Spectrum of density fluctuations in a particlefluid system—II. Polydisperse spheres

This paper presents an extension of the analysis shown in Part I to a polydisperse particle-fluid system. The density autocorrelation is shown to be a function of two quantities, a generalized Overlap function for which an analytical expression is derived, and the radial distribution function (RDF). In Fourier transform space, the density spectrum again appears to be a strong function of the mean particle size, and secondarily the mean particle separation distance. One unusual result is previously observed oscillations in the density spectrum of a monodisperse system of particles are severely dampened or even eliminated in the polydisperse case, depending on the width of the particle size distribution. Apparently contributions from different particle correlations interfere with each other, thereby reducing the coherent oscillations seen in the monodisperse particle-fluid system. Furthermore at large wavenumbers, the spectrum decays with a −2 power-law, independent of the shape of the particle size distribution. This behavior can be traced to the Overlap function which controls the behavior of the spectrum beyond the first peak. Remarkably the −2 power-law spectrum is determined by the shape of the particles (i.e. spheres) rather than their spatial distribution (RDF).

The effect of an asymptotically large pressure gradient on the correlation of several important higher-order moments is revisited for the polydisperse system. The relatively simple relationships developed for the monodisperse system are lost in the polydisperse case because particles of different sizes will be influenced differently by an applied pressure gradient. The result is moments that are of different order in velocity can no longer be related to each other (as they were in the monodisperse system), even in this idealized flow. A more comprehensive understanding of this phenomenon can only be achieved through direct numerical simulation or experiment.     

The effect of two-way coupling and inter-particle collisions on turbulence modulation in a vertical channel flow

Turbulence modulation due to its interaction with dispersed solid particles in a downward fully developed channel flow was studied. The Eulerian framework was used for the gas-phase, whereas the Lagrangian approach was used for the particle-phase. The steady-state equations of conservation of mass and momentum were used for the gas-phase, and the effect of turbulence on the flow-field was included via the standard k–ε model. The particle equation of motion included the drag, the Saffman lift and the gravity forces. Turbulence dispersion effect on the particles was simulated as a continuous Gaussian random field. The effects of particles on the flow were modeled by appropriate source terms in the momentum, k and ε equations. Particle–particle collisions and particle–wall collisions were accounted for in these simulations. Gas-phase velocities and turbulence kinetic energy in the presence of 2–100% mass loadings of two particle classes (50 μm glass and 70 μm copper) were evaluated, and the results were compared with the available experimental data and earlier numerical results. The simulation results showed that when the inter-particle collisions were important and was included in the computational model, the fluid turbulence was attenuated. The level of turbulence attenuation increased with particle mass loading, particle Stokes number, and the distance from the wall. When the inter-particle collisions were negligible and/or was neglected in the model, the fluid turbulence was augmented for the range of particle sizes considered.     

Permeability of periodic arrays of spheres

For periodic arrays of spheres the permeability is obtained numerically as a function of the dimensionless wave number kD in the flow direction, where D is the sphere diameter, k = 2π/λ is the wave number, and λ is the distance between the spheres in the flow direction. Our numerical results for the solids fraction of 0.45 show that for kD < 6.5 the permeability increases with increasing kD. But, it decreases for 6.5 < kD < 8.5 and reaches a local minimum at kD  8.5, and then increases again with increasing kD. Since the Fourier spectrum of the area fraction is zero for kD = 8.98, this result suggests that the area fraction plays an important role in determining the dependence of permeability on the distance between the spheres in the flow direction. For smaller solids fractions, the positions of the local maximum and minimum of permeability shift to slightly smaller kD’s.     

Überlegungen zur Scherviskosität von Suspensionen aufgrund einer Dimensionsanalyse

Dimensional analysis of the motion of solid particles suspended in a fluid phase shows that the macroscopic relative shear viscosity of suspensions generally depends not only on the volume concentration and particle shape but also on two Reynolds numbers and a dimensionless sedimentation number. These dimensionless numbers are formed using parameters characterizing the structure and motion of the suspension at the microscopic level. The analysis was based on the assumptions that the dispersed particles are rigid and sufficiently large that Brownian motion may be neglected, that the continuous fluid phase is Newtonian and that the interactions between particles and between particles and fluid phase are only hydrodynamic. The Reynolds numbers describe the influence of the inertial forces at the microscopic level, and the sedimentation number the influence of gravity. The dimensionless numbers can be neglected if their values are much smaller than one. For each of the dimensionless numbers both the shear rate and the particle size influence the shear viscosity. Thus sedimentation number is large for low shear rates, whereas the Reynolds numbers are large for high shear rates. The viscosity function for one suspension can be transformed into the viscosity function for another suspension with geometrically similar particles but of a different size. The scale-up rules are derived from the requirement that the relevant dimensionless numbers must be constant. The influence of non-hydrodynamic effects at the microscopic level on the shear viscosity can be detected by deviations from the derived scale-up rules.     

The instability of rhombic cell flows

Instability of two-dimensional periodic flows with rhombic cell structure represented by the stream function Ψ=cos kx+cosy is investigated. Stability characteristics are obtained for the Reynolds number R=1, 2, 3 and 4 and the ratio of the diagonals of the cell . Variation of the critical Reynolds number R_c with k is obtained, and the square cell flow (k=1) is found to be most stable (R_c=√2). It is found that R_c → 1 as k → 0, which leads to a finite gap between this limiting R_c and R_c=√2 for K=0 (Ψ=cos y).     

Particle collisions in a turbulent flow

An equation for the two-point probability density function of the two-particle the coordinate and velocity distribution is obtained. A closed system of equations for the first and second two-point moments of the velocity fluctuations of a pair of particles with allowance for the turbulent flow inhomogeneity is given. Boundary conditions for the equations of the particle concentration and the intensity of the relative random velocity during particle collision are obtained. A unified formula describing the interparticle collision process as a result of turbulent motion and the average relative particle velocity slip is obtained for the kernel of the coagulation equation. The effect of the average velocity slip of the particles and the carrier phase on the parameters of motion of the dispersed admixture and its coagulation is investigated on the basis of a two-point two-time velocity fluctuation autocorrelation function with two time and space scales representing the energy-bearing and small-scale motion of the fluid phase.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–116, March–April, 1996.     

Hydromagnetic flow of a dusty fluid over a stretching sheet

Analysis of hydromagnetic flow of a dusty fluid over a stretching sheet is carried out with a view to throw adequate light on the effects of fluid-particle interaction, particle loading, and suction on the flow characteristics. The equations of motion are reduced to coupled non-linear ordinary differential equations by similarity transformations. These coupled non-linear ordinary differential equations are solved numerically on an IBM 4381 with double precession, using a variable order, variable step-size finite-difference method. The numerical solutions are compared with their approximate solutions, obtained by a perturbation technique. For small values of β the exact (numerical) solution is in close agreement with that of the analytical (approximate) solution. It is observed that, even in the presence of a transverse magnetic field and suction, the transverse velocity of both the fluid and particle G phases decreases with an increase in the fluid-particle interaction parameter, β, or the particle-loading parameter, k. Moreover, the particle density is maximum at the surface of the stretching sheet, and the shearing stress increases with an increase in β or k.     

An unsymmetrical van der pol equation with stable subharmonics

Possible stable subharmonic solutions of the equation

ÿ − k(1 + 2cy − y²)ÿ + Y = bkμ cos μt, c > 0

, klarge, are discussed by the techniques used by J.E. Littlewood for van der Pol's equation in Acta Math. 97 (1957), that is the case of the above equation with c = 0 and

, k large. Their variation as c increases is also considered briefly.     

Experimental study of heat transfer characteristics due to confined impinging two-dimensional jets

Experimental results are presented for characteristics of impingement heat transfer caused by three slot jets. Experimental values were obtained for the dimensionless distance H = 0.5−3, dimensionless pitch P = 6−16, and Reynolds number Re = 500−8000. For laminar impinging flow, they were compared with numerical results. For turbulent impinging flow, two peaks of the local Nusselt number were obtained behind the second nozzle. The position of the second peak approached the nozzle as the space between nozzle and impinged surface decreased. The average Nusselt number between the central and second nozzles was determined from the ratio P/H and the Reynolds number based on the pitch of the nozzles.     

Stability of the plane Couette flow of a disperse medium with a finite volume fraction of the particles

The linear hydrodynamic stability of the plane Couette flow of a suspension with a finite volume fraction of the particles is considered. The two-phase medium flow is described within the framework of the model of mutually penetrating continua which allows for the finiteness of the volume occupied by the particles. In the main flow the phase velocities are the same, while gravity is not taken into account. The stability of disperse flows with both uniform and nonuniform particle distributions is studied. The linearized system of the equations of suspension motion with the no-slip boundary conditions imposed on solid walls is reduced to the eigenvalue problem for an ordinary differential fourth-order equation in the stream function. The eigenvalues are sought using the orthogonolization method. The parametric investigation of the stability characteristics of the disperse flow is performed. It is shown that in the case of the uniform spatial distribution of the particles in the main flow, the presence of an admixture in the flow leads to a slight variation in the wave decay rates, while the flow remains stable for any permissible combinations of the dimensionless governing parameters. In the case of nonuniform distribution of inclusions the flow loses stability already for low Reynolds numbers on a wide range of the dimensionless governing parameters.     

A local superposed-constituent volume-fraction mixture theory based on relative motion

This work presents a local formulation of kinematics and balance relations for a mixture composed of superposed-constituents as based on their motion relative to a model of the mixture as a moving region, with the motion of this region being determined by that of all the constituents. The relative motion of each constituent with respect to this moving mixture region can be interpreted as its diffusive motion in the mixture. A number of constituent kinematic properties in the mixture, in particular the constituent volume density (i.e. the infinitesimal volume-fraction) and constituent diffusion velocity, are determined by the constituent diffusive motion, and arise naturally in a formulation based upon it. In addition, this formulation yields in a natural fashion an evolution relation for the constituent volume density relative to the moving mixture region. The consequences of this kinematic model for constituent and mixture local balance relations, as well as the sum relations, are investigated in the last part of the work.     

Hydrodynamic interactions in quasi-two-dimensional colloidal suspensions

Dynamical correlations between colloidal particles, in a quasi-two-dimensional geometry, are measured by optical microscopy. The system consists of charged polystyrene spheres suspended in water and confined between two parallel plates. The long-range electrostatic interaction is screened and the interparticle direct interaction becomes an effective excluded volume interaction. Thus, the observed long-range correlations, between the motion of pairs of particles, are due to the solvent mediated hydrodynamic interactions. Such correlations are observed as a dependence in the interparticle distance of the diffusion coefficients for the normal modes, collective and relative, for which results are presented here for a range of particle concentrations.     

Measurement of inertial particle clustering and relative velocity statistics in isotropic turbulence using holographic imaging

We present the first measurements of relative velocity statistics of inertial particles in a homogeneous isotropic turbulent flow with three-dimensional holographic particle image velocimetry (holographic PIV). From the measurements we are able to obtain the radial relative velocity probability density function (PDF) conditioned on the interparticle separation distance, for distances on the order of the Kolmogorov length scale. Together with measurements of the three-dimensional radial distribution function (RDF) in our turbulence chamber, these statistics, in principle, can be used to determine interparticle collision rates via the formula derived by Sundaram and Collins (1997). In addition, we show temporal development of the RDF, which reveals the existence of an extended quasi-steady-state regime in our facility. Over this regime the measured two-particle statistics are compared to direct numerical simulations (DNS) with encouraging qualitative agreement. Statistics at the same Reynolds number but different Stokes numbers demonstrate the ability of the experiment to correctly capture the trends associated with particles of different inertia. Our results further indicate that even at moderate Stokes numbers turbulence may enhance collision rates significantly. Such experimental investigations may prove valuable in validating, guiding and refining numerical models of particle dynamics in turbulent flows.     

Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity

This paper concerns the Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid in their stream function formulations are used. The hydrodynamic drag force acting on each porous cylindrical particle in a cell and permeability of membrane built up by cylindrical particles with a porous shell are evaluated. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified. Variation of the drag coefficient and dimensionless hydrodynamic permeability with permeability parameter σ, particle volume fraction γ has been studied and some new results are reported. The flow patterns through the regions have been analyzed by stream lines. Effect of particle volume fraction γ and permeability parameter σ on flow pattern is also discussed. In our opinion, these results will have significant contributions in studying, Stokes flow through cylindrical swarms.     

Pipe flow of suspensions

This study shows that fully developed pipe flow of a particulate suspension is defined by four dimensionless parameters of particle-fluid interactions in addition to the Reynolds number. Effects accounted for include the Magnus effect due to fluid shear, electrostatic repulsion due to electric charges on the particles, and Brownian or turbulent diffusion. In the case of a laminar liquid-solid suspension, electrostatic effect is negligible but shear effect is prominent. Solution of the basic equations gives the density distribution of particles with a peak at the center (Einstein, Jeffery) or at other radii between the center and the pipe wall (Segré et al) depending on the magnitudes of the various flow parameters. In the case of a turbulent gas-solid suspension, the Magnus effect is significant only within the thickness of the laminar sublayer. However, charges induced on the particles by the impact of particles at the wall produce a higher density at the wall than at the center of the pipe. The velocity distribution of particles is characterized by a slip velocity at the wall and a lag in velocity in the core from the fluid phase. These results are verified by earlier measurements.     

Pipe flow of suspensions

This study shows that fully developed pipe flow of a particulate suspension is defined by four dimensionless parameters of particle-fluid interactions in addition to the Reynolds number. Effects accounted for include the Magnus effect due to fluid shear, electrostatic repulsion due to electric charges on the particles, and Brownian or turbulent diffusion. In the case of a laminar liquid-solid suspension, electrostatic effect is negligible but shear effect is prominent. Solution of the basic equations gives the density distribution of particles with a peak at the center (Einstein, Jeffery) or at other radii between the center and the pipe wall (Segré et al) depending on the magnitudes of the various flow parameters. In the case of a turbulent gas-solid suspension, the Magnus effect is significant only within the thickness of the laminar sublayer. However, charges induced on the particles by the impact of particles at the wall produce a higher density at the wall than at the center of the pipe. The velocity distribution of particles is characterized by a slip velocity at the wall and a lag in velocity in the core from the fluid phase. These results are verified by earlier measurements.     

Laminar free-convective heat transfer from a vertical isothermal plate to water at low temperatures with variable physical properties

All studies concerning laminar free convection along a vertical isothermal plate in water at low temperatures have been conducted assuming constant dynamic viscosity and thermal conductivity both taken at ambient or film temperature. In this study the problem has been treated taking into account the temperature dependence of all water physical properties. The results are obtained with the numerical solution of the boundary layer equations. The variation of μ and k with temperature has a small influence on wall heat transfer but a strong influence on wall shear stress. These quantities show a significant reduction at density extremum.     

Structural instability in the sedimentation of particulate suspensions through viscoelastic fluids

A theory is developed to describe a structural instability that has been observed during the sedimentation of particulate suspensions through viscoelastic fluids. The theory is based on the assumption that the influence of hydrodynamic interactions in viscoelastic fluids, which tend to cause particles to aggregate, is in competition with hydrodynamic dispersion, which acts to maintain a homogeneous microstructure. In keeping with the experimental observations, it predicts that the suspension structure will stratify into vertical columns when a dimensionless stability parameter exceeds a critical value. The column-to-column separation, measured in particle radii, is predicted to be proportional to the square root of the ratio of the dimensionless dispersion coefficient to the product of the particle volume fraction and the Deborah number. The time for the formation of the columns is predicted to scale with the inverse of the average volume fraction. These predictions are in agreement with experimental data reported in the literature.     

Modeling of dynamics, heat transfer, and combustion in two-phase turbulent flows: 1. Isothermal flows

This paper presents a review of authors' collective works in the field of two-phase flow modeling done in the past few decades. The paper is aimed at the construction of mathematical models for simulation of particle-laden turbulent flows. A kinetic equation was obtained for the probability density function (PDF) of the particle velocity distribution in turbulent flows. The proposed kinetic equation describes both the interaction of particles with turbulent eddies of the carrier phase and particle-particle collisions. This PDF equation is used for the derivation of different schemes describing turbulent momentum transfer in the dispersed particle phase. The turbulent characteristics of the gaseous phase are calculated on the basis of the k - turbulence model with a modulation effect of particles on the turbulence.

The constructed models have been applied to the calculation of various two-phase gas-particle turbulent flows in jets and channels as well as particle deposition in tubes and separators. For validating the theoretical and numerical results, a wide range of comparisons with experimental data from Russian and foreign sources has been done.