Particle dynamics in a turbulent particle–gas suspension at high Stokes number. Part 2. The fluctuating-force model

2010 ◽  
Vol 646 ◽  
pp. 91-125 ◽  
Author(s):  
PARTHA S. GOSWAMI ◽  
V. KUMARAN
Keyword(s):  
Relaxation Time ◽  
Boltzmann Equation ◽  
Velocity Distribution ◽  
Particle Velocity ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Force Distribution ◽  
Force Model ◽  
Gas Suspension ◽  
Velocity Fluctuations

A fluctuating-force model is developed for representing the effect of the turbulent fluid velocity fluctuations on the particle phase in a turbulent gas–solid suspension in the limit of high Stokes number, where the particle relaxation time is large compared with the correlation time for the fluid velocity fluctuations. In the model, a fluctuating force is incorporated in the equation of motion for the particles, and the force distribution is assumed to be an anisotropic Gaussian white noise. It is shown that this is equivalent to incorporating a diffusion term in the Boltzmann equation for the particle velocity distribution functions. The variance of the force distribution, or equivalently the diffusion coefficient in the Boltzmann equation, is related to the time correlation functions for the fluid velocity fluctuations. The fluctuating-force model is applied to the specific case of a Couette flow of a turbulent particle–gas suspension, for which both the fluid and particle velocity distributions were evaluated using direct numerical simulations by Goswami & Kumaran (2010). It is found that the fluctuating-force simulation is able to quantitatively predict the concentration, mean velocity profiles and the mean square velocities, both at relatively low volume fractions, where the viscous relaxation time is small compared with the time between collisions, and at higher volume fractions, where the time between collisions is small compared with the viscous relaxation time. The simulations are also able to predict the velocity distributions in the centre of the Couette, even in cases in which the velocity distribution is very different from a Gaussian distribution.

Particle dynamics in the channel flow of a turbulent particle–gas suspension at high Stokes number. Part 1. DNS and fluctuating force model

2011 ◽  
Vol 687 ◽  
pp. 1-40 ◽  
Author(s):  
Partha S. Goswami ◽  
V. Kumaran
Keyword(s):  
Particle Acceleration ◽  
Relaxation Time ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Time Correlation ◽  
Quantitative Agreement ◽  
Force Model ◽  
Velocity Fluctuations ◽  
Time Correlations ◽  
Non Gaussian

AbstractThe fluctuating force model is developed and applied to the turbulent flow of a gas–particle suspension in a channel in the limit of high Stokes number, where the particle relaxation time is large compared to the fluid correlation time, and low particle Reynolds number where the Stokes drag law can be used to describe the interaction between the particles and fluid. In contrast to the Couette flow, the fluid velocity variances in the different directions in the channel are highly non-homogeneous, and they exhibit significant variation across the channel. First, we analyse the fluctuating particle velocity and acceleration distributions at different locations across the channel. The distributions are found to be non-Gaussian near the centre of the channel, and they exhibit significant skewness and flatness. However, acceleration distributions are closer to Gaussian at locations away from the channel centre, especially in regions where the variances of the fluid velocity fluctuations are at a maximum. The time correlations for the fluid velocity fluctuations and particle acceleration fluctuations are evaluated, and it is found that the time correlation of the particle acceleration fluctuations is close to the time correlations of the fluid velocity in a ‘moving Eulerian’ reference, moving with the mean fluid velocity. The variances of the fluctuating force distributions in the Langevin simulations are determined from the time correlations of the fluid velocity fluctuations and the results are compared with direct numerical simulations. Quantitative agreement between the two simulations are obtained provided the particle viscous relaxation time is at least five times larger than the fluid integral time.

Particle dynamics in a turbulent particle–gas suspension at high Stokes number. Part 1. Velocity and acceleration distributions

2010 ◽  
Vol 646 ◽  
pp. 59-90 ◽  
Author(s):  
PARTHA S. GOSWAMI ◽  
V. KUMARAN
Keyword(s):  
Particle Velocity ◽  
Stokes Equations ◽  
Hard Spheres ◽  
Flow Direction ◽  
Fluid Velocity ◽  
Gaussian White Noise ◽  
Stokes Number ◽  
Navier Stokes ◽  
Velocity Fluctuations ◽  
Non Gaussian

The effect of fluid velocity fluctuations on the dynamics of the particles in a turbulent gas–solid suspension is analysed in the low-Reynolds-number and high Stokes number limits, where the particle relaxation time is long compared with the correlation time for the fluid velocity fluctuations, and the drag force on the particles due to the fluid can be expressed by the modified Stokes law. The direct numerical simulation procedure is used for solving the Navier–Stokes equations for the fluid, the particles are modelled as hard spheres which undergo elastic collisions and a one-way coupling algorithm is used where the force exerted by the fluid on the particles is incorporated, but not the reverse force exerted by the particles on the fluid. The particle mean and root-mean-square (RMS) fluctuating velocities, as well as the probability distribution function for the particle velocity fluctuations and the distribution of acceleration of the particles in the central region of the Couette (where the velocity profile is linear and the RMS velocities are nearly constant), are examined. It is found that the distribution of particle velocities is very different from a Gaussian, especially in the spanwise and wall-normal directions. However, the distribution of the acceleration fluctuation on the particles is found to be close to a Gaussian, though the distribution is highly anisotropic and there is a correlation between the fluctuations in the flow and gradient directions. The non-Gaussian nature of the particle velocity fluctuations is found to be due to inter-particle collisions induced by the large particle velocity fluctuations in the flow direction. It is also found that the acceleration distribution on the particles is in very good agreement with the distribution that is calculated from the velocity fluctuations in the fluid, using the Stokes drag law, indicating that there is very little correlation between the fluid velocity fluctuations and the particle velocity fluctuations in the presence of one-way coupling. All of these results indicate that the effect of the turbulent fluid velocity fluctuations can be accurately represented by an anisotropic Gaussian white noise.

Properties of a bidisperse particle–gas suspension Part 1. Collision time small compared with viscous relaxation time

1993 ◽  
Vol 247 ◽  
pp. 623-641 ◽  
Author(s):  
V. Kumaran ◽  
Donald L. Koch
Keyword(s):  
Relaxation Time ◽  
Stokes Number ◽  
Distribution Functions ◽  
Gas Suspension ◽  
Balance Equations ◽  
Mean Velocity ◽  
The Mean ◽  
Viscous Relaxation ◽  
The Difference ◽  
Velocity Variances

The properties of a dilute bidisperse particle–gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit τcτv using a perturbation analysis in the small parameter v, which is proportional to the ratio of timescales τc/τv. Here, τc is the time between successive collisions of a particle, and tv is the viscous relaxation time. The leading-order distribution functions for the two species are isotropic Gaussian distributions, and are identical to the molecular velocity distributions in a two-component gas at equilibrium. Balance equations are written for the mean and mean-square velocities, using a distribution function that is a small perturbation from the isotropic Gaussian. The collisional terms are calculated by performing an ensemble average over the relative configurations of the colliding particles, and the mean velocity and velocity variances are calculated correct to O(v2) by solving the balance equations. The difference in the mean velocities of the two species is O(v) smaller than the mean velocity of the suspension, and the fluctuating velocity is O(v½) smaller than the mean velocity.

scholarly journals Particle Velocity Fluctuations in Viscous Gas with Random Velocity as the Sum of Two Correlated Color Noises

2020 ◽  
pp. 33-49
Author(s):  
I. V. Derevich ◽  
A. K. Klochkov
Keyword(s):  
Relaxation Time ◽  
Turbulent Flows ◽  
Particle Velocity ◽  
Random Processes ◽  
Solid Particles ◽  
Gas Velocity ◽  
Dynamic Relaxation ◽  
Two Phase ◽  
Velocity Fluctuations ◽  
Stochastic Ordinary Differential Equations

The article focuses on methods for studying the phenomenon of two-phase turbulent flows. The turbulence effect on the movement of solid particles in a viscous gas is under study. Dynamics of particles movement in a gas is written in the Stokes approximation, which allows us to suppose the dynamic relaxation time to be a constant value.The random gas velocity is modeled by the sum of two correlated random noises. It is shown that this approach makes it possible to model noise of any structural complexity. The paper describes two research methods based on fundamentally different Euler and Lagrange approaches to the description of a continuous medium. The first approach uses a well-known generalization of the spectral analysis technique for random processes, a popular method for studying turbulence. The second approach implementation is based on the modern generalizations of the theory of numerical algorithms for solving stochastic ordinary differential equations. The spectral method is used to obtain analytical expressions of correlation functions and variance of random processes describing the velocity of gas and solid particles. The qualitative difference between the correlation of fluctuations of modulated random velocities and the behavior of correlations in the case of a single-component gas velocity composition is analyzed. A method of direct numerical simulation for studied processes based on the numerical solution of a stochastic ordinary differential equations system is proposed and analyzed in detail. An array of statistical data obtained as a result of direct numerical modeling is collected and processed. Analytical results are compared qualitatively with numerical results. The influence of input parameters on the character of turbulent flow is studied. The dynamic relaxation time has a significant effect on the complexity of the autocorrelation function of the particle velocity and the response function of particles to gas velocity fluctuations. It is shown that the obtained functions tend to the known results of the standard theory. The considered methods for describing two-phase turbulent flows hold promise for further research.

scholarly journals The effect of Stokes number on particle velocity and concentration distributions in a well-characterised, turbulent, co-flowing two-phase jet

2016 ◽  
Vol 809 ◽  
pp. 72-110 ◽  
Author(s):  
Timothy C. W. Lau ◽  
Graham J. Nathan
Keyword(s):  
Particle Velocity ◽  
Turbulent Jets ◽  
Axial Direction ◽  
Stokes Number ◽  
Particle Migration ◽  
Turbulent Regime ◽  
Exit Plane ◽  
Low Velocity ◽  
Two Phase ◽  
Velocity Fluctuations

Simultaneous measurements of particle velocity and concentration (number density) in a series of mono-disperse, two-phase turbulent jets issuing from a long, round pipe into a low velocity co-flow were performed using planar nephelometry and digital particle image velocimetry. The exit Stokes number,$Sk_{D}$, was systematically varied over two orders of magnitude between 0.3 and 22.4, while the Reynolds number was maintained in the turbulent regime ($10\,000\leqslant Re_{D}\leqslant 40\,000$). The mass loading was fixed at$\unicode[STIX]{x1D719}=0.4$, resulting in a flow that is in the two-way coupling regime. The results show that, in contrast to all previous work where a single Stokes number has been used to characterise fluid–particle interactions, the characteristic Stokes number in the axial direction is lower than that for the radial direction. This is attributed to the significantly greater length scales in the axial motions than in the radial ones. It further leads to a preferential response of particles to gas-phase axial velocity fluctuations,$u_{p}^{\prime }$, over radial velocity fluctuations,$v_{p}^{\prime }$. This, in turn, leads to high levels of anisotropy in the particle-phase velocity fluctuations,$u_{p}^{\prime }/v_{p}^{\prime }>1$, throughout the jet, with$u_{p}^{\prime }/v_{p}^{\prime }$increasing as$Sk_{D}$is increased. The results also show that the region within the first few diameters of the exit plane is characterised by a process of particle reorganisation, resulting in significant particle migration to the jet axis for$Sk_{D}\leqslant 2.8$and away from the axis for$Sk_{D}\geqslant 5.6$. This migration, together with particle deceleration along the axis, causes local humps in the centreline concentration whose value can even exceed those at the exit plane.

Particle dynamics in the channel flow of a turbulent particle–gas suspension at high Stokes number. Part 2. Comparison of fluctuating force simulations and experiments

2011 ◽  
Vol 687 ◽  
pp. 41-71 ◽  
Author(s):  
Partha S. Goswami ◽  
V. Kumaran
Keyword(s):  
Relaxation Time ◽  
Particle Velocity ◽  
Volume Fraction ◽  
Quantitative Agreement ◽  
Mean Square ◽  
Mass Loading ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
The Mean ◽  
Viscous Relaxation

AbstractThe particle and fluid velocity fluctuations in a turbulent gas–particle suspension are studied experimentally using two-dimensional particle image velocimetry with the objective of comparing the experiments with the predictions of fluctuating force simulations. Since the fluctuating force simulations employ force distributions which do not incorporate the modification of fluid turbulence due to the particles, it is of importance to quantify the turbulence modification in the experiments. For experiments carried out at a low volume fraction of $9. 15\ensuremath{\times} 1{0}^{\ensuremath{-} 5} $ (mass loading is 0.19), where the viscous relaxation time is small compared with the time between collisions, it is found that the gas-phase turbulence is not significantly modified by the presence of particles. Owing to this, quantitative agreement is obtained between the results of experiments and fluctuating force simulations for the mean velocity and the root mean square of the fluctuating velocity, provided that the polydispersity in the particle size is incorporated in the simulations. This is because the polydispersity results in a variation in the terminal velocity of the particles which could induce collisions and generate fluctuations; this mechanism is absent if all of the particles are of equal size. It is found that there is some variation in the particle mean velocity very close to the wall depending on the wall-collision model used in the simulations, and agreement with experiments is obtained only when the tangential wall–particle coefficient of restitution is 0.7. The mean particle velocity is in quantitative agreement for locations more than 10 wall units from the wall of the channel. However, there are systematic differences between the simulations and theory for the particle concentrations, possibly due to inadequate control over the particle feeding at the entrance. The particle velocity distributions are compared both at the centre of the channel and near the wall, and the shape of the distribution function near the wall obtained in experiments is accurately predicted by the simulations. At the centre, there is some discrepancy between simulations and experiment for the distribution of the fluctuating velocity in the flow direction, where the simulations predict a bi-modal distribution whereas only a single maximum is observed in the experiments, although both distributions are skewed towards negative fluctuating velocities. At a much higher particle mass loading of 1.7, where the time between collisions is smaller than the viscous relaxation time, there is a significant increase in the turbulent velocity fluctuations by ${\ensuremath{\sim} }1$–2 orders of magnitude. Therefore, it becomes necessary to incorporate the modified fluid-phase intensity in the fluctuating force simulation; with this modification, the mean and mean-square fluctuating velocities are within 20–30 % of the experimental values.

Collisional sheet flows of sediment driven by a turbulent fluid

1998 ◽  
Vol 370 ◽  
pp. 29-52 ◽  
Author(s):  
JAMES T. JENKINS ◽  
DANIEL M. HANES
Keyword(s):  
Particle Velocity ◽  
Particle Concentration ◽  
Fluid Velocity ◽  
Packed Bed ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Volume Flux ◽  
Velocity Fluctuations ◽  
Turbulent Fluid ◽  
The Mean

We consider a sheet flow in which heavy grains near a packed bed interact with a unidirectional turbulent shear flow of a fluid. We focus on sheet flows in which the particles are supported by their collisional interactions rather than by the velocity fluctuations of the turbulent fluid and introduce what we believe to be the simplest theory for the collisional regime that captures its essential features.We employ a relatively simple model of the turbulent shearing of the fluid and use kinetic theory for the collisional grain flow to predict profiles of the mean fluid velocity, the mean particle velocity, the particle concentration, and the strength of the particle velocity fluctuations within the sheet. These profiles are obtained as solutions to the equations of balance of fluid and particle momentum and particle fluctuation energy over a range of Shields parameters between 0.5 and 2.5. We compare the predicted thickness of the concentrated region and the predicted features of the profile of the mean fluid velocity with those measured by Sumer et al. (1996). In addition, we calculate the volume flux of particles in the sheet as a function of Shields parameter.Finally, we apply the theory to sand grains in air for the conditions of a sandstorm and calculate profiles of particle concentration, velocity, and local volume flux.

Properties of a bidisperse particle–gas suspension Part 2. Viscous relaxation time small compared with collision time

1993 ◽  
Vol 247 ◽  
pp. 643-660 ◽  
Author(s):  
V. Kumaran ◽  
Donald L. Koch
Keyword(s):  
Relaxation Time ◽  
Vertical Direction ◽  
Stokes Number ◽  
Terminal Velocity ◽  
Distribution Functions ◽  
Mean Square ◽  
Collision Time ◽  
Gas Suspension ◽  
The Mean ◽  
Viscous Relaxation

The properties of a dilute bidisperse particle–gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit τv [Lt ] τc, where τc is the time between successive collisions of a particle, and τv is the viscous relaxation time. In this limit, the particles relax close to their terminal velocity between successive collisions, and we use a perturbation analysis in the small parameter ε, which is proportional to τv/τc, about a base state in which all the particles settle at their terminal velocities. The mean velocities of the two species are O(ε) different from their terminal velocities, and the mean-square velocities are O(ε) smaller than the square of the terminal velocity. The distribution functions for the two species, which incorporate the first effects of collisions between particles settling at their terminal velocities, are derived. The velocity distribution is highly anisotropic in this limit, and the mean-square velocity in the vertical direction is twice that in the horizontal plane. The distribution function for each species is singular at its terminal velocity, and the distributions are non-zero in a finite region in velocity space between the two terminal velocities.

Particle response and turbulence modification in fully developed channel flow

1994 ◽  
Vol 277 ◽  
pp. 109-134 ◽  
Author(s):  
J. D. Kulick ◽  
J. R. Fessler ◽  
J. K. Eaton
Keyword(s):  
Particle Velocity ◽  
Transverse Direction ◽  
Laser Doppler Anemometry ◽  
Fluid Velocity ◽  
Stokes Number ◽  
Particle Mass ◽  
Wall Turbulence ◽  
Turbulence Energy ◽  
Streamwise Direction ◽  
Fluid Turbulence

The interactions between small dense particles and fluid turbulence have been investigated in a downflow fully developed channel in air. Particle velocities of, and fluid velocities in the presence of, 50 μm glass, 90 μm glass and 70 μm copper spherical beads were measured by laser Doppler anemometry, at particle mass loadings up to 80%. These particles were smaller than the Kolmogorov lengthscale of the flow and could respond to some but not all of the scales of turbulent motion. Streamwise mean particle velocity profiles were flatter than the mean fluid velocity profile, which was unmodified by particle loading. Particle velocity fluctuation intensities were larger than the unladen-fluid turbulence intensity in the streamwise direction but were smaller in the transverse direction. Fluid turbulence was attenuated by the addition of particles; the degree of attenuation increased with particle Stokes number, particle mass loading and distance from the wall. Turbulence was more strongly attenuated in the transverse than in the streamwise direction, because the turbulence energy is at higher frequencies in the transverse direction. Streamwise turbulence attenuation displayed a range of preferred frequencies where attenuation was strongest.

Sign in / Sign up

Export Citation Format

Share Document