Volume 13, Issue 1, January 2001
 LETTERS


Stabilization and destabilization of channel flow by location of viscositystratified fluid layer
View Description Hide DescriptionThe stability of the channel flow of two fluids of different viscosities with a mixed layer in between is demonstrated to be qualitatively different from both interface dominated flows and stratified flows. More important, this flow displays unexpected changes in stability when the mixed layer overlaps the critical layer of the disturbance: this feature can be exploited for flow control. When these layers are distinct, the flow is mildly destabilized when the less viscous fluid is in the outer region. When the layers overlap, however, there is an order of magnitude stabilization of the flow. The reverse occurs when the more viscous fluid is in the outer region. This behavior may be explained by the balance of stresses in the critical layer.

Granular jets
View Description Hide DescriptionWhen a solid sphere impacts on a deep layer of granular medium, it generates an ejecta sheet and a transient axisymmetric crater. The gravitydriven radial collapse of this crater generates a pressure spike, as the cavity closes up. This pressure spike drives up a narrow granular jet along the axis of symmetry. The maximum height of the jet is found to depend on the impact velocity, gravity as well as the effective viscosity of the granular medium, through a simple product of the Reynolds and Froude numbers. The presence of such granular jets, where surface tension is absent, may help pinpoint the role of surface tension for similar liquid jets.
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 ARTICLES


Effect of inertia on drop breakup under shear
View Description Hide DescriptionA spherical drop, placed in a second liquid of the same density and viscosity, is subjected to shear between parallel walls. The subsequent flow is investigated numerically with a volumeoffluid continuoussurfaceforce algorithm. Inertially driven breakup is examined. The critical Reynolds numbers are examined for capillary numbers in the range where the drop does not break up in Stokes flow. It is found that the effect of inertia is to rotate the drop toward the vertical direction, with a mechanism analogous to aerodynamic lift, and the drop then experiences higher shear, which pulls the drop apart horizontally. The balance of inertial stress with capillary stress shows that the critical Reynolds number scales inversely proportional to the capillary number, and this is confirmed with full numerical simulations. Drops exhibit selfsimilar damped oscillations towards equilibrium analogous to a onedimensional massspring system. The stationary drop configurations near critical conditions approach an inviscid limit, independent of the microphysical flow and fluidparameters.

Coating flows within a rotating horizontal cylinder: Lubrication analysis, numerical computations, and experimental measurements
View Description Hide DescriptionWe consider the flow within a rotating horizontal cylinder containing a small amount of a very viscous liquid which completely coats the cylinder surface. We show that, under creeping flow conditions, the addition of the hydrostatic pressure term to the standard lubrication equation leads to film thickness profiles which, over a broad range of parameters, are in close agreement with those obtained experimentally, as well as via the solution to the full Stokes equations.

An extended Rayleigh model of bubble evolution
View Description Hide DescriptionAn extended Rayleigh model for laser generated bubbles in water and soft tissue is presented. This model includes surface tension,viscosity, a realistic equation of state,material strength and failure, stress wave emission, and linear growth of interface instabilities. The model is validated by comparison to detailed compressible hydrodynamic simulations using the LATIS computer program. The purpose of this study is to investigate the use of the extended Rayleigh model as a much faster and simpler substitute for the detailed hydrodynamic simulations when only limited information is needed. It is also meant to benchmark the hydrosimulations and highlight the relevant physics. The extended Rayleigh model and the hydrosimulations are compared using both a 1D spherical geometry with a bubble in the center and a 2D cylindrical geometry of a laser fiber immersed in water with a bubble formed at the end of the fiber. Studies are done to test the validity of the material strength and failure, stress wave emission, and the interface instability terms in the extended Rayleigh model. The resulting bubble radii, material damage radii, the emitted stress wave energies, and the size of the interface distortions are compared. Many of the trends found in the hydrosimulations are illuminated by the extended Rayleigh model owing to its relative simplicity. The extended Rayleigh model is very useful since it is accurate over a large range of parameters and it is computationally much faster than the hydrosimulations.

Calculation of the Ewald summed farfield mobility functions for arbitrarily sized spherical particles in Stokes flow
View Description Hide DescriptionThe hydrodynamic interactions among an infinite suspension of arbitrarily sized spherical particles are calculated at low Reynolds number. Absolutely convergent expressions for the particle interactions are found that are cast into a form convenient for computational simulation of the motion of suspensions of spherical particles. The convergence of the periodically replicated form of these hydrodynamic expressions is accelerated by means of Ewald summation. The resulting farfield mobility functions that relate forces, torques, and stresslets to the particle velocities, angular velocities, and rate of strain are given for a suspension spherical particles of any size distribution. These mobility functions are used in simulations of particle suspensions that consider the dynamics of each particle. Specifically these mobility functions can be used in Stokesian dynamics simulations of polydisperse suspensions.

Model and experiments of a drop impinging on an immersed wall
View Description Hide DescriptionWe present a model describing the rebound of a drop impinging on a rigid plane wall immersed in water. This model is based on the resolution of the dropequation of motion in an unbounded fluid in which an additional pressure force is introduced accounting for the wall effect on the drop motion. This force is computed from a film drainage simulation model during the approach of the deformable particle to the wall. Results of the model have been compared with experimental trajectories of drops impinging vertically at terminal rise velocity against a horizontal wall immersed in water at rest. These trajectories have been obtained with the help of an image processing technique. A wide range of experimental conditions has been studied (drop diameter, interfacial tension, dropviscosity, and density). In most of the cases, the model predicts the experimental trajectories within a very good accuracy (height of bouncing, deformation, number of rebounds) even in the case of a significant deformation. The numerical results show that the rebound of a deformable inclusion against a wall in water is essentially governed by the balance between the added mass force and the film pressure force exerted on the drop during the impact. The model has also been successfully tested in the case of an impinging bubble at high particle Reynolds number, based on experimental data taken from Tsao and Koch [Phys. Fluids 9, 44 (1997)].

A particle model of rolling grain ripples under waves
View Description Hide DescriptionA simple model for the formation of rolling grain ripples on a flat sand bed by the oscillatory flow generated by a surface wave is presented. An equation of motion is derived for the individual ripples, seen as “particles,” on the otherwise flat bed. The model accounts for the initial appearance of the ripples, the subsequent coarsening of the ripples, and the final equilibrium state. The model is related to the physical parameters of the problem, and an analytical approximation for the equilibrium spacing of the ripples is developed. It is found that the spacing between the ripples scales with the squareroot of the nondimensional shear stress (the Shields parameter) on a flat bed. The results of the model are compared with measurements, and reasonable agreement between the model and the measurements is demonstrated.

Twodimensional direct numerical simulation of parametrically excited surface waves in viscous fluid
View Description Hide DescriptionStanding surface waves on a viscousfluid driven parametrically by a vertical harmonic oscillation are investigated, based on direct numerical simulations of the twodimensional Navier–Stokes equation, together with appropriate boundary conditions. The condition for the onset of the waves in the experiments by Lioubashevski et al. [Phys. Rev. Lett. 76, 3959 (1996)] is reproduced by our numerical simulation. The time evolution and the flow structures are investigated in detail. The form of the surface elevation is analyzed and the dependence of the saturated amplitude on the forcing strength shows a normal bifurcation. Instead of a localized state,spatially uniform standing waves are formed in an extended system. Using initial perturbations of the uniform state, numerical simulations show that the uniform standing waves are stable to twodimensional disturbances, which suggests that threedimensionality is essential for the spatiallylocalized state to occur.

Statistical mechanics with threedimensional particle tracking velocimetry experiments in the study of anomalous dispersion. I. Theory
View Description Hide DescriptionEulerian models developed to simulate dispersion in fluid mechanics often consider the flux of the contaminant species to be proportional to the concentration gradient via a constant or timedependent dispersion coefficient. These models are crude approximations for systems with velocity fluctuations evolving over a hierarchy of scales on the scale of observation. We say a system behaves in a Fickian fashion if the dispersion coefficient is constant, it is quasiFickian if the dispersion coefficient is time dependent, and it is convolutionFickian if the flux is a convolution. The fractional flux in the sense of fractional derivatives is a special case of a convolutionFickian flux. More general forms of the flux are possible, and in any case we call all fluxes anomalous if there is not a constant coefficient of proportionality between the flux and the gradient of concentration. In paper I of this twopart sequence we present a theory with statistical mechanical origins for simulating anomalous dispersion. Under appropriate limiting conditions the theory gives rise to Fickian, quasiFickian, convolutionFickian, and fractionalFickian fluxes. The primary result is a dispersive flux of integral type which in its most general form is not a convolution on time (it is nonMarkovian however), but it is always a convolution in space. The concentration is represented by the inverse Fourier transform of the selfpart of the intermediate scattering function. In paper II we present an experimental procedure that uses this theory to examine if and when the Fickian limit is reached in porous media homogeneous on the Darcyscale but heterogeneous on the porescale.

Statistical mechanics with threedimensional particle tracking velocimetry experiments in the study of anomalous dispersion. II. Experiments
View Description Hide DescriptionIn paper I [Phys. Fluids 13, 75 (2001)] we provided a theory for simulating anomalous dispersion which relied on the selfpart of the intermediate scattering function. Here we obtain Lagrangian trajectories for a conservative tracer in a porous medium and then use these trajectories to obtain the selfpart of the intermediate scattering function. We then use the scattering function as data for the inverse problem and obtain the generalized wavevector and frequency dependent dispersion tensor developed in paper I. The transverse components of this tensor are then examined as a function of wave vector to see if or when the dispersive process goes asymptotic (Fickian). The matched index (of refraction) technique has been used to obtain a transparent porous medium and three dimensional particle tracking has been used to obtain the trajectories. Over the life of the experiment the transverse dispersive process remained anomalous, though it was gradually approaching the Fickian limit.

Threedimensional Marangoni–Bénard flows in square and nearly square containers
View Description Hide DescriptionNumerical continuation is used to follow branches of steady solutions to the threedimensional Marangoni–Bénard problem in a zero gravity environment. The upper surface of the fluid is heated by a constant heat flux while the bottom is maintained at a constant temperature. Instability arises due to temperaturedependent surface tension effects but surface deflection is ignored. Containers with square and nearly square cross sections and noslip boundary conditions are analyzed, and the results interpreted in terms of predictions from equivariant bifurcation theory.

Exchange of instability modes for natural convection in a narrow horizontal annulus
View Description Hide DescriptionInstabilities and transitions of natural convection in a narrow horizontal concentric annulus are investigated theoretically by assuming twodimensional and incompressible flow fields. It is assumed that the inner cylinder is kept at a higher temperature than the outer cylinder. Steady solutions for the natural convection are obtained numerically by Newton–Raphson’s method for various values of Prandtl number and their linear stabilities are analyzed. It is found that there are two different instability modes for the natural convection depending on the Prandtl number, which exchange at a critical value of the Prandtl number. The origins of the two instabilities are clarified from the bifurcation and linear stability analyses of the steadystate solutions.

Oscillatory thermocapillary convection in liquid bridges with highly deformed free surfaces: Experiments and energystability analysis
View Description Hide DescriptionLaboratory experimentation, numerical simulation, and energystability theory are used to examine the effect of interface deformation on the onset of oscillatorythermocapillary convection in half zones. Experiments are performed to map the stability boundaries marking the onset of oscillatoryflow, modifying the freesurface deformation by adjusting the volume of liquid in the bridge. The stability results presented here along with those of other researchers [Monti et al., Proceedings of the 43rd Cong. Int. Artro. Fed. (1992); Hu et al., J. Cryst. Growth 142, 379 (1994)] show that freesurface curvature can have a pronounced influence on flow stability. Steady, axisymmetric flow simulations are computed using the commercial code FIDAP to model the conditions of the experiments, and reveal that flow structure near the stability boundary is sensitive to several parameters. Energy theory is applied to these simulations to determine sufficient conditions for stability. Comparisons between the theoretical and experimental results show nonconservative energy limits falling above the experimentally determined stability boundaries for bridges of various liquid volumes. While the trend of the experimental data is predicted for zones of large volume ratio (bulging zones), the same cannot be said for those with small volume ratio (neckeddown zones). In addition, energystability limits for some undeformedfreesurface cases were determined which are above the linearstability limits determined by other researchers, in clear contradiction of the roles of the respective theories.

Threedimensional centrifugalflow instabilities in the liddrivencavity problem
View Description Hide DescriptionThe classical rectangular liddrivencavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tangentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is twodimensional for small Reynolds numbers. By a linear stability analysis it is shown that this basic flow becomes unstable at higher Reynolds numbers to four different threedimensional modes depending on the aspect ratio of the cavity’s cross section. For shallow cavities the most dangerous modes are a pair of threedimensional short waves propagating spanwise in the direction perpendicular to the basic flow. The mode is localized on the strong basicstate eddy that is created at the downstream end of the moving lid when the Reynolds number is increased. In the limit of a vanishing layer depth the critical Reynolds number approaches a finite asymptotic value. When the depth of the cavity is comparable to its width, two different centrifugalinstability modes can appear depending on the exact value of the aspect ratio. One of these modes is stationary, the other one is oscillatory. For unit aspect ratio (square cavity), the critical mode is stationary and has a very short wavelength. Experiments for the square cavity with a large span confirm this instability. It is argued that this threedimensional mode has not been observed in all previous experiments, because the instability is suppressed by sidewall effects in smallspan cavities. For large aspect ratios, i.e., for deep cavities, the critical threedimensional mode is stationary with a long wavelength. The critical Reynolds number approaches a finite asymptotic value in the limit of an infinitely deep cavity.

Bifurcation phenomena in a Taylor–Couette flow with asymmetric boundary conditions
View Description Hide DescriptionWe present the results of an experimental and numerical study of bifurcation phenomena in the flow between a rotating inner cylinder and a stationary outer one, the socalled Taylor–Couette problem. Novel asymmetric boundary conditions have been used where one end plate is rotated with the inner cylinder while the other is held fixed. The steady cellular flows consist of one or three vortices in the aspect ratio range considered. We have investigated the selection procedure for the preferred steady states experimentally and numerically and good quantitative agreement is found between the two sets of results. The sequence involves a fold in the solution surface which produces a cusp in the bifurcation set. Hopf bifurcations to threedimensional oscillatory flow are also present and so the full bifurcation structure is revealed using the combined numerical and experimental approach in a complimentary manner.

Numerical simulation of a gas–liquid flow in a fixed bed
View Description Hide DescriptionA countercurrent gas–liquid flow through a fixed bed of spherical particles is examined numerically by solving the particlescale equations governing the gas and liquidflows. The liquid is assumed to flow along the surface of the particles forming a thin film. The case of small gas flow rates is examined in detail first. In this limit the presence of the liquid film increases the gas pressure drop over its value for a dry bed by three mechanisms: The liquid film makes the apparent size of the particles larger, decreases the pore space for the gas flow, and, with its velocity pointing opposite to the mean gas flow, increases the apparent velocity of the gas compared with the particle surface. The excess pressure drop is determined for both periodic and random arrangements of particles. Next, the case of high gas flow rates where the traction exerted by the gas at the gas–liquid interface is comparable to the weight of the liquid film is examined. In this regime the liquid holdup increases with the gas flow rate and the pressure dropgas velocity relation is nonlinear. The results of numerical simulations are compared with approximate models and it is shown that a simple capillary model yields reasonably accurate predictions for the liquid holdup and gas pressure drop.

Optimal twodimensional models for wake flows
View Description Hide DescriptionIn the case of nominally twodimensional (2D) cylinders of arbitrary cross section in cross flow, the threedimensionality of the wake manifests in the form of quasistreamwise vortices. These threedimensional (3D) features profoundly influence lift and drag forces. However, a twodimensional projection of such a flow, where the effects of threedimensionality are modeled, will be computationally very attractive. One can consider the twodimensional projection as the limiting case of large eddy simulation, where the spanwise direction has been completely averaged out. The transport equation for the spanaveraged spanwise component of vorticity, is considered; the 3D effects to be modeled appear as a subgrid scale flux of torque. It is shown that simple minded eddy viscosity type models that assume the flux vector to be proportional to the spatial gradient of are inadequate. Here we extend the optimal modeling formalism [Moser, Balachandar, and Adrian, Turbulence and Internal Flow/Unsteady Aerodynamics and Hypersonics Conference, Annapolis, MD, pp. 269–274 (1998); Langford and Moser, J. Fluid Mech. 398, 321 (1999)] to address issues pertaining to complex flows with multiple directions of inhomogeneity. We present optimal closures for subgrid flux modeled in terms of distribution, based on linear and quadratic stochastic approximations. These ideas are tested using the database of flow over a flat plate held normal to a cross flow. It is observed that even the optimal model has about 70% normalized error, indicating that the subgrid flux is only about 30% deterministic. Furthermore, it is observed that local models are inadequate, but there exists a region of nonlocality for model dependence, expanding beyond which does not improve the estimate. Higher order nonlinearities however do not seem to improve the model’s predictability.

Computation of threedimensional flows past circular cylinder of low aspect ratio
View Description Hide DescriptionResults are presented for finite element simulation of threedimensional unsteady flows past cylinders of low aspect ratio. The endconditions are specified to model the effect of a wall that may correspond to the flow in a wind tunnel, water channel or a towtank experiment with a cylinder having large endplates. Results are computed for Reynolds number and for a cylinder of aspect ratio The computations confirm that it is the end conditions for the finite cylinder that determine the mode of vortex shedding (parallel or oblique). Preliminary results for flow past a cylinder of aspect ratio are also presented. The and flows exhibit oblique mode of vortex shedding. The flow for is very organized, devoid of any vortex dislocations and is associated with only one cell along the cylinder span. The flow at is interspersed with vortex dislocations and the vortex shedding angle varies, both, temporally and spatially. The presence of vortex dislocations is responsible for the breakdown of spanwise coherence of vortex structures. Mode B pattern of vortex shedding is observed. Flow at results in flow patterns that correspond to the wake transition regime. Mode A and Mode B patterns of vortex shedding in addition to vortex dislocations are observed at different time instants. The present results indicate that the wake transition regime, that is known to occur in the range for large aspectratio cylinders, is either extended and/or delayed for a cylinder of small aspect ratio with “noslip” walls.

New results in the variational approach to turbulent Boussinesq convection
View Description Hide DescriptionThe new “background” variational formulation introduced by Doering and Constantin for bounding the heat transport in Boussinesq convection is related to the classical meanfluctuation approach pioneered by Malkus, Howard and Busse. Viewed from within a unified variational framework, these methods are seen to be complementary or dual principles. The Doering–Constantin approach seeks to estimate a saddle point value of a functional strictly from above as part of a minimization process whereas the Malkus–Howard–Busse approach estimates it from below as part of a maximization procedure. The best bounds on the Nusselt number available in each approach therefore coincide when the same dynamical constraints are imposed. The current best largeRayleighnumber bound, is improved fivefold so that the true bound corresponding to the dynamical constraints of global power balance, global entropy balance and mean heat balance is bracketed as follows: as The optimizing solution is shown to possess no mean flow even if additional momentum constraints are imposed. Finally, the effect of an extra plausible smoothness constraint on the Nusselt number bound is examined. If the minimum lengthscale allowed parallel to the plates in the velocity and temperature field is is the plate separation and is a constant), then as In particular, if the Kolmogorov length scale is the kinematicviscosity, is energy dissipation per unit mass, is the Prandtl number) is used as the minimum lengthscale allowed parallel to the plates in numerical simulations, the Nusselt number can never increase faster than as
