Effect of ${\text{Re}}_{\lambda }$ and Rouse numbers on the settling of inertial droplets in homogeneous isotropic turbulence

We present an experimental study on the settling velocity of dense sub-Kolmogorov particles in active-grid-generated turbulence in a wind tunnel. Using phase Doppler interferometry, we observe that the modifications of the settling velocity of inertial particles, under homogeneous isotropic turbulence and dilute conditions [i.e., small liquid fraction ${\varphi }_v\le O{\left(10\right)}^{-5}$], is controlled by the Taylor-Reynolds number ${\text{Re}}_{\lambda }$ of the carrier flow. Meanwhile, we did not find a strong influence of the ratio between the fluid and gravity accelerations on the particle settling behavior. Remarkably, we find that that the degree of hindering experienced by the particles (i.e., the measured particle settling velocity is smaller in magnitude than its respective one in still fluid conditions) increases with ${\text{Re}}_{\lambda }$. This observation is contrary to previous works at intermediate values of ${\text{Re}}_{\lambda }$ that report the opposite effect: settling enhancement. Nonetheless, our trend is observed at all particle sizes investigated, and when previous experimental data is included into the analysis, our data suggest that the particle settling behavior may be nonmonotonic with ${\text{Re}}_{\lambda }$: inducing enhancement at moderate values of ${\text{Re}}_{\lambda }$, and at promoting hindering at higher values of ${\text{Re}}_{\lambda }$. Moreover, at the highest ${\text{Re}}_{\lambda }$ studied, the settling enhancement regime ceases to exist. Finally, we find that the difference between the measured particle settling velocity $\left(V_p\right)$ and the particle terminal velocity in still fluid conditions $\left(V_T\right)$, normalized by the carrier phase rms fluctuations, $(V_p-V_T)/u$ scales linearly with the Rouse number $\text{Ro}=V_T/u$ (i.e., the ratio between the particles settling velocity and the fluid rms fluctuations). However, such behavior $(V_p-V_T)/u\sim -\text{Ro}$ appears only to be valid for moderate values of the Rouse number.

Figure 1

Sketch of our experimental setup (not to scale). The wind-tunnel cross-section is $75\times 75{\mathrm{cm}}^2$. Its center line is labeled as $\chi$ in the figure. The emitter and receiver components of the PDI are on the same horizontal plane. However, the receiver is positioned at 30 degrees (see $\alpha$ in the figure) with respect to the emitter to maximize the capture of the water droplets refraction [18, 37]. Two holes of approximately 10 cm were carved onto the walls to counteract the water accumulation on them. The measuring station was located at the position labeled as M1 on the wind-tunnel center line, and 3 m downstream of the droplets injection.

Figure 2

(a) Injector rack sketch. For the lowest volume fractions half of the injectors (filled markers) were utilized. (b) Spray characterization coming from Spraytec data from Sumbekova [39]. (c) Droplet histogram obtained with the PDI for all experimental datasets reported in this work.

Figure 3

(a) Example of an unidimensional power spectral density $F_{11}$ [42] from one of our hot-wire records at measuring station M1 (see Fig. 1). (b) Parameter space for the experiments conducted. The global liquid fraction was estimated as ${\varphi }_v\approx Q_W/Q_A$, where $Q_W$ and $Q_A$ are the volumetric flux of water and air, respectively.

Figure 4

Coordinate systems for the wind tunnel and the PDI device.

Figure 5

Particle vertical velocity measurements binned by diameter size against the binned Stokes number. Error bars have a size $\pm 5\times {10}^{-3}{\text{ms}}^{-1}$ (half of the PDI resolution).

Figure 6

Particle velocity over the carrier phase fluctuations against Rouse (left) and Stokes numbers (right). In the figures legend GEA refers to the data of Good et al. [29]. AEA refers to the data of Aliseda et al. [24], and SBK refers to the data of Sumbekova [18]. Error-bars denote the resolution uncertainty, and they are masked by the size of the markers on the plot.

Figure 7

Parameters computed from the data in Fig. 6 . The different line styles refer to different values of ${\text{Re}}_{\lambda }$.

Figure 8

(a) ${\text{Ro}}_{cr}$ cross over between enhancement and hindering against ${\gamma }_a=\sqrt{0.13{\text{Re}}_{\lambda }^{0.64}}\gamma$. The solid lines refer to the proposed scaling in Ref. [39]. (b) ${\text{Ro}}_{\text{cr}}$: crossover value between enhancement and hindering against ${\text{Re}}_{\lambda }$. (c) Slope of the settling velocity against Rouse number $(\mathrm{\Delta }V/u)/\text{Ro}$. (d) Maximum settling velocity, and Rouse value for these maxima.

Figure 9

Settling velocity normalized by different scales including the estimated settling velocity $V_{\text{cl}}$ following the approach of Obligado et al. [55]. The vertical axis in panels (a) and (b) is negative log, i.e., $-1$ $\times$ log. The black markers follow the legend found in Fig. 8

Figure 10

(a) Settling velocity in a relative frame. Error bars account for the velocity vertical resolution $\pm 0.005{\text{ms}}^{-1}$. (b) Scaling of Eq. (10) applied in the relative moving frame of reference. Black markers represent our experimental data, red hollow markers are data of Sumbekova [18]. The legend follows Fig. 8.

Figure 11

PDFs of the particles velocity for the different records. (a) Horizontal component. (b) Vertical component. The darker the color the larger ${\text{Re}}_{\lambda }$. In the figures, the normal distribution is plotted as a dashed line ($\text{−}\text{−}\text{−}$).

Figure 12

Particle velocity over the particle terminal speed. (a) Against Rouse. (b) Against stokes. The markers follow the legend of Fig. 8.

Figure 13

(a) Sumbekova et al. scaling [18]. (b) Combination of the velocity scales for the AG data. (c) Data from Fig.–16 of Rosa et al. [28]. In the legends, GEA corresponds to the data of Good et al. [29], AEA refers to the data of Aliseda et al. [24], and SBK refers to the data of Sumbekova [18].