Caustic formation in a non-Gaussian model for turbulent aerosols

Caustics in the dynamics of heavy particles in turbulence accelerate particle collisions. The rate $\mathcal{J}$ at which these singularities form depends sensitively on the Stokes number $\mathrm{St}$, the nondimensional inertia parameter. Exact results for this sensitive dependence have been obtained using Gaussian statistical models for turbulent aerosols. However, direct numerical simulations of heavy particles in turbulence yield much larger caustic-formation rates than predicted by the Gaussian theory. To understand the possible mechanisms explaining this difference, we analyze a non-Gaussian statistical model for caustic formation in the limit of small $\mathrm{St}$. We show that at small $\mathrm{St}$, $\mathcal{J}$ depends sensitively on the tails of the distribution of Lagrangian fluid-velocity gradients. This explains why different authors obtained different $\mathrm{St}$-dependencies of $\mathcal{J}$ in numerical-simulation studies. The most likely gradient fluctuation that induces caustics at small $\mathrm{St}$, by contrast, is the same in the non-Gaussian and Gaussian models. Direct numerical simulation results for particles in turbulence show that the optimal fluctuation is similar, but not identical, to that obtained by the model calculations.

Figure 1

Caustic formation in a non-Gaussian model for a turbulent aerosol. (a) Probability of $Q$ and $R$ at the onset of caustic formation (color-coded). Parameters: $\mathrm{St}=0.1$ and $M=1$. The thick dashed line is the right branch of the Vieillefosse line (see the text). The solid line is the threshold line for caustic formation, Eq. (4). (b) Rate of caustic formation for the non-Gaussian model for different values of $M$. Also shown are the limiting $\mathrm{St}$ scalings, dashed lines. (c) Collapse of the small-$\mathrm{St}$ data from panel (b) (markers) onto the scaling function $\mathcal{F}\left(a\right)$ given in Eq. (19) (thick black line), using the $\mathrm{St}\to 0$ limit of the scaling variable $a$ in Eq. (1). The inset shows the same, but using the finite Stokes correction for $a$ in Eq. (21) with $\delta {\lambda }_{\mathrm{th}}$ from the theory in Table 1.

Figure 2

One-dimensional model for caustic formation. (a) Optimal fluctuations of fluid-velocity gradients, $A^{*}\left(t\right)$, obtained from the shooting method for the Gaussian case (see text) for different $\mathrm{St}$ vs $t-t_{\text{th}}$. The thick horizontal line shows the threshold $-{\lambda }_{\mathrm{th}}=-1/4$. (b) Same as (a) but for simulations of the non-Gaussian model with $M=1$. The lines show the mean over realizations at each time step. (c) Optimal fluctuation as a function of $t-t_{\mathrm{c}}$ using $M=1$ and $\mathrm{St}=0.04$, where $t_{\mathrm{c}}$ is the time at which the caustic forms. The color map shows the probability distribution of $A\left(t\right)$, normalized at each time step by the maximal probability. The solid line is the mean obtained in panel (b) and the dashed line shows the result from the shooting method in panel (a). (d) The depth $\delta {\lambda }_{\text{th}}$ of the optimal fluctuation from simulations of the one-dimensional models with $M=1,2$, and $\infty$. Also shown are the results of the shooting method (see the text), shown as a dashed line.

Figure 3

Caustic formation using direct numerical simulations of turbulence to integrate Eqs. (2) and (3). (a) Color-coded probability density of $Q$ and $R$ at the onset of caustic formation $[(t_{\mathrm{th}}-t_{\mathrm{c}})/{\tau }_{\text{K}}=-1.53]$. (b) Probability density of the real cube root of the invariant $\text{Tr}\left(\mathbb{S}{\mathbb{O}}^2\right)$ before the formation of a caustic at $t=t_{\mathrm{c}}$. Dashed line shows the onset time for panel (a). (c) Same as panel (b), but for the statistical model with $M=1$ and $\mathrm{St}=0.1$ [the case in Fig. 1]. (d) Probability density of eigenvalues ${\lambda }_i$ of $\mathbb{S}$ minus their average values, $\langle {\lambda }_i\rangle$, evaluated along particle trajectories independent from the formation of caustics. The solid green line shows the $\mathrm{St}\to 0$ limit of the threshold, ${\lambda }_{\text{th}}(\mathrm{St}\to 0)=-1/4$. Solid blue lines show the optimal fluctuation according to theory for the eigenvalues, ${\lambda }_3\left(t\right)-\langle {\lambda }_3\rangle \sim {\lambda }_{\text{th}}\left(\mathrm{St}\right)f_S(t-t_{\text{th}})$ and ${\lambda }_1\left(t\right)-\langle {\lambda }_1\rangle ={\lambda }_2\left(t\right)-\langle {\lambda }_2\rangle \sim -\frac{1}{2}{\lambda }_{\text{th}}\left(\mathrm{St}\right)f_S(t-t_{\mathrm{th}})$, with a fitted threshold ${\lambda }_{\text{th}}\left(\mathrm{St}\right)\approx -0.23$. Parameters: ${\mathrm{Re}}_{\lambda }=185$ and $\mathrm{St}=0.3$. The maximal values of the densities are at each time scaled to unity. (e) Rate of caustic formation against Stokes number $\mathrm{St}$. Dashed lines indicate scalings ${\mathrm{St}}^{-1}$ and ${\mathrm{St}}^{-2}$. Data are obtained from Ref. [10] (${\mathrm{Re}}_{\lambda }=45,83$), Ref. [38] (${\mathrm{Re}}_{\lambda }=90,170$), and new results using the DNS underlying Refs. [39] (${\mathrm{Re}}_{\lambda }=185$) and [7] (${\mathrm{Re}}_{\lambda }=400$).