Volume 27, Issue 10, October 2015
 ARTICLES

 Micro and Nanofluid Mechanics

Flow patterning in HeleShaw configurations using nonuniform electroosmotic slip
View Description Hide DescriptionWe present an analytical study of electroosmotic flow in a HeleShaw configuration with nonuniform zeta potential distribution. Applying the lubrication approximation and assuming thin electric double layer, we obtain a pair of uncoupled Poisson equations for the pressure and depthaveraged stream function, and show that the inhomogeneous parts in these equations are governed by gradients in zeta potential parallel and perpendicular to the applied electric field, respectively. We obtain a solution for the case of a diskshaped region with uniform zeta potential and show that the flow field created is an exact dipole, even in the immediate vicinity of the disk. In addition, we study the inverse problem where the desired flow field is known and solve for the zeta potential distribution required in order to establish it. Finally, we demonstrate that such inverse problem solutions can be used to create directional flows confined within narrow regions, without physical walls. Such solutions are equivalent to flow within channels and we show that these can be assembled to create complex microfluidic networks, composed of intersecting channels and turns, which are basic building blocks in microfluidic devices.

Frequencydomain deviational Monte Carlo method for linear oscillatory gas flows
View Description Hide DescriptionOscillatory noncontinuum low Mach number gas flows are often generated by nanomechanical devices in ambient conditions. These flows can be simulated using a range of particle based Monte Carlo techniques, which in their original form operate exclusively in the timedomain. Recently, a frequencydomain weightbased Monte Carlo method was proposed [D. R. Ladiges and J. E. Sader, “Frequencydomain Monte Carlo method for linear oscillatory gas flows,” J. Comput. Phys. 284, 351–366 (2015)] that exhibits superior statistical convergence when simulating oscillatory flows. This previous method used the BhatnagarGrossKrook (BGK) kinetic model and contains a “virtualtime” variable to maintain the inherent timemarching nature of existing Monte Carlo algorithms. Here, we propose an alternative frequencydomain deviational Monte Carlo method that facilitates the use of a wider range of molecular models and more efficient collision/relaxation operators. We demonstrate this method with oscillatory Couette flow and the flow generated by an oscillating sphere, utilizing both the BGK kinetic model and hard sphere particles. We also discuss how oscillatory motion of arbitrary timedependence can be simulated using computationally efficient parallelization. As in the weightbased method, this deviational frequencydomain Monte Carlo method is shown to offer improved computational speed compared to the equivalent timedomain technique.
 Interfacial Flows

Instantaneous slip length in superhydrophobic microchannels having grooves with curved or dissimilar walls
View Description Hide DescriptionSuperhydrophobic (SHP) surfaces can be used to reduce the skinfriction drag in a microchannel. This is due to the peculiar ability of these surfaces to entrap air in their pores and thereby reduce the contact area between water and the solid surface. The favorable dragreduction effect, however, can quickly deteriorate if the surface geometry is not designed properly. The deterioration can be sudden, caused by exposure to excessive pressures, or gradual, due to the dissolution of the entrapped air into the ambient water. The formulations presented here provide a means for studying the timedependent dragreduction in a microchannel enhanced with transverse or longitudinal SHP grooves of varying wall profiles or wettabilities. Moreover, different mathematical approaches are developed to distinguish the performance of a sharpedged groove from that of a groove with round entrance. The work starts by deriving an equation for the balance of forces on the air–water interface (AWI) inside a groove and solving this differential equation, along with Henry’s law, for the rate of dissolution of the entrapped air into water over time. It was shown that the performance of a SHP groove depends mostly on the interplay between the effects of the apparent contact angle of the AWI and the initial volume of the groove. The instantaneous slip length is then calculated by solving the Navier–Stokes equations for flow in microchannels with SHP grooves. Our results are compared with the studies in the literature whenever available, and good agreement has been observed.

Coalescenceinduced jumping of droplet: Inertia and viscosity effects
View Description Hide DescriptionThe problem of coalescenceinduced selfpropelled jumping of droplet is studied using threedimensional numerical simulation. The focus is on the effect of inertia and in particular the effect of air density on the behavior of the merged droplet during jumping. A lattice Boltzmann method is used for two identical, static microdroplets coalescing on a homogeneous substrate with contact angle ranging from 0^{∘} to 180^{∘}. The results reveal that the effect of air density is significant on detachment of the merged droplet from the substrate at the later stage of the jumping process; the larger the air density, the larger the jumping height of the droplet. Analysis of streamlines and vorticity contours is performed for density ratios ranging from 60 to 800. These show a generation of vortical structures inside and around the droplet. The intensity of these structures gets weaker after droplet departure as the air inertia is decreased. The results are also presented in terms of phase diagrams of the merged droplet jumping for different Ohnesorge numbers (Oh) and surface wettabilities for both small and large density ratios. The critical value of contact angle where the merged droplet jumps away from the substrate is independent of density ratio and has a value around 150^{∘}. However, the critical value of Oh depends on both density ratio and wettability of the surface for contact angles greater than 150^{∘}. In this range of contact angle, the diagrams show two distinct dynamical regimes for different density ratios, namely, inertial and viscous regimes.

Spreading dynamics of a partially wetting water film atop a MHz substrate vibration
View Description Hide DescriptionA MHz vibration, or an acoustic wave, propagating in a solid substrate may support the convective spreading of a liquid film. Previous studies uncovered this ability for fully wetting silicon oil films under the excitation of a MHz Rayleigh surface acoustic wave (SAW), propagating in a lithium niobate substrate. Partially wetting deionized water films, however, appeared immune to this spreading mechanism. Here, we use both theory and experiment to reconsider this situation and show partially wetting water films may spread under the influence of a propagating MHz vibration. We demonstrate distinct capillary and convective (vibrational/acoustic) spreading regimes that are governed by a balance between convective and capillary mechanisms, manifested in the nondimensional number θ ^{3}/We, where θ is the three phase contact angle of the liquid with the solid substrate and We ≡ ρU ^{2} H/γ; ρ, γ, H, and U are the liquid density, liquid/vapour surface tension, characteristic film thickness, and the characteristic velocity amplitude of the propagating vibration on the solid surface, respectively. Our main finding is that the vibration will support a continuous spreading motion of the liquid film out of a large reservoir if the convective mechanism prevails (θ ^{3}/We < 1); otherwise (θ ^{3}/We > 1), the dynamics of the film is governed by the capillary mechanism.

Numerical investigation of phase relationships in an oscillating sessile drop
View Description Hide DescriptionForced linear oscillations of a viscous drop placed on a horizontal surface vibrating in perpendicular direction are investigated. The problem is solved for two cases: (1) constant contact angle, and (2) pinned contact line. Phasefrequency and amplitudefrequency characteristics of oscillations of the drop apex are found for the first axisymmetrical mode of oscillations. The independence of the difference of oscillation phases of the drop apex and the substrate on fluid density, viscosity, surface tension, and drop size as well as on presence or absence of the gravity force was demonstrated.
 Particulate, Multiphase, and Granular Flows

On the bubble rise velocity of a continually released bubble chain in still water and with crossflow
View Description Hide DescriptionThe rise velocities of inchain bubbles continually released from a single orifice in still water with and without crossflow are investigated in a series of laboratory experiments for wobbling ellipsoidal bubbles with moderate Reynolds number. For the limiting case in still water, that is, crossflow velocity = 0, the theoretical turbulent wake model correctly predicts the inchain bubble rise velocity. In this case, the bubble rise velocities VB are enhanced compared to the terminal velocities of the isolated bubbles V 0 due to wake drafting and are scaled with flow rate Q and bubble diameter D. Here, we also derive an updated wake model with consideration of the superposition of multiple upstream bubble wakes, which removes the nonlinear behavior of the nondistant (i.e., local) wake model. For the cases with crossflow, the enhancement of the inchain bubble rise velocity can be significantly reduced, and imaging of the experiments shows very organized paring and grouping trajectories of rising bubbles not observed in still water under different crossflow velocities. The inchain bubble rise velocities in crossflow are described by two models. First, an empirical model is used to correct the stillwater equation for the crossflow effect. In addition, a semitheoretical model considering the turbulent wake flow and the crossflow influence is derived and used to develop a theoretical normalization of bubble rise velocity, crossflow velocity, and the released bubble flow rate. The theoretical model suggests there are two different regimes of bubblebubble interaction, with strong interaction occurring for the nondimensional crossflow velocity less than 0.06 and weaker interaction occurring for greater than 0.06, where Uc is the crossflow velocity, g is the acceleration of gravity, and β is the mixing length coefficient.

Cavitation erosion prediction based on analysis of flow dynamics and impact load spectra
View Description Hide DescriptionCavitation erosion is the consequence of repeated collapseinduced high pressureloads on a material surface. The present paper assesses the prediction of impact load spectra of cavitating flows, i.e., the rate and intensity distribution of collapse events based on a detailed analysis of flow dynamics. Data are obtained from a numerical simulation which employs a densitybased finite volume method, taking into account the compressibility of both phases, and resolves collapseinduced pressure waves. To determine the spectrum of collapse events in the fluid domain, we detect and quantify the collapse of isolated vapor structures. As reference configuration we consider the expansion of a liquid into a radially divergent gap which exhibits unsteady sheet and cloud cavitation. Analysis of simulation data shows that global cavitation dynamics and dominant flow events are well resolved, even though the spatial resolution is too coarse to resolve individual vapor bubbles. The inviscid flow model recovers increasingly finescale vapor structures and collapses with increasing resolution. We demonstrate that frequency and intensity of these collapse events scale with grid resolution. Scaling laws based on two reference lengths are introduced for this purpose. We show that upon applying these laws impact load spectra recorded on experimental and numerical pressure sensors agree with each other. Furthermore, correlation between experimental pitting rates and collapseevent rates is found. Locations of high maximum wall pressures and high densities of collapse events near walls obtained numerically agree well with areas of erosion damage in the experiment. The investigation shows that impact load spectra of cavitating flows can be inferred from flow data that captures the main vapor structures and wave dynamics without the need for resolving all flow scales.

Particledriven gravity currents in nonrectangular cross section channels
View Description Hide DescriptionWe consider a highReynoldsnumber gravity current generated by suspension of heavier particles in fluid of density ρi , propagating along a channel into an ambient fluid of the density ρa . The bottom and top of the channel are at z = 0, H, and the cross section is given by the quite general −f 1(z) ≤ y ≤ f 2(z) for 0 ≤ z ≤ H. The flow is modeled by the onelayer shallowwater equations obtained for the timedependent motion which is produced by release from rest of a fixed volume of mixture from a lock. We solve the problem by the finitedifference numerical code to present typical height h(x, t), velocity u(x, t), and volume fraction of particles (concentration) ϕ(x, t) profiles. The methodology is illustrated for flow in typical geometries: powerlaw (f(z) = z ^{ α } and f(z) = (H − z)^{ α }, where α is positive constant), trapezoidal, and circle. In general, the speed of propagation of the flows driven by suspensions decreases compared with those driven by a reduced gravity in homogeneous currents. However, the details depend on the geometry of the cross section. The runout length of suspensions in channels of powerlaw cross sections is analytically predicted using a simplified depthaveraged “box” model. The present approach is a significant generalization of the classical gravity current problem. The classical formulation for a rectangular channel is now just a particular case, f(z) = const., in the wide domain of cross sections covered by this new model.
 Laminar Flows

Acoustic streaming, fluid mixing, and particle transport by a Gaussian ultrasound beam in a cylindrical container
View Description Hide DescriptionA computational study is reported of the acoustic streaming flow field generated by a Gaussian ultrasound beam propagating normally toward the end wall of a cylindrical container. Particular focus is given to examining the effectiveness of the acoustic streaming flow for fluid mixing within the container, for deposition of particles in suspension onto the bottom surface, and for particle suspension from the bottom surface back into the flow field. The flow field is assumed to be axisymmetric with the ultrasound transducer oriented parallel to the cylinder axis and normal to the bottom surface of the container, which we refer to as the impingement surface. Reflection of the sound from the impingement surface and sound absorption within the material at the container bottom are both accounted for in the computation. The computation also accounts for thermal buoyancy force due to ultrasonic heating of the impingement surface, but over the time period considered in the current simulations, the flow is found to be dominated by the acoustic streaming force, with only moderate effect of buoyancy force.

Time dependence and local structure of tracer dispersion in oscillating liquid HeleShaw flows
View Description Hide DescriptionPassive tracer dispersion in oscillating Poiseuille liquid flows of zero net velocity is studied experimentally in a HeleShaw cell and numerically by 2D simulations: this study is particularly focused on the time dependence and local properties of the dispersion. The dispersion mechanism is found to be controlled by the ratio τm /T of the molecular diffusion time across the gap and the oscillation period (when molecular diffusion parallel to the flow is negligible). The 2D numerical simulations complement the experiments by providing the local concentration c(x, z, t) at a given distance z from the cell walls (instead of only the average over z). Above a time lapse scaling like τm , the variation of c with the distance x along the flow becomes a Gaussian of width constant with z while the mean distance may depend both on z and t. For τm /T ≲ 2, the front spreads through Taylorlike dispersion and the normalized dispersivity scales as τm /T. The front oscillates parallel to the flow with an amplitude constant across the gap; its width increases monotonically at a rate modulated at twice the flow frequency, due to variations of the instantaneous dispersivity. For τm /T ≳ 20, the molecular diffusion distance during a period of the flow is smaller than the gap and the normalized dispersivity scales as (τm /T)^{−1}. The oscillations of the different points of the front follow the local fluid velocity: this produces a reversible modulation of the global front width at twice the flow frequency and in quadrature with that in the Taylorlike regime.
 Instability and Transition

Paper waves in the wind
View Description Hide DescriptionA flexible sheet clamped at both ends and submitted to a permanent wind is unstable and propagates waves. Here, we experimentally study the selection of frequency and wavenumber as a function of the wind velocity. These quantities obey simple scaling laws, which are analytically derived from a linear stability analysis of the problem and which also involve a gravityinduced velocity scale. This approach allows us to collapse data obtained with sheets whose flexible rigidity is varied by two orders of magnitude. This principle may be applied in the future for energy harvesting.

Effects of magnetic fields on magnetohydrodynamic cylindrical and spherical RichtmyerMeshkov instability
View Description Hide DescriptionThe effects of seed magnetic fields on the RichtmyerMeshkov instability driven by converging cylindrical and spherical implosions in ideal magnetohydrodynamics are investigated. Two different seed field configurations at various strengths are applied over a cylindrical or spherical density interface which has a singledominantmode perturbation. The shocks that excite the instability are generated with appropriate Riemann problems in a numerical formulation and the effect of the seed field on the growth rate and symmetry of the perturbations on the density interface is examined. We find reduced perturbation growth for both field configurations and all tested strengths. The extent of growth suppression increases with seed field strength but varies with the angle of the field to interface. The seed field configuration does not significantly affect extent of suppression of the instability, allowing it to be chosen to minimize its effect on implosion distortion. However, stronger seed fields are required in three dimensions to suppress the instability effectively.
 Turbulent Flows

Characteristics of backward and forward twoparticle relative dispersion in turbulence at different Reynolds numbers
View Description Hide DescriptionA new algorithm based on postprocessing of saved trajectories has been developed and applied to obtain wellsampled backward and forward relative dispersion statistics in stationary isotropic turbulence, over a range of initial separations ranging from Kolmogorov to energycontaining scales. Detailed results are obtained over a range of Taylorscale Reynolds numbers, up to 1000, which is higher than in recent work in the literature. Backward dispersion is faster, especially at intermediate times after the ballistic range and before longtime diffusive behavior is reached. Richardson scaling has been demonstrated for the meansquared separation, and forward and backward Richardson constants estimated to be gf = 0.55 and gb = 1.5, which are close to or comparable to other estimates. However, because of persistent dissipation subrange effects no corresponding scaling was observed for higher order moments of the separation. Analysis of the separation probability density function showed only transitory agreement with the wellknown Richardson prediction. The strong exponential growth of the separation on dissipation subrange scales was analyzed in terms of a central limit theory approximation. The resulting predictions for the ratio of the growth rates of the third and fourthorder moments are reasonably consistent with the theory. The backward growth rates, corresponding to the ratio of the magnitude of the smallest to largest Lyapunov exponents, are about 50% greater than the forward growth rates, somewhat higher than other estimates. The predicted asymmetry between backward and forward relative displacements at early times, manifested in a t ^{3} variation of the difference in the backward and forward meansquare relative displacement, was confirmed numerically and explicitly traced to Eulerian properties at the small scales. However, this t ^{3} growth is not simply connected to the t ^{3} growth in the Richardson regime and the asymmetry manifested there by the difference in the backward and forward Richardson constants. Asymmetry in time for higher order moments was also explained using a Taylorseries analysis at early times.

Turbulent structure at the midsection of an annular flow
View Description Hide DescriptionThe turbulent flow in the midsection of an annular gap between two concentric tubes at Reynolds number of 59 200–90 800 based on hydraulic diameter (dh = 57 mm) and average velocity is experimentally investigated. Measurements are carried out using particle tracking velocimetry (PTV) and planar particle image velocimetry (PIV) with spatial resolution of 0.0068dh (size of the binning window) and 0.0129dh (size of the interrogation window), respectively. Both PTV and PIV results show that the location of maximum mean streamwise velocity (yU ) does not coincide with the locations of zero shear stress (yuv ), minimum streamwise velocity fluctuation (yu 2), and minimum radial velocity fluctuation (yv 2). The separation between yU and yuv is 0.013dh based on PTV while PIV underestimates the separation distance as 0.0063dh . Conditional averages of turbulent fluctuations based on the four quadrants across the annulus demonstrate that the inner and outer wall flows overlap in the midsection. In the midsection, the flow is subject to opposing sweep/ejection events originating from both the inner and outer walls. The opposite quadrant events of the two boundary layers cancel out at yuv while the local minimum of spatial correlation of u (maximum mixing of the two wall flows) occurs at yU . Investigation of the budget of Reynolds shear stress showed that production and advection terms act towards the coincidence of the yU and yuv while the dissipation term works against the coincidence of the two points. The location of 〈U〉 max also overlaps with zero dissipation of 〈uv〉. The production of turbulent kinetic energy is slightly negative in the narrow region between yU and yuv . This negative production acts towards smoothing the mean velocity profile at the joint of the two wall flows by equalizing its curvature (∂^{2}〈U〉/∂y ^{2}) on the two sides of yU . The small separation distance of the yU and yuv is associated with slight deviation from the fully developed condition.

Linear stability of optimal streaks in the loglayer of turbulent channel flows
View Description Hide DescriptionThe importance of secondary instability of streaks for the generation of vortical structures attached to the wall in the logarithmic region of turbulent channels is studied. The streaks and their linear instability are computed by solving equations associated with the organized motion that include an eddyviscosity modeling the effect of incoherent fluctuations. Three friction Reynolds numbers, Re τ = 2000, 3000, and 5000, are investigated. For all flow cases, optimal streamwise vortices (i.e., having the highest potential for linear transient energy amplification) are used as initial conditions. Due to the liftup mechanism, these optimal perturbations lead to the nonlinear growth of streaks. Based on a Floquet theory along the spanwise direction, we observe the onset of streak secondary instability for a wide range of spanwise wavelengths when the streak amplitude exceeds a critical value. Under neutral conditions, it is shown that streak instability modes have their energy mainly concentrated in the overlap layer and propagate with a phase velocity equal to the mean streamwise velocity of the loglayer. These neutral loglayer modes exhibit a sinuous pattern and have characteristic sizes that are proportional to the wall distance in both streamwise and spanwise directions, in agreement with the Townsend’s attached eddy hypothesis (A. Townsend, the structure of turbulent shear flow, Cambridge university press, 1976 2nd edition). In particular, for a distance from the wall varying from y ^{+} ≈ 100 (in wall units) to y ≈ 0.3h, where h is half the height of the channel, the neutral loglayer modes are selfsimilar with a spanwise width of λz ≈ y/0.3 and a streamwise length of λx ≈ 3λz , independently of the Reynolds number. Based on this observation, it is suggested that compact vortical structures attached to the wall can be ascribed to streak secondary instabilities. In addition, spatial distributions of fluctuating vorticity components show that the onset of secondary instability is associated with the rollup of the shear layer at the edge of the lowspeed streak, similarly to a threedimensional mixing layer.

A minimal model of selfsustaining turbulence
View Description Hide DescriptionIn this work, we examine the turbulence maintained in a Restricted Nonlinear (RNL) model of plane Couette flow. This model is a computationally efficient approximation of the second order statistical state dynamics obtained by partitioning the flow into a streamwise averaged mean flow and perturbations about that mean, a closure referred to herein as the RNL∞ model. The RNL model investigated here employs a single member of the infinite ensemble that comprises the covariance of the RNL∞ dynamics. The RNL system has previously been shown to support selfsustaining turbulence with a mean flow and structural features that are consistent with direct numerical simulations (DNS). Regardless of the number of streamwise Fourier components used in the simulation, the RNL system’s selfsustaining turbulent state is supported by a small number of streamwise varying modes. Remarkably, further truncation of the RNL system’s support to as few as one streamwise varying mode can suffice to sustain the turbulent state. The close correspondence between RNL simulations and DNS that has been previously demonstrated along with the results presented here suggest that the fundamental mechanisms underlying wallturbulence can be analyzed using these highly simplified RNL systems.
 Others

Interaction between a vortex generator and a synthetic jet in a crossflow
View Description Hide DescriptionThe interaction of a combined vortex generator and a finitespan synthetic jet, i.e., a hybrid actuator, with a zero pressure gradient laminar boundary layer over a flat plate was explored experimentally using Stereoscopic Particle Image Velocimetry (SPIV). The free stream velocity was U ∞ = 10 m/s corresponding to a Reynolds number based on the local boundary layer thickness Re δ ≈ 2000. The synthetic jet was activated at multiple blowing ratios, and the vortex generator (placed either upstream or downstream of the synthetic jet) had a height of 1.6 times the local boundary layer thickness. When exposed to the crossflow, the pitched and skewed synthetic jet and vortex generator independently produced a single streamwise vortex in the far field. However, when the combined synthetic jet and vortex generator were placed together on the flat plate, the two streamwise vortices, associated with the two devices, did not combine. When the vortex generator was upstream of the synthetic jet, the jet pushed the vortex generator’s vortex upward into the free stream. When the vortex generator was placed downstream of the synthetic jet, the vortex associated with it was completely destroyed. Although the presence of the vortex generator did not impact the added enstrophy from the synthetic jet, it resulted in higher velocities near the surface when the vortex generator was upstream of the synthetic jet. It was shown that placing the vortex generator upstream of the synthetic jet was imperative for the performance of the hybrid actuator.