Paths to caustic formation in turbulent aerosols

The dynamics of small, yet heavy, identical particles in turbulence exhibits singularities, called caustics, that lead to large fluctuations in the spatial particle-number density, and in collision velocities. For large particle inertia, the fluid velocity at the particle position is essentially a white-noise signal and caustic formation is analogous to Kramers escape. Here we show that caustic formation at small particle inertia is different. Caustics tend to form in the vicinity of particle trajectories that experience a specific history of fluid-velocity gradients, characterized by low vorticity and a violent strain exceeding a large threshold. We develop a theory that explains our findings in terms of an optimal path to caustic formation that is approached in the small inertia limit.

Figure 1

Illustration of caustic formation in two spatial dimensions. Shown are three nearby particles. A caustic is formed when faster particles overtake slower ones. At a caustic, the area $\widehat{\mathcal{V}}$ of the parallelepiped spanned by separation vectors between the three particles, shown in gray, vanishes.

Figure 2

Path to caustic formation for $\mathrm{St}=0.31$. Numerical results using a DNS of two-dimensional turbulence. The path density is color coded, normalized to unity along each slice of constant $\mathcal{Z}$ to improve visibility. (a) Path density in the $\mathcal{Z}-{\mathcal{A}}^2$ plane. The dashed line shows the approximation, $\mathcal{Z}\sim {\mathcal{A}}^2$. The solid line shows the evolution of the stable fixed point according to Eq. (6). The dot marks the bifurcation to instability (see main text). (b) Individual contributions from strain ($\mathcal{Z}$ vs ${\sigma }^2$, positive axis) and vorticity ($\mathcal{Z}$ vs $-{\omega }^2$, negative axis). (c) Path densities for ${\sigma }^2$ (positive axis) and $-{\omega }^2$ (negative axis) as a function of time before caustic formation at $t=0$.

Figure 3

Optimal paths obtained from theory and path density for $\mathrm{St}=0.31$ from DNS. (a) $({\mathcal{Z}}_{\text{opt}},{\sigma }_{\text{opt}}^2)$ from theory (dashed line). The color-coded path density is the same as in Fig. 2. (b) ${\sigma }_{\text{opt}}^2\left(t\right)$ from theory (dashed line). The color-coded path density is the same as in Fig. 2.