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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:

  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



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

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