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Streamline diffusion



Given an advection-diffusion equation, streamline diffusion refers to all diffusion going on along the advection direction.

Additional recommended knowledge

Explanation

If we take an advection equation, for simplicity of writing we have assumed \nabla\cdot\mathbf{F}=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 on 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))
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Streamline_diffusion". A list of authors is available in Wikipedia.
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