Book Description

A solution to particle dispersion in an isotropic turbulence under Stokes drag, Basset force and gravitational force is obtained in closed form using the independence approximation. The Basset force has no effect on the fluid velocity structure seen by the particles or the long-time particle diffusivities. It does affect the intensities of particle motion for particles with large settling rate and with response time comparable to the turbulence integral time scale. A solution for particles dispersion in isotropic turbulence with non-Stokesian drag and gravitational force is obtained. The time constants of the particle fluctuation in the directions parallel and perpendicular to the gravity are anisotropic. Turbulence increases particle response time constants and reduces settling velocity. Influence of the nonlinear drag, particle response time constants and settling rate on particle dispersion are investigated. Monte-Carlo simulations are performed for particle motions in an isotropic turbulence with non-Stokesian drag. Pseudo-turbulence is generated using random Fourier modes representation. Statistical averages are obtained from more than 5000 particles. The results of the simulation validate the preceeding analysis in the non-Stokesian drag range. The influence of turbulence structure on the dispersions of fluid and particle is examined. In addition to the integral length and time scales, the functional form of the energy spectrum is also important in describing the dispersions of both fluid and particles. Numerical solution for unsteady flow over a sphere indicates that the added-mass force at finite Reynolds number is the same as in the creeping flow and the potential flow. The classical Stokes solution is not valid at small frequency, $omega$, and the corresponding Basset force is proportional to $omega$, instead of $sqrt{omega}$. The Basset-force term has a kernel decays faster than (t- $tau$)$sp{-1/2}$ at large time. The use of the steady state drag coefficient with the instantaneous velocity is justified to approximate the quasi-steady drag on particles. Limiting behavior of the unsteady drag on a sphere at small frequency and low Reynolds number is obtained using matched asymptotic expansions. The modified Basset-force term at finite Re is constructed. It has a kernel decays as (t- $tau$)$sp{-2}$ at large times.