Volume 18, Issue 3, March 2006
 LETTERS


Direct numerical simulation of turbulent channel flows with boundary roughened with virtual sandpaper
View Description Hide DescriptionA method to simulate the effects of a roughened surface on a turbulent boundary layer is introduced. The method is easy to implement, does not increase the numerical overhead of the code, and affects the mean velocity in an a priori predictable way. A single parameter is sufficient to fully characterize the roughness. The procedure has been tested in turbulent channel flows at , with roughness heights spanning the transitional regime. The properties of the rough flow agree well with experimental data.

“Forcefree” electrophoresis?
View Description Hide DescriptionWhen a colloidal particle is exposed to an externally applied electric field, it acquires an electrophoreticvelocity, resulting from fluid slip occurring across the Debye screening layer. When the field is uniformly applied, it is usually assumed that the net neutrality of the combined particlelayer system implies that the net electric force acting on it must vanish. This assumption of “forcefree” phoretic motion has been employed extensively to describe electrophoresis in both unbounded and bounded fluid domains [J. L. Anderson, Annu. Rev. Fluid Mech.21, 61 (1989)]. A careful inspection reveals here that this intuitive premise may fail when the fluid domain is bounded, in which case a nonzero electric force (resembling dielectrophoretic forces in nonuniformly applied fields) may actually exist. Such forces (represented via surface integrals of Maxwell stresses) result in particle motion above and beyond the one driven by the phoretic slip mechanism. A positive demonstration for the existence of a such a force is provided for a standard spherewall configuration, where the applied field acts parallel to the wall. In that scenario, particle motion consists of a (familiar) slipdriven contribution parallel to the wall, together with a superimposed forcedriven drift away from the wall. An analogy with pressure forces occurring at incompressible and inviscid potential flows is presented.

Flowinduced migration of polymers in dilute solution
View Description Hide DescriptionWe investigate the lateral migration of a confined polymer under pressure driven and uniform shear flows. We employ a hybrid algorithm which couples point particles to a fluctuatinglatticeBoltzmann fluid. We observe migration in both uniform shear and pressure driven flows, supporting the idea that migration is driven by a combination of shear and hydrodynamic interactions with the wall, rather than by the shear gradient. Recent numerical and theoretical investigations have suggested that polymers migrate toward the centerline when hydrodynamic interactions are included, but our simulations show that in sufficiently narrow channels there is a reversal of direction and the polymers move toward the wall.

Inertial migration of neutrally buoyant particles in a square duct: An investigation of multiple equilibrium positions
View Description Hide DescriptionInertial migration of neutrally buoyant particles in a square duct has been investigated by numerical simulation in the range of Reynolds numbers from 100 to 1000. Particles migrate to one of a small number of equilibrium positions in the crosssectional plane, located near a corner or at the center of an edge. In dilute suspensions, trains of particles are formed along the axis of the flow, near the planar equilibrium positions of single particles. At high Reynolds numbers, we observe particles in an inner region near the center of the duct. We present numerical evidence that closely spaced pairs of particles can migrate to the center at high Reynolds number.

Planar shock cylindrical focusing by a perfectgas lens
View Description Hide DescriptionWe document a gas lensing technique that generates a converging shock wave in a twodimensional wedge geometry. A successful design must satisfy three criteria at the contact point between the gas lens and the wedge leading edge to minimize nonlinear reflected and other wave effects. The result is a singlepoint solution in a multidimensional parameter space. The gas lens shape is computed using shockpolar analysis for regular refraction of the incident shock at the gas lens interface. For the range of parameters investigated, the required gaslens interface is closely matched by an ellipse or hyperbola. Nonlinear Euler simulations confirm the analysis and that the transmitted shock is circular. As the converging transmitted shock propagates down the wedge, its shape remains nearly uniform with less than 0.1% peak departures from a perfect circular cylinder segment. Departure from the design criteria leads to converging shocks that depart from the required shape. The sensitivity to incident shock Mach number, as well as the qualitative effects of the presence of boundary layers are also discussed.

A singletime twopoint closure based on fluid particle displacements
View Description Hide DescriptionA new singletime twopoint closure is proposed, in which the equation for the twopoint correlation between the displacement of a fluid particle and the velocity allows one to estimate a Lagrangian timescale. This timescale is used to specify the nonlinear damping of triple correlations in the closure. A closed set of equations is obtained without ad hoc constants. Taking advantage of the analogy between particle displacements and scalar fluctuations in isotropic turbulence subjected to a mean scalar gradient, the model is numerically integrated. Results for the energy spectrum are in agreement with classical scaling predictions. An estimate for the Kolmogorov constant is obtained.

Speed of sound from shock fronts in granular flows
View Description Hide DescriptionWe show, in two different experiments on stationary flow past an obstacle, that several features such as Mach cones and shock wave detachment usually observed in supersonic molecular fluids under extreme conditions are also observed for granular fluids. By pursuing this analogy, we measure the speed of sound in these experiments and find it in agreement with predictions from granular kinetic theories. Surprisingly, and in spite of this agreement, measuredvelocity distributions are far from being Gaussian and display algebraic tails.
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 ARTICLES

 Interfacial Flows

Effect of Marangoni and Coriolis forces on multicritical points in a Faraday experiment
View Description Hide DescriptionStability of the free surface of thin sheets of a metallic liquid on a vertically vibrating hot plate, in the presence of a uniform and small rigid body rotation about the vertical axis, is investigated. The inhomogeneity in the surface tension due to a uniform thermal gradient across the liquid sheet prefers subharmonic response, while the rigid body rotation prefers harmonic response at the fluid surface. The competition results in Marangoni and Coriolis forces acting as finetuning parameters in the selection of wave numbers corresponding to different instability tongues for subharmonic and harmonic responses of the fluid surface.Solutions corresponding to various pairs of tongues may be induced in a thin layer of metallic liquid at the onset of parametrically forced surface waves. These give rise to multicritical points involving standing waves of two or more different wave numbers. Bicritical points may involve both the solutions oscillating subharmonically, harmonically, or one oscillating subharmonically and the other harmonically with respect to the vertical forcing frequency. Two tricritical points involving different types of solutions are also possible in a thin layer of mercury. The effect of variation of the Galileo number on critical acceleration and wave number in very low Prandtl number liquids is also presented.

Wetting hysteresis of a dry patch left inside a flowing film
View Description Hide DescriptionWe investigate the influence of wetting hysteresis on the shape and stability of dry patches. Such patches are generated in a film flowing over an inclined plane in a situation of partial wetting. We performed experiments on silicon oil flowing over a glass plate coated with fluoropolymers in which direct visualizations are combined with refraction of a laser sheet to probe contact angle distribution around the patches. The shape evolution depends on the history of the flow, and is different for progressive increase or decrease of flow rate. For increasing flow rates, the shape is in agreement with a simple model based upon a balance between gravity and capillarity, and in which the contact angle is supposed to be uniform. Laser measurements indicate that its value coincides with the static advancing contact angle. For a decreasing flow rate the situation is much more complicated: the shape remains qualitatively that predicted by the model, but quantitatively, the curvature of the patch boundary is not reproducible. Laser measurements suggest that this is linked to wetting heterogeneities (wetting defects) that impose nonuniform contact angle distributions. Finally, the stability is explored in terms of two critical flow rates (expansion or advection of the patches).

Numerical simulations of fingering instabilities in surfactantdriven thin films
View Description Hide DescriptionWe study the surfactantinduced fingering phenomena in thin liquid films both below and beyond the critical micelle concentration using direct numerical simulations. Twodimensional numerical solutions of the coupled nonlinear lubrication equations for the film thickness and surfactant interfacial and bulk concentrations are obtained for different values of the deposited surfactant mass and underlying film thickness . We show that these parameters have a profound effect on the fingering characteristics. At low to intermediate , the deposited mound is relatively mobile and acts to “feed” the fingers that grow downstream efficiently; these fingers are essentially at the same elevation as the mound. At relatively high values, narrower fingers form near the foot of a less mobile mound in a thinned region; this retards the supply rate of fluid from the mound. We also show that increasing leads to less vigorous fingering. Our results are in agreement with trends observed experimentally.

Pearling instability of nanoscale fluid flow confined to a chemical channel
View Description Hide DescriptionWe investigate the flow of a nanoscale incompressible ridge of lowvolatility liquid along a “chemical channel”: a long, straight, and completely wetting stripe embedded in a planar substrate, and sandwiched between two extended less wetting solid regions. Molecular dynamics simulations, a simple longwavelength approximation, and a full stability analysis based on the Stokes equations are used, and give qualitatively consistent results. While thin liquid ridges are stable both statically and during flow, a (linear) pearling instability develops if the thickness of the ridge exceeds half of the width of the channel. In the flowing case, periodic bulges propagate along the channel and subsequently merge due to nonlinear effects. However, the ridge does not break up even when the flow is unstable, and the qualitative behavior is unchanged even when the fluid can spill over onto a partially wetting exterior solid region.

Convective flows in a twolayer system with a temperature gradient along the interface
View Description Hide DescriptionThe nonlinear stability of two superposed horizontal liquid layers bounded by two solid planes and subjected to a horizontal temperature gradient, is investigated. Two types of boundary conditions, periodic boundary conditions and heatinsulated lateral walls, are considered. The nonlinear simulations of the wavy convective regimes for a particular set of fluids, are performed. The dependence of the direction of the wave propagation depends on two factors, which are studied, the ratio of the layers thicknesses and the Marangoni number.

Simplicity and complexity in a dripping faucet
View Description Hide DescriptionContinuous emission of drops of an incompressible Newtonian liquid from a tube–dripping–is a much studied problem because it is important in applications as diverse as inkjet printing, microarraying, and microencapsulation, and recognized as the prototypical nonlinear dynamical system, viz., the leaky faucet. The faucet’s dynamics are studied in this paper by a combination of experiment, using highspeed imaging, and computation, in which the onedimensional slenderjet equations are solved numerically by finite element analysis, over ranges of the governing parameters that have heretofore been unexplored. Previous studies when the Bond number that measures the relative importance of gravitational to surface tension force is moderate, , and the Ohnesorge number that measures the relative importance of viscous to surface tension force is low, , have shown that the dynamics changes from (a) simple dripping, i.e., period1 dripping with or without satellites, to (b) complex dripping, where the system exhibits period doubling bifurcations and hysteresis, to (c) jetting, as the Weber number that measures the relative importance of inertial to surface tension force increases. New experiments and computations reveal that lowering the Bond number to while holding fixed results in profound simplification of the behavior of the faucet. At the lower value of , the faucet exhibits simply period1 dripping, period2 dripping, and jetting as increases. Experimental and computational bifurcation diagrams when and that depict the variation of drop length or volume at breakup with are reported and shown to agree well with each other. The range of over which the faucet exhibits complex dripping when is shown by both experiment and computation to shrink as increases. Computations are also used to develop a comprehensive phase diagram when that shows transitions between simple dripping and complex dripping, and those between dripping and jetting in space. Similar to the case of , dripping faucets of high viscosityliquids are shown to transition directly from simple dripping to jetting without exhibiting complex dripping when . When , computed values of that signal transition from dripping to jetting are further shown to accord well with estimates obtained from scaling analyses. By contrast, new computations in which the Bond number is increased to , while is held fixed at , reveal that the faucet’s response becomes quite complex for large . In such situations, the computations predict theoretical occurrence of (a) rare period3 dripping and period3 intermittence, which have previously been surmised solely by the use of ad hoc springmass models of dripping, and (b) chaotic attractors. Therefore, by combining insights from earlier studies and the detailed response of dripping which has been obtained here by varying (i) as , a range that is typical of most practical applications, (ii) from virtually zero to a value just exceeding that at which the system transitions from dripping to jetting, and (iii) from a small value to a value approaching that beyond which controlled formation of drops is prohibited, this paper provides a comprehensive understanding of the effect of the governing parameters on the nonlinear dynamics of dripping.

Thermocapillary structure formation in a falling film: Experiment and calculations
View Description Hide DescriptionThe present experimental and numerical study is focused on regular structure formation in a film falling down a vertical plate with a builtin rectangular heater. The data on critical Marangoni number and on the wavelength of the most unstable spanwise mode depending on Reynolds and Weber numbers are obtained. The influence of heating intensity on structure geometry is also investigated. The critical Marangoni number is shown to increase nonlinearly with the increase of Reynolds number. The wavelength of the most unstable spanwise mode appears to depend weakly on Reynolds number and much stronger on Weber number. We obtained that the height of the critical twodimensional bump is not invariant, but depends almost inversely on Reynolds number. The appearance of a reverse flow in the bump is not a criterium for instability onset. For the different fixed Reynolds number values, the structure width significantly grows with the increase of Marangoni number. Numerical results and experimental data are compared and show good enough agreement.

Numerical study of jet flow generated by impact on weakly compressible liquid
View Description Hide DescriptionThe early stage of twodimensional jet flow generated by an impulsive start of a wedge, initially floating on a free surface otherwise flat, is investigated taking into account liquid compressibility. During this short initial stage the flow close to the intersection points is selfsimilar. Velocity of the body is assumed much smaller than the sound speed in the liquid at rest. The acoustic solution of this problem reveals that the flowvelocity is singular about the intersection points. In order to obtain a uniformly valid description of the flow, an inner solution is derived by using stretched variables, which are dependent on the Mach number of the problem. It is shown that, for small Mach numbers, the inner flow is approximately potential and is governed by the Laplace equation. The solution of the boundaryvalue problem is achieved numerically through an iterative procedure. A modified velocity potential, which significantly simplifies the boundary conditions on the free surface, is introduced. Accurate solutions are presented in terms of free surface shapes and jet lengths for different deadrise angles of the wedge. The analogy between this problem and that of jetting flow caused by shock wave impact on a wedgeshaped cavity is discussed as well.
 Viscous and NonNewtonian Flows

Twoscale modeling in porous media: Relative permeability predictions
View Description Hide DescriptionWe present a numerical analysis of fluid flow through a porous medium with two distinct characteristic scales. The system considered is a monodisperse matrix with porosity and permeability with an embedded second phase, characterized by a phase content or saturation and phase length scales and . Both two and threedimensional simulations are performed to compute the mobile fluid phase relative permeability and its dependence on and . The relative permeability is found to vary as a power law of saturation, with a quasilinear behavior for low permeability, and increasing values of the exponent as increases. For media with low permeability, the linearity of is attributed to the drag force, whereas for high , the decrease of with is due primarily to viscous forces. An analytical model for is also presented to aid the interpretation and to corroborate the simulation results. In the second part, in order to elucidate the role of the length scales on , simulations explicitly resolving both porous media and secondphase scales are performed. The relative permeability is found to drop rapidly when both scales are of the same order or when . Three regimes (Darcy, Brinkman, Stokes) are consequently identified based on the length scales.

Quantifying transport within a porous medium over a hierarchy of length scales
View Description Hide DescriptionMagnetic resonance techniques are used to probe transport within a porous medium over length scales of microns to centimeters. In particular, the apparent discrepancy between estimates of dispersion within porous media determined by pulsed field gradient magnetic resonance techniques and a conventional elution analysis is addressed. The model porous medium considered is a packed bed of height and internal diameter 22.5 and , respectively, packed with highly porous crosslinked dextran particles approximately in diameter. Experiments were performed for Peclet numbers in the range . First, a nonspatially resolved displacement encoding Alternating Pulsed Field Gradient Stimulated Echo Nuclear Magnetic Resonance (APGSTE NMR) measurement was used to yield estimates of bed porosity , mobile phase volume fraction , intraparticle diffusion coefficient , and characteristic time, Te, for exchange between the intra and interparticle pore space . The value of porosity was in excellent agreement with that obtained by elution analysis. However, values of the axial dispersion coefficient obtained using the two approaches did not agree well. For example, at , the dispersion coefficients measured by APGTSE NMR and elution analysis were and , respectively. These results suggest that whilst the micro/mesolength scale properties of the porous medium are well characterized using the APGSTE NMR measurement, the technique is unable to probe the millimeter length scales in the bed over which heterogeneities in the flow may exist and therefore contribute significantly to the macroscopic dispersion characteristic of the bed, as determined by elution analysis. This is confirmed by demonstrating that the contribution of mechanical mixing to dispersion within the porous medium extends to the longest time scales studied . To identify the dominant influences on the macroscopic dispersion characteristics of the porous medium,magnetic resonance flow velocity images within the packed bed were acquired. Numerical reconstructions of the residence time distribution of the fluid within the bed using these data yielded a value of the dispersion coefficient of , in far better agreement with the elution analysis, thereby demonstrating that it is the millimeterscale heterogeneity in the flow field within the bed that is the dominant contribution to the macroscopic dispersion. Extension of the model to incorporate the effect of maldistribution of the input pulse further improves agreement with the elution analysis.
 Particulate, Multiphase, and Granular Flows

A multiscale model for dilute turbulent gasparticle flows based on the equilibration of energy concept
View Description Hide DescriptionThe objective of this study is to improve EulerianEulerian models of particleladen turbulent flow. We begin by understanding the behavior of two existing models—one proposed by Simonin [von Kármán Institute of Fluid Dynamics Lecture Series, 1996], and the other by Ahmadi [Int. J. Multiphase Flow16, 323 (1990)]—in the limiting case of statistically homogeneous particleladen turbulent flow. The decay of particlephase and fluidphase turbulent kinetic energy (TKE) is compared with direct numerical simulation results. Even this simple flow poses a significant challenge to current models, which have difficulty reproducing important physical phenomena such as the variation of turbulent kinetic energy decay with increasing particle Stokes number. The model for the interphase TKE transfer time scale is identified as one source of this difficulty. A new model for the interphase transfer time scale is proposed that accounts for the interaction of particles with a range of fluid turbulence scales. A new multiphase turbulence model—the equilibration of energy model (EEM)—is proposed, which incorporates this multiscale interphase transfer time scale. The model for Reynolds stress in both fluid and particle phases is derived in this work. The new EEM model is validated in decaying homogeneous particleladen turbulence, and in particleladen homogeneous shear flow. The particle and fluid TKE evolution predicted by the EEM model correctly reproduce the trends with important nondimensional parameters, such as particle Stokes number.
 Laminar Flows

Effect of timedependent piston velocity program on vortex ring formation in a piston/cylinder arrangement
View Description Hide DescriptionAn analytical model describing laminar vortex ring formation in a nozzle flow generator (piston/cylinder arrangement) proposed previously by the authors is extended to timedependent velocity programs. The predictions of the model are in good agreement with the available numerical data for impulsive, linear, and trapezoidal velocity programs. We also show that properly scaled vortex circulation is another universal quantity, in addition to the dimensionless energy, related to vortex rings and verify this by comparing with available numerical simulations and experimental results.
 Instability and Transition

Linear stability analysis and dynamic simulations of free convection in a differentially heated cavity in the presence of a horizontal magnetic field and a uniform heat source
View Description Hide DescriptionThe steady state and stability of twodimensional free convectionflow in a square cavity is examined, in the presence of a uniform internal heat source and a uniform magnetic field that is perpendicular to gravity and parallel to an imposed temperature gradient. The finite element method is used for calculating the steady and dynamic state of the system in the parameter space defined by the dimensionless numbers, Gr, Ha, Pr, and S. The trapezoidal rule is used for time integration. Linear stability analysis is performed by solving a generalized eigenvalue problem. The Arnoldi method is used for the calculation of eigenvalues with significant savings in storage and CPU time requirements. The base solution normally exhibits two recirculation regions when the heat production term is large enough. Stability analysis predicts a Hopf bifurcation to a periodic branch. A neutral stability diagram is constructed for a range of values of Ha, Gr, and S for liquid lithium,. Internal heat generation, i.e,. increasing S, enhances instability by decreasing the critical value of Grashof, , determining the onset of the Hopf branch, whereas intensifying the magnetic field, i.e., increasing Ha, stabilizes the flow by increasing . Dynamic simulations confirm the above structure, identify the oscillatory solution branch as a supercritical Hopf bifurcation for the entire parameter range that was examined, and recover the time constants predicted by stability analysis. As Gr increases or as Ha decreases symmetric arrangement of the two rolls is eliminated and the steady flow configuration loses stability when . Subsequently, time periodicity sets in leading to more or less efficient heat removal in terms of lowering or increasing the average cavity temperature, in comparison with the steady flow configuration for the same Gr, depending on whether Ha lies below or above a critical value, respectively, for fixed S and Pr.