Statistical Lagrangian evaporation rate of droplets released in a homogeneous quasi-isotropic turbulence

Turbulence has long been suspected to increase the evaporation rate of droplets via the convective effects it generates. The experimental data reported in this paper provide evidence of this increase and statistically quantify these effects. Ether droplets have been released in a quasi-isotropic homogeneous turbulence generated by synthetic jets and tracked using inline digital holography. Their Schmidt number is typically of the order of 2 and their Reynolds number is moderate $(\le 3)$. Their instantaneous positions and diameters have been measured by processing the holograms with an inverse problem approach that has been implemented in Python language for a high-performance computer. This has allowed us to drastically reduce the processing time and to reconstruct a high number of trajectories for various turbulence conditions. The Lagrangian statistics computed from these trajectories, totaling 1.3 million samples, show that the relative mean motion and turbulence seen by the droplets on average increases their evaporation rate. Within the parameter range investigated, we find that this increase is not well predicted when estimating the convective effect in the Sherwood number with the norm of the instantaneous relative velocity seen by the droplets. In contrast, this increase is very well predicted when the Sherwood number is calculated using a Reynolds number based on the norm of the mean relative velocity plus its root-mean-square fluctuation.

Figure 1

Sketch of the woofers (red spheres) and of the holographic optical setup axes (blue axes); distance between woofers of each pair: 680 mm; the angle between each pair is ${90}^{\circ }$; $z$ is the optical axis of the inline holographic setup; $y\text{−}z$ are the two axes of the two-velocity component LDV system used to measure the velocity in the turbulence box; the green sphere represents the $50{\mathrm{mm}}^3$ turbulence box; the injector is positioned on the $y$ axis.

Figure 2

A sequence illustrating the droplets jet behavior at the top of the turbulence box; the large circle materializes the edge of the turbulence box; the sequence was recorded with a 30 fps camera. (a) No entering, (b) partial entering, (c) full entering and dispersion.

Figure 3

Holographic optical setup. The coordinate axis are the same as in Fig. 1.

Figure 4

Inline hologram of three evaporating droplets. The recorded hologram (a) after background correction is compared to the synthetic hologram that best matches it (b) and to the residual (c) obtained by subtracting the best fit hologram from the recorded hologram.

Figure 5

Probability density functions of (a) the normalized droplets velocity $\mathrm{components},+$: $x$ direction, $▵$: $y$ direction, $○$: $z$ direction, dashed line: normal distribution; (b) the normalized acceleration components, same symbols and colors, dashed line: Eq. (1) with $s=0.62$.

Figure 6

Probability density functions of the normalized relative velocity components. $+$: $x$ direction, $▵$: $y$ direction, $○$: $z$ direction, dashed line: Eq. (1) with $s=0.62$.

Figure 7

Probability density function of evaporation rate $K_S$ (a) and normalized evaporation rate (b) for the runs RMS 0.2 ($+$), RMS 0.4 ($+$), RMS 0.6 ($+$), RMS $0.8^1$ ($+$), RMS $0.8^2$ ($+$), RMS 1.0 ($+$). The black dashed line is the PDF of Eq. (1) with $s=0.62$.

Figure 8

Mean statistical Lagrangian surface evaporation rate as a function of the mean Reynolds number based on the instantaneous relative velocity. $\circ$: measurements, red solid line: calculation with $|{\mathbf{U}}_{\mathbf{d}}-\mathbf{U}|=U_r$; $▿$: calculation with $|{\mathbf{U}}_{\mathbf{d}}-\mathbf{U}|=\overline{U_r}$; $▵$: calculation with $|{\mathbf{U}}_{\mathbf{d}}-\mathbf{U}|=\overline{U_r}+u_r^{\prime }$; blue solid line: calculation with Eq. (9).

Figure 9

The mean statistical Lagrangian evaporation rate $\overline{K_s}$ normalized by the value without convective effect $2K_{s0}$ as a function of the box turbulent kinetic energy. $\circ$: measurements; blue line: $\overline{K_s}$ calculated with Eq. (9); red line: $\overline{K_s\left(\overline{U_r}\right)}/2K_s^0$.