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Volume 9, Issue 5, May 1997

An intermittency model for passivescalar turbulence
View Description Hide DescriptionA phenomenological model for the inertial range scaling of passivescalar turbulence is developed based on a bivariate logPoisson model. An analytical formula of the scaling exponent for threedimensional passivescalar turbulence is deduced. The predicted scaling exponents are compared with experimental measurements, showing good agreement.

On recent intermittency models of turbulence
View Description Hide DescriptionTwo recent intermittencymodels by Chen and Cao [Phys. Rev. E 52, R5757 (1995)] and Nelkin [Phys. Rev. E 52, R4610 (1995)] stemming from Novikov’s criticism [Phys. Rev. E 50, R3303 (1994)] on the She–Lévêque model [Phys. Rev. Lett. 72, 336 (1994)] are discussed. Both models are rewritten using (deviation from Kolmogorov 1941 of the sixth order structure function exponent) values computed numerically and using longitudinal and lateral velocity differences). Scaling exponents are compared.

Conduction in the small gap between two spheres
View Description Hide DescriptionA solution to the conduction equation has been developed for two equal size, nonconducting spheres with the line between centers perpendicular to the applied field. The solution, valid when the gap between the spheres is small compared to their radius, is based on a matched asymptotic expansion. For the case when the conductivity is uniform everywhere (i.e., Laplace’s equation), the solution agrees well with numerical results obtained from an infinite series solution in bispherical coordinates. An example with a nonuniform conductivity in the gap is presented to demonstrate how the method can be extended to more general conduction problems.

Analytic solutions for Stokes flow past a partially encapsulated droplet
View Description Hide DescriptionThe Stokes flow past the unusual geometry of two fused (overlapping) spheres, somewhat surprisingly, has an exact analytic solution, provided that the spheres intersect orthogonally. The problem of flow past a partially encapsulated droplet is considered, to illustrate the basic idea. The simple nature of the analytical solution is of some interest in biophysical models of molecular assemblies involving a combination of stick and slip surfaces.

The conformation change of model polymers in stochastic flow fields: Flow through fixed beds
View Description Hide DescriptionSimulations of the conformation change of model polymers in various steady, anisotropic Gaussian random flow fields are presented. These flow fields have been chosen because they are models for the flow through porous media and have been predicted to be “stochastic strong flows” according to the criteria developed by Shaqfeh and Koch [J. Fluid Mech. 244, 17 (1992)]. To be specific, beyond a certain Deborah number (based on the sampling time of a velocity fluctuation), large average conformation change in the polymer is predicted. In our simulations, the polymers are modeled as dumbbells, but beyond this restriction, the assumptions of the theory by Shaqfeh and Koch are removed. Many realizations of the Gaussian fields are synthesized spectrally following a modified version of the method developed by Kraichnan [Phys. Fluids 13, 22 (1970)]. Moreover, the ratio of the mean “plug” flow to the amplitude of the fluctuations is varied from meandominant to fluctuationdominant flows. The simulated conformation change shows that, in fact, these flows are “strong” in the sense that the average second moment of the endtoend distance becomes large (relative to equilibrium) beyond a critical value of the fluctuation Deborah number. Although qualitatively capturing these trends, the theory by Shaqfeh and Koch underestimates the strength of the flows and thus overestimates the critical Deborah number. We present a new theory which includes spring relaxation and Brownian motion in the sampling of a velocity fluctuation (two factors which were neglected in the existing theory), thereby breaking the fore–aft symmetry of the sampling, thus increasing the average polymer stretch. The new theory quantitatively predicts the simulation results. To the authors knowledge, this is the first evidence via direct simulation that these random flows can produce large conformation change in model polymer molecules, even when the mean flow would produce no such change.

Local similarity solutions in the presence of a slip boundary condition
View Description Hide DescriptionThe local solution behavior near corners formed by the intersection of a slip surface with either a noslip or a shearfree boundary is analyzed by finite element calculations of the twodimensional flow of an inertialess Newtonian fluid in several model flow geometries; these flows are the flow in a tapered contraction, a sudden expansion and the extrudate swell from a planar die. Local finite element mesh refinement based on irregular, embedded elements is used to obtain extremely fine resolution of the velocity and pressure fields near the region where there is a sudden change in boundary condition. The calculations accurately reproduce the expected asymptotic behavior for a shearfree surface intersecting a noslip boundary, where the solution is given by a selfsimilar form for the velocity and pressure fields. Replacing the shearfree condition with a slip condition yields a similar form for the local velocity and pressure fields and indicates that the slip boundary behaves, to leading order, as a shearfree surface. Calculations for a slip boundary intersecting a shearfree surface yield similar results, with the local behavior being given by asymptotic analysis for two shearfree surfaces intersecting to form a wedge. These results suggest that replacing the noslip boundary condition in planar Newtonian die swell with a slip boundary condition can give rise to local behavior of velocity gradients and pressure which is more singular than the flow created with noslip boundary conditions. This prediction is confirmed by calculations of Newtonian die swell with slip. These calculations also demonstrate that the local solution in Newtonian die swell is sensitive to the details of the numerical method.

Measurement of the inertial lift on a moving sphere in contact with a plane wall in a shear flow
View Description Hide DescriptionThe translational and rotational velocities of a positively buoyant sphere suspended in a parallel plate device are measured, and related to the inertial lift of the sphere from the upper plate. The experimental results obtained agree well with the corrected theoretical predictions of Krishnan and Leighton [Phys. Fluids 7, 11 (1995)]. The motion of the sphere principally depends on the shear Reynolds number, where is the shear rate, is the particle radius and the kinematicviscosity, and the sedimentationReynolds number where is the Stokes sedimentation velocity of the sphere. The transition from translation without slip to translation with slip along the plane is observed to scale as Re/ and occurs at order . The separation of the sphere from the plane due to inertial lift at low Reynolds number is found to have the scaling as predicted by theory. In experiments with two particles of diameter 1.5 and 2 mm in two glycerin/water solutions of different densities, the particles were found to lift off the plane at . This value is consistent with measurements of the particle surface roughness.

Wave damping by a thin layer of viscous fluid
View Description Hide DescriptionThe rate of damping of surface gravity–capillary waves is investigated, in a system which consists of a thin layer of a Newtonian viscous fluid of thickness floating on a Newtonian fluid of infinite depth. The surface and interfacial tensions, elasticities and viscosities are taken into account. In particular, an approximate dispersion relation is derived for the case where and are both small, where is the wavenumber, is the angular frequency and is the kinematicviscosity of the upper fluid. If while remains finite, published dispersion relations for viscoelasticsurface films of extremely small (e.g., monomolecular) thickness are reproduced, if we add the surface and interfacial tensions, elasticities and viscosities together, and then add an additional to the surfaceviscosity, where is the density of the upper fluid. A simple approximation is derived for the damping rate and associated frequency shift when their magnitudes are both small. An example is given of what may happen with a slick of heavy fuel oil on water: a slick 10 thick produces a damping rate only slightly different from that of a film of essentially zero thickness, but the effect of the finite thickness becomes very noticeable if it is increased to 0.1–1 mm.

Effects of lift on the motion of particles in the recessed regions of a slider
View Description Hide DescriptionParticle contamination in a cavity, or a recessed region, on the slider surface is an important issue in slider designs for magnetic recording. In this paper, a model is developed for the simulation of particle motion in a recessed region, in which various forces such as viscous drag, Saffman lift, Magnus lift, and gravity are considered. It is found that the latter two forces have a very small effect on the vertical motion of the particles compared with the Saffman lift, and they can be neglected in the analysis. It is also found, through the simulation for various cases, that the Saffman lift is not so important for the motion of the smaller particles as for the motion of larger particles, and that the magnitude of the velocity of particles relative to the air flow affects the Saffman lift, or vertical motion of the particles, significantly. The air flow velocity is relatively low in a recessed region, and a particle often has a large relative velocity with positive sign when entering it, which causes the Saffman lift force to point to the surface of the slider. Therefore, particle contamination on the slider occurs at the edges of the recessed region. A similar result can also be obtained for the particles entering an air bearing under a taper, which explains why the contamination is concentrated on the surface of tapers.

Absolute instability of RayleighBénard convection in a timeperiodic shear flow
View Description Hide DescriptionThe onset of thermal convection in an oscillatory shear flow is considered as an initialvalue problem. For the case with zero mean shear, numerical results indicate that the instability is absolute in nature, just as for the case without any shear. If a mean flow component also exists, then the instability tends to have the same nature as that which occurs in the flow when no oscillation is present.

Combined thermocapillarybuoyancy convection in a cavity. Part II. An experimental study
View Description Hide DescriptionThe problem of buoyantthermocapillary convection in cavities is governed by a relatively large number of nondimensional parameters, and there is consequently a large number of different types of flow that can be found in this system. Previous results give disjoint glimpses of a wide variety of qualitatively and quantitatively different results in widely different parts of parameter space. In this study, we report experiments on the primary and secondary instabilities for acetone as the working fluid with a Prandtl number of 4.44, and in a geometry with equal aspect ratios in the range from 1 to 8 in both the direction along and perpendicular to the applied temperature gradient. We thus complement previous work that mostly involved either fluid layers of large extent in both directions, or consisted of investigations of strictly twodimensional disturbances. We investigate the qualitative and quantitative features of the fluid velocity field by flow visualization and particle tracking techniques. We observe the primary transition from an essentially twodimensional flow to steady threedimensional longitudinal rolls. The critical Marangoni number for this first transition is found to depend on the aspect ratios of the system, and varies from at aspect ratio 2.0 to at aspect ratio 3.5. The structure of the steady threedimensional flow far above the transition is found to involve a nonintuitive reverse flow against the temperature driving due to the strongly nonlinear threedimensional flow associated with the longitudinal rolls. Further, we have investigated the stability of this threedimensional flow at larger Marangoni numbers, and find a novel oscillatory flow at critical Marangoni numbers of the order of We suggest possible mechanisms that give rise to the oscillations.

Transitional regimes of lowPrandtl thermal convection in a cylindrical cell
View Description Hide DescriptionThe transitions from the onset of convection to fully developed turbulence of a Rayleigh–Bénard flow, in a lowaspectratio cell and in mercury, are studied through threedimensional numerical simulation of the Navier–Stokes equations. The calculation of the growth rate of the azimuthal energy modes permitted the accurate determination of the critical Rayleigh number for the establishment of the convective regime which is in good agreement with analytical and other numerical results. Increasing the Rayleigh number, the flow remained steady up to when an oscillatory instability was observed. Further increases in the Rayleigh produced a chaotic state through the period doubling mechanism and finally the turbulent state was achieved. It is shown that for the mean flow consists of a largescale convective cell which persists in the whole range of studied Rayleigh numbers . The dependence of the Nusselt number over the Rayleigh number is also analyzed and, for , when the turbulent state is reached, a power law in quantitative agreement with previous results at higher Ra is observed.

Chaotic heat transfer enhancement in rotating eccentric annularflow systems
View Description Hide DescriptionThermal Taylor dispersion theory for timeperiodic systems was used to study the extent of chaotic laminar heat transfer enhancement and axial thermal dispersion occurring during combined transverse and axial annular flow between two nonconcentric circular cylinders undergoing alternate rotations. A local Newton’s “law of cooling” heat transferboundary condition was used on the outer cylinder, whereas the inner cylinder was supposed insulated. The effective heat transfer coefficient describing the global rate of heat loss from the system (differing in general from the true microscale Newton’s law heat transfer coefficient on the outer cylinder) was calculated as a function of the system parameters, thereby serving to quantify the extent of chaotic heat transfer enhancement. The axial thermal Taylor dispersivity provided an independent measure of the effects of chaotic mixing, as too did the axial thermal velocity. Calculations were performed for three different cases: (i) concentric cylinder rotation (for which case the resulting circular transverse flow has no effect upon the effective transport properties); (ii) nonconcentric counterrotating circular cylinders, each undergoing a steady rotation, thereby creating a timeindependent transverse flow field; (iii) nonconcentric counter and corotating circular cylinders, each undergoing timeperiodic alternate rotation while the other remains at rest. A “regular” enhancement of the heat transfer rate over the concentric cylinder case was observed in case (ii), arising from the presence of a secondaryflow recirculation region. Enhancement due to chaotic advection was observed in case (iii) [about 50% more than that of case (ii) and more than double that of case (i), all other things being equal]. Concomitant values of the axial thermal Taylor dispersivity and axial thermal velocity confirmed the existence of enhanced transverse transport due to chaotic advection. It was observed that the functional dependence of the enhanced heat transfer rate upon the system parameters does not consistently display the same trends as are qualitatively suggested by the “degree of chaoticity” of the comparable Poincaré plots. This observation signals the need for caution in simply assuming that the greater the degree of chaotic “mixing” implicit in the Poincaré plot the greater will be the corresponding global transport rate. By simple redefinition of the symbols used in the present paper, our energy transport results may be reinterpreted so as to apply to the case of reactivespecies transport involving a firstorder irreversible chemical reaction occurring on the outercylinder surface; explicitly, the Nusselt number quantifying the local heat transfer coefficient rate is simply replaced by a comparable Damköhler number quantifying the local kinetics of the surface reaction.

On the threedimensional instabilities of plane flows subjected to Coriolis force
View Description Hide DescriptionLinear stability of twodimensional flows in a frame rotating with angular velocity vector perpendicular to their plane is considered. Sufficient conditions for instability have been derived for simple inviscid flows, namely parallel shear flows (characterized by the “Pedley” or “BradshawRichardson” number), circular vortices (by the “generalized Rayleigh” discriminant) and unbounded flows having a quadratic streamfunction (with elliptical, rectilinear or hyperbolic streamlines). These exact criteria are reviewed and contrasted using stability analysis for both threedimensional disturbances and oversimplified “pressureless” versions of the linear theory. These suggest that one defines a general inviscid criterion for rotation and curvature, based on the sign of the second invariant of the “inertial tensor,” and stating that, in a Cartesian coordinate frame: a sufficient condition for instability is that somewhere in the flow domain. It involves the “tilting vorticity” [Cambon et al., J. Fluid Mech. 175 (1994)] and the symmetric part of the velocity gradient of the basic flow.

On the effects of suction and injection on the absolute instability of the rotatingdisk boundary layer
View Description Hide DescriptionIn this paper we are concerned with the theoretical behavior of the laminar von Kármán boundarylayer flow, extending the work presented by Lingwood [J. Fluid Mech. 299, 17 (1995); 314, 373 (1996)] to the flow with mass transfer at the surface of the disk. It is known that, within specific regions of the parameter space, the flow is absolutely unstable in the radial direction, i.e. disturbances grow in time at every radial location within these regions. Uniform suction through the disk is shown to delay the onset of absolute instability, while uniform injection promotes the onset. By comparing suction and injection velocities of the same magnitude, it is shown that suction has a greater stabilizing effect on the absolute instability than the destabilizing effect of injection. Suction is also strongly stabilizing to both stationary and travelling inviscidly unstable branch1 modes; injection is destabilizing. Stationary viscously unstable branch2 modes are strongly stabilized and destabilized by suction and injection, respectively, but travelling branch2 modes are shown to be much less sensitive to mass transfer through the disk.

A particle scheme for the numerical solution of the Enskog equation
View Description Hide DescriptionIt is shown that the kinetic equation proposed by Enskog for a dense hard sphere gas can be solved numerically by a particle simulation method. The technique can be considered an extension of the well known DSMC method used to solve the Boltzmann equation. Unlike a recently proposed Nanbulike particle scheme, the present method exactly preserves momentum and energy. The calculation of the density profile in a dense gas in equilibrium near a hard wall is presented as an example.

Coarsening of solidliquid mixtures in a random acceleration field
View Description Hide DescriptionThe effects of flow induced by a random acceleration field (gjitter) are considered in two related situations that are of interest for microgravity fluid experiments: the random motion of isolated buoyant particles, and diffusion driven coarsening of a solidliquid mixture. We start by analyzing in detail actual accelerometer data gathered during a recent microgravity mission, and obtain the values of the parameters defining a previously introduced stochastic model of this acceleration field. The diffusive motion of a single solid particle suspended in an incompressible fluid that is subjected to such random accelerations is considered, and mean squared velocities and effective diffusion coefficients are explicitly given. We next study the flow induced by an ensemble of such particles, and show the existence of a hydrodynamically induced attraction between pairs of particles at distances large compared with their radii, and repulsion at short distances. Finally, a mean field analysis is used to estimate the effect of gjitter on diffusion controlled coarsening of a solidliquid mixture. Corrections to classical coarsening rates due to the induced fluid motion are calculated, and estimates are given for coarsening of Snrich particles in a SnPb eutectic fluid, an experiment to be conducted in microgravity in the near future.

Transition flow in an impact pressure probe: Relevance of gas–wall interaction
View Description Hide DescriptionImpact pressure probes may be used to measure the stagnation pressure of a spacecraft vehicle during reentry into the atmosphere. We model the flow in an impact pressure probe in the collisionfree and transition regime using a kinetic simulation scheme of the Boltzmann equation. It is shown that, depending on the gasdynamicproperties density, velocity, temperature of the inflow into the impact pressure probe, the measuredpressure may deviate by almost a factor 2 from the stagnation pressure. Furthermore, we study the dependence of the flow in the impact pressure probe on the gas–wall interaction. The models by Maxwell, Cercignani and Lampis, and by Lord are implemented. The influence of the gas–wall interaction is sensible for small accommodation coefficients; it is more pronounced, however, for temperature and density than for the pressure.

Passive scalar conditional statistics in a model of random advection
View Description Hide DescriptionWe study numerically a model of random advection of a passive scalar by an incompressible velocity field of different prescribed statistics. Our focus is on the conditional statistics of the passive scalar and specifically on two conditional averages: the averages of the time derivative squared and the second time derivative of the scalar when its fluctuation is at a given value. We find that these two conditional averages can be quite well approximated by polynomials whose coefficients can be expressed in terms of scalar moments and correlations of the scalar with its time derivatives. With the fitted polynomials for the conditional averages, analytical forms for the probability density function (pdf) of the scalar are obtained. The variation of the coefficients with the parameters of the model result in a change in the pdf. Three different kinds of velocity statistics, (i) Gaussian, (ii) exponential, and (iii) triangular, are studied, and the same qualitative results are found demonstrating that the onepoint statistics of the velocity field do not affect the statistical properties of the passive scalar.

Statistics of turbulence in a generalized randomforcedriven Burgers equation
View Description Hide DescriptionThe statistics of solutions to a family of onedimensional randomforcedriven advectiondiffusion equations is studied using high resolution numerical simulations. The equation differs from the usual Burgers equation by the nonlocal form of the nonlinear interaction term mimicking the nonlocality of the Navier–Stokes equation. It is shown that under an appropriate choice of random forcing the statistical properties of the solution (energy spectrum and scaling exponents of structure functions) coincide with those of Kolmogorov turbulence. Also, a generalization is proposed which allows intermittency effects to be modeled.