Experimental evidence of settling retardation in a turbulence column

Settling experiments were conducted in a turbulence column to investigate the effect of turbulence on the effective fall velocity of solid particles slightly denser than the fluid $({\rho }_p\gtrsim {\rho }_f)$. Five types of particles of different materials and shapes were tested, their size ranging between $o\left(1\right)\eta$ and $o\left(10\right)\eta$, where $\eta$ is the Kolmogorov viscous length scale. Thus, the particles were of finite size with an unknown analytical form for the fluid-particle forces. The density ratio ranged as $({\rho }_p-{\rho }_f)/{\rho }_f=\{0.13:1.6\}$, and the still-fluid particle Reynolds number as ${\text{Re}}_p^0=\{75:981\}$. The turbulence levels characterized with the integral-scale Reynolds number ranged as ${\text{Re}}_L=\{34:510\}$. Two-dimensional (2D) particle image velocimetry was used to obtain flow statistics, the residual mean circulation, and the turbulence statistics, while 2D particle tracking was performed to measure particle settling velocities. For all types of particles tested, settling retardation is observed as the turbulence intensity is increased. It is found that if both the effective fall velocity $W_s$ and the turbulent fluid velocity $W_{f,\text{rms}}$ are nondimensionalized by the still-fluid particle terminal velocity $W_0$, the settling retardation can be described by a unique relation independent of the particle type, $W_s/W_0=f(W_{f,\text{rms}}/W_0)$, for the given range of flow regimes. Using analytical descriptions of the loitering and nonlinear drag effects, this scaling is shown to have a solid physical basis.

Figure 1

Sketch of the turbulence column. The dashed red parallelepiped corresponds to the measurement volume, where the settling of a solid particle was tracked. The two green rectangles correspond to the two planes where fluid velocity measurements were performed using PIV. Both sides of the tank are equipped with five grids each, all oscillating independently at random phase. The top two are not activated in this study. The blue arrows show the direction of the grids' oscillation.

Figure 2

PIV results for the stirring frequency $F=5$ Hz. (a) The time-averaged velocity vector field $(\overline{U},\overline{W})$, with the velocity scale given at the top of the plot. (b) The horizontal turbulence intensity $U_{f,\text{rms}}$. (c) The vertical turbulence intensity $W_{f,\text{rms}}$. (d) The local turbulent correlation coefficient $C$. The black rectangle represents the region used for the settling velocity analysis (size $B\times H$).

Figure 3

(a) The ratio between the particle size and dissipation scale, $d_p/\eta$, and (b) the particle Stokes number, ${\text{St}}_{\eta }={\tau }_p/{\tau }_{\eta }$, as a function of the local dimensionless vertical turbulence intensity, $W_{f,\text{rms}}/W_0$, for particle types S1–S5. The dashed lines correspond to $d_p/\eta =1$ and ${\text{St}}_{\eta }=1$.

Figure 4

Scatter plots of the instantaneous particle settling velocities as functions of the local vertical turbulence intensities $W_{f,\text{rms}}$ for particle types S1–S5. The solid blue lines correspond to the linear regressions models. Black and red lines show the probability distribution functions of $W_s$ for intervals of $\mathrm{\Delta }{W}_{f,\text{rms}}=0.33\mathrm{cm}/\mathrm{s}$ from 0 to 3.00 cm/s. (a) S1 ($\square$), (b) S2 ($+$), (c) S3 ($\circ$), (d) S4 ($\diamond$), and (e) S5 ($\nabla$).

Figure 5

Particle velocity data averaged within eight levels of turbulence intensity with a step of $\mathrm{\Delta }{W}_{f,\text{rms}}=0.33$ cm/s. (a) Particle settling velocity as a function of the vertical turbulence intensity. (b) Particle horizontal velocity as a function of the vertical turbulence intensity. (c) Particle settling velocity as a function of the vertical turbulence intensity normalized by the still-water settling velocity. S1 ($\square$), S2 ($+$), S3 ($\circ$), S4 ($\diamond$), and S5 ($\nabla$).

Figure 6

(a) rms of the vertical particle velocity $W_{s,\text{rms}}$ and (b) rms of the horizontal particle velocity $U_{s,\text{rms}}$ as functions of the corresponding local turbulent velocities. $W_{0,\text{rms}}$ and $U_{0,\text{rms}}$ are the rms of the vertical and horizontal particle velocities in still water. The solid lines correspond to the equations $W_{s,\text{rms}}-W_{0,\text{rms}}=W_{f,\text{rms}}$ and $U_{s,\text{rms}}-U_{0,\text{rms}}=U_{f,\text{rms}}$. S1 ($\square$), S2 ($+$), S3 ($\circ$), S4 ($\diamond$), and S5 ($\nabla$).