# Velocity-pressure coupling

### From CFD-Wiki

(3 intermediate revisions not shown) | |||

Line 6: | Line 6: | ||

provide an useful means of doing this for segregated solvers. However it is possible to solve the system of Navier-Stokes equations in coupled manner, taking care of inter equation coupling in a single matrix. | provide an useful means of doing this for segregated solvers. However it is possible to solve the system of Navier-Stokes equations in coupled manner, taking care of inter equation coupling in a single matrix. | ||

+ | ==Formulation== | ||

+ | we have at each cell descretised equation in this form, <br> | ||

+ | :<math> a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{{\nabla p}}{V} </math> ; Where V = Volume of cell.<br> | ||

+ | According to [[Rhie-Chow interpolation]], we have <br> | ||

+ | :<math> \vec v_P = \frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }} - \frac{{\nabla p}}{{a_p V}} </math> <br> | ||

+ | |||

+ | For continuity : <br> | ||

+ | :<math> \sum\limits_{faces} {\vec v_f \bullet \vec A} = 0 </math> <br> | ||

+ | so we get: <br> | ||

+ | :<math> \sum\limits_{faces} \left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} - \sum\limits_{faces} \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face} = 0 </math> <br> | ||

+ | this gives us: <br> | ||

+ | :<math> \sum\limits_{faces} \left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} = \sum\limits_{faces} \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face} </math><br> | ||

+ | defining <math> H = \sum\limits_{neighbours} {a_l } \vec v_l </math> <br> | ||

+ | :<math> \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} </math> <br> | ||

+ | from this a pressure correction equation could be formed as: <br> | ||

+ | :<math> \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} = \sum\limits_{faces} \left[ {\frac{1}{{a_p }}\frac{{\nabla p^' }}{V}} \right]_{face} </math> <br> | ||

+ | This is a poisson equation. | ||

+ | |||

+ | Here the gradients could be used from previous iteration. | ||

Line 21: | Line 40: | ||

---- | ---- | ||

- | <i> Return to <br> | + | <i> Return to: <br> |

- | # [[Numerical methods | Numerical Methods]] </i> | + | # [[Numerical methods | Numerical Methods]] |

+ | # [[Solution of Navier-Stokes equation]] | ||

+ | </i> |

## Latest revision as of 05:50, 24 October 2005

If we consider the discretised form of the Navier-Stokes system, the form of the equations shows linear dependence of velocity on pressure and vice-versa. This inter-equation coupling is called velocity pressure coupling. A special treatment is required in order to velocity-pressure coupling. The methods such as:

- SIMPLE
- SIMPLER
- SIMPLEC
- PISO

provide an useful means of doing this for segregated solvers. However it is possible to solve the system of Navier-Stokes equations in coupled manner, taking care of inter equation coupling in a single matrix.

## Contents |

## Formulation

we have at each cell descretised equation in this form,

- ; Where V = Volume of cell.

According to Rhie-Chow interpolation, we have

For continuity :

so we get:

this gives us:

defining

from this a pressure correction equation could be formed as:

This is a poisson equation.

Here the gradients could be used from previous iteration.

## SIMPLE

See SIMPLE algorithm

## SIMPLER

## SIMPLEC

## PISO

See PISO algorithm

* Return to:
*