Volume 19, Issue 2, February 2007
 LETTERS


Rapidly sheared homogeneous stratified turbulence in a rotating frame
View Description Hide DescriptionRapid distortion theory is applied to stratified homogeneous turbulence that is sheared in a rotating frame. Insight into the stabilizing and destabilizing effects of the combined stratification and frame rotation is gained by considering initial fields that are twodimensional, with the axis of independence aligned with the flow direction. For these conditions, we derive solutions for the Fourier components of the flow variables, and for onepoint statistics in physical space. The analytical results are in qualitative agreement with the exact numerical solution for initially isotropic homogeneous turbulence, and they could be a reference point for the development of turbulencemodels.
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 ARTICLES

 Interfacial Flows

Boundary conditions for the moving contact line problem
View Description Hide DescriptionThe physical processes near a moving contact line are investigated systematically using molecular dynamics and continuum mechanics. Constitutive relations for the friction force in the contact line region, the fluidfluid interfacial force, and the stresses in the fluidsolid interfacial region are studied. Verification of force balance demonstrates the importance of the normal stress jump across the contact line region. Effective boundary conditions are derived using force balance. It is found that in the flow regime studied, the deviation of the wall contact angle from the equilibrium contact angle is proportional to the velocity of the contact line. The effective continuum model is solved numerically and the behavior of the apparent contact angle and the wall contact angle is studied. It is found that the fluidfluidinterface near the wall exhibits a universal behavior. The onset of the nonlinear response for the contact line motion is studied within the framework of Blake’s molecular kinetic theory.

Fixed contact line helical interfaces in zero gravity
View Description Hide DescriptionFluid interfaces supported in microgravity by a helical structure are shown to have a more robust stability than more common structures such as liquid bridges. In particular, helical interfaces can take the form of infinite right circular cylinders over a broad range of configurations. In the case of a single fixed contact line support, the infinite cylinder is stable for all cases in which the pitch to diameter ratio is less than (more tightly coiled interfaces). When there are two or more equally spaced fixed contact line supports, the infinite cylinder is stable for all configurations. Furthermore, in the two support case (the double helix), stability persists for all volumes from the cylinder to zero volume, when the pitch to diameter ratio is greater than (more loosely coiled interfaces). The equivalent to the axisymmetric YoungLaplace equation is derived for helical interfaces. Interfacial stability is determined from equilibrium branch structure following the application of Maddocks’ method by Lowry and Steen [Proc. R. Soc. London, Ser. A449, 411 (1995)]. Perturbations to finite wavelength disturbances are considered for the case of a single helical support. Overall stability envelopes are presented for single and multiple support cases. Limited experimental results verify the infinite length stability limit for the single helical support case.
 Viscous and NonNewtonian Flows

Ericksen number and Deborah number cascade predictions of a model for liquid crystalline polymers for simple shear flow
View Description Hide DescriptionWe consider the behavior of the DoiMarrucciGreco (DMG) model for nematic liquid crystalline polymers in planar shear flow. We found the DMG model to exhibit dynamics in both qualitative and quantitative agreement with experimental observations reported by Larson and Mead [Liq. Cryst.15, 151 (1993)] for the Ericksen number and Deborah number cascades. For increasing shear rates within the Ericksen number cascade, the DMG model displays three distinct regimes: stable simple shear, stable roll cells, and irregular structure accompanied by disclination formation. In accordance with experimental observations, the model predicts both and disclinations. Although defects form via the ridgesplitting mechanism first identified by Feng, Tao, and Leal [J. Fluid Mech.449, 179 (2001)], a new mechanism is identified for the formation of defects. Within the Deborah number cascade, with increasing Deborah number, the DMG model exhibits a streamwise banded texture, in the absence of disclinations and roll cells, followed by a monodomain wherein the mean orientation lies within the shear plane throughout the domain.

Experimental investigation of the effects of copolymer surfactants on flowinduced coalescence of drops
View Description Hide DescriptionFlowinduced coalescence of a pair of polymeric drops was studied at the level of individual drops, using a fourroll mill, to understand how the process is affected by the presence of a copolymer at the drop interface. The experimental system consisted of polybutadiene (PBd) drops suspended in polydimethylsiloxane(PDMS).Copolymers were produced at the drop interface by a reaction between functionalized homopolymers (PBdCOOH and PDMS). The experiments were carried out over wider ranges of parameters than our earlier studies of Hu et al. [Phys. Fluids12, 484 (2000)] and Ha et al. [Phys. Fluids15, 849 (2003)], in an attempt to understand the puzzling results found in our earlier studies. The experimental results were consistent with a qualitative mechanism of immobilization of the boundaries of the thin film between drops due to a flowinduced Marangoni effect. A critical or minimum copolymer interfacial coverage exists, above which the copolymer effect becomes independent of the coverage or the viscosity ratio. Using selfconsistent mean field theory, the was found to be approximately corresponding to , which is around 30% of the saturation concentration, . However, a whole new set of phenomena was discovered when the copolymer coverage is smaller . In this case, we found that there was a strong surfactant effect at small values, but that there was a transition capillary number above which the Marangoni effect apparently becomes negligible. In this case, two critical capillary numbers for coalescence ( and ) exist, and there are two ranges of and offset where coalescence is possible. The first is for and small offsets. The second is for , where has almost the same values as the critical capillary number for a clean interface system. Between and , coalescence is not possible. For the copolymer systems, coalescence at occurred at the angle just prior to the apparent separation of the drops in the extensional quadrant. The nonmonotonic change in with copolymer concentration, found in our earlier study, is due to the fact that the separation angle increases with increased concentration, as can be seen by examination of the collision trajectory data. A copolymer with a smaller molecular weight was also used to probe the potential significance of nonhydrodynamic effects related to the molecular weight. We observed the same saturated limit for the copolymer effect (when ) as in the case of the higher molecular weight copolymer system. We conclude that the Marangoni effect is the main mechanism for the suppression of coalescence in the current polymer/copolymer system.

Vortex shedding in confined swirling flow of polymer solutions
View Description Hide DescriptionThe unsteady swirling flow of an aqueous polymer solution due to a rotating disk in a cylindrical casing was investigated using the flow visualization technique. As the aqueous polymer solution, polyacrylamide solutions whose concentrations were 0.025, 0.1, 0.2, 0.5, 0.8, and were used. The unsteady secondary flow patterns were classified using the Reynolds and elastic numbers. We found a new phenomenon of vortex shedding in which the ring vortex formed near the rotating disk was periodically shed away from the rotating disk in the unsteady flow regime. The nonaxisymmetric ring vortex was also observed for the higher Reynolds number compared to that of the axisymmetric ring vortex. The dependence of the period of vortex shedding on the Reynolds number was clarified.
 Particulate, Multiphase, and Granular Flows

Leukocyte margination in a model microvessel
View Description Hide DescriptionThe physiological inflammation response depends upon the multibody interactions of blood cells in the microcirculation that bring leukocytes (white blood cells) to the vessel walls. We investigate the fluid mechanics of this using numerical simulations of 29 red blood cells and one leukocyte flowing in a twodimensional microvessel, with the cells modeled as linearly elastic shell membranes. Despite its obvious simplifications, this model successfully reproduces the increasingly blunted velocity profiles and increased leukocyte margination observed at lower shear rates in actual microvessels. Red cell aggregation is shown to be unnecessary for margination. The relative stiffness of the red cells in our simulations is varied by over a factor of 10, but the margination is found to be much less correlated with this than it is to changes associated with the blunting of the mean velocity profile at lower shear rates. While velocity around the leukocyte when it is near the wall depends upon the red cell properties, it changes little for strongly versus weakly marginating cases. In the more strongly marginating cases, however, a red cell is frequently observed to be leaning on the upstream side of the leukocyte and appears to stabilize it, preventing other red cells from coming between it and the wall. A wellknown feature of the microcirculation is a nearwall cellfree layer. In our simulations, it is observed that the leukocyte’s most probable position is at the edge of this layer. This wall standoff distance increases with velocity following a scaling that would be expected for a lubrication mechanism, assuming that there were a nearly constant force pushing the cells toward the wall. The leukocyte’s nearwall position is observed to be less stable with increasing mean standoff distance, but this distance would have potentially greater effect on adhesion since the range of the molecular binding is so short.

Numerical simulations of granular materials flow around obstacles: The role of the interstitial gas
View Description Hide DescriptionA twofluid model was used to determine the influence of the gas phase on granular flow interaction with obstacles. The governing equations of twodimensional, twophase flow were solved numerically using a finitevolumebased numerical code. The numerical results are qualitatively compared to experimental observations from various sources, and good agreement is found. Several cases were tested in order to examine the role of the velocities of both phases, solid volume fraction, particle diameter, and gravity. The results show that under certain conditions, particularly for dilute flows, a bow granular shock wave clearly forms in the front of the obstacle. Alternatively, for dense flows, granular shock waves were not observed. Based on the present observations, it appears that the formation and the shape of the bow granular shock wave are influenced by the interaction between the solid and gas phases and the gravity force. The results indicate that the common assumptions of neglecting both the gravity and the influence of the gas phase on the granular flow, may be appropriate only in the vicinity of the obstacle, where granular creeping flow takes place.
 Instability and Transition

Hydrodynamic stability of rockets with headwall injection
View Description Hide DescriptionWe investigate the hydrodynamicinstability of the fulllength, cylindrical models of solid and hybrid rockets with headwall injection. Our baseline is the rotational incompressible flowfield proposed in a recent study (Majdalani and Vyas, “Inviscid models of the classic hybrid rocket,” AIAA Paper 20043474). The local nonparallel approach is implemented in which the amplitude functions are assumed to be radially dependent at fixed streamwise locations. The usual singularity along the chamber axis is eliminated using Taylor series expansions. As a result, three compatibility relations are derived and substituted for the local boundary conditions along the axis. These depend on whether the tangential wave number is 0, 1, or larger. Our rotational model is shown to exhibit a range of instability that broadens with successive increases in headwall injection. The lowest frequency below which the flow remains unconditionally stable is observed at regardless of the headwall injection rate. As usual, the zeroth order tangential mode is found to be the most amplified. Using a representative headwall injection velocity for hybrid rockets, we identify a range of frequencies along which large excursions in pressure and velocity amplitudes are possible. These surges signal the presence of a resonantlike mechanism that is akin to an acoustic instability response. The most excited frequencies vary between 387 and 415 in the vicinity of the headwall. These frequencies are spatially delayed and lowered to 93.8–163.5 when the headwall injection rate is reduced to the level associated with solid rockets. For the most critical streamwise stations, these resurging wave amplitudes are quantified and shown to exhibit spectra that mimic the waterfall data acquired in acoustic instability tests.

Linear impulse response in hot round jets
View Description Hide DescriptionThe linear impulse response of axisymmetric jets is examined for a family of variabletemperature profiles typical of the potential core. The influence of jet heating, shear layer thickness, and Reynolds and Mach numbers on the spatiotemporal stability of both axisymmetric and helical modes is investigated. The linear impulse response is retrieved from a numerical solution of the spatial eigenvalue problem, which is derived from the fully compressible equations of motion. Changes in the spatiotemporal stability of heated versus isothermal jets are shown to arise solely from the effect of the baroclinic torque. By considering the full linear impulse response, the competition between jet column modes and shear layer modes is characterized. Jet column modes are only found to occur for axisymmetric disturbances. In thin shear layer jets, the jet column mode is shown to prevail at low group velocities, whereas axisymmetric and helical shear layer modes dominate at high group velocities. The absolute mode of zero group velocity is found to always be of the jet column type. Although only convectively unstable, the maximum growth rates of the shear layer modes greatly exceed those of the jet column modes in thin shear layer jets. In thick shear layer jets, axisymmetric modes of mixed jet column/shear layer type arise. The weakened maximum growth rate of mixed modes accounts for the dominance of helical modes in temporal stability studies of thick shear layer jets.

Stochastic estimation of a separatedflow field using wallpressurearray measurements
View Description Hide DescriptionConcurrent, surfacepressure and planar, particle image velocimetry(PIV)measurements were obtained in the separating/reattaching flow region downstream of an axisymmetric, backwardfacing step at a Reynolds number of 8081, based on step height. The surfacepressure and PIVmeasurements were used to investigate the evolution of coherent structures in the flow field by employing proper orthogonal decomposition (POD) and multipoint, linear, stochastic estimation (mLSE) analysis techniques. POD was used to determine the dominant modes in the pressure signature, while mLSE was used to estimate the dominant flow structures above the wall from the wallpressure POD modes over a series of time steps. It was found that a largescale, coherent structure develops in place (i.e., temporally) at approximately half the reattachment distance. Once this structure reaches a height equivalent to the step, it sheds and accelerates downstream. This growth in place, and then shedding, resembles the evolution of the flow structure in the wake of bluff bodies. Such a “wake mode” has been observed in numericalsimulation studies of long cavities and backwardfacing steps, where flow two dimensionality is controllable. The present study shows for the first time evidence for the existence of a wake mode in an experimental study of a backwardfacing step. This is believed to relate to the quality of the two dimensionality (i.e., axisymmetry) of the test geometry and the ability to track the temporal evolution of structural features through mLSE.

Highresolution simulations and modeling of reshocked singlemode RichtmyerMeshkov instability: Comparison to experimental data and to amplitude growth model predictions
View Description Hide DescriptionThe reshocked singlemode RichtmyerMeshkov instability is simulated in two spatial dimensions using the fifth and ninthorder weighted essentially nonoscillatory shockcapturing method with uniform spatial resolution of points per initial perturbation wavelength. The initial conditions and computational domain are modeled after the singlemode, Mach air(acetone)/shock tubeexperiment of Collins and Jacobs [J. Fluid Mech.464, 113 (2002)]. The simulation densities are shown to be in very good agreement with the corrected experimental planar laserinduced fluorescence images at selected times before reshock of the evolving interface. Analytical, semianalytical, and phenomenological linear and nonlinear, impulsive, perturbation, and potential flow models for singlemode RichtmyerMeshkov unstable perturbation growth are summarized. The simulation amplitudes are shown to be in very good agreement with the experimental data and with the predictions of linear amplitude growth models for small times, and with those of nonlinear amplitude growth models at later times up to the time at which the driverbased expansion in the experiment (but not present in the simulations or models) expands the layer before reshock. The qualitative and quantitative differences between the fifth and ninthorder simulation results are discussed. Using a local and global quantitative metric, the prediction of the Zhang and Sohn [Phys. Fluids9, 1106 (1997)] nonlinear Padé model is shown to be in best overall agreement with the simulation amplitudes before reshock. The sensitivity of the amplitude growth model predictions to the initial growth rate from linear instability theory, the postshock Atwood number and amplitude, and the velocity jump due to the passage of the shock through the interface is also investigated numerically.

Stability of gravitydriven freesurface flow past a deformable solid at zero and finite Reynolds number
View Description Hide DescriptionThe linear stability of Newtonian liquidflow down an inclined plane lined with a deformableelasticsolid layer is analyzed at zero and finite Reynolds number. There are two qualitatively different interfacial modes in this composite system: the freesurface or gasliquid (“GL”) mode which becomes unstable at low wave numbers and nonzero Reynolds number in flow down a rigid plane, and the liquidsolid (“LS”) mode which could become unstable even in the absence of inertia at finite wave numbers when the solid layer is deformable. The objectives of this work are to examine the effect of solid layer deformability on the GL and LS modes at zero and finite inertia, and to critically assess prior predictions concerning GL mode instability suppression at finite inertia obtained using the linear elasticmodel by comparison with the more rigorous neoHookean model for the solid. In the creepingflow limit where the GL mode instability is absent in a rigid incline, we show that for both solidmodels, the GL and LS modes become unstable at finite wavelengths when the solid layer becomes sufficiently soft. At finite wavelengths, the labeling of the two interfacial modes as GL and LS becomes arbitrary because these two modes get “switched” when the solid layer becomes sufficiently deformable. The critical strain required for instability becomes independent of the solid thickness (at high enough values of thickness) for both GL and LS modes in the linear elasticsolid, while it decreases with the thickness of the neoHookean solid. At finite Reynolds number, it is shown for both the solidmodels that the freesurface instability in flow down a rigid plane can be suppressed at all wavelengths by the deformability of the solid layer. The neutral curves associated with this instability suppression are identical for both linear elastic and neoHookean models. When the solid becomes even more deformable, both the GL and LS modes become unstable for finite wave numbers at nonzero inertia, but the corresponding neutral curves obtained from the two solidmodels differ significantly in detail. At finite inertia, for both the solidmodels, there is a significant window in the shear modulus of the solid for moderate values of solid thickness where both the GL and LS modes are stable at all wave numbers. Thus, using the neoHookean model, the present study reaffirms the prediction that soft elastomeric coatings offer a passive route to suppress and control interfacialinstabilities.
 Turbulent Flows

Helicity in the formation of turbulence
View Description Hide DescriptionHelicity in vortex structures and spectra is studied in the developmental stages of a numerical simulation of the NavierStokes equations using threedimensional visualizations and spectra. First, time scales are set using the growth and decay of energy dissipation, the peak value of vorticity, and the helicity. Then, two stages between the early time, nearly inviscid Euler dynamics with vortexsheets and a final state of fully developed turbulence with vortex tubes, are described. In the first stage, helicity fluctuations develop in Fourier space during a period still dominated by vortexsheets and rapidly growing peak vorticity. At the end of this period the strongest structure consists of transverse vortexsheets with mixed signs of helicity. During the second stage, a dissipative interaction propagates along one of these vortices as the sheets roll each other into vortex tubes.

Rapid shear of initially anisotropic turbulence in a rotating frame
View Description Hide DescriptionIn the present study, we investigate, using inviscid rapid distortion theory, the evolution of sheared turbulence in a rotating frame as a function of the rotation rate (including stable, transitional, and unstable regimes), and examine the sensitivity of the results for various nonisotropic initial conditions. Analytical solutions are derived for the evolution of the stresses and the structure dimensionality tensor components for three onedimensional and three twodimensional initializations. From these solutions, we calculate the asymptotic states of the turbulence, which are compared to the exact numerical solution of the threedimensional initially isotropic case. From the investigation it is shown that the qualitative characteristics of the isotropic solution in the unstable regime are represented quite accurately when the initial turbulence is dependent at least on the axis of the rotation of the frame. For the transitional and the stable regimes, though, the initial dependence of the turbulence on the axis of the mean flow is also crucial.

Vortex shedding from a circular cylinder near a moving ground
View Description Hide DescriptionThe flow and force characteristics have been experimentally investigated of a circular cylinder with an aspect ratio of 8.33, with and without endplates, placed near and parallel to a moving ground, on which substantially no boundary layer developed to interfere with the cylinder. Mean drag and lift measurements, surface oil flow visualization, and particle image velocimetry (PIV) measurements were carried out at two uppersubcritical Reynolds numbers of 0.4 and (based on the cylinder diameter ) to investigate the mechanisms of the ground effect, i.e., the effect of the gaptodiameter ratio , where is the gap between the cylinder and the ground. For the cylinder with endplates, on which the oil flow patterns were observed to be essentially twodimensional, the drag rapidly decreased as decreased to less than 0.5 but became constant for of less than 0.35, unlike that usually observed near a fixed ground. This critical drag behavior was found to be directly related to a global change in the near wake structure of the cylinder; as decreased the Kármántype vortex shedding became intermittent at , and then totally ceased and instead two nearly parallel shear layers were formed behind the cylinder at and below. For the cylinder without endplates, however, no such critical change in drag was observed as the Kármántype vortices were not generated in the near wake region at all investigated. Based on the experimental results obtained, further discussions are also given to the essential cause of the cessation of the Kármán vortex shedding in the ground effect.

Inertial range scaling of scalar flux spectra in uniformly sheared turbulence
View Description Hide DescriptionA model based on twopoint closure theory of turbulence is proposed and applied to study the Reynolds number dependency of the scalar flux spectra in homogeneous shear flow with a crossstream uniform scalar gradient. For the crossstream scalar flux, in the inertial range the spectral behavior agrees with classical predictions and measurements. The streamwise scalar flux is found to be in good agreement with the results of atmospheric measurements. However, both the model results and the atmospheric measurements disagree with classical predictions. A detailed analysis of the different terms in the evolution equation for the streamwise scalar flux spectrum shows that nonlinear contributions are governing the inertial subrange of this spectrum and that these contributions are relatively more important than for the crossstream flux. A new expression for the scalar flux spectra is proposed. It allows us to unify the description of the components in one single expression, leading to a classical inertial range for the crossstream component and to a new scaling for the streamwise component that agrees better with atmospheric measurements than the prediction of J. C. Wyngaard and O. R. Coté [Quart. J. R. Met. Soc.98, 590 (1972)].
 Geophysical Flows

Decay and return of internal solitary waves with rotation
View Description Hide DescriptionThe effect of rotation on the propagation of internal solitary waves is examined. Wave evolution is followed using a new rotating extension of a fully nonlinear, weakly nonhydrostatic theory for waves in a twolayer system. When a solitary wavesolution of the nonrotating equations is used as the initial condition, the wave initially decays by radiation of longer inertiagravity waves. The radiated inertiagravity wave always steepens, leading to the formation a secondary solitarylike wave. This decay and reemergence process then repeats. Eventually, a nearly localized wave packet emerges. It consists of a longwave envelope and shorter, faster solitarylike waves that propagate through the envelope. The radiation from this mature state is very weak, leading to a robust, longlived structure that may contain as much as 50% of the energy in the initial solitary wave. Interacting packets may either pass through one another, or merge to form a longer packet. The packets appear to be modulated, fully nonlinear versions of the steadily translating quasicnoidal waves.
 Others

On thermal diffusion and convection in multicomponent mixtures with application to the thermogravitational column
View Description Hide DescriptionThe theoretical framework for describing the multicomponent mixtures with the Soret effect is revised and extended. The separation ratio, a fundamental parameter characterizing the influence of thermal diffusion on convective phenomena, is generalized to the multicomponent case. It is shown how to define this parameter for a particular component of the mixture. To characterize multicomponent system as a whole, the net separation ratio , which does not depend on the choice of solvent, is introduced. Based on these results, the dimensionless equations for convection in multicomponent mixture are derived. The proposed formulation is applied to analyzing the steady state separation in the thermogravitational column (TGC). The approximation neglecting vertical diffusion in the column is employed and conditions for its validity are analyzed. The distributions of velocity, temperature, and composition in a multicomponent system are found. The relevant parameters here are the solutal Rayleigh numbers, which characterize the vertical separation of species, and the net solutal Rayleigh number , which quantifies the overall effect of this separation on the system behavior. It is shown that the key relation of TGC theory, which relates the vertical compositional gradients to the thermal diffusion coefficients, reduces to the dependence of on . A detailed study of this dependence is performed. The evolution of vertical velocity and density profiles with changing the net separation ratio is investigated. It is shown that the solution is unique for , nonunique for , and does not exist for . A particular ternary mixture is considered and its behavior in the column is analyzed.

Heat transfer between parallel plates: An approach based on the linearized Boltzmann equation for a binary mixture of rigidsphere gases
View Description Hide DescriptionAn analytical version of the discreteordinates method is used to develop a concise and particularly accurate solution of the heattransfer problem for a binary gas mixture confined between two parallel plates. The formulation of the problem allows general (speculardiffuse) Maxwellboundary conditions for each of the two types of particles and is based on a form of the linearized Boltzmann equation that incorporates recently established analytical expressions for the relevant rigidsphere kernels. Numerical results are reported for the density, the temperature, and the heatflow profiles relative to each species in and mixtures.