Showing papers on "Reynolds number published in 1994"

Numerical simulations of particulate suspensions via a discretized boltzmann equation: part 1. theoretical foundation

Abstract: A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

A New K-epsilon Eddy Viscosity Model for High Reynolds Number Turbulent Flows: Model Development and Validation

Abstract: A new k-epsilon eddy viscosity model, which consists of a new model dissipation rate equation and a new realizable eddy viscosity formulation, is proposed. The new model dissipation rate equation is based on the dynamic equation of the mean-square vorticity fluctuation at large turbulent Reynolds number. The new eddy viscosity formulation is based on the realizability constraints: the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. We find that the present model with a set of unified model coefficients can perform well for a variety of flows. The flows that are examined include: (1) rotating homogeneous shear flows; (2) boundary-free shear flows including a mixing layer, planar and round jets; (3) a channel flow, and flat plate boundary layers with and without a pressure gradient; and (4) backward facing step separated flows. The model predictions are compared with available experimental data. The results from the standard k-epsilon eddy viscosity model are also included for comparison. It is shown that the present model is a significant improvement over the standard k-epsilon eddy viscosity model.

Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results

Abstract: A new and very general technique for simulating solid–fluid suspensions has been described in a previous paper (Part 1); the most important feature of the new method is that the computational cost scales linearly with the number of particles. In this paper (Part 2), extensive numerical tests of the method are described; results are presented for creeping flows, both with and without Brownian motion, and at finite Reynolds numbers. Hydrodynamic interactions, transport coefficients, and the short-time dynamics of random dispersions of up to 1024 colloidal particles have been simulated.

Dynamics of Drop Deformation and Breakup in Viscous Fluids

Abstract: This article describes the dynamics of drop deformation and breakup in viscous flows at low Reynolds numbers. An attempt has been made to bring together a wide range of studies in the drop deformation literature, as well as to provide a large number of references to potential applications. In particular, a summary is provided of experimental, numerical, and theoretical investigations that examine drop breakup in externally­ imposed flows, e.g. uniaxial extensional fluid motion or more complicated time-periodic flows. For well-characterized flow conditions that lead to breakup, the effects of flow and material parameters on the drop size distribution are summarized. Also, a short discussion is given of the stability of the shapes of translating drops. The subject of deformation of neutrally buoyant drops in viscous shear flows at low particle Reynolds numbers was summarized by Acrivos ( 1983) and was reviewed in this series by Rallison ( 1 984). The Acrivos and Rallison papers present (a) theoretical descriptions of steady, nearly spheri­ cal shapes and steady, long slender shapes, (b) a description of efficient boundary integral numerical methods, and (e) a summary of the experi­ mental work performed prior to 1984. As documented in these review articles, many of the important ideas necessary for understanding drop

Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet

Abstract: The turbulent flow resulting from a top-hat jet exhausting into a large room was investigated The Reynolds number based on exit conditions was approximately 105 Velocity moments to third order were obtained using flying and stationary hot-wire and burst-mode laser-Doppler anemometry (LDA) techniques The entire room was fully seeded for the LDA measurements The measurements are shown to satisfy the differential and integral momentum equations for a round jet in an infinite environmentThe results differ substantially from those reported by some earlier investigators, both in the level and shape of the profiles These differences are attributed to the smaller enclosures used in the earlier works and the recirculation within them Also, the flying hot-wire and burst-mode LDA measurements made here differ from the stationary wire measurements, especially the higher moments and away from the flow centreline These differences are attributed to the cross-flow and rectification errors on the latter at the high turbulence intensities present in this flow (30% minimum at centreline) The measurements are used, together with recent dissipation measurements, to compute the energy balance for the jet, and an attempt is made to estimate the pressure-velocity and pressure-strain rate correlations

Local isotropy in turbulent boundary layers at high Reynolds number

Abstract: To test the local-isotropy predictions of Kolmogorov's universal equilibrium theory, we have taken hot-wire measurements of the velocity fluctuations in the test-section-ceiling boundary layer of the 80×120 foot Full-Scale Aerodynamics Facility at NASA Ames Research Center, the world's largest wind tunnel. The maximum Reynolds numbers based on momentum thickness, R θ , and on Taylor microscale, R λ , were approximately 370 000 and 1450 respectively. These are the largest ever attained in laboratory boundary-layer flows. The boundary layer develops over a rough surface, but the Reynolds-stress profiles agree with canonical data sufficiently well for present purposes

Pressure-driven flow of suspensions : simulation and theory

Abstract: Dynamic simulations of the pressure-driven flow in a channel of a non-Brownian suspension at zero Reynolds number were conducted using Stokesian Dynamics. The simulations are for a monolayer of identical particles as a function of the dimensionless channel width and the bulk particle concentration. Starting from a homogeneous dispersion, the particles gradually migrate towards the centre of the channel, resulting in an homogeneous concentration profile and a blunting of the particle velocity profile. The time for achieving steady state scales as (H/a)3a/[left angle bracket]u[right angle bracket], where H is the channel width, a the radii of the particles, and [left angle bracket]u[right angle bracket] the average suspension velocity in the channel. The concentration and velocity profiles determined from the simulations are in qualitative agreement with experiment. A model for suspension flow has been proposed in which macroscopic mass, momentum and energy balances are constructed and solved simultaneously. It is shown that the requirement that the suspension pressure be constant in directions perpendicular to the mean motion leads to particle migration and concentration variations in inhomogeneous flow. The concept of the suspension ‘temperature’ – a measure of the particle velocity fluctuations – is introduced in order to provide a nonlocal description of suspension behaviour. The results of this model for channel flow are in good agreement with the simulations.

Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment

Abstract: Direct numerical simulations (DNS) and experiments are carried out to study fully developed turbulent pipe flow at Reynolds number Rec ≈ 7000 based on centreline velocity and pipe diameter The agreement between numerical and experimental results is excellent for the lower-order statistics (mean flow and turbulence intensities) and reasonably good for the higher-order statistics (skewness and flatness factors) To investigate the differences between fully developed turbulent flow in an axisymmetric pipe and a plane channel geometry, the present DNS results are compared to those obtained from a channel flow simulation Beside the mean flow properties and turbulence statistics up to fourth order, the energy budgets of the Reynolds-stress components are computed and compared The present results show that the mean velocity profile in the pipe fails to conform to the accepted law of the wall, in contrast to the channel flow This confirms earlier observations reported in the literature The statistics on fluctuating velocities, including the energy budgets of the Reynolds stresses, appear to be less affected by the axisymmetric pipe geometry Only the skewness factor of the normal-to-the-wall velocity fluctuations differs in the pipe flow compared to the channel flow The energy budgets illustrate that the normal-to-the-wall velocity fluctuations in the pipe are altered owing to a different ‘impingement’ or ‘splatting’ mechanism close to the curved wall

An experimental investigation of the flow around a circular cylinder : influence of aspect ratio

Abstract: The investigation is concentrated on two important quantities – the Strouhal number and the mean base suction coefficient, both measured at the mid-span position Reynolds numbers from about 50 to 4 × 104 were investigated Different aspect ratios, at low blockage ratios, were achieved by varying the distance between circular end plates (end plate diameter ratios between 10 and 30) It was not possible, by using these end plates in uniform flow and at very large aspect ratios, to produce parallel shedding all over the laminar shedding regime However, parallel shedding at around mid-span was observed throughout this regime in cases when there was a slight but symmetrical increase in the free-stream velocity towards both ends of the cylinder At higher Re, the results at different aspect ratios were compared with those of a ‘quasi-infinite cylinder’ and the required aspect ratio to reach conditions independent of this parameter, within the experimental uncertainties, are given For instance, aspect ratios as large as L/D = 60–70 were needed in the range Re ≈ 4 × 103–104 With the smallest relative end plate diameter and for aspect ratios smaller than 7, a bi-stable flow switching between regular vortex shedding and ‘irregular flow’ was found at intermediate Reynolds number ranges in the subcritical regime (Re ≈ 2 × 103)

Direct Simulation of Initial Value Problems for the Motion of Solid Bodies in a Newtonian Fluid Part 1. Sedimentation

Abstract: This paper reports the result of direct simulations of fluid–particle motions in two dimensions. We solve the initial value problem for the sedimentation of circular and elliptical particles in a vertical channel. The fluid motion is computed from the Navier–Stokes equations for moderate Reynolds numbers in the hundreds. The particles are moved according to the equations of motion of a rigid body under the action of gravity and hydrodynamic forces arising from the motion of the fluid. The solutions are as exact as our finite-element calculations will allow. As the Reynolds number is increased to 600, a circular particle can be said to experience five different regimes of motion: steady motion with and without overshoot and weak, strong and irregular oscillations. An elliptic particle always turn its long axis perpendicular to the fall, and drifts to the centreline of the channel during sedimentation. Steady drift, damped oscillation and periodic oscillation of the particle are observed for different ranges of the Reynolds number. For two particles which interact while settling, a steady staggered structure, a periodic wake-action regime and an active drafting–kissing–tumbling scenario are realized at increasing Reynolds numbers. The non-linear effects of particle–fluid, particle–wall and interparticle interactions are analysed, and the mechanisms controlling the simulated flows are shown to be lubrication, turning couples on long bodies, steady and unsteady wakes and wake interactions. The results are compared to experimental and theoretical results previously published.

Direct Simulation of a Self-Similar Turbulent Mixing Layer

Abstract: Three direct numerical simulations of incompressible turbulent plane mixing layers have been performed. All the simulations were initialized with the same two velocity fields obtained from a direct numerical simulation of a turbulent boundary layer with a momentum thickness Reynolds number of 300 computed by Spalart (J. Fluid Mech. 187, 61, 1988). In addition to a baseline case with no additional disturbances, two simulations were begun with two-dimensional disturbances of varying strength in addition to the boundary layer turbulence. After a development stage, the baseline case and the case with weaker additional two-dimensional disturbances evolve self-similarly, reaching visual thickness Reynolds numbers of up to 20 000. This self-similar period is characterized by a lack of large-scale organized pairings, a lack of streamwise vortices in the 'braid' regions, and scalar mixing that is characterized by 'marching' Probability Density Functions (PDFs). The case begun with strong additional two-dimensional disturbances only becomes approximately self-similar, but exhibits sustained organized large-scale pairings, clearly defined braid regions with streamwise vortices that span them, and scalar PDFs that are 'nonmarching.' It is also characterized by much more intense vertical velocity fluctuations than the other two cases. The statistics and structures in several experiments involving turbulent mixing layers are in better agreement with those of the simulations that do not exhibit organized pairings.

Effects of the computational time step on numerical solutions for turbulent flow

Abstract: Effects of large computational time steps on the computed turbulence were investigated using a fully implicit method. In turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined. Turbulence fluctuations could not be sustained if the computational time step was near or larger than the Kolmogorov time scale.

Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows

Abstract: This paper reports the results of a two-dimensional finite element simulation of the motion of a circular particle in a Couette and a Poiseuille flow. The size of the particle and the Reynolds number are large enough to include fully nonlinear inertial effects and wall effects. Both neutrally buoyant and non-neutrally buoyant particles are studied, and the results are compared with pertinent experimental data and perturbation theories. A neutrally buoyant particle is shown to migrate to the centreline in a Couette flow, and exhibits the Segre-Silberberg effect in a Poiseuille flow. Non-neutrally buoyant particles have more complicated patterns of migration, depending upon the density difference between the fluid and the particle. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and, in the case of Poiseuille flow, a lift caused by the velocity profile curvature. These forces are analysed by examining the distributions of pressure and shear stress on the particle. The stagnation pressure on the particle surface are particularly important in determining the direction of migration.

Numerical Simulation of Viscous Flow by Smoothed Particle Hydrodynamics

Abstract: present a new SPH method that can be used to solve the Navier-Stokes equations for constant viscosity. The method is applied to two-dimensional Poiseuille flow, three-dimensional Hagen­ Poiseuille flow and two-dimensional isothermal flows around a cylinder. In the former two cases, the temperature of fluid is assumed to be linearly dependent on a coordinate variable x along the flow direction. The numerical results agree well with analytic solutions, and we obtain nearly uniform density distributions and the expected parabolic and paraboloid velocity profiles. The density and ·velocity field in the latter case are compared with the results obtained using a finite difference method. Both methods give similar results for Reynolds number Re=6, 10, 20, 30 and 55, and the differences in the total drag coefficients are about 2~4%. Our numerical simulations indicate that SPH is also an effective numerical method for calculation of viscous flows.

The influence of a mean magnetic field on three- dimensional magnetohydrodynamic turbulence

Abstract: Building on results from two-dimensional magnetohydrodynamic (MHD) turbulence (Shebalin, Matthaeus & Montgomery 1983), the development of anisotropic states from initially isotropic ones is investigated numerically for fully three-dimensional incompressible MHD turbulence. It is found that when an external d.c. magnetic field (B0) is imposed on viscous and resistive MHD systems, excitations are preferentially transferred to modes with wavevectors perpendicular to B0). The anisotropy increases with increasing mechanical and magnetic Reynolds numbers, and also with increasing wavenumber. The tendency of B0 to inhibit development of turbulence is also examined.

Numerical experiments on the flow past A circular cylinder at sub-critical reynolds number

Abstract: : The performance of the dynamic subgrid-scale eddy-viscosity model and the suitability of high-order accurate, upwind-biased numerical methods for large eddy simulations of complex flows are investigated in the case of the turbulent wake behind a circular cylinder at Reynolds number 3,900, based on freestream velocity and cylinder diameter The numerical method consists of high-order upwind-biased finite difference techniques applied to the compressible Navier-Stokes equations written in generalized coordinates Integration in time is done using a fully implicit, second-order accurate iterative technique The results of three fifth-order accurate simulations performed on identical grids with the least-squares version of the dynamic model, the fixed-coefficient Smagorinsky model, and with no subgrid-scale model are compared in the first 10 diameters of the wake The impact of three-dimensionality is also examined via two and three-dimensional calculations without a subgrid-scale model The effect of numerical dissipation is investigated by comparing two simulations using upwind-biased schemes, the first being fifth-order, and the second seventh-order accurate It is found that the near-wake is highly three-dimensional at this Reynolds number It contains pairs of counter-rotating streamwise vortices, the effect of which cannot be reproduced in two-dimensional calculations Three-dimensional computations are essential for predicting flow statistics of engineering interest (AN)

A note on the history force on a spherical bubble at finite Reynolds number

Abstract: An approximate expression for the history force on a spherical bubble is proposed for finite Reynolds number, Re. At small time, the history‐force kernel is a constant, which decreases with increasing Re, and the kernel decays as t−2 for large time. For an impulsively started flow over a bubble, accurate finite difference results show that the history force on the bubble decays as t−2 at large time. Satisfactory agreement is observed between the presently proposed history force and the numerical solution.

Active vorticity control in a shear flow using a flapping foil

Abstract: It is shown experimentally that free shear flows can be substantially altered through direct control of the large coherent vortices present in the flow.First, flow-visualization experiments are conducted in Kalliroscope fluid at Reynolds number 550. A foil is placed in the wake of a D-section cylinder, sufficiently far behind the cylinder so that it does not interfere with the vortex formation process. The foil performs a heaving and pitching oscillation at a frequency close to the Strouhal frequency of the cylinder, while cylinder and foil also move forward at constant speed. By varying the phase of the foil oscillation, three basic interaction modes are identified. (i) Formation of a street of pairs of counter-rotating vortices, each pair consisting of one vortex from the initial street of the cylinder and one vortex shed by the foil. The width of the wake is then substantially increased. (ii) Formation of a street of vortices with reduced or even reverse circulation compared to that of oncoming cylinder vortices, through repositioning of cylinder vortices by the foil and interaction with vorticity of the opposite sign shed from the trailing edge of the foil. (iii) Formation of a street of vortices with circulation increased through merging of cylinder vortices with vortices of the same sign shed by the foil. In modes (ii) and (iii) considerable repositioning of the cylinder vortices takes place immediately behind the foil, resulting in a regular or reverse Karman street. The formation of these three interaction patterns is achieved only for specific parametric values; for different values of the parameters no dominant stable pattern emerges.Subsequently, the experiments are repeated in a different facility at larger scale, resulting in Reynolds number 20000, in order to obtain force and torque measurements. The purpose of the second set of experiments is to assess the impact of flow control on the efficiency of the oscillating foil, and hence investigate the possibility of energy extraction. It is found that the efficiency of the foil depends strongly on the phase difference between the oscillation of the foil and the arrival of cylinder vortices. Peaks in foil efficiency are associated with the formation of a street of weakened vortices and energy extraction by the foil from the vortices of the vortex street.

A Lattice Boltzmann Subgrid Model for High Reynolds Number Flows

Abstract: A subgrid turbulence model for the lattice Boltzmann method is proposed for high Reynolds number fluid flow applications. The method, based on the standard Smagorinsky subgrid model and a single-time relaxation lattice Boltzmann method, incorporates the advantages of the lattice Boltzmann method for handling arbitrary boundaries and is easily implemented on parallel machines. The method is applied to a two-dimensional driven cavity flow for studying dynamics and the Reynolds number dependence of the flow structures. The substitution of other subgrid models, such as the dynamic subgrid model, in the framework of the LB method is discussed.

Optimal energy density growth in Hagen-Poiseuille flow

Abstract: Linear stability of incompressible flow in a circular pipe is considered. Use is made of a vector function formulation involving the radial velocity and radial vorticity only. Asymptotic as well as transient stability are investigated using eigenvalues and e-pseudoeigenvalues, respectively. Energy stability is probed by establishing a link to the numerical range of the linear stability operator. Substantial transient growth followed by exponential decay has been found and parameter studies revealed that the maximum amplification of initial energy density is experienced by disturbances with no streamwise dependence and azimuthal wavenumber n = 1. It has also been found that the maximum in energy scales with the Reynolds number squared, as for other shear flows. The flow field of the optimal disturbance, exploiting the transient growth mechanism maximally, has been determined and followed in time. Optimal disturbances are in general characterized by a strong shear layer in the centre of the pipe and their overall structure has been found not to change significantly as time evolves. The presented linear transient growth mechanism which has its origin in the non-normality of the linearized Navier–Stokes operator, may provide a viable process for triggering finite-amplitude effects.

The Physics of Supersonic Turbulent Boundary Layers

Abstract: When a vehicle travels at Mach numbers greater than one, a significant temperature gradient develops across the boundary layer due to the high levels of viscous dissipation near the wall. In fact, the static-temperature variation can be very large even in an adiabatic flow, resulting in a low­ density, high-viscosity region near the wall. In turn, this leads to a skewed mass-flux profile, a thicker boundary layer, and a region in which viscous effects are somewhat more important than at an equivalent Reynolds number in subsonic flow. Intuitively, one would expect to see significant dynamical differences between subsonic and supersonic boundary layers. However, many of these differences can be explained by simply accounting for the fluid-property variations that accompany the temperature

Reynolds Number Effects in Wall-Bounded Turbulent Flows

Abstract: This paper reviews the state of the art of Reynolds number effects in wall-bounded shear-flow turbulence, with particular emphasis on the canonical zero-pressure-gradient boundary layer and two-dimensional channel flow problems. The Reynolds numbers encountered in many practical situations are typically orders of magnitude higher than those studied computationally or even experimentally. High-Reynolds number research facilities are expensive to build and operate and the few existing are heavily scheduled with mostly developmental work. For wind tunnels, additional complications due to compressibility effects are introduced at high speeds. Full computational simulation of high-Reynolds number flows is beyond the reach of current capabilities. Understanding of turbulence and modeling will continue to play vital roles in the computation of high-Reynolds number practical flows using the Reynolds-averaged Navier-Stokes equations. Since the existing knowledge base, accumulated mostly through physical as well as numerical experiments, is skewed towards the low Reynolds numbers, the key question in such high-Reynolds number modeling as well as in devising novel flow control strategies is: what are the Reynolds number effects on the mean and statistical turbulence quantities and on the organized motions? Since the mean flow review of Coles (1962), the coherent structures, in low-Reynolds number wall-bounded flows, have been reviewed several times. However, the Reynolds number effects on the higher-order statistical turbulence quantities and on the coherent structures have not been reviewed thus far, and there are some unresolved aspects of the effects on even the mean flow at very high Reynolds numbers. Furthermore, a considerable volume of experimental and full-simulation data have been accumulated since 1962. The present article aims at further assimilation of those data, pointing to obvious gaps in the present state of knowledge and highlighting the misunderstood as well as the ill-understood aspects of Reynolds number effects.

A global stability analysis of the steady and periodic cylinder wake

Abstract: A global, three-dimensional stability analysis of the steady and the periodic cylinder wake is carried out employing a low-dimensional Galerkin method. The steady flow is found to be asymptotically stable with respect to all perturbations for Re less than 54. The onset of periodicity is confirmed to be a supercritical Hopf bifurcation which can be modeled by the Landau equations. The periodic solution is observed to be only neutrally stable for 54 less than Re less than 170. While two-dimensional perturbations of the vortex street rapidly decay, three-dimensional perturbations with long spanwise wavelengths neither grow nor decay. The periodic solution becomes unstable at Re = 170 by a perturbation with the spanwise wavelength of 1.8 diameters. This instability is shown to be a supercritical Hopf bifurcation in the spanwise coordinate and leads to a three-dimensional periodic flow. Finally the transition scenario for higher Reynolds numbers is discussed.

A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient

Abstract: The mixing of a passive scalar in the presence of a mean gradient is studied in three dimensions by direct numerical simulations. The driving velocity field is either a solution of the three‐dimensional (3‐D) Navier–Stokes equations, at a microscale Reynolds number in between 20 and 70, and with a Prandtl number varying between 1/8 and 1, or a solution of the Euler equation restricted to a shell of wave numbers, which formally corresponds to an infinite Prandtl number. The probability distribution function (PDF) of the scalar gradients parallel and perpendicular to the direction of the mean gradient are studied. The gradients parallel to the mean gradient have a skewness of order 1 in the range of Peclet number considered. The PDFs are sharply peaked and their maxima correspond to a perfect mixing of the scalar. The PDF of the scalar gradient perpendicular to the mean gradient are reasonably well fit by stretched exponentials. Similar properties are observed for the restricted Euler model. In physical space, the scalar is well mixed in large domains, separated by narrow regions, where very large gradients concentrate. These ‘‘cliffs’’ are found to sit in regions where the flow is hyperbolic, whereas the scalar gradients are much weaker where the flow is elliptic. The present results are generally in agreement with the conclusions reached in a comparable study in two dimensions by Holzer and Siggia (to appear in Phys. Fluids). The stretching acting on the scalar is studied by computing various correlations between scalar gradient and velocity derivatives, as well as the correlations between vorticity and scalar gradient.

Settling Velocity of Irregularly Shaped Particles

Abstract: A new correlation has been developed to predict the settling velocity of irregularly shaped particles in Newtonian and non-Newtonian fluids for all types of slip regimes The correlation was derived from extensive data on the drag coefficients and particle Reynolds numbers of irregularly shaped particles The effective fluid viscosity at the settling shear rate is used in the correlation A trial-and-error or numerical iteration method is required to predict the settling velocity for non-Newtonian fluids The correlation predicted and verified the effects of fluid properties, particle properties, and operation parameters on the settling velocity

Development of a k-ε Model for Bubbly Two-Phase Flow

Abstract: An extension of the k-[epsilon] 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-[epsilon] 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.