Volume 25, Issue 12, December 2013
Index of content:

The convection velocity of large and intermediate scale velocity fluctuations in a nominally twodimensional planar mixing layer, and its dependence upon the length scale, is explored by carrying out particle image velocimetry (PIV) experiments. A “global” convection velocity, containing the convection of all the length scales present in the flow, is produced by examining the autocorrelation functions between velocity fluctuations in successive PIV records across the mixing layer. This “global” convection velocity is found to be similar to the mean flow, although fluctuations on the low speed side of the mixing layer on average convect at speeds greater than the mean and fluctuations on the high speed side of the mixing layer are observed to convect at speeds less than the mean. Scale specific convection velocity profiles are then produced by examining the phase difference between the spectral content specific to one wavenumber in streamwise velocity fluctuation traces in successive PIV records, offset by time τ. Probability density functions (pdfs) are produced of this phase difference, which is subsequently converted into a convection displacement, and these show that the convection of single length scale fluctuations exhibits a significant variance, particularly so for larger scale fluctuations and in the high speed side of the mixing layer. Convection velocity profiles are produced from these pdfs using both the mean convection distance and the modal convection distance. It is observed that the convection velocity is relatively insensitive to the length scale of the fluctuations considered, particularly when the mean convection distance is used. A slight sensitivity to length scale is, however, observed for convection velocities based on the modal convection distance. This dependence is primarily observed in the high speed side of the mixing layer, in which smaller lengthscale fluctuations convect more quickly than larger lengthscale fluctuations. It is also observed that the magnitude of the fluctuation itself affects the convection velocity with larger magnitude fluctuations convecting less rapidly than lower magnitude ones at the largest length scales investigated with this behaviour being reversed at more intermediate length scales.
 ARTICLES


Biofluid Mechanics

Locomotion of a flapping flexible plate
View Description Hide DescriptionThe locomotion of a flapping flexible plate in a viscous incompressible stationary fluid is numerically studied by an immersed boundarylattice Boltzmann method for the fluid and a finite element method for the plate. When the leadingedge of the flexible plate is forced to heave sinusoidally, the entire plate starts to move freely as a result of the fluidstructure interaction. Mechanisms underlying the dynamics of the plate are elucidated. Three distinct states of the plate motion are identified and can be described as forward, backward, and irregular. Which state to occur depends mainly on the heaving amplitude and the bending rigidity of the plate. In the forward motion regime, analysis of the dynamic behaviors of the flapping flexible plate indicates that a suitable degree of flexibility can improve the propulsive performance. Moreover, there exist two kinds of vortex streets in the downstream of the plate which are normal and deflected wake. Further the forward motion is compared with the flappingbased locomotion of swimming and flying animals. The results obtained in the present study are found to be consistent with the relevant observations and measurements and can provide some physical insights into the understanding of the propulsive mechanisms of swimming and flying animals.

Micro and Nanofluid Mechanics

Direct numerical simulation of electrokinetic instability and transition to chaotic motion
View Description Hide DescriptionA new type of instability—electrokinetic instability—and an unusual transition to chaotic motion near a chargeselective surface (semiselective electric membrane, electrode, or system of micro/nanochannels) was studied by the numerical integration of the NernstPlanckPoissonStokes system and a weakly nonlinear analysis near the threshold of instability. A special finitedifference method was used for the space discretization along with a semiimplicit step RungeKutta scheme for the integration in time. Two kinds of initial conditions were considered: (a) whitenoise initial conditions to mimic “room disturbances” and subsequent natural evolution of the solution, and (b) an artificial monochromatic ion distribution with a fixed wave number to simulate regular wave patterns. The results were studied from the viewpoint of hydrodynamic stability and bifurcation theory. The threshold of electroconvective movement was found by the linear spectral stability theory, the results of which were confirmed by numerical simulation of the entire system. Our weakly nonlinear analysis and numerical integration of the entire system predict possibility of both kinds of bifurcations at the critical point, supercritical and subcritical, depending on the system parameters. The following regimes, which replace each other as the potential drop between the selective surfaces increases, were obtained: onedimensional steady solution, twodimensional steady electroconvective vortices (stationary point in a proper phase space), unsteady vortices aperiodically changing their parameters (homoclinic contour), periodic motion (limit cycle), and chaotic motion. The transition to chaotic motion does not include Hopf bifurcation. The numerical resolution of the thin concentration polarization layer showed spikelike charge profiles along the surface, which could be, depending on the regime, either steady or aperiodically coalescent. The numerical investigation confirmed the experimentally observed absence of regular (nearsinusoidal) oscillations for the overlimiting regimes. There is a qualitative agreement of the experimental and the theoretical values of the threshold of instability, the dominant size of the observed coherent structures, and the experimental and theoretical volt–current characteristics.

Thermodiffusion in concentrated ferrofluids: A review and current experimental and numerical results on nonmagnetic thermodiffusion
View Description Hide DescriptionFerrofluids are colloidal suspensions consisting of magnetic nanoparticles dispersed in a carrier liquid. Their thermodiffusive behaviour is rather strong compared to molecular binary mixtures, leading to a Soret coefficient (S T ) of 0.16 K^{−1}. Former experiments with dilute magnetic fluids have been done with thermogravitational columns or horizontal thermodiffusion cells by different research groups. Considering the horizontal thermodiffusion cell, a former analytical approach has been used to solve the phenomenological diffusion equation in one dimension assuming a constant concentration gradient over the cell's height. The current experimental work is based on the horizontal separation cell and emphasises the comparison of the concentration development in different concentrated magnetic fluids and at different temperature gradients. The ferrofluid investigated is the kerosenebased EMG905 (Ferrotec) to be compared with the APG513A (Ferrotec), both containing magnetite nanoparticles. The experiments prove that the separation process linearly depends on the temperature gradient and that a constant concentration gradient develops in the setup due to the separation. Analytical one dimensional and numerical three dimensional approaches to solve the diffusion equation are derived to be compared with the solution used so far for dilute fluids to see if formerly made assumptions also hold for higher concentrated fluids. Both, the analytical and numerical solutions, either in a phenomenological or a thermodynamic description, are able to reproduce the separation signal gained from the experiments. The Soret coefficient can then be determined to 0.184 K^{−1} in the analytical case and 0.29 K^{−1} in the numerical case. Former theoretical approaches for dilute magnetic fluids underestimate the strength of the separation in the case of a concentrated ferrofluid.

Interfacial Flows

The role of the droplet deformations in the bouncing droplet dynamics
View Description Hide DescriptionDroplets bouncing on a vibrated liquid bath open ways to methods of manipulating droplets, creating double emulsion, and performing pilot wave model experiments. In this work, we focus on the role of the droplet deformations in the vertical bouncing dynamics by neglecting the deformation of the surface of the bath. To be under this favorable condition, low viscous oil droplets are dropped over a highly viscous oil bath that is vibrated. These droplets bounce vertically on the surface of the bath and exhibit many periodic trajectories and resonant modes when tuning the forcing parameters, i.e., the oscillation of the bath. This complex dynamics emphasizes the interplay between elastic energy storage and energy dissipation in droplets at each bounce. We propose to model droplets using a bouncing massspringdamper system that mimics a deformable droplet bouncing on a nondeformable liquid bath. From the experimental measurements, we constructed bifurcation diagrams of the bouncing trajectories and challenged our bouncing spring model. The agreement between experiment and the spring model reveals that this model can be used to rationalize and predict a variety of bouncing droplets behaviors involving multiperiodicities.

Application of Floquet theory to the stability of liquid films on structured surfaces
View Description Hide DescriptionWe consider instability of a liquid film on a substrate structured by an array of gasfilled grooves. The instability is driven by disjoining pressure, while the effect of structuring on viscous flow in the film is modeled by a squarewave variation of the slip length along the substrate. Linear stability criteria are established analytically using Floquet theory and compared with the predictions of a straightforward numerical approach, all in the framework of a lubricationtype model. Then, stability is analyzed for a more general model based on Stokes flow approximation; validity of the lubricationtype approach is discussed. The structuring is found to enhance the instability for a wide range of conditions. Resonant interaction between the interfacial deformations and the substrate structuring pattern leads to discontinuities in the dispersion curves, a situation analogous to appearance of gaps in the energy spectra seen in the applications of Floquet theory in solid state physics.

Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water
View Description Hide DescriptionUsing a Boussinesq model with improved linear dispersion, we show numerical evidence that bottom nonuniformity can provoke significantly increased probability of freak waves as a wave field propagates into shallower water, in agreement with recent experimental results [K. Trulsen, H. Zeng, and O. Gramstad, “Laboratory evidence of freak waves provoked by nonuniform bathymetry,” Phys. Fluids24, 097101 (2012)]. Increased values of skewness, kurtosis, and probability of freak waves can be found on the shallower side of a bottom slope, with a maximum close to the end of the slope. The increased probability of freak waves is typically seen to endure some distance into the shallower domain, before it decreases and reaches a stable value depending on the depth. The maxima of the statistical parameters are observed both in the case where there is a region of constant depth after the slope, and in the case where the uphill slope is immediately followed by a downhill slope. In the case that waves propagate over a slope from shallower to deeper water, however, we do not find any increase in freak wave occurrence.

An experimental investigation of fingering instabilities and growth dynamics in inclined countercurrent gasliquid channel flow
View Description Hide DescriptionThe results of an experimental study involving low Reynolds number, countercurrent flows of glycerol and air on an inclined glass substrate inside a rectangular channel are presented. The interface forms a thickened front immediately upstream of a thin, precursor layer region. This front is vulnerable to spanwise perturbations, which, under certain conditions, grow to acquire the shape of “fingers.” Decreasing the inclination angle has a stabilizing effect on the front; complete stability is achieved below a critical angle whose value depends on the remaining system parameters. Regions of transient finger formation are also observed. It is also found that increasing the ratio of the precursor to the inlet film thickness, and increasing the liquid and air flowrates also exerts a stabilizing effect on the interface. Analyses of the initial finger growthrate corroborate the findings of previous theoretical work, showing this growthrate to be independent of inclination angle and liquid film Reynolds number, and weaklydependent on the air flowrate for low inclination angles. Both qualitative and quantitative agreement with theoretical studies from the literature was also found, in terms of the effects of flow parameters and the observed dynamics of the developing fingers.

Viscous and NonNewtonian Flows

Analysis of energy balance during collision of an air bubble with a solid wall
View Description Hide DescriptionThe dynamics of bubblewall collision is studied by means of numerical simulations to elucidate the mechanism of bubble rebound at a solid, noslip wall in viscous liquid. Results obtained are compared with experimental data as well as data reported in the literature. Similarities and differences are discussed. Bubble trajectory, shape deformation, added mass variation as a function of distance from the wall, and relations between various forms of energy in the system during bubble impact, liquid film formation, and rebound are presented and analyzed. On the basis of this, collision time is quantitatively defined as a time interval during which pronounced changes of kinetic energy are observed. For a rising bubble colliding with a horizontal wall, series of collisions are observed, each associated with dissipation of kinetic energy, mainly in the thin film formed between the bubble and the wall.

Particulate, Multiphase, and Granular Flows

Integral momentum balance on a growing bubble
View Description Hide DescriptionThe integral momentum balance on a growing boiling bubble is investigated. All forces acting on the bubble are detailed, and the methods and assumptions used to calculate their integral resultants are discussed. The momentum balance computation is then performed using experimental data of bubbles growing on an artificial nucleation site in a controlled environment. The relative magnitude of each force component is compared, showing negligible dynamic forces, upwards forces composed mainly of the buoyancy and contact pressure components, and downwards forces being exclusively due to surface tension and adhesion. The difficulty encountered in measuring the apparent contact angle due to mirage effects has been highlighted; a new method, fitting numerically simulated bubble profile to the contour measurements has been proposed and used to correct the effects of refraction on the bubble profile determination. As all forces acting on the bubble were measured, it was possible to estimate the residuals of the momentum balance. Their small value validated both the expressions used for the forces and the methodology to evaluate their value.

A hybrid stochasticdeconvolution model for largeeddy simulation of particleladen flow
View Description Hide DescriptionWe develop a hybrid model for largeeddy simulation of particleladen turbulent flow, which is a combination of the approximate deconvolution model for the resolved scales and a stochastic model for the subgrid scales. The stochastic model incorporates a priori results of direct numerical simulation of turbulent channel flow, which showed that the parameters in the stochastic model are quite independent of Reynolds and Stokes number. In order to correctly predict the flux of particles towards the walls an extra term should be included in the stochastic model, which corresponds to the term related to the wellmixed condition in Langevin models for particle dispersion in inhomogeneous turbulent flow. The model predictions are compared with results of direct numerical simulation of channel flow at a frictional Reynolds number of 950. The inclusion of the stochastic forcing is shown to yield a significant improvement over the approximate deconvolution model for the particles alone when combined with a Stokes dependent weightfactor for the wellmixed term.

Dry granular avalanche down a flume: Choice of discrete element simulation parameters
View Description Hide DescriptionThis paper presents a method to assign softsphere contact model parameters in a discreteelement simulation with which we can reproduce the experimentally measured avalanche dynamics of finite dry granular mass down a flume. We adopt the simplest linear model in which interaction force is decomposed along or tangent to the contact normal. The model parameters are chosen uniquely to satisfy theoretical models or to meet experimental evidences at either the particle or the bulk size level. The normal mode parameters are chosen specifically to ensure Hertzian contact time (but not its forcedisplacement history) and the resulting loss of particle kinetic energy, characterized by a measured coefficient of restitution, for each pair of colliding surfaces. We follow the literature to assign the tangential spring constant according to an elasticity model but propose a method to assign the friction coefficient using a measured bulk property that characterizes the bulk discharge volume flow rate. The linear contact model with the assigned parameters are evaluated by comparing the simulated bulk avalanche dynamics down three slopes to the experimental data, including instantaneous particle trajectories and bulk unsteady velocity profile. Satisfying quantitative agreement can be obtained except at the free surface and the earlytime front propagation velocity.

The effect of neutrally buoyant finitesize particles on channel flows in the laminarturbulent transition regime
View Description Hide DescriptionThe presence of finitesize particles in a channel flow close to the laminarturbulent transition is simulated with the Force Coupling Method which allows twoway coupling with the flow dynamics. Spherical particles with channel heighttoparticle diameter ratio of 16 are initially randomly seeded in a fluctuating flow above the critical Reynolds number corresponding to single phase flow relaminarization. When steadystate is reached, the particle volume fraction is homogeneously distributed in the channel crosssection (ϕ ≅ 5%) except in the nearwall region where it is larger due to inertiadriven migration. Turbulence statistics (intensity of velocity fluctuations, smallscale vortical structures, wall shear stress) calculated in the fully coupled twophase flow simulations are compared to singlephase flow data in the transition regime. It is observed that particles increase the transverse r.m.s. flow velocity fluctuations and they break down the flow coherent structures into smaller, more numerous and sustained eddies, preventing the flow to relaminarize at the singlephase critical Reynolds number. When the Reynolds number is further decreased and the suspension flow becomes laminar, the wall friction coefficient recovers the evolution of the laminar singlephase law provided that the suspension viscosity is used in the Reynolds number definition. The residual velocity fluctuations in the suspension correspond to a regime of particulate shearinduced agitation.

Laminar Flows

Flow induced vibration of two rigidly coupled circular cylinders in tandem and sidebyside arrangements at a low Reynolds number of 150
View Description Hide DescriptionFlow induced vibration of two rigidly coupled identical circular cylinders in tandem and sidebyside arrangements at a low Reynolds number of 150 is studied numerically. The two cylinders vibrate in the crossflow direction and have the same displacement. The NavierStokes equations are solved by the finite element method and the equation of motion of the cylinders is solved by the fourthorder RungeKutta algorithm. Simulations are conducted for a constant mass ratio of 2 and the gap ratios (defined as the ratio of the centretocentre distance between the two cylinders L to the cylinder diameter D) of 1.5, 2, 4, and 6. The reduced velocities range from 0.5 to 15 with an increment of 0.5 for the tandem arrangement and from 0.5 to 30 with an increment of 0.5 for the sidebyside arrangement. It is found that the gap between the two cylinders has significant effect on the response. For a tandem arrangement, the lockin regime of the reduced velocity is narrower than that of a single cylinder for L/D = 1.5 and 2 and wider than later for L/D = 4 and 6. If the two cylinders are allowed to vibrate in the crossflow direction, the vortex shedding from the upstream cylinder occurs at L/D as small as 2. The most interesting phenomenon found in the sidebyside arrangement is the combination of vortexinduced vibration (VIV) and galloping at L/D = 1.5 and 2. For L/D = 1.5 and 2, the response is dominated by VIV as V r<15 and by galloping as V r>15. At reduced velocities close to 15, the response is a combination of VIV and galloping.

Revisiting the analytical solution for temperature in Poiseuille flow of an ideal gas
View Description Hide DescriptionThe discrepancy between the numerical solution to the NavierStokes equations for a pressuredriven Poiseuille flow in an ideal gas and the classical analytical solution for temperature is examined. A minimum temperature at the center is observed in the numerical solution to the NavierStokes equations but the analytical solution predicts a maximum temperature at the center. It is found that the term should not be neglected in the reduced energy equation in the analytical method when the fluid under consideration is an ideal gas. When the term is retained, a minimum instead of a maximum temperature arises at the center in the new analytical solution as well. On the other hand, when the fluid is a liquid, the reduced energy equation and the solution assume the classical form, and thus the solution in fact has a maximum at the center.

A pressuregradient mechanism for vortex shedding in constricted channels
View Description Hide DescriptionNumerical simulations of the unsteady, twodimensional, incompressible Navier–Stokes equations are performed for a Newtonian fluid in a channel having a symmetric constriction modeled by a twoparameter Gaussian distribution on both channel walls. The Reynolds number based on inlet halfchannel height and mean inlet velocity ranges from 1 to 3000. Constriction ratios based on the halfchannel height of 0.25, 0.5, and 0.75 are considered. The results show that both the Reynolds number and constriction geometry have a significant effect on the behavior of the postconstriction flow field. The Navier–Stokes solutions are observed to experience a number of bifurcations: steady attached flow, steady separated flow (symmetric and asymmetric), and unsteady vortex shedding downstream of the constriction depending on the Reynolds number and constriction ratio. A sequence of events is described showing how a sustained spatially growing flow instability, reminiscent of a convective instability, leads to the vortex shedding phenomenon via a proposed streamwise pressuregradient mechanism.

Instability and Transition

Global and Koopman modes analysis of sound generation in mixing layers
View Description Hide DescriptionIt is now well established that linear and nonlinear instability waves play a significant role in the noise generation process for a wide variety of shear flows such as jets or mixing layers. In that context, the problem of acoustic radiation generated by spatially growing instability waves of twodimensional subsonic and supersonic mixing layers are revisited in a global point of view, i.e., without any assumption about the base flow, in both a linear and a nonlinear framework by using global and Koopman mode decompositions. In that respect, a timestepping technique based on disturbance equations is employed to extract the most dynamically relevant coherent structures for both linear and nonlinear regimes. The present analysis proposes thus a general strategy for analysing the nearfield coherent structures which are responsible for the acoustic noise in these configurations. In particular, we illustrate the failure of linear global modes to describe the noise generation mechanism associated with the vortex pairing for the subsonic regime whereas they appropriately explain the Mach wave radiation of instability waves in the supersonic regime. By contrast, the Dynamic Mode Decomposition (DMD) analysis captures both the nearfield dynamics and the farfield acoustics with a few number of modes for both configurations. In addition, the combination of DMD and linear global modes analyses provides new insight about the influence on the radiated noise of nonlinear interactions and saturation of instability waves as well as their interaction with the mean flow.

Suppression of purely elastic instabilities in the torsional flow of viscoelastic fluid past a soft solid
View Description Hide DescriptionExperiments are performed to explore the role of a soft, deformable solid layer on the purely elastic instability in the torsional flow of polymer solutions between two circular discs. The gel layer is placed on the stationary bottom plate of a rheometer, and the polymer solution is placed between the gel and the rotating top disc. The observed variation of viscosity with shear rate (or shear stress) is correlated with the presence or absence of purely elastic instability in the viscometric flow. Earlier work has shown that with increase in shear rate, the torsional flow of a polymer solution between rigid discs undergoes transition from the simple viscometric flow state to elastic turbulence via a sequence of instability modes. We combine rheological observations and flow visualization to show that the deformable solid has a profound effect on the stability of the torsional flow. In marked contrast to flow between rigid plates (where the fluid shows apparent shearthickening at the onset of instability), the apparent viscosity continues to decrease up to a much larger value of shear rate with the presence of a soft gel. At a fixed shear rate, for flow past a soft gel, the measured stress does not exhibit marked temporal fluctuations that would otherwise be present without the soft gel. Using flow visualization, we show that secondary flow patterns that form after the instability for a rigid surface disappear for flow on soft gel surfaces. In the case of rigid surfaces, the instability is subcritical and exhibits hysteresis behavior, which again is absent when the flow occurs past a soft solid layer. Our results show that the role of the soft deformable solid is to suppress the purely elastic instability in torsional flows of polymeric liquids for intermediate shear rates. While it is known that soft deformable solids destabilize the flow of Newtonian liquids in the absence of inertial effects, our study shows that the effect of deformability can be opposite in the torsional flow of viscoelastic liquids.

Stability of two immiscible leakydielectric liquids subjected to a radial electric field in an annulus duct
View Description Hide DescriptionIn this paper, we investigated the stability of a two coaxial leaky dielectric fluid system flowing in an annulus duct. A constant pressure gradient was applied to drive the flow in the duct. A radial electric field was imposed between the outer and inner surfaces of the duct. Linear stability analysis was employed to discuss the influences of electric field on the capillary and interface wave instabilities. The former instability is caused by surface tension and the latter is caused by viscosity stratification at the interface. It was found that, depending on the electrical permittivities and conductivities of the two liquids, the electric field either stabilized or destabilized the flow system. Apart from that, it was found that an external electric field could impede the capillary and interface wave instabilities. Influences of the inner radius of the duct, viscosity ratio, thickness ratio, and Reynolds number on the stability of the system were discussed as well.

On the transient phase of the Faraday instability
View Description Hide DescriptionThis study pertains to the threedimensional direct numerical simulation (DNS) of a vertically oscillating vessel containing an incompressible Newtonian liquid, surrounded by air at rest and ambient conditions. Squire's theorem was extended and shown to apply in this case, allowing for the theory of linear stability to be implemented and a comparison to be made with the DNS results. It was further discovered that the method by which a fluid instability is initiated in the numerical simulation affects the initial development of the instability. This phenomenon was confirmed through an optimal perturbations analysis. A possible physical explanation of this effect is also presented.

Localized rotating convection with noslip boundary conditions
View Description Hide DescriptionLocalized patches of stationary convection embedded in a background conduction state are called convectons. Multiple states of this type have recently been found in twodimensional Boussinesq convection in a horizontal fluid layer with stressfree boundary conditions at top and bottom, and rotating about the vertical. The convectons differ in their lengths and in the strength of the selfgenerated shear within which they are embedded, and exhibit slanted snaking. We use homotopic continuation of the boundary conditions to show that similar structures exist in the presence of noslip boundary conditions at the top and bottom of the layer and show that such structures exhibit standard snaking. The homotopic continuation allows us to study the transformation from slanted snaking characteristic of systems with a conserved quantity, here the zonal momentum, to standard snaking characteristic of systems with no conserved quantity.
