Structural Diffusion in 2D and 3D Random Flows

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_* k^−α 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.