Index of content:
Volume 7, Issue 11, November 1995

Photographic study of the shock‐induced dispersion of microscopic gas bubbles
View Description Hide DescriptionSeparated and attached microscopic air bubbles excited by a sequence of expansion and compression (shock) waves are considered. It was found out that the expanded bubbles break up into clusters of air nuclei, under certain conditions, due to the shock wave‐induced collapses.

Columnar vortex generation and interaction with a clean or contaminated free surface
View Description Hide DescriptionThe interaction of a columnar vortex with a free surface is studied experimentally. The vortex is generated by a pair of corotating flaps and is visualized by planar laser‐induced fluorescence. When the surface is clean, the vortex is unaffected and remains essentially two dimensional. However, when the surface is contaminated by a relatively viscous monomolecular layer, the vortex dynamics can be significantly altered, resulting in a downward axial flow followed by vortex breakdown.

Enclosure gas flows driven by non‐isothermal walls
View Description Hide DescriptionThe phenomenon of thermal creep, according to which a rarefied gas will ‘‘slip’’ at a fluid/solid interface in the presence of appreciable temperature gradients along the interface, has been previously treated by analytical techniques in the one‐dimensional, semi‐infinite, linearized case. The more general problem of multidimensional gas flows at high ΔT/T values typically defies analytical or semi‐analytical solution. We employ a direct simulation Monte Carlo method to a hard‐sphere gas in order to study the characteristics of thermal creep convective motion in a Cartesian, two‐dimensional, confined flow geometry which captures important features of ampoules used in microgravity experiments on crystal growth.Vortex roll formation is indeed observed in these numerical experiments even for zero‐gravity conditions, driven by thermal creep at the non‐isothermal boundaries and in close resemblance to the classical fluid dynamic problem of the wall‐driven cavity.

Inertial lift on a moving sphere in contact with a plane wall in a shear flow
View Description Hide DescriptionIn this paper we calculate the lift force on a smooth sphere rotating and translating in a simple shear flow in contact with a rigid wall. The calculation involves only known creeping flow solutions and is presented in terms of six different coefficients, each arising as a result of a pairwise combination of the translational velocity, rotational velocity, and the imposed shear flow. The results obtained agree well with those of Cherukat and McLaughlin [J. Fluid Mech. 263, 1 (1994a); and (personal communication, 1994b)], extrapolated for the case of zero separation distance. The calculated lift is further integrated into a force and torque balance on a non‐neutrally buoyant rough sphere moving in contact with a plane. It is found that if the shear Reynolds number Re is sufficiently large, the lift force exceeds the gravitational force and the sphere separates from the plane. The increased separation is accompanied by an increase in the translational velocity U of the sphere and a corresponding decrease in the lift force due to the negative shear‐translation coefficient, ultimately resulting in the sphere acquiring some steady separation distance. The equilibrium separation distance and velocity are plotted as a function of the parameter Re^{2}/Re_{ s }, where Re_{ s } is the sedimentationReynolds number.

Extensional flow of semidilute suspensions of rod‐like particles through an orifice
View Description Hide DescriptionThis paper deals with the extensional flow of semidilute suspensions of non‐Brownian rod‐like particles through an orifice. The experimental setup allows simultaneously visualizations of the flow pattern and measurements of the flow rate‐pressure drop relation. The flow pattern in the converging region is characterized by the appearance of a vortex ring which grows with the volume fraction of the particles and the aspect ratio. At the same time, the pressure drop experienced by the suspension as it flows through the orifice increases. The experimental results are interpreted in terms of a competition between the shear stress due to the solvent and the extensional stress generated by the particles. This interpretation is supported by the scaling of the vortex size and of the pressure drop with the volume fraction and the aspect ratio of the particles.

Correlation structure dependence of the effective permeability of heterogeneous porous media
View Description Hide DescriptionA theory is given in which the effective permeability tensorK _{ eff } of heterogeneous porous media is derived by a perturbation expansion of Darcy’s law in the variance σ^{2} of the log‐permeability ln[κ(ub;;‐45r ubx)]. The only assumption is that the spatially varying permeability κ(ub;;‐45r ubx) is a expressed in terms of the moments of the distribution of ln[κ(ub;;‐45r ubx)], i.e. K _{ eff } can formally be computed for any given distribution of the fluctuations of the log‐permeability. The explicit dependence of K _{ eff } on multi‐point statistics is given for non‐gaussian log‐permeability fluctuations up to order σ^{6}. As a special case of the theory, we examine K _{ eff } for a normal distribution function for both isotropic and anisotropic media. In the case of three‐dimensional isotropic porous media, a conjecture has been made in the past according to which the scalar effective permeability κ_{ eff }=K _{ G }exp[σ^{2}/6] where K _{ G } is the geometric mean of the log‐permeability. It is shown here that this conjecture is incorrect as the σ^{6}‐order term of κ_{ eff } contains additional terms than those corresponding to the development of the above formula. Moreover, these additional terms depend on the structure of the two‐point correlation function of ln[κ]. The resulting κ_{ eff } computed for both Gaussian and exponentially decaying covariances lies below the exponential formula. This result might suggest the exponential formula as being an upper bound for κ_{ eff }. For anisotropic systems, K _{ eff } is given up to the σ^{4}‐order for the general case where the mean flow is arbitrarily oriented with regard to the axes of stratification.

On the permeability of unidirectional fibrous media: A parallel computational approach
View Description Hide DescriptionThe problems of viscous and inertial flows through unidirectional fibrous porous media are addressed using an entirely parallel computational approach. The pertinent partial differential equations, derived from homogenization theory, are solved by a parallel finite element method in conjunction with Monte Carlo techniques to predict the statistical permeability coefficient. A nip‐element method, which mitigates the frequent geometry‐induced numerical difficulties, while providing both accurate approximations for the permeability coefficient, and rigorous error estimates, is also presented. The seepage permeability coefficient is determined for a wide range of fiber concentration. It is shown to deviate markedly at low porosities from the behavior predicted by earlier cell models, while exhibiting generally good agreement at high, and moderate porosities with the cell models, and with the limited available experimental and analytical results. Limited but illustrative inertial flow results at moderate Reynolds numbers are also presented for both regular and random arrays. For regular arrays, the flow is found to be unsteady for Reynolds numbers greater than approximately 150 at which traveling waves characterized by distinct periods and amplitudes are observed. Some modest discrepancy is found in comparison with available data which is attributed to the unsteady effects and other numerical issues. For random arrays, several configuration permeability values are calculated and compared satisfactorily against the Ergun correlation.

A numerical Eulerian approach to mixing by chaotic advection
View Description Hide DescriptionResults of numerical simulation of the advection‐diffusion equation at large Péclet number are reported, describing the mixing of a scalar field under the action of diffusion and of a class of steady, bounded, three‐dimensional flows, which can have chaotic streamlines. The time evolution of the variance of scalar field is calculated for different flow parameters and shown to undergo modulated exponential decay, with a decay rate which is a maximum for certain values of the flow parameters, corresponding to cases in which the streamlines are chaotic everywhere. If such global chaos is present, the decay rate tends to oscillate, whereas the presence of regular regions produces a more constant decay rate. Significantly different decay rates are obtained depending on the detailed properties of the chaotic streamlines. The relationship between the decay rate and the characteristic Lyapunov exponents of the flow is also investigated.

Instability of a deformed liquid drop in an acoustic field
View Description Hide DescriptionThe flattening and breakup of an axially symmetric liquid drop in an acoustic standing wave field in air have been studied using a boundary integral method. The interaction between the drop and sound field are crucial to this approach. Our computations are focused on the threshold beyond which the drop loses its static equilibrium, and on the dynamic behavior after it loses its equilibrium up to the point when it breaks up. The numerical results are given in terms of drop size and the strength of sound field and are found to be in good agreement with the measurements of others.

The collapse of a cavitation bubble in shear flows—A numerical study
View Description Hide DescriptionThe collapse of a cavitation bubble is examined by direct numerical simulations of the Navier–Stokes equations, using a finite difference/front tracking technique. Bubbles in both a quiescent fluid as well as shear flows are examined. For quiescent fluid, the results are compared with theoretical and previous computational results. For bubbles in a shear flow it is shown that large shear can increase the rate of collapse, and for bubbles near boundaries shear can eliminate the re‐entrant jet seen for bubbles in a quiescent flow.

Drop formation in liquid–liquid systems before and after jetting
View Description Hide DescriptionThe formation of drops resulting from the breakup of an axisymmetric Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from startup to breakup into drops was simulated numerically by solving the time‐dependent axisymmetric equations of motion and continuity using a combination of the volume‐of‐fluid (VOF) and continuous‐surface‐force (CSF) methods. The numerical simulation results compare well with previous experimental data and are significantly more accurate than previous simplified analyses based on drop formation before and after jetting over a wide range of conditions.

The breakdown of asymptotic hydrodynamic models of liquid spreading at increasing capillary number
View Description Hide DescriptionComplex hydrodynamics near the moving contact line control spreading of a fluid across a solid surface. In the confined region near the contact line, velocity gradients in the fluid are large and viscous forces control the shape of the fluid/fluid interface. The present model for liquid spreading describes the viscous effect on the dynamic interface shape to lowest order in capillary number, Ca. Using videomicroscopy and image analysis techniques, we have examined the shape of liquid/air interfaces very near moving contact lines for Ca≥0.10 where the interfaces are in capillary depression. We find that the theory correctly describes the data up to Ca=0.10 for distances from 20 to 400 μm from the contact line. As Ca increases, the model fails to describe the data in a region near the contact line, which grows as Ca increases. In this expanding region, the model predicts too large a curvature for the interface. We explore the origins of this breakdown by examining the fundamental assumptions of the model. The geometry‐dependent part of the solution to O(1) in Ca is sufficient even at Ca=0.44. The breakdown of the model arises from the low order of the geometry‐free part of the perturbation solution and/or contributions to the interface shape from the unique hydrodynamics very near the moving contact line.

Surfactant‐driven spreading of a liquid on a vertical surface
View Description Hide DescriptionThe spreading behavior of a liquid on the surface of a solid substrate is greatly changed by the presence of a molecular layer of organic material (a surfactant) on the liquid surface. In this work, we studied the spreading of water covered by a monolayer of valinomycin on a vertical glass slide, using an apparatus for Langmuir–Blodgett film deposition. The rate of spreading strongly depends on the surfactant concentration, and the spreading front is highly unstable: it bifurcates while spreading, forming tree‐like patterns.

A three‐dimensional instability in mixed convection with streamwise periodic heating
View Description Hide DescriptionThree‐dimensional numerical simulations of flow through a channel with spatially periodic temperature boundary conditions in the streamwise direction have been carried out. A spectral method employing Fourier series expansion in the streamwise and spanwise directions and Chebyshev expansion in the vertical direction is used. A supercritical bifurcation to three‐dimensional flow is found to occur before the onset of unsteadiness. The bifurcation to unsteady three‐dimensional flow is also found to be supercritical. The unsteady three‐dimensional flow oscillates at a slightly lower frequency than the two‐dimensional flow, and introduces a slightly higher incommensurate frequency causing quasiperiodic flow and a slow spanwise wave motion.

Two‐phase Rayleigh–Bénard instabilities
View Description Hide DescriptionThe influence of particles on the first Rayleigh–Bénard bifurcation from the motionless to the convective state has been studied in a suspension layer heated from below and cooled from above, and effective physical properties have been measured. The particle parameters are the diameter and the concentration. The experimental setup allows one to measure accurately the temperature and the heat flux and to computerize the data recording. The suspensions consisted of polystyrene particles (diameters 0.1, 0.2, 3, and 100 μm and solid volume fraction from 1% to 10%) in water with the addition of surfactants and sodium sulphate for particles greater than 1 μm. The onset of convection was detected by a change of the slope of the curves showing the heat flux versus the temperature difference. The experimental results show (i) a good agreement between the experimental values of effective thermal conductivity and the theoretical formulas, (ii) an increase of the convective threshold with a volume fraction that will be attributed to the stabilizing effects of the particles on the fluid (growth of the thermal relaxation time and decrease of the buoyancy time), and (iii) a nonmonotonous subcritical bifurcation for particles smaller than 1 μm and supercritical bifurcation otherwise.

The role of time‐varying gravity on the motion of a drop induced by Marangoni instability
View Description Hide DescriptionWe study the influence of a time‐dependent force on the translatory convective instability of a drop due to surface stresses (Marangoni effect). This effect and its associate flows inside and outside the drop are induced by solute transfer from within the drop to the drop surface and its consumption there in an isothermal chemical reaction. In particular, we show that for a gravity field sinusoidally varying in time, the drop rather than aligning with the direction of the force tends to move in a direction orthogonal to it.

Nonlinear waves and turbulence in Marangoni convection
View Description Hide DescriptionIn the paper we present a numerical study of a new type of nonlinear waves in Marangoni convection. The waves are caused by nonlinear interaction between long‐scale deformational instability and short‐scale convection. It is shown that, due to a nonlinear coupling with the deformation of the free liquid–gas interface, the primary convectionpattern can undergo oscillatory instability generating various kinds of long surface waves which modulate the short‐scale convection. The numerical analysis of the system of nonlinear coupled equations describing these waves confirms the predictions of weakly nonlinear analysis and shows the existence of either standing or travelling waves in the proper parametric regions, at low supercriticality. With increasing supercriticality, the waves undergo various transformations leading to the formation of pulsating travelling waves, aharmonic standing waves as well as irregular wavy behavior resembling ‘‘interfacial turbulence.’’ We map regions in the parameter space where various kinds of waves can be observed, and describe some characteristics of irregular wavy behavior.

Numerical simulations of thermal convection in a rotating spherical fluid shell at high Taylor and Rayleigh numbers
View Description Hide DescriptionIn this study, we carry out numerical simulations of thermal convection in a rapidly rotating spherical fluid shell at high Taylor number Ta and Rayleigh number R with a nonlinear, three‐dimensional, time‐dependent, spectral‐transform code. The parameters used in the simulations are chosen to be in a range which allows us to study two different types of convection, i.e., single column and multi‐layered types, and the transition between them. Numerical solutions feature highly time‐dependent north–south open columnar convective cells. The cells occur irregularly in longitude, are quasi‐layered in cylindrical radius, and maintain alternating bands of mean zonal flow. The complex convective structure and the banded mean zonal flow are results of the high Taylor and Rayleigh numbers. The transition between the two types of convection appears to occur gradually with increasing Rayleigh and Taylor numbers. At a Taylor number of 10^{7} the differential rotation pattern consists of an inner cylindrical region of subrotation and an outer cylindrical shell of superrotation manifest at the outer boundary as an equatorial superrotation and a high latitude subrotation. The differential rotation pattern is similar at Ta=10^{8} and low Rayleigh number. Cylindrical shells of alternately directed mean zonal flow begin to develop at Ta=10^{8} and R=50R _{ c } and at Ta=10^{9} and R=25R _{ c }. This pattern is seen on the outer surface as a latitudinally‐banded zonal flow consisting of an equatorial superrotation, a middle and high latitude subrotation, and a polar superrotation. At Ta=10^{9} and R=50R _{ c } the differential rotation appears at the surface as a broad eastward flow in the equatorial region with alternating bands of westward and eastward flow at high latitudes.

Unsteady swirling flow in an enclosed cylinder with reflectional symmetry
View Description Hide DescriptionA numerical investigation of the multiple stable solutions found in confined swirling flows is presented. The flows consist of fluid in a completely filled cylinder driven by the constant corotation of the two end walls. When reflectional symmetry at the cylinder half‐plane is imposed, the flow corresponds to that in a cylinder of half the height driven by the bottom end wall, with the top surface being flat and stress‐free. Comparisons with available experiments in this case are made and the observed toroidal recirculation zones attached to the free surface are described in terms of secondary motions induced by the bending of vortex lines. Calculations are also presented where the reflectional symmetry is not imposed and the possibility of the flow breaking this symmetry is discussed.

Fractal characteristics of isoconcentration surfaces in plumes dispersing in the atmospheric surface layer
View Description Hide DescriptionThe geometrical properties of isoconcentration surfaces in a plume dispersing in the atmospheric surface layer are studied using a generalized box‐counting method applied to a limited random point set. This method yields the hierarchy of generalized dimensions D _{ q } that can be used to characterize the fractal nature of the plume concentration level sets. The dimension spectra for the concentration level sets are computed from one‐dimensional cuts of the concentration field. The concentration level sets are found to be monofractals that can be characterized by one scaling exponent or fractal dimension. The fractal dimension of the level sets is independent of the concentration threshold over a wide range of threshold values. The evolution of the fractal dimension of plume concentration level sets with distance x downwind from the source, cross‐wind distance y from the lateral mean‐plume centerline, and vertical height z above the ground is examined. At a fixed plume height, the fractal dimension is essentially independent of either x or y. The fractal dimension of the plume isoconcentration surface decreases roughly linearly from a value of 0.7±0.05 near the surface (z≲1 m) to 0.45±0.05 higher up in the plume (z≳8 m). The increased wrinkling of the plume isoconcentration surface near the ground is most likely the result of the increased mean velocity shear and blocking by the surface.