Volume 17, Issue 11, November 2005
 LETTERS


Multiparticle dispersion in fully developed turbulence
View Description Hide DescriptionThe statistical geometry of dispersing Lagrangian clusters of four particles (tetrahedra) is studied by means of highresolution direct numerical simulations of threedimensional homogeneous isotropic turbulence. We give evidence of a selfsimilar regime of shape dynamics characterized by almost twodimensional, strongly elongated geometries. The analysis of fourpoint velocitydifference statistics and orientation shows that inertialrange eddies typically generate a straining field with a strong extensional component aligned with the elongation direction and weak extensional/compressional components in the orthogonal plane.

Subharmonic mechanism of the mode C instability
View Description Hide DescriptionThe perturbation field of the recently discovered subharmonic mode C instability in the wake behind a ring is compared via a sidebyside comparison to the perturbation fields of the modes A and B instabilities familiar from past studies of the vortex street behind a circular cylinder. Snapshots of the wake are presented over a full shedding cycle, along with evidence from a linear stabilityanalysis, to verify and better understand how the subharmonic instability is sustained.
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 ARTICLES

 Interfacial Flows

Local effects of on small particle motion in a linear flow field
View Description Hide DescriptionSmall particle motion in a linearized flow field is investigated analytically. The aim is to study the effects of relatively large Stokes numbers . It is assumed that the only dominant hydrodynamic force is the quasisteady Stokes drag where all other effects are neglected. An exact solution for a general, threedimensional, unsteady flow field, including gravity, is obtained. Emphasis is placed on the role taken by the spatially varying flow field and the particle inertia. Although an immediate consequence of the linearization is that the longterm particle velocity is insensitive to initial conditions, no matter the value of , the shortterm behavior is quite intriguing. This behavior is found to be highly sensitive to the initial particle velocity when exceeds critical values depending on the flow; this happens, for example, in shear flows. The exponential approach of the particle velocity to the fluid velocity, predicted by the asymptotic solution for , may under certain flow conditions transform into damped oscillations for critical values of the number. An explanation is provided by showing the equivalence of the particle equation to that of a damped pendulum.

Temporal instability of a capillary jet with a source of mass
View Description Hide DescriptionThe temporal linear instability of an inviscid capillary liquid jet with a source of mass is investigated. Two different spatial mass distributions are considered. In the first case, mass is added uniformly everywhere in the jet. In the second case, mass is added uniformly in the radial direction, but allowed to change along the axial direction. The problem is solved using a onedimensional approximation for the capillary jet. An analytical solution for the steady basic flow of a jet with a small source of mass is obtained. The dispersion relation and the growth rate of the disturbances are obtained analytically for small source terms. The influence of the mass addition on the instability characteristics, the breakup time, and the breakup length of a jet are presented. It is shown that the mass addition decreases the rate of growth of the disturbances. The mass addition increases the breakup time and decreases the breakup length. Also, the sizes of the generated drops increase with increasing the source of mass.

Numerical simulation of bubble growth in film boiling using a coupled levelset and volumeoffluid method
View Description Hide DescriptionA coupled levelset and volumeoffluid method is presented for modeling incompressible twophase flows with surface tension. The coupled algorithm conserves mass and captures the complicated interfaces very accurately. A planar simulation of bubble growth is performed in water at near critical pressure for different degrees of superheat. The effect of superheat on the frequency of bubble formation has been analyzed. In addition, simulation of film boiling and bubble formation is performed in refrigerant R134a at near critical and far critical pressures. The effect of saturation pressure on the frequency of bubble formation has also been studied. A deviation from the periodic bubble release is observed in the case of superheat beyond in water. The effect of heat flux on the instability has also been analyzed. It is found that for water at near critical condition, a decrease in superheat from leads to oscillations with subharmonics influencing the time period of the ebullition cycle.

Hydrothermal waves in a liquid bridge with aspect ratio near the Rayleigh limit under microgravity
View Description Hide DescriptionA 2 cSt silicone oil liquid bridge of length and radius (aspect ratio ) was established under microgravity during the flight of the sounding rocket MAXUS4. Four different temperature differences , , , and were applied between the ends, each for sufficient time to reach steadystate thermocapillary flow conditions. The aim of the experiment—to observe the onset of hydrothermal waves and to measure their features such as the wave phase speed and the angle between the wave vector and the applied temperature gradient—was reached. We used microgravity in this experiment in a twofold manner: (1) a liquid bridge with can be established only under microgravity; (2) it was possible to study hydrothermal waves without the influence of gravity and without the aspect ratio restrictions at normal gravity.
 Viscous and NonNewtonian Flows

Building a better snail: Lubrication and adhesive locomotion
View Description Hide DescriptionMany gastropods, such as slugs and snails, crawl via an unusual mechanism known as adhesivelocomotion. We investigate this method of propulsion using two mathematical models: one for direct waves and one for retrograde waves. We then test the effectiveness of both proposed mechanisms by constructing two mechanical crawlers. Each crawler uses a different mechanical strategy to move on a thin layer of viscous fluid. The first uses a flexible flapping sheet to generate lubrication pressures in a Newtonian fluid, which in turn propel the mechanical snail. The second generates a wave of compression on a layer of Laponite, a nonNewtonian, finiteyield stress fluid with characteristics similar to those of snail mucus. This second design can climb smooth vertical walls and perform an inverted traverse.

Mean drift velocity in viscous flow over a corrugated bottom
View Description Hide DescriptionThe mean, steadystate particle velocity in pressuredriven viscousflow in a horizontal channel with a sinusoidal bottom and a plane top is computed using a Lagrangian description of motion. The bottom corrugations are longitudinal, i.e., having crests aligned along the basic flow. At the bottom a noslip condition is applied. At the upper boundary two different cases are considered: one stress free and one no slip. The nonlinear interaction between the basic Poiseuille flow and the bottom corrugations yields a mean Lagrangian drift velocity that is directed along the basic flow for small nondimensional wave numbers, and against the basic flow for larger wave numbers. More specifically, the entire mean drift velocity is negative for stressfree and noslip upper boundaries when the wave number for the two cases exceeds 1.35 and 2.28, respectively. It is demonstrated that the total nonlinear wall drag is zero for both these cases. The results for longitudinal corrugations are compared with earlier calculations for bottom corrugations that are transverse to the basic flow. In that case the nonlinear mean drift current is always directed against the basic flow. It is found that for the same wave number, transverse corrugations induce a stronger negative mean drift velocity than do longitudinal corrugations.
 Particulate, Multiphase, and Granular Flows

On the stability of the MaxeyRiley equation in nonuniform linear flows
View Description Hide DescriptionIn this work, we address the stability of equilibrium points for particle motion in an important class of flows for which a consistent Lagrangianequation of motion for the particle can be derived and solved exactly. This class of flows includes all nonuniform, steady solenoidal flows that are characterized by constant vorticity and constant strain rate. We demonstrate that equilibrium points for particle motion exist for such flow fields and that, under some specific conditions, these equilibrium points are stable and therefore represent the observable steadystate solution of the Lagrangianequation of motion for the particle. We show that there exists an exact solution procedure for the MaxeyRiley equation of motion in all linear solenoidal flows. We also show that for twodimensional linear solenoidal velocity fields characterized by closed streamlines, the stability of the equilibrium point for the particle motion is determined by the relative density of the particle with respect to the fluid: heavy particles produce unstable equilibrium points and light particles produce stable equilibrium points. For flows that are not characterized by closed streamlines, the equilibrium points for the particle motion for both heavy and light particles are unstable. Finally, we completely characterize threedimensional linear solenoidal velocity fields that are solutions of the NavierStokes equations and establish the necessary and sufficient conditions for the stability of particle equilibrium points in such flows.

Threedimensional simulations of a plateletshaped spheroid near a wall in shear flow
View Description Hide DescriptionWe have investigated the threedimensional motion of an oblate spheroidshaped particle or “platelet” close to an infinite planar wall in shear flow. The completed double layerboundary integral equation method modified to include a flat surface boundary was used to compute the effects of the wall on the flow behavior of a platelet of aspect ratio 0.25. Platelet simulations were initiated with the platelet having its axis of symmetry normal to the surface. The platelet is observed to demonstrate three distinct regimes of flow, the dominant regime of flow being dependent on its initial height from the surface. Platelets further than 1.2 platelet radii from the surface display a “modified” Jeffery orbit with periodic rotational motion in the direction of flow (regime I). The presence of the wall retards platelet flow and more importantly introduces periodic lateral motion normal to the wall. When the platelet starts flowing at a height between 1.1 and 0.75 platelet radii from the surface (regime II), it is found to dip down and then “pole vault” off of the surface. Periodic platelet rotation and contact with the surface then ensues with no drift of the platelet away from the surface. In the third regime of flow, a platelet with an initial height of 0.7 platelet radii and below demonstrates wobble motion. The platelet neither rotates nor contacts the surface, but instead, moves laterally in a periodic manner as it translates in the direction of flow.Flow behavior typical of regime III is not observed at all spheroid aspect ratios. The range of initial heights that demonstrate regime III flow diminishes with increasing aspect ratio and regime III disappears completely above an aspect ratio of 0.41. If a platelet located at heights relevant to regime III is given an initial tilt about the axis or axis (where is the flow direction), the resulting flow may either continue in regime III or shift to regime II, depending on the tilt angle. There is a small critical angle of tilt, the magnitude of which depends on the initial platelet height, above which the platelet flow behavior immediately transitions from regime III flow to that of regime II. Tilts about the axis induce fully threedimensional flow with rotational motion occurring about all three axes. Therefore, even small tilts about the axis that are below the critical tilt angles can cause platelet flow to transform from regime III to II eventually at some point downstream. Thus, a platelet flowing close to the surface in linear shear has ample opportunity to contact the surface, an important event for initiating receptorligand binding of blood platelets to the injured vascular endothelium during the onset of hemostasis.

Interface patterns between still gas and liquid in horizontal tubes
View Description Hide DescriptionThe pattern of the interface between a liquid and a gas at rest in a small horizontal tube is investigated both experimentally and theoretically. The experimental study was carried out for different liquid fractions with air and fluids of various surface tension and contact angle. When the contact angle is zero, a hysteresis is observed at the transition between the stratified and bubble patterns. The theory shows that the bubbletostratified transition occurs when the energy of the stratified pattern becomes smaller than that of the bubble pattern, and that the stratifiedto bubbletransition occurs when the energy of the linearly perturbed interface becomes smaller than the energy of the unperturbed one. The theory for the bubbletostratified transition is extended to nonzero contact angle. It shows that its influence is significant when it is greater than 90°. Experiments carried out with mercury confirm the theory.

Reformulating and quantifying the generalized added mass in filtered gassolid flow models
View Description Hide DescriptionTo account for mesoscale phenomena in coarse grid simulations, Reynolds stress terms appearing in the filtered gassolidflowequations have to be modeled. A generalized added mass approach previously proposed by Zhang and VanderHeyden [Int. J. Multiphase Flow28, 805 (2002)] to model the acoustic gassolid interaction Reynolds stress term is analyzed. Theoretically, it is shown that a generalized added mass term appears directly from the filtered acoustic gassolid interaction term and that it corresponds to a redistribution of the filtered gas phase pressure gradient over the phases. This direct contribution scales according to the mean square of the solid volume fraction fluctuations. Twodimensional dynamic mesoscale simulations over a broad solid volume fraction range and for two domain sizes and two grid resolutions are carried out to calculate the magnitude of the generalized added mass effect. Calculated values of the mean square of the solid volume fraction fluctuations are qualitatively in agreement with the experimental observations of Zenit and Hunt [Int. J. Multiphase Flow26, 763 (2000)]. A second, indirect contribution to the generalized added mass term from the filtered acoustic gassolid interaction term is shown to be statistically significant, but one order of magnitude smaller than the direct contribution. A further quantification of the maximum generalized added mass effect as a function of the filter frequency is obtained from a mixture speed of sound test. The results show that a large generalized added mass coefficient, as previously reported by Zhang and VanderHeyden [Int. J. Multiphase Flow28, 805 (2002)], is justified only in case the filter frequency is low , i.e., if the grid is, either spatially or temporally, sufficiently coarse.
 Laminar Flows

Numerical experiments on flapping foils mimicking fishlike locomotion
View Description Hide DescriptionThe results of numerical experiments aimed at investigating the topology of the vortex structures shed by an oscillating foil of finite span are described. The motion of the foil and its geometry are chosen to mimic the tail of a fish using the carangiform swimming. The numerical results have been compared with the flow visualizations of Freymuth [J. Fluids Eng.111, 217 (1989)] and those of von Ellenrieder et al. [J. Fluid Mech.490, 129 (2003)]. The results show that a vortex ring is shed by the oscillating foil every half a cycle. The dynamics of the vortex rings depends on the Strouhal number . For relatively small values of , the interaction between adjacent rings is weak and they are mainly convected downstream by the free stream. On the other hand, for relatively large values of , a strong interaction among adjacent rings takes place and the present results suggest the existence of reconnection phenomena, which create pairs of longitudinal counterrotating vortices.

Flow visualization of cavitating flows through a rectangular slot microorifice ingrained in a microchannel
View Description Hide DescriptionMultifarious hydrodynamic cavitating flow patterns have been detected in the flow of deionized water through a wide and deep rectangular slot microorifice established inside a wide and long microchannel. This article presents and discusses the flow patterns observed at various stages of cavitation in the aforementioned micrometersized silicondevice.Cavitation inception occurs with the appearance of inchoate bubbles that emerge from two thin vapor cavities that emanate from the boundaries of the constriction element. A reduction in the cavitation number beyond inception results in the development of twin coherent unsteady large vapor cavities, which appear just downstream of the microorifice and engulf the liquid jet. The shedding of both spherical and nonspherical vapor bubbles and their subsequent collapse into vapor plumes downstream of the orifice occurs intermittently. A further reduction in the exit pressure only aids in the elongation of the two coherent cavities and produces two stable vapor pockets. Additionally, interference fringes are clearly observed, showing that the vapor pocket has a curved interface with liquid. At low cavitation numbers, the flow undergoes a flip downstream and the two vapor pockets coalesce and form a single vapor pocket that is encircled by the liquid and extends until the exit of the microchannel. The cavitating flow patterns are unique and are markedly different from those reported for their macroworld counterparts. Evidence of pitting due to cavitation has been observed on the silicon just downstream of the microorifice. It is therefore apparent that cavitation will continue to influence/impact the design of highspeed MEMS hydraulic machines, and the pernicious effects of cavitation in terms of erosion, choking, and a reduction in performance will persist in microfluidic systems if apposite hydrodynamic conditions develop.

Slender body method for slender prolate spheroids and hemispheroids on planes in linearized oscillatory flow
View Description Hide DescriptionThis paper extends the slender body theory of Stokes flow[G. K. Batchelor, J. Fluid Mech.44, 419 (1970)] to slender prolate hemispheroids on infinite planes that are undergoing rotational or translational oscillatory motion. Unsteady stokeslets and doublets are placed along the focal length axis to represent the hydrodynamics, with strengths or weightings that are determined numerically from the boundary conditions. The hydrodynamictorque is determined using two different methods. Method 1 uses Green’s functions for torque that are derived from unsteady stokeslets and doublets. Method 2 uses the stokeslet weightings, the boundary velocity, and the spheroidal shape to compute the torque, in a similar way as was done [C. Pozrikidis, Phys. Fluids A1, 1508 (1989)] in computing the force on a prolate spheroid undergoing translational oscillatory motion. The results agree with those of the singularity and boundary element methods.
 Instability and Transition

Parametric study of streak breakdown mechanism in Hartmann flow
View Description Hide DescriptionOur goal in this study is the identification of the regions of stability and instability of laminar Hartmann flow against the transition to turbulence. The Hartmann flow is characterized by two dimensionless parameters: the Reynolds number,, based on the thickness of the Hartmann layer and the Hartmann number, . In a previous paper [Krasnov et al., J. Fluid Mech.504, 183 (2004)] it has been shown that the transition takes place by a twostep mechanism—optimal twodimensional (2D) perturbation plus a 3D random perturbation at maximum grown 2D perturbation—leading to the streak breakdown scenario. By numerical simulations based on this twostep mechanism, a parametric study is carried out and three major results are established: (1) the critical Reynolds number is determined for higher Hartmann numbers to reach the parameter values of the recent experiment of Moresco and Alboussière [J. Fluid Mech.504, 167 (2004)], a very good agreement is found. (2) For moderate Hartmann numbers the actual minimum size of 2D optimal and 3D random perturbations necessary to trigger a transition to turbulence is determined depending on the Reynolds number. (3) For a higher Reynolds number, it is shown that the transition is possible by random 3D perturbations alone, and the minimum size of the perturbations is determined.

Linear stability of spiral Poiseuille flow with a radial temperature gradient: Centrifugal buoyancy effects
View Description Hide DescriptionFor spiral Poiseuille flow with a radial temperature gradient and radius ratio , we have computed complete linear stability boundaries for several values of the rotation rate ratio , where and are the inner and outer cylinder radii, respectively, and and are the corresponding (signed) angular speeds. The effects of gravity are neglected, but the variation of density with temperature induces radial buoyancy effects through the centripetal acceleration term in the radial momentum equation. The analysis extends previous results with no axial flow(Reynolds number) to the range of for which the annular Poiseuille flow is stable and accounts for arbitrary disturbances of infinitesimal amplitude. For and a temperature gradient consistent with the Boussinesq approximation, we show that over the entire range of considered the stability boundaries do not differ significantly from those found for the isothermal case by Cotrell and Pearlstein [“The connection between centrifugal instability and TollmienSchlichtinglike instability for spiral Poiseuille flow,” J. Fluid. Mech.509, 331 (2004)]. For and , we show for the first time that the flow is destabilized for any positive value of the radial buoyancy parameter (i.e., ), where is the temperature difference between the outer and inner cylinder walls, respectively, and is the coefficient of thermal expansion. This differs significantly from the isothermal results which show that there is no linear instability for in the absence of centrifugal buoyancy effects, where is the turning point value in the isothermal results.

Instability inside a rotating gas cylinder subject to axial periodic strain
View Description Hide DescriptionWe study numerically the instability of a confined rotating gas flow subject to periodic strain along the axis of rotation, under the low Mach number approximation. An axisymmetric timestepping spectral Galerkintype code is used to investigate the viscous basic flow and its stability. Parametric resonance can lead to instability of this flow via the growth of inertial modes coupled by the oscillating strain. The marginal stability curve compares well with earlier experimental and (asymptotic) analytical results in the case of the axisymmetric inertial mode (1,1,0). The resulting flow is dominated by a timeoscillating toroidal vortex and differs very little from the theoretical mode. Two different nonlinear regimes are found, one with saturation to a constant modal amplitude, the other with weak periodic modulation. We also show evidence of the presence of an azimuthal circulation, apparently responsible for the observed modulation.

Transition thresholds in the asymptotic suction boundary layer
View Description Hide DescriptionEnergy thresholds for transition to turbulence in an asymptotic suction boundary layer is calculated by means of temporal direct numerical simulations. The temporal assumption limits the analysis to periodic disturbances with horizontal wave numbers determined by the computational box size. Three well known transition scenarios are investigated: oblique transition, the growth and breakdown of streaks triggered by streamwise vortices, and the development of random noise. Linear disturbance simulations and stability diagnostics are also performed for a base flow consisting of the suction boundary layer and a streak. The scenarios are found to trigger transition by similar mechanisms as obtained for other flows. Transition at the lowest initial energy is provided by the oblique wave scenario for the considered Reynolds numbers 500, 800, and 1200. The Reynolds number dependence on the energy thresholds are determined for each scenario. The threshold scales like for oblique transition and like for transition initiated by streamwise vortices and random noise, indicating that oblique transition has the lowest energy threshold also for larger Reynolds numbers.
 Turbulent Flows

Lagrangian statistics of particle pairs in homogeneous isotropic turbulence
View Description Hide DescriptionWe present a detailed investigation of the particle pair separation process in homogeneous isotropic turbulence. We use data from direct numerical simulations up to following the evolution of about two million passive tracers advected by the flow over a time span of about three decades. We present data for both the separation distance and the relative velocity statistics. Statistics are measured along the particle pair trajectories both as a function of time and as a function of their separation, i.e., at fixed scales. We compare and contrast both sets of statistics in order to gain insight into the mechanisms governing the separation process. We find very high levels of intermittency in the early stages, that is, for travel times up to order ten Kolmogorov time scales. The fixed scale statistics allow us to quantify anomalous corrections to Richardson diffusion in the inertial range of scales for those pairs that separate rapidly. It also allows a quantitative analysis of intermittency corrections for the relative velocity statistics.