Volume 15, Issue 11, November 2003
 LETTERS


Fractal clustering of inertial particles in random flows
View Description Hide DescriptionIt is shown that preferential concentrations of inertial (finitesize) particle suspensions in turbulent flows follow from the dissipative nature of their dynamics. In phase space, particle trajectories converge toward a dynamical fractalattractor. Below a critical Stokes number (nondimensional viscousfriction time), the projection on position space is a dynamical fractal cluster; above this number, particles are space filling. Numerical simulations and semiheuristic theory illustrating such effects are presented for a simple model of inertial particle dynamics.

A new approach to turbulent transport of a mean scalar
View Description Hide DescriptionWe develop a simple mean field approach to the transport of a passive scalar for which the fundamental equation is a second order differential equation in the transported quantity, not a first order equation. Triple correlations are included, as they must be for any realistic description of turbulence. No correlation time enters the theory, only an eddy turnover time. The approach is simpler than standard approaches which incorporate triple correlations, but more realistic than Gaussian or short correlation time closures which do not. A similar approach has proven useful in magnetohydrodynamics.

Chaotic advection in a braided pipe mixer
View Description Hide DescriptionChaotic advection is investigated in a “braided pipe mixer” (BPM). This static mixing device consists of an outer pipe containing several intertwining internal pipes, with fluid pumped down the gap. It has recently been proposed, by analogy with the twodimensional theory of topological chaos, that the BPM should be an effective mixing device, specifically that (i) good mixing might be achieved with only very thin internal pipes, and (ii) the quality of the mixing should be improved if the internal pipes form a mathematical braid. Our results suggest that neither (i) nor (ii) is the case for the BPM studied.
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 ARTICLES


Drop impact on a liquid–liquid interface
View Description Hide DescriptionThe effects of Reynolds number (Re) on the gravitydriven impact of a single drop onto a liquid–liquid interface were studied experimentally using the particle imagevelocimetry method. The liquid beneath the interface was identical to the drop liquid. Two liquids with different viscosities were used as the ambient above the interface resulting in viscosity ratios (drop to ambient) of 0.14 and 0.33. Index matching and a slight camera inclination were employed to eliminate optical distortion. Image planes were captured at a rate of 500 Hz, and velocity fields were determined from consecutive images. The flow Reynolds numbers based on drop impact velocity and ambient viscosity were 20 and 68 for the lower and higher viscosity ratios, respectively. During the approach toward the interface, the drop shape was more oblate for the higher Re case. At the same time, viscous stresses generated a vortex ring inside each drop and a wake behind it. Each wake contained a detached ring of similar sign to the ring inside the drop. The subsequent deformation of the drop and the interface due to impact was observed to be more radical in the higher Re case. The impingement and shearing of the trailing wake on the upper surface of each drop played a significant role in dissipating the vorticity inside both drops, and the vorticity dissipated faster for lower Re.

Predicting mixing microstructure in threedimensional chaotic systems
View Description Hide DescriptionThis paper explores the application of asymptotic directionality to threedimensional (3D) chaotic periodic flows by examining flow in a tank agitated by four impellers. Numerical simulations and experimental methods are employed to reveal the spatial structure of the evolving mixing patterns and its statistical properties. It is demonstrated that there exists an invariant field of orientations in the system that creates selfsimilar mixing structures. As a result, the frequency distributions of stretching can be collapsed onto an invariant curve by a simple homogeneous scaling. This statistical scaling behavior is a direct consequence of the asymptotic directionality property. It is also shown that striation thickness distributions (STDs) can be predicted directly from the stretching distributions in fully 3D chaotic systems, thus providing a method for calculation of STDs in complex flows.

Reconstruction subgrid models for nonpremixed combustion
View Description Hide DescriptionLargeeddy simulation of combustion problems involves highly nonlinear terms that, when filtered, result in a contribution from subgrid fluctuations of scalars, Z, to the dynamics of the filtered value. This subgrid contribution requires modeling. Reconstruction models try to recover as much information as possible from the resolved field based on a deconvolution procedure to obtain an intermediate field The approximate reconstruction using moments (ARM) method combines approximate reconstruction, a purely mathematical procedure, with additional physicsbased information required to match specific scalar moments, in the simplest case, the Reynoldsaveraged value of the subgrid variance. Here, results from the analysis of the ARM model in the case of a spatially evolving turbulent plane jet are presented. A priori and a posteriori evaluations using data from direct numerical simulation are carried out. The nonlinearities considered are representative of reacting flows: power functions, the dependence of the density on the mixture fraction (relevant for conserved scalar approaches) and the Arrhenius nonlinearity (very localized in Z space). Comparisons are made against the more popular beta probability density function (PDF) approach in the a priorianalysis, trying to define ranges of validity for each approach. The results show that the ARM model is able to capture the subgrid part of the variance accurately over a wide range of filter sizes and performs well for the different nonlinearities, giving uniformly better predictions than the beta PDF for the polynomial case. In the case of the density and Arrhenius nonlinearities, the relative performance of the ARM and traditional PDF approaches depends on the size of the subgrid variance with respect to a characteristic scale of each function. Furthermore, the sources of error associated with the ARM method are considered and analytical bounds on that error are obtained.

Influence of fluid thermal sensitivity on the thermomechanical stability of the Taylor–Couette flow
View Description Hide DescriptionRecent theoretical [AlMubaiyedh et al., Phys. Fluids 11, 3217 (1999); J. Fluid Mech. 462, 111 (2002)] and experimental [White and Muller, Phys. Rev. Lett. 84, 5130 (2000); J. Fluid Mech. 462, 133 (2002)] studies have revealed that viscous heating causes significant destabilization of the Taylor–Couette flow of highly viscous and thermally sensitive fluids. In this work, the roles of thermal sensitivity of fluid properties and corotation on the thermomechanical stability of Taylor–Couette flow are investigated theoretically. In turn, our theoretical findings are compared with the recent experimental ones by White and Muller [Phys. Fluids 14, 3880 (2002)]. It is shown that a finite gap temperature is necessary to predict the timedependent transitions observed in the experiments. A universal scaling between the critical Reynolds number and the Nahme number is obtained for intermediate values of Nahme number ranging from 0.01 to 1.0. Studies are also performed to determine the influence of corotation of the outer cylinder relative to the inner one on the thermomechanical stability. Overall, a very favorable comparison between theoretical and experimental results is obtained.

Dilute granular flow around an immersed cylinder
View Description Hide DescriptionIn this paper we investigate a twodimensional dilute granular flow around an immersed cylinder using discrete element computer simulations. Simulation measurements of the drag force acting on the cylinder, are expressed in terms of a dimensionless drag coefficient, where ρ is the upstream particle mass density, is the upstream solid fraction, is the upstream velocity, and is the sum of the cylinder diameter, and surrounding particle diameter, The drag coefficient increases rapidly with decreasing Mach number for subsonic Mach numbers, but remains insensitive to Mach number for supersonic values. The drag coefficient is also a strong function of the flow Knudsen number, with the drag coefficient increasing with increasing Knudsen number and approaching an asymptotic value for very large Knudsen numbers. The drag coefficient decreases with decreasing normal coefficient of restitution and is relatively insensitive to the friction coefficient. Bow shock structures and expansion fans are also observed in the simulations and are compared to similar structures observed in compressible gas flows.

Immiscible front evolution in randomly heterogeneous porous media
View Description Hide DescriptionThe evolution of a sharp interface between two immiscible fluids in a randomly heterogeneous porous medium is investigated analytically using a stochastic moment approach. The displacing fluid is taken to be at constant saturation and to have a much larger viscosity than does the displaced fluid, which is therefore effectively static. Capillary pressure at the interface is related to porosity and permeability via the Leverett Jfunction. Whereas porosity is spatially uniform, permeability forms a spatially correlated random field. Displacement is governed by stochastic integrodifferential equations defined over a threedimensional domain bounded by a random interface. The equations are expanded and averaged in probability space to yield leading order recursive equations governing the ensemble mean and variance of interface position, rate of propagation and pressure gradient within the displacing fluid. Solutions are obtained for onedimensional head and fluxdriven displacements in statistically homogeneous media and found to compare well with numerical Monte Carlo simulations. The manner in which medium heterogeneity affects the mean pressure gradient is indicative of how it impacts the stability of the mean interface. Capillary pressure at the interface is found to have a potentially important effect on its mean dynamics and stability.

Diagnosing transport and mixing using a tracerbased coordinate system
View Description Hide DescriptionThe advectiondiffusion equation for the concentration of a tracer may be transformed into a pure diffusion equation by using the area inside concentration contours as a coordinate. The corresponding effective diffusivity depends on the geometry of the tracer field, which is determined by the underlying flow. Recent studies have used effective diffusivity, calculated from a suitable tracer, as a qualitative indicator of the transport and mixing properties of a given flow. Here we show that the effective diffusivity may further be used as a quantitative diagnostic of transport and mixing. We use a family of incompressible twodimensional timeperiodic flows as a testbench and compare the calculated effective diffusivity with other diagnostics. The results demonstrate how the effective diffusivity accurately captures the location and character of barrier and mixing regions. We also show that the effective diffusivity parameterizes the transport of particles relative to the tracerbased coordinates. These results support the use of effective diffusivity as a quantifier of transport and mixing, and thereby strengthen the conclusions of previous qualitative studies.

Intermittent dry granular flow in a vertical pipe
View Description Hide DescriptionThe intermittent compact flow of glass beads in a vertical glass pipe of small diameter is studied experimentally by combining particle fraction, pressure, and air, and grain flow rates measurements with a spatiotemporal analysis of the flow. At the onset of the flow, a decompaction front is observed to propagate from the bottom to the top of the tube at a velocity much larger than that of the grains. The blockage front also propagates upward and at a still higher velocity. The decompaction induces a decreasing pressure wave strongly amplified as it propagates upward toward the top of the tube. Pressure variations of 3000 Pa or more are detected in this region while particle fraction variations are of the order of 0.02. Grain velocities during the flow period also increase strongly at the top of the tube while the corresponding fraction of total time decreases. A onedimensional numerical model based on a simple relation between the effective acceleration of the grains and the particle fraction variations reproduces the amplification effect and provides predictions for its dependence on the permeability of the packing.

Interfacial instability in nonNewtonian fluid layers
View Description Hide DescriptionSuperposed layers of fluid flowing down an inclined plane are prone to interfacialinstability even in the limit of zero Reynolds number. This situation can be explored by making use of a lubricationstyle approximation of the governing fluid equations. Two versions of the lubrication theory are presented for superposed layers of nonNewtonian fluid with powerlaw rheology. First, the fluids are assumed to have comparable effective viscosities. The approximation then furnishes a simplified model for which the linear stability problem can be solved analytically and concisely. Weakly nonlinear analysis and numerical computations indicate that instabilities saturate at low amplitude beyond onset and form steady wavetrains. Further from onset, secondary instabilities arise that destroy trains of widely spaced wave trains. Patterns of closely spaced waves, on the other hand, coarsen due to wave merger events. The two mechanisms select steady wavetrains with a characteristic spatial scale. The second lubrication theory assumes that the upper layer is far more viscous than the lower layer. As a result, the upper fluid flows almost rigidly, and extensional stresses can become promoted into the leadingorder balance of forces. Interfacialinstability still arises in Newtonian fluid layers, and the nonlinear dynamics is qualitatively unchanged. Significant complications arise when the upper fluid is nonNewtonian due to the behavior of the viscosity at zero strain rate.

A class of steady solutions to twodimensional free convection
View Description Hide DescriptionWe obtained steady solutions to the twodimensional Boussinesq approximation equations without a mean temperature gradient. This system is referred to as free convection in this paper. Under an external flow described by the stream function steady solutions are found. They are kept steady by the balance between the strain of Ψ and the diffusion. In this sense, they are similar to the Burgers vortex layer solution. Two examples other than are shown to have steady solutions. We discuss the relation between these solutions and longlived fine scale coherent structures observed in direct numerical simulations of twodimensional free convectionturbulence.

Resonance gas oscillations in closed tubes: Numerical study and experiments
View Description Hide DescriptionPeriodic gas oscillations in closed tubes are investigated experimentally and numerically. At resonance, these oscillations are accompanied by shock waves traveling back and forth along the tube. Results of gas temperature and pressure measurements are reported. It is found that the gas temperature changes substantially along the tube. A twodimensional numerical model of turbulent gas oscillations is developed and verified by a comparison with experiments. It is found that the experimental data of temperature and pressure inside the resonance tube are well correlated by the numerical model. Using the numerical model,turbulence and acoustic streaming at resonance are investigated. It is shown that the direction of gas streaming at resonance is opposite to that in nonresonant oscillations. A parametric investigation of resonant flow is performed in terms of the tube parameter ε, acoustic Reynolds number and dimensionless tube length Λ. In particular, it is found that the normalized pressure amplitude, as well as other flow characteristics, are functions of a single parameter Useful correlations for flow characteristics are proposed in terms of tube dimensionless parameters.

Pinchoff and satellite formation in compound viscous threads
View Description Hide DescriptionThe breakup of viscous compound threads in the presence of insoluble surfactant at both interfaces is investigated. We use asymptotic methods in the limit of long axisymmetric waves to derive a coupled system of five onedimensional (1D) partial differential equations governing the evolution of the outer and inner interfaces, the surfactant concentrations there, and the leading order axial velocity component in the jet. The linear, and nonlinear, stability of these equations is then investigated for a wide range of outer to inner viscosity ratio, m, outer to inner surface tension ratio, γ, the ratio of initial outer to inner radii, α, initial surfactant concentrations at the outer and inner interfaces, and surfactant activities, and and the Schmidt numbers, and defined as the ratio of the kinematicviscosity to the surfactantsurfacediffusion coefficient. We also show that if these results are recovered via solution of 1D evolution equations governing the dynamics of an effective singlesurfactant covered thread, which are obtained through appropriate rescalings; these rescalings are detailed herein.

On the frontal collision of two round jets in water
View Description Hide DescriptionIn our experiment two laminar round jets collide in water forming a zeromomentum toroidal vortex and this flow is modeled theoretically. First, the linearized time dependent basic solution for the starting round jet is derived in a straightforward manner. Then a superposition of these solutions is used to model the frontal collision of two round jets. The resulting flow patterns are calculated and compared with the experiments. The comparison shows good qualitative agreement.

Fourthorder statistical moments of the velocity gradient tensor in homogeneous, isotropic turbulence
View Description Hide DescriptionA compact expression of fourthorder statistical moments of the velocity gradient tensor in homogeneous, isotropic, incompressible turbulence is obtained as a function of its invariants and of generic components of the velocity gradient. This single, compact expression is in full agreement with the four different expressions previously obtained by Siggia as functions of the same invariants and of generic components of the vorticity vector and the strain tensor; however, some discrepancies arise with respect to a similar, single expression obtained by PhanThien and Antonia. The used algorithm may be easily extended to handle higher order statistical moments of the velocity gradient.

Formation and separation of merged liquid sheets developed from the mixing of coaxial swirling liquid sheets
View Description Hide DescriptionLiquid–liquid coaxial swirl atomizers are used in liquid rocket engines to achieve an efficient mixing between the fuel and oxidizer sprays. The characteristics of the mixed spray are mainly controlled by the flow behavior of merged liquid sheet originating at the contact point of inner and outer swirling liquid sheets. With an intention of identifying various flow regimes of merged liquid sheet at different conditions of inner and outer liquid sheets, we report here a fundamental experimental investigation on the characteristics of merged liquid sheets using water as the experimental liquid. The physical processes involved in the formation and separation of a merged liquid sheet are described from the experimental measurements. For a given outer liquid sheet condition, the merged liquid sheet forms and separates at specific inner liquid sheet flow conditions. At low outer liquid sheet flow conditions with Weber number less than 50, the merged liquid sheet exhibits a selfsustaining periodic separation process, whose frequency increases with increasing inner liquid sheet Weber number for a given outer liquid sheet Weber number. Experimental measurements are presented to show that the dynamics of the contact point plays a major role in governing the characteristics of merged liquid sheets.

Viscosity of argon at temperatures >2000 K from measured shock thickness
View Description Hide DescriptionMottSmith’s kinetic theory of shock structure [Phys. Rev. 82, 885 (1951); C. Muckenfuss, Phys. Fluids 5, 1325 (1962)] suggests that, for any intermolecular potential, the average number of collisions undergone by a molecule as it crosses the shock approaches a limit as the Mach number increases. We check this with direct simulation Monte Carlo calculations and use it to estimate the gas viscosity at high temperatures from measurements of shock thickness. We consider a monatomic gas for five different collision models and hence five different viscosity laws The collision models are: the variable hard sphere, with three values of the generalized hard sphere; and the Maitland–Smith potential. For shock Mach numbers all these collision models predict a shock thickness where is a suitably defined “shock length scale” which depends on a collision cross section derived from the viscosity of the gas at a temperature characteristic of the collisions between upstream and downstream molecules. Using and the experimental measurements of shock thickness in argon given by Alsmeyer [J. Fluid Mech. 74, 498 (1976)], we estimate the viscosity of argon at high values of These estimates agree with the values recommended by the CRCHandbook of Chemistry and Physics, 82nd ed. (CRC Press, Boca Raton, FL, 2001) at For for which there appear to be no reliable direct measurements of viscosity, our estimated values lie between those recommended by the CRCHandbook and those predicted by the simple power law with and Taking the error in the experimental measurements of Δ as the scatter in the results of Alsmeyer (±2%), we estimate the uncertainty in the viscosity predictions as less than ±5%. To this accuracy, our results agree with the power law predictions and disagree with the CRCHandbook values, for

Effects of vibration method and wall boundaries on size segregation in granular beds
View Description Hide DescriptionThis paper investigates the rise rate of a large impurity in an otherwise homogeneous granular bed subject to discrete and continuous vertical oscillations. Experiments and twodimensional discrete element computer simulations reveal that the impurity rise rate depends upon the oscillation method when sidewall convection effects are negligible. For discrete oscillations the rise rate increases monotonically with the oscillation velocity amplitude while for continuous oscillations the rise rate reaches a maximum value at a critical velocity amplitude. The rise rate trends correlate with changes in the bed void fraction. When sidewall convection effects are present, the rise rate of the impurity increases linearly with the oscillation velocity amplitude regardless of the vibration method. The results from the studies presented here reconcile the seemingly inconsistent results reported previously in the literature.
