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SIMPLE

  • Assume Navier-Stokes equations for steady-state incompressible fluid flow
  • Mass conservation equation (continuity equation)

    (11.8)

     

  • Momentum conservation equation (velocity equation)

    (11.9)

     

  • When we discretize momentum conservation equation, we get:

    (11.10)

     

  • Matrix linear system:

    (11.11)

     

  • Matrix  is diagonal, with diagonal of original matrix 
  •  is a vector corresponding to ()
  • Rewriting () with substitution from ():

    (11.12)

     

  • Express from equation (), i.e. momentum correction equation :

    (11.13)

     

  • To momentum correction equation () applying continuity equation () getting:

    (11.14)

     

  • SIMPLE algorithm description:
  1. Computation starts, initial  and  are chosen
  2. Time step starts
  3. Boundary conditions are updated
  4. Linear system is solved 
  5. Relaxation 
  6. Obtained new 
  7. Building matrix  and vector 
  8. Mass flow over cell faces is computed
  9. Pressure correction is applied ()
  10. Relaxation 
  11. Obtained new 
  12. Mass flow over cell faces is corrected
  13. Momentum correction is applied ()
  14. Obtained new corrected 
  15. Boundary conditions are updated
  16. Steps 5 – 15 are repeated with respect to nCorrectors value
  17. Time step ends

  • Source code of SIMPLE algorithm is implemented e.g. for solver simpleFoam:
    $FOAM_SOLVERS/incompressible/simpleFoam