Volume 13, Issue 11, November 2001
 LETTERS


Selfsimilar concentration profiles in buoyant mixing of miscible fluids in a vertical tube
View Description Hide DescriptionThe influence of the density contrast (characterized by the Atwood number At) on gravityinduced mixing between two miscible fluids in a long vertical tube has been studied experimentally. Crosssection averaged fluid concentration profiles along the tube are measured optically: for large enough At values, they display a selfsimilar dependence in a broad range of times and distances and verify a diffusion law with an effective diffusivity times higher than for molecular diffusion. At lower At values, this diffusive domain is limited by a sharp front moving at a velocity increasing with At. Below a threshold At value the diffusive behavior disappears.
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 ARTICLES


Capillary flow in interior corners: The infinite column
View Description Hide DescriptionCapillary flow of a sinusoidally perturbed liquid column in an interior corner of infinite extent is solved using lubrication theory. Due primarily to the length scales selected to nondimensionalize the momentum equation, an analytic time scale governing the settling of the perturbation is determined. The time scale, which is shown to be independent of a steady base state flow, proves useful in rapidly predicting transients for surface settling in certain liquidbearing tanks of spacecraft employing interior corners for fluids management purposes. The asymptotic analysis is extended to address flows along interior corners whose faces are slightly nonplanar. The generalized formulation is presented for the case of a perfectly wetting fluid in a secondorder polynomial corner. A leadingorder analytic solution for small corner angles is provided. It is shown that a “convex corner” decreases the response time of the liquid and increases the capillary flow rate along the corner by increasing both the driving force and crosssectional area of the flow. Gravity acting normal to the corner axis along the bisector of the corner angle is also considered and is found to accelerate, decelerate, or destabilize such flows depending on its sign and magnitude.

Timeaveraged hydrodynamic roughness of a noncolloidal sphere in low Reynolds number motion down an inclined plane
View Description Hide DescriptionA system consisting of a spherical particle in motion down an inclined planar surface in a viscous liquid was investigated theoretically and experimentally to examine the effects of surface roughness on the interactions between the sphere and the plane. Two characteristic roughness scales were used to describe the microscopic surface roughness of the sphere. The smallest roughness elements are assumed to dominate the surface, and the largest roughness elements are more sparse. The timeaveraged nominal separation between the sphere and the plane was found to increase as the planar surface was made steeper. This apparent hydrodynamic roughness is governed by the heights of the smallest roughness when the sphere resides on a horizontal plane, whereas the largest roughness elements govern the apparent hydrodynamic roughness when the plane is inclined at a steep angle. On a steep incline, the normal component of the gravitational force that drives the sphere toward the plane is relatively weak. Hence, as the sphere migrates toward the plane after contact with a large asperity ends, its rotation may result in another large asperity forcing the sphere away from the plane before contact with the smaller asperities occurs. The timeaveraged separation at intermediate angles increases with increasing surface coverage by the largest roughness elements. The method of Smart and Leighton [Phys. Fluids A 1, 526 (1989)] was modified to determine the hydrodynamic separation between the sphere and the plane during its motion down the incline. The apparent hydrodynamic roughness values obtained in the experiments increase as the angle of inclination of the plane was increased, and provide a satisfactory validation of the model. The relatively large but sparse roughness elements have a disproportionate effect on the timeaveraged hydrodynamic roughness, especially at high angles of inclination. These findings may be important in the interaction of pairs of spherical particles in viscoussuspensions, where the effective angle of inclination varies significantly. For example, the presence of a low concentration of relatively large roughness elements should result in significantly higher levels of hydrodynamic diffusion.

Wavelength selection of fingering instability inside Hele–Shaw cells
View Description Hide DescriptionFingering instabilities involving fluids confined between two plates sometimes give rise to a typical wavelength λ proportional to the gap h. This unexplained behavior is investigated for the case of the Rayleigh–Taylor instability between two liquids of the same viscosity. Using qualitative scaling arguments and linear stability analysis for a simplified model of hydrodynamics, we show that, in the miscible case, h becomes a natural cutoff when diffusion is negligible, i.e., when the Péclet number is large (η viscosity, g gravitational acceleration, D diffusivity, density difference). The same result holds in the immiscible case for large capillary number (γ surface tension). In this saturation regime, the dominant wavelength is given by while in the opposite limit (low Pe or low Ca) λ scales, respectively, as or These results are in agreement with a recent experimental study.

Axisymmetric flow due to a porous sphere sedimenting towards a solid sphere or a solid wall: Application to scavenging of small particles
View Description Hide DescriptionA classic difficulty in applying a boundary condition is handled by a new mathematical approximation, correct to second order, which is used to reduce to a double set of difference equations the creeping flow problem posed by the sedimentation of a solid sphere and a porous sphere within which the Brinkman equation is assumed to be valid. It is shown that the permeability allows heavier small solid particles to be captured and markedly reduces the force experienced by a porous sphere approaching a solid plane.

Singularity induced exterior and interior Stokes flows
View Description Hide DescriptionIn this paper, the twodimensional Stokes flow inside and outside a circular cylinder induced by a pair of line singularities (rotlet and stokeslet) is studied. Analytical solutions for the flow field are obtained by straightforward application of the Fourier method. The streamline patterns are sketched for a number of special cases where the cylinder is either stationary or rotating about its own axis. In particular, some interesting flow patterns are observed in the parameter space which may have potential significance in studies of various flows including flows in journal bearing, mixing flows, chaotic flows, etc. We also investigate into the way the streamline topologies change as the parameters are varied.

Stability of return thermocapillary flows under gravity modulation
View Description Hide DescriptionThe effect of gravity modulation on the stability of a timeharmonic parallel flow in a slot geometry is studied. A constant temperature gradient applied along the length of a slot with a flat free interface at the top drives a steady thermocapillary return flow. A vertical timeharmonic gravity modulation drives a timeperiodic buoyant return flow. The relative strength of the two components is characterized by the Marangoni number Ma and the Rayleigh number Ra, respectively. There is potential for Rayleigh–Bénard, shear, and hydrothermal wave instabilities. The linear stability for twodimensional rolls is studied and the stability boundaries are obtained by Floquet theory. Stability diagrams in the Ra–Ma plane are obtained for fixed modulation frequency and various Prandtl numbers and a disturbance kineticenergy analysis is used to determine the instability mechanism. Buoyant and hydrothermal instabilities are found at small Ma and small Ra, respectively, whereas shear instabilities are found not to be important in the parameter regime studied. The two active mechanisms are found to either reinforce or oppose each other in different parameter regimes. This leads to regions of stability in the Ra–Ma plane where flows are stable due to the combined effect of buoyancy and thermocapillarity whereas a purely buoyant or a purely thermocapillary flow at the same Ra or Ma would be unstable. These results are contrasted with those obtained by Nield [J. Fluid Mech. 19, 341 (1964)] in the case of steady gravity, for which thermocapillarity and buoyancy reinforce each other.

Pattern formation in the flow of thin films down an incline: Constant flux configuration
View Description Hide DescriptionWe present fully nonlinear timedependent simulations of a thin liquid film flowing down an inclined plane. Within the lubrication approximation, and assuming complete wetting, we find that varying the inclination angle considerably modifies the shape of the emerging patterns: Fingershaped patterns result for the flow down a vertical plane, while sawtooth patterns develop for the flows down an inclined plane. However, in all of our simulations, the roots always move, indicating that the shape of the patterns is not necessarily related to the surface coverage, a technologically important feature of the flow. Furthermore, we find that triangular steadystate patterns may be produced for the flow down an incline, while the fingers typically grow in length for all explored times. We find quantitative agreement with reported experiments, and suggest new ones.

Threedimensional instability of a twolayer Dean flow
View Description Hide DescriptionStability of a twolayer Dean flow in a cylindrical annulus with respect to threedimensional perturbations is studied by a global Galerkin method. It is shown that for large inner radius of the annulus (i) the instability becomes threedimensional if one of the fluid layers is thin, (ii) its onset is not affected by possible small deformations of the interface, and (iii) multiple threedimensional flow states are expected in a slightly supercritical flow regime. Stability diagrams and patterns of the threedimensional perturbations are reported. It is concluded that even when the axisymmetric perturbation is the most dangerous, the resulting supercritical flow is expected to be threedimensional. Possible multiplicity of supercritical threedimensional states is predicted. The basis functions of the global Galerkin method are constructed so as to satisfy analytically the boundary conditions on noslip walls and at the liquid–liquid interface. A modification of the numerical approach, accounting for small deformations of the interface which is subject to the action of the capillary force, is proposed. The results are of potential importance for development of novel bioseparators employing Dean vortices for enhancement of mass transfer of a passive scalar (say, a protein) through the interface. The developed numerical approach can be used for stability analysis in other twofluid systems.

Viscous fingering in a magnetic fluid. II. Linear Hele–Shaw flow
View Description Hide DescriptionViscous fingering phenomenon in a linear channel is studied for a magnetic fluid subjected to an external magnetic field. The competition between the hydrodynamic effects and the capillary effects leads to the formation of an interface between the air and the fluid which has a finger shape. It is the socalled Saffman–Taylor instability (STI). The influence of the magnetic effects depends on the direction of the applied field: it is possible either to enhance or to reduce the destabilizing phenomena. We study the onset of the STI and compare the experimental results with the linear analysis including the magnetic contribution. In the nonlinear regime, the measurement of the width of the finger as a function of the direction and the amplitude of the magnetic field is understood using a phenomenological approach.

Loworder models for the flow in a differentially heated cavity
View Description Hide DescriptionThe proper orthogonal decomposition (P.O.D.) is applied to the flow in a differentially heated cavity. The fluid considered is air, and the aspect ratio of the cavity is 4. At a fixed Rayleigh number, P.O.D. empirical functions are extracted, and lowdimensional models are built and compared to the numerical simulation. Generally speaking, lowD models provide a coarse picture of the flow, which is also quick, cheap, and easy to understand. They can help pinpoint leading instability mechanisms. They are potentially key players in a number of applications such as optimization and control. Our goal in this study is to determine how well the flow can be represented by very lowdimensional models. Two moderately complex situations are examined. In the first case, at some distance from the bifurcation point, the dynamics can still be reduced down to two modes, although it is necessary to account for the effect of higherorder modes in the model. In the second case, farther away from the bifurcation, the flow is chaotic. A tendimensional model successfully captures the essential dynamics of the flow. The procedure was seen to be robust. It clearly illustrates the power of the P.O.D. as a reduction tool.

Scattering of surface waves by a semiinfinite floating elastic plate
View Description Hide DescriptionA new inner product is developed based on the Fourier analysis to study the scattering of surface waves by a floating semiinfinite elastic plate in a twodimensional water domain of finite depth. The eigenfunctions for the platecovered region are orthogonal with respect to this new inner product. The problem is studied for various wave and geometrical conditions. Especially, the influence of different edge conditions on the hydrodynamic behavior is investigated and compared. The edge conditions considered in the present study involve (i) a free edge, (ii) a simply supported edge, and (iii) a builtin edge. The hydrodynamic performance of an elastic plate is characterized for various conditions in terms of wave reflection and transmission, plate deflection, and surface strain. It is observed that the hydrodynamic behavior depends on the wave conditions, the geometrical settings, and the edge conditions. The builtin edge condition induces the maximum wave reflection and the minimum wave transmission. The free edge condition leads to the maximum plate deflection.

Internal waves generated by the wake of Gaussian hills
View Description Hide DescriptionDepending on their structure and dynamics, the wakes of various obstacles can generate different kinds of internal waves in stratified fluids. Experiments on waves emitted by threedimensional Gaussian models towed uniformly in a linearly stratified fluid were carried out. Beyond a critical value of the Froude number Fr, the developed wake was observed to radiate an internal wave field shorter than the lee waves of the hill. The emergence of such waves is correlated with the periodic shedding of three dimensional vortical structures at Measurements show that the wavelengths of these short waves are constant in space and time, and proportional to Fr. Their phase and group velocities are proportional to the coherent structure velocity which is estimated from velocity measurements inside the wake. All those spatiotemporal characteristics prove that these short waves are generated by the displacement of the coherent structure inside the wake.

Onset of thermohaline convection in a rectangular porous cavity in the presence of vertical vibration
View Description Hide DescriptionDouble diffusive convection in a twodimensional rectangular cavity with imposed vertical differences in temperature and concentration is considered in the presence of highfrequency vertical vibrations. Linear stability analysis permits the domains of oscillatory and stationary convection thresholds to be delimited. Depending on the governing parameters, vibrations are found to delay or speed up the onset of convection. The effect of vibration on the flow structure near the bifurcation is also analyzed, showing different possible behaviors. The stationary bifurcation is then studied by means of a weakly nonlinear analysis. In agreement with the symmetries present in the problem, this bifurcation is found to be pitchfork. The parametric ranges where it can be subcritical or supercritical are precisely delimited. Finally, direct numerical simulations have been performed to confirm and illustrate these findings.

Rayleigh–Bénard convection in liquid metal layers under the influence of a vertical magnetic field
View Description Hide DescriptionThe influence of a vertical magnetic field on the integral heat transfer and the temporal dynamics of liquid metal Rayleigh–Bénard convection is studied in an experiment using a small Prandtl number (Pr≈0.02) sodiumpotassium alloy as a test fluid. The test section is a rectangular box of large aspect ratio 20 : 10 : 1 that covers a parameter range of Rayleigh numbers, and Chandrasekhar numbers, The integral heat transfer across the layer is evaluated from the measuredtemperatures at the upper and the lower boundary and the applied heat flux. Local, timedependent temperatures are obtained from a fourelement temperature probe placed in the middle of the liquid metal layer. The noncoplanar arrangement of the thermocouples enables the evaluation of the timedependent temperature gradient vector that allows us to estimate the local isotropy properties of the timedependent flow. From the damping effect of Joule dissipation, the convective heat transport decreases monotonically with increasing Chandrasekhar numbers. Fluctuations of the temperature field are damped significantly by the magnetic field. However, this effect is selective with respect to frequency. Long period fluctuations are strongly damped whereas short period fluctuations are less damped or may even be amplified. The observations show that significant convective heat transport is practically always associated with timedependent flow. The fluctuating part of the local temperature gradient confirms the horizontal isotropy of the velocity field; no predominant orientation of timedependent flow structures is established either with or without a magnetic field.

Energy amplification in channel flows with stochastic excitation
View Description Hide DescriptionWe investigate energy amplification in parallel channel flows, where background noise is modeled as stochastic excitation of the linearized Navier–Stokes equations. We show analytically that the energy of threedimensional streamwiseconstant disturbances achieves amplification. Our basic technical tools are explicit analytical calculations of the traces of solutions of operator Lyapunov equations, which yield the covariance operators of the forced random velocity fields. The dependence of these quantities on both the Reynolds number and the spanwise wave number are explicitly computed. We show how the amplification mechanism is due to a coupling between wallnormal velocity and vorticity disturbances, which in turn is due to nonzero mean shear and disturbance spanwise variation. This mechanism is viewed as a consequence of the nonnormality of the dynamical operator, and not necessarily due to the existence of near resonances or modes with algebraic growth.

Comparison of experiments and a nonlinear model equation for spatially developing flame instability
View Description Hide DescriptionWe extend the Michelson–Sivashinsky equation, for arbitrary gas density ratio, to model the dynamics of an inclined anchored flame front. A flow velocity component tangential to the flame front transforms temporally developing structures into spatially developing ones. Closer investigation shows that a transition from absolute to convective instability should occur for a finite flow velocity. Using pole decomposition, we give a particular solution to this equation in the limit of high flow velocity compared to laminar flame speed. The analytical results are compared to measurements on a twodimensional laminar flame. The growth rate, shape, and amplitude of the saturated structures, as well as the width of the crossover region are all well described. However, we experimentally observe the presence of a large scale transverse gas circulation not predicted by the analysis. We attribute this effect to the streamwise variation of the pressure jump across the flame brush.

Application of power laws to low Reynolds number boundary layers on smooth and rough surfaces
View Description Hide DescriptionScaling laws for the overlap region of nearwall turbulent flows are of particular interest to turbulence researchers and engineers. For the mean flow at sufficiently high Reynolds numbers, the classical boundary layer theory proposes a logarithmic law for the overlap region. On the other hand, at low Reynolds numbers, refined measurements and direct numerical simulation results indicate that the log law region becomes negligibly small. Instead, power laws have received increasing attention as an alternative formulation for the overlap region at low Reynolds numbers. In the present study, we use open channel flow measurements to assess the ability of the power laws proposed by Barenblatt [J. Fluid Mech. 248, 513 (1993)] and George and Castillo [Appl. Mech. Rev. 50, 689 (1997)] to describe the overlap region in low Reynolds number boundary layers on smooth and rough surfaces. The skin friction laws derived from the power laws are also used to estimate the friction velocity, which values are then compared to measurements obtained by other reliable techniques. The results indicate that at low Reynolds numbers the power law formulations can model a wider extent of the flow than the classical logarithmic profile. Both Barenblatt’s and George and Castillo’s power laws give an excellent prediction of the friction velocities for flows over a smooth surface, but only the skin friction law proposed by George and Castillo gives good prediction for the rough wall data.

Control of circular cylinder wakes using base mass transpiration
View Description Hide DescriptionThe effect of steady base suction and blowing on the stability and dynamics of the cylinder wake is investigated at low Reynolds numbers, using numerical simulation and stabilityanalysis. Simulation results show that, in the supercritical Reynolds number regime slight blowing or high enough suction stabilizes the wake; in the subcritical regime, suction destabilizes the wake and results in vortex shedding, whereas blowing has no detectable effect on the flowstability. At supercritical Reynolds numbers, the transition from unsteady to steady flow at a critical suction flow rate is accompanied by simultaneous symmetry breaking, resulting in strongly asymmetric steady flow. For finite flow domain, the flow undergoes another transition from steady asymmetric to steady symmetric flow at even higher suction flow rates. The dynamics of vortex shedding in the controlledflow can be strongly modified, in comparison to the uncontrolled flow. Global stabilityanalysis confirms the results of numerical simulation, and yields detailed information on the dynamics of linear global modes. Computational domain size can significantly affect the flow transitions, in particular, at high suction flow rates.

Effects of nonreacting solid particle and liquid droplet loading on an exothermic reacting mixing layer
View Description Hide DescriptionNumerical simulations are conducted of twodimensional (2D) exothermic reacting mixing layers laden with either solid particles or evaporating liquid droplets. An irreversible reaction of the form fuel+r Oxidizer→(1+r) Products with exothermic Arrhenius kinetics is considered. The temporally developing mixing layers are formed by the merging of parallel flowing oxidizer and fuel streams, each uniformly laden with nonreacting particles or droplets. The gaseous phase is governed by the compressible form of the Navier–Stokes equations together with transport equations for the fuel, oxidizer, product, and evaporated vapor species concentrations. Particles and droplets are assumed smaller than the gasphase length scales and are tracked individually in the Lagrangian reference frame. Complete “twoway” couplings of mass, momentum, and energy between phases are included in the formulation. The simulation parameters are chosen to study the effects of the mass loading ratio, particle Stokes number, vaporization, flow forcing, and reaction Zeldovich number on the flame evolutions. Quasionedimensional simulations reveal that the asymptotic state of the laminar flames is independent of the particle or droplet loading. For forced 2D simulations, both particles and droplets are preferentially concentrated into the highstrain braid regions of the mixing layer. Cold solid particles entrained into the mixing zone cool the flame in the braid regions due to their finite thermal inertia. This results in flame suppression and, under certain conditions, local flame extinction in the braids. The potential for flame extinction is substantially enhanced by evaporating droplets through the latent heat, and also by the addition of nonreacting evaporated vapor which locally dilutes the reactant concentrations. In contrast, combustion proceeds robustly within vortex cores which have relatively dilute droplet distributions due to preferential concentration; particularly at moderate Stokes numbers. The extent of flame suppression and local extinction are increased with increasing reaction activation energy, dispersed phase mass loading, and also by decreasing particle or droplet Stokes number.
