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Volume 18, Issue 7, July 2006
 LETTERS


Dynamics of spectrally truncated inviscid turbulence
View Description Hide DescriptionThe evolution of the turbulent energy spectrum for the inviscid spectrally truncated Euler equations is studied by closure calculations. The observed behavior is similar to the one found in direct numerical simulations [Cichowlas, Bonaïtiti, Debbasch, and Brachet, Phys. Rev. Lett.95, 264502 (2005)]. A Kolmogorov spectral range and an equipartition range are observed simultaneously. Between these two ranges a “quasidissipative” zone is present in the kinetic energy spectrum. The time evolution of the wave number that marks the beginning of the equipartition range is analyzed and it is shown that spectral nonlocal interactions are governing this evolution.

Electrokinetic particle aggregation patterns in microvortices due to particlefield interaction
View Description Hide DescriptionA complex and dynamic particle banding phenomenon in an electric field is reported. A single cylindrical vortexflow is generated by a dcbiased acelectroosmotic flow on parallel electrodes. Charged particles are attracted to the vortex by positive dielectrophoresis(DEP) to form rotating cylindrical structures. As the particle concentration increases, the cylinder undergoes longitudinal symmetry breaking, producing concentrated rotating bands caused by field screening effects. The focusing of the particles into bands is shown to obey negative diffusion dynamics of a longwave instability. Funnels and butterflylike patterns also form because of secondary longitudinal DEP forces from nonuniform screening effects.

Rossiter’s formula: A simple spectral model for a complex amplitude modulation process?
View Description Hide DescriptionSpectral analysis of pressure signals recorded for high subsonic compressible flows past open cavities generally reveals frequency components of strong magnitudes (Rossiter modes). These components are not multiples of each other in the case of shallow cavities. This nonharmonic property is examined through a nonlinear modeling of the spectral distributions. For that purpose, a signal processing interpretation of the Rossiter formula is developed, which combines nonlinear distortion and amplitude modulation. The consequences of this new interpretation on the prediction of frequencies, on mode identification, and on mode interactions are discussed as well as the underlying physical mechanisms.

 ARTICLES

 Interfacial Flows

On the paradox of thermocapillary flow about a stationary bubble
View Description Hide DescriptionWhen a stationary bubble is exposed to an external temperature gradient, Marangoni stresses at the bubble surface result in fluid motion. A straightforward attempt to calculate the influence of this thermocapillary flow upon the temperature distribution fails to provide a wellbehaved solution [Balasubramaniam and Subramanian, Phys. Fluids16, 3131 (Year: 2004)]. This problem is revisited here using a regularization procedure which exploits the qualitative disparity in the longrange flow fields generated by a stationary bubble and a moving one. The regularization parameter is an (exponentially small) artificial bubble velocity, which reflects the inability of any asymptotic expansion to satisfy the condition of exact bubble equilibrium. The solution is obtained using asymptotic matching of two separate Reynoldsnumber expansions: an inner expansion, valid at the bubble neighborhood, and a remote outer expansion, valid far beyond the familiar Oseen region. This procedure provides a wellbehaved solution, which is subsequently used to evaluate the convectioninduced correction to the hydrodynamic force exerted on the bubble. The independence of that correction upon the artificial velocity confirms the adequacy of the regularization procedure to describe the stationarybubble case. The ratio of the calculated force to that pertaining to the classical pureconduction limit [Young, Goldstein, and Block, J. Fluid Mech.6, 350 (Year: 1959)] is given by , where Ma is a radiusbased Marangoni number.

An experimental study of dropondemand drop formation
View Description Hide DescriptionThe dynamics of dropondemand (DOD) drop formation have been studied experimentally using an imaging system with an interframe time of and a spatial resolution of . Using a piezoelectrical actuated inkjet printhead with the nozzle orifice diameter of , experiments were conducted over a range of viscosities and surface tensions. The effects of the driving signal, which controls the piezoelectric transducer that produces the pressure pulse to drive the liquid from the reservoir through the orifice, have been examined along with those of liquid properties. The main stages of DOD drop formation, including ejection and stretching of liquid, pinchoff of liquid thread from the nozzle exit, contraction of liquid thread, breakup of liquid thread into primary drop and satellites, and recombination of primary drop and satellites, are analyzed based on the experimental results. The breakup time of liquid threads was found to be dependent mainly on the capillary time based on the length scale of the nozzle orifice and the growth rate of the most unstable disturbance normalized by the inverse of the capillary time. However, a welldesigned waveform of driving signal can initiate an abrupt pinchoff of the liquid thread from the nozzle exit. During the contraction of the liquid thread after it has pinched off from the nozzle, two modes of breakup were observed: endpinching where the liquid thread pinches off from an almost spherical head, and multiple breakup due to capillary waves. The effects of liquid and system parameters on the formation and recombination of the primary drop and satellites were investigated. Based on experimental observations, a necessary condition for the recombination of the primary drop and satellite and the limit for liquid thread length without breakup during contraction are proposed. The primary drop size increases slightly with increasing surface tension and/or decreasing viscosity. The driving voltage to the piezoelectric transducer mainly determines whether satelliteformation will occur and the size of satellites, and it has insignificant effect on primary drop size.

Stationary regimes of axisymmetric thermal wake interaction of two buoyant drops at low Reynolds and high Peclet number
View Description Hide DescriptionAxisymmetric motion of a leading fluid drop and a trailing gas bubble (or thermally nonconducting drop) in a viscous fluid under the combined effect of gravity and thermocapillarity is considered under the assumption of negligible inertia effects and of nondeformable interfaces. The ambient fluid far from the inclusions is isothermal and the temperature of the leading particle differs from that of the continuous medium. At large Peclet number, thermal boundary layers are present along the fluidliquid and the gasliquid interfaces, and thermal wakes are formed downstream from the particles. The interaction of the thermal wake, shed from the leading inclusion, with the thermal boundary layer on the surface of the trailing one causes a nonuniform temperature distribution on the surface of the latter. The induced Marangoni flow results in the change of the flow pattern, the velocity of both particles, and the equilibrium separation distance. In the present paper, the influence of the Marangoni effect on the drag force and the rate of heat transfer from the drops translating at given velocity is studied as well as on the equilibrium velocities and separation distance of the drops freely migrating under the action of gravity. The analysis considers particles at arbitrary separation distance and takes full account of thermal and hydrodynamic interactions.

Translation and oscillation of a bubble under axisymmetric deformation
View Description Hide DescriptionIn this work the nonlinear interactions between the axisymmetric shape distortions, the axial translational motion, and the volume oscillations of a gas bubble in an inviscid, incompressible liquid are considered. Representing the surface deformation by a complete set of Legendre polynomials and assuming that both this deformation and the translational motion are small, a Lagrangian energy formulation is used to derive a system of equations valid to third order in these interaction terms. The effects of surface tension and pressure are also accounted for to this order. No restriction is placed on the size of the volume oscillations. Examination of the translational motionequation indicates that while at second order the interaction of two neighboring shape modes can cause the bubble to move, at third order any combination of three odd shape modes or one odd and two even shape modes can result in the bubble moving. Third order interactions between any three even shape modes or one even and two odd modes will contribute to the volume oscillations of the bubble. Restricting attention to the first fifteen shape modes, a truncated subset of equations is obtained. For this system, the bubble is given an initial static deformation in shape and the process by which the other modes become excited is considered in detail for some chosen examples. The role of second and third order interactions is identified and general relations which determine which terms can interact to excite a given shape mode are given.

Drop coalescence through planar surfaces
View Description Hide DescriptionA highspeed digital camera is used to investigate the important parameters of drop coalescence at a planar interface. A series of experiments have been performed to observe coalescingdrops at the interface between two immiscible fluids. A variety of fluids have been used to fully investigate the effects of physical properties of fluids involved in this phenomenon. It has been shown that the important dimensionless parameter in this process is the Ohnesorge number, , which dictates the regime of coalescence. We have also shown that for , drops fully coalesce but when , drops partially coalesce and a secondary drop is created. We show the dependence of the ratio of secondary drop radius to the primary drop on the regime of coalescence (inertia or viscous). Using the scaling arguments, we developed a relationship between the drop ratio and the Ohnesorge number that shows good agreement with our experimental results.

Nonlinear dynamics of a twodimensional viscous drop under shear flow
View Description Hide DescriptionThe dynamics of a viscousdrop moving along a substrate under the influence of shear flow in a parallelwalled channel is investigated. A front tracking numerical method is used to simulate a drop with moving contact lines. A Navier slip boundary condition is applied to relax the contact line singularity. Steady state solutions are observed for small Reynolds and capillary number. Unsteady solutions are obtained with increasing Reynolds or capillary number. For large values of the parameters, the interface appears to rupture, but for intermediate parameter values, time periodic drop interface oscillations are possible as the drop is moving along the bottom channel wall. These different states are identified in the Reynolds number–capillary number plane for a specific range of physical parameters. The effects of density and viscosity ratio are also illustrated.
 Viscous and NonNewtonian Flows

Multicomponent diffusion revisited
View Description Hide DescriptionThe derivation of the multicomponent diffusion law is revisited. Following Furry [Am. J. Phys.16, 63 (1948)], Williams [Am. J. Phys.26, 467 (1958); Combustion Theory, 2nd ed.(Benjamin/Cummings , Menlo Park, CA,1985)] heuristically rederived the classical kinetic theory results using macroscopic equations, and pointed out that the dynamics of the mixture fluid had been assumed inviscid. This paper generalizes the derivation, shows that the inviscid assumption can easily be relaxed to add a new term to the classical diffusion law, and the thermal diffusion term can also be easily recovered. The nonuniqueness of the multicomponent diffusion coefficient matrix is emphasized and discussed.
 Particulate, Multiphase, and Granular Flows

A comparison of the predictions of a simple kinetic theory with experimental and numerical results for a vibrated granular bed consisting of nearly elastic particles of two sizes
View Description Hide DescriptionA comparison of the predictions of a simple kinetic theory with experimental and numerical results for a vibrated granular bed consisting of nearly elastic particles of two sizes has been performed. The results show good agreement between the data sets for a range of numbers of each size of particle, and are particularly good for particle beds containing similar proportions of each species. The agreement suggests that such a model may be a good starting point for describing polydisperse systems of granular flows.

Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field
View Description Hide DescriptionMicroscale inertia is found to break the degenerate closedstreamline configuration that occurs in a shearing flow past a neutrally buoyant torquefree spherical particle in the inertialess limit. The broken symmetry at small but finite allows heat or mass to be convected away in an efficient manner in sharp contrast to the inertialess diffusionlimited scenario. Inertial forces scale with the particle Reynolds number, defined as , where is the radius of the particle, is the characteristic magnitude of the velocity gradient, and is the kinematic viscosity of the suspending fluid. The dimensionless heat or mass transfer rate is then given by when and , the constant being a function of the flow in the vicinity of the particle. Here, is the Nusselt number defined as , where is the dimensional heat/mass flux, the appropriate transport coefficient, and the driving force viz. the temperature or concentration difference between the particle and the ambient fluid; for pure diffusion, . The Peclét number is a dimensionless measure of the relative dominance of the convective and diffusive transfer mechanisms. It is shown that equals for a twodimensional linear flow, where measures the relative magnitudes of extension and vorticity. For simple shear , knowledge of the inertial velocity field to enables one to determine the next term in the asymptotic expansion for ; one finds in the limit . It is argued that the convective enhancement at finite via symmetrybreaking streamline bifurcations will occur in generic shearing flows with nonlinear velocity profiles; the degenerate Stokes streamline pattern around a neutrally buoyant torquefree particle in a quadratic flow serves to reinforce this assertion. The above mechanism represents a possible means for heat or mass transfer enhancement from the dispersed phase in multiphase systems. Implications for particles in turbulent flows are also discussed.

The influence of different species’ granular temperatures on segregation in a binary mixture of dissipative grains
View Description Hide DescriptionWe employ a kinetic theory for binary mixtures of slightly inelastic, frictionless particles to study segregation in a uniformly agitated system under gravity, where large particles are dilute in a dense gas of small particles. We take the initial motion of large particles as an indicator of segregation. By incorporating the effects of two different granular temperatures, we show that, although the temperature differences can be significant with increasing differences in the mass or size of the particles, the segregation is not greatly affected. However, we find that there is a small range in size and material density ratios where the initial direction of segregation from a wellmixed state is reversed due to nonequipartition.
 Laminar Flows

Physical effects in laminar microconvection due to variations in incompressible fluid properties
View Description Hide DescriptionIn this investigation we report the identification of laminar microconvection physical effects due to the variation of viscosity and thermalconductivity of liquid. Viscosity variation significantly distorts the axial velocity profile and varies this distortion along the microflow, thereby inducing radial flow due to flow continuity. The resulting induced radial heat convection can be a significant percentage of the axial heat convection, especially in microconvection. Also, axial conduction is induced due to fluid thermalconductivity variation along the flow. The effect of distorted axial velocity profile and the induced radial flow on microconvection due to fluid viscosity variation are opposite. However, fluid thermalconductivity variation along and across the flow have the same effect on microconvection. Thus, a deviation in convection due to thermalconductivity variation exceeds the deviation due to viscosity variation, although the dimensionless temperature sensitivity of viscosity is higher.

Mass transfer to reactive boundaries from steady threedimensional flows in microchannels
View Description Hide DescriptionThis paper presents a numerical study of the effect of transverse secondary flows on mass transfer to reactive boundaries in microchannels. The geometry considered is relevant to surface catalyzed reactions, fuel cells, biochemical sensors, and other microreactor applications. The 3D flows that we consider approximate flows that are experimentally achievable through topographical patterning of one wall of a microchannel, as in the Staggered Herringbone Mixer (SHM) and similar geometries. We simulate a mass transfer process using passive tracers to model reactivesolute molecules in a Stokes flow (Reynolds number, ) over a range of Péclet number, , with instantaneous kinetics at the reactive boundary. Our simulation allows for the evaluation of the local Sherwood number produced by a uniaxial Poiseuille flow and several chaotic and nonchaotic 3D flows. In chaotic flows, the local Sherwood number evolves in a simple manner that shares features with the classic Graetz solution for transfer from a uniaxial pipe flow: an entrance region with cuberoot scaling in the Graetz number and a constant asymptotic value. This “Modified Graetz” behavior also differs in important ways from the standard case: the entrance length is independent and the asymptotic rate of transfer is dependent and potentially much greater than in the uniaxial case. We develop a theoretical model of the transfer process; the predictions of this model compare well with simulation results. We use our results to develop a correlation for the mass transfer in laminar channel flows, to elucidate the importance of chaos in defining transfer in these flows, and to provide design rules for microreactors with a single reactive wall.
 Instability and Transition

Experiments on the threedimensional incompressible RichtmyerMeshkov instability
View Description Hide DescriptionThe threedimensional (3D) RichtmyerMeshkov instability of incompressible, miscible liquids with a 3D singlemode initial perturbation is investigated. This study uses the apparatus of the earlier experiments of Niederhaus and Jacobs [J. Fluid Mech.485, 243 (2003)] in which the instability is generated by impulsively accelerating a tank containing the two liquids. However, the present investigation uses a tank with square cross section allowing the generation of a squaremode 3D initial perturbation by lateral oscillation along the tank’s diagonal. Amplitude measurements of the 3D instability are found to be effectively collapsed by the dimensionless scaling used in the twodimensional (2D) study and to be in good agreement with linear stability theory up until a dimensionless time , later than is found for the 2D flow. Latetime 3D amplitude measurements show faster growth than 2D as is predicted by popular bubble models. However, latetime growth rate measurements are found to deviate from model predictions at the latest times showing a constant growth rate instead of the dependence given by the models. The constant latetime growth rate is the result of the observed vorticity distribution which takes the form of an array of upward and downward traveling vortex rings. This fundamental difference between existing models and observation indicates that bubble models may not be suitable for predicting the behavior of the low Atwood number instability which is vortex dominated.

Formation of a pitchfork bifurcation in thermal convection flow inside an isosceles triangular cavity
View Description Hide DescriptionTransient, laminar thermal convection of air confined to an isosceles triangular cavity heated from the base and symmetrically cooled from the upper inclined walls has been investigated numerically. The system of conservation equations, subject to the proper boundary conditions, along with the equation of state assuming the air behaves as a perfect gas are solved with the finite volume method. In the conservation equations, the secondorderaccurate QUICK scheme was used for the discretization of the convective terms and the SIMPLE scheme for the pressurevelocity coupling. The maximum heighttobase aspect ratio is fixed at 0.5, while the Grashof number extends from a low to a high . The influence of Gr on the flow and temperature patterns is analyzed and discussed for two opposing scenarios, one corresponding to increasing Gr and the other corresponding to decreasing Gr. It is found that two steadystate solutions are possible, excluding their solutionimages through a vertical mirror plane. The symmetrical solution prevails for relatively low Grashof numbers. However, as the Gr is gradually increased, a transition occurs at a critical value of Gr. Above this critical value of Gr, an asymmetrical solution exhibiting a pitchfork bifurcation arises and eventually becomes steady. The existing ranges of these unsteady and steady solutions are reported for the two opposing scenarios. Also, issues related to the observed hysteresis phenomenon are discussed in detail.

Dynamics of vortex lockon in a perturbed cylinder wake
View Description Hide DescriptionThe dynamics of vortex lockon, downstream of a circular cylinder in an oscillatory flow with nonzero mean velocity, is investigated by applying a timeresolved particle image velocimetry technique at the Reynolds number 360. Since the lockon occurs when the near wake behind a cylinder is perturbed at twice the natural shedding frequency, we interrogate the wake regions of mean recirculation and vortexformation, and analyze the dynamic behavior of the shed vortices, the coherent structure, the phases of vortex evolution, and the Reynolds stress fields. It is shown that due to vortex lockon the mean recirculation and vortexformation regions are considerably reduced, which is consistent with previous studies. Besides, the kinetic energy in the near wake is more concentrated near the cylinder base. A dramatic change is demonstrated in the trajectory of the shed vortices, which is evaluated from a hybrid method based on complex eigenvalue and vortex centroid. A novel method to identify the phases of the vortex evolution is proposed in accordance with the distribution of the coherent Reynolds shear stress. Then, it is shown that two flow fields in the natural shedding and lockon states have a phase difference of about . As a result of vortex lockon, the distributions of the Reynolds stresses exhibit not only a stronger Kármán vortex, but also a more synchronized pattern with a shortened wake region. From the streamwise force balance on the wake bubble, it is noted that an outstanding decrease (a reversal) of the shear stress by the lockon results in an increase of the drag force and that the energy transfer from random component to coherent component decreases the three dimensionality of the cylinder wake, inducing the abovementioned shedding of stronger Kármán vortices.

The role of streamwise perturbations in pipe flow transition
View Description Hide DescriptionThe phenomenon of subcritical transition in HagenPoiseuille or pipe flow is explored for a wide range of Reynolds numbers within the interval by means of a computational method that numerically resolves the transitional dynamics with nearly degrees of freedom on a medium aspectratio domain of length . The aim of this exploration is to provide a theoretical characterization of the basin of attraction of the basic regime by measuring the minimal amplitude of an initial global perturbation leading to transition. The analysis is based on a particular theoretical scenario that considers streamwiseindependent finite amplitude initial vortical perturbations that trigger global transition via optimal inflectional instabilities of streamwisedependent modes with selected axial wave numbers. Disturbances consisting of 1, 2, and 3 pairs of vortices are investigated. Special attention is given to relaminarization phenomena that is frequently observed for low Reynolds numbers. Long lasting turbulent regimes and relaminarized flows are distinguished by means of time integrations of suitable length between and advective time units. Some transitional runs are specifically analyzed to exemplify the transition scenario under investigation and its independence of pipe length is verified with a few computations on a longer pipe of length ( degrees of freedom). For large values of the Reynolds number, a theoretical scaling law for the threshold amplitude of a perturbation required to trigger transition is provided. Different types of perturbations seem to respond to different scaling laws.

Transitional and weakly turbulent flow in a rotating magnetic field
View Description Hide DescriptionThe early stage of turbulent flow driven by a rotating magnetic field is studied via direct numerical simulations and electric potential measurements for the case of a cylindrical geometry. The numerical results show that the undisturbed flow remains stable up to the linear stability limit , whereas small perturbations may initiate a nonlinear transition at subcritical Taylor numbers. The observed instabilities occur randomly as isolated pairs of TaylorGörtler vortices, which grow from spots to long tubes until they are dissipated in the lid boundary layers. At , the flow is governed by largescale threedimensional fluctuations and may be characterized as weakly turbulent. TaylorGörtler vortices provide the major turbulence mechanism, apart from oscillations of the rotation axis. As the vortices tend to align with the azimuthal direction, they result in a locally twodimensional turbulence pattern.