Inertial particles driven by a telegraph noise

We present a model for the Lagrangian dynamics of inertial particles in a compressible flow, where fluid velocity gradients are modeled by a telegraph noise. The model allows for an analytic investigation of the role of time correlation of the flow in the aggregation-disorder transition of inertial particles. The dependence on the Stokes number St and the Kubo number Ku of the Lyapunov exponent of particle trajectories reveals the presence of a region in parameter space (St, Ku), where the leading Lyapunov exponent changes sign, thus signaling the transition. The asymptotics of short- and long-correlated flows are discussed, as well as the fluid-tracer limit.

Figure 1

Parameter space (St, Ku). Shape transitions in the probability density function (PDF) of $x$ occur for $\mathrm{St}={\mathrm{St}}^{*}=1∕4$ (vertical solid line), $\tilde{m}=1$ (dashed line), and $m=1$ (dash-dotted line). Labels refers to the PDFs shown in Figs. 2, 3. The gray region is not accessible by the model because Ku does not fulfill condition (12). Crosses represent the boundary of the chaotic region.

Figure 2

PDF of $x$ in the small Stokes regime $(\mathrm{St}=1∕8)$ for different value of the Kubo number: $\mathrm{Ku}=1∕8\left(a\right)$, $\mathrm{Ku}=1∕16\left(b\right)$, and $\mathrm{Ku}=1∕24\left(c\right)$.

Figure 3

PDF of $x$ in the large Stokes regime $(\mathrm{St}=1)$ for different values of Kubo number, $\mathrm{Ku}=0.1\left(d\right)$, $\mathrm{Ku}=1\left(e\right)$.

Figure 4

The isolines of the Lyapunov exponent $\lambda$ on the plane of Stokes and Kubo numbers. The isolines are spaced every $0.02\mathrm{s}$ (dotted and solid lines). The boundary of the chaotic region $(\lambda >0)$ is represented by the dot-dashed line.

Figure 5

The Lyapunov exponent $\lambda$ as a function of the Kubo number. An interval of positive Lyapunov exponents appears for $\mathrm{St}\gtrsim 4.3$. The dashed line represents the asymptotic behavior for $\mathrm{Ku}={\mathrm{Ku}}^{*}$.

Figure 6

Lyapunov exponent $\lambda$ in the short-correlated limit.

Figure 7

Lyapunov exponent $\lambda$ for $\mathrm{Ku}=1$ (solid line) and on the line $\mathrm{Ku}={\mathrm{Ku}}^{*}$ (dashed line), corresponding to the longest correlation of velocity gradients achievable within the model. Upper inset: asymptotic behavior for large Stokes number $\lambda s^{-1}\sim {\mathrm{St}}^{-2∕3}$ (dotted line), $(\mathrm{Ku}=1)$. Lower inset: asymptotic behavior for small Stokes number $\mid \lambda -{\lambda }_0\mid s^{-1}\sim {\mathrm{St}}^{\zeta }$ (dotted line).