# Rhie-Chow interpolation

(Difference between revisions)
 Revision as of 11:28, 23 October 2005 (view source)Zxaar (Talk | contribs)← Older edit Latest revision as of 06:14, 27 August 2012 (view source)Michail (Talk | contribs) (7 intermediate revisions not shown) Line 1: Line 1: we have at each cell descretised equation in this form,
we have at each cell descretised equation in this form,
:$a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{{\nabla p}}{V}$ ;
:$a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{{\nabla p}}{V}$ ;
- we have
+ For continuity we have
- :$\vec v_P = \frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }} - \frac{{\nabla p}}{{a_p V}}$
+ :$\sum\limits_{faces} \left[ {\frac{1}{{a_p }}H} \right]_{face} = \sum\limits_{faces} \left[ {\frac{1}{{a_p }}\frac{{\nabla p}}{V}} \right]_{face}$
- For continuity :
+ where
- :$\sum\limits_{faces} {\vec v_f \bullet \vec A} = 0$
+ :$H = \sum\limits_{neighbours} {a_l } \vec v_l$
- 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. + This interpolation of variables H and ${\nabla p}$ based on coefficients $a_p$ for [[Velocity-pressure coupling | pressure velocity coupling  ]] is called Rhie-Chow interpolation. + + the Rhie-Chow interpolation is the same as adding a pressure term, which is proportional to a third derivative of the pressue + + ---- + Return to:
+ # [[Numerical methods | Numerical Methods]] + # [[Solution of Navier-Stokes equation]] +

## Latest revision as of 06:14, 27 August 2012

we have at each cell descretised equation in this form,

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

For continuity we have

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

where

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

This interpolation of variables H and ${\nabla p}$ based on coefficients $a_p$ for pressure velocity coupling is called Rhie-Chow interpolation.

the Rhie-Chow interpolation is the same as adding a pressure term, which is proportional to a third derivative of the pressue