Streamline diffusion

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

Explanation

In an advection equation, for simplicity assume: $\nabla\cdot{\bold u}=0$ , and $||{\mathbf {u}}||=1$

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

Adding a diffusion term, again for simplicity, assume the diffusion to be constant over the entire field.

D\nabla ^{2}\psi

,

Giving an equation of the form:

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

The equation may now be rewritten in the following form:

{\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

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

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

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.