Heavy particles in a persistent random flow with traps

We study a one-dimensional model for heavy particles in a compressible fluid. The fluid-velocity field is modeled by a persistent Gaussian random function, and the particles are assumed to be weakly inertial. Since one-dimensional fluid-velocity fields are always compressible, the model exhibits spatial trapping regions where particles tend to accumulate. We determine the statistics of fluid-velocity gradients in the vicinity of these traps and show how this allows one to determine the spatial Lyapunov exponent and the rate of caustic formation. We compare our analytical results with numerical simulations of the model and explore the limits of validity of the theory. Finally, we discuss implications for higher-dimensional systems.

Figure 1

Steady-state distributions for $Z, P_s(Z=z)$, as a function of $z$. Shown are the persistent-flow model results (14) for $\kappa =0.25$ (solid line) and the corresponding distribution for the white-noise model (dashed-dotted line). The dotted lines show the power-law tails $\sim \mathcal{J}{z}^{-2}$.

Figure 2

Spatial Lyapunov exponent $\lambda$ and rate of caustic formation $\mathcal{J}$ in the different limits. (a) Persistent-flow limit results for $\lambda$, Eq. (17) (solid line), and $\mathcal{J}$, Eq. (19) (dashed-dotted line), plotted as functions of $\kappa$. The dotted lines show the asymptotic behaviors for small and large values of $\kappa$ (see main text). (b) Corresponding results for the white-noise limit: $\lambda$ (solid line) and $\mathcal{J}$ (dashed-dotted line) plotted as functions of $\varepsilon$. The dotted lines show the asymptotic behaviors for small values of $\varepsilon$ (see main text).

Figure 3

Spatial Lyapunov exponents and rate of caustic formation as functions of $\kappa$ for different values of $\text{Ku}$. (a) Results for $\lambda$ obtained from direct numerical simulations of Eq. (20) plotted as pentagons ($\text{Ku}={10}^1$), half circles ($\text{Ku}={10}^2$), and solid circles ($\text{Ku}={10}^3$). The corresponding results for ${\lambda }_A$ (Sec. 3d) are shown as diamonds ($\text{Ku}={10}^1$), triangles ($\text{Ku}={10}^2$), and squares ($\text{Ku}={10}^3$). The solid line is the persistent-flow model result, Eq. (17). (b) Results for $\mathcal{J}$ obtained from direct numerical simulations of Eq. (21) plotted as pentagons ($\text{Ku}={10}^1$), half circles ($\text{Ku}={10}^2$), and solid circles ($\text{Ku}={10}^3$). The corresponding results for ${\mathcal{J}}_A$ (Sec. 3d) are shown as diamonds ($\text{Ku}={10}^1$), triangles ($\text{Ku}={10}^2$), and squares ($\text{Ku}={10}^3$). The solid line shows the persistent-flow model result, Eq. (19).

Figure 4

Probability density $P(A_n=a)$ of fluid-velocity gradients for $\kappa =0.12$ and different $\text{Ku}$. Shown are results for $\text{Ku}={10}^1$ (dotted line), $\text{Ku}={10}^2$ (dashed line), and $\text{Ku}={10}^3$ (dashed-dotted line). The solid line corresponds to the distribution (7) that is approached in the limit $\text{Ku}\to \infty$.