Streamline diffusion

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


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.


Giving us an equation of the form:

\frac{\partial\psi}{\partial t}
+{\bold u}\cdot\nabla\psi

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))

The term below is called streamline diffusion.

{\bold u}({\bold u}\cdot D\nabla^2\psi)

Crosswind diffusion[edit]

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))