Volume 21, Issue 11, November 2009

Twodimensional statistically stationary isotropic turbulence with an imposed uniform scalar gradient is investigated. Dimensional arguments are presented to predict the inertial range scaling of the turbulent scalar flux spectrum in both the inverse cascade range and the enstrophy cascade range for small and unity Schmidt numbers. The scaling predictions are checked by direct numerical simulations and good agreement is observed.
 LETTERS


Localized edge states in plane Couette flow
View Description Hide DescriptionThe dynamics at the threshold of transition in plane Couette flow is investigated numerically in a large spatial domain for a certain type of localized initial perturbation, for Re between 350 and 1000. The corresponding edge state is an unsteady spotlike structure, localized in both streamwise and spanwise directions, which neither grows nor decays in size. We show that the localized nature of the edge state is numerically robust, and is not influenced by the size of the computational domain. The edge trajectory appears to transiently visit localized steady states. This suggests that basic spatiotemporally intermittent features of transition to turbulence, such as the growth of a turbulent spot, can be described as a dynamical system.

Spectrum of a passive scalar in moderate Reynolds number homogeneous isotropic turbulence
View Description Hide DescriptionFor moderate Reynolds numbers, the spectral scaling exponents of both velocity and the transported passive scalar fields exhibit departures from the asymptotic prediction . However, at the same Reynolds number, the passive scalar spectrum for homogeneous isotropic turbulence is closer to the universal asymptotic state than the dynamic velocity field that transports it. This paper provides a possible explanation for this behavior, in the case of a gaseous mixing with Prandtl (or Schmidt) number . A scenario of the scalar energy transfer toward higher wavenumbers is proposed and validated using experimental data, in which the velocity field itself is actively involved via its characteristic time. A direct relationship between velocity and scalar spectra and therefore between and is thus established.

Comparison of largescale amplitude modulation in turbulent boundary layers, pipes, and channel flows
View Description Hide DescriptionRecent investigations by Monty et al. [J. Fluid Mech.632, 431 (2009)] showed that important modal differences exist between channels/pipes and boundary layers, mainly in the largest energetic scales. In addition, Mathis et al. [J. Fluid Mech.628, 311 (2009)] recently reported and quantified a nonlinear scale interaction in zeropressure gradient turbulent boundary layers, whereby the largescale motion amplitude modulates the smallscale motions. In this study, a comparison of this modulation effect of the streamwise velocity component is undertaken for all three flows for matched Reynolds number and measurement conditions. Despite the different largescale phenomena in these internal and external wallbounded flows, the results show that their amplitude modulation influence remains invariant in the inner region with some differences appearing in the outer region.

Experimental vortex breakdown topology in a cylinder with a free surface
View Description Hide DescriptionThe free surface flow in a circular cylinder driven by a rotating bottom disk is studied experimentally using particle imagevelocimetry. Results are compared with computational results assuming a stressfree surface. A dye visualization study by Spohn et al. [“Observations of vortex breakdown in an open cylindrical container with a rotating bottom,” Exp. Fluids14, 70 (1993)], as well as several numerical computations, has found a range of different vortex breakdown structures in this flow. We confirm the existence of a transition where the top of the breakdown bubble crosses from the axis to the surface, which has previously only been found numerically. We employ a technique by Brøns et al. [“Topology of vortex breakdown bubbles in a cylinder with rotating bottom and free surface,” J. Fluid Mech.428, 133 (2001)] to find the corresponding bifurcation curve in the parameter plane, which has hitherto only been used on numerical data. The bifurcation curve located here agrees well with previous numerical simulations. For low values of the Reynolds number we find a regime with vortex breakdown that has not been previously identified. Experiments deviate substantially from computations, indicating the importance of surface effects in this regime.

 ARTICLES

 Micro and Nanofluid Mechanics

Note on the relation between thermophoresis and slow uniform flow problems for a rarefied gas
View Description Hide DescriptionA relation between the problem of thermophoresis of a sphere and that of a uniform flow past a sphere is discussed on the basis of the linearized Boltzmann equation. First pointed out is the disagreement between the relation predicted by the representation theorem recently developed by the author and that of the existing theory by Sharipov. The two contradicting predictions are assessed by the asymptotic theory for small Knudsen numbers, which results in showing the failure of the latter. The reason of this failure is also explained. Finally, new data of a slip coefficient, which is predominantly responsible for the thermal polarization in a slightly rarefied gas, are obtained for the hardsphere Boltzmann equation by the use of the correct relation.

Quantifying effective slip length over micropatterned hydrophobic surfaces
View Description Hide DescriptionWe employ microparticle image velocimetry to investigate laminar microflows in hydrophobic microstructured channels, in particular the slip length. These microchannels consist of longitudinal microgrooves, which can trap air and prompt a shearfree boundary condition and thus slippage enhancement. Our measurements reveal an increase in the slip length when the width of the microgrooves is enlarged. The result of the slip length is smaller than the analytical prediction by Philip [Z. Angew. Math. Phys.23, 353 (1972)] for an infinitely large and textured channel comprised of alternating shearfree and noslip boundary conditions. The smaller slip length (as compared with the prediction) can be attributed to the confinement of the microchannel and the bending of the meniscus(liquidgas interface). Our experimental studies suggest that the curvature of the meniscus plays an important role in microflows over hydrophobic microridges.
 Interfacial Flows

Spray and microjets produced by focusing a laser pulse into a hemispherical drop
View Description Hide DescriptionWe use highspeed video imaging to study laser disruption of the free surface of a hemispheric drop. The drop sits on a glass surface and the Nd:YAG (yttrium aluminum garnet) laser pulse propagates through the drop and is focused near the free surface from below. We focus on the evolution of the cylindrical liquid sheet and spray which emerges out of the drop and resembles typical impact crowns. The tip of the sheet emerges at velocities over 1 km/s. The tip of the crown breaks up into fine spray some of which is sucked back into the growing cavity at about 100 m/s. We measure the size of the typical spray droplets to be about . We also show the formation of fine microjets, which are produced when the laser is focused inside the drop and the shock front hits small bubbles sitting under the free surface. For water these microjets are in diameter and exit at 100–250 m/s. For higher viscositydrops, these jets can emerge at over 500 m/s.

Convective instabilities in liquid layers with free upper surface under the action of an inclined temperature gradient
View Description Hide DescriptionWe present the results of an experimental study of convective instabilities in a horizontal liquid layer with free upper surface under the action of an inclined temperature gradient, i.e., when horizontal and vertical temperature gradients are applied at the same time. Silicone oil of 10 cSt (Prandtl number ) was employed as the test fluid. We investigated the layers with different thicknesses to examine the influence of gravity on the formation of the convective patterns. It is found out that the system behavior appreciably depends on the dynamic Bond number, which shows a relation of buoyancy and thermocapillary forces. In the case of small dynamic Bond numbers, when the influence of buoyancy is minimal, four different flow patterns, according to the combination of the vertical and horizontal Marangoni numbers, have been found: steady parallel flow, Bénard–Marangoni cells, drifting Bénard–Marangoni cells, and longitudinal rolls. At larger dynamic Bond number, when the influence of buoyancy becomes considerable, new convective structures, named by us the “surface longitudinal rolls” and the “surface drifting cells,” appear in addition to the patterns listed above. These instabilities exist only in the surface part of the thermocapillary flow, whereas the return flow remains stable. Under large enough dynamic Bond number these patterns become the dominating ones, forcing out the classical Bénard–Marangoni instability. We give a phenomenological description of the obtained convective patterns and present the stability diagram in the plane of the vertical and the horizontal Marangoni numbers.

Onset of flow separation for the oblique water impact of a wedge
View Description Hide DescriptionFor the oblique impact of a wedge on a liquid half space, the limiting angles of the entry velocity and the wedge orientation corresponding to flow separation from the wedge vertex during the initial stage of the impact are investigated on the basis of an analytical solution of the problem. The liquid is assumed to be ideal and incompressible; gravity, surface tension, and air cushioning effects are ignored. The flow generated by the impact is two dimensional and potential. The solution is presented in terms of two governing expressions, which are the complex velocity and the derivative of the complex potential defined in a parameter region. These expressions are obtained using generalized integral formulas for solving mixed and uniform boundaryvalue problems for the first quadrant. They include two unknown functions, which are the velocity magnitude and angle to the free surface determined from the dynamic and kinematic boundary conditions. The obtained system of integral equations is solved by using the method of successive approximations. The effect of the horizontal component of the entry velocity is studied for various wedge orientations. The analysis of the computations revealed configurations of the impact such that the pressure along the whole length of one side of the wedge becomes less than the pressure on the free surface. Although air effects are not included in the analysis, such a pressure distribution provides conditions for the ventilation of the wedge side, which, in the presence of the air, starts from the contact point on the free surface and extends suddenly along the whole length of the wedge side, thus leading to flow separation from the wedge vertex. The theoretical predictions of flow separation and the experimental data on flow separation by Judge et al. [“Initial water impact of a wedge at vertical and oblique angles,” J. Eng. Math.48, 279 (2004)] are remarkably close to each other.

Timeresolved proper orthogonal decomposition of liquid jet dynamics
View Description Hide DescriptionNew insight into the mechanism of liquid jet in crossflow atomization is provided by an analysis technique based on proper orthogonal decomposition and spectral analysis. Data are provided in the form of highspeed videos of the jet near field from experiments over a broad range of injection conditions. For each condition, proper orthogonal modes (POMs) are generated and ordered by intensity variation relative to the time average. The feasibility of jet dynamics reduction by truncation of the POM series to the first few modes is then examined as a function of crossflow velocity for laminar and turbulentliquid injection. At conditions where the jet breaks up into large chunks of liquid, the superposition of specific orthogonal modes is observed to track long waves traveling along the liquid column. The temporal coefficients of these modes can be described as a bandpass spectrum that shifts toward higher frequencies as the crossflow velocity is increased. The dynamic correlation of these modes is quantified by their crosspower spectrum density. Based on the frequency and wavelength extracted from the videos, the observed traveling waves are linked to the linearly fastest growing wave of Kelvin–Helmholtz instability. The gas boundary layer thickness at the gasliquid shear layer emerges at the end of this study as the dominant length scale of jet dynamics at moderate Weber numbers.

Numerical simulation of liquid sloshing in a partially filled container with inclusion of compressibility effects
View Description Hide DescriptionA numerical scheme of study is developed to model compressible twofluid flows simulating liquid sloshing in a partially filled tank. For a twofluid system separated by an interface as in the case of sloshing, not only a Machuniform scheme is required but also an effective way to eliminate unphysical numerical oscillations near the interface. By introducing a preconditioner, the governing equations expressed in terms of primitive variables are solved for both fluids (i.e., water, air, gas, etc.) in a unified manner. In order to keep the interface sharp and to eliminate unphysical numerical oscillations in unsteady fluid flows, the nonconservative implicit split coefficient matrix method is modified to construct a fluxdifference splitting scheme in the dualtime formulation. The proposed numerical model is evaluated by comparisons between numerical results and measured data for sloshing in an 80% filled rectangular tank excited at resonance frequency. Through similar comparisons, the investigation is further extended by examining sloshing flows excited by forced sway motions in two different rectangular tanks with 20% and 83% filling ratios. These examples demonstrate that the proposed method is suitable to capture induced free surface waves and to evaluate sloshing pressure loads acting on the tank walls and ceiling.

Failure of thermocapillarydriven permanent nonwetting droplets
View Description Hide DescriptionA droplet may be prevented from molecular contact with a solid surface by providing a thin, lubricating film of surrounding fluid between the solid and liquid surfaces. In this study, we exploit thermocapillary convection, caused by a temperature difference maintained between the droplet and the unwetted surface, to provide this lubricating film. This state may be sustained indefinitely (permanent nonwetting) if the load applied to the droplet does not exceed a threshold. Failure of such systems may be categorized as either film or pinning failures, depending on whether the lubrication film is breached, resulting in a molecular contact between the droplet and the solid surface, or the droplet is forced from its support by losing its pinning contact line. In this work, loads that trigger film and pinning failures are quantified, and their mechanisms explained. Results show that larger loads can be sustained for systems with an elevated temperature difference and for droplets of higher viscosity.
 Viscous and NonNewtonian Flows

The friction of a meshlike superhydrophobic surface
View Description Hide DescriptionWhen a liquiddroplet is located above a superhydrophobic surface, it only barely touches the solid portion of the surface, and therefore slides very easily on it. More generally, superhydrophobic surfaces have been shown to lead to significant reduction in viscousfriction in the laminar regime, so it is of interest to quantify their effective slipping properties as a function of their geometric characteristics. Most previous studies considered flows bounded by arrays of either long grooves, or isolated solid pillars on an otherwise flat solid substrate, and for which therefore the surrounding air constitutes the continuous phase. Here we consider instead the case where the superhydrophobic surface is made of isolated holes in an otherwise continuous noslip surface, and specifically focus on the meshlike geometry recently achieved experimentally. We present an analytical method to calculate the friction of such a surface in the case where the mesh is thin. The results for the effective slip length of the surface are computed, compared to simple estimates, and a practical fit is proposed displaying a logarithmic dependence on the area fraction of the solid surface.

Optimal shapes for best draining
View Description Hide DescriptionThe container shape that minimizes the volume of draining fluid remaining on the walls of the container after it has been emptied from its base is determined. The film of draining fluid is assumed to wet the walls of the container, and is sufficiently thin so that its curvature may be neglected. Surface tension is ignored. The initial value problem for the thickness of a film of Newtonian fluid is studied, and is shown to lead asymptotically to a similarity solution. From this, and from equivalent solutions for powerlaw fluids, the volume of the residual film is determined. The optimal container shape is not far from hemispherical, to minimize the surface area, but has a conical base to promote draining. The optimal shape for an axisymmetric mixing vessel, with a hole at the center of its base for draining, is also optimal when inverted in the manner of a washed wine glass inverted and left to drain.
 Particulate, Multiphase, and Granular Flows

A unified sweepstick mechanism to explain particle clustering in two and threedimensional homogeneous, isotropic turbulence
View Description Hide DescriptionOur work focuses on the sweepstick mechanism of particle clustering in turbulent flows introduced by Chen et al. [L. Chen, S. Goto, and J. C. Vassilicos, “Turbulent clustering of stagnation points and inertial particles,” J. Fluid Mech.553, 143 (2006)] for twodimensional (2D) inverse cascading homogeneous, isotropic turbulence (HIT), whereby heavy particles cluster in a way that mimics the clustering of zeroacceleration points. We extend this phenomenology to threedimensional (3D) HIT, where it was previously reported that zeroacceleration points were extremely rare. Having obtained a unified mechanism we quantify the Stokes number dependency of the probability of the heavy particles to be at zeroacceleration points and show that in the inertial range of Stokes numbers, the sweepstick mechanism is dominant over the conventionally proposed mechanism of heavy particles being centrifuged from high vorticity regions to high strain regions. Finally, having a clustering coincidence between particles and zeroacceleration points, both in 2D and 3D HIT, motivates us to demonstrate the sweep and stick parts of the mechanism in both dimensions. The sweeping of regions of low acceleration regions by the local fluidvelocity in both flows is demonstrated by introducing a velocity of the acceleration field. Finally, the stick part is demonstrated by showing that heavy particles statistically move with the same velocity as zeroacceleration points, while moving away from any nonzeroacceleration region, irrespective of their Stokes number. These results explain the clustering of inertial particles given the clustering of zeroacceleration points.

Multibubble cavitation inception
View Description Hide DescriptionThe inception of cavitation in multibubble cases is studied numerically and theoretically to show that it is different from that in singlebubble cases in several aspects. Using a multibubble model based on the Rayleigh–Plesset equation with an acoustic interaction term, we confirmed that the recently reported suppression of cavitation inception due to the interaction of nonidentical bubbles can take place not only in liquidmercury but also in water, and we found that a relatively large bubble can significantly decrease the cavitation threshold pressure of a nearby small bubble. By examining in detail the transition region where the dynamics of the suppressed bubble changes drastically as the interbubble distance changes, we determined that the explosive expansion of a bubble under negative pressure can be interrupted and turn into collapse even though the farfield liquidpressure well exceeds the bubble’s threshold pressure. Numerical results suggest that the interruption of expansion occurs when the bubble radius is exceeded by the instantaneous unstable equilibrium radius of the bubble determined using the total pressure acting on the bubble. When we extended the discussion to systems of larger numbers of bubbles, we found that a larger number of bubbles have a stronger suppression effect. The present findings would be useful in understanding the complex behavior of cavitationbubbles in practical applications where, in general, many cavitation nuclei exist and may interact with each other.
 Laminar Flows

Flow in a channel with accelerating or decelerating wall velocity: A comparison between selfsimilar solutions and Navier–Stokes computations in finite domains
View Description Hide DescriptionWe investigate computationally whether a class of selfsimilar solutions of the Navier–Stokes equations in infinite channels driven by accelerating or decelerating walls, arises anywhere in a channel restricted to finite length. The selfsimilar solutions satisfy a nonlinear partial differential equation involving time and the vertical coordinate and have been studied previously. Admissible selfsimilar solutions include stable and unstable steady branches, timeperiodic branches emerging from Hopf bifurcations, as well as chaotic solutions following a perioddoubling Feigenbaum cascade. The objective here is to explore whether such solutions emerge when restricted to finite, although long, channels. The problem is addressed numerically using fast algorithms to solve the Navier–Stokes equations. It is established that all branches of the selfsimilar solutions (including timeperiodic ones) can occur in finite channels provided the Reynolds numbers are not too high (approximately 500 for accelerating and 33 for decelerating wall flows, respectively). As the Reynolds number increases it becomes more difficult to recover the selfsimilar branches with general inflow conditions, and this has been overcome by utilizing the selfsimilar solutions as inflow Dirichlet conditions. At sufficiently high Reynolds numbers, the selfsimilar inflow conditions are incapable of producing the exact solution in the interior, but instead new steady or timeperiodic states emerge which depend on the Reynolds number and the channel length. It is established numerically by solving the linearized Navier–Stokes equations that such phenomena are due to a spatial instability of the selfsimilar states to perturbations which are not of selfsimilar form. The results indicate that caution must be exercised in drawing conclusions from a stability theory restricted to selfsimilar perturbations and examples are given where such analysis is erroneous. In the case of decelerating wall flows the selfsimilar solutions are recovered on finite domains but for a much smaller range of Reynolds numbers as compared to accelerating wall flows. The results also show that timeperiodic selfsimilar states are possible in finite channels (the Navier– Stokes equations are solved subject to the spatiotemporal selfsimilar solution at inflow) for accelerating wall flows but not for decelerating ones. Of special interest are computed aperiodic states of the Navier–Stokes equations in the case of accelerating walls and relatively high Reynolds numbers. These solutions behave intermittently and bounce between longlived states that are congruent to the selfsimilar solutions to nonselfsimilar aperiodic ones. This is a complicated phenomenon due to the dimensionality and nonlinear nature of the system but the results indicate that the selfsimilar states can be strong attractors in some regions of the phase space.
 Instability and Transition

Effect of electric field on the stability of an oscillatory contaminated film flow
View Description Hide DescriptionThe stability of the viscous liquid film on an oscillating plane is investigated in the presence of both insoluble surfactant on the filmsurface and uniform electric field, acting normal to the plane. In this problem main motivation is to study the combine effect of surfactant and electric field on the stability of the liquid film. Here liquid is treated as a perfect conductor and the air above the liquid film is also treated as a perfect dielectric. The linear stability analysis is performed using the longwave perturbation method based on Floquet theory. It is observed that two Floquet modes exist due to the presence of surfactant and both modes can be unstable. The growth rate corresponding to the Floquet modes increase with the presence of an electric field and decrease with the presence of surfactant.

Global mode analysis of axisymmetric bluffbody wakes: Stabilization by base bleed
View Description Hide DescriptionThe flow around a slender body with a blunt trailing edge is unstable in most situations of interest. Usually the flow instabilities are generated within the wake behind the bluff body, inducing fluctuating forces and introducing the possibility of resonance mechanisms with modes of the structure. Base bleed is a simple and wellknown means of stabilizing the wake. In the present research, we investigate the global instabilityproperties of the laminarincompressible flow that develops behind a cylinder with sharp edges and axis aligned with the free stream using a spectral domain decomposition method. In particular, we describe the flow instabilitycharacteristics as a function of the Reynolds number,, and the bleed coefficient, defined as the bleedtofreestream velocity ratio, , where is the diameter of the body and and the density and viscosity of the free stream, respectively. For a truncated cylinder of aspect ratio , where is the length of the body, our calculations reveal the presence of a first steady bifurcation in the wake at , as well as a second oscillatory one at with an associated Strouhal number for the most unstable azimuthal mode . In addition, we report the existence of two critical values of the bleed coefficient and , which vary with the aspect ratio of the body, needed to stabilize both the first and second bifurcations in the range of Reynolds numbers under study, . Finally, the numerical results for the oscillatory mode obtained for a bulletlike body of aspect ratio without base bleed are compared with experiments performed in a wind tunnel using hotwire anemometry, showing the limitations of using an axisymmetric basic flow at Reynolds numbers higher than the critical one corresponding to the first steady bifurcation in the global stabilityanalysis.

Smallamplitude perturbations in the threedimensional cylindrical Richtmyer–Meshkov instability
View Description Hide DescriptionWe first study the linear stability of an interface between two fluids following the passage of an imploding or exploding shock wave. Assuming incompressible flow between the refracted waves following shock impact, we derive an expression for the asymptotic growth rate for a threedimensional combination of azimuthal and axial perturbations as a function of the Atwood ratio, the axial and azimuthal wave numbers, the initial radial position and perturbation amplitude of the interface, and the interface velocity gain due to the shock interaction. From the linearized theory, a unified expression for the impulsive asymptotic growth rate in plane, cylindrical, and spherical geometries is obtained which clearly delineates the effects of perturbation growth due to both geometry and baroclinic vorticity deposition. Several different limit cases are investigated, allowing recovery of Mikaelian’s purely azimuthal theory and Richtmyer’s plane model. We discuss the existence of threedimensional perturbations with zero growth, typical of curvilinear geometries, as first observed by Mikaelian. The effect of shock proximity on the interface growth rate is studied in the case of a reflected shock. Analytical predictions of the effect of the incident shock strength and the perturbation wave numbers are then compared with results obtained from highly resolved numerical simulations of cylindrical imploding Richtmyer–Meshkov instability for ideal gases. A parallel is made with the instability growth in spherical and plane geometry. In particular, we propose a representation of the perturbation growth by considering the volume of the perturbed layer. This volume is found to grow faster in the plane case than in the imploding cylindrical geometry, among other results.