Volume 20, Issue 3, March 2008

In this work, the temporal evolution of a low swirlnumber turbulent Batchelor vortex is studied using pseudospectral direct numerical simulations. The solution of the governing equations in the vorticityvelocity form allows for accurate application of boundary conditions. The physics of the evolution is investigated with an emphasis on the mechanisms that influence the transport of axial and angular momentum. Excitation of normal modeinstabilities gives rise to coherent large scale helical structures inside the vortical core. The radial growth of these helical structures and the action of axial shear and differential rotation results in the creation of a polarized vortex layer. This vortex layer evolves into a series of hairpinshaped structures that subsequently breakdown into elongated fine scale vortices. Ultimately, the radially outward propagation of these structures results in the relaxation of the flow towards a stable highswirl configuration. Two conserved quantities, based on the deviation from the laminar solution, are derived and these prove to be useful in characterizing the polarized vortex layer and enhancing the understanding of the transport process. The generation and evolution of the Reynolds stresses is also addressed.
 AWARD AND INVITED PAPERS


Turbulent scaling^{a)}
View Description Hide DescriptionThis article is a review and extension of several papers which presented a model for inertial range intermittency and anomalous scaling of velocity difference structure functions. The method of matched asymptotic expansions is used with the Navier–Stokes equation to derive a basic law for the instantaneous velocity difference between neighboring points in a turbulent flow. This is a power law with an undetermined exponent which must be specified statistically by making a physical hypothesis. (The simplest hypothesis, mean dissipation independent of Reynolds number makes the exponent be 1/3, the Kolmogorov 1941 result.) The resulting general theory is used to calculate structure functions, anomalous exponents and the probability density function of velocity differences and is extended to include the Reynolds number dependence of the instantaneous dissipation at a point and a theory for passive scalars. (A parameter in the instantaneous dissipation is adjusted to make the mean dissipation approximately independent of Reynolds number.)

The use of global modes to understand transition and perform flow control^{a)}
View Description Hide DescriptionThe stability of nonparallel flows is considered using superposition of global modes. When perturbed by the worst case initial condition, these flows often exhibit a large transient growth associated with the development of wave packets. The global modes of the systems also provide a good starting point for the design of reduced order models used to control the growing disturbances. Three recent investigations are reviewed. The first example is the growth of a wave packet on a falling liquid sheet. The optimal perturbation analysis shows that the worst case initial condition is a localized disturbance that creates a propagating wave packet that hits the downstream end, regenerating a wave packet upstream through a global pressure pulse. Second, we consider twodimensional disturbances in the Blasius boundary layer. It is found that a wave packet is optimally excited by an initial condition consisting of localized backward leaning Orr structures. Finally, the control of a globally unstable boundarylayer flow along a shallow cavity is considered. The disturbance propagation is associated with the development of a wave packet along the cavity shear layer, unstable to the Kelvin–Helmholtz mechanism, followed by a global cycle related to the two unstable global modes. Direct numerical simulations of this flow are coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and provides the actuation at the upstream lip. A reduced order model for the control is obtained by a projection on the least stable global eigenmodes. The linearquadraticGaussian controller is run in parallel to the Navier–Stokes time integration and it is shown to damp out the global oscillations.

Vortex formation in shallow flows^{a)}
View Description Hide DescriptionHighly coherent vortex formation can occur in a variety of shallow, separated shear layers. Spacetime imaging is employed to gain insight into representative physical concepts, including vortex formation due to an inherent instability, forced vortex formation due to controlled inflow, and vortex formation coupled with gravitywave resonance.

 LETTERS


Wave celerity on a shearthinning fluid film flowing down an incline
View Description Hide DescriptionThis letter presents a phenomenological model predicting the celerity of long surface waves on a nonNewtonian fluid flowing down an inclined plane. We show that, for a shearthinning fluid, the celerity is greater than the wellknown value . The developed model points at the significant effect of the viscosity disturbance and also provides a likely explanation for the decrease in threshold for the instability.

Electric field effect on a twofluid interface instability in channel flow for fast electric times
View Description Hide DescriptionThe application of an electric field to a twofluid layer in channel flow has been shown to be an effective way to destabilize microscale interfacial flows. Here, we perform a linear stability analysis of a flat interface between two leaky dielectric liquids flowing in a channel while also subjected to an electric field parallel to the interface. It is shown that the analysis simplifies for fast electric chargerelaxation times, in which case conditions for the electric field to be either stabilizing or destabilizing are derived analytically. These results are compared to those previously obtained for a normal electric field.

A note on the fluctuation of dissipative scale in turbulence
View Description Hide DescriptionWe present an application of the multifractal formalism able to predict the whole shape of the probability density function (pdf) of the dissipative scale, . We discuss both intense velocity fluctuations, leading to dissipative scales smaller than the Kolmogorov scale, where the formalism gives a pdf decaying as a superposition of stretched exponential, and smooth velocity fluctuations, where the formalism predicts a powerlaw decay. Both trends are found to be in good agreement with recent direct numerical simulations [J. Schumacher, “SubKolmogorovscale fluctuations in fluid turbulence,” Europhys. Lett.80, 54001 (2007)].

 ARTICLES

 Interfacial Flows

Hydrodynamic forces involving deformable interfaces at nanometer separations
View Description Hide DescriptionA model is developed to describe the dynamic forces acting between two deformable drops, or between one drop and a solid surface, when they are in relative axisymmetric motion at separations of in a Newtonian liquid. Forces arise from hydrodynamic pressure in the draining liquid film that separates the interfaces and from disjoining pressure due to repulsive or attractive surface forces. Predictions of the model are successfully compared with recent experimental measurements of the force between two micrometerscale surfactant stabilized decane drops in water in an atomic force microscope [S. L. Carnie, D. Y. C. Chan, C. Lewis, R. Manica, and R. R. Dagastine, Langmuir21, 2912 (2005); R. R. Dagastine, R. Manica, S. L. Carnie, D. Y. C. Chan, G. W. Stevens, and F. Grieser, Science313, 210 (2006)] and with subnanometer resolution measurements of timedependent deformations of a millimeterscale mercurydrop approaching a flat mica surface in a modified surface force apparatus [J. N. Connor and R. G. Horn, Faraday Discuss.123, 193 (2003); R. G. Horn, M. Asadullah, and J. N. Connor, Langmuir22, 2610 (2006)]. Special limits of the model applicable to small and moderate deformation regimes are also studied to elucidate the key physical ingredients that contribute to the characteristic behavior of dynamic collisions involving fluid interfaces.

A thin conducting viscous film on an inclined plane in the presence of a uniform normal electric field: Bifurcation scenarios
View Description Hide DescriptionA theory for two dimensional long and stationary waves of finite amplitude on a thin viscousliquid film down an inclined plane in the presence of uniform electric field at infinity is investigated. A set of exact averaged equations for the filmflow system is described and linearized stability analysis of the uniform flow is performed using normalmode formulation and the critical condition for linear instability is obtained. The linearized instability for the permanent wave equation, consistent to the second order in , is examined and the eigenvalue properties of the fixed points are classified in various parametric regimes. Numerical integration of the permanent wave equation as a thirdorder dynamical system is carried out. Different bifurcation scenarios leading to multiplehump solitary waves or leading to chaos are exhibited in the parametric space.

Characterization of the electrosprays of 1ethyl3methylimidazolium bis(trifluoromethylsulfonyl) imide in vacuum
View Description Hide DescriptionThe electrosprays of 1ethyl3methylimidazolium bis(trifluoromethylsulfonyl) imide are composed of a complex mixture of ions and chargeddroplets, which can be analyzed to determine the structure of the beam and infer significant features of the electrohydrodynamic atomization. In particular, we use a combination of retarding potential and time of flight techniques to study these beams and are able to quantify the voltage drop along the cone jet, together with the velocity and diameter of the jet at the breakup location, confirm the strong influence of viscosity and electrification in the breakup, show that the electric field in and near the Taylor cone tip is insensitive to external electrostatic parameters, and study the spatial distribution of ions and droplets, whereby the paradoxical absence of ions in the outmost region of the beam is established. The research described in this article can be exploited in the modeling of capillary instability of chargedjets: testing the results of these models is difficult, especially when nanojets are involved, and our findings and techniques provide the experimental support required by the theoretical activity. The present research is also applicable to the modeling of colloid thruster beams.
 Viscous and NonNewtonian Flows

A note on oblique stagnationpoint flow
View Description Hide DescriptionPrevious analyses of oblique stagnationpoint flow at a plane wall are discussed and unified with reference to a free parameter. The oblique flow consists of orthogonal stagnationpoint flow to which is added a shear flow whose vorticity is fixed at infinity. Physically the free parameter may be viewed as altering the structure of the shear flow component by varying the magnitude of the pressure gradient. For large adverse pressure gradients, the shear component has a region of reversed flow near the wall. Remarkably, combining an orthogonal flow with shear flows featuring different levels of reversed flow always produces the same oblique flow but with the stagnationpoint of attachment shifted along the wall by a predictable amount.

Settling of an isolated spherical particle in a yield stress shear thinning fluid
View Description Hide DescriptionWe visualize the flow induced by an isolated nonBrownian spherical particle settling in a shear thinningyield stress fluid using particle image velocimetry. With , we show a breaking of the foreaft symmetry and relate this to the rheological properties of the fluid. We find that the shape of the yield surface approximates that of an ovoid spheroid with its major axis approximately five times greater than the radius of the particle. The disagreement of our experimental findings with previous numerical simulations is discussed.
 Particulate, Multiphase, and Granular Flows

Scaling the final deposits of dry cohesive granular columns after collapse and quasistatic fall
View Description Hide DescriptionThis paper reports on laboratory experiments that were designed to investigate the collapse and quasistatic fall of dry cohesive granular columns. These experiments were compared with similar experiments that were performed with noncohesive dry sand columns. A powder of gypsum (calcium sulphate dihydrate) was used to represent cohesive granular material. In all the experiments, the cohesive granular columns fractured and flowed in coherent blocks but, while faults remained steep in the quasistatic fall experiments, they flattened in the collapse experiments as the initial aspect ratio of the columns increased. Dilation was seen in the quasistatic fall experiments, while some air entrapment within the columns occurred in the collapse experiments. The final deposits of the cohesive granular columns were found to satisfy power law relationships as a function of the initial aspect ratio of the columns. Two asymptotes were found for the lower and higher range of initial aspect ratios, which varied between 0.5 and 8, respectively. When compared with the power law relationships found for dry noncohesive columns, the power dependence of the ratio of initial to final height and final runout to initial length with the aspect ratio of the columns was found to be similar. The prefactors of the power laws were found to slightly decrease with the increase of the cohesion or, equivalently, the decrease in grain size. Similar to the dry noncohesive case, the prefactors for the runout length were found to increase by a factor 2 with the increase of flow rate. When the collapse experiments were compared with the quasistatic fall experiments, a shift towards higher aspect ratios of the transition between the two asymptotic power laws was found.
 Laminar Flows

Vortex shedding from a circular cylinder of finite length at low Reynolds numbers
View Description Hide DescriptionFlow past a circular cylinder of finite spanlength with two free ends is investigated based on direct numerical solutions of the threedimensional unsteady incompressible Navier–Stokes equations. Special attention is paid to the effect of lengthtodiameter ratio on vortex shedding from the cylinder into its wake, where is the spanlength and is the diameter of the cylinder. The lengthtodiameter ratio is prescribed in the range of , and the Reynolds number Re which is based on is . Results show that vortex shedding from, and thus wake pattern behind, the cylinder changes drastically depending on both and Re. Five basic patterns of vortex shedding are found to exist: (i) Periodic oblique vortex shedding at relatively large and at Re beyond a critical Reynolds number, (ii) quasiperiodic oblique vortex shedding also at relatively large but at Re below the critical Reynolds number, (iii) periodic shedding of hairpinshaped vortices that occurs when is moderate, (iv) steady two counterrotating vortex pairs that appear when both and Re are small, and (v) alternate shedding of a counterrotating vortex pair from two flat ends, which occurs when is small but Re is high. Transition processes between the basic patterns are also examined. Threedimensional vortical structures in the wake are clarified in some detail.

An experimental investigation and a simple model of a valveless pump
View Description Hide DescriptionWe construct a valveless pump consisting of a section of elastic tube and a section of rigid tube connected in a closed loop and filled with water. By periodically squeezing the elastic tube at an asymmetric location, a persistent flow around the tubes is created. This effect, called the Liebau phenomenon or valveless pumping, has been known for some time but is still not completely understood. We study the flow rates for various squeezing locations, frequencies, and elastic tube rigidities. To understand valveless pumping, we formulate a simple model that can be described by ordinary differential equations. The time series of flow velocities generated by the model are qualitatively and quantitatively similar to those seen in the experiment. The model provides a physical explanation of valveless pumping, and it allows us to identify the essential pumping mechanisms.

Quantification of energy dissipation for laterally oscillating microstructures
View Description Hide DescriptionFluid film damping in laterally oscillating microstructures is investigated in the entire Knudsen regime and a wide Stokes number range using analytical and numerical results from Park et al. [“Rarefaction effects on shear driven oscillatory gas flows: A direct simulation Monte Carlo study in the entire Knudsen regime,” Phys. Fluids16, 317 (2004)] and Hadjiconstantinou [“Oscillatory shear driven gas flows in the transition and freemolecularflow regimes,” Phys. Fluids17, 100611 (2005)]. An energy dissipation parameter per oscillation cycle that allows a consistent interpretation of the quality factor utilized in the design of microelectromechanical systemdevices is presented.
 Instability and Transition

Frictional drag reduction in bubbly Couette–Taylor flow
View Description Hide DescriptionFrictional drag reduction due to the presence of small bubbles is investigated experimentally using a Couette–Taylor flow system; i.e., shear flow between concentric cylinders. Torque and bubble behavior are measured as a function of Reynolds number up to while air bubbles are injected constantly and rise through an array of vortical cells. Silicone oil is used to avoid the uncertain interfacial property of bubbles and to produce nearly monosized bubble distributions. The effect of drag reduction on sensitivity and power gain are assessed. The sensitivity exceeds unity at , proving that the effect of the reduction in drag is greater than that of the reduction in mixture density. This is due to the accumulation of bubbles toward the rotating inner cylinder, which is little affected by turbulence. The power gain, which is defined by the power saving from the drag reduction per the pumping power of bubble injection, has a maximum value of at higher numbers around 2500. An image processing measurement shows this is because of the disappearance of azimuthal waves when the organized bubble distribution transforms from toroidal to spiral modes. Moreover, the axial spacing of bubble clouds expands during the transition, which results in an effective reduction in the momentum exchange.

Spatially localized states in natural doubly diffusive convection
View Description Hide DescriptionNumerical continuation is used to compute a multiplicity of stable spatially localized steady states in doubly diffusive convection in a vertical slot driven by imposed horizontal temperature and concentration gradients. The calculations focus on the socalled opposing case, in which the imposed horizontal thermal and solutal gradients are in balance. Noslip boundary conditions are used at the sides; periodic boundary conditions with large spatial period are used in the vertical direction. The results demonstrate the presence of homoclinic snaking in this system, and can be interpreted in terms of a pinning region in parameter space. The dynamics outside of this region are studied using direct numerical simulation.

Modeling of transitional channel flow using balanced proper orthogonal decomposition
View Description Hide DescriptionWe study reducedorder models of threedimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from threedimensional simulations in physical space as opposed to the traditional singlewavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods suggests that the use of a nonorthogonal projection with adjoint modes is most likely the main reason for the superior performance of BPOD. We also demonstrate that for singlewavenumber perturbations, loworder BPOD models reproduce the dominant eigenvalues of the full system better than POD models of the same order. These features indicate that the simple, yet accurate BPOD models are a good candidate for developing modelbased controllers for channel flow.

Directional effect of a magnetic field on oscillatory lowPrandtlnumber convection
View Description Hide DescriptionThe directional effect of a magnetic field on the onset of oscillatory convection is studied numerically in a confined threedimensional cavity of relative dimensions 4:2:1 (length:width:height) filled with mercury and subject to a horizontal temperature gradient. The magnetic field suppresses the oscillations most effectively when it is applied in the vertical direction, and is the least efficient when applied in the longitudinal direction (parallel to the temperature gradient). In all cases, however, exponential growths of the critical Grashof number, ( , ratio of buoyancy to viscous dissipation forces) with the Hartmann number ( , ratio of magnetic to viscous dissipation forces) are obtained. Insight into the damping mechanism is gained from the fluctuating kinetic energy budget associated with the timeperiodic disturbances at threshold. The kinetic energy produced by the vertical shear of the longitudinal basic flow dominates the oscillatory transition, and when a magnetic field is applied, it increases in order to balance the stabilizing magnetic energy. Moreover, subtle changes in the spatial distribution of this shear energy are at the origin of the exponential growth of . The destabilizing effect of the velocity fluctuations strongly decreases when is increased (due to the decay of the velocity fluctuations in the bulk accompanied by the appearance of steep gradients localized in the Hartmann layers), so that an increase of the shear of the basic flow at is required in order to sustain the instability. This yields an increase in , which is reinforced by the fact that the shear of the basic flow naturally decreases at constant with the increase of , particularly when the magnetic field is applied in the vertical direction. For transverse and longitudinal fields, the decay of the velocity fluctuations is combined with an increase of the shear energy term due to a sustained growth in stabilizing magnetic energy with .

Stability of the boundary layer flow on a long thin rotating cylinder
View Description Hide DescriptionThe development and stability of the boundary layer flow over a long thin cylinder aligned with the main flow and which rotates around its axis is considered. Numerical results show that the introduction of rotation has an important effect on the behavior of the basic flow. When the swirl increases, the shear stress at the wall also increases due to the changes in the pressure distribution along the cylinder surface. A nonparallel linear stability analysis of the basic flow is performed using parabolized stability equations. Even at moderately low rotation, we find the existence of unstable centrifugal modes, in addition to the shear ones found in previous stability analysis of the boundary layer flow on a cylinder with no rotation. These centrifugal instabilities develop at Reynolds numbers, based on the cylinder radius and external axial velocity, much smaller than those required for the growing of the shear instabilities. Our analysis shows that nonparallel effects play a key role in the onset and development of these instabilities, being the spiral mode with azimuthal wavenumber , the first to become unstable as the Reynolds number is increased in most cases of interest. We characterize the critical Reynolds number for convective instability as a function of the axial distance to the leading edge for several values of the swirl parameter.