Volume 6, Issue 12, December 1994
Index of content:

Three dimensionalization of the stratified mixing layer
View Description Hide DescriptionA theoretical analysis of the stability of a stratified, two‐dimensional Kelvin–Helmholtz billow against three‐dimensional perturbations is presented. This predicts the three‐dimensional spectrum to be dominated by a shear aligned convective instability, which is localized in the region surrounding the billow core. The results of a direct numerical simulation of the evolution of the three‐dimensional stratified mixing layer fully verify the dominance of this convective mode in the mixing transition. The origin of the streamwise streaks of vorticity, which precede turbulent collapse in a stratified shear layer, is thereby explained.

On the decay of a turbulent vortex ring
View Description Hide DescriptionThe spatiotemporal evolution of a turbulentvortex ring with an initial Reynolds number of 7500 is experimentally investigated using the technique of digital particle imagevelocimetry. The flow is initially characterized by the laminar/turbulent transition via azimuthal bending instabilities. After transition, the shedding of vorticity from peripheral regions of the ring is found to be responsible for the formation of a wake region. This shedding process results in the staircase‐like decay of the circulation and propagation speed of the vortex ring.

A study of the sedimentation of noncolloidal bidisperse, concentrated suspensions by an acoustic technique
View Description Hide DescriptionThis paper uses an acoustic technique to determine the concentration profile developing during the sedimentation of noncolloidal bidisperse suspensions of glass beads in a Newtonian fluid. Various bead diameter ratios have been used and a wide range of relative concentrations is covered. From the shock front velocities and the concentrations in different zones, the sedimentation velocities of small and large particles in a homogeneous suspension of respective concentrations c _{ s0} and c _{ l0} have been determined. The semidilute regime (c _{0}=s _{ s0}+c _{ l0}<20%) has many similarities with the dilute regime, where large particles provide the dominant hydrodynamic hindrance to settling. In the concentrated regime (c _{0}≳35%), a mutual hindrance leads to a velocity reduction of large particles and to an enhancement of small ones, as compared to a monodisperse suspension. The data clearly demonstrate that size segregation in the concentrated regime disappears at a critical concentration, which for the size ratio 1.68 is equal to c _{0}=45%.

Isotropic Cartesian tensors of arbitrary even orders and velocity gradient correlation functions
View Description Hide DescriptionA recursive relation for the unit isotropic tensor of an arbitrary even order is derived from a rotational Brownian motion of the unit vector. An expression for the eighth‐order velocity gradient correlation function is derived from the eighth‐order isotropic tensors, after preserving the solenoidal property of the velocity field. This correlation function is compared with measurements in a turbulent wake.

Modulation effects along stability border in Taylor–Couette flow
View Description Hide DescriptionThe stability of time modulated Taylor–Couette flow with co‐ and counter‐rotating cylinders and modulated inner cylinder velocity is investigated. The mean velocities of the inner and outer cylinders are chosen to be in constant relative distance to the stability border of nonmodulated Taylor–Couette flow and the inner cylinder velocity is periodically modulated. The critical modulation amplitude, which leads to linear instability of the modulated laminar flow, is calculated with numerically integrated Floquet theory, as well as with perturbation theory. The critical modulation amplitude as a function of the outer cylinder Reynolds number has a rich structure for counter‐rotating cylinders, which is open to experimental test. The critical modulation amplitude decreases monotonically with increasing rotation frequency for corotating cylinders.

On the solution of Stokes’ equations between confocal ellipses
View Description Hide DescriptionThe analytical solution of Stokes’ equations between two concentric, confocal ellipses is derived here. This bounded flow, similar in certain respects to the journal bearing flow, was imagined in order to investigate two‐dimensional mixing and Lagrangianchaos in a bounded flow with two symmetry axis. The derived streamfunction is in the form of a Fourier cosine series and, when the eccentricity ratio of the inner ellipse is not very low, the solution converges very rapidly. When the ellipses turn in opposite directions, there are cases where two saddle points are connected by two different streamlines, a necessary and sufficient condition for structural instability according to Peixoto’s theorem. This flow geometry could be particularly effective for mixing of viscous fluids since the number of low period hyperbolic and elliptical points during time periodic boundary motion is greater than for the eccentric rotating cylinder system. The Poincaré sections obtained with a discontinuous velocity protocol suggest that the size of regions of poor mixing can be reduced by increasing the inner ellipse motion per period. For this geometry, the Poincaré sections indicate that counter‐rotation yields a more chaotic long term behavior than co‐rotation.

Reverse flow in channel‐effect of front and rear obstructions
View Description Hide DescriptionThe occurrence of reverse flow in a channel when a bluff body is kept at the entry is already known. In the earlier investigations, attention was focused on the generation of the reverse flow with bluff bodies, such as flat plate and other geometries, having the same width as the channel. The separation of the shear layers from the obstruction at the front end and the interaction of the shear layers at the rear end are mainly responsible for the reverse flow. To gain further insight into the phenomenon, the effects of the width of the obstruction at the front and that of placing another at the rear end in tandem with the front one are examined in this study. It is observed that the reverse flow occurs even when the width of the flat plate (b) is less than the channel width (w); the lower limit being b/w=0.6. At this b/w the reverse flow velocity is small, but it increases progressively with b/w until a maximum of about 30% of the forward velocity is attained for b/w≥2.0. However, reverse flow as high as 0.6 times the free‐stream velocity is obtained when another plate is kept close to the rear end in addition to the front plate. Further increase in the reverse flow to 0.83 times the free‐stream velocity has been achieved by replacing the flat plate model at the rear with a semicircular scoop.

Direct simulation Monte Carlo for thin‐film bearings
View Description Hide DescriptionThe direct simulation Monte Carlo (DSMC) scheme is used to study the gas flow under a read/write head positioned nanometers above a moving disk drive platter (the slider bearing problem). In most cases, impressive agreement is found between the particle‐based simulation and numerical solutions of the continuum hydrodynamic Reynolds equation which has been corrected for slip. However, at very high platter speeds the gas is far from equilibrium, and the load capacity for the slider bearing cannot be accurately computed from the hydrodynamicpressure.

Liquid film flow in a fracture between two porous blocks
View Description Hide DescriptionLiquid filmflow in a fracture between two porous bodies is mainly driven by pressure. The pressure drop across such a small distance could be significant. The flow of a liquid film is governed by the pressure drop across the fracture space, and therefore, understanding of liquid filmflow in a single liquid bridge along a solid wall between two porous bodies is needed. The shape of the liquid bridge between the porous blocks is an unknown. The first step is to determine the shape of the free surface. Due to the nature of the problem, a boundary integral technique is found to provide the solution to the whole problem. Solutions are reported for a range of capillary numbers observed in cracked porous media. Pressure drop is correlated using a dimensionless capillary number group. Using analogy from the Darcy flow in porous media, a correlation for the equivalent Darcy permeability is developed.

The internal structure of lateral intrusions in a continuously stratified heat/salt system
View Description Hide DescriptionLaboratory experiments have been performed for a double diffusive system in which opposing vertical gradients of temperature and salinity are heated from one side. Details of the internal structure of the intrusions that form along the heated endwall are discussed. Fingering motions and convective overturns are prominent characteristics of the internal intrusion structure, particularly when the rate of lateral heating is high relative to the strength of the ambient vertical density gradient. Analysis of the overturning scale indicates that the RMS size of the overturns is typically 10%–30% of the total layer thickness. Comparison of the flow inside the convective layers with that which develops in a long box heated and cooled at opposite sides based on the analysis of Jeevaraj and Imberger [J. Fluid Mech. 222, 565 (1991)] shows poor agreement when molecular values of the diffusivity and viscosity are used in the theory. However, moderate increases in the diffusivity values (to account for increased vertical mixing) give good agreement between the experiments and theory. Building on the results of Schladow et al. [J. Fluid Mech. 236, 127 (1992)], further evidence of the ability of the intrusions to continue propagating following removal of the endwall heating is presented. Modification of the flow field ahead of the intrusion fronts can create conditions for which the stability ratio, R _{ρ}, drops below a critical value, resulting in continued propagation of the intrusions.

Stability of the O(2)‐symmetric flow past a sphere in a pipe
View Description Hide DescriptionThe linear stability of the O(2)‐symmetric laminar flow past a centrally located sphere in a pipe for a range of sphere sizes is examined. For all values of the blockage ratio studied, the first instability observed as the flow rate increased, occurred at a steady symmetry‐breaking bifurcation point. A transformation technique was used to solve the generalized eigenvalue problem which arises when using a mixed finite‐element method to determine linear stability. The present results support the conclusion of Natarajan and Acrivos [J. Fluid Mech. 254, 323 (1993)] who have computed the stability of the flow past a sphere in an unbounded domain. The primary instability in three dimensions differs qualitatively from that of the analogous two‐dimensional flow past a centrally located cylinder in a channel, which Chen et al. (to appear in J. Fluid Mech.) have shown to occur at a Z _{2} symmetry breaking Hopf bifurcation point.

Linear instability of a two‐layer flow with differential particle loading
View Description Hide DescriptionThe linear instability of an unbounded, two‐layer particle‐laden flow subjected to infinitesimal perturbations was investigated. The two‐phase flow model employed in this study is dissipative because of a viscous mechanism through which momentum is transferred between the perturbed interstitial fluid and the suspended particles. The momentum interchange was found to depend critically on the ratio of the particle response time and the time scales of the perturbation. The dispersion equation for a flowsystem of large time scale ratio was solved analytically. Two unstable modes were found to coexist. The effects of three nondimensional parameters, a mean particle loading parameter, a differential loading parameter, and a relative offset between shear and the two layer interface, on two unstable modes were examined. In particular, if the velocity boundary layer is shifted toward the layer of low particle loading and the mean particle loading parameter increases, the most unstable mode ‘‘switches’’ from a long‐wave instability to a short‐wave instability.

Linear stability theory of two‐layer fluid flow in an inclined channel
View Description Hide DescriptionThe linear stability of the two‐layer flow of immiscible, incompressible fluids in an inclined channel is considered. In the long‐wave limit, mechanisms for linear instability, and the consequences of competition between mechanisms, are identified. For arbitrary wave numbers, air–water and olive oil–water systems are considered, in order to determine the influence of the channel thickness and the mean interfacial height on the stability of the flow. This paper characterizes those physical situations in which the primary instability is to long‐wave interfacial disturbances. The odd Orr–Sommerfeld shear mode within the water layer, which is necessarily stable in plane Poiseuille flow, is found to grow and even be the dominant mode of instability for the olive oil–water system. The consequences beyond linear stability are discussed.

Traveling wave instability in sustained double‐diffusive convection
View Description Hide DescriptionExperiments on buoyancy‐driven double‐diffusive convection sustained by imposed vertical concentration gradients (one stabilizing, the other destabilizing) have been conducted in a thin (Hele–Shaw) isothermal rectangular cell. Novel gel‐filled membranes were used to sustain the concentrations at the boundaries. When the destabilizing solute diffuses more rapidly than the stabilizing one, the primary instability leads to traveling waves with a high reflection coefficient at the ends of the cell. The measured critical Rayleigh numbers and frequencies are in reasonable accord with a stability analysis that includes corrections for the finite thickness of the cell and cross‐diffusion effects. The weakly nonlinear waves that appear at onset do not stabilize, even very close to the transition, but continue to evolve, eventually becoming a packet of large amplitude plumes. The packet travels back and forth along the cell in a nearly periodic manner. This behavior and the absence of measurable hysteresis are consistent with the present weakly nonlinear analysis which predicts tricritical scaling (∼ε^{1/4} rather than the usual ε^{1/2}) all along the instability boundary. However, the range of this scaling in ε was found to be less than 0.005, which is inaccessible in the present experiments.

Absolute and convective instabilities of fluidized beds
View Description Hide DescriptionThe stability of a fluidized bed is investigated with respect to spatially growing disturbances. A general linearized model is derived from the theories of Anderson and Jackson and of Batchelor. The absolute and convective nature of the instability is analyzed using the mathematical framework of the open flow linear stability theory. The results of the analysis provide the domains of absolute and convective instabilities.

Effect of pressure gradient on first mode of instability in compressible boundary layers
View Description Hide DescriptionThe effect of a pressure gradient on the first mode of instability of compressible subsonic and supersonic boundary layers is investigated using linear stability theory. A pressure gradient is studied that generates potential‐flow Mach number distributions at the edge of the boundary layer of the form M _{ e }=cx ^{ n }, where c is a constant and x is the dimensionless streamwise distance. Variations are calculated for the maximum growth rates of three‐dimensional first‐mode waves with different edge Mach numbers and different levels of both adverse and favorable pressure gradients. A favorable pressure gradient is shown to have a stabilizing effect on first‐mode waves. However, at high edge Mach numbers, a favorable pressure gradient becomes less effective in stabilizing first‐mode waves. The frequencies and streamwise and spanwise wave numbers that correspond to the maximum growth rates of first‐mode waves decrease as the pressure gradient becomes more favorable at all Mach numbers when the Reynolds numberR=1500 and at M _{ e }≥2 when R=600. Setting the Prandtl number to unity significantly increases the maximum growth rates of first‐ and second‐mode waves at high Mach numbers compared with setting it to the realistic value for air of 0.72. Predicted transition in flow over a flat plate using the N‐factor criterion is found to be due to first‐mode waves up to free‐stream Mach numbers of 6–6.5.

Vortex stripping and the erosion of coherent structures in two‐dimensional flows
View Description Hide DescriptionThis paper studies the erosion of a monotonically distributed vortex by the joint action of inviscid stripping, induced by an externally imposed adverse shear, and viscousdiffusion, either in the form of Newtonian viscosity or hyperviscosity. It is shown that vortexerosion is greatly amplified by the presence of diffusion; abrupt vortex breakup or gradual quasi‐equilibrium evolution depend crucially on the strain to peak vorticity ratio and on the Reynolds number. Peculiar, unexpected effects are observed when hyperviscosity is used in place of Newtonian viscosity.

Maximum entropy states for rotating vortex patches
View Description Hide DescriptionThe statistical equilibrium theory of coherent structures in two‐dimensional turbulence is used to study the coalescence and symmetrization of rotating vortex patches. In this theory, most probable states, which are local probability distributions on the fluctuating vorticity field, are characterized by maximizing entropy subject to constraints derived from the conserved quantities for the Euler equations. The merger of a symmetric pair of patches and the axisymmetrization of an elliptical patch are computed by solving the constrained maximum entropy problem. In this way, these generic phenomena, which are usually simulated by direct time integration, are shown to be equilibration processes for the statistical mechanics model. Moreover, macroscopic features of the filamentation generated in these processes are captured by the statistical equilibrium theory. Namely, the delicate balance between the coalescence into a vortex core and the migration of filamentary vorticity away from the core, which is dictated by the conservation of angular impulse, is exhibited by the equilibrium solutions.

Small‐scale properties of scalar and velocity differences in three‐dimensional turbulence
View Description Hide DescriptionTurbulent flows are known to concentrate strong vorticity in vortex tubes, giving rise to large velocity jumps across the tubes. When a passive scalar is advected by the flow, very steep scalar fronts separate well‐mixed regions, and result in large scalar differences. The properties of these large jumps are investigated by studying the probability distribution functions of velocity, scalar differences as a function of the separation between the points, of the Reynolds and of the Prandtl number. Over the range of parameters covered by the direct numerical simulations reported here (20≤R _{λ}≤90 and 1/32≤Pr≤1), it is found that the widths of the velocity (respectively, the scalar) jumps scale like the Kolmogorov length (respectively, like the Batchelor length). For both the scalar and the velocity, the large differences over small distance become rarer as the Reynolds number increases.

Inertial range statistics of Burgers turbulence
View Description Hide DescriptionVelocity structure function of Burgers turbulence in the inertial range is studied by mapping closure. It is shown that in the inertial range 〈‖u(x+l,t)−u(x,t)‖^{ q }〉= n(t)〈‖Δu(t)‖^{ q }〉‖l‖ and the spectrum E(k,t)=(1/2π)n(t)〈‖Δu(t)‖^{2}〉k ^{−2} for initially support broad spectrum, where n(t) is the average of the number density of shocks and 〈‖Δu(t)‖^{ q }〉 is qth moment of shock strength. Agreement with DNS is found to be good for initially Gaussian fields. Statistics at large time is also studied by the mapping closure using time dependent reference field. It is found that the length scale of the field and the decay law of the total kinetic energy are self‐similar in time and various quantities are given as functions of the exponent n of the initial energy spectrum E(k,0) ∝ k ^{ n } at low wave‐number range. Extension to more general initial conditions with non‐Gaussian statistics is also discussed.