Connecting finite-time Lyapunov exponents with supersaturation and droplet dynamics in a turbulent bulk flow

The impact of turbulent mixing on an ensemble of initially monodisperse water droplets is studied in a turbulent bulk that serves as a simplified setup for the interior of a turbulent ice-free cloud. A mixing model was implemented that summarizes the balance equations of water vapor mixing ratio and temperature to an effective advection-diffusion equation for the supersaturation field $s(\mathit{x},t)$. Our three-dimensional direct numerical simulations connect the velocity and scalar supersaturation fields in the Eulerian frame of reference to an ensemble of cloud droplets in the Lagrangian frame of reference. The droplets are modeled as point particles with and without effects due to inertia. The droplet radius is subject to growth by vapor diffusion. We report the dependence of the droplet size distribution on the box size, initial droplet radius, and the strength of the updraft, with and without gravitational settling. In addition, the three finite-time Lyapunov exponents ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ are monitored which probe the local stretching properties along the particle tracks. In this way, we can relate regions of higher compressive strain to those of high local supersaturation amplitudes. For the present parameter range, the mixing process in terms of the droplet evaporation is always homogeneous, while it is inhomogeneous with respect to the relaxation of the supersaturation field. The probability density function of the third finite-time Lyapunov exponent, ${\lambda }_3<0$, is related to the one of the supersaturation $s$ by a simple one-dimensional aggregation model. The probability density function (PDF) of ${\lambda }_3$ and the droplet radius $r$ are found to be Gaussian, while the PDF of the supersaturation field shows sub-Gaussian tails.

Figure 1

Contour plots of two-dimensional slice cuts of (a) the magnitude of the velocity field, (b) the supersaturation field, and (c) the logarithm of the dissipation rate of turbulent kinetic energy. It is given by $\epsilon =2\nu S_{ij}{S}_{ij}$, $S_{ij}=({\partial }_j{u}_i+{\partial }_i{u}_j)/2$ with the rate-of-strain tensor $S_{ij}$. Data are for the simulation run 2 with $L=256$ mm at $t=0.045{\tau }_L$. The lower panels (d)–(f) show corresponding plots for simulation run 3 with $L=512$ mm at $t=0.032{\tau }_L$. See also Table 1. The characteristic time unit, the large-scale eddy turnover time ${\tau }_L$ is specified in Sec. 2b; see also Table 1.

Figure 2

(a) Sub-Gaussian PDF of rescaled supersaturation field taken from Eulerian approach (blue line) and Gaussian PDF for reference. (b) Instantaneous snapshot of isosurfaces of the scalar dissipation rate field ${\epsilon }_s(\mathit{x},t)$, which is defined in Eq. (11). The level of the isosurfaces is ${log}_{10}{\epsilon }_s=-3.3$ in this plot.

Figure 3

Droplet size distribution at different time instants for computational domains with (a) $L=128$ mm (run 1), (b) $L=256$ mm (run 2), and (c) $L=512$ mm (run 3). All times are expressed in units of the corresponding large-scale eddy turnover time ${\tau }_L$ which are provided in Table 1.

Figure 4

Time evolution of two individual droplet radii in (a) and the corresponding supersaturation field in (b). Solid and dashed lines correspond to a particular particle. Time is given in seconds. (c) Probability density function (PDF) of the rescaled supersaturation field, $s/s_{\mathrm{rms}}$, experienced by the chosen individual particle for the whole calculation period, and (d) PDF of rescaled supersaturation field for all particles averaged in time. The dashed line shows the corresponding Gaussian PDF for comparison.

Figure 5

Parameter space which is spanned by the two Damköhler numbers ${\mathrm{Da}}_d$ and ${\mathrm{Da}}_s$. The operating points of the three simulation runs are inserted; see the legend and Table 1. Solid lines show $\text{Da}=1$ for both. They mark the corresponding transition between inhomogeneous and homogeneous mixing. Damköhler numbers greater and smaller than 1 characterize inhomogeneous and homogeneous mixing regimes, respectively.

Figure 6

The impact of different initial sizes of the cloud droplets on the droplet size distribution. Panel (a) is taken at $t=1$ s, (b) at $t=2$ s, and (c) $t=3$ s. Data correspond to runs 2a and 2e in Table 2.

Figure 7

Droplet radius vs time for four randomly chosen individual droplets in run 2e from Table 2.

Figure 8

Comparison of the time evolution of the cloud droplet size distribution for nonenhanced prefactor, $A_1=5\times {10}^{-4}$, (solid line) and enhanced factor, $A_1=0.2$ (dashed line), for cloud droplets with initial radius $r_0=20$ $\mu \text{m}$. These are runs 2d and 2e from Table 2. Panel (a) is taken at $t=1$ s, (b) at $t=5$ s, and (c) $t=15$ s.

Figure 9

Comparison of the time evolution of the cloud droplet size distribution for nonenhanced prefactor, $A_1=5\times {10}^{-4}$, for cloud droplets with dynamics as gravity settling particles (dash-dotted and dotted lines) and as Lagrangian tracers (solid and dashed lines) for droplets with initial radius $r_0=20$ $\mu \text{m}$ (dash-dotted and dashed lines) and $r_0=10$ $\mu \text{m}$ (solid and dashed lines). These are runs 2a, 2b, 2e, and 2f from Table 2. Panel (a) is taken at $t=1$ s, (b) at $t=5$ s, and (c) $t=15$ s.

Figure 10

Contour plots of horizontal slice cuts of ${\lambda }_3(\mathit{x},t)$ at different times in units of the large-scale eddy turnover time. (a) $t=0.03{\tau }_L$, (b) $t=0.09{\tau }_L$, and (c) $t=0.32{\tau }_L$. The magnitudes are given by the corresponding color bars. Data are for run 3.

Figure 11

Probability density functions of the finite-time Lyapunov exponents at different times in units of the large-scale eddy turnover time ${\tau }_L$. (a) Run 2 with $L=256$ mm and (b) run 3 with $L=512$ mm.

Figure 12

Probability density function of the magnitude of the gradient of the supersaturation field, $\chi =\left|\mathbf{\nabla }s\right|/{\left|\mathbf{\nabla }s\right|}_{\mathrm{rms}}$ for runs 1 (solid line), 2 (dashed line), and 3 (dash-dotted line). We indicate the corresponding Reynolds numbers in the legend.

Figure 13

Joint probability density functions of the third FTLE with (a) the droplet radius, $p({\lambda }_3,r)$, (b) the supersaturation field at the particle positions, $p({\lambda }_3,s)$, and (c) the normalized gradient magnitude of the supersaturation field, $\tilde{p}({\lambda }_3,\chi )$; see (18) for a specific definition. Data are for run 3 at $t=19.4{\tau }_L$; see also Fig. 11.

Figure 14

Probability density functions of supersaturation $s$ for times (a) $t=4.86{\tau }_L$ and (b) $t=9.75{\tau }_L$, which are obtained from the FTLEs (solid line) from run 3 via the aggregation model. They are compared to the directly evaluated ones (dash-dotted line). The dashed line is a Gaussian PDF.