Volume 13, Issue 6, June 2001
 LETTERS


Evolution of particlevelocity correlations in sedimentation
View Description Hide DescriptionThe velocity field of a dilute monodisperse sedimenting suspension is measured as a function of time with the Eulerian particle image velocimetry technique. The sedimentation process is observed to be dominated by vortices of the size of the container in the initial moments after the cessation of mixing, which diminish with time and seem to reach an ultimate size mean interparticle distances.

Chainlinkfence structures produced in a plane jet
View Description Hide DescriptionVortical structures of the chainlinkfence type have been experimentally produced in a plane jet. The jet was excited by temporal periodic disturbances with spanwise phase variations added in the initial shear layer. Chainlinkfencelike structures, which significantly differed from ordinary vortices such as “roller and rib” structures, are observed experimentally by stereoscopic particle image velocimetry when the temporal phase difference of disturbances with the spanwise direction is 180°. Some of the original vortices remain downstream as stretched lambdatype vortices, although the destruction of the orderly vortices into complex turbulence was significant.

Universal spectrum of zonal flows on giant planets
View Description Hide DescriptionThe energy spectra of the observed zonal flows on Jupiter and Saturn are shown to obey the scaling law in the range of total wave numbers n not affected by large scale friction (here, and R are the rotation rate and the radius of the planet, and is an orderone constant). These spectra broadly resemble their counterpart in recent simulations of turbulent flows on the surface of a rotating sphere [Huang et al., Phys. Fluids 13, 225 (2001)] that represents a strongly anisotropicflow regime evoked by the planetary vorticity gradient. It is conjectured that this regime governs the large scale circulations and the multiple zonal jets on giant planets. The observed strong equatorial jets that were not produced in the nearly inviscid simulations by Huang et al. are attributed to the combined effect of the energy condensation in the lowest zonal modes and the large scale friction.
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 ARTICLES


A swarm of Stokeslets with interfacial tension
View Description Hide DescriptionA formal analogy between sedimenting drops in Stokes flow and a swarm of Stokeslets [Machu et al., J. Fluid Mech. (in press)] is extended to include interfacial tension. Using a cohesive potential, mean curvature is extended as a meaningful quantity off the interface, allowing the boundaryintegral formulation to be rewritten in volumetric form. A prescription for assigning forces to the Stokeslets comprising the swarm incorporates the action of interfacial tension without having to identify a boundary surface. Numerical simulations agree with linear smalldeformation theory, and reproduce the spontaneous coalescense of two touching drops.

Secondary breakup of axisymmetric liquid drops. II. Impulsive acceleration
View Description Hide DescriptionThe secondary breakup of impulsively accelerated liquid drops is examined for small density differences between the drops and the ambient fluid. Two cases are examined in detail: a density ratio close to unity and a density ratio of 10. A finite difference/front tracking numerical technique is used to solve the unsteady axisymmetric Navier–Stokes equations for both the drops and the ambient fluid. The breakup is governed by the Weber number, the Reynolds number, the viscosity ratio, and the density ratio. The results show that Weber number effects are dominant. In the higher density ratio case, different breakup modes—oscillatory deformation, backwardfacing bag mode, and forwardfacing bag mode—are seen as the Weber number increases. The forwardfacing bag mode observed at high Weber numbers is an essentially inviscid phenomenon, as confirmed by comparisons with inviscid flow simulations. At the lower density ratio, the backwardfacing bag mode is absent. The deformation rate also becomes larger as the Weber number increases. The Reynolds number has a secondary effect, changing the critical Weber numbers for the transitions between breakup modes. The increase of the dropviscosity reduces the drop deformation. The results are summarized by “breakup maps” where the different breakup modes are shown in the We–Re plane for different values of the density ratios.

Flow fields with constant rate of strain along streamlines
View Description Hide DescriptionIn order to study the effect of strain rate on entrained bubbles it would be useful to produce a flow in which the bubbles are subjected to uniform rates of strain. However it is shown that the only steady twodimensional or axisymmetric flows with constant components of the rate of strain tensor in streamline coordinates are the trivial cases of rigid body motion.

An experimental study of drop deformation and breakup in extensional flow at high capillary number
View Description Hide DescriptionAn experimental study is reported of the flowinduced stretching of drops in a fourroll mill at low Reynolds number, but for capillary numbers that are large compared to the critical capillary number for onset of stretching. We mainly consider Newtonian drops in a Newtonian suspending fluid, but also present a brief study of Newtonian drops in a viscoelastic (Boger fluid) suspending fluid. The stability of the drops following cessation of flow is determined, in either case, by the ratio of their extended length to the undeformed radius. If this ratio is large enough, the drops will break into two or more parts via the capillary flow process known as endpinching. However, for Newtonian drops in a Newtonian suspending fluid, it is shown that the critical degree of stretch for breakup increases sharply with increase of the capillary number that characterizes the stretching process. Furthermore, it is shown, in this case, that the critical stretch ratio is not unique, but that there can be a discrete range of stretch ratios above the first (or smallest) critical value where the drop is again stable before it encounters a second larger “critical” stretch ratio. This “restabilization” is associated with the transition from two to three drops in the breakup process. Newtonian drops in the viscoelastic Boger fluid are found to be slightly more stable than the same drops in a Newtonian fluid when stretched at strain rates just exceeding the critical value. By this we mean that the critical elongation ratio necessary for the drops to break upon cessation of flow is increased by about 20%. When stretched at a higher strain rate, approximately 2.15 times the critical value, large drops in the viscoelastic fluid (above 100 microns in radius for this particular suspending fluid) are destabilized relative to their counterpart in a Newtonian suspending fluid, while smaller drops are strongly stabilized.

Slow viscous flow in a partitioned channel
View Description Hide DescriptionTwodimensional slow viscousflow in a partitioned channel is investigated based on the Stokes approximation. The partitioned channel is composed of two infinite parallel planes and a semiinfinite plate located midway between the infinite planes. The flow is allowed to turn around the semiinfinite plate by a proper pressure gradient. By solving a Wiener–Hopf equation, an exact analytic solution for the stream function is obtained. From the streamline patterns shown, it is found that a Moffatt’s infinite sequence of viscouseddies develops in the channel. The pressure and shear stress distribution on the boundaries are illustrated, and a local analysis for the flow near the edge of the plate is performed. By superposing the symmetric and the antisymmetric flow, general flows in this partitioned channel are obtained and typical flow patterns are shown.

Nonlinear phenomena in hybrid Couette flow composed of planar and circular shear
View Description Hide DescriptionResults are presented from an experimental investigation of a novel shear flow. Two parallel sections of planar Couette flow are connected by two semicircular sections of circular Couette flow to give a flow domain with the shape of a running track. Driving the flow with a moving inner boundary leads to centrifugal instability in the curved regions as in conventional Taylor–Couette flow. This is in contrast to the planar regions, which are linearly stable and are characterized instead by finiteamplitude instability. In the steady regime, the entire flow field is dominated by structures akin to Taylor vortices. The mechanism of exchange between a fourcell and a sixcell flow over a range of aspect ratio is qualitatively the same as for the standard Taylor–Couette problem. In the unsteady regime, the flow is characterized by various spatiotemporal modes, the selection of which is dependent on the manner of flow evolution. Quasistatic increase of the Reynolds number from zero typically results in flow with a banded spatial structure and lowdimensional dynamics, both of which are associated with instability in the semicircular regions. However, an abrupt steplike increase of Reynolds number produces a persistent flow state with strong spatial disorder and a broadband dynamical spectrum. The results of this study have implications for the conventional distinctions between the properties of open and closed flows, and suggest the possibility of intermediate flows which are worthy of investigation in their own right.

Thermocapillary migration of long bubbles in polygonal tubes. I. Theory
View Description Hide DescriptionThermocapillary migration of long gas bubbles in cylinders of regularpolygonal or rectangular cross sections is studied. An imposed axial temperature gradient produces a gradient of surface tension leading to a steady migration of the bubble towards the hotter region. A leading order approximation for the bubble migration speed is found by computing the volume flux of liquid through the corner regions of the cylinder, which provide a parallel channel for the flow of fluid driven by the Marangoni stress. A global mass balance is used to relate the dimensionless bubble speed to the modified capillary number where β is the temperature gradient, a the tube length scale, σ the mean surface tension, and the temperature coefficient of surface tension. The dimensionless bubble speed is found to be linear in this parameter at leading order. The approximation is improved by accounting for the deposition of a thin film on the cylinder walls at small capillary number. A modified Landau–Levich equation governs the thin film profile, the solution of which allows the calculation of the additional flux so produced. It is found that corrections in the bubble velocity are nonlinear in Δσ^{*}. For regularpolygonal tubes with small numbers of sides or low aspect ratio rectangular cross sections, most of the flux passes through the corner regions, while for larger numbers of sides or large aspect ratios, flow in the thin film regions dominates and the results tend toward those for cylindrical tubes.

Stability and evolution of a dry spot
View Description Hide DescriptionThe motion of a thin viscous layer of fluid on a horizontal solid surface bounded laterally by a dry spot and a vertical solid wall is considered. A lubrication model with contact line motion is studied. We find that for a container of fixed length the axisymmetric equilibrium solutions with small dry spots are unstable to axisymmetric disturbances. As the size of the dry spot increases, the equilibrium solutions become unstable to nonaxisymmetric disturbances. In addition, we present numerical solutions of the nonlinear evolution equations in the axisymmetric and nonaxisymmetric cases for different values of the parameters. The axisymmetric results show good agreement with existing experimental results.

Thermodynamics modeling for moving contact line in gas/liquid/solid system: Capillary rise problem revisited
View Description Hide DescriptionThe nonequilibrium thermodynamics framework is tailored in the present paper to formulate the gas/liquid/solid system. In this system, there are two important issues, namely, the contact line motion and contact angle change, and shear stress singularity, during the dynamic evolution. Traditionally, the fluid mechanics approach has been applied to model these two issues. In the present paper, we applied a thermodynamics formulation to reexamine the first issue, i.e., the dynamic motion of the contact line and angle. It provides a new angle to understand the fundamentals of this classical problem. In order to verify the reliability of the present thermodynamics formulation, the capillary rise problem is revisited by using the formulation. A numerical result obtained based on the thermodynamics formulation is then compared with experimental test data. Excellent agreement between our analytical and experimental results gives us confidence for the future works on this approach.

Evolution of small scale regular patterns generated by waves propagating over a sandy bottom
View Description Hide DescriptionIn this paper the evolution of a sandy bottom subject to a wave generated flow has been analyzed using an image acquisition technique. In particular, ripple formation was observed starting from a flat bed till a stable configuration was attained. The experimental findings showed that rolling grain ripples never appeared as a stable configuration but only as a transition toward the equilibrium, represented by vortex ripples. The latter stage was reached after about 100 cycles if lightweight sediments were used, or about 400 cycles if quartz sediments were adopted. It was also observed that when the bedforms appear their wavelength is much smaller (about one half) than that at the equilibrium stage, this result being in contrast with most of the analytical models on bedform evolution. Finally, it was observed that ripples migrate as soon as they appear. The measuredvelocity of migration seems to confirm, from a qualitative point of view, the theoretical findings obtained for rolling grain ripples by Blondeaux et al. [Eur. J. Mech. B/Fluids 19, 285 (2000)].

On the nonparallel stability of the injection induced twodimensional Taylor flow
View Description Hide DescriptionIn this article, the nonparallel stability of the socalled Taylor flow induced by wall injection is investigated. This flow is unstable due to nonparallel effects which have not been rigorously treated yet. Only inconsistent methods have been used. They give good results in comparison to the experiments but they suffer from a lack of justification and from the dependency of the results on the formulation. The necessity of using inconsistent methods instead of the Orr–Sommerfeld approach is justified in this article and a much more rigorous asymptotic analysis is performed in the limit of the large streamwise distances.

Selfsimilarity of strongly stratified inviscid flows
View Description Hide DescriptionIt is wellknown that strongly stratified flows are organized into a layered pancake structure in which motions are mostly horizontal but highly variable in the vertical direction. However, what determines the vertical scale of the motion remains an open question. In this paper, we propose a scaling law for this vertical scale when no vertical lengthscales are imposed by initial or boundary conditions and when the fluid is strongly stratified, i.e., when the horizontal Froude number is small: where U is the magnitude of the horizontal velocity, N the Brunt–Väisälä frequency and the horizontal lengthscale. Specifically, we show that the vertical scale of the motion is by demonstrating that the inviscid governing equations in the limit without any a priori assumption on the magnitude of are selfsimilar with respect to the variable where z is the vertical coordinate. This selfsimilarity fully accounts for the layer characteristics observed in recent studies reporting spontaneous layering from an initially vertically uniform flow. For such a fine vertical scale, vertical gradients are large, Therefore, even if the magnitude of the vertical velocity is small and scales like the leading order governing equations of these strongly stratified flows are not twodimensional in contradiction with a previous conjecture. The selfsimilarity further suggests that the vertical spectrum of horizontal kinetic energy of pancake turbulence should be of the form giving an alternative explanation for the observed vertical spectra in the atmosphere and oceans.

On the breaking of standing waves by falling jets
View Description Hide DescriptionBy direct numerical time stepping, using two different methods of calculation, we follow the development of a standing gravity wave on the surface of deep water when it is given an initial energy exceeding that of the most energetic periodic wave of the same wavelength. Two aspects of the motion are studied in detail: the sequence of wave profiles close to the instant when the maximum surface slope angle is close to 45° and, second, the conditions under which a sharp cusp is formed at the free surface. When a cusp is formed, it can fall vertically into the wave trough, enclosing “bubbles of air.”

A numerical study of the evolution of a solitary wave over a shelf
View Description Hide DescriptionUsing Reynolds averaged Navier–Stokes (RANS)equations, we have conducted a series of numerical experiments to investigate the evolution of a solitary wave propagating over a step. Both nonbreaking and breaking solitary waves are studied. To study breaking waves, the RANSequations are coupled with turbulenceequations. For the nonbreaking wave case the numerical solutions are compared with available experimental data. The numerical experiments demonstrate that both nonbreaking and breaking solitary waves disintegrate into several solitons over the step. However, the fission processes for generating the second and third soliton are quite different for nonbreaking and breaking solitary waves.

Linear stability and receptivity analyses of the Stokes layer produced by an impulsively started plate
View Description Hide DescriptionThe stability and the receptivity of the boundary layer produced by the impulsive motion of a flat plate in its plane is studied. The evolution of twodimensional traveling disturbance waves for this physical situation, known as Stokes’ first problem, is treated by integrating directly the (parabolic in time) linearized Navier–Stokes equation and by a multiplescale approach. In the asymptotic analysis, the Orr–Sommerfeld equation is found at leading order. Through the compatibility condition for the equation at next order, an correction to growth rates and frequencies is achieved. Such corrections are found to be very mild. After having established that the leadingorder results are adequate when looking at the stability characteristics of the flow for a given (large) time, a receptivity analysis is performed. The adjoint of the parabolic system is obtained, and through its backwardintime integration, the initial and wall Green’s functions are obtained. These are then compared to the results of the multiplescale receptivity analysis.

Precessing vortex breakdown mode in an enclosed cylinder flow
View Description Hide DescriptionThe flow in a cylinder driven by the rotation of one endwall for height to radius ratios around three is examined. Previous experimental observations suggest that the first mode of instability is a precession of the central vortex core, whereas a recent linear stabilityanalysis to general threedimensional perturbations suggests a Hopf bifurcation to a rotating wave at lower rotation rates than those where the precession mode was first detected. Here, this apparent discrepancy is resolved with the aid of fully nonlinear threedimensional Navier–Stokes computations.

Investigating a stretched vortex with ultrafast twodimensional ultrasonic speckle velocimetry
View Description Hide DescriptionUltrasonicspecklevelocimetry (USV) was recently introduced as a new ultrafast tool for measuringfluidvelocities in two dimensions by analyzing the speckle signal backscattered from particles moving with the flow [Sandrin et al., Appl. Phys. Lett. 78, 1155 (2001)]. An experimental study of a stationary stretched vortex is presented, which demonstrates the efficiency of ultrafast 2D USV in terms of quantitative assessment of vortical flows. The full velocity field is recovered and compared to the Burgers model for two different fluids: water and milk. The dependence of the vortex characteristics on the main control parameter is investigated.
