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SIMPLE algorithm

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SIMPLE [Semi-Implicit Method for Pressure-Linked Equations]

If a steady-state problem is being solved iteratively, it is not necessary to fully resolve the linear pressure-velocity coupling, as the changes between consecutive solutions are no longer small. The SIMPLE algorithm:

  • An approximation of the velocity field is obtained by solving the momentum equation. The pressure gradient term is calculated using the pressure distribution from the previous iteration or an initial guess.
  • The pressure equation is formulated and solved in order to obtain the new pressure distribution.
  • Velocities are corrected and a new set of conservative fluxes is calculated.

SIMPLE Solver Algorithm

The algorithm may be summarized as follows:

The basic steps in the solution update are as follows:

  1. Set the boundary conditions.
  2. Compute the gradients of velocity and pressure.
  3. Solve the discretized momentum equation to compute the intermediate velocity field .
  4. Compute the uncorrected mass fluxes at faces .
  5. Solve the pressure correction equation to produce cell values of the pressure correction .
  6. Update the pressure field:  p^{k + 1}  = p^k  + urf \bullet p^' where urf is the under-relaxation factor for pressure.
  7. Update the boundary pressure corrections  p_b^' .
  8. Correct the face mass fluxes: \dot m_f^{k + 1}  = \dot m_f^*  + \dot m_f^'
  9. Correct the cell velocities:  \vec v^{k + 1}  = \vec v^*  - \frac{{Vol\nabla p^' }}{{\vec a_P^v }}  ; where  {\nabla p^' } is the gradient of the pressure corrections,  {\vec a_P^v } is the vector of central coefficients for the discretized linear system representing the velocity equation and Vol is the cell volume.
  10. Update density due to pressure changes.

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