Volume 12, Issue 3, March 2000
 LETTERS


Free decay of turbulence and breakdown of selfsimilarity
View Description Hide DescriptionIt has been generally assumed, since the work of von Kármán and Howarth in 1938, that free decay of fullydeveloped turbulence is selfsimilar. Here we present a simple phenomenological model of the decay of threedimensional incompressible turbulence, which predicts breakdown of selfsimilarity for lowwavenumber spectral exponents n in the range where is some threshold wavenumber. Calculations with the eddydamped quasinormal Markovian approximation give the value as The energy spectrum for this range of exponents develops two lengthscales, separating three distinct wavenumber ranges.
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 ARTICLES


Spreading of a wetting film under the action of van der Waals forces
View Description Hide DescriptionThe profiles of a spreading wetting film are computed using a variable grid implicit scheme. The form of Tanner’s law is deduced from the scaling, and the dependence of its coefficient on ratio of the van der Waals to capillary length and on the inclination angle is determined.

Drop deformation, breakup, and coalescence with compatibilizer
View Description Hide DescriptionThe effect of copolymers on the breakup and coalescence of polybutadiene (PB) drops in polydimethylsiloxane(PDMS) is studied using a fourroll mill flow cell. Copolymers are produced at the interface by a reaction between functionalized homopolymers. They reduce the interfacial tension and thus enhance breakup; they also inhibit coalescence of drops. Under the conditions of our experiments, the latter effect is much more significant than the former. For example, the addition of copolymer sufficient to reduce the interfacial tension by only 3% relative to the bare interface value is found to reduce the critical capillary number for coalescence by a factor of 6. The critical capillary number for coalescence in the absence of copolymer is also measured for the first time. It is found to scale with the drop radius a as and with the viscosity ratio λ as

Homogeneous pattern selection and director instabilities of nematic liquid crystal polymers induced by elongational flows
View Description Hide DescriptionWe characterize homogeneous patterns, their stability, and phase transitions in nematic liquid crystalpolymers(LCPs) with imposed elongational flows. We combine the flowinduced analysis of order parameters by See et al. [J. Chem. Phys. 92, 792 (1990)], Bhave et al. [J. Rheol. 37, 413 (1993)], Rey [Macromol. Theory Simul. 4, 857 (1995)], and Wang [J. NonNewtonian Fluid Mech. 22, 147 (1997)], with the pure nematic, full tensor analysis of Shimada et al. [J. Chem. Phys. 88, 7181 (1988)]. To make contact with these seminal studies, we select a momentaveraged Doi kinetic model for flows of rodlike nematic LCPs with a quartic shortrange intermolecular potential; the connection with alternative kinetic and continuum models for flows of LCPs is noted. New elongationinduced director instabilities are revealed for patterns previously identified as candidates for stable pattern selection. From a full tensor analysis, we determine the complete phase diagram for homogeneous patterns in the parameter space of LCP concentration and elongation rate. With respect to experimental predictions, in axial extension, biaxial patterns exist but they are all unstable and the only stable patterns are uniaxial; in planar extension, above a moderate concentration the only stable nematic patterns are biaxial.

Numerical investigation of boundary conditions for moving contact line problems
View Description Hide DescriptionWhen boundary conditions arising from the usual hydrodynamic assumptions are applied, analyses of dynamic wetting processes lead to a wellknown nonintegrable stress singularity at the dynamic contact line, necessitating new ways to model this problem. In this paper, numerical simulations for a set of representative problems are used to explore the possibility of providing materialboundary conditions for predictive models of inertialess moving contact line processes. The calculations reveal that up to Capillary number Ca=0.15, the velocity along an arc of radius is an inner, microscopic length scale) from the dynamic contact line is independent of the macroscopic length scale a for and compares well to the leading order analytical “modulatedwedge” flow field [R. G. Cox, J. Fluid Mech. 168, 169 (1986)] for Capillary number Ca<0.1. Systematic deviations between the numerical and analytical velocity field occur for 0.1<Ca<0.15, caused by the inadequacy of the leading order analytical solution over this range of Ca. Meniscus shapes produced from calculations in a truncated domain, where the modulatedwedge velocity field [R. G. Cox, J. Fluid Mech. 168, 169 (1986)] is used as a boundary condition along an arc of radius from the dynamic contact line, agree well with those using two inner slip models for Ca<0.1, with a breakdown at higher Ca. Computations in a cylindrical geometry reveal the role of azimuthal curvature effects on velocity profiles in the vicinity of dynamic contact lines. These calculations show that over an appropriate range of Ca, the velocity field and the meniscus slope in a geometryindependent region can potentially serve as material boundary conditions for models of processes containing dynamic contact lines.

Twodimensional stagnation flow around a vertical plate above a plane wall
View Description Hide DescriptionTwodimensional slow viscous stagnation flow around a vertical plate above a plane wall is investigated based on the Stokes approximation. An exact formal expression of the stream function is obtained by use of the threepart Wiener–Hopf technique. From the formal expression obtained, the streamline patterns around the plate are shown and the force exerted on the plate is calculated. The stress distributions on the boundaries are also calculated. By examining the limiting case of very small distance between the wall and the plate, the formation of the Moffatt’s viscouseddies near the corner is explained.

The interdependence of friction, pressure gradient, and flow rate in unsteady laminar parallel flows
View Description Hide DescriptionSolutions to the Laplacetransformed Navier–Stokes equations are developed that describe transients in fully developed channel and pipe flow. The relative ease with which inverse Laplace transforms can be carried out numerically makes it straightforward to find the form of new expressions relating flow rate, pressure gradient, and wall friction, for flows of arbitrary unsteadiness in time. In particular, expressions for flow rates and cumulative throughflows are derived in terms of wall shear stress and pressure–gradient histories, together with a channelflow counterpart to Zielke’s pipeflow friction law [J. Basic Eng. 90, 109 (1968)] expressing wall shear stress as a functional of flowrate history. It is shown how these results can be expressed as the unsteady counterparts to wellknown steadyflow relationships. The relative importance of unsteady effects to quasisteady ones is determined by a dimensionless parameter of the form where R is the span of the duct. When departures from quasisteady forms of these relations exist, the memory of the Navier–Stokes equations of earlier transients, comparable in size to present ones, extends roughly seconds backward in time. The derived relationships are used to illustrate how pathdependent quantities such as flow work vary with different unsteady flow histories.

Instability of a liquid jet emerging from a droplet upon collision with a solid surface
View Description Hide DescriptionA linear perturbation theory is developed to investigate the interface instabilities of a radiallyexpanding, liquid jet in cylindrical geometries. The theory is applied to rapidly spreading droplets upon collision with solid surfaces as the fundamental mechanism behind splashing. The analysis is based on the observation that the instability of the liquid sheet, i.e., the formation of the fingers at the spreading front, develops in the extremely early stages of droplet impact. The shape evolution of the interface in the very early stages of spreading is numerically simulated based on the axisymmetric solutions obtained by a theoretical model. The effects that factors such as the transient profile of an interface radius, the perturbation onset time, and the Weber number have on the analysis results are examined. This study shows that a large impact inertia, associated with a high Weber number, promotes interface instability, and prefers high wave number for maximum instability. The numbers of fingers at the spreading front of droplets predicted by the model agree well with those experimentally observed.

Thermocapillary migration of long bubbles in cylindrical capillary tubes
View Description Hide DescriptionWe consider the problem of the migration of a long bubble in a tube with a prescribed axial temperature gradient. The resulting thermocapillary stress moves the bubble toward hotter regions and we are interested in determining the speed of the bubble. Assuming small Peclet, Reynolds, Bond, and capillary numbers, Ca allows the uncoupling of the temperature field from the flow field, the use of creeping flow and lubricationtheory, the assumption of axisymmetry, and the use of matched expansions in Ca, respectively. The structure of the solution is that of a constant thickness film bounded by constant curvature cap regions, with transition layers in between. A modified Landau–Levich equation governing the film profile in the transition regions is solved numerically, establishing the relationship between the unknown film thickness and the unknown bubble speed. A global mass conservation relation is then used to complete the solution and relate the bubble speed to the thermophysical properties. The solution is a function of a single dimensionless parameter, where is the temperature gradient, a the tube radius, the mean surface tension, and the temperature coefficient of surface tension. It is found that the speed increases with the temperature gradient as expected, and has a nonlinear dependence on that is ultimately connected with the nonlinearity of the fluid dynamics in the transition regions.

The motion of a falling liquid filament
View Description Hide DescriptionWhen a liquid drop falls from a fluid source with a slow flow rate, it remains attached to the source by an elongating liquid filament until the filament pinches off. For many fluids, this pinchoff occurs first near the end of the filament, where the filament joins to the liquid drop. For other fluids, the filament pinches off at one or more interior points. In this paper, we study the motion of this filament, and we make two points. First, the flow in this filament is not that of a uniform jet. Instead, we show experimentally that a different solution of the Navier–Stokes equations describes the motion of this filament before it pinches off. Second, we propose a criterion for the location of the first pinchoff. In particular, we analyze the linearized stability of the exact solution, both for an inviscid fluid and for a very viscous fluid. Our criterion for pinchoff is based on this stability analysis. It correctly predicts whether a given filament pinches off first near its ends or at points within its interior for all of our experimental data.

Twodimensional nuclear magnetic resonance measurements and numerical simulations of fluid transport in porous rocks
View Description Hide DescriptionPulsed magnetic field gradient nuclear magnetic resonance (PFGNMR) measurements have been performed for water flowing through porous Fontainebleau sandstones and are compared with flow through a packed bed of monodisperse glass beads. Pulsed gradients were applied both parallel and perpendicular to the main flow axis simultaneously to obtain the twodimensional displacement joint probability density of the moving spins. The evolution of as a function of encoding time Δ and flow rate is investigated. Good agreement is found between experimental and those obtained by numerical simulations of flow through computergenerated structures of equivalent statistical properties to those studied. The simulations are employed to compare a wider range of flow parameters than those accessible by experiment. In addition to averaged quantities, such as dispersion coefficients and moments of displacement distributions, the correlations between displacements in both directions are presented. The average transverse dispersion, for a subset of particles possessing a given axial displacement, at any encoding time Δ is found to scale with for flow rates and times discussed in this study, a power law relation is observed with the spreading exponent γ being characteristic of the connectivity and statistical geometric features of the pore space. The correlation coefficient is found to be positive in all cases and strongly influenced by the ratio of convective to diffusive contributions to the total particle displacements, expressed by the Péclet number. A maximum in the correlation coefficient occurs at a time scale dependent on the Péclet number and in the structures studied here, this corresponds to a characteristic lengthscale of the systems, approximated by their average pore size.

Point vortex motion on a sphere with solid boundaries
View Description Hide DescriptionWe consider the motion of a point vortex on the surface of a sphere with solid boundaries. This problem is of interest in oceanography, where coherent vortex structures can persist for long times, and move over distances large enough so that the curvature of the Earth becomes important (see Gill [1982], Chaos [1994]). In this context, the boundary is a first step in modeling the presence of coastlines and shores using inviscid theory. Using the equations of motion for the vortex projected onto the stereographic plane, we construct the appropriate Green’s function using classical image method ideas, as long as the domain has certain symmetry properties. After the solution is obtained in the stereographic plane, it is projected back down to the sphere, yielding the sought after solution to the problem. We demonstrate the utility of the method by solving for the vortex trajectories and streamlines for several canonical examples, including a spherical cap, longitudinal wedge, half longitudinal wedge, channel, and rectangle. The results are compared with the corresponding ones in the physical plane in order to highlight the effect of the spherical geometry.

Effects of trailing jet instability on vortex ring formation
View Description Hide DescriptionNumerical simulations of an impulsively started jet were performed in order to investigate the effects of trailing jet instability on axisymmetric vortex ring formation. The predictions were compared to experimental results reported in the literature and to recently published numerical results. The total and vortex ring circulations were found to be in good agreement with both the experimental and the numerical results. The presence of a universal formation time scale was confirmed. The results also highlighted an important interaction between an instability which develops in the trailing jet for large discharge times and the dynamics of the head vortex ring. This interaction accelerates the process by which the vortex ring detaches from the trailing jet and has a significant effect on the vortex ring circulation.

Infrared imaging of the surface temperature field of water during film spreading
View Description Hide DescriptionDeposition of a spontaneouslyspreading film on a clean water surface creates a front which propagates radially outward from the point of deposition. This rapidly spreading film was used as a tool to quickly change the boundary condition of a water surface from one which is shearfree, to a boundary condition which supports shear. Infrared images of a water surface experiencing evaporative convection were recorded as this film spread. These images were converted to surface temperature fields. The amount of turbulent structure present in these fields changes dramatically across the front. Ahead of the front, significant variations at large and small spatial scales are evident, while behind the front the small scale structures are eliminated. The time scale at which this damping occurs is short and has not been reported on heretofore. In addition to being relevant to free surfaceturbulence, these results demonstrate the utility of infrared imaging in the study of spreading films.

Numerical simulation of bubbletype vortex breakdown within a tubeandvane apparatus
View Description Hide DescriptionNumerical calculations for threedimensional, unsteady, laminar, bubbletype vortex breakdown within a tubeandvanetype apparatus at a Reynolds number of 2000 and circulation number Ω=1.41 are presented. This study is unique in that rather than specifying the inlet swirl velocity through a fit to experimental data (or a Burgers profile), the swirl was induced by directing the fluid through an array of 16 turning vanes, the arrangement being similar to that employed in the original experimental works of Sarpkaya [J. Fluid Mech. 45, 545 (1971); AIAA J. 12, 602 (1974)]. The interior of the resulting breakdown bubble consisted of one primary torroidal recirculation cell, which was tilted from, and found to gyrate about, the bubble centerline. The dominant frequency of gyration was identified, and the mechanism of fluid exchange examined. Subsequent calculations were performed using fixed inlet swirl and axial velocity profiles that were obtained from the results computed using the full geometry (including the turning vanes). Results revealed no significant difference in the downstream breakdown location or structure.

Threedimensional numerical experiments on thermal convection in a very viscous fluid: Implications for the dynamics of a thermal boundary layer at high Rayleigh number
View Description Hide DescriptionIn this study we present results from threedimensional numerical experiments on thermal convection in a volumetrically heated, infinite Prandtl number fluid cooled from above. At high Rayleigh number, a thin thermal boundary layer forms adjacent to the cold top boundary. On the basis of our numerical results we study the thermal structure and dynamics of this boundary layer and the population of plumes that it creates. Cold thermal plumes that develop by boundary layer instability form continuous nearly vertical columns that migrate horizontally sweeping off the unstable boundary layer. A plume usually persists until it coalesces with another plume. The average spacing of plumes, inferred from the variation of the observed number of plumes with Rayleigh number, is proportional to where and d are the thermal boundary layer and fluid layer thicknesses, respectively. Based on a “kinetic theory” of plume populations, we show that this is consistent with an equilibrium plume population in which the creation of plumes by boundary layer instability and their disappearance by coalescing with other plumes are balanced. This scaling of average plume spacing is a consequence of the width of velocity plumes in a very viscous (infinite Prandtl number) fluid comparable to the fluid layer depth. For finite Prandtl number, the same analysis but with temperature and velocity plumes of comparable width predicts a plume spacing proportional to the boundary layer thickness.

Energy and velocity of a forming vortex ring
View Description Hide DescriptionIt is known that vortex rings formed by large stroke ratios (in a piston/cylinder arrangement) pinch off from their generating jets at a fairly constant universal time scale. In this paper we show that the hypothesis that at the pinch off the translational velocity of the ring equals the jet flow velocity near the ring is equivalent to the recently proposed idea based on a variational principle by Kelvin and Benjamin that the pinch off occurs when the apparatus is no longer able to deliver energy at a rate required for steady vortex ring existence. A formula for the propagation velocity of a thick vortex ring is also proposed and compared with available experimental data and empirical relations.

Scott Russell’s wave generator
View Description Hide DescriptionIn this paper we study the dynamics of a system consisting of a heavy box sinking vertically into water. The classic configuration is due to Scott Russell who used the sinking box in 1844 to illustrate the formation of a solitary wave in a long rectangular tank. We use a combination of computer simulation and experiment to clarify details of the wave formation and the dynamics of the sinking box. We find that as the box sinks the water is heaved up to form both the solitary wave and a reverse plunging wave which forms a vortex. This vortex follows the wave down the tank. The computer simulation uses the particle method smoothed particle hydrodynamics (SPH) which allows us to follow the formation of the waves and the dynamics of the box. The simulation results are in satisfactory agreement with the experiments. We derive scaling relations for the velocity of the box and for the amplitude of the solitary wave. These scaling relations are in reasonable agreement with the experiments and the simulations.

Turbulent boundary layer control utilizing the Lorentz force
View Description Hide DescriptionDirect numerical simulations (DNS) of a turbulent channel flow at low Reynolds number where is the Reynolds number based on the wallshear velocity and channel halfwidth) are carried out to examine the effectiveness of using the Lorentz force to reduce skin friction. The Lorentz force is created by embedding electrodes and permanent magnets in the flat surface over which the flow passes. Both openloop and closedloop control schemes are examined. For openloop control, both temporally and spatially oscillating Lorentz forces in the nearwall region are tested. It is found that skinfriction drag can be reduced by approximately 40% if a temporally oscillating spanwise Lorentz force is applied to a channel flow. However, the power to generate the required Lorentz force is an order of magnitude larger than the power saved due to the reduced drag. Simulations were carried out at higher Reynolds numbers to determine whether efficiency, defined as the ratio of the power saved to the power used, improves with increasing Reynolds number. We found that the efficiency decreases with increasing Reynolds number. An idealized wall–normal Lorentz force is effected by detecting the nearwall turbulent events responsible for highskin friction. It is found that the drag can be significantly reduced with a greater efficiency than that produced by the spanwise openloop control approach. This result suggests that, when employed with a closedloop control scheme, the Lorentz force might result in a net decrease of power required to propel objects through viscous conducting fluids.

Direct numerical simulation and subgrid analysis of a transitional droplet laden mixing layer
View Description Hide DescriptionDirect numerical simulations of a temporally developing, droplet laden mixing layer undergoing transition to mixingturbulence are conducted. The formulation includes complete twoway couplings of mass, momentum, and energy. As many as grid points are used to discretize the Eulerian gas phase equations and up to initially polydisperse evaporating droplets are tracked in the Lagrangian reference frame. The complete transition to mixingturbulence is captured for several of the higher Reynolds number simulations and it is observed that increasing the droplet mass loading ratio results in a more “natural” turbulence characterized by increased rotational energy and less influence of the initial forcing perturbations. An increased mass loading also results in increased droplet organization within the layer. An a priori subgrid analysis is then conducted which shows that neglecting subgrid velocity fluctuations in the context of large eddy simulations may result in significant errors in predicting the droplet drag force for Stokes numbers (with the flow time scale based on the mean velocity difference and initial vorticity thickness). Similar possible errors of lesser magnitude are also observed for the droplet heat flux and evaporation rate when thermodynamic subgrid fluctuations are neglected. An extension of the eddy interaction model commonly used in Reynoldsaveraged simulations is then proposed in order to account for the missing subgrid information. Probability density functions(PDFs) of the subgrid fluctuations calculated across homogeneous planes are shown to be highly intermittent, particularly near the laminar–turbulent boundaries of the mixing layer. However, the actual subgrid PDFs calculated locally are much less intermittent and may be adequately modeled by the Gaussian distribution throughout the majority of the mixing layer. A scale similarity model is then employed to predict both the velocity and thermodynamic subgrid variances. The similarity model is well correlated with the actual subgrid variances and shows good agreement in predicting the local fluctuation intensities when a filter widthdependent model constant is used. The subgrid fluctuation variances acting on the droplets are then shown to be well modeled if the Eulerian subgrid variance model is interpolated to the droplet locations.
