Impact of diffusion on surface clustering in random hydrodynamic flows

Buoyant material clustering in a stochastic flow, which is homogeneous and isotropic in space and stationary in time, is addressed. The dynamics of buoyant material in three-dimensional hydrodynamic flows can be considered as the motion of passive tracers in a compressible two-dimensional velocity field. The latter is of interest in the present study. It is well known that the clustering of the density of passive tracers occurs in this case. We evaluate the impact of diffusion on the clustering process by using a numerical model. In general, the effect of diffusion is negligible in the very beginning of the evolution of initially uniformly distributed passive tracers. Therefore, the clustering of the density of passive tracers can emerge in accordance with the general theory. We analyze the long time clustering affected by diffusion and show that the emerged cluster structure persists in time in spite of the diffusion effect. However, the clusters split in time.

Figure 1

A realization of the area and the mass of the clusters at $\overline{\rho }=2.0$ for the parameters $l=0.02,{\sigma }_{\mathbf{u}}=0.166$, marked by circles (magenta); $l=0.04,{\sigma }_{\mathbf{u}}=0.333$, not marked (blue); $l=0.08,{\sigma }_{\mathbf{u}}=0.333$, marked by squares (green); $l=0.16,{\sigma }_{\mathbf{u}}=0.666$, marked by triangles (red). The bold lines indicate the asymptotic solutions (9), (10).

Figure 2

The density distribution at the time $t=\tau$ for the parameters of the velocity field: $l=0.04,{\sigma }_{\mathbf{u}}=0.333$ (top), $l=0.08,{\sigma }_{\mathbf{u}}=0.666$ (center), $l=0.16,{\sigma }_{\mathbf{u}}=0.666$ (bottom).

Figure 3

The area and mass of all the clusters for $\overline{\rho }=2.0$. (Top) $l=0.08,{\sigma }_{\mathbf{u}}=0.333$. The diffusion coefficients: $\kappa =2\times {10}^{-4}$, marked by rhombus (purple); $\kappa =2\times {10}^{-5}$, marked by circles (magenta); $\kappa =2\times {10}^{-6}$, marked by triangles (red); $\kappa =2\times {10}^{-7}$, marked by squares (green); $\kappa =2\times {10}^{-8}$, not marked (blue). The curves with a greater value of $\kappa$ are higher for the area and lower for the mass. (Bottom) The diffusion coefficient is $\kappa =2\times {10}^{-4}$. The parameters of the velocity field are $l=0.04,{\sigma }_{\mathbf{u}}=0.333$, not marked (blue); $l=0.08,{\sigma }_{\mathbf{u}}=0.333$, marked by squares (green); $l=0.16,{\sigma }_{\mathbf{u}}=0.666$, marked by triangles (red). The bold lines correspond to the asymptotic values (9), (10).

Figure 4

The density distribution at the time $t=27.3\tau$ for $l=0.08,{\sigma }_{\mathbf{u}}=0.333,\kappa =0$ (top). The enlarged part $\left(0/0.2,0/0.2\right)$ (center). The same part calculated for a finer resolution in the space (bottom).

Figure 5

The density distribution at the time $t=27.3\tau$ for $l=0.08,{\sigma }_{\mathbf{u}}=0.333$, and $\kappa =2\times {10}^{-4}$ (top). The enlarged part $\left(0/0.2,0/0.2\right)$ (center). The same part calculated for a finer resolution in the space (bottom).

Figure 6

The same as in the bottom panel of Fig. 5 for a range $\left(0.04/0.06,0.1/0.12\right)$ with a finer resolution.