of assessing turbulent dispersion models. Sufficiently far away from the inlet of the channel, where the turbulence was fully developed, these turbulent dispersion models were able to predict the DNS results when a Schmidt number, Scb=0.83, was used. This result highlights the fact that even bubbles of diameter, Db=42 μm, considerably smaller than the Kolmogorov length scale, η=75 μm, do not behave as passive scalars for which Scb=1. In addition, these models were also assessed against a bubbly mixing layer flow having a low Stokes number, St

This paper reviews the state-of-the-art in the prediction of multidimensional multiphase flow and heat transfer phenomena using a four field, two-fluid model. It is shown that accurate mechanistic computational fluid dynamic (CFD) predictions are possible for a wide variety of adiabatic and diabatic flows using this computational model. In particular, the model is able to predict the bubbly air/water upflow data of Serizawa (Serizawa, A., 1974. Fluid dynamic characteristics of two-phase flow. Ph.D. thesis, (Nuclear Engineering), Kyoto University, Japan), the downflow data of Wang et al. (Wang, S.K., Lee, S.J., Lahey Jr., R.T., Jones, O.C., 1987. 3-D turbulence structure and phase distribution measurements in bubbly two-phase flows. Int. J. Multiphase Flow 13 (3), 327–343), the isosceles triangle upflow data of Lopez de Bertodano et al. (Lopez de Bertodano, M., Lahey Jr., R.T., Jones, O.C., 1994b. Phase distribution in bubbly two-phase flow in vertical ducts. Int. J. Multiphase Flow 20 (5), 805–818), the heated annular R-113 subcooled boiling data of Velidandala, et al. (Velidandla, V., Pulta, S., Roy, P., Kaira, S.P., 1995. Velocity field in turbulent subcooled boiling flow. ASME Preprint HTD-314, 107–123) and the R-113 CHF data of Hino and Ueda (Hino, R., Ueda, T., 1985. Studies on heat transfer and flow characteristics in subcooled boiling-part 2, flow characteristics. Int. J. Multiphase Flow 11, 283–297). It can also predict external two-phase flows, such as those for spreading two-phase jets (Bonetto, F., Lahey Jr., R.T., 1993. An experimental study on air carryunder due to a plunging liquid jet. Int. J. Multiphase Flow 19 (2), 281–294) and multiphase flows around the hull of naval surface ships (Carrica, P.M., Bonetto, F., Drew, D.A., Lahey, R.T., 1999. A polydispersed model for bubbly two-phase flow around a surface ship. Int. J. Multiphase Flow 25 (2), 257–305).

This paper presents experimental data for dispersed liquid/liquid upflows. Water was the continuous phase and mineral oil was the dispersed droplet phase. For this flow regime reduced gravity bubbly flow phenomena was simulated because the mineral oil and water had almost the same density. The mean velocity and turbulence fields, the size distributions of the oil droplets, the volume fraction, and interfacial area density distribution were measured using fiber optic Laser Doppler Anemometer (LDA) and phase Doppler Anemometer (PDA) systems. Significantly, the results presented in this paper are similar to those for bubbly air/water flows in microgravity conditions (Kamp el at., 1995).

A spherically-symmetric problem is considered in which a small gas bubble at the centre of a spherical flask filled with a compressible liquid is excited by small radial displacements of the flask wall. The bubble may be compressed, expanded and made to undergo periodic radial oscillations. Two asymptotic solutions have been found for the low-Mach-number stage. The first one is an asymptotic solution for the field far from the bubble, and it corresponds to the linear wave equation. The second one is an asymptotic solution for the field near the bubble, which corresponds to the Rayleigh–Plesset equation for an incompressible fluid. For the analytical solution of the low-Mach-number regime, matching of these asymptotic solutions is done, yielding a generalization of the Rayleigh–Plesset equation. This generalization takes into account liquid compressibility and includes ordinary differential equations (one of which is similar to the well-known Herring equation) and a difference equation with both lagging and leading time. These asymptotic solutions are used as boundary conditions for bubble implosion using numerical codes which are based on partial differential conservation equations. Both inverse and direct problems are considered in this study. The inverse problem is when the bubble radial motion is given and the evolution of the flask wall pressure and velocity is to be calculated. The inverse solution is important if one is to achieve superhigh gas temperatures using non-periodic forcing (Nigmatulin et al. 1996). In contrast, the direct problem is when the evolution of the flask wall pressure or velocity is given, and one wants to calculate the evolution of the bubble radius. Linear and nonlinear periodic bubble oscillations are analysed analytically. Nonlinear resonant and near-resonant periodic solutions for the bubble non-harmonic oscillations, which are excited by harmonic pressure oscillations on the flask wall, are obtained. The applicability of this approach bubble oscillations in experiments on single-bubble sonoluminescence is discussed.

Local measurements of the velocity and phase distribution were made for solid/liquid two-phase flows in a horizontal pipe using a multidimensional laser Doppler anemometer (LDA) system. The pipe used had a 30.6 mm inner diameter and was made of a special optically clear material, Fluorinated Ethylene Propylene (FEP), which has the same index of refraction as water. The test section and laser probes were submerged in water so that no optical corrections were necessary. Simultaneous measurements of the liquid and particle phases were obtained and discriminated based on differences in the residence time that each particle had when crossing the LDA's measurement volume. Positive buoyant and negative buoyant solid spherical particles, about 2 mm in diameter, were used in this study. A complete data set of the phasic mean velocities, turbulence and volume fractions were taken for each particle type. The results presented herein are complete and self-consistent, and can be used to assess the predictive ...

A three-dimensional two-fluid model has been developed using ensemble-averaging techniques. The two-fluid model was closed for adiabatic two-phase bubbly flows using cell averaging which accounted for the dispersed phase distribution in the region of the averaging volume. The phasic interfacial momentum exchange includes the surface stress developed on the interface which is induced by the relative motion of the phases. The surface stress has been obtained by treating the interface as an elastic spherical shell. A characteristic analysis revealed that the one-dimensional system of two-fluid conservation equations which were derived is well-posed over a range of void fractions with increased value of the interfacial pressure. The propagation of void fraction disturbances (i.e. the void wave) has also been analyzed by performing a dispersion analysis. The speed, stability and damping of the linear void waves have been obtained. To study finite amplitude void waves, the system of equations has been transformed into a moving coordinate system, and asymptotic solutions of the transformed nonlinear void wave equation have been obtained. The speed and the stability of different types of nonlinear void waves have been found to be sensitive to the closure relations of the two-fluid model. Among the different constitutive parameters, the interfacial pressure difference in the continuous phase and the void fraction gradient in the non-drag force are found to be the most significant in determining behavior of void waves in bubbly flows. The derived void wave speed agrees well with the void wave data of bubbly air–water flow.

The lateral force on a tethered rigid sphere submerged in a turbulent, uniform shear flow of water was measured. Periodic and non-periodic motions of the sphere were observed depending on flowrate, shear and sphere density. The direction of the observed lateral force was opposite to that predicted by inviscid theory and increased in magnitude as the sphere’s Reynolds numbers based on relative velocity, Re ,a nd average shear, Rer, increased. The lateral force was found to correlate with the product Re Rer. The data suggests that a sign reversal occurs at relatively small values of the product Re Rer, where the lateral force is dominated by inviscid eAects. The results are explained assuming that the lateral forces on rigid spheres are a consequence of two competing factors: namely, inviscid lift forces and the vortex shedding-induced lateral forces which are dominant for higher Reynolds numbers. An estimate of the kinetic energy in the wake was used to show that the vortex shedding-induced lateral forces correlate with the product Re Rer and are in a direction opposite to the inviscid lift force. Combining the experimental data of this study with similar data a correlation for the lift coeAcient of spheres in turbulent shear flows was developed. This correlation is applicable to turbulent multiphase flows having a spherical dispersed phase. # 1999 Elsevier Science Ltd. All rights reserved.

An analysis of dilute, turbulent particle/liquid two-phase flow is presented. The three-dimensional conservation equations that govern turbulent motion in solid/fluid flows are derived using ensemble averaging, and the unknown terms in these equations are constituted to achieve closure. These closure terms include the shear stress due to interparticle collisions, the corresponding terms in the Reynolds stress equation, the force and dissipation due to particle-wall collisions, and the interfacial work due to particle/turbulent eddy interaction. The resultant two-fluid model was then evaluated using a computational fluid dynamic (CFD) solver and the predictions were compared with experimental data. Good agreement was observed for a variety of flow conditions.

A three dimensional polydisperse model for bubbly two-phase flow around a surface ship is presented. The Boltzmann equation for the bubble mass probability density function is evaluated using a multigroup approach with groups of constant bubble mass. The intergroup transfer mechanisms are bubble breakup, coalescence and the dissolution of air into the ocean, and their effects on the two-phase flow field are analyzed. A three dimensional two-fluid model is used for each bubble mass group to calculate the group’s average gas velocity, resulting in four scalar equations per group. The air entrainment process is modeled using simulated breaking bow waves and the steady-state evolution of the gas bubbles for zero Froude number is obtained. It was found that intergroup transfer is very important in these flows. Some of the research areas that need further improvement for the numerical prediction of polydisperse two-phase flow around a ship have been identified and are discussed.

A two-fluid model suitable for the calculation of the two-phase flow field around a naval surface ship is presented. This model couples the Reynolds-averaged Navier–Stokes (RANS) equations with equations for the evolution of the gas-phase momentum, volume fraction and bubble number density, thereby allowing the multidimensional calculation of the two-phase flow for monodisperse variable size bubbles. The bubble field modifies the liquid solution through changes in the liquid mass and momentum conservation equations. The model is applied to the case of the scavenging of wind-induced sea-background bubbles by an unpropelled US Navy frigate under non-zero Froude number boundary conditions at the free surface. This is an important test case, because it can be simulated experimentally with a model-scale ship in a towing tank. A significant modification of the background bubble field is predicted in the wake of the ship, where bubble depletion occurs along with a reduction in the bubble size due to dissolution. This effect is due to lateral phase distribution phenomena and the generation of an upwelling plume in the near wake that brings smaller bubbles up to the surface. © 1998 John Wiley & Sons, Ltd.

The spherically-symmetric problem of the oscillations of a small gas bubble in the center of a spherical flask filled with a compressible liquid that is excited by small radial displacement of the flask's wall is considered. Two asymptotic solutions have been found for the low Mach number stage. The first one is an asymptotic solution for the field far from the bubble, and it corresponds to linear wave theory. The second one is an asymptotic solution for the boundary layer near the bubble and it corresponds to an incompressible fluid. In the analytical solution of the low Mach number step matching of these asymptotic solutions is done. A generalization of the Rayleigh = Plesset equation for a compressible liquid is given in the form of two ordinary difference-differential equations that take into account the pressure waves which are reflecting from the bubble and those that are incident on the bubble from the flask wall. The initial value problem for the initiation of the bubble oscillations due ...

An analytic model for the investigation of non-linear dynamics in boiling systems has been developed. This model is comprised of a nodal formulation that uses one-dimensional homogeneous equilibrium assumptions for diabatic two-phase flow, a lumped parameter approach for heated wall dynamics, and point neutron kinetics for the consideration of nuclear feedback in a boiling water reactor (BWR) loop. This model indicates that a boiling channel coupled with a riser may experience chaotic oscillations. In contrast, a boiling channel without a riser that is subjected to a constant pressure drop (i.e. a parallel channel) may undergo a supercritical bifurcation (i.e. may experience a limit cycle), but chaos was not found. Flow instabilities in a two-phase natural circulation loop have been verified using the model presented in this paper. The predictions of the effects of the channel inlet resistance, outlet resistance and liquid level in the downcomer agree with the data of Kyung and Lee. Finally, an analysis of nuclear-coupled density-wave instabilities in a simplified BWR (SBWR) was performed. Significantly, even for low pressure conditions, a simplified SBWR appears to be stable during start-up and normal operations; however, a limit cycle may occur for abnormal operating conditions.

A method for superhigh compression of a gas bubble in a liquid due to forced non-linear, non-periodic resonance oscillations with moderate to small amplitudes of forcing pressure is proposed and analyzed. The key feature of the method is the coordination of the liquid pressure change with the change of bubble volume. The corresponding regime of forced oscillations is called the “basketball dribbling regime”. Analytical solutions describing the process are obtained.

Plunging liquid jets are commonly encountered in nature and are widely used in industrial applications (e.g., in waterfalls, waste-water treatment, the oxygenation of chemical liquids, etc.). Despite numerous experimental studies that have been devoted to this interesting problem, there have been very few two-phase flow simulations. The main difficulty is the lack of a quantitative subgrid model for the air entrainment process, which plays a critical role in this problem. In this paper, we present in detail a computational multiphase fluid dynamics (CMFD)-based approach for analyzing this problem. The main ingredients of this approach are a comprehensive subgrid air entrainment model that predicts both the rate and location of the air entrainment and a two-fluid transport model, in which bubbles of different sizes are modeled as a continuum fluid. Using this approach, a Reynolds-averaged Navier Stokes (RaNS) two-way coupled two-phase flow simulation of a plunging liquid jet with a diameter of 24 mm and a liquid jet velocity around 3.5 m/s was performed. We have analyzed the simulated void fraction and bubble count rate profiles at three different depths beneath the average free surface and compared them with experimental data in literature. We observed good agreement with data at all locations. In addition, some interesting phenomena on the different movements of bubbles with different sizes were observed and discussed.

The lateral phase distribution in bubbly flows in vertical ducts was analyzed using a three-dimensional two-fluid model. The constitutive relations of the model are based on analytic and experimental information on the behavior of a single bubble, and on the assumption of linear superposition of shear-induced and bubble-induced turbulence. The experiments chosen to test the model include available data for pipes and new data obtained in an isosceles triangular duct. While most of the data could be reproduced satisfactorily by the model, some could not. This is attributed to certain physical mechanisms that are still not well understood and therefore were not included in the constitutive relations.

The propagation of void fraction disturbances (i.e. void waves) in bubbly two-phase air-oil flow was investigated experimentally. The void wave data often showed two simultaneous void wave propagations; in particular, the classical kinematic void wave and a faster void wave propagation associated with bubble clusters. Significantly, the bubbly-to-slug flow regime transition was found to be associated with amplification of the void waves associated with the propagating bubble clusters. These data should be valuable for the assessment of the closure laws used in two-fluid models and for the development of mechanistic models for the bubbly-slug flow regime transition.

The phase distribution and turbulence structure for solid/fluid upflow in a vertical pipe were investigated. Spherical particles, approximately 2 mm in diameter, were used and runs were made with particles having two different specific gravities. In particular, ceramic particles, which were heavier than water, and expanded polystyrene particles, which were lighter than water, were used. A new method is presented for the measurement of the volume fraction in solid/fluid two-phase flows using a laser-Doppler anemometer (LDA). The measured local time fractions obtained with the LDA must be corrected, because bias is produced by the presence of natural seeding, the finite size of the measurement volume and interruptions of the laser beams by the dispersed particles. An analytical method has been developed which accounts for these effects. A single-beam traversing γ-ray densitometer was used as a reference against which to assess the volume fraction correction method. Good agreement between the the corrected LDA and γ-ray densitometer results was obtained. The volume fraction profiles show that at low flow rates the ceramic particles have an almost uniform distribution, while increasing the flow rate causes coring. In contrast, the phase distribution of the light polystyrene particles had wall peaking for both the low and high flow rates. However, wall peaking was flattened as liquid flow rate was increased.

An extension of the k-ε model for bubbly two-phase flow is proposed and tested against experimental data. The basic assumption made is that the shear-induced turbulence and bubble-induced turbulence may be linearly superposed. This assumption results in a model with two time constants that matches both homogeneous two-phase turbulence data (Lance and Bataille, 1991) and pipe data (Serizawa, 1986). The coefficients of the single-phase k-ε model have not been modified and only one additional coefficient is required: the virtual volume coefficient of the bubbles, which may be determined from first principles. This model not only agrees with the data trends, but it also predicts the turbulence suppression which has been measured for high Reynolds number bubbly air/water flows in pipes.

The temperature profile for an N- component mixture of particles in a fluid flowing through a heated tube has been evaluated using a numerical method for the fluid and an analytical solution for the particles. The results for the time-at-temperature of the particles depend on the drift-flux parameters, which quantify the local slip. In particular, the more slip the better the heat transfer and the shorter the heat exchanger required for aseptic food processing. Two diagrams are given, which cover all parameters of interest for laminar flow.

Multidimensional bubbly two-phase flow data was obtained in an isosceles triangular duct having D h = 40 mm and L / D = 73. The data include local measurements of phase distribution, velocity and Reynolds stresses at 10,000 Re 1 bubble = 1250. The measurements were performed with single sensor and X-sensor cylindrical hot film probes of 0.025 mm diameter. These small probes have good bubble penetration characteristics so they are capable of the simultaneous measurement of the liquid phase instantaneous velocity and the phase indicator function. These data are useful for the validation of multidimensional two-fluid CFD models.

This paper details experiments and analyses regarding the phenomenon of liquid discharge into a gaseous atmosphere from the bottom of a vessel with an unvented, upper gas space. The primary goal is the development of a simple model that predicts the rate of liquid discharge under the prevailing unvented condition. A literature survey of previous work on this phenomenon yielded only simple experiments and analyses that were limited in scope. Experiments were subsequently undertaken with an air-water system, using a larger volume and a wide range of drain line diameters. In addition to flowrate data, visual information was acquired regarding the physical mechanism possibly governing the prevalent flow regimes. The governing physical mechanism is identified as the stability of a gas-liquid interface, perturbed by buoyancy, at the drain line entrance. G.I. Taylor's fundamental analysis of interfacial stability lead to the determination of criteria for flow regime transition among the three prevalent flow regimes, corresponding to so-called small, medium, and large diameters. Also, analysis of the growth of interfacial instabilities lead to the application of flooding models for drainage rates within each regime. The models for moderate and large diameters were then compared against data, which confirmed their success in predicting discharge rates under the unvented condition. The motivation for this effort, besides the basic scientific significance of studying such a fundamental phenomenon, was its numerous applications, one of which is commercial nuclear reactor systems. Specifically, the phenomenon prevails in liquid coolant discharge from a PWR pressurizer, with an unvented steam volume, into a steam atmosphere existing in the adjoining hot coolant leg. Such a phenomenon could occur as part of a transient, or severe accident, scenario, entailing saturated conditions and steam production in the normally subcooled primary heat transport loop. The developed model was implemented in the Modular Accident Analysis Program (MAAP), a computer code designed to predict reactor system behavior in response to postulated off-normal conditions, including severe accident scenarios.

This paper presents an analysis on void wave phenomena in bubbly flows. Both linear and nonlinear analyses are performed. A dispersion relation is derived and the criterion for the formation of void shocks and solitons is given. It is shown that void wave phenomena is quite sensitive to the closure laws used in the two-fluid models. As a consequence void wave data appears to be very useful for two-fluid model assessment.

Bubbly air/water two-phase flow data have been taken as an isosceles triangle using hot film probes. It was found that a 3-D two-fluid model was able to predict these data and those taken previously in circular conduits. It appears that mechanically-based CFD predictions of bubbly two-phase flows is possible for many cases of practical concern.

This paper presents experimental data on the air entrainment process for a plunging liquid jet. Local data, including the size distribution of the bubble diameter, void fraction, bubble velocities and turbulent liquid velocities were obtained using a fiber optics phase Doppler anemometer system. A dual element conductivity probe was also used to measure the spatial distribution of the local void fraction. For the larger bubbles, the turbulence of the plunging jet was parametrically controlled using a specially designed nozzle and the turbulence intensity was measured using a laser Doppler anemometer system.

Plunging liquid jets are commonly encountered in nature and are widely used in industrial applications (e.g., in waterfalls, waste-water treatment, the oxygenation of chemical liquids, etc.). Despite numerous experimental studies that have been devoted to this interesting problem, there have been very few two-phase flow simulations. The main difficulty is the lack of a quantitative model to simulate the air entrainment process, which plays a critical role in this problem. In this paper, we present a computational multiphase fluid dynamics (CMFD) approach for solving this problem. The main ingredients of this approach are a comprehensive subgrid air entrainment model that predicts the rate and location of the air entrainment and a two-fluid transport model in which bubbles of different sizes are modeled as a continuum fluid. Using this approach, a Reynolds-averaged Navier Stokes (RaNS) two-way coupled two-phase flow simulation of a plunging liquid jet with a diameter of 24mm and a liquid jet velocity around 3.5m/s was performed. We analyzed the simulated void fraction and bubble count rate profiles at three different depths beneath the average free surface, and compared them with experimental data. We observed good agreement at all locations.

New data is presented for horizontal air/water two-phase flow having various flow regimes. It is shown that drift-flux models are able to correlate these data and that the drift velocity, V g j , is normally finite.

Three-phase flow regimes for an air/water/oil system flowing in a horizontal pipe line were observed and flow regime maps were constructed. Superficial velocities, ranging from 15 to 5000 cm/s for air ( j a ) and from 0.4 to 66 cm/s for water ( j w ), were investigated. The superficial velocity of the immiscible oil “phase” ( j o ) was kept constant at three discrete values, 4.3, 9 and 24 cm/s. Water- and oil-based wavy, plug, slug and stratifying-annular flow regimes were observed. Many interesting new flow regime configurations not seen in two-phase flow were found.