# Velocity-pressure coupling

(Difference between revisions)
 Revision as of 00:07, 24 October 2005 (view source)Zxaar (Talk | contribs)← Older edit Latest revision as of 05:50, 24 October 2005 (view source)Zxaar (Talk | contribs) Line 8: Line 8: ==Formulation== ==Formulation== we have at each cell descretised equation in this form,
we have at each cell descretised equation in this form,
- :$a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{{\nabla p}}{V}$ ;
+ :$a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{{\nabla p}}{V}$ ;  Where V = Volume of cell.
According to [[Rhie-Chow interpolation]], we have
According to [[Rhie-Chow interpolation]], we have
:$\vec v_P = \frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }} - \frac{{\nabla p}}{{a_p V}}$
:$\vec v_P = \frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }} - \frac{{\nabla p}}{{a_p V}}$
Line 15: Line 15: :$\sum\limits_{faces} {\vec v_f \bullet \vec A} = 0$
:$\sum\limits_{faces} {\vec v_f \bullet \vec A} = 0$
so we get:
so we get:
- :$\left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} - \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face} = 0$
+ :$\sum\limits_{faces} \left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} - \sum\limits_{faces} \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face} = 0$
this gives us:
this gives us:
- :$\left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} = \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face}$
+ :$\sum\limits_{faces} \left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} = \sum\limits_{faces} \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face}$
defining $H = \sum\limits_{neighbours} {a_l } \vec v_l$
defining $H = \sum\limits_{neighbours} {a_l } \vec v_l$
- :$\left[ {\frac{1}{{a_p }}H} \right]_{face} = \left[ {\frac{1}{{a_p }}\frac{{\nabla p}}{V}} \right]_{face}$
+ :$\sum\limits_{faces} \left[ {\frac{1}{{a_p }}H} \right]_{face} = \sum\limits_{faces} \left[ {\frac{1}{{a_p }}\frac{{\nabla p}}{V}} \right]_{face}$
from this a pressure correction equation could be formed as:
from this a pressure correction equation could be formed as:
- :$\left[ {\frac{1}{{a_p }}H} \right]_{face} - \left[ {\frac{1}{{a_p }}\frac{{\nabla p^* }}{V}} \right]_{face} = \left[ {\frac{1}{{a_p }}\frac{{\nabla p^' }}{V}} \right]_{face}$
+ :$\sum\limits_{faces} \left[ {\frac{1}{{a_p }}H} \right]_{face} - \sum\limits_{faces} \left[ {\frac{1}{{a_p }}\frac{{\nabla p^* }}{V}} \right]_{face} = \sum\limits_{faces} \left[ {\frac{1}{{a_p }}\frac{{\nabla p^' }}{V}} \right]_{face}$
- This is a poisson equation. + This is a poisson equation. Here the gradients could be used from previous iteration. Here the gradients could be used from previous iteration.

## 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:

1. SIMPLE
2. SIMPLER
3. SIMPLEC
4. 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.

## Formulation

we have at each cell descretised equation in this form,

$a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{{\nabla p}}{V}$ ; Where V = Volume of cell.

According to Rhie-Chow interpolation, 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:

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

this gives us:

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

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

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

from this a pressure correction equation could be formed as:

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

This is a poisson equation.

Here the gradients could be used from previous iteration.