Volume 19, Issue 5, May 2007
Index of content:

We present experimental observations of microjetting from an oscillating drop. The jet is generated by the focusing of axisymmetric capillary waves that overturn and collide at an apex of the drop. These jets are up to two orders of magnitude smaller than the original drops. We present two widely different configurations that produce such microjets. The first occurs on a satellitedrop, produced by the pinchoff of a water drop from a vertical nozzle. The large oscillations following the contraction of the satellite bridge focus waves at the bottom, sending out a jet at . The second jet arises when a water drop, containing surfactants, falls onto and passes through a hemispherical soap film. The gentle deformation of the drop creates a surface wave that focuses at its top, shooting out a tiny jet and entrapping a small bubble inside the drop. This jet is in diameter and emerges at . In this configuration, the soap film wraps around the drop and acts as a sensor of the air flow, revealing that the liquid jet is preceded by a localized fastermoving air jet. The jetting in both configurations is quite robust and occurs even for slightly asymmetric conditions. These microjets appear for much lower values of the Reynolds and Weber numbers than previously observed, suggesting that freesurface jetting is not limited to the inviscid capillaryinertial regime, which has been the focus of much of the theoretical work.
 LETTERS


On the effective temperature concept in the problem of laminar vortex shedding behind a heated circular cylinder
View Description Hide DescriptionPrediction of the nonisothermal vortex shedding based on the effective temperature is always linked with the specific material properties of the working fluid, while a desirable universal prediction is unknown. Here we show that the thermal effect in diluted gases, where cylinder heating stabilizes wake flow (thus delaying the onset of vortex shedding), can be quantified in the extraordinary compact form of the proportionality between the critical Reynolds number and the dimensionless film temperature . The theoretically derived value for any dilute gas is related to an exponent of the kinematic viscositytemperature power law by a simple formula . The universal character of the proportionality has been proven; the linear increase of the critical Reynolds number is independent of the specific material properties for all diluted gases.

The effect of subgridscale models on the near wall vortices: A priori tests
View Description Hide DescriptionDirect numerical simulation of a turbulent channel flow at is used to analyze the resolved enstrophy dissipation caused by the subgridscale (SGS) models. The SGS enstrophy dissipation attains a minimum (forward scatter) at about and a backscatter region for , and is correlated with the highspeed streaks. A priori tests show that in the buffer layer, of all the models considered, the dynamic Smagorinsky, filtered structure function, and scalesimilarity models display the smallest amount of resolved enstrophy dissipation, whereas the Smagorinsky and mixed models are the most dissipative.

 ARTICLES


Interfacial Flows

Microjetting from wave focusing on oscillating drops
View Description Hide DescriptionWe present experimental observations of microjetting from an oscillating drop. The jet is generated by the focusing of axisymmetric capillary waves that overturn and collide at an apex of the drop. These jets are up to two orders of magnitude smaller than the original drops. We present two widely different configurations that produce such microjets. The first occurs on a satellitedrop, produced by the pinchoff of a water drop from a vertical nozzle. The large oscillations following the contraction of the satellite bridge focus waves at the bottom, sending out a jet at . The second jet arises when a water drop, containing surfactants, falls onto and passes through a hemispherical soap film. The gentle deformation of the drop creates a surface wave that focuses at its top, shooting out a tiny jet and entrapping a small bubble inside the drop. This jet is in diameter and emerges at . In this configuration, the soap film wraps around the drop and acts as a sensor of the air flow, revealing that the liquid jet is preceded by a localized fastermoving air jet. The jetting in both configurations is quite robust and occurs even for slightly asymmetric conditions. These microjets appear for much lower values of the Reynolds and Weber numbers than previously observed, suggesting that freesurface jetting is not limited to the inviscid capillaryinertial regime, which has been the focus of much of the theoretical work.

Threedimensional disturbances in channel flows
View Description Hide DescriptionIn this paper we conduct a linear stability analysis of threedimensional twofluid flows and use an energy method to comment on its stability. The governing equations are solved using a Chebyshevtau method that reduces the order of the coupled governing OrrSommerfeld and Squire equation and hence achieves more accurate results. A new norm, called the norm, is defined to overcome the problem of nonconvergence of the disturbance energy. The maximum amplification of is achieved for streamwise independent disturbances due to the “liftup effect,” as is the case of threedimensional singlefluid flow. In contrast to twodimensional flows, where the adjoint of the leading mode influences the growth, the threedimensional singlefluid flow growth is influenced by the adjoint of the second mode. Although most growth in threedimensional twofluid flow is due to the contribution of the adjoint of the second mode, at large time the interfacial mode contributes to most growth.

Crown behavior in drop impact on wet walls
View Description Hide DescriptionAxisymmetric computations of drop impingement on walls with a preexisting liquid film are reported. A highdensityratio latticeBoltzmann model is employed for the computations. The focus of the work is on the behavior of the crown that forms as a result of impingement. When the crown forms, its base radius and height grow with time. Subsequently, it may break up. The influence of wall liquid film thickness, and surrounding gas density and viscosity on crown behavior is investigated. When the liquid film is thin, it is observed that the rate of increase of the radius and height of the crown increases with increasing film thickness. The breakup of the crown is delayed. On thicker films, the rate of increase decreases with increasing film thickness, but the breakup is further delayed. When either gas density or viscosity is increased, the rate of increase of the radius and height decreases and breakup is delayed.

Viscous and NonNewtonian Flows

Streaming potential generated by twophase flow in a capillary
View Description Hide DescriptionThe streaming potential generated by pressuredriven twophase flow in a circular capillary differs from that generated by singlephase flow. Three model problems are considered, in which the dispersed phase consists of either (i) a rigid spherical particle (possibly charged), (ii) an uncharged spherical bubble, and (iii) a long, uncharged Bretherton bubble. In all three cases, the particle or bubble is assumed to lie on the center line of the capillary tube, so that the problem is axisymmetric, and is assumed to be of almost the same diameter as the internal diameter of the capillary, so that lubrication theory can be used. The electrical potentials on the surface of the particle and on the walls of the capillary are and , respectively, and the Debye length is assumed much smaller than the gap between the particle and the walls of the capillary. If the flow rate is held constant, the presence of the rigid particle increases the pressure drop between the ends of the capillary, and also changes the streaming potential by an amount proportional to . This change in potential will in general be small compared to the total streaming potential developed between the two ends of a long capillary. However, if the capillary is filled with a large number of rigid particles, not only will the changes in pressure drop and streaming potential between the two ends of the capillary be large, but there will be a significant change in the coefficient of proportionality between pressure drop and streaming potential. The presence of an uncharged spherical bubble or Bretherton bubble changes the pressure drop between the ends of the capillary (for a given flow rate) but does not change the linear relation between pressure drop and streaming potential. However, the linear relation between flow rate and streaming potential is modified for the spherical bubble, and becomes nonlinear when a Bretherton bubble is present.

Anomalous reduction in thrust/reaction of water jets issuing from microapertures
View Description Hide DescriptionThrusts and reactions of water jets issuing through small orifices and thin slots were measured and compared with the predictions from numerical solutions of the NavierStokes equations. Reasonable agreements were obtained between the experimental and numerical results for orifices and slots with openings of the order of size, but not with those of or less. The experimental results were found to be well below the predictions for apertures of the order of . The difference between the numerically calculated and the measured thrusts/reactions for the small apertures was found to be proportional to the square of the mean velocity. Several possible causes for the observed reduction in jet thrust/reaction in small apertures were examined, but none of them could adequately explain the flow anomaly.

Penalty immersed boundary method for an elastic boundary with mass
View Description Hide DescriptionThe immersed boundary (IB) method has been widely applied to problems involving a moving elastic boundary that is immersed in fluid and interacting with it. Most of the previous applications of the IB method have involved a massless elastic boundary and used efficient Fourier transform methods for the numerical solutions. Extending the method to cover the case of a massive boundary has required spreading the boundary mass out onto the fluid grid and then solving the NavierStokes equations with a variable mass density. The variable mass density of this previous approach makes Fourier transform methods inapplicable, and requires a multigrid solver. Here we propose a new and simple way to give mass to the elastic boundary and show that the method can be applied to many problems for which the boundary mass is important. The method does not spread mass to the fluid grid, retains the use of Fourier transform methodology, and is easy to implement in the context of an existing IB method code for the massless case. Two verifications of the method are given. One is a numerical convergence study that shows that our numerical scheme is secondorder accurate for a particular test problem. The other is direct comparison with experimental data of vortexinduced vibrations of a massive cylinder, which shows that the results obtained by the present method are quite comparable to the experimental data.

Elastic turbulence in von Karman swirling flow between two disks
View Description Hide DescriptionWe discuss the role of elastic stress in the statistical properties of elasticturbulence, realized by the flow of a polymer solution between two disks. The dynamics of the elastic stress are analogous to those of a smallscale fast dynamo in magnetohydrodynamics, and to those of the turbulent advection of a passive scalar in the Batchelor regime. Both systems are theoretically studied in the literature, and this analogy is exploited to explain the statistical properties, the flow structure, and the scaling observed experimentally. The following features of elasticturbulence are confirmed experimentally and presented in this paper: (i) The rms of the vorticity (and that of velocity gradients) saturates in the bulk of the elasticturbulent flow, leading to the saturation of the elastic stress. (ii) The rms of the velocity gradients (and thus the elastic stress) grows linearly with in the boundary layer, near the driving disk. The rms of the velocity gradients in the boundary layer is one to two orders of magnitude larger than in the bulk. (iii) The PDFs of the injected power at either constant angular speed or torque show skewness and exponential tails, which both indicate intermittent statistical behavior. Also the PDFs of the normalized accelerations, which can be related to the statistics of velocity gradients via the Taylor hypothesis, exhibit wellpronounced exponential tails. (iv) A new length scale, i.e., the thickness of the boundary layer, as measured from the profile of the rms of the velocity gradient, is found to be relevant for the boundary layer of the elastic stresses. The velocity boundary layer just reflects some of the features of the boundary layer of the elastic stresses (rms of the velocity gradients). This measured length scale is much smaller than the vessel size. (v) The scaling of the structure functions of the vorticity, velocity gradients, and injected power is found to be the same as that of a passive scalar advected by an elasticturbulent velocity field.

Particulate, Multiphase, and Granular Flows

Viscous resuspension in a tube: The impact of secondary flows resulting from second normal stress differences
View Description Hide DescriptionThe viscous resuspension of nonneutrally buoyant particles has been modeled as a competition between the rate of shearinduced diffusion and sedimentation or as a balance between gradients in the particle stress and gravity. Typically, however, the rheology of the suspension has been modeled using a simple concentrationdependent Newtonian viscosity. In this paper, we demonstrate through theory and comparison with existing experimental results that the anisotropy of the total stress tensor must be included to accurately describe the resuspension process in a tube. At steady state, the isotropic model predicts a secondary current within the tube cross section that flows downwards at the center and upwards near the sidewalls, resulting in a concave upward interface between the clear suspending fluid and the particles [K. Zhang and A. Acrivos, Int. J. Multiphase Flow20, 579 (1994)]. In contrast, with the inclusion of the known nonNewtonian suspension rheology, the secondary current profile is reversed: upwards near the center and downwards near the walls. This leads to a concave downward shape of the interface between the suspending fluid and the suspension and is in quantitative agreement with the experimental measurements of Altobelli et al. [J. Rheol.35, 721 (1991)].

Rapid flow of dry granular materials down inclined chutes impinging on rigid walls
View Description Hide DescriptionWe performed laboratory experiments of dry granular chute flows impinging an obstructing wall. The chute consists of a wide rectangular channel, inclined by 50° relative to the horizontal, which, downslope abruptly changes into a horizontal channel of the same width. of quartz chips are released through a gate with the same width as the chute and a gap of height, respectively. Experiments are conducted for two positions of the obstructing wall, (i) below the exit gate and perpendicular to the inclined chute, and (ii) into the horizontal runout and then vertically oriented. Granular material is continuously released by opening the shutter of the silo. The material then moves rapidly down the chute and impinges on the obstructing wall. This leads to a sudden change in the flow regime from a fast moving supercritical thin layer to a stagnant thick heap with variable thickness and a surface dictated by the angle of repose typical for the material. We conducted particle imagevelocimetry(PIV) experiments by recording the moving material from the side with charge coupled devices(CCD)cameras. The experiment was also video recorded. From the CCD data velocities were also deduced using the PIV technique. In order to compare the results here we describe the experiments for the same material and the same gap width of the silo gate but for the two positions of the obstructing wall. Analysis of the shock front formation and propagation upslope, evolution of the height of the supercritical flow, impact velocity and momentum are presented and discussed in detail. Computed and derived shock front heights match well.

Laminar Flows

Computational study of scalar mixing in the field of a gaseous laminar line vortex
View Description Hide DescriptionA computational study of scalar mixing in a laminar vortex is presented for vortices generated between two gas streams (one seeded and another unseeded) flowing parallel to each other in a rectangular flow channel. An isolated line vortex is initiated by momentarily increasing one of the stream velocities in relation to the other in otherwise equal velocity, coflowing streams separated upstream by a splitter plate. A detailed parametric study was conducted to determine the effects of vortex strength, convection time, and nonuniform temperature on scalar mixing characteristics. A qualitative relationship was developed between the vortex and the convectionReynolds numbers to obtain a welldefined vortical structure. As it is wellknown in the literature on mixing layers, the vortex initiation process creates an abundance of the fluid in the vortex core from the pulsed (or high speed) stream. Spatial mixing statistics are obtained in the vortex interaction domain by determining the scalar concentration probability density functions as well as the mean mixed fluid concentration and its variance. Computational results are found to be in excellent agreement with the experiments conducted in the same configuration by one of the authors. Both computations and experiments suggest that the interfacial area generation as a result of vortex interaction is primarily responsible for mixing augmentation at high vortexReynolds numbers. Effects of molecular diffusion become more important for weak vortices and at short convection times. Temperature (or density) ratio between coflowing streams clearly affects the rate of growth of the vortices and this is consistent with the findings in nonuniform density mixing layers. Nonuniform temperatures result in a decrease of the mean mixed fluid concentration regardless of the stream from which the vortex is generated. The mean mixed fluid concentration in the vortex interaction region scales with the product of vortexReynolds number and nondimensional convective time scale (or degree of stirring) and inversely with temperature ratio. Empirical correlations were developed for this parameter for vortices generated from either stream.

Instability and Transition

Transition to turbulence in a tall annulus submitted to a radial temperature gradient
View Description Hide DescriptionWe have investigated the transition to turbulence in a water flow confined inside a tall vertical cylindrical annulus submitted to a radial temperature gradient using the spacetime diagrams technique. As soon as a small radial temperature gradient is applied to the annular flow, the radial stratification of density induces a torque that produces a large convection cell. The first instability of this flow occurs via a supercritical bifurcation and gives rise to axisymmetric rolls localized in the middle of the system. Just above the onset, the pattern contains spatiotemporal defects. For large values of the control parameter, we have observed a coexistence of turbulent bursts and laminar domains. We have measured the turbulent fraction and have performed a statistical analysis of the laminar and turbulent zones, and have found that they bear the main characteristics of spatiotemporal intermittency.

Thermal convection in a rotating porous layer using a thermal nonequilibrium model
View Description Hide DescriptionLinear stability of a rotating fluidsaturated porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy model, which includes the time derivative and Coriolis terms, is employed as a momentum equation. A twofield model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that a small interphase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through the oscillatory mode rather than the stationary one. The rotation inhibits the onset of convection in both stationary and oscillatory mode. In addition, the effect of porosity modified conductivity ratio, DarcyPrandtl number, and the ratio of diffusivities on the stability of the system is investigated. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the Nusselt number. The effect of thermal nonequilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is also investigated by solving the finite amplitude equations using RungeKutta method.

Time scales for transition in TaylorCouette flow
View Description Hide DescriptionThe time scale for onset and decay of vortices in a TaylorCouette system cannot be predicted from linear stability analysis, yet it is important from a practical standpoint. A twodimensional pseudospectral direct numerical simulation was used to examine the time scales for subcriticaltosupercritical transition and supercriticaltosubcritical transition for a variety of aspect ratios and radius ratios ( , 0.7, and 0.9). A viscous time scale incorporating both the gap width, , and the distance between the endwalls of the system, , is most appropriate for the onset of Taylor vortices, although no time scale collapses the data for all aspect ratios and radius ratios. For decay, a viscous time scale using the gap width as the length scale collapses the data as the aspect ratio gets large. These results indicate that the onset of vortices is a consequence of the propagation of vortical structures related to the endwalls, while decay is related to viscous dissipation from the sidewalls.

The effect of Mach number on unstable disturbances in shock/boundarylayer interactions
View Description Hide DescriptionThe effect of Mach number on the growth of unstable disturbances in a boundary layer undergoing a strong interaction with an impinging oblique shock wave is studied by direct numerical simulation and linear stability theory (LST). To reduce the number of independent parameters, test cases are arranged so that both the interaction location Reynolds number (based on the distance from the plate leading edge to the shock impingement location for a corresponding inviscid flow) and the separation bubble length Reynolds number are held fixed. Smallamplitude disturbances are introduced via both whitenoise and harmonic forcing and, after verification that the disturbances are convective in nature, linear growth rates are extracted from the simulations for comparison with parallel flow LST and solutions of the parabolized stability equations (PSE). At Mach 2.0, the oblique modes are dominant and consistent results are obtained from simulation and theory. At Mach 4.5 and Mach 6.85, the linear NavierStokes results show large reductions in disturbance energy at the point where the shock impinges on the top of the separated shear layer. The most unstable second mode has only weak growth over the bubble region, which instead shows significant growth of streamwise structures. The two higher Mach number cases are not well predicted by parallel flow LST, which gives frequencies and spanwise wavenumbers that are significantly different from the simulations. The PSE approach leads to good qualitative predictions of the dominant frequency and wavenumber at Mach 2.0 and 4.5, but suffers from reduced accuracy in the region immediately after the shock impingement. Threedimensional NavierStokes simulations are used to demonstrate that at finite amplitudes the flow structures undergo a nonlinear breakdown to turbulence. This breakdown is enhanced when the obliquemode disturbances are supplemented with unstable Mack modes.

Marginally unstable Holmboe modes
View Description Hide DescriptionMarginally unstable Holmboe modes for smooth density and velocity profiles are studied. For a large family of flows and stratification that exhibit Holmboe instability, it is shown that the modes with phase velocity equal to the maximum or the minimum velocity of the shear are marginally unstable. This allows us to determine the critical value of the control parameter (expressing the ratio of the velocity variation length scale to the density variation length scale) above which Holmboe instability is present, . Furthermore, systems for which the parameter is very close to this critical value are examined. For this case, an analytical expression for the dispersion relation of the complex phase speed in the unstable region is derived. The growth rate and the width of the region of unstable wavenumbers has a very strong (exponential) dependence on the deviation of from the critical value. Two specific examples are examined and a physical interpretation of the results is described.

An intrinsic stabilization scheme for proper orthogonal decomposition based lowdimensional models
View Description Hide DescriptionDespite the temporal and spatial complexity of common fluid flows, model dimensionality can often be greatly reduced while both capturing and illuminating the nonlinear dynamics of the flow. This work follows the methodology of direct numerical simulation (DNS) followed by proper orthogonal decomposition (POD) of temporally sampled DNS data to derive temporal and spatial eigenfunctions. The DNS calculations use Chorin’s projection scheme; twodimensional validation and results are presented for driven cavity and square cylinder wake flows. The flow velocity is expressed as a linear combination of the spatial eigenfunctions with timedependent coefficients. Galerkin projection of these modes onto the NavierStokes equations obtains a dynamical system with quadratic nonlinearity and explicit Reynolds number dependence. Truncation to retain only the most energetic modes produces a lowdimensional model for the flow at the decomposition . We demonstrate that although these lowdimensional models reproduce the flowdynamics, they do so with small errors in amplitude and phase, particularly in their long term dynamics. This is a generic problem with the POD dynamical system procedure and we discuss the schemes that have so far been proposed to alleviate it. We present a new stabilization algorithm, which we term intrinsic stabilization, that projects the error onto the POD temporal eigenfunctions, then modifies the dynamical system coefficients to significantly reduce these errors. It requires no additional information other than the POD. The premise that this method can correct the amplitude and phase errors by finetuning the dynamical system coefficients is verified. Its effectiveness is demonstrated with lowdimensional dynamical systems for driven cavity flow in the periodic regime, quasiperiodic flow at , and the wake flow. While derived in a POD context, the algorithm has broader applicability, as demonstrated with the Lorenz system.

Gyrotactic bioconvection in three dimensions
View Description Hide DescriptionThe bioconvectionequations, based on the continuum model of Pedley et al. [J. Fluid Mech.195, 223 (1988)], consist of the NavierStokes equations for an incompressible fluid coupled with a microorganism conservation equation. These equations are solved efficiently using a semiimplicit secondorder accurate conservative finitedifference method. The structure and stability of a threedimensional plume in deep rectangular boxes with stressfree sidewalls are investigated. Comparisons are made with the twodimensional and axisymmetric bioconvection. In deep chambers, the threedimensional plume that forms initially along the central axis of the chamber typically breaks down via a meandering instability.

Frequency selection in globally unstable round jets
View Description Hide DescriptionThe selfsustained formation of synchronized ring vortices in hot subsonic jets is investigated by direct numerical simulation of the axisymmetric equations of motion. The onset of global instability and the global frequency of synchronized oscillations are examined as functions of the ambienttojet temperature ratio and the initial jet shear layer thickness. The numerical results are found to follow the predictions from nonlinear global instabilitytheory; global instability sets in as the unperturbed flow is absolutely unstable over a region of finite streamwise extent at the inlet, and the global frequency near the global instability threshold corresponds to the absolute frequency of the inlet profile. In strongly supercritical thin shear layer jets, however, the simulations display global frequencies well above the absolute frequency, in agreement with experimental results. The inner structure of rolledup vortices in hot jets displays fine layers of positive and negative vorticity that are produced and maintained by the action of the baroclinic torque.
