CFD Online Logo CFD Online URL
Home > Wiki > Rhie-Chow interpolation

Rhie-Chow interpolation

From CFD-Wiki

Revision as of 11:28, 23 October 2005 by Zxaar (Talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

we have at each cell descretised equation in this form,

 a_p \vec v_P  = \sum\limits_{neighbours} {a_l } \vec v_l  - \frac{{\nabla p}}{V}  ;

we have

 \vec v_P  = \frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }} - \frac{{\nabla p}}{{a_p V}}

For continuity :

 \sum\limits_{faces} {\vec v_f  \bullet \vec A}  = 0

so we get:

\left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face}  - \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face}  = 0

this gives us:

 \left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face}  = \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face}

defining  H = \sum\limits_{neighbours} {a_l } \vec v_l

 \left[ {\frac{1}{{a_p }}H} \right]_{face}  = \left[ {\frac{1}{{a_p }}\frac{{\nabla p}}{V}} \right]_{face}

from this a pressure correction equation could be formed as:

 \left[ {\frac{1}{{a_p }}H} \right]_{face}  - \left[ {\frac{1}{{a_p }}\frac{{\nabla p^* }}{V}} \right]_{face}  = \left[ {\frac{1}{{a_p }}\frac{{\nabla p^' }}{V}} \right]_{face}

This is a poisson equation.

Here the gradients could be used from previous iteration.

My wiki