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Streamline diffusion

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

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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:

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

Categories: Fluid dynamics | Diffusion

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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