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Volume 7, Issue 3, March 1995

Transition of viscous fingering patterns in polymer solutions
View Description Hide DescriptionViscous fingering patterns of aqueous hydroxypropyl methyl cellulose (HPMC) solutions pushed by air in the Hele–Shaw cell were observed as a function of isopropyl alcohol content under a constant pressure of 15 cm H_{2}O. A morphological transition from side branching patterns to tip splitting ones with increasing isopropyl alcohol content, accompanied with a decrease in surface tension and an increase in viscosity is found. The observed morphology transition was correlated with the dimension of the fingering pattern, as well as the average tip velocity in the fingering.

Meniscus shape experiments in forward roll coating
View Description Hide DescriptionIn the present paper, experiments are reported for the measurement of the complete meniscus profile in asymmetric forward roll coating. A thin laser sheet technique is used to recover accurately the shape and position of the interface for an extensive database of operating conditions. Knowledge of the interface profile and position is then used to measure the total flow rate and its distribution on each cylinder. The results provide for the first time validation of numerical predictions for the extended shape of the meniscus. Comparison with existing theoretical and numerical predictions validates both these approaches, but reveals the need to model the whole flow region, including gravity, instead of restricting to the region downstream of the nip separating the rotating cylinders.

The effects of permeability heterogeneity on miscible viscous fingering: A three‐dimensional magnetic resonance imaging analysis
View Description Hide DescriptionThe three‐dimensional evolution of the viscous fingering instability has been visualized directly with magnetic resonance imaging(MRI).Miscible displacement of thin solute bands by aqueous solvent was investigated in packed beds of 30 μm chromatographic particles. Fingering behavior into samples of glycerol and a protein, bovine serum albumin (BSA), with viscosity ratios ranging from 1 to approximately 4, were compared. The three‐dimensional morphology and dynamics of fingers were monitored to approximately millimeter spatial resolution using MRI. Linear and nonlinear fingering behavior were observed. Permeability heterogeneities with length scales on the order of the finger wavelength induced complex three‐dimensional fingering patterns. Sample and column boundary effects on fingering dynamics were also noted. The differences in fingering behavior observed between albumin and glycerol samples are consistent with the wavelength predictions of linear stability analysis and the large differences in molecular diffusivity.

Tracer diffusion under shear flow for general repulsive interactions
View Description Hide DescriptionTracer diffusion in a steady shear flow state is analyzed. A kinetic model incorporating a temperature dependence in the collision frequencies is used. This allows for the consideration of a general repulsive intermolecular interaction. A perturbative scheme is applied to get the shear rate dependence of the tracer diffusiontensor in terms of the mass ratio, the force constants ratio, and a parameter characterizing the interaction potential considered. In addition, the heat flux arising from the concentration gradient of the tracer species is also evaluated. The results are illustrated for the two extreme cases of Maxwell molecules and hard spheres.

The effect of hydrodynamic interactions on the orientation distribution in a fiber suspension subject to simple shear flow
View Description Hide DescriptionA single, non‐Brownian fiber suspended in a viscous, Newtonian fluid undergoing simple shear flow rotates in one of a set of closed orbits known as Jeffery orbits. In a fiber suspension, the hydrodynamic interactions among the fibers determine the distribution of fibers among these orbits. The hydrodynamic interactions in dilute and semidilute suspensions have been studied using slender‐body theory. Hydrodynamic, orientational diffusivities were obtained from an ensemble average of the fiber–fiber interactions. The steady‐state fiber orientation distribution is controlled by the anisotropy and orientation dependence of the diffusivities. The steady‐state and transient fiber orientation distributions are derived using a perturbation analysis for weak hydrodynamic orientational diffusion that is an extension of the work of Leal and Hinch [J. Fluid Mech. 46, 685 (1972)] for weak, isotropic, rotary Brownian motion. In the dilute regime, the steady‐state experimental distributions of Anczurowski and Mason [J. Colloid Interface Science 23, 522 (1967)] do not agree with the theoretical predictions. An explanation for these discrepancies accompanied with new experimental results is presented in this work. The theoretical predictions for the steady‐state orientation distribution, and the temporal orbit constant correlation function in the semidilute regime are in good agreement with the experimental results of Stover et al. [J. Fluid Mech. 238, 277 (1992)]. The correlation time for the fiber orientation is approximately inversely proportional to fiber concentration in both the dilute and semidilute regimes.

Hydrodynamics of a one‐dimensional granular medium
View Description Hide DescriptionThe question whether one‐dimensional granular systems can be described by hydrodynamic equations is the main theme of the present work. Numerical simulations are used to create a database with which theory is compared. The system investigated in the numerical work is that of a one‐dimensional collection of point particles colliding inelastically. The dependence of the dynamical properties on both the degree of inelasticity and the number of particles is investigated. A hydrodynamictheory which describes the large‐scale motion of such systems has been developed. It is shown that the standard set of hydrodynamic fields (density, velocity, and granular temperature) is insufficient for this purpose and that an additional hydrodynamic field corresponding to the third moment of the fluctuating velocity field must be added to that set. The results of a linear stability analysis of the derived hydrodynamic equations are in a close agreement with those of the numerical simulations. The question of the effects of velocity correlations on the hydrodynamics is addressed as well. It is shown that these correlations, though not negligible, do not affect the hydrodynamic equations. The form of the single particle initial distribution function is shown to slightly affect the form of the hydrodynamic equations for transient times. Except for this minor effect the hydrodynamic equations possess a universal form. Possible implications for higher dimensional systems are mentioned.

On the unsteady separated flow past a semi‐infinite plate: Exact solution of the Brown and Michael model, scaling, and universality
View Description Hide DescriptionTwo‐dimensional unsteady separated flow past a semi‐infinite plate is considered. The rolling up of the separated shear layer is modeled by a point vortex, whose time‐dependent circulation is predicted by an unsteady Kutta condition. The equation of motion for the starting vortex is derived and solved in closed form for any free‐stream condition. A time‐dependent scaling that captures the universality of the flow is proposed.

Sprays in radially spreading shear‐layer flows
View Description Hide DescriptionTwo radially spreading adjacent streams, which differ in their radial velocities and thus form a radially spreading shear‐layer flow, are considered here. The theory presented by the authors for sprays suspended in unidirectional [Katoshevski and Tambour, Phys. Fluids A 5, 3085 (1993)] shear layers is extended here for radially spreading shear layer flows. The behavior of a multisize (polydisperse) evaporating spray, which is suspended in one of the streams, is studied. The spray spreads in the lateral direction towards the other coflowing stream, resulting in lateral changes in spray densities and in local droplet size distributions across the shear layer. These effects are analyzed here via similarity solutions of the governing equations. A comparison between the behavior of the multisize sprays and their vapors in radially spreading versus unidirectional shear‐layer flows is also presented and discussed here. The dynamics of the radially spreading spray is essentially different from that of the unidirectional spray. In the radial case, streamlines of the host‐gas flow become more crowded with radial distance, and thus, it is shown here how lateral evolution in size histograms and lateral Sauter mean diameter (SMD) profiles are affected by this feature of the radially spreading flow. The effects of initial drop‐size histograms on lateral distributions of: droplet SMD, overall spray densities, and vapor are also studied here for three basic initial drop‐size distributions: monodisperse, bimodal, and polydisperse. It is shown how the behavior exhibited by polydisperse (and bimodal) sprays differs intrinsically from the behavior of monodisperse sprays. For example, for sprays which are initially monodisperse the lateral profile of the spray’s SMD across the shear layer always decreases, whereas for polydisperse or bimodal sprays it may increase or assume an ‘‘S’’ shaped curve.

High gradient phenomena in two‐dimensional vortex interactions
View Description Hide DescriptionHyperdiffusion, a simple linear eddy diffusivity scheme, is commonly used in atmospheric and oceanic simulations because it increases the range of inertially behaving spatial scales for a given model resolution. Compared with molecular diffusion (which is utterly negligible in the atmosphere and oceans), hyperdiffusion more sharply confines the dissipation to the smallest scales of the numerical model. But is this all that hyperdiffusion does? In this paper, the inelastic interaction of two distributed vortices of unequal size is examined. Contour surgery (CS) simulations are compared with pseudospectral (PS) simulations employing hyperdiffusion or molecular diffusion. The example illustrates what is believed to be the most fundamental characteristic of two‐dimensional (2‐D) (and layerwise‐2‐D) vortex dynamics, namely, the formation of exceedingly high vorticity gradients. There is an excellent agreement between the hyperdiffusive PS and CS calculations at early times (i.e., for a few vortex rotation periods). Thereafter, significant discrepancies develop, beginning abruptly from the time when vorticity‐gradient intensification is arrested by diffusion. A rapid inward erosion of the smaller of the two vortices then takes place. This erosion takes place under the joint action of (hyper) diffusion and stripping (the peeling of the vortex periphery by the external flow). With hyperdiffusion, the erosion is accompanied by a serious numerical artifact: a climb in the peak vorticity by 30% in this example. Eventually, the erosion reaches the vortex center and the vortex is sheared into a filament. In the CS calculation, there is no erosion, no climb in peak vorticity, and the vortex appears to last indefinitely.
In the PS calculations, the viscosity or hyperdiffusion is adjusted according to the resolution to give the largest possible inertial range while ensuring numerical stability. It is found that vortices that are spanned by fewer than 10–20 grid points are eroded away in only a few vortex rotation periods (a time scale that is very much shorter than one would estimate from pure viscous decay). These findings bring into question the results of many 2‐ turbulence simulations using hyperdiffusion, for hyperdiffusion simulates neither inviscid dynamics nor molecular‐diffusive dynamics.

Numerical simulation of vortices with axial velocity deficits
View Description Hide DescriptionAxial velocity deficit is a source of instability in vortices that may otherwise be stable. Temporal large‐eddy simulation is performed to study the response of vortices with axial velocity deficits to random and controlled disturbances at high Reynolds numbers. The qvortex [Batchelor, J. Fluid Mech. 20, 321 (1964)] is used as a model of such vortices. When the vortex is linearly unstable, the disturbances grow and result in the appearance of large‐scale helical sheets of vorticity. Later, these large‐scale helical structures break up into small‐scale filaments. Associated with the formation of the large‐scale structures is a redistribution of both angular and axial momentum between the core and the surroundings. The redistribution weakens the axial velocity deficit in the core while strengthens the rigid‐body‐like rotation of the core. The emerging mean velocity profiles drive the vortex core to a stable configuration. The vortex eventually returns to a laminar state, with an insignificant decay in the tangential velocity, but with a much weakened axial velocity deficit. A direct numerical simulation obtained at a lower Reynolds number confirms the above conclusions.

Heat transfer in a strongly nonlinear spatially developing longitudinal vorticity system
View Description Hide DescriptionIn the present paper, the heat‐transfer problem in a nonlinearly developing longitudinal vorticity system that arose from upstream weak Görtler vortices is considered. The general situation in which the Prandtl number (Pr) is different from unity is studied. The heat transportequation must be integrated for each Pr. The present paper presents results for gaseous (Pr<1) and liquid [Pr=O(10)] media, with the downstream integration carried out for three‐dimensional advecting velocities in a spatially developing longitudinal vorticity system [Phys. Fluids A 4, 95 (1992)]. The behavior of the gaseous system is not significantly different from the Pr=1 situation; for a liquid system the isotemperature structures develop into thin, palm‐tree‐like figures in the cross‐sectional plane. It is shown that for larger Pr, local surface heat‐transfer rates are significantly enhanced over the local Blasius–Pohlhausen flat‐plate value (by about 400% in the example computed for an initially amplified Görtler vortex for Pr=7.07 and about 340% for Pr=0.72), with an accompanying similar (about 350%) but considered moderate increase in the skin friction relative to the Blasius value. Detailed analyses of the heat transport and conversion mechanisms before and after Reynolds averaging are presented. For a fixed flow‐parameter range, it is shown that the initial velocity perturbation level, not the initial scalar perturbation level, is an important controlling parameter for heat‐transfer enhancement. The well‐known analogy between heat and mass transfer is used to interpret flow visualization studies; in particular, the isoconcentration lines are typified by palm‐tree‐like structures in water, for which the Schmidt number Sc≫1.

The decay of axisymmetric homogeneous turbulence
View Description Hide DescriptionThe decay of a homogeneous turbulence generated by an axisymmetric distribution of random impulsive forces acting at the initial instant is studied by means of large‐eddy simulations. The impulsive forces may be either parallel or perpendicular to the symmetry axis. For impulsive forces, which result in a k ^{4} low wave number energy spectrum of the turbulence, it is determined that the flow approaches isotropy on all scales of motion at long times, provided the Reynolds number is large. However, for the type of impulsive forces originally proposed by Saffman [J. Fluid Mech. 27, 581 (1967)], in which a k ^{2} low wave number energy spectrum is produced, the turbulence approaches isotropy only at the smallest scales, and remains significantly anisotropic at the largest and energy‐containing scales. Nevertheless, a similarity state of the flow field establishes itself asymptotically, in which the kinetic energy per unit mass of the turbulence decays as t ^{−6/5}.

On the representation of backscatter in dynamic localization models
View Description Hide DescriptionThe dynamic localization model is a recently developed method that allows one to compute rather than prescribe the unknown coefficients in a subgrid scale model as a function of position at each time‐step. A realistic subgrid scale model should describe both the direct and reverse (backscatter)energy transfers at the local level. A previously developed dynamic localization model accounted for backscatter by means of a (deterministic) eddyviscosity that could locally assume positive as well as negative values. Here this paper presents an alternative stochastic model of backscatter in the context of the dynamic procedure. A comparative discussion of the merits of stochastic versus deterministic modeling of backscatter is presented. These models are applied to a large eddy simulation of isotropic decaying and forced turbulence. Tests are also performed with versions of the model that do not account for backscatter. The results are compared to experiments and direct numerical simulation. It is shown that the models correctly predict the energy and three‐dimensional (3D) energy spectra in decaying turbulence. In the forced case the Kolmogorov 5/3 law seems better predicted by models accounting for backscatter. A relative evaluation of the various versions of the model in terms of predictive capability and cost is provided.

Scaling and dissipation in the GOY shell model
View Description Hide DescriptionThis is a paper about multifractal scaling and dissipation in a shell model of turbulence, called the Gledzer–Ohkitani–Yamada (GOY) model. This set of equations describes a one‐dimensional cascade of energy towards higher wave vectors. When the model is chaotic, the high‐wave‐vector velocity is a product of roughly independent multipliers, one for each logarithmic momentum shell. The appropriate tool for studying the multifractal properties of this model is shown to be the energy flux on each shell rather than the velocity on each shell. Using this quantity, one can obtain better measurements of the deviations from Kolmogorov scaling (in the GOY dynamics) than were available up to now. These deviations are seen to depend upon the details of inertial‐range structure of the model and hence are not universal. However, once the conserved quantities of the model are fixed to have the same scaling structure as energy and helicity, these deviations seem to depend only weakly upon the scale parameter of the model. The connection between multifractality in the velocity distribution and multifractality in the dissipation is analyzed. Arguments suggest that the connection is universal for models of this character, but the model has a different behavior from that of real turbulence. Also, the scaling behavior of time correlations of shell velocities, of the dissipation, and of Lyapunov indices are predicted. These scaling arguments can be carried over, with little change, to multifractal models of real turbulence.

Characterization of the low‐pressure filaments in a three‐dimensional turbulent shear flow
View Description Hide DescriptionIn this paper results on the low‐pressure filaments that appear spontaneously in three‐dimensional turbulent flows are presented. An individual characterization of the filaments is first obtained by studying the correlations between the flow visualization and local measurements of the pressure and the velocity. Then, a statistical study of the time recordings of the pressure that exhibits intermittent short and deep depressions is presented. It is shown that the pressure histograms depend only on the square of the injection velocity, and that the rate of production of strong depressions is independent of the Reynolds number. These results impose severe constraints on the possible mechanisms of formation of the filaments; they are consistent with a simple model, in which the formation of the filaments results primarily from the partial rollup of stretched shear layers. In this model there is a difference between the hierarchies of pressure and vorticity filaments: the filaments with the largest depression are the thickest (and the longest), while the filaments with the strongest vorticity are likely to be the thinnest (and shortest).

Direct numerical simulation of particle entrainment in turbulent channel flow
View Description Hide DescriptionParticle entrainment process in a turbulent channel flow is studied. The time history of the instantaneous turbulent velocity vector field is generated by the direct numerical simulation of the Navier–Stokes equation with the aid of a pseudospectral code. The equation of motion of submicrometer particles including Stokes drag and Brownian diffusion is used, and typical entrained particle trajectories are evaluated and statistically analyzed. It is shown that the wall coherent structure plays a dominant role on the particle entrainment process. Particles are removed from the wall region by being captured in the high speed streams moving away from the wall, which are formed by the flow structure. Furthermore, single streamwise vortices are shown to be more frequent than pairs of counter‐rotating ones at every instance of time. Temporal average of the vorticity field, however, shows roughly periodic sequence of counter‐rotating vortices in the wall region.

A multiple‐scale model for compressible turbulent flows
View Description Hide DescriptionA multiple‐scale model for compressible turbulent flows is proposed in this paper. It is assumed that turbulent eddy shocklets are formed primarily by large energetic eddies. The extra straining of the large eddy, due to their interactions with shocklets, enhances the energy cascade to smaller eddies.Modeltransportequations are developed for the turbulent kinetic energies and the energy transfer rates of the different scale. The turbulent eddyviscosity is determined by the total turbulent kinetic energy and the rate of energy transfer from the large scale to the small scale, which is different from the energy dissipation rate. The model coefficients in the modeled turbulent transportequations depend on the ratio of the turbulent kinetic energy of the large scale to that of the small scale, which renders the model more adaptive to the characteristics of individual flow. The model is tested against compressible free shear layers, boundary layers, and a compression ramp flow. The results agree satisfactorily with measurements.

Interactions between two electrified liquid columns pinned on a dielectric solid surface
View Description Hide DescriptionAs a preliminary to understanding the complicated interactions between two electrified drops, this paper analyzes the simpler but instructive problem of the electrohydrostatic interactions between two parallel, translationally symmetric supported liquid columns. The columns are taken to be electrically conducting, surrounded by an insulating fluid, and pinned at their contact lines on the surface of an insulating solid support. The issue of the shapes and stability of the columns is mathematically a two‐dimensional, nonlinear free boundary problem, which is solved here by means of the Galerkin/finite element method. Despite the generality of the formulation of the problem and that of the numerical scheme used to solve it, attention is focused here on situations in which the two columns have the same volume per unit length and their undeformed cross sections correspond to semicircles bounded by a straight line that represents the solid plane on which the columns are pinned. Computational results are reported that show different behaviors of electrified columns obtained by varying the direction in which the external field is applied or connecting/disconnecting the columns from power supplies that maintain them at fixed electric potentials. When the potentials of the columns are held fixed, an externally applied field always tends to pull them apart. For isolated columns, however, an externally applied field within a large range of oblique angles tends to pull the columns together. In the absence of an externally applied field, electrostatic interactions can also arise when the columns bear net electrical charges. Whether the interaction forces are attractive or repulsive then depends on the relative amounts of charge on the columns. An attractive force can result even when the two columns bear net charges of the same sign, provided that the relative difference in the amounts of charge is large enough.

The structure of (linearly) stable double diffusive flow patterns in a laterally heated stratified liquid
View Description Hide DescriptionLayered double diffusive flow patterns in a laterally heated stably stratified liquid are considered in a configuration which allows for steady states to exist. For the heat/salt system, these flows are characterized by the thermal and solutal Rayleigh numbers Ra _{ T } and Ra _{ S }, or equivalently by Ra _{ T } and the buoyancy ratio R _{ρ}. The bifurcation structure of steady patterns with respect to Ra _{ T } is computed for two cases: fixed Ra _{ S } and fixed R _{ρ}. For the first case, results in N. Tsitverblit and E. Kit [Phys. Fluids A 5, 1062 (1993)], are computed and extended, and it is shown that many of the previously found flow patterns are unstable; only in a small interval of Ra _{ T }, multiple (linearly) stable steady states exist. For the second case, the physical relevance of the unstable steady states with respect to the evolution of the flow toward a stable steady state is demonstrated.