Enhanced settling of nonheavy inertial particles in homogeneous isotropic turbulence: The role of the pressure gradient and the Basset history force

The Stokes drag force and the gravity force are usually sufficient to describe the behavior of sub-Kolmogorov-size (or pointlike) heavy particles in turbulence, in particular when the particle-to-fluid density ratio ${\rho }_p/{\rho }_f\gtrsim {10}^3$ (with ${\rho }_p$ and ${\rho }_f$ the particle and fluid density, respectively). This is, in general, not the case for smaller particle-to-fluid density ratios, in particular not for ${\rho }_p/{\rho }_f\lesssim {10}^2$. In that case the pressure gradient force, added mass effects, and the Basset history force also play important roles. In this study we focus on the understanding of the role of these additional forces, all of hydrodynamic origin, in the settling of particles in turbulence. In order to qualitatively elucidate the complex dynamics of such particles in homogeneous isotropic turbulence, we first focus on the case of settling of such particles in the flow field of a single vortex. After having explored this simplified case we extend our analysis to homogeneous isotropic turbulence. In general, we found that the pressure gradient force leads to a decrease in the settling velocity. This can be qualitatively understood by the fact that this force prevents the particles from sweeping out of vortices, a mechanism known as preferential sweeping which causes enhanced settling. Additionally, we found that the Basset history force can both increase and decrease the enhanced settling, depending on the particle Stokes number. Finally, the role of the nonlinear Stokes drag has been explored, confirming that it affects settling of inertial particles in turbulence, but only in a limited way for the parameter settings used in this investigation.

Figure 1

Schematic picture of the trajectory of a settling particle in homogeneous and isotropic turbulence. The blue and red circles represent, schematically, vortices with clockwise $(+)$ and counterclockwise $(-)$ circulation. Due to preferential sweeping, particles can obtain an enhanced settling velocity by preferentially selecting flow regions with downward velocities.

Figure 2

Setup for the study of the influence of the different hydrodynamic forces on the settling of inertial particles in the vicinity of a single isolated vortex. (a) Schematic setup of the flow domain, where a single vortex is placed. We locate the origin of our coordinate system at the center of this vortex. The particles start with their terminal settling velocity on the top of the domain (at $y=L)$. In order to make sure that the particles are also in a force equilibrium when leaving the domain, the domain is made longer in the $y$ direction (domain size of $L$ above and $3L$ below the vortex). (b) Azimuthal velocity $v_{\theta }$ as a function of the distance $r$ to the center of the vortex, with ${\omega }_0=1$ and $R=1$; see Eq. (9).

Figure 3

Inertial particles with $\beta =0$ settling in a stationary velocity field induced by an isolated vortex with positive core vorticity; see Fig. 2. The particle parameters are $\text{St}=0.5$ and $\text{Sv}=0.2$. (a) Visualization of the inertial particle trajectories. (b) Delay time $\mathrm{\Delta }t$ of the particles (plotted as a function of the starting position $x)$ between the actual settling time compared with the time needed for the particle when it falls with the terminal settling velocity $U_s$; see Eq. (8).

Figure 4

Inertial particles with $R_{\rho }=1.2$ settling in a stationary velocity field [see Eq. (2)] induced by an isolated vortex with positive core vorticity. The particle parameters are ${\text{St}}^{*}=2$ and ${\text{Sv}}^{*}=0.4$. (a) Visualization of inertial particle trajectories. (b) The delay time $\mathrm{\Delta }t$ as a function of the starting position $x$ when different combinations of hydrodynamic forces are included, indicated by the different colors. ST, Stokes drag force; G, gravity force; Press or P, pressure gradient force; Basset or B, Basset history force.

Figure 5

The normalized enhanced settling velocity $U_{es}/u_{r.m.s.}$ of inertial particles as a function of ${\text{St}}^{*}$ and ${\text{Sv}}^{*}$. Two density ratios are shown: (a) $R_{\rho }=\infty$ (i.e., $\beta =0$; for this particular case ${\text{St}}^{*}=\text{St}$ and ${\text{Sv}}^{*}=\text{Sv})$ and (b) $R_{\rho }=10$. For $R_{\rho }=10$ not all Stokes numbers have been simulated as the particle radius becomes bigger than the smallest length scale in the flow, as $a>\eta$ for ${\text{St}}^{*}\gtrsim 1$.

Figure 6

The relative enhanced settling velocity $U_{es}/U_s$. In this figure the same data set as in Fig. 5 has been used.

Figure 7

Relative enhanced settling velocity for different density ratios (a) as a function of ${\text{St}}^{*}$ with ${\text{Sv}}^{*}=0.4$ (horizontal cut of Fig. 6) and (b) as a function of ${\text{Sv}}^{*}$ with ${\text{St}}^{*}=0.3$ (vertical cut of Fig. 6).

Figure 8

Relative enhanced settling velocity is shown as a function of ${\text{St}}^{*}$ with ${\text{Sv}}^{*}=0.4$ (horizontal cut of Fig. 6). The different lines correspond to different combinations of hydrodynamic forces (in particular, the pressure gradient force and the Basset history force), in order to investigate the influence of these forces. Two different density ratios are used, and shown in (a) $R_{\rho }=100$ and (b) $R_{\rho }=10$.

Figure 9

Relative enhanced settling velocity $U_{es}/U_t$. Both plots are taken with $R_{\rho }=100$ and as functions of ${\text{St}}^{*}$ and ${\text{Sv}}^{*}$. (a) Linear Stokes drag and (b) nonlinear Stokes drag. Although the differences in the relative settling velocity between the plots are small, the settling velocity itself changes more drastically, because $U_t$ is changed.

Figure 10

Relative enhanced settling velocity for different density ratios. The solid lines are obtained excluding the nonlinear drag, while the dashed lines include the nonlinear drag. For $R_{\rho }=\infty$ the dashed and solid lines are exactly the same (here, $\frac{a}{\eta }\to 0$ resulting in the linear drag regime). (a) $U_{es}/U_t$ as a function of ${\text{St}}^{*}$ with ${\text{Sv}}^{*}=0.4$ (horizontal cut of Fig. 9), and (b) $U_{es}/U_t$ as a function of ${\text{Sv}}^{*}$ with ${\text{St}}^{*}=0.3$ (vertical cut of Fig. 9).