Dispersion of Droplet Clouds in Turbulence

We measure the absolute dispersion of clouds of monodisperse, phosphorescent droplets in turbulent air by means of high-speed image-intensified video recordings. Laser excitation allows the initial preparation of well-defined, pencil-shaped luminous droplet clouds in a completely nonintrusive way. We find that the dispersion of the clouds is faster than the dispersion of fluid elements. We speculate that preferential concentration of inertial droplet clouds is responsible for the enhanced dispersion.

Figure 1

(a) Experimental setup, not to scale. It consists of a closed chamber in which turbulence is generated by loudspeaker-driven synthetic jets. For clarity, only one loudspeaker is shown. The chamber contains a fog of approximately monodisperse droplets which are made from a water-based phosphorescent solution. A pencil-shaped volume is delineated by exciting phosphorescence with a pulsed focused laser beam. The glowing droplets are followed using a fast intensified camera. (b) Uniformity of the turbulent velocity $v$ (vertical $y$ component) inside the chamber. The initially delineated pencil-shaped volume which is imaged by the camera ($50.1\mathrm{mm}$ long) is indicated. (c) Turbulent velocity $v$ along the delineated pencil. (d) Volume-weighted droplet size probability distribution function (PDF) for two rotation rates of the spinning disk, corresponding to mean droplet diameters of $d_p=12\mu \mathrm{m}$ and $d_p=46\mu \mathrm{m}$, respectively. These PDFs are normalized; the PDF of the $46\mu \mathrm{m}$ droplets has been shifted vertically for clarity.

Figure 2

Dispersion of pencil-shaped volumes delineated at $t=0$ in a turbulent cloud. (a) Single tagged volume at initial time. (b) Initially tagged volume, averaged over 775 cycles. (c) Tagged volume after $t=0.8\mathrm{ms}$ (corresponding to $t=2{\tau }_{\eta }$). The cross-sectional intensity profiles of the phase-averaged images are indicated.

Figure 3

Cross-sectional profiles of tagged volume at initial time ($t=3\mu \mathrm{s}$) and after $t=0.4$, 1.0, and 1.6 ms after tagging. The initial profile is used in Eq. (1) for a fit of the widening $\sigma (t)$. The fits are indicated by the thick gray lines.

Figure 4

Line dispersion as a function of Stokes number. (a) Time dependence of linewidth $\sigma (t)$ for three different Stokes numbers $\text{St}=2.1$, 8.8, and 20 for the lines labeled by 1, 2, and 3, respectively. The symbols are the measurements, the full lines are fits, and ${\sigma }^2(t)=2v_d^2{t}^2$, with $v_d$ the dispersion velocity. (b) Closed dots represent the dispersion velocity of narrow lines, starting with an initial Gaussian width ${\sigma }_0=10\eta$; open circles correspond to ${\sigma }_0=30\eta$. The horizontal error bars were computed from the width of the volume-weighted droplet diameter PDF [see Fig. 1]; the vertical error bars were computed using a decimation procedure. Where the vertical error bars are absent, no decimation of the data was possible. The thick dashed line is the turbulent velocity $v$, and the thick solid line is to guide the eye to the results with the narrowest initial cloud. The numbered data points correspond to the numbered lines in (a).