Streamline diffusion

Streamline diffusion, given an advection-diffusion equation, refers to all diffusion going on along the advection direction.

Explanation

If we take an advection equation, for simplicity of writing we have assumed \(\nabla\cdot{\bold u}=0\), and \(||{\bold u}||=1\) \[ \frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi=0. \]

we may add a diffusion term, again for simplicty, we assume the diffusion to be constant over the entire field.

\[D\nabla^2\psi\],

Giving us an equation of the form:

\[ \frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi +D\nabla^2\psi =0 \]

We may now rewrite the equation on the following form:

\[ \frac{\partial\psi}{\partial t} +{\bold u}\cdot \nabla\psi +{\bold u}({\bold u}\cdot D\nabla^2\psi) +(D\nabla^2\psi-{\bold u}({\bold u}\cdot D\nabla^2\psi)) =0 \]

The term below is called streamline diffusion. \[{\bold u}({\bold u}\cdot D\nabla^2\psi)\]

Crosswind diffusion

Any diffusion orthogonal to the streamline diffusion is called crosswind diffusion, for us this becomes the term: \[ (D\nabla^2\psi-{\bold u}({\bold u}\cdot D\nabla^2\psi)) \]