Volume 21, Issue 4, April 2009

The effect of nonlinear interaction of instabilityeigenmodes on jet flow transition and its near acoustic field for a highsubsonic round jet at a Reynolds number of and a Mach number of is investigated using largeeddy simulations. At the inflow, helical perturbations of azimuthal wavenumbers determined from linear stability theory are superimposed on a laminar base flow in order to trigger transition to turbulence. The disturbance amplitude is varied parametrically in the range from 1.5% to 4.5% of the jet exit velocity . Thereby we aim to characterize sources of noise generation and, in particular, underlying mode interactions. With increasing forcing amplitude, the transitional behavior of the jet changes which affects the mean flow and also the acoustic nearfield, which are both analyzed in detail. As the forcing amplitude is increased, the axial rootmeansquare peak levels along the jet centerline are reduced by approximately 7%. Simultaneously, pronounced dualpeak distributions are generated along the jet lip line which are related to the localization of vortex pairings of the jet column mode. For lowamplitude excitation the azimuthal turbulent kinetic energy spectra show that the unexcited, naturally least stable axisymmetric mode and the helical mode dominate the early nonlinear regimes between and where is the jet radius. An analysis of the Fourier mode amplitude clarifies that this energy rise is linked to the helical mode . For higher forcing amplitudes, in addition to the varicose mode interactions between the excited even mode and higher azimuthal harmonics thereof dominate the azimuthal energy spectra. These differences in the early nonlinear development of the eigenmodes are found to alter the acoustic nearfield. At small angles from the downstream jet axis, the peak acoustic frequency occurs at a Strouhal number based on the angular frequency and the jet diameter of . For lowamplitude forcing sound pressure levels are slightly enhanced which can be linked to the dominant low azimuthal wavenumbers identified in the transitional region. In the sideline direction, regardless of the excitation level, broadbanded spectra with maxima in the band are found which is maintained at intermediate observer angles. For high forcing amplitude, however, a tonal component outside the initially excited frequency range is observed. This peak at can be explained by weakly nonlinear interactions of initially forced eigenmodes and together with the jet column mode.
 ANNOUNCEMENTS
 LETTERS


On the unsteady behavior of turbulence models
View Description Hide DescriptionPeriodically forced turbulence is used as a test case to evaluate the predictions of twoequation and multiplescale turbulencemodels in unsteady flows. The limitations of the twoequation model are shown to originate in the basic assumption of spectral equilibrium. A multiplescale model based on a picture of stepwise energy cascade overcomes some of these limitations, but the absence of nonlocal interactions proves to lead to poor predictions of the time variation of the dissipation rate. A new multiplescale model that includes nonlocal interactions is proposed and shown to reproduce the main features of the frequency response correctly.
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 ARTICLES

 Biofluid Mechanics

Extracting energetically dominant flow features in a complicated fish wake using singularvalue decomposition
View Description Hide DescriptionWe developed a method to extract the energetically dominant flow features in a complicated fish wake according to an energetic point of view, and applied singularvalue decomposition (SVD) to twodimensional instantaneous fluid velocity, vorticity and (vortexdetector) data. We demonstrate the effectiveness and merits of the use of SVD through an example regarding the wake of a fish executing a faststart turn. The energy imparted into the water by a swimming fish is captured and portrayed through SVD. The analysis and interpretation of complicated data for the fish wake are greatly improved, and thus help to characterize more accurately a complicated fish wake. The velocity vectors and Galilean invariants (i.e., vorticity and ) resulting from SVD extraction are significantly helpful in recognizing the energetically dominant largescale flow features. To obtain successful SVD extractions, we propose useful criteria based on the Froude propulsion efficiency, which is biologically and physically related. We also introduce a novel and useful method to deduce the topology of dominant flowmotions in an instantaneous fish flow field, which is based on combined use of the topological criticalpoint theory and SVD. The concept and approach proposed in this work are useful and adaptable in biomimetic and biomechanical research concerning the fluid dynamics of a selfpropelled body.
 Micro and Nanofluid Mechanics

The influence of hydrodynamic slip on the electrophoretic mobility of a spherical colloidal particle
View Description Hide DescriptionRecent theoretical studies have suggested a significant enhancement in electroosmotic flows over hydrodynamically slipping surfaces, and experiments have indeed measured enhancements. In this paper, we investigate whether an equivalent effect occurs in the electrophoretic motion of a colloidal particle whose surface exhibits hydrodynamic slip. To this end, we compute the electrophoreticmobility of a uniformly charged spherical particle with slip length as a function of the zeta (or surface) potential of the particle and diffuselayer thickness . In the case of a thick diffuse layer, (where is the particle size), simple arguments show that slip does lead to an enhancement in the mobility, owing to the reduced viscous drag on the particle. On the other hand, for a thindiffuse layer , the situation is more complicated. A detailed asymptotic analysis, following the method of O’Brien [J. Colloid Interface Sci.92, 204 (1983)], reveals that an increase in the mobility occurs at lowtomoderate zeta potentials (with measured on the scale of thermal voltage ). However, as is further increased, the mobility decreases and ultimately becomes independent of the slip length—the enhancement is lost—which is due to the importance of nonuniform surface conduction within the thindiffuse layer, at large and large, but finite, . Our asymptotic calculations for thick and thindiffuse layers are corroborated and bridged by computation of the mobility from the numerical solution of the full electrokinetic equations (using the method of O’Brien and White [J. Chem. Soc., Faraday Trans. 274, 1607 (1978)]). In summary, then, we demonstrate that hydrodynamic slip can indeed produce an enhancement in the electrophoreticmobility; however, such enhancements will not be as dramatic as the previously studied limit would suggest. Importantly, this conclusion applies not only to electrophoresis but also to electroosmosis over highly charged surfaces, wherein any inhomogeneities (e.g., due to curvature, roughness, charge patterning, or a variation in slip length) will drive nonuniform surface conduction, which prevents the significant slipdriven flow enhancements predicted for a uniform highly charged surface.

Electroosmotic flow in a sector microchannel
View Description Hide DescriptionThe study presents analytical solutions of electroosmotic flow in a sector microchannel with particular emphasis on the corner effects. The analytical solutions are obtained by the method of eigenfunction expansions in sequel for steady state, starting transient flow and oscillatory flow. It is shown that the smaller the opening angle is, the faster the flow approaches to steady state. For large nondimensional electrokinetic lengths, especially near an obtuse corner, a core of maximum velocity is soon established after the flow is started. The oscillatory flow exhibits phase lags between different modes that depend on their eigenvalues as well as the frequency of the externally applied voltage, hence distinguishing itself significantly from the steady state solution. This fact has an important physical implication about fast mixing of electrolytes in microchannels.

Swirl flow focusing: A novel procedure for the massive production of monodisperse microbubbles
View Description Hide DescriptionA volumeoffluid numerical method is used to predict the dynamics of microbubbleformation in an axisymmetric flowfocusing microfluidic device for a gasliquid system. Numerical results show that, in all the cases analyzed, the introduction of swirl into the focusing liquid stabilizes the tapering gasliquidmeniscus from which a steady gas ligament issues. Consequently, a drastic reduction in the size of bubbles generated by the device is achieved under similar gas and liquidflow rates.
 Interfacial Flows

On the planar extensional motion of an inertially driven liquid sheet
View Description Hide DescriptionWe derive a timedependent exact solution of the free surface problem for the Navier–Stokes equations that describes the planar extensional motion of a viscoussheet driven by inertia. The linear stability of the exact solution to one and twodimensional symmetric perturbations is examined in the inviscid and viscous limits within the framework of the longwave or slender body approximation. Both transient growth and longtime asymptotic stability are considered. For onedimensional perturbations in the axial direction, viscous and inviscid sheets are asymptotically marginally stable, though depending on the Reynolds and Weber numbers transient growth can have an important effect. For onedimensional perturbations in the transverse direction, inviscid sheets are asymptotically unstable to perturbations of all wavelengths. For twodimensional perturbations, inviscid sheets are unstable to perturbations of all wavelengths with the transient dynamics controlled by axial perturbations and the longtime dynamics controlled by transverse perturbations. The asymptotic stability of viscoussheets to onedimensional transverse perturbations and to twodimensional perturbations depends on the capillary number (Ca); in both cases, the sheet is unstable to longwave transverse perturbations for any finite Ca.

Analytical solution for Stokes flow inside an evaporating sessile drop: Spherical and cylindrical cap shapes
View Description Hide DescriptionExact analytical solutions are derived for the Stokes flows within evaporating sessile drops of spherical and cylindrical cap shapes. The results are valid for all contact angles. Solutions are obtained for arbitrary evaporative flux distributions along the free surface as long as the flux is bounded at the contact line. Specific results and computations are presented for evaporation corresponding to uniform flux and to purely diffusive gas phase transport into an infinite ambient. Wetting and nonwetting contact angles are considered with the flow patterns in each case being illustrated. For the spherical cap with evaporation controlled by vapor phase diffusion, when the contact angle lies in the range , the mass flux of vapor becomes singular at the contact line. This condition requires modification when solving for the liquidphase transport. Droplets in all of the above categories are considered for the following two cases: the contact lines are either pinned or free to move during evaporation. The present viscousflow behavior is compared to the inviscid flow behavior previously reported. It is seen that the streamlines for viscousflow lie farther from the substrate than the corresponding inviscid ones.

Direct numerical simulation of the nearfield dynamics of annular gasliquid twophase jets
View Description Hide DescriptionDirect numerical simulation has been used to examine the nearfield dynamics of annular gasliquid twophase jets. Based on an Eulerian approach with mixed fluid treatment, combined with an adapted volume of fluid method and a continuum surface force model, a mathematical formulation for the flow system is presented. The swirl introduced at the jet nozzle exit is based on analytical inflow conditions. Highly accurate numerical methods have been utilized for the solution of the compressible, unsteady, Navier–Stokes equations. Two computational cases of gasliquid twophase jets including swirling and nonswirling cases have been performed to investigate the effects of swirl on the flow field. In both cases the flow is more vortical at the downstream locations. The swirling motion enhances both the flow instability resulting in a larger liquid spatial dispersion and the mixing resulting in a more homogeneous flow field with more evenly distributed vorticity at the downstream locations. In the annular nonswirling case, a geometrical recirculation zone adjacent to the jet nozzle exit was observed. It was identified that the swirling motion is responsible for the development of a central recirculation zone, and the geometrical recirculation zone can be overwhelmed by the central recirculation zone leading to the presence of the central recirculation region only in the swirling gasliquid case. Results from a swirling gas jet simulation were also included to examine the effect of the liquid sheet on the flow physics. The swirling gas jet developed a central recirculation region, but it did not develop a precessing vortex core as the swirling gasliquid twophase jet. The results indicate that a precessing vortex core can exist at relatively low swirl numbers in the gasliquid twophase flow. It was established that the liquid greatly affects the precession and the swirl number alone is an insufficient criterion for the development of a precessing vortex core.

Linear stability analysis and numerical simulation of miscible twolayer channel flow
View Description Hide DescriptionThe stability of miscible twofluid flow in a horizontal channel is examined. The flowdynamics are governed by the continuity and Navier–Stokes equations coupled to a convectivediffusion equation for the concentration of the more viscous fluid through a concentrationdependent viscosity. Our analysis of the flow in the linear regime delineates the presence of convective and absolute instabilities and identifies the vertical gradients of viscosity perturbations as the main destabilizing influence in agreement with previous work. Our transient numerical simulations demonstrate the development of complex dynamics in the nonlinear regime, characterized by rollup phenomena and intense convective mixing; these become pronounced with increasing flow rate and viscosity ratio, as well as weak diffusion.

Thermocapillary migration of interfacial droplets
View Description Hide DescriptionWe study the thermocapillary driven motion of a droplet suspended at an interface of two fluid layers subjected to an imposed temperature gradient parallel to the interface. We compute the temperature and velocity fields inside and outside of the droplet using a boundary collocation numerical scheme in the limit of small capillary and thermal Péclet numbers and compare the results with the classical problem of thermocapillary migration of a droplet in the bulk. In particular, we find that, for typical values of parameters, interfacial droplets migrate in the direction opposite to the temperature gradient, while in the classical problem migration is always in the direction of the gradient. Furthermore, we find that a rich variety of flow structures can emerge inside interfacial droplets. We also confirm that for parameters matching a recent experimental study of mixing inside interfacial microdroplets [R. O. Grigoriev, V. Sharma, and M. F. Schatz, Lab Chip6, 1369 (2006)] the interior flow can be approximated with reasonable accuracy by assuming the droplet to be completely submerged in the bottom layer.

Gravity effects on capillary flows in sharp corners
View Description Hide DescriptionWe analyze the effect of gravity on capillary flows in sharp corners. We consider gravity perpendicular and parallel to the channel axis. We analyze both steady and unsteady flows. In the steady analysis the main result is a closed form expression for the flow rate as a function of the two gravity components. Good agreement with steady experiments is offered as support of the model. The unsteady analysis is restricted to “small” values of the two gravity parameters and is accomplished using a similarity formulation. The similarity coefficients of the gravity corrections are fully determined by the coefficients of the gravityless problem. The main result of the unsteady analysis is the gravity corrections to the flow rate (or rate of advance) of the liquid in the channel. In addition, we obtain corrections for the liquid height as a function of position and time. We address in detail unsteady problems with select boundary conditions that are representative of typical flow types. In Appendix A we present a new exact solution to one of the gravityless similarity cases, which is analogous to a nonlinear heat conduction equation. In Appendix B we offer dimensional formulas for all the unsteady flow results, which are valuable for systems design and analysis.

Merging and colliding bores
View Description Hide DescriptionInteractions between two undular bores in a long rectangular channel are studied by solving the twodimensional Laplace equation with fully nonlinear freesurface conditions. Each bore is generated by a steady bottom source or sink that is turned on impulsively. Two types of nonlinear interaction between bores are investigated: (1) The merging of two bores that move in the same direction. (2) The collision of two bores that move in opposite directions.

Numerical investigation of the stability of bubble train flow in a square minichannel
View Description Hide DescriptionThe stability of a train of equally sized and variably spaced gas bubbles that move within a continuous wetting liquid phase through a straight square minichannel is investigated numerically by a volumeoffluid method. The flow is laminar and cocurrent upward and driven by a pressure gradient and buoyancy. The simulations start from fluid at rest with two identical bubbles placed on the axis of the computational domain, the size of the bubbles being comparable to that of the channel. In vertical direction, periodic boundary conditions are used. These result in two liquid slugs of variable length, depending on the initial bubbletobubble distance. The time evolution of the length of both liquid slugs during the simulation indicates if the bubble train flow is “stable” (equal terminal length of both liquid slugs) or “unstable” (contact of both bubbles). Several cases are considered, which differ with respect to bubble size, domain size, initial bubble shape, and separation. All cases lead to axisymmetric bubbles with the capillary number in the range of 0.11–0.23. The results show that a recirculation pattern develops in the liquid slug when its length exceeds a critical value that is about 10%–20% of the channel width. If a recirculation pattern exists in both liquid slugs, then the bubble train flow is stable. When there is a recirculation pattern in one liquid slug and a bypass flow in the other, the bubble train flow may be stable or not depending on the local flow field in the liquid slugs close to the channel centerline. These results suggest that a general criterion for the stability of bubble train flow cannot be formulated in terms of the capillary and Reynolds number only, but must take into account the length of the liquid slug.

Spatial evolution of a film flowing down a fiber
View Description Hide DescriptionWe report the response to a forcing at the inlet of a film flowing down a vertical fiber. Parameters are chosen in order to examine the effects of both inertia and surface tension. The spatial response of the film to inlet forcing depends on the ratio of the forcing frequency to the frequency corresponding to the maximum linear growth rate. At , the primary instability leads directly to the formation of a saturated wavetrain at the forcing frequency, whereas at low forcing frequencies , this formation is preceded by a sequence of periodic coalescence events. The amplitude, speed, profiles, and inner flow pattern of traveling waves have been characterized and compared to the solutions to the twoequation model obtained by RuyerQuil et al. [J. Fluid Mech.603, 431 (2008)], showing a remarkable agreement. A steepening of the waves is observed when inertia becomes dominant. An excellent correlation between data is observed when the amplitude and speed of the waves are made dimensionless with reference to the substrate thickness and freesurface velocity.
 Particulate, Multiphase, and Granular Flows

Flowing grains in an inclined duct
View Description Hide DescriptionLarge scale, threedimensional computer simulations were performed to investigate flow dynamics of monosized, viscoelastic, spherical solid particles past a stationary wedge located in the middle of an inclined duct. At low flow rates of solid particles, a continuous flow was observed similar to that excited by steadily and rapidly adding particles to the top of a heap. However, at high flow rates, a totally different situation arises, where a flow with a different nature was established in the duct. In this case, the granular flow within the upper part of the duct accelerates adjacent to the pointed tip of the wedge, and develops into vast masses of solid particles thrust and folded over each other. This is similar to the supercritical nappes in an openchannel flow of a liquid. In addition, some experimental evidences have been presented that suggest the existence of supercritical nappes in flowing grains over a stationary wedge within an inclined duct at high flow rates.

Locating an atmospheric contamination source using slow manifolds
View Description Hide DescriptionFinitesize particle motion in fluids obeys the Maxey–Riley equations, which become singular in the limit of infinitesimally small particle size. Because of this singularity, finding the source of a dispersed set of small particles is a numerically illposed problem that leads to exponential blowup. Here we use recent results on the existence of a slow manifold in the Maxey–Riley equations to overcome this difficulty in source inversion. Specifically, we locate the source of particles by projecting their dispersed positions on a timevarying slow manifold, and by advecting them on the manifold in backward time. We use this technique to locate the source of a hypothetical anthrax release in an unsteady threedimensional atmospheric wind field in an urban street canyon.

Simulation of a particleladen turbulent channel flow using an improved stochastic Lagrangian model
View Description Hide DescriptionThe purpose of this paper is to examine the Lagrangian stochastic modeling of the fluid velocity seen by inertial particles in a nonhomogeneous turbulent flow. A new Langevintype model, compatible with the transport equation of the drift velocity in the limits of low and high particle inertia, is derived. It is also shown that some previously proposed stochastic models are not compatible with this transport equation in the limit of high particle inertia. The drift and diffusion parameters of these stochastic differential equations are then estimated using direct numerical simulation (DNS) data. It is observed that, contrary to the conventional modeling, they are highly space dependent and anisotropic. To investigate the performance of the present stochastic model, a comparison is made with DNS data as well as with two different stochastic models. A good prediction of the first and second order statistical moments of the particle and fluid seen velocities is obtained with the three models considered. Even for some components of the triple particle velocity correlations, an acceptable accordance is noticed. The performance of the three different models mainly diverges for the particle concentration and the drift velocity. The proposed model is seen to be the only one which succeeds in predicting the good evolution of these latter statistical quantities for the range of particle inertia studied.

Fingerprints of random flows?
View Description Hide DescriptionWe consider the patterns formed by small rodlike objects advected by a random flow in two dimensions. An exact solution indicates that their direction field is nonsingular. However, we find from simulations that the direction field of the rods does appear to exhibit singularities. First, “scar lines” emerge where the rods abruptly change direction by . Later, these scar lines become so narrow that they “heal over” and disappear, but their ends remain as point singularities, which are of the same type as those seen in fingerprints. We give a theoretical explanation for these observations.
 Laminar Flows

Secondary flow behavior in a double bifurcation
View Description Hide DescriptionSecondary flows in the form of multivortex structures can occur in bifurcationmodels as the result of upstream influence. Results from numerical modeling of steady inspiratory flows indicate that, for the case of a symmetric planar double bifurcation, four counterrotating vortices develop in each of the grand daughter branches. In this paper, experimental visualization and verification is provided by particle imagevelocimetrymeasurements on a modified single bifurcationmodel. A splitter plate was positioned in the mother tube so that secondary vorticity was introduced into the fluid core. The axial velocity profile before the bifurcation junction resembles the Mshaped velocity profile commonly observed in bifurcated tube flows. The result of this manipulation is the development of a physically observable fourvortex configuration in the cross sections of the daughter branches, thus demonstrating the strong influence of upstream secondary vorticity. Through numerical visualization of vortex lines, it is shown that secondary vorticity is amplified by the extension of vortex lines due to secondary flow within the daughter tube. Orderofmagnitude arguments have been applied to the vorticity transport equation; and key dimensionless parameters have been obtained, accounting for curvature effects. Results indicate that the secondary vorticity goes through a maximum with increasing downstream distance, as a result of the interplay between vortex stretching and viscous effects.