Volume 11, Issue 8, August 1999
 LETTERS


Direct numerical simulations of threedimensional bubbly flows
View Description Hide DescriptionDirect numerical simulations of the motion of many buoyant bubbles are presented. The Navier–Stokes equation is solved by a front tracking/finite difference method that allows a fully deformable interface. The evolution of 91 nearly spherical bubbles at a void fraction of 6% is followed as the bubbles rise over 100 bubble diameters. While the individual bubble velocities fluctuate, the average motion reaches a statistical steady state with a rise Reynolds number of about 25.
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 ARTICLES


Parametrically driven surface waves in viscoelastic liquids
View Description Hide DescriptionWhen a container of liquid is subject to vertical sinusoidal oscillation, the free surface becomes unstable at a critical driving acceleration and gives rise to standing waves. Here, we consider containers of finite depth, neglect the influence of lateral boundaries, and perform a linear stability analysis for viscoelastic liquids. Floquet theory is applied to transform the linearized governing equations into a recursion relation for the temporal modes of the free surface deformation. In the absence of external forcing, the recursion relation yields the dispersionequation for free surface waves on viscoelastic liquids. In the presence of external forcing, the recursion relation is studied both numerically and analytically for the case where the polymer stresses are described by a singlemode Maxwell model. When surface tension forces are sufficiently strong relative to elastic forces, the numerical results show that the standing waves respond subharmonically to the driving frequency. However, the instability threshold increases less rapidly with the driving frequency than that for a Newtonian liquid of the same zeroshear viscosity. When elastic forces become sufficiently strong relative to surface tension forces, the standing waves can respond harmonically within certain ranges of the driving frequency if the product of the driving frequency and the liquid relaxation time is not too large or too small. Dramatic changes are seen in the behavior of the neutral stability curves, dispersion relations, and instability thresholds. In the case where the viscous boundary layer thickness is much less than the disturbance wavelength, the viscoelastic recursion relation is simplified to yield a Mathieu equation which is nonlocal in time. The method of multiple scales is then applied to determine the instability threshold analytically. The results of this study indicate that the behavior of parametrically driven surface waves is very sensitive to the liquid relaxation time, and suggest that such waves may serve as a useful tool for the measurement of rheological properties.

Imaging of particle shear migration with electrical impedance tomography
View Description Hide DescriptionElectrical impedance tomography(EIT) is used to investigate the net migration of particles in a suspension undergoing pressuredriven flow through a tube at low Reynolds number. A low frequency electrical current is applied to the flowing suspension by two flushmounted electrodes on the pipe wall to create a potential field which is sampled by other pairs of flushmounted electrodes. A numerical inversion of the data, which takes into account the geometry and symmetry of the problem, gives the conductivity variation within the flow. An image of the suspension particle volume fraction in the tube is then formed using a relationship between the local conductivity and local suspension concentration. Consistent with particle shear migration of a concentrated suspension at low Reynolds number, the images from experimental data show a net migration of particles toward the centerline of the tube. The images of a 0.25 volume fraction suspension at the lowest Reynolds number examined compare favorably to an existing continuum theory of particle shear migration. Other images from experiments at higher, but still small Reynolds numbers, and at volume fractions of 0.25 and 0.40 are also presented. The particle migration measured with the relatively inexpensive EIT at the latter condition is in very good agreement with the particle distribution measured with magnetic resonance imaging by Hampton et al. [J. Rheol. 41, 621 (1997)].

Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle
View Description Hide DescriptionThe dynamic contact angle formed when a liquid curtain impinges onto a moving solid is measured for aqueous glycerol solutions in different flow regimes. It is usually assumed that the dynamic contact angle is simply a function of the contactline speed and the material properties of the contacting media. The new results show that this is not the case. For a given gas/liquid/solid combination and a given contactline speed, the dynamic contact angle can be varied by varying the flow rate of the liquid and/or the curtain height, that is by varying the flow field near the contact line. The possibility of attributing this effect merely to freesurface bending and interpreting the results in terms of the socalled “apparent” contact angle is discussed and ruled out on the basis of some general qualitative arguments and analysis of the characteristic length scales involved. A probable connection between the observed effect and the physical mechanism of interface disappearance and formation incorporated in a recently developed theory of wetting is discussed.

The dynamics of vapor bubbles in acoustic pressure fields
View Description Hide DescriptionIn spite of a superficial similarity with gas bubbles, the intimate coupling between dynamical and thermal processes confers to oscillating vapor bubbles some unique characteristics. This paper examines numerically the validity of some asymptotictheory predictions such as the existence of two resonant radii and a limit size for a given sound amplitude and frequency. It is found that a small vapor bubble in a sound field of sufficient amplitude grows quickly through resonance and continues to grow thereafter at a very slow rate, seemingly indefinitely. Resonance phenomena therefore play a role for a few cycles at most, and reaching a limit size—if one exists at all—is found to require far more than several tens of thousands of cycles. It is also found that some small bubbles may grow or collapse depending on the phase of the sound field. The model accounts in detail for the thermofluidmechanic processes in the vapor. In the second part of the paper, an approximate formulation valid for bubbles small with respect to the thermal penetration length in the vapor is derived and its accuracy examined. The present findings have implications for acoustically enhanced boiling heat transfer and other special applications such as boiling in microgravity.

Effects of granular additives on transition boundaries between flow states of rimming flows
View Description Hide DescriptionAn experimental study of the rimming flow established inside a partially fluidfilled cylinder rotating around a horizontal axis of rotation is described. For the first time effects of granular additives on transition boundaries between flow states adopted by the fluid for different experimental conditions are studied. For the granulefree fluid and low filling levels we confirm results of previous authors showing that the ratio of viscous stresses and gravitational force remains constant along the transition boundaries considered. For higher filling levels our new data indicate, however, that the gravitational force becomes increasingly more important. For the solid–liquid twophase flow our data reveal that even small amounts of granular additives can have a significant effect on a suitable parameter defined to characterize the transition boundaries. Granular additives can lead to the stabilization of states and to the extension of the parameter range over which certain states can be observed. It is shown that the origin of the observed effects appears to be associated with an increased bulk density of the solid–liquidflow. For high granule concentrations a pattern of equallyspaced circumferential granular bands is observed to form on the inner cylinder wall. It is speculated that these bands form as a consequence of the mechanism which has been referred to as shearinduced migration/diffusion in the literature in the past. It appears that the granuleband pattern has not been observed previously for the flow investigated here.

Bifurcation study of flow through rotating curved ducts
View Description Hide DescriptionThe bifurcation structure of the twodimensional pressuredriven flow through a curved rotating duct is studied. In this study we add to the rich literature that already exists on this problem [J. Fluid Mech. 262, 353 (1994)], revealing even more intricate details of the solution structure. The problem depends on the Reynolds number, the Rotation number, the aspect ratio, and the radius ratio (or curvature ratio), here U is the velocity scale, b is the duct width in the spanwise direction, Ω is the rotational speed, are the inner and outer radii of the duct, and ν is the kinematicviscosity of the fluid. For a curvature ratio continuation on Re is used to trace the bifurcation diagram for zero rotation Then, for continuation on is used from the solutions at zero rotation to generate bifurcation diagrams for positive and negative rotational number for the purpose of studying the effect of rotation. Extended systems are used to solve for limit points and symmetry breaking points. These points are then traced as functions of the Reynolds number.Eigenvalue systems are solved to determine the stability properties of the multiple solutions to twodimensional perturbations.Bifurcation diagrams reveal more intricate solution structures than those found in earlier studies, raising the question whether it is ever possible to construct a complete bifurcation diagram. New solution branches are found even for the wellstudied case of a system with no rotation.

Universal properties of chaotic transport in the presence of diffusion
View Description Hide DescriptionThe combined, finite time effects of molecular diffusion and chaotic advection on a finite distribution of scalar are studied in the context of time periodic, recirculating flows with variable stirring frequency. Comparison of two disparate frequencies with identical advective fluxes indicates that diffusive effects are enhanced for slower oscillations. By examining the geometry of the chaotic advection in both high and low frequency limits, the flux function and the width of the stochastic zone are found to have a universal frequency dependence for a broad class of flows. Furthermore, such systems possess an adiabatic transport mechanism which results in the establishment of a “Lagrangian steady state,” where only the asymptotically invariant core remains after a single advective cycle. At higher frequencies, transport due to chaotic advection is confined to exchange along the perimeter of the recirculating region. The effects of molecular diffusion on the total transport are different in these two cases and it is argued and demonstrated numerically that increasing the diffusion coefficient (in some prescribed range) leads to a dramatic increase in the transport only for low frequency stirring. The frequency dependence of the total, long time transport of a limited amount of scalar is more involved since faster stirring leads to smaller invariant core sizes.

On the stability of the Hartmann layer
View Description Hide DescriptionIn this paper we are concerned with the theoreticalstability of the laminar Hartmann layer, which forms at the boundary of any electrically conducting fluid flow under a steady magnetic field at high Hartmann number. We perform both linear and energetic stability analyses to investigate the stability of the Hartmann layer to both infinitesimal and finite perturbations. We find that there is more than three orders of magnitude between the critical Reynolds numbers from these two analyses. Our interest is motivated by experimental results on the laminar–turbulent transition of ducted magnetohydrodynamicsflows. Importantly, all existing experiments have considered the laminarization of a turbulent flow, rather than transition to turbulence. The fact that experiments have considered laminarization, rather than transition, implies that the threshold value of the Reynolds number for stability of the Hartmann layer to finiteamplitude, rather than infinitesimal, disturbances is in better agreement with the experimental threshold values. In fact, the critical Reynolds number for linear instability of the Hartmann layer is more than two orders of magnitude larger than experimentally observed threshold values. It seems that this large discrepancy has led to the belief that stability or instability of the Hartmann layer has no bearing on whether the flow is laminar or turbulent. In this paper, we give support to Lock’s hypothesis [Proc. R. Soc. London, Ser. A 233, 105 (1955)] that “transition” is due to the stability characteristics of the Hartmann layer with respect to largeamplitude disturbances.

Threedimensional stability of a vortex pair
View Description Hide DescriptionThis paper investigates the threedimensional stability of the Lamb–Chaplygin vortex pair. Shortwavelength instabilities, both symmetric and antisymmetric, are found. The antisymmetric mode possesses the largest growth rate and is indeed the one reported in a recent experimental study [J. Fluid Mech. 360, 85 (1998)]. The growth rates, wave numbers of maximum amplification, and spatial eigenmodes of these shortwavelength instabilities are in good agreement with the predictions from elliptic instability theory. A longwavelength symmetric instability similar to the Crow instability of a pair of vortex filaments is also recovered. Oscillatory bulging instabilities, both symmetric and antisymmetric, are identified albeit their growth rates are lower than for the shortwavelength instabilities. Their behavior and eigenmodes resemble those of the oscillatory bulging instability occurring in the mixing layer.

On the onset of convective instabilities in cylindrical cavities heated from below. I. Pure thermal case
View Description Hide DescriptionThreedimensional steady flows are simulated in a circular cylindrical cavity of aspect ratio where H is the height and D the diameter of the cavity. The cavity is heated from below and its sidewalls are considered to be adiabatic. The effect of the geometry of the cavity on the onset of convection and on the structure and symmetries of the flow is analyzed. The nonlinear evolution of the convection beyond its onset is presented through bifurcation diagrams for two typical aspect ratios and Axisymmetric and asymmetric ( and ) azimuthal modes are observed. For the axisymmetric solution loses its stability to a threedimensional solution at a secondary bifurcation point. Better understanding of the mechanisms leading to this instability is obtained by analyzing the energy transfer between the basic state and the critical mode. To study the influence of the Prandtl number on the flow pattern and on the secondary bifurcation, three values of the Prandtl number are investigated: Pr=0.02 (liquid metal), Pr=1 (transparent liquids), and Pr=6.7 (water).

On the onset of convective instabilities in cylindrical cavities heated from below. II. Effect of a magnetic field
View Description Hide DescriptionThe effect of a constant and uniform magnetic field on electrically conductingliquidmetal flow, in cylindrical cavities heated from below, is numerically analyzed by using a spectral element method to solve the threedimensional Navier–Stokes and Ohm equations. The cavity is characterized by its aspect ratio defined as The lateral surfaces are adiabatic and all the boundaries are electrically insulating. The flow with a vertical magnetic field has the same symmetries as that without a magnetic field, so that similar convective modes ( and ) occur, but they are not equally stabilized. Here m is the azimuthal wave number. For for sufficiently large values of the Hartmann number Ha, the mode becomes the critical mode in place of The horizontal magnetic field breaks some symmetries of the flow. The axisymmetric mode disappears giving an asymmetric mode i.e., a combination of the and modes, whereas the asymmetric modes and which were invariant by azimuthal rotation without a magnetic field, now have two possible orientations, either parallel or perpendicular to the applied magnetic field B . These five modes are differently stabilized, weakly if the axis of the rolls is parallel to B and strongly if the axis is perpendicular. Beyond the primary thresholds, the secondary bifurcation, found in the pure thermal case for becomes an imperfect bifurcation consisting of two disconnected branches.

On the thermal offset in turbulent rotating convection
View Description Hide DescriptionA simple mechanistic model to explain the thermal offset observed in highly turbulent rotating convectionexperiments is presented. The experiments indicate that for sufficiently high Taylor number the system develops an asymmetry that is counter to the expected behavior of a Boussinesq fluid: The temperature at the midplane of the experiment is higher than the average of the top and bottom isothermal boundaries. The magnitude of this bias increases with the basic rotation and with the density difference between the top and bottom boundaries. Our model considers this to be a result of a mean circulation induced by centrifugal buoyancy. An analytical calculation of this motion field, postulated to be a laminar perturbation to the turbulenceinduced mean stratification, generates a reasonably accurate description of the observed thermal offset.

Comparison of Burnett and DSMC predictions of pressure distributions and normal stress in onedimensional, strongly nonisothermal gases
View Description Hide DescriptionThe Burnett equations have been shown to provide improved descriptions, relative to the Navier–Stokes equations, of flow structure in highvelocity (i.e., hypersonic) gases. We examine here the accuracy of the Burnett constitutive equation for fluid stress as applied to stationary gases. Specifically, we investigate the effects of “thermal stress” (fluid stress induced by a temperature gradient), as predicted by the Burnett equation, on the pressure distributions and normal stress in a stationary, buoyancyfree, hardsphere gas for the case of onedimensional heat transfer. We show, using firstlaw principles and the Burnett equation, that thermal stress results in a reduction in normal stress in the nonisothermal gas relative to that in the equilibrium state. The normal stress, in turn, can be obtained as an eigenvalue to a secondorder ordinary differential equation, representing the Burnett equation, for the pressure distribution in the gas. Simple asymptotic solutions to the Burnett equation are developed, and are used in combination with orderpressure slip relations to formulate pressureboundary conditions at the heated and cooled surfaces. The approximate solutions, as well as exact numerical calculations, are compared with pressure distributions generated from the directsimulation Monte Carlo (DSMC) method. The Burnett and DSMC predictions of pressure are in good agreement for effective Knudsen numbers (based on the temperature gradient in the gas) less than 0.1. In particular, the Burnett equations can provide a reasonable description of the Knudsen (or rarefaction) layers adjacent to the heated and cooled surfaces that bound the gas, and can also describe the variation in pressure in the bulk gas. In addition, theoretical predictions of the reduction in normal stress correspond well to DSMCderived values.

Vibrational–translational energy exchange models for the direct simulation Monte Carlo method
View Description Hide DescriptionThe model which controls the distribution of energy among the different molecular modes is a crucial component of accurate simulation of nonequilibrium rarefied flows. Two new models for the direct simulation Monte Carlo method that govern energy redistribution between the translational and vibrational modes are presented here. The first model is a modified form of the phenomenological Borgnakke–Larsen model. The probability of inelastic collision is evaluated using the relative velocity of collision. The second energy exchange model considered in this study is the multiple quantumstep transition model. The process of vibrational relaxation occurs through transitions between the different energy levels, allowing jumps of more than one level. Probabilities of activation and deactivation which depend on the relative velocity are used here. The new models are compared with existing schemes for several conditions. Significant differences are found for the vibrational energy distribution function computed in a hypersonic bowshock wave.

Evolution and convection of largescale structures in supersonic reattaching shear flows
View Description Hide DescriptionDoublepulsed Mie scattering studies were performed to characterize the evolution of largescale structures embedded within a planar supersonic base flow.Images were obtained at several streamwise stations along the shear layers, at reattachment, and in the nearwake regions. From these timecorrelated images, the evolution characteristics of the largescale structures were examined over a range of nondimensional time delays, as defined by local integral length and velocity scales. The doublepulsed images indicated that for short time delays (i.e., less than the representative eddy rollover time), the structures exhibited a simple translation in the streamwise direction. As the time delay was increased, rotation and elongation of the structures were observed in addition to the translation feature. Time delays that appreciably exceeded the local eddy rollover time generally resulted in a dramatic loss of structure identity. No eddyinteractions, such as pairing, were observed at any of the imaging locations. Images obtained near reattachment provided evidence of shocklets moving in concert with the local eddies. In the initial portions of the shear layers, the mean convectionvelocity was measured to be significantly higher than the isentropic estimate, which is consistent with the results of previous convectionvelocity studies using mixing layers composed of supersonic/subsonic freestream combinations. The eddies decelerate through the recompression and reattachment regions, presumably due to the influence of the adverse pressure gradient. Downstream of reattachment, the largescale structures accelerate as the wake develops.

Kinetic instabilities of threelayer thermocapillary creeping flows
View Description Hide DescriptionIt is shown that plane viscosity stratified system under longitudinal temperature gradient may exhibit longwavelength instabilities of a purely kinetic nature which persist at arbitrary small Reynolds number. The first instability is socalled alpha effect, and the second one is a new surfacetensioninduced instability. The weakly nonlinear equations for the evolving interfaces are derived and simulated.
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 SYMPOSIUM PAPERS


Probability density functions in steadystate Burgers turbulence
View Description Hide DescriptionProbability density functions(PDFs) for steadystate Burgers turbulence supported by whiteintime random forcing at low wave numbers are studied by direct numerical simulation and compared to theoretical predictions. The velocity PDFs decay slightly faster than a Gaussian at large amplitudes. The putative power law exponent of the PDF of velocity gradient is examined at large Reynolds number and found to be approximately 3 or slightly greater. The tail of behaves like at large negative where is a forcing parameter. The exponent is near unity, which is smaller than predicted by theory. It decreases slowly with the Reynolds numberR up to The central parts of the PDFs of higher velocity space derivatives are found to be cusplike, and the cusp exponents are measured. The PDF tails are stretched exponentials.

On the statistical solution of the Riemann equation and its implications for Burgers turbulence
View Description Hide DescriptionThe statistics of the multivalued solutions of the forced Riemann equation, is considered. An exact equation for the signed probability density function of these solutions and their gradient is derived, and some properties of this equation are analyzed. It is shown in particular that the tails of the signed probability density function generally decay as for large Further considerations give bounds on the cumulative probability density function for the velocity gradient of the solution of Burgers equation.

Taylor expansions in powers of time of Lagrangian and Eulerian twopoint twotime velocity correlations in turbulence
View Description Hide DescriptionA method is developed for generating the Taylor expansions in powers of the time difference of the Lagrangian and Eulerian twopoint twotime velocity correlations in turbulence. The expansions are based on the Taylor series of the Eulerian and Lagrangian velocity fields subject to given dynamics along with initial and boundary conditions. The lowest few coefficients in the expansions enable us to construct approximations to the correlations. An application of the method to turbulence obeying the NavierStokes dynamics yields approximations, particularly Padé approximations that agree well with direct numerical simulations of homogeneous isotropic turbulence at moderate Reynolds numbers. The ratios of the secondorder to the zerothorder coefficients of the Taylor series of the Lagrangian and Eulerian correlations give, respectively, the estimates for the Lagrangian and Eulerian micro time scales and An analysis of a high resolution ( grid points) direct numerical simulation database at large Reynolds number suggests the scalings and for wave numbers in the inertial subrange. The role of flow structures in turbulence in determining the time scales is also discussed.
