Settling of finite-size particles in turbulence at different volume fractions

Abstract

We study the settling of finite-size rigid spheres in quiescent fluid and in sustained homogeneous isotropic turbulence (HIT) by direct numerical simulations using an immersed boundary method to account for the dispersed solid phase. We consider semi-dilute and dense suspensions of rigid spheres with solid volume fractions $\phi =0.5{-}10\%$, solid-to-fluid density ratio $R=1.02$, and Galileo number (i.e., the ratio between buoyancy and viscous forces) $Ga=145$. In HIT, the nominal Reynolds number based on the Taylor microscale is $Re_{\lambda } \simeq 90$, and the ratio between the particle diameter and the nominal Kolmogorov scale is $(2a)/\eta \simeq 12$ (being a the particle radius). We find that in HIT the mean settling speed is less than that in quiescent fluid for all $\phi$. For $\phi =0.5\%$, the mean settling speed in HIT is $8\%$ less than in quiescent fluid. However, by increasing the volume fraction the difference in the mean settling speed between quiescent fluid and HIT cases reduces, being only $1.7\%$ for $\phi =10\%$. Indeed, while at low $\phi$ the settling speed is strongly altered by the interaction with turbulence, at large $\phi$ this is mainly determined by the (strong) hindering effect. This is similar in quiescent fluid and in HIT, leading to similar mean settling speeds. On the contrary, particle angular velocities are always found to increase with $\phi$. These are enhanced by the interaction with turbulence, especially at low $\phi$. In HIT, the correlations of particle lateral velocity fluctuations oscillate around zero before decorrelating completely. The time period of the oscillation seems proportional to the ratio between the integral lengthscale of turbulence and the particle characteristic terminal velocity. Regarding the mean square particle displacement, we find that it is strongly enhanced by turbulence in the direction perpendicular to gravity, even at the largest $\phi$. Finally, we investigate the collision statistics for all cases and find the interesting result that the collision frequency is larger in quiescent fluid than in HIT for $\phi =0.5{-}1\%$. This is due to frequent drafting–kissing–tumbling events in quiescent fluid. The collision frequency becomes instead larger in HIT than in still fluid for $\phi =5{-}10\%$, due to the larger relative approaching velocities in HIT, and to the less intense drafting–kissing–tumbling events in quiescent fluid. The collision frequency also appears to be almost proportional to the estimate for small inertial particles uniformly distributed in space, though much smaller. Concerning the turbulence modulation, we find that the mean energy dissipation increases almost linearly with $\phi$, leading to a large reduction of $Re_{\lambda }$.