A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

/pdf/percolation-statistical-topography-and-transport-in-random-1ddemlupey.pdf

Percolation, statistical topography, and transport in random media

https://www.math.nyu.edu/faculty/majda/pdfFiles/PhysRep%201999%20Majda%20Kramer.pdf

Simplified models for turbulent diffusion : theory, numerical modelling, and physical phenomena

The small scale properties of a passive scalar mixed by a turbulent velocity field show departures from Komogorov theory at least as prominent as for the velocity field1. While one might have imagined that the scalar merely inherits the intermittency of the velocity field itself, a series of recent numerical experiments with a Gaussian but multiscale velocity field argue otherwise.2 A trivial but multiscale velocity produces scalar statistics in some cases quantitatively similar to those obtained in laboratory experiments.3

http://www.physics.rockefeller.edu/siggia/Publications/1990-9_files/holzerPhyOfFlu1994.pdf

Turbulent Mixing of a Passive Scalar

This paper studies the influence of convection by periodic or cellular flows on the effective diffusivity of a passive scalar transported by the fluid when the molecular diffusivity is small. The flows are generated by two-dimensional, steady, divergence-free, periodic velocity fields.

/pdf/convection-enhanced-diffusion-for-periodic-flow-h2tmajzk0z.pdf

Convection enhanced diffusion for periodic flow

We investigate the spreading of passive tracers in closed basins. If the characteristic length scale of the Eulerian velocities is not very small compared with the size of the basin the usual diffusion coefficient does not give any relevant information about the mechanism of spreading. We introduce a finite size characteristic time τ(δ) which describes the diffusive process at scale δ. When δ is small compared with the typical length of the velocity field one has τ(δ)∼λ−1, where λ is the maximum Lyapunov exponent of the Lagrangian motion. At large δ the behavior of τ(δ) depends on the details of the system, in particular the presence of boundaries, and in this limit we have found a universal behavior for a large class of system under rather general hypothesis. The method of working at fixed scale δ makes more physical sense than the traditional way of looking at the relative diffusion at fixed delay times. This technique is displayed in a series of numerical experiments in simple flows.

Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient

The effective diffusion coefficient D* of a passive component, such as test particles, dye, temperature, magnetic flux, etc, is derived for motion in periodic two‐dimensional incompressible convective flow with characteristic velocity v and size d in the presence of an intrinsic local diffusivity D Asymptotic solutions for effective diffusivity D*(P) in the large P limit, with P∼ vd/D, is shown to be of the form D*=cDP1/2 with c being a coefficient that is determined analytically The constant c depends on the geometry of the convective cell and on an average of the flow speed along the separatrix The asymptotic method of evaluation applies to both free boundary and rough boundary flow patterns and it is shown that the method can be extended to more complicated patterns such as the flows generated by rotating cylinders, as in the problem considered by Nadim, Cox, and Brenner [J Fluid Mech 164, 185 (1986)] The diffusivity D* is readily calculated for small P, but the evaluation for arbitrary P requires numerical methods Monte Carlo particle simulation codes are used to evaluate D* at arbitrary P, and thereby describe the transition for D* between the large and small P limits

Effective diffusion in laminar convective flows

The effective diffusion coefficient D* of a passive component, such as test particles, dye, temperature, magnetic flux, etc., is derived for motion in periodic two‐dimensional incompressible convective flow with characteristic velocity v and size d in the presence of an intrinsic local diffusivity D. Asymptotic solutions for effective diffusivity D*(P) in the large P limit, with P∼ vd/D, is shown to be of the form D*=cDP1/2 with c being a coefficient that is determined analytically. The constant c depends on the geometry of the convective cell and on an average of the flow speed along the separatrix. The asymptotic method of evaluation applies to both free boundary and rough boundary flow patterns and it is shown that the method can be extended to more complicated patterns such as the flows generated by rotating cylinders, as in the problem considered by Nadim, Cox, and Brenner [J. Fluid Mech. 164, 185 (1986)]. The diffusivity D* is readily calculated for small P, but the evaluation for arbitrary P requires numerical methods. Monte Carlo particle simulation codes are used to evaluate D* at arbitrary P, and thereby describe the transition for D* between the large and small P limits.

I. Doxas

Papers

Effective diffusion in laminar convective flows

Effective diffusion in laminar convective flows