Volume 22, Issue 8, August 2010
Index of content:
We report on our direct numerical simulation of an incompressible, nominally zero-pressure-gradient flat-plate boundary layer from momentum thickness Reynolds number 80–1950. Heat transfer between the constant-temperature solid surface and the free-stream is also simulated with molecular Prandtl number . Skin-friction coefficient and other boundary layer parameters follow the Blasius solutions prior to the onset of turbulent spots. Throughout the entire flat-plate, the ratio of Stanton number and skin-friction 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 wall-pressure 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 near-wall 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 well-resolved signatures of hairpin vortices, which indicates the persistence of the hairpin forests.
22(2010); http://dx.doi.org/10.1063/1.3474703View Description Hide Description
Consider a two-dimensional 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 two-dimensional inverse cascade are briefly discussed.
22(2010); http://dx.doi.org/10.1063/1.3478308View Description Hide Description
This letter reports on the pronounced turbulence modulations and the accompanying drag reduction observed in a two-way coupled simulation of particle-laden 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.
22(2010); http://dx.doi.org/10.1063/1.3478311View Description Hide Description
Inertial focusing in a pressure-driven 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 cross-sections, 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 cross-section and the particle volume fraction. These results are rationalized using simulations of the particle-laden 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.
- Biofluid Mechanics
22(2010); http://dx.doi.org/10.1063/1.3467494View Description Hide Description
We 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 self-similar 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.
22(2010); http://dx.doi.org/10.1063/1.3469786View Description Hide Description
A 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 flagella-based 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
Electro-osmotic flow in a wavy microchannel: Coherence between the electric potential and the wall shape function22(2010); http://dx.doi.org/10.1063/1.3467035View Description Hide Description
The electro-osmotic 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 second-order 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 electro-osmotic flow can be enhanced at long-wavelength 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 electro-osmotic 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 electro-osmotic 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 out-of-phase corrugations with long wavelength and large amplitude, (ii) small zeta potential, and (iii) slight overlapping of electric double layers.
22(2010); http://dx.doi.org/10.1063/1.3480561View Description Hide Description
Microfluidic 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
22(2010); http://dx.doi.org/10.1063/1.3464343View Description Hide Description
Axisymmetric boundary-integral (BI) simulations were made for buoyancy-induced squeezing of a deformable drop through a ring constriction. The algorithm uses the Hebeker representation for the solid-particle contribution. A high-order, near-singularity subtraction technique is essential for near-critical squeezing. The drop velocity and minimum drop-solid 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 100-fold or more, and the drop-solid 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 drop-to-hole 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 near-critical 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 drop-to-hole ratio and is a weak function of the ratio of the ring cross-sectional 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 drop-to-total 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 boundary-integral method for slightly supercritical Bond numbers. For very large ratios of the drop-to-hole 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.
22(2010); http://dx.doi.org/10.1063/1.3473922View Description Hide Description
A two-dimensional Stokes flow due to a pair of counter-rotating vortices of equal strength below the free surface is analyzed, and the streamline pattern and free-surface 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 [“Free-surface 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, trough-shaped 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.
22(2010); http://dx.doi.org/10.1063/1.3475527View Description Hide Description
The impact and spreading of a compound viscousdroplet on a flat surface are studied computationally using a front-tracking method as a model for the single cell epitaxy. This is a technology developed to create two-dimensional and three-dimensional tissue constructs cell by cell by printing cell-encapsulating droplets precisely on a substrate using an existing ink-jet 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 cell-encapsulating 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 air-solution interface to the solution-cell 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.
22(2010); http://dx.doi.org/10.1063/1.3480394View Description Hide Description
In 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 round-off 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.
22(2010); http://dx.doi.org/10.1063/1.3483558View Description Hide Description
The dynamics of the triple gas-liquid-solid 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 Non-Newtonian Flows
22(2010); http://dx.doi.org/10.1063/1.3467040View Description Hide Description
With the advent of microtechnologies, manufacturing of swimming microrobots that mimic the motion of micro-organisms 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 cross-sections were examined, the latter being the most commonly fabricated microtail. It was observed that for a given cross-section area and propagating wave velocity the trapezoidal cross-section yields the highest tail velocity, whereas the elliptic tail results in the lowest one. Generally, it was obtained that if the cross-section 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 cross-section 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 cross-section 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 numbers22(2010); http://dx.doi.org/10.1063/1.3466577View Description Hide Description
The 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 shear-induced 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 droplet-droplet 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 shear-induced 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 time-dependent average velocity obeys a self-similar solution, provided the thickness of the droplet-depleted region near the walls is always nonzero.
- Laminar Flows
22(2010); http://dx.doi.org/10.1063/1.3475818View Description Hide Description
This 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 through-flows and the global meridional circulation. One of the through-flows has U-shape geometry. It is shown that swirl decay due to fluid-wall friction induces both the U-shape through-flow and the circulation. The circulation does not deteriorate particle separation. The solution illustrates how the swirl-induced pressure distribution drives the counterflow and results in the paradoxical centrifugal stratification where the high-density fluid located at the periphery is hot while the low-density fluid located near the axis is cold.
22(2010); http://dx.doi.org/10.1063/1.3459157View Description Hide Description
Flow 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 step-cylinder wake: a single vortex shedding cell in the wake of the small cylinder (the S-cell) and two vortex shedding cells in the wake of the large cylinder, one in the region downstream of the step (the N-cell) and the other away from the step (the L-cell). 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 S-cell and the N-cell was relatively narrow and its spanwise extent did not fluctuate significantly. In this region, vortex dislocations manifested as half-loop connections between two S-cell vortices of opposite sign. In contrast, the region at the boundary between the N-cell and the L-cell exhibited transient behavior, with large scale vortex dislocations causing cyclic variation in the extent of N-cell vortices. Spectral analysis of velocity data showed that the presence of the N-cell 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 step-cylinder 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 N-cell vortices was shown to be linked to downwash fluctuations near the step.
- Instability and Transition
22(2010); http://dx.doi.org/10.1063/1.3478877View Description Hide Description
Direct numerical simulations using a high-order finite-difference 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.
22(2010); http://dx.doi.org/10.1063/1.3466661View Description Hide Description
We present numerically determined traveling-wave solutions for pressure-driven flow through a straight duct with a square cross section. This family of solutions represents typical coherent structures (a staggered array of counter-rotating streamwise vortices and an associated low-speed streak) on each wall. Their streamwise average flow in the cross-sectional plane corresponds to an eight-vortex 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 scale-dependent statistics22(2010); http://dx.doi.org/10.1063/1.3457167View Description Hide Description
This 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. Wavelet-based directional energies capture the properties of velocity gradients. The intermittency of the flow in different directions is quantified with scale-dependent directional flatness. Scale-dependent 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 two-dimensionalization 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.
22(2010); http://dx.doi.org/10.1063/1.3466658View Description Hide Description
A 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 fixed-point averages of the Eulerian velocity field. The equations apply to general turbulent flow in which fixed-point Eulerian velocity statistics are non-Gaussian 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.