We investigate the relative velocity of inertial particles induced by turbulent motions, extending our earlier work on equal-size particles to the bidisperse case for different particles of arbitrary sizes. The model of Pan & Padoan (PP10) shows that the relative velocity between different particles has two contributions, named the generalized shear and acceleration terms, respectively. The generalized shear term represents the particles’ memory of the spatial flow velocity difference across the particle distance at given times in the past, while the acceleration term is associated with the temporal flow velocity difference on individual particle trajectories. The latter vanishes for equal-size particles. Using a simulation, we compute the root-mean-square (rms) relative velocity, <w^2>^1/2, as a function of the particle friction times, tau_p1 and tau_p2, and show that the prediction of the PP10 model is in satisfactory agreement with the data, confirming the validity of its physical picture. For a given tau_p1 below the Lagrangian correlation time of the flow, T_L, the rms relative velocity as a function of tau_p2 shows a dip at tau_p2 = tau_p1, indicating tighter velocity correlation between similar particles. Defining a friction time ratio as f=tau_p,l/tau_p,h, with tau_p,l and tau_p,h the friction times of the smaller and larger particles, respectively, we find that <w^2>^1/2 increases with decreasing f due to the generalized acceleration contribution, which starts to dominate at f <1/4. At a fixed f, our model predicts the rms relative velocity scales as tau_p,h^1/2 for tau_p,h in the inertial range of the flow, stays roughly constant for T_L < tau_p,h < T_L/f, and finally decreases as tau_p,h^-1/2 for tau_p,h >> T_L/f (or tau_p,l>>T_L). The predicted inertial-range scaling, <w^2>^1/2 \propto tau_ p,h^1/2}, at any f needs to be verified by simulations at high resolutions.
L. Pan, P. Padoan and J. Scalo
Tue, 18 Mar 14