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Volume 7, Issue 2, February 1995

Evidence of shocklets in a counterflow supersonic shear layer
View Description Hide DescriptionIn this Letter, experimental evidence of shocklets in a counterflow Mach 2 shear layer is presented. Schlieren photography reveals shock waves emanating from the turbulent structure; they are normal in the vicinity of the structure and weaken into Mach waves a short distance away from the shear layer. The slope of the Mach waves suggests that the turbulent structures are nonstationary, even though the shear layer is symmetric.

Wetting effects on the spreading of a liquid droplet colliding with a flat surface: Experiment and modeling
View Description Hide DescriptionIn this paper an experimental and theoretical study of the deformation of a spherical liquid droplet colliding with a flat surface is presented. The theoretical model accounts for the presence of inertia, viscous,gravitation,surface tension, and wetting effects, including the phenomenon of contact‐angle hysteresis. Experiments with impingement surfaces of different wettability were performed. The study showed that the maximum splat radius decreased as the value of the advancing contact angle increased. The effect of impact velocity on droplet spreading was more pronounced when the wetting was limited. The experimental results were compared to the numerical predictions in terms of droplet deformation, splat radius, and splat height. The theoretical model predicted well the deformation of the impacting droplet, not only in the spreading phase, but also during recoiling and oscillation. The wettability of the substrate upon which the droplet impinges was found to affect significantly all phases of the spreading process, including the formation and development of a ring structure around the splat.

The spreading of volatile liquid droplets on heated surfaces
View Description Hide DescriptionA two‐dimensional volatile liquid droplet on a uniformly heated horizontal surface is considered. Lubrication theory is used to describe the effects of capillarity, thermocapillarity, vapor recoil, viscous spreading, contact‐angle hysteresis, and mass loss on the behavior of the droplet. A new contact‐line condition based on mass balance is formulated and used, which represents a leading‐order superposition of spreading and evaporative effects. Evolution equations for steady and unsteady droplet profiles are found and solved for small and large capillary numbers. In the steady evaporation case, the steady contact angle, which represents a balance between viscous spreading effects and evaporative effects, is larger than the advancing contact angle. This new angle is also observed over much of the droplet lifetime during unsteady evaporation. Further, in the unsteady case, effects which tend to decrease (increase) the contact angle promote (delay) evaporation. In the ‘‘large’’ capillary number limit, matched asymptotics are used to describe the droplet profile; away from the contact line the shape is determined by initial conditions and bulk mass loss, while near the contact‐line surface curvature and slip are important.

Scaling of temperature‐ and stress‐dependent viscosity convection
View Description Hide DescriptionSimple scaling analysis of temperature‐ and stress‐dependent viscosityconvection with free‐slip boundaries suggests three convective regimes: the small viscosity contrast regime which is similar to convection in a fluid whose viscosity does not depend on temperature, the transitional regime characterized by self‐controlled dynamics of the cold boundary layer and the asymptotic regime in which the cold boundary becomes stagnant and convection involves only the hottest part of the lid determined by a rheological temperature scale. The first two regimes are usually observed in numerical experiments. The last regime is similar to strongly temperature‐dependent viscosityconvection with rigid boundaries studied in laboratory experiments.

Lagrangian self‐diffusion of Brownian particles in periodic flow fields
View Description Hide DescriptionThe steady transport of Brownian particles convected by a periodic flow field is studied by following the motion of a randomly chosen tagged particle in an otherwise uniform solute concentration field. A nonlocal, Fickian constitutive relation is derived, in which the steady mass flux of Brownian particles equals a convolution integral of the concentration gradient times a (tensorial) diffusion function D _{ L }(R). In turn, the diffusion function is uniquely determined via the nth diffusivities, which are determined analytically in terms of the nth cumulants of the probability distribution by exploiting the translational symmetry of the velocity field. The Lagrangian, long‐time self‐diffusion function D _{ L }(R) is shown to be equal to the symmetric part of the Eulerian, gradient diffusion function D _{ E }(R). Since the latter characterizes the dissipative steady‐state mass transport, while D _{ L }(R) describes the fluctuations of the concentration field about its uniform equilibrium value, the equality between D _{ E }(R) and D _{ L }(R) can be seen as an aspect of the fluctuation–dissipation theorem. Finally, the present results are applied to study the transport of solute particles immersed in a fluid flowing in rectilinear pipes and through periodic fixed beds of spheres at low Péclet number. In the first case, the first six nth diffusivities are determined; in the second, the first two diffusivities are calculated, showing that the enhancement to the second diffusivity due to convection is eight times larger in the direction parallel to the fluid flow than in the transversal direction.

Stokes drag on conglomerates of spheres
View Description Hide DescriptionThe Stokes drag coefficients for conglomerates of between two and 167 spheres are obtained from a recently developed scheme for numerical calculations of hydrodynamic interactions [J. Chem. Phys. 100, 3780 (1994)]. Experimental data for these conglomerates were provided by Lasso and Weidmann [Phys. Fluids 29, 3921 (1986)]. The numerical results and the experimental data agree very well. It is shown that in numerical calculations of hydrodynamic interactions, all long‐range contributions must be included exactly.

Cavity flow induced by a fluctuating acceleration field
View Description Hide DescriptionBuoyancy driven convection induced by a fluctuating acceleration field is studied in a two dimensional square cavity. This is a simplified model of, for example, fluid flow in a directional solidification cell subject to external accelerations, such as those encountered in a typical microgravity environment (g‐jitter). The effect of both deterministic and stochastic acceleration modulations normal to the initial density gradient are considered. In the latter case, the acceleration field is modeled by narrow band noise defined by a characteristic frequency Ω, a correlation time τ, and an intensity G ^{2}. If the fluid is quiescent at t=0 when the acceleration field is initiated, the ensemble average of the vorticity at the center of the cavity remains zero for all times. The mean squared vorticity 〈ξ^{2}〉, however, is seen to exhibit two distinct regimes: For t≪τ, 〈ξ^{2}〉 oscillates in time with frequency Ω. For t≫τ, 〈ξ^{2}〉 grows linearly in time with an amplitude equal to R^{2}Pr/(1+(Ωτ)) ^{2}, where R is a new dimensionless number which reduces to the Rayleigh number in the case of a constant gravity, and Pr is Prandtl number. At yet later times, viscous dissipation at the walls of the cavity leads to saturation, with 〈ξ^{2}〉_{sat}={(Pr τ+1)R^{2}/[(Pr τ+1)^{2}+Ω^{2}τ^{2}]}.

Large‐scale and periodic modes of rectangular cell flow
View Description Hide DescriptionLinear stability of the rectangular cell flow: Ψ=cos kx cos y (0<k<1), is studied, both numerically and analytically. Owing to its spatial periodicity, the disturbances are characterized by the Floquet exponents (α,β). Based on numerical results, it is found that two types of the critical modes with vanishingly small exponents exist. One type (large‐scale mode) has an almost uniform spatial structure. The other type (periodic mode) has a structure with the same periodicity as the main flow. The large‐scale mode gives the critical Reynolds number in a more isotropic case (i.e., k≳0.6), while the periodic mode does so in the less isotropic case (i.e., k<0.6). Asymptotic expansions from (α,β)=(0,0) agree with the numerical results. Using the periodic mode, a possible explanation is given for the merging process of a pair of counter‐rotating vortices observed in the experiments of a linear array of vortices by Tabeling et al. [J. Fluid Mech. 213, 511 (1990)].

Effect of film elasticity on the drift velocity of capillary–gravity waves
View Description Hide DescriptionThe effect of an insoluble, elastic surface film on the drift velocity of capillary–gravity waves is studied theoretically on the basis of a Lagrangian description of motion. There is no forcing from the atmosphere, and the wave amplitude is taken to attenuate in time. Defining a nondimensional parameter α, which combines film elasticity, fluid viscosity, and wave frequency, maximum damping of the linear waves occurs when α=1 (the Marangoni effect). In this case the frequency of capillary–gravity waves nearly coincides with that of elastic film waves. The nonlinear drift velocity is obtained for general values of α. In particular, it is found that the absolute maximum of the transient drift current is located below the surface when α≳2/3. At the surface, maximum drift velocity (in time domain) occurs for values of α that are somewhat less than one.

Anomalous sideband instabilities of thermal Rossby waves at low Prandtl numbers
View Description Hide DescriptionConvection in the form of thermal Rossby waves is studied in the low Prandtl number regime. An anomalous Eckhaus instability is found in the range between 10^{−3} and 10^{−2} for a moderately strong Coriolis parameter, where no supercritical stable solution can exist near the onset. Numerical results for higher Rayleigh numbers show detachment of the stability limit from the neutral curve, shrinkage and distortion of the stable region, and the onset of a second sideband mode of instability, which is characterized by a maximum growth rate at a finite modulation parameter. The new mode is more pronounced at higher Rayleigh numbers, and appears to destabilize all spatially periodic waves for the whole supercritical band for all Prandtl numbers below the anomalous range.

Transient, nonaxisymmetric modes in the instability of unsteady circular Couette flow. Laboratory and numerical experiments
View Description Hide DescriptionLaboratory and numerical experiments were conducted to quantitatively determine the modal structure of transient, nonaxisymmetric modes observed during the instability of an impulsively initiated circular‐Couette flow. The instability develops initially as an axisymmetric, Görtler‐vortex state and persists ultimately as a steady, axisymmetric Taylor‐vortex state of different wavelength. The transition between these two states results from the instability of the Görtler mode combined with the underlying developing swirl flow and is dominated by nonaxisymmetric modes. The laboratory experiments employed flow visualization coupled with digital video and image‐processing techniques; numerical experiments were performed using the spectral‐element code, nekton.

Finite amplitude perturbation and spots growth mechanism in plane Couette flow
View Description Hide DescriptionThe plane Couette flow, a shear flow linearly stable for all values of the Reynolds number,R, is experimentally studied. A finite amplitude perturbation, local in both time and space, is created in order to destabilize the flow. A critical amplitude, A _{ c }(R), below which disturbances are not sustained is measured. Above this amplitude, a turbulent spot grows to a spatially‐bounded turbulent state, persistent over times long compared to its typical growth time. The critical amplitude, A _{ c }(R), is seen to diverge when R approaches the nonlinear critical Reynolds numberR _{ NL }=325±5 from above. Below this value of the Reynolds number, no destabilization occurs with this kind of perturbation, whatever its amplitude. The divergent behavior on approaching R _{ NL } is characterized in terms of a power law. This result sheds light on the discrepancies previously observed between critical Reynolds number measurements. The spot is then analyzed in terms of its inside structure, spreading rates, as well as waves and velocity profiles close to the spot, in order to compare it to plane Poiseuille and boundary layer spots. The spot evolution appears to be very similar to that observed for the plane Poiseuille spot. It is shown that the growth of the plane Couette spot can be described by the mechanism of ‘‘growth by destabilization.’’

Instability in a spatially periodic open flow
View Description Hide DescriptionLaboratory experiments and numerical computations are conducted for plane channel flow with a streamwise‐periodic array of cylinders. Well‐ordered, globally stable flow states emerge from primary and secondary instabilities, in contrast with other wall‐bounded shear flows, where instability generally leads directly to turbulence. A two‐dimensional flow resembling Tollmien–Schlichting waves arises from a primary instability at a critical value of the Reynolds number,R _{1}=130, more than 40 times smaller than for plane Poiseuille flow. The primary transition is shown to be a supercritical Hopf bifurcation arising from a convectiveinstability. A numerical linear stability analysis is in quantitative agreement with the experimental observations, and a simple one‐dimensional model captures essential features of the primary transition. The secondary flow loses stability at R _{2}≊160 to a tertiary flow, with a standing wave structure along the streamwise direction and a preferred wave number in the spanwise direction. This three‐dimensional flow remains stable for a range of R, even though the structures resemble the initial stages of the breakdown to turbulence typically displayed by wall‐bounded shear flow. The results of a Floquet stability analysis for the onset of three‐dimensional flow are in partial agreement with experiment.

The Rayleigh–Taylor and Kelvin–Helmholtz stability of a viscous liquid–vapor interface with heat and mass transfer
View Description Hide DescriptionLinear stability analysis of a liquid–vapor interface under adverse gravitational field and velocity streaming is considered. The liquid is assumed viscous, incompressible, and motionless over a vapor layer with a uniform horizontal velocity. It is shown that while the coupled viscosity‐phase change mechanism of former studies adds considerably to the stability of the Rayleigh–Taylor problem, it has a deleterious effect on the Kelvin–Helmholtz mode of stability.

Instabilities of the Type I Long’s vortex at large flow force
View Description Hide DescriptionThe temporal inertial instabilities of the Type I Long’s vortex for large values of the flow force, M, are explored. It is found that growth rates and wave speeds obtained numerically for the exact vortex profiles agree very well with those obtained for large‐M profiles found by asymptotic means. Agreement is excellent, so the large‐M structure and inertial instability modes of the Type I Long’s vortex are now well established. It is determined that the modes computed here are ring modes. Both exact and asymptotic solutions display a spurious instability mode, which vanishes for increased resolution.

Decay of dipolar vortex structures in a stratified fluid
View Description Hide DescriptionIn this paper the viscous decay of dipolar vortex structures in a linearly stratified fluid is investigated experimentally, and a comparison of the experimental results with simple theoretical models is made. The dipoles are generated by a pulsed horizontal injection of fluid. In a related experimental study by Flór and van Heijst [J. Fluid Mech. 279, 101 (1994)], it was shown that, after the emergence of the pancake‐shaped vortex structure, the flow is quasi‐two‐dimensional and decays due to the vertical diffusion of vorticity and entrainment of ambient irrotational fluid. This results in an expansion of the vortex structure. Two decay models with the horizontal flow based on the viscously decaying Lamb–Chaplygin dipole, are presented. In a first model, the thickness and radius of the dipole are assumed constant, and in a second model also the increasing thickness of the vortex structure is taken into account. The models are compared with experimental data obtained from flow visualizations and from digital analysis of particle‐streak photographs. Although both models neglect entrainment and the decay is modeled by diffusion only, a reasonable agreement with the experiments is obtained.

Steadily translating vortices in a stratified fluid
View Description Hide DescriptionThe existence of steadily translating vortices in a semi‐infinite barotropic fluid stratified by a constant gravitational field is considered. Assuming that the flow field of the vortex is subsonic and contains finite total kinetic energy, it is found that steadily translating vortices do not exist in three dimensions, but do exist in two dimensions. An analogy between a subsonic, barotropic, stratified fluid, and a uniform fluid with a free‐slip planar boundary is exploited to show that the same result applies in a semi‐infinite uniform fluid.

Drag and lift forces on microscopic bubbles entrained by a vortex
View Description Hide DescriptionThe forces that act on bubbles as they are entrained by a vortex are measured using particle imagevelocimetry. Triple exposure images are used to measure the velocity and acceleration of the fluid and the bubbles simultaneously. Distinction between phases is achieved by using fluorescent particles as liquid flow tracers. The buoyancy, pressure, and inertia forces are computed from the data, while the drag and the lift forces are determined from a force balance on each bubble. It is found that in the present range of bubble diameters, 500 μm<d<800 μm, and Reynolds numbers, 20<Re<80, the drag on a bubble is similar to that on a solid body. Vorticity does not have a significant effect on the drag coefficient. The lift coefficients are significantly higher than currently available analytical and numerical estimates. The coefficients are independent of the Reynolds number and are proportional to the fourth root of the local vorticity. Estimates of the Bassett force show that it can be neglected in the present experiment. Computed bubble trajectories, based on the measuredlift and drag coefficients, compare well with experimental observations.

Dynamics of heavy particles in a Burgers vortex
View Description Hide DescriptionThis paper presents a linear stabilityanalysis as well as some numerical results for the motion of heavy particles in the flow field of a Burgers vortex, under the combined effects of particle inertia, Stokes drag, and gravity. By rendering the particle motion equations dimensionless, the particle Stokes number, a Froude number, and a vortexReynolds number are obtained as the governing three parameters. In the absence of gravity, the vortex center represents a stable equilibrium point for particles up to a critical value of the Stokes number, as the inward drag overcomes the destabilizing centrifugal force on the particle. Particles exceeding the critical Stokes number value asymptotically approach closed circular orbits. Under the influence of gravity, one or three equilibrium points appear away from the vortex center. Both their locations and their stability characteristics are derived analytically. These stability characteristics can furthermore be related to the nature of the critical points in a related directional force field. These findings are expected to be applicable to the coupling between the small‐scale turbulent flow structures and the motion of suspended particles.

Nonlocal nature of vortex stretching in an inviscid fluid
View Description Hide DescriptionThree‐dimensional Euler equations are studied numerically and analytically to characterize intense vortex stretching in an inviscid fluid. Emphasis is put on the nonlocal effects stemming from the pressure term. The purpose of this paper is twofold. One is to give numerically a detailed characterization of vortex structures on the basis of previously proposed two eigenvalue problems associated with vorticity. The other is to give some mathematical analyses which highlight the role of the pressure Hessian in vortex dynamics, especially in connection with a possible singularity. Also discussed are the differences in local and global (possible) blowups. The blowup problem is not directly discussed by the present numerics at moderate resolution.