Volume 22, Issue 8, August 2010
Index of content:

We report on our direct numerical simulation of an incompressible, nominally zeropressuregradient flatplate boundary layer from momentum thickness Reynolds number 80–1950. Heat transfer between the constanttemperature solid surface and the freestream is also simulated with molecular Prandtl number . Skinfriction coefficient and other boundary layer parameters follow the Blasius solutions prior to the onset of turbulent spots. Throughout the entire flatplate, the ratio of Stanton number and skinfriction deviates from the exact Reynolds analogy value of 0.5 by less than 1.5%. Mean velocity and Reynolds stresses agree with experimental data over an extended turbulent region downstream of transition. Normalized rms wallpressure fluctuation increases gradually with the streamwise growth of the turbulent boundary layer. Wall shear stress fluctuation,, on the other hand, remains constant at approximately 0.44 over the range, . Turbulent Prandtl number peaks at around 1.9 at the wall, and decreases monotonically toward the boundary layer edge with no nearwall secondary peak, in good agreement with previous boundary layer heat transfer experiments. In the transitional region, turbulent spots are tightly packed with numerous hairpin vortices. With the advection and merging of turbulent spots, these young isolated hairpin forests develop into the downstream turbulent region. Isosurfaces of temperature up to are found to display wellresolved signatures of hairpin vortices, which indicates the persistence of the hairpin forests.
 LETTERS


On the energy of elliptical vortices
View Description Hide DescriptionConsider a twodimensional axisymmetric vortex with circulation . Suppose that this vortex is isovortically deformed into an elliptical vortex. We show that the reduction in energy is , where is the ratio of the major to the minor axis of any particular elliptical vorticity contour. It is notable that is independent of the details of vorticity profile of the axisymmetric vortex and, in particular, independent of its average radius. The implications of this result for the twodimensional inverse cascade are briefly discussed.

Turbulence modulation and drag reduction by spherical particles
View Description Hide DescriptionThis letter reports on the pronounced turbulence modulations and the accompanying drag reduction observed in a twoway coupled simulation of particleladen channel flow. The present results support the view that drag reduction can be achieved not only by means of polymeric or fiber additives but also with spherical particles.

Axial and lateral particle ordering in finite Reynolds number channel flows
View Description Hide DescriptionInertial focusing in a pressuredriven flow refers to the positioning of particles transverse to the mean flow direction that occurs as a consequence of a finite particle Reynolds number. In channels with rectangular crosssections, and for a range of channel aspect ratios and particle confinement, experimental results are presented to show that both the location and the number of focusing positions depend on the number of particles per unit length along the channel. This axial number density is a function of both the channel crosssection and the particle volume fraction. These results are rationalized using simulations of the particleladen flow to show the manner in which hydrodynamic interactions set the preferred locations in these confined flows. A criterion is presented for the occurrence of a stepwise transition from one to two or more trains of particles.

 ARTICLES

 Biofluid Mechanics

Selfsimilar bending in a flow: The axisymmetric case
View Description Hide DescriptionWe study how sheets roll up into conical configurations when exposed to fluid flows using simulations and analysis. The simulations couple the bending of thin sheets to axisymmetric flows with vortex shedding. We find quasisteady flows with vortex ring wakes in which the radii of the rings scale with the radii of the cone bases. The cone angles scale with the dimensionless flow speed raised to the power −1/3. The drag coefficients for the cones scale with flow speed to the power −1. We find good agreement with the previously published experimental results. The scalings we have found result from a selfsimilar behavior of the flow at the outer edges of the cones, with length scales set by the radii of the vortex rings in the wakes.

Jet propulsion without inertia
View Description Hide DescriptionA body immersed in a highly viscous fluid can locomote by drawing in and expelling fluid through pores at its surface. We consider this mechanism of jet propulsion without inertia in the case of spheroidal bodies and derive both the swimming velocity and the hydrodynamic efficiency. Elementary examples are presented and exact axisymmetric solutions for spherical, prolate spheroidal, and oblate spheroidal body shapes are provided. In each case, entirely and partially porous (i.e., jetting) surfaces are considered and the optimal jetting flow profiles at the surface for maximizing the hydrodynamic efficiency are determined computationally. The maximal efficiency which may be achieved by a sphere using such jet propulsion is 12.5%, a significant improvement upon traditional flagellabased means of locomotion at zero Reynolds number, which corresponds to the potential flow created by a source dipole at the sphere center. Unlike other swimming mechanisms which rely on the presentation of a small cross section in the direction of motion, the efficiency of a jetting body at low Reynolds number increases as the body becomes more oblate and limits to approximately 162% in the case of a flat plate swimming along its axis of symmetry. Our results are discussed in the light of slime extrusion mechanisms occurring in many cyanobacteria.
 Micro and Nanofluid Mechanics

Electroosmotic flow in a wavy microchannel: Coherence between the electric potential and the wall shape function
View Description Hide DescriptionThe electroosmotic flow through a wavy microchannel is studied under the Debye–Hückel approximation. An analytic solution by perturbation with appropriate averaging is carried out up to the secondorder in terms of the small amplitude of corrugation. It is shown that the wavelength and phase difference of the corrugations can be utilized to control the flow relative to the case of flat walls. In particular, for thick electric double layers the electroosmotic flow can be enhanced at longwavelength corrugations because of the coherence between the electric potential and the wall shape function. Notably, these findings are not restricted to small amplitudes of corrugation. By applying the Ritz method to solve for the electroosmotic flow, it is found that the enhancement becomes even greater (up to 30%) with increases in corrugation. Moreover, the nonlinear Poisson–Boltzmann equation is solved by finite difference to study the electroosmotic flow in terms of the relative strength of the zeta potential. The issue of overlapped electric double layers when they are very thick is also discussed. The relative flow rate is shown to increase under the following conditions: (i) completely outofphase corrugations with long wavelength and large amplitude, (ii) small zeta potential, and (iii) slight overlapping of electric double layers.

Corrugated interfaces in multiphase coreannular flow
View Description Hide DescriptionMicrofluidic devices can be used to produce highly controlled and monodisperse double or multiple emulsions. The presence of inner drops inside a jet of the middle phase introduces deformations in the jet, which leads to breakup into monodisperse double emulsions. However, the ability to generate double emulsions can be compromised when the interfacial tension between the middle and outer phases is low, leading to flow with high capillary and Weber numbers. In this case, the interface between the fluids is initially deformed by the inner drops but the jet does not break into drops. Instead, the jet becomes highly corrugated, which prevents formation of controlled double emulsions. We show using numerical calculations that the corrugations are caused by the inner drops perturbing the interface and the perturbations are then advected by the flow into complex shapes.
 Interfacial Flows

Buoyancyinduced squeezing of a deformable drop through an axisymmetric ring constriction
View Description Hide DescriptionAxisymmetric boundaryintegral (BI) simulations were made for buoyancyinduced squeezing of a deformable drop through a ring constriction. The algorithm uses the Hebeker representation for the solidparticle contribution. A highorder, nearsingularity subtraction technique is essential for nearcritical squeezing. The drop velocity and minimum dropsolid spacing were determined for different ring and hole sizes, viscosity ratios, and Bond numbers, where the latter is a dimensionless ratio of gravitational to interfacial forces. The drop velocity decelerates typically 100fold or more, and the dropsolid spacing reduces to typically 0.1%–1% of the nondeformed drop radius as the drop passes through the constriction. The critical Bond number (below which trapping occurs) was determined for different conditions. For supercritical conditions, the nondimensional time required for the drop to pass through the ring increases for a fixed droptohole size with increasing viscosity ratio and decreasing Bond number, but it has a nonmonotonic dependence on the ratio of the radii of the drop and ring cross section. Numerical results indicate that the square of the drop squeezing time is inversely proportional to the Bond number minus the critical Bond number for nearcritical squeezing. The critical Bond number, determined from dynamic BI calculations, compares favorably to that obtained precisely from a static algorithm. The static algorithm uses the Young–Laplace equation to calculate the pendant and sessile portions of the drop interface coupled through the conditions of global pressure continuity and total drop volume conservation. Over a limited parameter space, the critical Bond number increases almost linearly with the droptohole ratio and is a weak function of the ratio of the ring crosssectional radius to the hole radius. Another dynamic phenomenon, in addition to drop squeezing, is a drop “dripping” around the outer edge of the ring constriction, and a critical Bond number maximum versus the droptototal ring radius ratio is caused by the transitions from squeezing to dripping for the loss of a drop steady state on a constriction. The initial stages of drop dripping are numerically simulated using a boundaryintegral method for slightly supercritical Bond numbers. For very large ratios of the droptohole radii, however, a sharp maximum in the critical Bond number is reached, as there is a transition from the drop passing through the inside hole to dripping over the outside edge of the ring for Bond numbers above the critical line. Drop squeezing and trapping mechanisms are also observed experimentally, and the measured critical Bond numbers and trapped drop shapes compare favorably to theoretical calculations from the Young–Laplace algorithm.

Twodimensional Stokes flow due to a pair of vortices below the free surface
View Description Hide DescriptionA twodimensional Stokes flow due to a pair of counterrotating vortices of equal strength below the free surface is analyzed, and the streamline pattern and freesurface deformation are discussed. Two vortices are placed at a fixed depth and an arbitrary distance between each other. In the analysis, Stokes’ approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained by using conformal mapping and complex function theory. From the solution, typical flow patterns are seen, depending on the capillary number, Ca, and the distance between the two vortices, and some interesting results are obtained. For separation distances below a critical distance, a cusp occurs at the center of the free surface as , following the results of Jeong and Moffatt [“Freesurface cusps associated with flow at low Reynolds number,” J. Fluid Mech.241, 1 (1992)] for no separation (distance of zero). However, above the critical distance, the cusp disappears and a smooth, troughshaped interface is formed. At even greater separation distances, a pair of viscous eddies exists near the free surface beyond some critical values of Ca. As the capillary number vanishes, the solution is reduced to that of a linearized potential flow.

Impact of a compound droplet on a flat surface: A model for single cell epitaxy
View Description Hide DescriptionThe impact and spreading of a compound viscousdroplet on a flat surface are studied computationally using a fronttracking method as a model for the single cell epitaxy. This is a technology developed to create twodimensional and threedimensional tissue constructs cell by cell by printing cellencapsulating droplets precisely on a substrate using an existing inkjet printing method. The success of cell printing mainly depends on the cell viability during the printing process, which requires a deeper understanding of the impact dynamics of encapsulated cells onto a solid surface. The present study is a first step in developing a model for deposition of cellencapsulating droplets. The inner droplet representing the cell, the encapsulating droplet, and the ambient fluid are all assumed to be Newtonian. Simulations are performed for a range of dimensionless parameters to probe the deformation and rate of deformation of the encapsulated cell, which are both hypothesized to be related to cell damage. The deformation of the inner droplet consistently increases: as the Reynolds number increases; as the diameter ratio of the encapsulating droplet to the cell decreases; as the ratio of surface tensions of the airsolution interface to the solutioncell interface increases; as the viscosity ratio of the cell to encapsulating droplet decreases; or as the equilibrium contact angle decreases. It is observed that maximum deformation for a range of Weber numbers has (at least) one local minimum at . Thereafter, the effects of cell deformation on viability are estimated by employing a correlation based on the experimental data of compression of cells between parallel plates. These results provide insight into achieving optimal parameter ranges for maximal cell viability during cell printing.

Accurate series solutions for gravitydriven Stokes waves
View Description Hide DescriptionIn the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of roundoff error, allowing a much greater number of terms to be found with confidence. It is shown that a higher order behavior of the series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Padé approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Padé approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Dynamics of the triple contact line on a nonisothermal heater at partial wetting
View Description Hide DescriptionThe dynamics of the triple gasliquidsolid contact line is analyzed for the case where the gas is the saturated vapor corresponding to the liquid. For partial wetting conditions, a nonstationary contact line problem where the contact line motion is caused by evaporation or condensation is treated. It is shown that the Navier slip condition alone is not sufficient to relax the hydrodynamic contact line singularity: the Marangoni term is equally important when the heat transfer is involved. The transient heat conduction inside the heater is accounted for. A multiscale problem of drop evaporation with freely moving contact line is solved in the lubrication approximation as an illustration of the proposed approach.
 Viscous and NonNewtonian Flows

The action of waving cylindrical tails with noncircular crosssection in propelling microrobots
View Description Hide DescriptionWith the advent of microtechnologies, manufacturing of swimming microrobots that mimic the motion of microorganisms has become feasible. Based upon the work of Taylor [“The action of waving cylindrical tails in propelling microscopic organisms,” Proc. R. Soc. London, Ser. A209, 225 (1951)], the creeping flow induced by a noncircular swimming tail waving in a plane or in spirals was investigated. Tails with rectangular, elliptic, and trapezoidal crosssections were examined, the latter being the most commonly fabricated microtail. It was observed that for a given crosssection area and propagating wave velocity the trapezoidal crosssection yields the highest tail velocity, whereas the elliptic tail results in the lowest one. Generally, it was obtained that if the crosssection deviation from circularity is expressed by a Fourier series expansion only the symmetric second harmonic affects the propulsion of the tail provided that the wave amplitude is smaller than the crosssection mean radius and of the order of the deviation from circularity. It was also shown that for a planar wave propagating velocity, a higher swimming velocity is obtained if the wider side of the noncircular crosssection faces the waving motion. For helical tails, first order effects of noncircularity on the swimming velocity vanish.
 Particulate, Multiphase, and Granular Flows

A constitutive equation for droplet distribution in unidirectional flows of dilute emulsions for low capillary numbers
View Description Hide DescriptionThe concentration distribution of droplets in the unidirectional flow of an emulsion for small capillary numbers (Ca) can be written as a balance between the drift flux arising from dropletdeformation and the flux due to shear induced migration. The droplet drift flux is modeled using the O(Ca) theoretical results of Chan and Leal [J. Fluid Mech.92, 131 (1979)], while the flux due to shearinduced migration is modeled using the suspension balance approach of Nott and Brady [J. Fluid Mech.275, 157 (1994)], whereby particle migration is ascribed to normal stress gradients in the flowing dilute emulsion. In the limit of vanishingly small capillary numbers, the leading order contribution of the normal stresses in dilute emulsions arises from dropletdroplet interaction and thus scales as , where is the droplet volume fraction and is the local shear stress. In our model, the normal stress calculations of Zinchenko [Prikl. Mat. Mekh.47, 56 (1984)] are connected to our gradient diffusivity data computed from droplet trajectories [M. Loewenberg and E. J. Hinch, J. Fluid Mech.338, 299 (1997)] via a reduced droplet mobility to derive the droplet flux due to shearinduced migration. As an example, the model is applied to the tube Poiseuille flow of a dilute emulsion at small Ca. It is demonstrated that the unsteady concentration distribution of droplets resulting from arbitrary timedependent average velocity obeys a selfsimilar solution, provided the thickness of the dropletdepleted region near the walls is always nonzero.
 Laminar Flows

Nature of counterflow and circulation in vortex separators
View Description Hide DescriptionThis paper focuses on the physical mechanism of elongated counterflows occurring in vortex tubes and hydrocyclones. To this end, a new solution to the Navier–Stokes equations is obtained which describes a flow pattern consisting of two throughflows and the global meridional circulation. One of the throughflows has Ushape geometry. It is shown that swirl decay due to fluidwall friction induces both the Ushape throughflow and the circulation. The circulation does not deteriorate particle separation. The solution illustrates how the swirlinduced pressure distribution drives the counterflow and results in the paradoxical centrifugal stratification where the highdensity fluid located at the periphery is hot while the lowdensity fluid located near the axis is cold.

Vortex shedding in the wake of a step cylinder
View Description Hide DescriptionFlow past a circular cylinder with a single stepwise discontinuity in diameter was investigated numerically for the diameter ratio and two Reynolds numbers, and 300. The primary focus was on vortex shedding and vortex interactions occurring in the cylinder wake. In agreement with previous experimental findings, three distinct spanwise vortexcells were identified in the stepcylinder wake: a single vortex shedding cell in the wake of the small cylinder (the Scell) and two vortex shedding cells in the wake of the large cylinder, one in the region downstream of the step (the Ncell) and the other away from the step (the Lcell). Due to the differences in vortex shedding frequencies, complex vortex connections occurred in two vortex interaction regions located between the adjacent cells. However, distinct differences in vortex splitting and vortex dislocations were identified in the two regions. The region at the boundary between the Scell and the Ncell was relatively narrow and its spanwise extent did not fluctuate significantly. In this region, vortex dislocations manifested as halfloop connections between two Scell vortices of opposite sign. In contrast, the region at the boundary between the Ncell and the Lcell exhibited transient behavior, with large scale vortex dislocations causing cyclic variation in the extent of Ncell vortices. Spectral analysis of velocity data showed that the presence of the Ncell was continuous through all simulations. For , small scale streamwise vortices forming in the wake of the large cylinder weaken the primary spanwise vortices and vortex connections, complicating vortex dynamics in the stepcylinder wake. However, no significant Reynolds number effect on the average spanwise extent of the vortexcells and the two transition regions between neighboring cells was observed. Finally, formation of Ncell vortices was shown to be linked to downwash fluctuations near the step.
 Instability and Transition

Marginal turbulent magnetohydrodynamic flow in a square duct
View Description Hide DescriptionDirect numerical simulations using a highorder finitedifference method were performed of the turbulent flow in a straight square duct in a transverse magnetic field. Without magnetic field the turbulence can be maintained for values of the bulk Reynolds number above approximately [M. Uhlmann et al., “Marginally turbulent flow in a square duct,” J. Fluid Mech.588, 153 (2007)]. In the magnetohydrodynamic case this minimal value of the bulk Reynolds number increases with the Hartmann number. The flow is laminar at when the Hartmann number is larger than and the flow is turbulent for . The secondary mean flow structure at consists of eight vortices located mainly at the Hartmann walls.

Travelingwaves consistent with turbulencedriven secondary flow in a square duct
View Description Hide DescriptionWe present numerically determined travelingwave solutions for pressuredriven flow through a straight duct with a square cross section. This family of solutions represents typical coherent structures (a staggered array of counterrotating streamwise vortices and an associated lowspeed streak) on each wall. Their streamwise average flow in the crosssectional plane corresponds to an eightvortex pattern much alike the secondary flow found in the turbulent regime.
 Turbulent Flows

On the structure and dynamics of sheared and rotating turbulence: Anisotropy properties and geometrical scaledependent statistics
View Description Hide DescriptionThis study is based on a series of nine direct numerical simulations of homogeneous turbulence, in which the rotation ratio of Coriolis parameter to shear rate is varied. The presence of rotation stabilizes the flow, except for a narrow range of rotation ratios . The main mechanism for the flow’s destabilization is an increased turbulence production due to increased anisotropy.Reynolds stress and the dissipation rate anisotropytensors have been evaluated and provide a reference for newly defined anisotropy measures. Waveletbased directional energies capture the properties of velocity gradients. The intermittency of the flow in different directions is quantified with scaledependent directional flatness. Scaledependent helicity probability distribution functions allow one to statistically characterize the geometry of the motion at different scales. Small scales are found locally to be predominantly helical, while large scales are not since they tend to twodimensionalization for cases with growing turbulent kinetic energy. Joint probability distribution functions show that the signs of velocity helicity and vorticity helicity are strongly correlated. This indicates that vorticity helicity tends to diminish velocity helicity.

Langevin and diffusion equation of turbulent fluid flow
View Description Hide DescriptionA derivation of the Langevin and diffusionequations describing the statistics of fluid particle displacement and passive admixture in turbulent flow is presented. Use is made of perturbation expansions. The small parameter is the inverse of the Kolmogorov constant , which arises from Lagrangian similarity theory. The value of in high Reynolds number turbulence is 5–6. To achieve sufficient accuracy, formulations are not limited to terms of leading order in including terms next to leading order in as well. Results of turbulencetheory and statistical mechanics are invoked to arrive at the descriptions of the Langevin and diffusionequations, which are unique up to truncated terms of in displacement statistics. Errors due to truncation are indicated to amount to a few percent. The coefficients of the presented Langevin and diffusionequations are specified by fixedpoint averages of the Eulerian velocity field. The equations apply to general turbulent flow in which fixedpoint Eulerian velocity statistics are nonGaussian to a degree of . The equations provide the means to calculate and analyze turbulent dispersion of passive or almost passive admixture such as fumes, smoke, and aerosols in areas ranging from atmospheric fluid motion to flows in engineering devices.