Volume 18, Issue 2, February 2006
 LETTERS


Freesurface flows with large slopes: Beyond lubrication theory
View Description Hide DescriptionThe description of freesurface flows can often be simplified to thinfilm (or lubrication)equations, when the slopes of the liquidgas interface are small. Here, we present a longwavelength theory that remains fully quantitative for steep interface slopes, by expanding about Stokes flow in a wedge. For small capillary numbers, the variations of the interface slope are slow and can be treated perturbatively. This geometry occurs naturally for flows with contact lines: we quantify the difference with ordinary lubricationtheory through a numerical example and analytically recover the full CoxVoinov asymptotic solution.

The lowest oscillation mode of a pendant drop
View Description Hide DescriptionThe lowest oscillation mode of a pendant drop has long been conceived to be the longitudinal vibration, i.e., periodic elongation and contraction along the longitudinal direction. However, here we experimentally show that the rotation of the drop about the longitudinal axis is the oscillation mode of the lowest resonance frequency. This rotational mode can be invoked by periodic acoustic forcing and is analogous to the pendulum rotation, having the frequency independent of the drop density and surface tension but inversely proportional to the square root of the drop size.
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 ARTICLES

 Interfacial Flows

Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer
View Description Hide DescriptionThe breakup of surfactantladen drops and jets is of technological interest and fundamental scientific importance. Surfactants are routinely used to control the breakup of drops and jets in applications ranging from inkjet printing to crop spraying. Accurate computation of breakup of surfactantladen drops and jets is often the key to the development of new applications and to providing a rational fundamental understanding of both existing and emerging applications. While highly accurate algorithms for studying the breakup of surfactantfree drops and jets are well documented and much is now known about the dynamics in such situations, little is known by contrast about the closely related problem of interface rupture when surfactant effects cannot be neglected. The deformation and breakup of a stretching liquid bridge of an incompressible Newtonian fluid whose surface is covered with an insoluble surfactantmonolayer are analyzed here experimentally and computationally. In the experiments, highspeed visualization is used to capture the transient deformation of a bridge. The dynamic shapes of bridges (captive between two rods of 3.15 mm diameter) are captured and analyzed with a time resolution of 1 ms. The bridge lengths are 3.15 mm initially and about 4–7 mm at breakup, which occurs after stretching for about 0.1–0.2 s, depending on the volume and viscosity of the liquid and the surface density of spread monolayers. The dynamics of a surfactantcovered bridge is governed by the NavierStokes and convectiondiffusion equations. First, these equations are solved with a threedimensional, but axisymmetric, or twodimensional (2D), finite element algorithm using elliptic mesh generation. Second, the governing set of 2D equations is reduced to a set of onedimensional (1D) equations by means of the slenderjet approximation and the resulting set of 1D equations is solved with a 1D finite element algorithm. The presence of surfactant results not only in the lowering of surface tension and the capillary pressure, but also in surface tension gradients and Marangoni stresses, both of which affect the transient dynamics leading to breakup. In particular, the role of Marangoni stresses in delaying bridge breakup and on formation of satellite droplets is investigated as a function of the initial surface density and surface activity of the surfactant, and surface Peclet number that measures the importance of convection relative to diffusion. The predictions of the 2D algorithm are confirmed to be faithful to the physics by demonstrating that the computed results accord well with the experiments and existing scaling theories. In the pinchoff region, the surfactant is swept out of a thinning neck by strong convection. The calculations thus reveal that the scaling behavior in the presence of surfactant parallels that observed in the absence of surfactant, in accordance with recent reports by others. The 2D computations and the experiments are used in tandem to identify regions in the space of governing parameters where the 1D equations can be used with confidence.

Theoretical analysis of the effect of insoluble surfactant on the dip coating of chemically micropatterned surfaces
View Description Hide DescriptionMicrofluidic flow on chemically heterogeneous surfaces is a useful technique with applications ranging from selective material deposition to the selfassembly of nanostructures. The recent theoretical analysis by Davis [Phys. Fluids17, 038101 (2005)] of the dip coating of a pure fluid onto vertical, wetting stripes surrounded by nonwetting regions quantified the experimentally observed deviations from the classical LandauLevich result due to lateral confinement of the fluid by chemical surface patterning. In this present work, the analysis of dip coating of these heterogeneous surfaces is extended to a liquid containing an insoluble surfactant. Using matched asymptotic expansions based on lubrication theory in the limit of a small capillary number, the thickness of the deposited liquid film and the surfactant concentration in the deposited monolayer are predicted for a wide range of fluid properties and process parameters. The increase in the deposited film thickness is shown analytically to be limited by a multiplicative factor of times the result for a pure liquid. Numerical results demonstrate that the thickening due to Marangoni effects is nonmonotonic in the capillary number because of the competition between viscous stresses, Marangoni stresses, and surface diffusion.

The detachment of a viscous drop in a viscous solution in the presence of a soluble surfactant
View Description Hide DescriptionWhen a buoyant viscous drop is injected into a viscous fluid, it evolves to form a distended shape that detaches via the rapid formation and pinching of a neck. The effects of surfactants in altering this process are studied numerically. In the absence of surfactants, surface contraction is fastest in the vicinity of the neck. Thus, when surfactants are present, they accumulate there and alter the ensuing dynamics by reducing the surface tension that drives the contraction. The surface tension is described by a nonlinear surface equation of state that accounts for the maximum packing of surfactant in a monolayer. When surfactant adsorptiondesorption is very slow, interfaces dilute significantly during drop expansion, and drops form necks which are only slightly perturbed in their dynamics from the surfactantfree case. When adsorptiondesorption dynamics are comparable to the rate of expansion, drops thin to form a primary neck at low surfactant coverage, to form both primary and secondary necks at moderate coverages, form only a secondary neck at higher coverages, or fail to neck at elevated coverages. When surfactant adsorptiondesorption kinetics are rapid, the surface remains in equilibrium with the surrounding solution, and drops behave like surfactantfree drops with a uniform surface tension. These arguments are used to construct a phase diagram of drop neck shapes as a function of surfactant coverage. A map of neck/noneck thresholds is also constructed as a function of surfactant coverage and sorption dynamics, suggesting that drop detachment can be used as a means of characterizing surfactant dynamics.

Numerical analysis of the Rayleigh instability in capillary tubes: The influence of surfactant solubility
View Description Hide DescriptionA twodimensional (2D) free surface flow model already used to study the Rayleigh instability of thin films lining the interior of capillary tubes under the presence of insoluble surfactants [D. M. Campana, J. Di Paolo, and F. A. Saita, “A 2D model of Rayleigh instability in capillary tubes. Surfactant effects,” Int. J. Multiphase Flow30, 431 (2004)] is extended here to deal with soluble solutes. This new version that accounts for the mass transfer of surfactant in the bulk phase, as well as for its interfacial adsorption/desorption, is employed in this work to assess the influence of surfactantsolubility on the unstable evolution. We confirm previously reported findings: surfactants do not affect the system stability but the growth rate of the instability [D. R. Otis, M. Johnson, T. J. Pedley, and R. D. Kamm, “The role of pulmonary surfactant in airway closure,” J. Appl. Physiol.59, 1323 (1993)] and they do not change the successive shapes adopted by the liquid film as the instability develops [S. Kwak and C. Pozrikidis, “Effects of surfactants on the instability of a liquid thread or annular layer. Part I: Quiescent fluids,” Int. J. Multiphase Flow27, 1 (2001)]. Insoluble surfactants delay the instability process, and the time needed to form liquid lenses disconnecting the gas phase—i.e., the closure time—is four to five times larger than for pure liquids. This retardation effect is considerably reduced when the surfactants are somewhat soluble. For a typical system adopted as a reference case, detailed computed predictions are shown; among them, curves of closure time versus adsorption number are given for solubility values ranging from insoluble to highly soluble conditions. In addition, the evolution of the four mass transport terms appearing in the interfacial mass balance equation—normal and tangential convection,diffusion and sorption—is scrutinized to uncover the mechanisms by which surfactantsolubility affects the growth rate of the instability.
 Particulate, Multiphase, and Granular Flows

Thermal transport in sheared electro and magnetorheological fluids
View Description Hide DescriptionThermal energy transport in sheared electrorheological and magnetorheological (ER and MR) fluids is analyzed. Although energy production by viscous dissipation can be significant, energy transport on the particle length scale can be analyzed by ignoring viscous dissipation. For typical situations, energy transport normal to the flow direction is dominated by conduction. Particlelevel simulations were employed to determine the suspension structure as a function of Mason number and volume fraction. A selfconsistent meanfield dipole model is used to estimate the effective thermal conductivities for these simulated structures. The fieldinduced chainlike aggregates that form at small Mason number result in a larger effective thermal conductivity at small Mason number than at large Mason number. Effects of higherorder multipoles are estimated by analyzing effective thermal conductivities of model structures. For highly conducting particles, the effective thermal conductivity of a sheared ER or MR suspension is predicted to roughly double as the Mason number is decreased from the large to the small Mason number limits.
 Instability and Transition

Nonlinear instabilities in a vertical pipe flow discharging from a cylindrical container
View Description Hide DescriptionWe report results from threedimensional numerical simulations of the incompressible flow in a vertical pipe of circular crosssection discharging from a cylindrical container. Natural Coriolis forces due to Earth rotation trigger the instability of the axisymmetric flow, and nonlinear spiral waves with azimuthal wave number are formed above a critical Reynolds number based on the pipe flow rate . We characterize this critical Reynolds number as a function of the Coriolis parameter , that is proportional to the square of the radius of the container. As a difference with previous numerical works on nonlinear instabilities and transition in a pipe flow, here the nonlinear disturbances needed to trigger the instabilities are not artificially introduced inside the pipe flow, but naturally produced by Coriolis forces, the amplitude of these disturbances being characterized by a nondimensional Coriolis parameter. We find that the pipe flow can be unstable for as low as 300 for the largest value of considered. We also discuss the relevance of the residual swirl introduced by natural Coriolis forces in triggering the nonlinear traveling waves.

Nonlinear mechanics of wavy instability of steady longitudinal vortices and its effect on skin friction rise in boundary layer flow
View Description Hide DescriptionWavy secondary instability of steady longitudinal vortices in boundary layer flow is studied. The nonlinear interaction problem is parabolized through scaling guided by observations. Emphasis is placed on the nonlinear modification of the steady problem by the Reynolds stresses of the wavy disturbance. It is found that the skin friction in such a modification process increases well above the local turbulent boundary layer value.

Rayleigh’s instability of LennardJones liquid nanothreads simulated by molecular dynamics
View Description Hide DescriptionA liquid thread of radius will break up into drops if the axial wavelength of the surface perturbation . If , the thread is stable and will remain intact. This is Rayleigh’s stability criterion that was derived using a continuum model. We use molecular dynamics to simulate the evolution of LennardJones liquid threads with , , and , where is the equilibrium distance between two atoms. We find that Rayleigh’s stability criterion holds, even at the molecular scale.
 Turbulent Flows

An efficient wall model for largeeddy simulation based on optimal control theory
View Description Hide DescriptionLargeeddy simulation is currently very expensive for high Reynolds number attached flows due to the need to resolve the wall layer. In order to reduce this expense, wall modeling has been proposed to provide approximate boundary conditions to the LES that allow the wall layer to be unresolved. Unfortunately, subgrid scale modeling errors and numerical errors are important in this region when a coarse grid is used, necessitating a wall model that can compensate for their effects. Optimal control theory has been used to provide such models in the past, but is impractical for complex flows due to the need to provide a target mean velocity profile and the computational expense involved with solving gradientbased optimization problems. In this paper we address the latter issue by reformulation of the optimization problem to include only data near the wall. Further approximations have been made to the NavierStokes and adjoint equations used in the optimization process that significantly reduce the computational cost. Results will be presented comparing this method with other controlbased and standard wall models.

Mechanisms for deposition and resuspension of heavy particles in turbulent flow over wavy interfaces
View Description Hide DescriptionIt has been long recognized that turbulent flow over steep waves can produce coherent flowstructures of different temporal and spatial scales. In particular, quasistreamwise vortices grow up on the upslope side of the wave and interact with geometrydependent vortical structures, aligned spanwise and located within the recirculation bubble in the wave trough, thus creating the conditions for the development of a threedimensional highly turbulent flow field. In this work, we have analyzed the trajectories of small dense particles (either in solid form or in the form of liquid droplets) released into a turbulent air flow over waves precisely to clarify the role of coherent vortical structures in controlling particle deposition and resuspension. The threedimensional timedependent flow field at is calculated using largeeddy simulation, and the dynamics of individual differentsized particles is described using a Lagrangian approach. Drag, gravity, and lift are used in the equation of motion for particles that have no influence on the flow field. Particlewall interaction is fully elastic. Our findings show that differentsized particles interact selectively with vortical flowstructures, producing different distribution patterns and dispersion rates qualitatively depending on the particletofluid timescale ratio. Specifically, we find that quasistreamwise vortices on the upslope side of the wave control particle dispersion and eventual segregation in the flow separation region downstream the wave crest as well as in the shear layer forming behind the wave, just above the separation region. These vortices generate momentum mixing events that also entrain and move particles towards and away from the wavy wall. This process is similar to the sweep/ejection cycle occurring in turbulent flow over a flat boundary layer.

Stretching of polymers around the Kolmogorov scale in a turbulent shear flow
View Description Hide DescriptionWe present numerical studies of stretching of Hookean dumbbells in a turbulent NavierStokes flow with a linear mean profile, . In addition to the turbulence features beyond the viscous Kolmogorov scale , the dynamics at the equilibrium extension of the dumbbells significantly below is well resolved. The variation of the constant shear rate causes a change of the turbulent velocity fluctuations on all scales and thus of the intensity of local stretching rate of the advecting flow. The latter is measured by the maximum Lyapunov exponent which is found to increase as , in agreement with a dimensional argument. The ensemble of up to passively advected dumbbells is advanced by Brownian dynamics simulations in combination with a pseudospectral integration for the turbulentshear flow.Anisotropy of stretching is quantified by the statistics of the azimuthal angle which measures the alignment with the mean flow axis in the shear plane, and the polar angle which determines the orientation with respect to the shear plane. The asymmetry of the probability density function (PDF) of increases with growing shear rate . Further, the PDF becomes increasingly peaked around mean flow direction . In contrast, the PDF of the polar angle is symmetric and less sensitive to changes of .

Drag reduction in turbulent channel flow with periodically arrayed heating and cooling strips
View Description Hide DescriptionA new technique giving significant drag reduction in turbulent shear flows has been proposed by using the buoyancy effect to generate periodic spanwise motion. Such spanwise motion can be obtained by arranging heating and cooling strips periodically aligned in the spanwise direction of a vertical channel, where the streamwise mean flow is perpendicular to the gravity vector. The strip size has been changed in order to obtain the optimum size corresponding to the maximum drag reduction. Series of direct numerical simulation have been performed where the bulk Reynolds number, is fixed at 2270 whereas the Grashof number is changed to between and . At the lowest Grashof number, the buoyancy forces are not strong enough to disturb the flow and no apparent variation of drag compared to the plane channel is observed. However, as the Grashof number increases, considerable drag reduction can be obtained. At the highest Grashof number, an optimum strip size of about 250 wall units gives drag reduction of about 35%. The greater the Grashof number, the smaller the strip size which attains the maximum drag reduction. The similarity between the induced lateral motions by the buoyancy forces and those induced by the spanwise wall oscillations is highlighted.

Damping rate of magnetohydrodynamic vortices at low magnetic Reynolds number
View Description Hide DescriptionThe damping rate of vortices in an electrically conducting fluid submitted to a uniform magnetic field is analyzed for a large Hartmann number . The fluid is contained in a layer of constant thickness , bounded by two insulating walls that are perpendicular to the magnetic field. The damping times and the eigenfunctions along the magnetic field are obtained from a linear eigenvalue problem. According to the damping times and these eigenfunctions, vortices are classified into several classes by the range of combinations of the mode number in the magnetic field direction and the wave number in the plane perpendicular to the magnetic field. It is found that the damping rate of vortices in the range of and is of the same order as that of largescale twodimensional vortices. This fact suggests that actual quasitwodimensional magnetohydrodynamicturbulent flows include not only but also highermode eigenfunctions of this wavenumber range, although the eigenfunction of has a 30% variation and the highermode eigenfunctions change their sign along the magnetic field.

Octant analysis based structural relations for threedimensional turbulent boundary layers
View Description Hide DescriptionA flow structure based tripleproduct correlation model developed by Nagano and Tagawa [J. Fluid Mech.215, 639 (1990)] has been expanded to threedimensional turbulent flows. Threedimensional turbulent boundary layer data obtained away from the vortex in a wingbody junction flow are analyzed to calculate the contributions from eight velocity octants to the stresses and higherorder products. The analysis showed that the sweep and ejection modes dominate the flow physics of some shear stresses and some triple products, while the interaction modes are negligible away from the wall. These experimental observations are used together with the extended NaganoTagawa mathematical model to obtain relations among the triple products in threedimensional turbulent boundary layers that can simplify the turbulent diffusionmodeling used in Reynoldsaveraged NavierStokes equations. Results show that , , and triple product correlations can be modeled if an appropriate turbulencemodel is described for the triple product correlation, and that triple products correlations can be modeled if an appropriate turbulencemodel is described for the triple product correlation.

Synthetic turbulence inflow conditions for largeeddy simulation
View Description Hide DescriptionDirect numerical or largeeddy simulations of the majority of spatially inhomogeneous turbulent flows require turbulent inflow boundary conditions. A potential implication is that any results computed may be strongly influenced by the prescribed instantaneous inlet velocity profiles. Such profiles are practically never available, and a usual practice is to generate synthetic inflow data satisfying certain statistical properties, which may, for example, be known from experimental data or empirical correlations. The present paper describes a new method for generating turbulent inflow data based on digital filters that is capable of reproducing specified statistical data. Two variants of the approach are presented: a simple method in which the Reynolds stresses and a single length scale are prescribed, and a more detailed approach that is able to reproduce the complete Reynoldsstress tensor as well as any given, locally defined, spatial and temporal correlation functions. The application of the methods to a plane jet flow and to a developing wall boundary layer serve to demonstrate the applicability of the approach.

Subgrid modeling in particleladen channel flow
View Description Hide DescriptionDirect numerical simulation (DNS) and largeeddy simulation(LES) of particleladen turbulent channel flow, in which the particles experience a drag force, are investigated for two subgrid models and several Reynolds and Stokes numbers. In this flow, turbophoresis leads to an accumulation of particles near the walls. The objectives of the work are to investigate the accuracy of the subgrid models studied with respect to particle behavior and to explain the observed particle behavior predicted by the different models. The focus is on particle dispersion and mean particle motion in the direction normal to the walls of the channel. For a low Reynolds number, it is shown that the turbophoresis and particle velocityfluctuations are reduced compared to DNS, if the filtered fluid velocity calculated in the LES is used in the particle equation of motion. This is a combined effect of the disregard of the subgrid scales in the fluid velocity and the inadequacy of the subgrid model. Better agreement with DNS is obtained if an inverse filtering model, which was recently proposed, is incorporated into the particle equation. This model is shown to enhance turbophoresis and particle velocityfluctuations in actual LES. The results of the approximate deconvolution model(ADM) agree better with DNS results than results of the dynamic eddyviscosity model. This can be explained from the better prediction of the fluid velocity statistics by ADM and the better correspondence of the subgrid models adopted in the fluid and particle equations. Although the differences between the two subgrid models become smaller, similar conclusions are obtained at a higher Reynolds number. Compared to fourthorder interpolation of the fluid velocity to the particle position, secondorder interpolation approximately cancels the effect of the subgrid model in the particle equation of motion.
 Compressible Flows

Time evolution and mixing characteristics of hydrogen and ethylene transverse jets in supersonic crossflows
View Description Hide DescriptionWe report an experimental investigation that reveals significant differences in the nearflowfield properties of hydrogen and ethylene jets injected into a supersonic crossflow at a similar jettofreestream momentum flux ratio. Previously, the momentum flux ratio was found to be the main controlling parameter of the jet’s penetration. Current experiments, however, demonstrate that the transverse penetration of the ethylene jet was altered, penetrating deeper into the freestream than the hydrogen jet even for similar jettofreestream momentum flux ratios. Increased penetration depths of ethylene jets were attributed to the significant differences in the development of largescale coherent structures present in the jet shear layer. In the hydrogen case, the periodically formed eddies persist long distances downstream, while for ethylene injection, these eddies lose their coherence as the jet bends downstream. The large velocity difference between the ethylene jet and the freestream induces enhanced mixing at the jet shear layer as a result of the velocity induced stretchingtiltingtearing mechanism. These new observations became possible by the realization of high velocity and high temperature freestream conditions which could not be achieved in conventional facilities as have been widely used in previous studies. The freestream flow replicates a realistic supersonic combustor environment associated with a hypersonic airbreathing engine flying at Mach 10. The temporal evolution, the penetration, and the convection characteristics of both jets were observed using a fastframingrate (up to 100 MHz) camera acquiring eight consecutive schlieren images, while OH planar laserinduced fluorescence was performed to verify the molecular mixing.
 Geophysical Flows

Vortex cores, strain cells, and filaments in quasigeostrophic turbulence
View Description Hide DescriptionWe present numerical simulations of decaying twodimensional (2D) and threedimensional quasigeostrophic (3D QG) turbulence. The resulting vorticity fields are decomposed into three components: the vortex cores, the strain cells, and the background. In 2D, the vortex cores induce five times the energy as the background, while in 3D QG the background plays a more dominant role and induces the same amount of energy as the vortex cores, quantifying previous observations that 3D QG has a more active filamentary background. The probability density function of the total velocity field is nearly Gaussian in 3D QG but significantly less so in 2D. In both 2D and 3D QG, the velocities induced by the vortex cores and the strain cells are nonGaussian. In both 2D and 3D QG turbulence, the enstrophy spectrum of the background is close to predicted by inverse cascade theories.