Particle Dynamics In A Turbulent ParticleGas Suspension At High Stokes Number
Abstract
Particle laden turbulent flows find applications in many industrial processes such as energy conversion, air pollution control etc. In these types of flows, there are strong coupling between the turbulent fluctuations in the fluid velocity fields, and the fluctuating velocities of the particles. In order to analyze the stresses and the heat and mass transfer properties in turbulent suspensions, it is necessary to have a good understanding of not just the mean flow of the gas and particles, but also of the fluctuations in the two phases. The coupling is a twoway coupling; the fluid turbulence contributes to the velocity fluctuations in the particles, and conversely, the particle velocity fluctuations generate fluctuations in the fluid. Twophase flow models capture these interactions only in an indirect way, usually through a ‘particle pressure’ term for the particle phase.
In the present work the effect of fluid velocity fluctuations on the dynamics of the particles in a turbulent gassolid suspension is analyzed in the low Reynolds number and high Stokes number limit, where the particle relaxation time is long compared to the correlation time for the fluid velocity fluctuations. The direct numerical simulation (DNS) is used for solving the NavierStokes equations for the fluid, the particles are modeled as hard spheres which undergo elastic collisions. A oneway 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. This is because the main focus of our study is to examine the effect of the fluid turbulence on the particle fluctuations, and we are interested in examining whether a Langevin model with random forcing can accurately capture the effect of fluid turbulence on the particle phase.
First, the turbulent flow in a plane Couette is analyzed. Though this is a model flow which is not encountered often in applications, it is easier to analyze because the turbulent velocity fluctuations are maximum at the center of the channel, in contrast to the Poiseuille flow, where the velocity fluctuations are maximum at a location between the center and the wall. Also, in a Couette flow, the wallnormal and the spanwise root mean square velocities are nearly a constant in the central region in the channel, and the percentage variation in the streamwise velocity fluctuations is also less than that in a pressure driven Poiseuille flow. Therefore, it is possible to treat the central region as a region with homogeneous, but anisotropic, fluid velocity fluctuations and with a linear mean velocity variation. The particle mean and root mean square fluctuating velocities, as well as the probability distribution function for the fluid velocity fluctuations and the distribution of acceleration of the particles in the central region of the Couette, which comprises about 20% of the entire channel have been studied. It is found that the distribution of particle velocities is very different from a Gaussian, especially in the spanwise and wallnormal 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 nonGaussian nature of the fluid velocity fluctuations is found to be due to interparticle collisions induced by the large particle velocity fluctuations in the flow direction.
Another interesting result is a comparison of the distribution of the acceleration on a particle due to the fluid velocity fluctuation at the particle position, and the distribution of the ratio of fluid velocity fluctuation to the viscous relaxation time in the fluid. The comparison shows that these two distributions are almost identical, indicating that the fluid velocity fluctuations are not correlated over time scales comparable to the relaxation time of a particle. This result is important because it indicates that in order to model the fluctuating force on the particle, it is sufficient to obtain the variance of the force distribution from the variance of the fluid velocity distribution function.
Finally, the correlation time for the acceleration correlations is calculated along the trajectory of a particle. The correlation time is found to be of the same magnitude as the correlation time for the fluid velocity in an Eulerian reference frame, and much smaller than the viscous relaxation time and the time between collisions of the particles. All of these results indicate that the effect of the turbulent fluid velocity fluctuations can be accurately represented by an anisotropic Gaussian white noise.
The above results are used to formulate a ‘fluctuating force’ model for the particle phase alone, where the force exerted by the fluid turbulent velocity fluctuations is modeled as random Gaussian white noise, which is incorporated into the equation of motion for the particles. The variance of the distribution function for the fluctuating force distribution is obtained from the variance of the local turbulent fluid velocity fluctuations, assuming linear Stokes drag law. The force distribution is anisotropic, and it has a nonzero correlation between the flow and gradient directions. It is found that the results of the fluctuating force simulations are in quantitative agreement with the results of the complete DNS, both for the particle concentration and variances of the particle velocity fluctuations, at relatively low volume fractions where the viscous relaxation time is small compared to the time between collisions, as well as at higher volume fractions where the time between collisions is small compared to the viscous relaxation time. The simulations are also able to predict the velocity distributions in the center of the Couette, even in cases where the velocity distribution is very different from a Gaussian distribution.
The fluctuating force model is applied to the turbulent flow of a gasparticle suspension in a vertical channel in the limit of high Stokes number. In contrast to the Couette flow analyzed the fluid velocity variances in the different directions in the channel are highly nonhomogeneous, and they exhibit a significant variation across the channel. First, we analyze the fluctuating particle velocity and acceleration distributions at different locations across the channel using direct numerical simulation. The distributions are found to be nonGaussian near the center of the channel, and they exhibit significant skewness. The time correlations of the fluid velocity fluctuations and the acceleration fluctuations on the particles are evaluated and compared. Unlike the case of Couette flow it is found that the time correlation functions for the fluid in the fixed Eulerian frame are not in agreement with the time correlation of the acceleration on the particles. However, the time correlations of the particle acceleration are in good agreement with the velocity time correlations in the fluid in a ‘moving Eulerian’ reference frame, moving with the mean velocity of the fluid. The fluctuating force simulations are used to model the particle phase, where the force on the particles due to the fluid velocity fluctuations are substituted by random white noise in the equations for the particle motion. The random noise is assumed to be Gaussian and anisotropic. The variances of the fluctuating force are calculated form the fluid velocity fluctuations in a moving Eulerian reference frame using DNS. The results from the fluctuating force simulations are then compared with the results obtained from DNS. 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.
The interactions between the solid particles and the fluid turbulence have been investigated experimentally in a vertical fully developed channel flow of air and solid particles. Experiments are conducted at low volume fraction for which viscous relaxation time of the particle is expected to be lower than the particle particle collision time, as well as at moderately high volume fraction where the particle particle collision time is expected to be lower than the particle relaxation time. Velocity statistics of both the particle and gas phases are obtained using high spatial resolution Particle Image Velocimetry (PIV) system. It is observed that at low solid volume fraction, the particle root mean square velocities and the velocity distribution are in good agreement with those predicted by the fluctuating force simulation, provided the polydispersity in the particle size distribution is incorporated in the fluctuating force simulations. In this case, the modification of turbulence in the center of the channel due to the particles is small. At much higher volume fraction, the mean gas flow is significantly affected by the presence of particles, and the mean flow is no longer symmetric about the center line of the channel. Simultaneously, there is also a significant change in the volume fraction across the channel, and the volume fraction is also not symmetric about the center line. This seems to indicate that there is a spontaneous instability of the symmetric volume fraction and velocity profiles, giving rise to a region of high fluid velocity and high particle volume fraction coexisting with a region of low gas velocity and low particle volume fraction. There is some recirculation of the gas within the channel, and the gas phase turbulence intensity is significantly enhanced when the velocity and volume fraction profiles become asymmetric. As we have considered only one way coupling in the computation of the particle laden flow it is expected that the particle statistics obtained for this condition can not be predicted by our fluctuating force model due to modification of the gas phase statistics.
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