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Velocity-pressure coupling

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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: <br>
 +
# 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.
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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 pressure-velocity coupling.
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==Formulation==
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we have at each cell descretised equation in this form, <br>
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:<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>
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According to [[Rhie-Chow interpolation]], we have <br>
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:<math> \vec v_P  = \frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }} - \frac{{\nabla p}}{{a_p V}} </math> <br>
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For continuity : <br>
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:<math> \sum\limits_{faces} {\vec v_f  \bullet \vec A}  = 0 </math> <br>
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so we get: <br>
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:<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>
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this gives us: <br>
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:<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>
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defining <math> H = \sum\limits_{neighbours} {a_l } \vec v_l </math> <br>
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:<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>
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from this a pressure correction equation could be formed as: <br>
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:<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>
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This is a poisson equation.
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Here the gradients could be used from previous iteration.
 +
 
 +
 
 +
==SIMPLE==
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See [[SIMPLE algorithm]]
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== SIMPLER==
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See [[SIMPLER algorithm]]
 +
 
 +
== SIMPLEC==
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See [[SIMPLEC algorithm]]
 +
 
 +
== PISO ==
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See [[PISO algorithm]]
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 +
 
 +
----
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<i> Return to: <br>
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# [[Numerical methods | Numerical Methods]]
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# [[Solution of Navier-Stokes equation]]
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</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:

  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.

Contents

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.


SIMPLE

See SIMPLE algorithm

SIMPLER

See SIMPLER algorithm

SIMPLEC

See SIMPLEC algorithm

PISO

See PISO algorithm


Return to:

  1. Numerical Methods
  2. Solution of Navier-Stokes equation

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