A Stochastic Model for High Stokes Number Particle Pair Dynamics in Isotropic Turbulence
Sarma Rani (Univ of Alabama in Huntsville)
Complex Fluids: Suspensions, Emulsions, and Gels
Wed 3:10 - 4:30
Barus-Holley 160
In this study, we derived the Fokker-Planck equation governing the joint PDF of pair separation and relative velocity vectors of high Stokes number particles in isotropic turbulence. The Fokker-Planck equation contains a particle-pair diffusion coefficient in relative velocity space, which is shown to be $1/\tau_v^2$ times the time-integral of the two-time correlation of fluid relative velocities along the pair trajectory; here, $\tau_v$ is the particle viscous relaxation time. We developed an analytical theory to predict this pair diffusion coefficient in the limit of particle ${\rm Stokes~number} \gg 1$. Using the diffusion coefficient, Langevin-equation-based stochastic simulations were performed to evolve pair relative velocities and separations in isotropic turbulence for Kolmogorov time scale ($\tau_\eta$)-based particle Stokes numbers, $St_\eta = 1,~2,~4,~10, \mbox{ and,}~20$ at a Taylor micro-scale Reynolds number, $\Re_\lambda = 75$. The Langevin simulations capture the transition of the relative velocity PDF from a Gaussian PDF at separations of the order of the integral length scale to a non-Gaussian PDF at smaller separations. The pair radial distribution functions (RDFs) computed from our Langevin simulations show reasonable quantitative agreement with the RDF predictions of Zaichik \& Alipchenkov (2003). However, even at higher Stokes numbers ($St_\eta = 4,~10, \mbox{ and,}~20$), our RDFs continue to exhibit preferential concentration marked by a power law dependence on separation at the sub-Kolmogorov length scale, while the moments-approach-based RDFs of Zaichik \& Alipchenkov (2003) plateaued at small separations. This behaviour of our RDFs can be attributed to our theory's ability to capture the non-Gaussian relative velocity PDF at separations smaller than the integral length scale, as seen in the DNS of Sundaram \& Collins (1997).