# 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.

## 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.