Index of content:
Volume 15, Issue 4, April 2003
 ARTICLES


Strong temperaturedependentviscosity effects on a rivulet draining down a uniformly heated or cooled slowly varying substrate
View Description Hide DescriptionWe use the lubrication approximation to investigate the steady locally unidirectional gravitydriven draining of a thin rivulet of Newtonian fluid with temperaturedependentviscosity down a slowly varying substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere. We consider the situation in which the Biot number (and hence the variation of temperature across the rivulet) is small, but in which the variation of viscosity with temperature is sufficiently strong that thermoviscosity effects appear at leading order in the limit of small Biot number. Three different models for the dependence of viscosity on temperature (specifically, the linear, exponential and Eyring models) are considered, but our attention is concentrated on the more realistic exponential and Eyring models (which coincide at leading order in the limit of small Biot number). We show that the effect of cooling the atmosphere is always to widen and deepen the rivulet, while the effect of heating the atmosphere is always to narrow and shallow it. We interpret our results as describing a slowly varying rivulet draining in the azimuthal direction from the top to the bottom of a large horizontal circular cylinder, and find that the behavior of the rivulet is rather different on the upper and lower parts of the cylinder (i.e., for sessile and pendent rivulets). Specifically, the effect of strong cooling of the atmosphere is to produce a wide rivulet with finite uniform thickness on the upper part of the cylinder, but a deep rivulet with finite semiwidth on the lower part of the cylinder. On the other hand, the effect of strong heating of the atmosphere is to produce a narrow and shallow rivulet everywhere except near the top and the bottom of the cylinder.

Effects of initial conditions on the passive and active scalar fluxes in unsteady stably stratified turbulence
View Description Hide DescriptionPassive and active scalar fluxes in unsteady strongly stratified turbulence are analyzed using the rapid distortion theory (RDT). Analytical solutions of the RDT equations show that the initial conditions, i.e., the initial cross correlation between passive scalar and active scalar (density) and the initial potential energy, make a difference between the passive and active scalar flux, giving the difference between the turbulent diffusion coefficients for passive and active scalars. In other words even the initial zero correlation gives the complete correlation in the long time development, in contrast to the previous discussions in the literature based on the equations for stationaryturbulence. The difference appears in the components slowly oscillating at half frequency N (N: Brunt–Väisälä frequency) in the passive scalar flux, while the active scalar flux has only the rapidly oscillating components which oscillate at frequency It is also found that the correlation coefficient between passive and active scalars is not a good measure of identifying the agreement or disagreement of the turbulent diffusion coefficients for these scalars, since the initially large passive scalar variance can lead to a final small correlation coefficient even when the turbulent diffusion coefficients are equal for the active and passive scalars. The results show the importance of unsteady analysis of the initial value problem in the diffusion problem in stratified turbulence.

The effect of compatibilizer on the coalescence of two drops in flow
View Description Hide DescriptionThis paper reports results from an experimental study of the effects of copolymer/compatibilizer on the coalescence of two equal size drops in the flow field produced by a fourroll mill. The data encompass two different fluid systems, both with PDMS as the suspending fluid and PBd as the drops, and an acidbase complex of adsorbed at the interface that we shall refer to as a copolymer. The two systems differ in the ratio of viscosities (λ) of the drop to the suspending fluid, one having λ=0.19 and the other λ=1.3. For the lower viscosity ratio system, as the amount of adsorbed copolymer is increased, the drainage time for coalescence in a headon collision is increased monotonically and the critical capillary number for coalescence in a glancing collision is also reduced monotonically in a manner that appears qualitatively consistent with a slowing of the film drainage process due to Marangoni stresses. Detailed trajectory measurements for drops with copolymer show agreement with predicted theoretical results for spherical drops without copolymer, but with an increased viscosity ratio. With copolymer present, we also find that coalescence occurs for the largest capillary numbers only after the drops begin to be pulled apart by the external flow. For the higher viscosity ratio system, the effect of increasing the copolymer concentration is nonmonotonic. For very small concentrations, there is a major decrease in the critical capillary number for coalescence and a corresponding increase in the drainage time prior to coalescence, but as the copolymer concentration is further increased, the film drainage time decreases and the critical capillary number increases to a value that is intermediate between the clean interface result, and the result for the smallest copolymer concentration. This is shown to be due to a dependence of the critical coalescence angle on copolymer concentration that was not present in the lower viscosity ratio system. We conclude by speculating about mechanisms, in addition to the Marangoni effect, that might “explain” these observations.

The settling velocity of heavy particles in an aqueous nearisotropic turbulence
View Description Hide DescriptionThe ensembleaverage settling velocity, of heavy tungsten and glass particles with different mean diameters in an aqueous nearisotropic turbulence that was generated by a pair of vertically oscillated grids in a water tank was measured using both particle tracking and particle image velocimetries. Emphasis is placed on the effect of the Stokes number, St, a time ratio of particle response to the Kolmogorov scale of turbulence, to the particle settling rate defined as where is the particle terminal velocity in still fluid. It is found that even when the particle Reynolds number is as large as 25 at which where is the Kolmogorov velocity scale of turbulence, the mean settling rate is positive and reaches its maximum of about 7% when St is approaching to unity, indicating a good trend of DNS results by Wang and Maxey (1993) and Yang and Lei (1998). This phenomenon becomes more and more pronounced as values of decrease, for which DNS results reveal that the settling rate at and can be as large as 50% when However, the present result differs drastically with Monte Carlo simulations for heavy particles subjected to nonlinear drag in turbulence in which the settling rate was negative and decreases with increasing St. Using the wavelet analysis, the fluid integral time the Taylor microscale and two heavy particles’ characteristic times are identified for the first time. For and whereas and for This may explain why the settling rate is a maximum near because the particle motion is in phase with the fluid turbulent motion only when where the relative slip velocities are smallest. These results may be relevant to sediment grains in rivers and aerosol particles in the atmosphere.

Complex flow transitions in a homogeneous, concentrated emulsion
View Description Hide DescriptionResults are presented for the flow of homogeneous, concentrated, oilinwater emulsions subjected to a shear flow between rotating, horizontal concentric cylinders. Nuclear magnetic resonance imaging(NMRI) was used to measurevelocity profiles. This technique allows velocity profiles to be measured noninvasively within a flowing, concentrated emulsion. It was observed that below a critical velocity, in a portion of the gap, the fluid moves in a direction opposite to the outer, rotating cylinder. Above this critical velocity, the emulsion corotates with the outer cylinder. Theoretical analysis suggests that the transitions are driven by buoyancy effects. The corotating and counterrotating flow states at different rotation speeds can be characterized by a single dimensionless parameter which relates buoyancy and viscous effects.

Front and back instability of a liquid film on a slightly inclined plate
View Description Hide DescriptionWe study the transverse instability of a liquid ridge on horizontal and inclined substrates using a film evolution equation based on a long wave approximation. The equation incorporates an additional pressure term—the disjoining pressure—accounting for the effective interaction of the film with the substrate. On a horizontal substrate the dominant instability mode is varicose, but may turn into a zigzag mode on a slightly inclined substrate depending on the inclination angle and the ridge volume. For larger angles or volumes the instabilities at the front and back decouple. The linear stability properties of a onedimensional transverse ridgelike state are studied in detail, and an energy analysis is used to demonstrate that the disjoining pressure provides the dominant instability mechanism at both the front and the back, while the body force is responsible for the main differences between these two instabilities. An amplitude equation for the time evolution of perturbations with small transverse wave numbers is derived that predicts correctly the linear crossing of the most dangerous eigenvalues at zero wave number in the inclined case, in contrast to the situation on a horizontal substrate.

Analysis of heat transfer from single wires close to walls
View Description Hide DescriptionTwodimensional numerical investigations of the forced heat convection from a microcylinder in laminar crossflow, both in free stream and in nearwall flow, were carried out aiming at a better understanding of the physics behind the wall effects on hotwire nearwall measurements. In the physical model, an infinitely thin plate with the same properties as the fluid (air) was used as an artificial wall. The conjugate heat transfer between the flow regions on both sides of the plate was taken into account. The effect of the conjugate thermal conditions (temperature distribution and diffusive heat flux) at the interface of the two flow regions on the heat transfer from the wire was investigated by varying the flow conditions on the side opposite to the wire location. Careful energy balanceanalysis was performed for both the freestream case and the nearwall case. This enabled the authors to verify their own understanding of the physical mechanism responsible for the wall effect on hotwire measurements and to examine other mechanisms proposed in the literature. The numerical results showed that the heat diffusion from the wire is significantly enhanced in the case of small wiretowall distances This is mainly caused by modifications of the thermal boundary condition (diffusive effect) at the fluid–wall interface. In contrast, the flow distortion (enhanced convection) was shown not to be the most important influencing factor for the heat transfer of a hot wire. Although the present model study was performed for a laminar flow, the results obtained are applicable to hotwire measurements in turbulent flows, as stated in the literature.

Drop dynamics on the beadsonstring structure for viscoelastic jets: A numerical study
View Description Hide DescriptionIt is well known that a viscoelastic jet breaks up much more slowly than a Newtonian jet. Typically, it evolves into the socalled beadsonstring structure, where large drops are connected by thin threads. The slow breakup process provides the viscoelastic jet sufficient time to exhibit some new phenomena. The aim of this paper is to investigate the drop dynamics of the beadsonstring structure. This includes drop migration, drop oscillation, drop merging and drop draining. We will use a 1D OldroydB model for the viscoelastic jet, and solve this model numerically by an explicit finite difference method. Close to exponential draining of the filament, we found that the variation of the axial elastic force in the filament is roughly four times larger than the variation of the capillary force with opposite sign. This fact implies that the elastic force is responsible for the drop migration and oscillation. Our study of the drop draining process shows that the elastic force also plays an important role here, allowing the liquid to flow from smaller drops into larger drops through the filament.

Radial source flows in porous media: Linear stability analysis of axial and helical perturbations in miscible displacements
View Description Hide DescriptionLinear stability results are presented for axial and helical perturbation waves in radial porous media displacements involving miscible fluids of constant density. A numerical eigenvalue problem is formulated and solved in order to evaluate the relevant dispersion relations as functions of the Peclet number and the viscosity ratio. In contrast to the constant algebraic growth rates of purely azimuthal perturbations [C. T. Tan and G. M. Homsy, Phys. Fluids 30, 1239 (1987)], axial perturbations are seen to grow with a timedependent growth rate. As a result, there exists a critical time up to which the most dangerous axial wavenumbers are larger, and beyond which the most dangerous azimuthal wavenumbers have higher values. This raises the possibility that early on, the smaller flow scales appear in the axial direction, whereas the later flow stages are dominated by smaller azimuthal features. By rescaling the axial wavenumber, the explicit appearance of time can be eliminated. The maximum growth rate of axial perturbations, as well as their most dangerous and cutoff wavenumbers, are seen to increase with the Peclet number and the viscosity ratio. The most dangerous wavenumber is observed to shift towards the lower end of the spectrum as the Peclet number increases. With increasing viscosity contrast, it first moves towards the lower part of the spectrum, only to shift towards the higher end later on. In the limit of large Pe, asymptotic solutions are obtained for the growth of axial disturbances. Numerical solutions of the full eigenvalue problem generally show good agreement with these asymptotic solutions for large Peclet numbers. Over the entire range of wave vector directions between the purely axial and azimuthal extrema, helical waves display an approximately constant maximum growth rate. The wavenumber of maximum growth as well the maximum growth rate of helical waves can be evaluated from the corresponding purely azimuthal and axial problems. This suggests that in threedimensional flows the nature of the initial conditions plays an important role.

Threedimensional velocity field for wavy Taylor–Couette flow
View Description Hide DescriptionThe stability of wavy supercritical cylindrical Couette flow has been studied extensively, but few measurements of the velocity field in flow have been made. Particle image velocimetry was used to measure the azimuthal and radial velocities in latitudinal planes perpendicular to the axis of rotation for wavy cylindrical Couette flow in the annulus between a rotating inner cylinder and a fixed outer cylinder. These measurements were matched to previous measurements of the axial and radial velocity measured in several meridional planes resulting in an experimentally measured, timeresolved, threedimensional, threecomponent velocity field for wavy cylindrical Couette flow. Using this complete velocity field it is possible to evaluate details of the flow field. The vortical motion transports azimuthal momentum radially while the axial exchange of fluid between vortices in wavy flow transports azimuthal momentum axially. As the Reynolds number increases, these effects strengthen. Streams of net axial flow stretch axially along the length of the annulus and wind around the vortices from the inner cylinder to the outer cylinder and back while also winding azimuthally in the annulus. The azimuthal velocity measured at the center of a vortex is similar to the azimuthal wave speed. Measurements of the azimuthal velocity in cylindrical surfaces concentric with the axis of rotation suggest that the origin of the waviness is related to a jetlike azimuthal velocity profile rather than the radial outflow jet. Near both cylinder walls, the shear stress is quite large, decreasing to near zero at the middle of the annular gap.

Improved Lagrangian mixing models for passive scalars in isotropic turbulence
View Description Hide DescriptionLagrangian data for velocity, scalars, and energy and scalar dissipation from direct numerical simulations are used to validate Lagrangian mixing models for inert passive scalars in stationary isotropic turbulence. The scalar fluctuations are nearly Gaussian, and, as a result of production by uniform mean gradients, statistically stationary. Comparisons are made for Taylorscale Reynolds numbers in the range 38 to about 240 and Schmidt numbers in the range 1/8 to 1. Model predictions for onepoint, onetime Eulerian statistics (Eulerian correspondence) and oneparticle, twotime Lagrangian statistics (Lagrangian correspondence) are examined. Two scalar mixing models, namely the Lagrangian Fokker–Planck model and the Lagrangian colorednoise (LCN) model, are proposed and written in terms of stochastic differential equations (SDE) with specified drift and diffusion terms. Both of these models rely on statistics of the scalar field conditioned upon the energy dissipation, as provided by the Lagrangian spectral relaxation (LSR) model. With the exception of the scalar dissipation, the models are shown to capture the Reynolds and Schmidtnumber dependence of the Lagrangian integral time scales. However, the LCN model provides a more realistic description of the Lagrangian scalar fluctuations as differentiable time series having the correct form of the scalar autocorrelation function. Further extensions of the new mixing models to nonGaussian scalars are conceptually straightforward, but require a closure for the scalarconditioned scalar dissipation rate matrix. Likewise, accurate prediction of joint statistics for differential diffusion between different scalars with unequal molecular diffusivities will require the formulation of a multiscale SDE similar to the LSR model.

A quantitative study of the interaction of two Richtmyer–Meshkovunstable gas cylinders
View Description Hide DescriptionWe experimentally investigate the evolution and interaction of two Richtmyer–Meshkovunstable gas cylinders using concentration field visualization and particle image velocimetry. The heavygas cylinders have an initial spanwise separation of (where D is the cylinder diameter) and are simultaneously impacted by a planar, Mach 1.2 shock. The resulting flow morphologies are highly reproducible and highly sensitive to the initial separation, which is varied from to 2.0. The effects of the cylinder–cylinder interaction are quantified using both visualization and highresolution velocimetry. Vorticity fields reveal that a principal interaction effect is the weakening of the inner vortices of the system. We observe a nonlinear, thresholdtype behavior of inner vortex formation around A correlationbased ensembleaveraging procedure extracts the persistent character of the unstable flow structures, and permits decomposition of the concentration fields into mean (deterministic) and fluctuating (stochastic) components.

An equality about the velocity derivative skewness in turbulence
View Description Hide DescriptionWe study velocity derivative skewness S of incompressible homogeneous isotropic turbulence. By using exact relations of isotropic turbulence and various typical models of secondorder structure function and energy spectrum it is found that when Taylormicroscale Reynolds number is high. Here, C is a coefficient, is the center wavenumber of energy dissipation spectrum, and is the Kolmogorov wavenumber. Therefore, the problem of Reynolds number dependence of S becomes the problem of Reynolds number dependence of In the inertial range, we have scaling and is the secondorder inertialrange scaling exponent. Equality is valid in the case of (intermittencymodels of Kolmogorov’s 1962 theory) as well as in the case of (Kolmogorov’s 1941 theory).

Inverse velocity statistics in twodimensional turbulence
View Description Hide DescriptionWe present a numerical study of twodimensional turbulent flows in the enstropy cascade regime, with different largescale energy sinks. In particular, we study the statistics of morethandifferentiable velocity fluctuations by means of two sets of statistical estimators, namely inverse statistics and secondorder differences. In this way, we are able to probe statistical fluctuations that are not captured by the usual spectral analysis. We show that a new set of exponents associated to morethandifferentiable fluctuations of the velocity field exists. We also present a numerical investigation of the temporal properties of umeasured in different spatial locations.

Interaction of two equal vortices on a β plane
View Description Hide DescriptionThe interaction of two equal vortices under the influence of a gradient of background vorticity (β) is studied numerically and experimentally. If the initial shape and vorticity distribution of the vortices is fixed, two parameters determine the evolution: the normalized intercentroid distance where is the radius of the vortex; and the normalized gradient of background vorticity where ω is the peak vorticity of the vortex. Alternate ways of identifying regimes of behavior in the parameter plane are presented. These are applied to numerical simulations of interaction of vortices with steplike, steep and smooth vorticity profiles. It is found that the critical distance for merger decreases with increasing for all vortex types, and that vortices with smooth vorticity profile are the most mergerprone vortices.Laboratory experiments were done in a rotating water tank with a flat sloping bottom providing the β effect. The vortices produced have a smooth vorticity profile and show the same behavior observed in the simulations, except that, as a result of viscous effects, the critical merger distance is shifted towards larger values of

Evolution and instability of monopolar vortices in a stratified fluid
View Description Hide DescriptionThe evolution of initially axisymmetric shielded pancakelike vortices in a nonrotating linearly stratified fluid has been investigated experimentally and numerically. The evolution process and the shape of tripoles in laboratory experiments depend on the experimental parameter values. In order to investigate this phenomenon we have considered the influence of the Reynolds (Re) and Froude (F) numbers on the tripole formation process. Also, the role of the (absolute) ratio γ between the vorticity values of the satellite vortices and the core vortex on the tripole evolution has been investigated. Additionally, a set of numerical simulations has been performed to enable an examination of the role of Reynolds numbers (up to and Froude numbers 0.2, 0.4, and 0.8) outside the experimentally accessible range. The steepness parameter α was varied between 2 and 8 in order to estimate the relative importance of the different modes constituting the perturbation. From this study we conclude that tripole formation and dipole splitting in a linearly stratified fluid can be well described in terms of the parameter set

Growth and mutual interference of protein seeds under reduced gravity conditions
View Description Hide DescriptionThis analysis deals with new models and computational methods as well as with novel results on the relative importance of “controlling forces” in macromolecular crystal growth. The attention is focused in particular on microgravity fluiddynamic aspects and on the case of the simultaneous growth of different seeds. A “kineticcoefficientbased” volume of fraction method is specifically and carefully developed according to the complex properties and mechanisms of macromolecular proteincrystal growth. It is shown that the size and the shape of the growing crystals play a “critical role” in the relative importance of surface effects and in determining the intensity of convection. Convective effects, in turn, are found to impact growth rates, macroscopic structures of precipitates, particle size and morphology as well as the mechanisms driving growth. The face growth rates in particular depend on the complex multicellular structure of the convective field and on associated “pluming phenomena.” The relative importance of mass transport in liquid phase and surface attachment kinetics is investigated. The simulations show that it does not behave as a “fixed” parameter and that different crystallization conditions may occur in the protein chamber due to mutual interference of the growing seeds, complex convective effects and the “finite size” of the reactor.

Effect of the Schmidt number on the diffusion of axisymmetric pancake vortices in a stratified fluid
View Description Hide DescriptionAn asymptotic analysis of the equations for quasitwodimensional flow in stratified fluids is conducted, leading to a model for the diffusion of pancakelike vortices in cyclostrophic balance. This analysis permits one to derive formally the model for the diffusion of an axisymmetric monopole proposed by Beckers et al. [J. Fluid Mech. 433, 1 (2001)], and to extend their results. The appropriate parameter for the perturbation analysis is identified as the square of the vertical Froude number where U is the horizontal velocity scale, N is the Brunt–Väisälä frequency, and the vertical length scale. The physical mechanisms involved in the vortex decay are examined under the light of the asymptotic analysis results. In particular we discuss the effects of the Schmidt number, Sc, which measures the balance between the diffusion of momentum and the diffusion of the stratifying agent. Remarkably, the vertical transport due to the slow cyclostrophic adjustment is shown to slowdown the velocity decay when Sc is larger than unity whereas it accelerates it when Sc is smaller than unity.

Dynamics of the local entanglement on two vortex filaments described by the Korteweg–de Vries equation
View Description Hide DescriptionAn entanglement equation is found to reduce a Korteweg–de Vries (KdV) equation. An entanglement equation is the localized induction approximation of the Biot–Savart equation describing the dynamics of a corotating vortex pair. A soliton solution together with a cnoidal wave solution of the KdV equation shows that vortex filaments make a local entanglement when the distance between a pair of vortex filaments locally decreases.

Kolmogorov turbulence by matched asymptotic expansions
View Description Hide DescriptionThe Kolmogorov [Dokl. Akad. Nauk. SSSR 30, 299 (1941), hereafter K41] inertial range theory is derived from first principles by analysis of the Navier–Stokes equation using the method of matched asymptotic expansions without assuming isotropy or homogeneity and the Kolmogorov (K62) [J. Fluid Mech. 13, 82 (1962)] refined theory is analyzed. This paper is an extension of Lundgren [Phys. Fluids 14, 638 (2002)], in which the second and thirdorder structure functions were determined from the isotropic Karman–Howarth [Proc. R. Soc. London, Ser. A 164, 192 (1938)] equation. The starting point for the present analysis is an equation for the difference in velocity between two points, one of which is a Lagrangian fluid point and the second, slaved to the first by a fixed separation r, is not Lagrangian. The velocity difference, so defined, satisfies the Navier–Stokes equation with spatial variable r. The analysis is carried out in two parts. In the first part the physical hypothesis is made that the mean dissipation is independent of viscosity as viscosity tends to zero, as assumed in K41. This means that the mean dissipation is finite as Reynolds number tends to infinity and leads to the K41 inertial range results. In the second part this dissipation assumption is relaxed in an attempt to duplicate the K62 theory. While the K62 structure is obtained, there are restrictions, resulting from the analysis which shows that there can be no inertial range intermittency as Reynolds number tends to infinity, and therefore the mean dissipation has to be finite as Reynolds number tends to infinity, as assumed in part one. Reynolds numberdependent corrections to the K41 results are obtained in the form of compensating functions of which tend to zero slowly like as
