Volume 21, Issue 6, June 2009
Index of content:

Twodimensional turbulence admits two different ranges of scales: a direct enstrophy cascade from the injection scale to the small scales and an inverse energy cascade at large scales. It has already been shown in previous papers that vortical structures are responsible for the transfers of energy upscale while filamentary structures are responsible for the forward transfer of the enstrophy. Here we propose an original mathematical tool, the interaction function, for studying the space localization of the enstrophy fluxes. It is defined using an orthogonal twodimensional wavelet decomposition.
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Twowave structure of liquid film and wave interrelation in annular gasliquid flow with and without entrainment
View Description Hide DescriptionThe wavy structure of liquid film in downward annular gasliquidflow with and without liquid entrainment is examined for a wide range of gas velocities using highspeed laserinduced fluorescence technique. It is shown that the wavy structure is always represented by two types of waves, longlived and shortlived waves with the latter always generated at the back slopes of the longlived waves. Main regularities of shortlived wave generation and evolution are also described.

 ARTICLES


Micro and Nanofluid Mechanics

Thermophoresis of an aerosol spheroid along its axis of revolution
View Description Hide DescriptionThe thermophoretic motion of a spheroidal particle freely suspended in a gaseous medium prescribed with a uniform temperature gradient along the axis of revolution of the particle is studied theoretically in the steady limit of small Peclet and Reynolds numbers. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the particle surface. The general solutions in prolate and oblate spheroidal coordinates can be expressed in infiniteseries forms of separation of variables for the temperature distribution and of semiseparation of variables for the stream function. The jump/slip boundary conditions on the particle surface are applied to these general solutions to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundarycollocation method or explicit formulas derived analytically. Numerical results for the thermophoreticvelocity of the spheroidal particle are obtained in a broad range of its aspect ratio with good convergence behavior for various cases. For the axisymmetric thermophoresis of an aerosol spheroid, prolate, or oblate, with no temperature jump and frictional slip at its surface, our results agree excellently with the analytical solution obtained previously. The agreement between our results and the available numerical solutions obtained by using a singularity method is also very good. For most practical cases of a spheroid with a specified aspect ratio, the thermophoretic mobility of the particle is not a monotonic function of its relative jump/slip coefficients and thermal conductivity. The axisymmetric thermophoretic mobilities of a prolate spheroid and of an oblate spheroid with large aspect ratios can be much greater and smaller, respectively, than that of a sphere with the same equatorial radius.

Interfacial Flows

On the effect of the atmosphere on the evaporation of sessile droplets of water
View Description Hide DescriptionAn experimental and theoretical study of the effect of the atmosphere on the evaporation of pinned sessile droplets of water is described. The experimental work investigated the evaporation rates of sessile droplets in atmospheres of three different ambient gases (namely, helium, nitrogen, and carbon dioxide) at reduced pressure (from 40 to 1000 mbars) using four different substrates (namely, aluminum, titanium, Macor, and polytetrafluoroethylene) with a wide range of thermal conductivities. Reducing the atmospheric pressure increases the diffusion coefficient of water vapor in the atmosphere and hence increases the evaporation rate. Changing the ambient gas also alters the diffusion coefficient and hence also affects the evaporation rate. A mathematical model that takes into account the effect of the atmospheric pressure and the nature of the ambient gas on the diffusion of water vapor in the atmosphere and the thermal conductivity of the substrate is developed, and its predictions are found to be in encouraging agreement with the experimental results.

Thermocapillary instability of a liquid layer under heat flux modulation
View Description Hide DescriptionThe parametric excitation of the Marangoni instability in a horizontal liquid layer is analyzed in the case of a heat flux periodically varying at the deformable interface. Two response modes of the convective system to an external periodic stimulation, synchronous and subharmonic ones, have been found. The cellular and longwave instability thresholds are compared. The neutral stability curves are presented for a variety of external conditions. It is shown that contrary to the classical parametric resonance, the synchronous disturbances may become most dangerous for the stability of the base state, and the longwave mode may cause the instability prior to the cellular mode within a definite range of parameters.

Computational investigation on bubble detachment from submerged orifice in quiescent liquid under normal and reduced gravity
View Description Hide DescriptionIn this work numerical simulations have been carried out to study the problem of dynamic air bubble formation from a submerged orifice in quiescent liquid, under constant inflow condition, at normal and reduced gravity levels. A coupled levelset and volumeoffluid method is used to simulate the bubble formation, bubble detachment, and the bubble rise above the orifice. For the described study, the authors have mainly focused on low and medium air flow rate for simulation of bubble formation at the orifice. The employed gravity levels are in the range of , , and . The influence of buoyancy on the bubble shape has been studied. The study includes the bubble volume, formation frequency, pinchoff rate, detached bubble diameter, and the bubble growth history for different air flow rates. Even for the static contact angle , it is observed that at low gravity levels the bubble base spreads along the surface of the orifice plate away from the orifice rim during the expansion stage, and during the detachment stage the bubble base again comes back to the orifice rim. As the air flow rate is increased under normal and low gravity conditions, coalescence between the rising bubbles or between the detached bubble and the forming bubble at the orifice is observed. It is shown that the increasing trend of bubble size at detachment, with increasing air flow rate under normal gravity is reversed in the case of reduced gravity .

Decomposition driven interface evolution for layers of binary mixtures. II. Influence of convective transport on linear stability
View Description Hide DescriptionWe study the linear stability with respect to lateral perturbations of free surfacefilms of polymer mixtures on solid substrates. The study focuses on the stability properties of the stratified and homogeneous steady film states studied in Part I [U. Thiele, S. Madruga, and L. Frastia, Phys. Fluids19, 122106 (2007)]. To this aim, the linearized bulk equations and boundary equations are solved using continuation techniques for several different cases of energetic bias at the surfaces corresponding to linear and quadratic solutal Marangoni effects. For purely diffusive transport, an increase in the film thickness either exponentially decreases the lateral instability or entirely stabilizes the film. Including convective transport leads to a further destabilization as compared to the purely diffusive case. In some cases the inclusion of convective transport and the related widening of the range of available film configurations (it is then able to change its surface profile) change the stability behavior qualitatively. We furthermore present results regarding the dependence of the instability on several other parameters, namely, the Reynolds number, the surface tension number, and the ratio of the typical velocities of convective and diffusive transport.

Instability in gravitydriven flow over uneven surfaces
View Description Hide DescriptionWe consider the gravitydriven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography and surface tension on the stability of the flow. The equations of motion are approximations to the Navier–Stokes equations which exploit the assumed relative shallowness of the fluid layer. Included in these equations are diffusive terms that are second order relative to the shallowness parameter. These terms, while small in magnitude, represent an important dependence of the flow dynamics on the variation in bottom topography and play a significant role in theoretically capturing important aspects of the flow. Some of the secondorder terms include normal shear contributions, while others lead to a nonhydrostatic pressure distribution. The explicit dependence on the crossstream coordinate is eliminated from the equations of motion by means of a weighted residual approach. The resulting mathematical formulation constitutes an extension of the modified integralboundarylayer equations proposed by RuyerQuil and Manneville [Eur. Phys. J. B15, 357 (2000)] for flows over even surfaces to flows over variable topography. A linear stability analysis of the steady flow is carried out by taking advantage of Floquet–Bloch theory. A numerical scheme is devised for solving the nonlinear governing equations and is used to calculate the evolution of the perturbed equilibrium flow. The simulations are used to confirm the analytical predictions and to investigate the interfacial wave structure. The bottom profile considered in this investigation corresponds to periodic undulations characterized by measures of wavelength and amplitude. Conclusions are drawn on the combined effect of bottom topography and surface tension.

Calculation of resonant shortcrested waves in deep water
View Description Hide DescriptionSolving the problem of resonant shortcrested waves is very challenging because the appearance of small divisors causes the classical perturbation methods to fail. From a numerical point of view, the case of resonant gravity shortcrested waves has been studied, but as far as we know, there are very few results in the case of resonant capillarygravity shortcrested waves. In fact, to the best of our knowledge, the most related study which has been made is the one of Craig and Nicholls [SIAM J. Math. Anal.32, 323 (2000)] who gave existence theorems for the case where the surface tension is supposed not to be too small. There is a need for such an investigation, and the work considered herein therefore provides a calculation technique and presents new results on resonant shortcrested gravitycapillary waves. We overcome the technical problems associated with small divisors by using a method derived from Whitham’s variational formulation of the classical problem of shortcrested waves. Whitham’s method is not modified in essence, but computations are carried out and organized to obtain a method that has been applied to series of cases demonstrating the robustness and flexibility of the approach. In particular, numerical solutions corresponding to threedimensional Wilton ripples have been obtained. Moreover, these waves are also obtained for long wave configurations. This method is able to handle the case of small or zero surface tension, including the resonant cases, and works well very near the limiting twodimensional cases.

Energy integral method model for the nonlinear dynamics of an axisymmetric thin liquid film falling on a vertical cylinder
View Description Hide DescriptionThe nonlinear dynamics of an axisymmetric liquid film falling on the outer surface of a vertical cylinder is investigated in this paper. Using the energy integral method we derive a set of two coupled evolution equations which constitute a firstorder approximation to the original hydrodynamic equations. We carry out the linear stability analysis of the axially uniform axisymmetric flow of a liquid film in the framework of the evolution equations. The results of the linear theory are compared to experimental results and with several other relevant linear stability theories available in literature. Traveling wave solutions of the evolution equations are investigated and favorably compared to experiments and other nonlinear theories describing the dynamics of falling films on a vertical cylinder. A bifurcation structure of traveling waveflows is studied for the cases of fluids with both low and relatively large Kapitza number. Both  and waves are found and investigated.

Investigation of coupled airwater turbulent boundary layers using direct numerical simulations
View Description Hide DescriptionWe perform direct numerical simulation of air and water turbulence in a Couette flow. The airwater interface is kept flat, and the coupling between the two fluids is through continuity of velocity and shear stress at the interface. True airtowater ratios of density and viscosity are used in the simulation to illustrate features of airwater coupled boundary layers. Our analysis of statistics of the velocity, vorticity, and turbulent kinetic energy budget confirms known features, notably the similarity of the airside interface boundary layer to wall boundary layer. Our study obtains new insights on the characteristics of the waterside motions. Compared to the airside, waterside turbulence structures are more persistent and larger in scale, which dominate the interface signatures. The interface boundary layer on the waterside possesses unique features that are intermediate between but qualitatively different from wall boundary layer and freeslip surface layer. On the waterside, as the interface is approached, enstrophy and viscous dissipation first decrease together, with higher turbulence mixing and production than the airside, and then increase sharply in response to airside stress fluctuations. The waterside turbulence is characterized by interfaceconnected, hairpin, and quasistreamwise vortices that are closely related to each other in their evolutions. Conditional average using a variableinterval space averaging approach illustrates the strong airwater coupling in splat, hairpin vortex, and quasistreamwise vortex events. It is found that waterside velocity gradients are amplified significantly across the interface, with the presence of horizontal jets at splat and quasistreamwise vortex regions. We also study the transport of passive scalars near the airwater coupled boundary layers. As expected, vertical advection associated with coupling airwater coherent structures greatly enhances interfacial transfer of scalars. By considering scalars of different diffusivity and solubility values, we investigate the partition of interfacial transfer between the waterside and airside. We illustrate statistical characteristics of gaslike and heatlike scalars and perform detailed analysis of the scalar variance budget for these to reveal unique features that are substantially different from wall boundary layer or freeslip surface layer.

Viscous and NonNewtonian Flows

A mathematical model of the Leidenfrost effect on an axisymmetric droplet
View Description Hide DescriptionA simple mathematical model is developed for the Leidenfrost effect on an axisymmetric droplet. We first examine the practice of describing large droplets as cylindrical and small droplets as almost spherical, by solving the Young–Laplace equation. Best fit formulas are presented relating the dropletsurface area to volume. The full model for Leidenfrost is then discussed. This shows that the calculation may be reduced to the solution of three first order ordinary differential equations involving the volume, evaporation rate, and vapor film thickness. Comparison with experimental results shows good agreement provided the model incorporates two components neglected in previous theoretical studies. First, a finite heat transfer coefficient between the vapor and substrate is required. Second evaporation from the upper surface of the droplet must be included, even for large droplets.

Oscillatory shear induced droplet deformation and breakup in immiscible polymer blends
View Description Hide DescriptionDeformation and breakup of droplets in polybutadiene/polydimethylsiloxane blends subject to oscillatoryshear flow were investigated experimentally using an optical shear flow cell. The apparent major axis and the minor axis in the vorticity direction of the droplets were measured as functions of time. From the time series of and and the deformation parameter, , we define the deformation amplitudes as onehalf the differences between the maximum and minimum values. The deformation amplitude parameters generally decrease with increasing viscosity ratio, time scale ratio, and dropletelasticity. The dependences of the deformation amplitude parameters on capillary number are generally linear up to a certain value for Newtonian droplets regardless of viscosity ratio and time scale ratio. The dependences become totally nonlinear with increasing dropletelasticity.Dropletviscosity and elasticity generally impede breakup under oscillatory shear. Critical capillary number for breakup, the number of resultant daughter droplets, and the number of cycle required for breakup to occur increase with time scale ratio. The apparent breakup pattern changes from the dumbbell type to the endpinching type as time scale ratio increases.

Particulate, Multiphase, and Granular Flows

The inertial lift on a spherical particle settling in a horizontal viscous flow through a vertical slot
View Description Hide DescriptionThe inertial lift force exerted on a small rigid sphere settling due to gravity in a horizontal channel flow between vertical walls is investigated. The method of matched asymptotic expansions is used to obtain solutions for the disturbance flow on the length scales of the particle radius and the channel width (inner and outer regions, respectively). The channel Reynolds number is finite, while the particle Reynolds numbers that are based on the slip velocity and the mean shear rate are small. The inner flow is described by the linear Stokes equations. The outer problem is governed by the linear Oseenlike equations with the particle effect approximated by a point force. The outer equations are solved numerically using the twodimensional Fourier transform of the disturbance velocity field. The lift coefficient is evaluated as a function of governing dimensionless parameters: the particle coordinate across the channel, the channel Reynolds number, and the slip parameter. The particle always migrates away from the walls, with an equilibrium position being on the channel centerline. Close to the walls, the lift coefficient is the same regardless of the slip velocity and the channel Reynolds number. At large channel Reynolds numbers, a local maximum of the migration velocity forms near the channel centerline due to a combined effect of the slip, the linear shear, and the curvature of the undisturbed velocity profile. The results obtained are extended to the case when the drag on a particle has components both parallel and perpendicular to the undisturbed flow. One of primary applications of the results is modeling of the crossflow migration of settling particles during particle transport in a hydraulic fracture.

Pairwise interactions between deformable drops in free shear at finite inertia
View Description Hide DescriptionInteractions between a pair of equalsize viscous drops in shear are numerically investigated at finite Reynolds number. At low Reynolds number the simulation compares well with a previous experimental observation. Apart from the usual pairwise motion where drops driven by shear pass over each other (type I trajectory), finite inertia introduces a new type (type II) of reversed trajectory where drops approaching each other reverse their initial trajectories. The new trajectory is explained by a reversed streamline pattern observed around a single drop in an imposed shear, and is similar to what is also observed for rigid spheres at finite inertia. However, drop deformability introduces a nonuniform transition from one to the other type of trajectory—drops display type I trajectory for high and low capillary numbers and type II for intermediate capillary numbers. The phenomenon is explained by noting that increasing capillary number gives rise to competing effects—while it increases drop deformation and therefore increases resistance to sliding motion, it also increases drop flexibility, decreases inclination angle, and overall effect of the drop’s presence is reduced, all helping them to slide by. The nonuniform behavior—type II trajectory for an intermediate range of capillary numbers—occurs only at Reynolds number above a critical value. Further increase in Reynolds number increases the range of capillary numbers for type II trajectory. For type I trajectory, terminal crossstream separation increases linearly with increasing inertia indicating an enhanced shear induced diffusion. Increasing initial streamwise separation aids in reversed (type II) trajectory due to increased overlap with the reversed streamline zone. Increasing crossstream distance expectedly facilitates (type I) sliding motion. For passing drops (type I trajectory), terminal crossstream separation is not appreciably affected by capillary number and initial drop separation.

The impact of bubble diffusivity on confined oscillated bubbly liquid
View Description Hide DescriptionWe consider the dynamics of monodisperse bubbly liquid confined by two plane solid walls and subject to smallamplitude highfrequency transverse oscillations. The period of these oscillations is assumed small in comparison with typical relaxation times for a single bubble but comparable with the period of volume eigenoscillations. The timeaveraged description accounting for the twoway coupling between the liquid and the bubbles and for the diffusivity of bubbles is applied. We find nonuniform steady states with the liquid quiescent on average. At relatively low frequencies, accumulation of bubbles either at the walls or in planes parallel to the walls is detected. These onedimensional states are shown to be unstable. At relatively high frequencies, this accumulation is found at the central plane and the solution is stable.

Instability and Transition

The threedimensional wake of a cylinder undergoing a combination of translational and rotational oscillation in a quiescent fluid
View Description Hide DescriptionPrevious twodimensional numerical studies have shown that a circular cylinder undergoing both oscillatory rotational and translational motions can generate thrust so that it will actually selfpropel through a stationary fluid. Although a cylinder undergoing a single oscillation has been thoroughly studied, the combination of the two oscillations has not received much attention until now. The current research reported here extends the numerical study of Blackburn et al. [Phys. Fluids11, L4 (1999)] both experimentally and numerically, recording detailed vorticity fields in the wake and using these to elucidate the underlying physics, examining the threedimensional wake development experimentally, and determining the threedimensional stability of the wake through Floquet stability analysis. Experiments conducted in the laboratory are presented for a given parameter range, confirming the early results from Blackburn et al. [Phys. Fluids11, L4 (1999)]. In particular, we confirm the thrust generation ability of a circular cylinder undergoing combined oscillatory motions. Importantly, we also find that the wake undergoes threedimensional transition at low Reynolds numbers to an instability mode with a wavelength of about two cylinder diameters. The stability analysis indicates that the base flow is also unstable to another mode at slightly higher Reynolds numbers, broadly analogous to the threedimensional wake transition mode for a circular cylinder, despite the distinct differences in wake/mode topology. The stability of these flows was confirmed by experimental measurements.

A Mach number study of the Richtmyer–Meshkov instability in a varicose, heavygas curtain
View Description Hide DescriptionA varicoseperturbed, thin, heavygas curtain is impulsively accelerated by a planar shock wave of varying strength and investigated experimentally using concentration field visualization. Experiments were performed with Mach 1.2 and 1.5 incident shock waves, acquiring images of the initial conditions, and 18 different times after shock interaction in each case. Repeatability of the initial conditions allows for visualization of flow feature development over time for both Mach numbers despite capturing only one dynamic, postshock image per run of the experiment. Good agreement between integral width experimental data and a mixing width model is demonstrated for early to intermediate times in the flow. Integral width growth rates for Mach 1.2 and 1.5 are shown to collapse using a scaling based upon the convection velocity of the curtain. The diffusion driven instantaneous mixing rate, , is also estimated and compared between experiments. Results from this gradient based metric show differences in mixing trends between Mach numbers that do not scale in the same way as integral width, suggesting that integral width alone is insufficient for completely describing the flow. An experiment with a Mach 2.0 incident shock was carried out for the first time in the experimental facility. The resulting image provides further evidence for the mixing trends observed in this paper as Mach number is increased.

Absolute stability mechanism of a swept cylinder laminar boundary layer with imposed spanwise periodic conditions
View Description Hide DescriptionThe linear impulse response of a laminar threedimensional boundary layer with imposed spanwise periodic conditions is theoretically analyzed. Because of the imposed spanwise periodicity, a unidirectional absolute instability is physically significant. The base flow is obtained through a direct numerical simulation of the flow along a swept cylinder, with periodic boundary conditions enforced at the spanwise boundaries. It is shown that this flow is absolutely stable. However, it exhibits crossflow modes of zero group velocity whose damping rate in time is quite slow. Moreover, these crossflow modes are spatially amplified. A direct numerical simulation of the same configuration but in the presence of surface roughness has enabled the observation of these traveling crossflow waves. The introduction of the roughness element in the computation creates initial transients that play the role of an impulselike forcing, which generates the traveling crossflow waves as predicted by the theoretical analysis. Depending on the roughness chordwise location, the frequency and damping rate in time of the crossflow waves change in agreement with the theoretical results.

Stability of plane Couette–Poiseuille flow of shearthinning fluid
View Description Hide DescriptionA linear stability analysis of the combined plane Couette and Poiseuille flow of shearthinning fluid is investigated. The rheological behavior of the fluid is described using the Carreau model. The linearized stability equations and their boundary conditions result in an eigenvalue problem that is solved numerically using a Chebyshev collocation method. A parametric study is performed in order to assess the roles of viscosity stratification and the Couette component. First of all, it is shown that for shearthinning fluid, the critical Reynolds number for a twodimensional perturbation is less than for a three dimensional. Therefore, it is sufficient to deal only with a modified Orr–Sommerfeld equation for the normal velocity component. The influence of the velocity of the moving wall on the critical conditions is qualitatively similar to that for a Newtonian fluid. Concerning the effect of the shear thinning, the computational results indicate that this behavior leads to a decrease in the phase velocity of the traveling waves and an increase in stability, when an appropriate viscosity is used in the definition of the Reynolds number. Using a longwave version of the Orr–Sommerfeld equation, the cutoff velocity is derived. The mechanisms responsible for the changes in the flow stability are discussed in terms of the location of the critical layers, Reynolds stress distribution, and the exchange of energy between the base flow and the disturbance.

Temperature modulation in ferrofluid convection
View Description Hide DescriptionConvective instability is suppressed by sinusoidal variation in temperature of horizontal boundaries of a ferrofluid layer subjected to vertical magnetic field. On increasing the amplitude of modulation, instability may arise in the form of oscillations which may have time period equal to that of the temperature modulation or double of it. The effect of frequency of modulation, applied magnetic field, and Prandtl number on the onset of a periodic flow in the ferrofluid layer has been investigated numerically using the Floquet theory. Some theoretical results have also been obtained to discuss the limiting behavior of the underlying instability with the temperature modulation. Depending upon the parameters, the flow patterns at the onset of instability have been found to consist of timeperiodically oscillating vertical magnetoconvective rolls.
