Uniform Diffusion


This page describes a “process”. Refer to Physical Processes for general information about processes and their implementation.

The molecular diffusion process represents transport along a gradient of some quantity $q$, also known as Fickian diffusion. The diffusion coefficient $D$ is assumed to be constant in space and time, resulting in

\[\frac{∂q}{∂t} = … + D \frac{∂²q}{∂x_i²} \,.\]

The process is discretized with

\[\begin{aligned} f_i(\hat{q}^{κ₁κ₂ζ}) = &D \left( ∂₁(κ₁)² + ∂₂(κ₂)² \right) \hat{q}^{κ₁κ₂ζ} + \\ &D ∂₃\left(ζ\right) \left( ∂₃\left(ζ⁺\right) \left(\hat{q}^{κ₁κ₂(ζ+δζ)} - \hat{q}^{κ₁κ₂ζ}\right) − ∂₃\left(ζ¯\right) \left(\hat{q}^{κ₁κ₂ζ} − \hat{q}^{κ₁κ₂(ζ−δζ)}\right) \right) \,. \end{aligned}\]

This requires boundary conditions for $q$ at $ζ=0$ and $ζ=1$.

For a vector quantity like the velocity/momentum field, an instance of the diffusion process should be included for each component $u_i$.

MolecularDiffusion(field, diffusivity)

Diffusive transport of a scalar quantity $q$ with a diffusion coefficient $D$ that is constant in space and time.


  • field::Symbol: The name of the quantity $q$.
  • diffusivity::Real: The diffusion coefficient $D$.

Contributions to Budget Equations

Contribution to the instantaneous momentum equation:

\[\frac{\partial}{\partial t} u_i = … + \nu \frac{\partial^2 u_i}{\partial x_j^2}\]

Contribution to the mean momentum equation:

\[\frac{\partial}{\partial t} \overline{u_i} = … + \nu \frac{\partial^2 \overline{u_i}}{\partial x_j^2}\]

Contribution to the turbulent momentum equation:

\[\frac{\partial}{\partial t} u_i^\prime = … + \nu \frac{\partial^2 u_i^\prime}{\partial x_j^2}\]

Contribution to the mean kinetic energy equation:

\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} &= … + \nu \frac{\partial^2}{\partial x_j^2} \frac{\overline{u_i}^2}{2} - \frac{\partial \overline{u_i}}{\partial x_j} \frac{\partial \overline{u_i}}{\partial x_j} \\ &= … + \frac{\partial}{\partial x_j} \left( 2 \nu \overline{u_i} \overline{S_{ij}} \right) - 2 \nu \overline{S_{ij}} \, \overline{S_{ij}} \end{aligned}\]

Contribution to the turbulent kinetic energy equation:

\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} &= … + \nu \frac{\partial^2}{\partial x_j^2} \frac{\overline{u_i^\prime u_i^\prime}}{2} \underbrace{ - \nu \overline{ \frac{\partial u_i^\prime}{\partial x_j} \frac{\partial u_i^\prime}{\partial x_j} } }_{\text{pseudo-dissipation}} \\ &= … + \nu \frac{\partial^2}{\partial x_j^2} \frac{\overline{u_i^\prime u_i^\prime}}{2} + \nu \frac{\partial^2 \overline{u_i^\prime u_j^\prime}}{\partial x_i \partial x_j} - 2 \nu \overline{S_{ij}^\prime S_{ij}^\prime} \\ &= … + \frac{\partial}{\partial x_j} \left( 2 \nu \overline{u_i^\prime S_{ij}^\prime} \right) \underbrace{ - 2 \nu \overline{S_{ij}^\prime S_{ij}^\prime} }_{\text{dissipation}} \end{aligned}\]