# Structural Diffusion in 2D and 3D Random Flows

Conference paper
Part of the Fluid Mechanics and its Applications book series (FMIA, volume 36)

## Abstract

We investigate one- and two-particle (relative) diffusion of fluid particles in incompressible 2D and 3D steady and unsteady random flows with prescribed self-similar power spectra of the form E(k) = C α ε * 2/3 kα for k min k(=|k|) ≤ k max and zero otherwise, for 1 < α < 3. C α is a constant, and ε* = εL1(5−3α)/2 where ε is the rate of energy dissipation per unit mass, and L1 is a length scale — in our work we choose L1 = 2π/k min . The role played by the streamline topology is of special interest. The velocity fields are generated using Kinematic Simulation (KS), Fung, Hunt, Malik & Perkins 1992, JFM 236, 281, viz
$$\text{u}\left( {\text{x},t} \right) = \sum\limits_{n = 1}^{N_k } {\left\{ {\text{a}_n \cos \left( {k_n \cdot \text{x + }\omega _\text{n} t + \phi _n^a } \right) + \text{b}_n \sin \left( {k_\text{n} \cdot \text{x + }\omega _\text{n} t + \phi _n^b } \right)} \right\}}$$
and
$$\text{a}_\text{n} \cdot \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{k} _n = \text{b}_n \cdot \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{k} _n = 0$$
. The unsteadiness is proportional to the eddy-turnover frequency,
$$\omega \left( k \right) = \lambda \in _*^{1/3} k^{\left( {3 - \alpha } \right)/2}$$
, and λ is a non-dimensional unsteadiness factor. The method of selecting the modes is described in the previous reference.
$$\left\langle {u_i^2 } \right\rangle = 1$$
, i = 1, 2 and also for i = 3 in 3D.

## Keywords

Fluid Particle Inertial Range Previous Reference Structural Diffusion Kinematic Simulation
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