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 ${\displaystyle ||{\mathbf {u}}||=1}$

${\displaystyle {\frac {\partial \psi }{\partial t}}+{\mathbf {u}}\cdot \nabla \psi =0.}$

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

${\displaystyle D\nabla ^{2}\psi }$,

Giving us an equation of the form:

${\displaystyle {\frac {\partial \psi }{\partial t}}+{\mathbf {u}}\cdot \nabla \psi +D\nabla ^{2}\psi =0}$

We may now rewrite the equation on the following form:

${\displaystyle {\frac {\partial \psi }{\partial t}}+{\mathbf {u}}\cdot \nabla \psi +{\mathbf {u}}({\mathbf {u}}\cdot D\nabla ^{2}\psi )+(D\nabla ^{2}\psi -{\mathbf {u}}({\mathbf {u}}\cdot D\nabla ^{2}\psi ))=0}$

The term below is called streamline diffusion.

${\displaystyle {\mathbf {u}}({\mathbf {u}}\cdot D\nabla ^{2}\psi )}$

Crosswind diffusion

Any diffusion orthogonal to the streamline diffusion is called crosswind diffusion, for us this becomes the term:

${\displaystyle (D\nabla ^{2}\psi -{\mathbf {u}}({\mathbf {u}}\cdot D\nabla ^{2}\psi ))}$