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Pressure Velocity
Coupling


            Arvind Deshpande
Semi-Implicit Method for Pressure
Linked Equations
       Patankar and Spalding - Guess and Correct procedure
        for calculation of pressure on staggered grid
        arrangement
1.      Initial guess for velocity and pressure field.
2.      Convective mass flux per unit area F is evaluated from
        guessed velocity components.
3.      Guessed pressure field is used to solve momentum
        equations to get velocity components.
4.      Values of velocity components are substituted in
        continuity equation to get a pressure correction equation.
5.      Values of pressure and velocity are updated.
6.      The process is iterated until convergence of pressure
        and velocity fields.
4/11/2012                    Arvind Deshpande(VJTI)              2
SIMPLE
                           
     aeue   anbunb  PP  PE Ae  be
     a v   a v  P  P A  b
        n n           nb nb        P          N        n      n

     a u   a u  P  P A  b
          *
        e e
                         *
                      nb nb
                                   *
                                   P
                                              *
                                              E        e     e

     a v   a v  P  P A  b
          *
        n n
                         *
                      nb nb
                                   *
                                   P
                                             *
                                             N         n     n

     a u  u    a u  u   P  P   P  P A
        e     e
                  *
                  e           nb        nb
                                                  *
                                                  nb          P
                                                                  *
                                                                  P   E
                                                                          *
                                                                          E   e

     a v  v    a v  v   P  P   P  P A
        n     n
                  *
                  n           nb    nb
                                                  *
                                                  nb         P
                                                                  *
                                                                  P   N
                                                                          *
                                                                          N   n

     a u   a u  P  P A
          '
        e e
                         '
                      nb nb        P
                                    '
                                             E
                                              '
                                                       e

     a v   a v  P  P A
          '
        n n
                         '
                      nb nb        P
                                    '        '
                                             N         n

4/11/2012                                Arvind Deshpande(VJTI)                   3
Omit        a  '
                 u & a v
             nb nb
                                                      '
                                                   nb nb

                      Ae        An
                 de     & dn 
                      ae        an
                              
                 ue  PP'  PE' d e
                  '

                  '
                 vn    P  P d
                            P
                             '       '
                                     N        n

                 ue    u  d P  P 
                         *
                         e       e       P
                                          '
                                                    E
                                                     '


                 vn    v  d P  P 
                        *
                        n        n       P
                                          '         '
                                                    N

                 uw    u  d P  P 
                         *
                         w           w        W
                                               '
                                                         P
                                                          '


                 vs    v  d P  P 
                        *
                        s        s       S
                                          '
                                                    P
                                                     '


                      Aw        As
                 dw     & ds 
                      aw        as
4/11/2012               Arvind Deshpande(VJTI)                4
Continuity equation
uA        i 1, J                                    
                         uAi , J   vAI , j 1  vAI , j  0                                
 A u
  e    e
                *
                e                                     
                     d e PP'  PE'   w Aw u w  d w PW  PP'
                                               *         '
                                                                                                    
 A v
  n     n
                *
                n    d P    n       P
                                       '
                                            P    A v
                                                    '
                                                    N             s       s
                                                                                 *
                                                                                 s    ds   P  P   0
                                                                                                  S
                                                                                                   '
                                                                                                           P
                                                                                                            '


dAe  dAw  dAn  dAs PP' 
dAe PE'  dAw PW'  dAn PN'  dAs PS' 
u A  u A  v A  v A 
      *
                w
                                  *
                                           e
                                                             *
                                                                      s
                                                                                     *
                                                                                          n

aP P  aE P  aW P  a N P  aS P  b
      P
       '
                         E
                          '
                                               W
                                                '                 '
                                                                  N                  S
                                                                                      '       '
                                                                                              P
  4/11/2012                                             Arvind Deshpande(VJTI)                                  5
aE  dAe
      aW  dAw
      a N  dAn
      aS  dAs
      aP  aE  aW  a N  aS
                                 
      bP  u * A w  u * A e  v* A s  v* A n
       '
                                                          
      PP  PP*  PP'
                      
      ue  ue  d e PP'  PE'
            *
                               
      vn  vn  d n
            *
                      P
                       P
                        '
                            P 
                              '
                              N

4/11/2012                          Arvind Deshpande(VJTI)       6
Discussion of Pressure Correction
Equation
1. Omission of  a u &  a v
                         '                      '
                      nb nb                  nb nb

2. Semi-Implicit
3. Justification of omission
4. Mass source is useful indicator of
   convergence
5. Pressure correction equation is intermediate
   step to get correct pressure field



4/11/2012           Arvind Deshpande(VJTI)           7
Under-relaxation
 Pressure correction equation is susceptible to divergence unless
  some under-relaxation factor is used during iterative process.
 αp, αu, αv,are under relaxation factors for pressure, u-velocity and
  v-velocity. u and v are corrected values without under relaxation
  and un-1 and vn-1 are values at the end of previous iteration.


                  P new  P*   p P'
                  u new   u u  (1   u )u n 1
                  v new   v v  (1   v )v n 1


4/11/2012                    Arvind Deshpande(VJTI)                      8
Under-relaxation
 A correct choice of these factors is important for cost effective
     simulation. Large value of α leads to oscillatory behavior or even
     divergence and small value cause extremely slow convergence.
    There are no general rules for choosing the best value for α.
     Optimum values depends on nature of the problem, the number
     of grid points, grid spacing, and iterative procedures used.
    Suitable value of α can be found by experience and from
     exploratory computations for the given problem.
    Suggested values are 0.5 for α and 0.8 for αp
    X-momentum and Y-momentum equations are modified
     considering under-relaxation factors instead of applying under-
     relaxing velocity correction as velocity values are continuity
     satisfying.

4/11/2012                    Arvind Deshpande(VJTI)                  9
Under-relaxation
                                  
            aeue   anbunb  PP*  PE* Ae  be
                         *
                                                    
            ue 
                             
                     anbunb  PP*  PE* Ae  be
                         *
                                               
                                  ae

            ue  u e  
                    *
                               *
                                          
                         anbunb  PP*  PE Ae  be
                                           *
                                                        * 
                                                      ue 
                                                            
                       
                                   ae                    
                                                          

            ue  u e   u 
                    *
                                   *           *
                                                
                             anbunb  PP*  PE Ae  be    * 
                                                          ue 
                                                                
                           
                                       ae                    
                                                              
            ae
            
                                                    
                                                             a  *
               ue   anbunb  PP*  PE Ae  be  (1   ) e u e
                              *          *

                                                              
                                                     
                                                   an  *
            an
            
                          nb nb          P
                                                    *

                                                     
                                                    N   
              vn   a v  P  P An  bn  (1   )  v e
                             *            *

                                           
                                                      
4/11/2012                                     Arvind Deshpande(VJTI)   10
SIMPLE algorithm
1)      Initial guess P*,u*,v*,φ*
2)     Solve discretized momentum equations and calculate u*,v*
                                            1    *
       ue   anbunb  PP*  PE Ae  be  
     ae *          *             *

                                                 aeue

                                             1    *
        vn   anbvnb  PP*  PN An  bn  
     an *          *            *

                                                    an u n
                                               
3)      Solve pressure correction equation and calculate P’
      aP P'P  aW P'W aE P'E aS P'S aN P' N b'P
4)     Correct Pressure and velocities                      PP  PP   P PP'
                                                                  *


                                                                            
                                                            ue  u *  d e PP'  PE'
                                                                   e
                                                                                        
                                                            vn  vn  d n
                                                                  *
                                                                            P
                                                                             P
                                                                                 '
                                                                                     P 
                                                                                       '
                                                                                       N

4/11/2012                          Arvind Deshpande(VJTI)                                  11
SIMPLE algorithm
5) Solve all other discretized transport equations
                aPP  aW W  aEE  aSS  aNN  b

6) Check for convergence. If converged, stop. Otherwise set
    P*  P, u*  u, v*  v,  *  

7) Goto step 2




  4/11/2012                       Arvind Deshpande(VJTI)      12
SIMPLER (SIMPLE Revised) -
Patankar
 Discretised continuity equation is used to
  derive discretised equation for pressure,
  instead of pressure correction equation as in
  simple.
 Pressure field is obtained without correction.
 Velocities are obtained through velocity
  corrections as in SIMPLE.



4/11/2012          Arvind Deshpande(VJTI)      13
SIMPLER Algorithm
            ue      
                      a    u  be
                           nb nb
                                          
                                            Ae
                                               PP  PE               
                            ae              ae

            vn      
                      a    v  bn
                           nb nb
                                         
                                              An
                                                      
                                                 PP  PN              
                            an                an

            u   ^
                    
                      a     v  be
                           nb nb
                e
                            ae

            v   ^
                    
                      a    v  bn
                           nb nb
                n
                            an
                          
            ue  ue^  d e PP  PE            
            vn  vn
                  ^
                       A Pn      P    PN   
4/11/2012                                         Arvind Deshpande(VJTI)   14
Continuity equation
uAe  uAw   vAn  vAs   0
 A u
  e    e
              ^
                                            
                   d e PP  PE   w Aw u w  d w PW  PP        ^
                                                                                     
 A v                                                                      P  P   0
              e


  n     n
              ^
              n    dn   P  P
                                  P    A v
                                     N              s       s
                                                                  ^
                                                                  s    ds        S   P


dAe  dAw  dAn  dAs PP 
dAe PE  dAw PW  dAn PN  dAs PS                                         
u A  u A  v A  v A 
      ^
              w
                         ^
                                 e
                                              ^
                                                        s
                                                                      ^
                                                                          n

aP PP  aW P  aE PE  a N PN  aS PS  bP
            W
  4/11/2012                              Arvind Deshpande(VJTI)                           15
SIMPLER Algorithm
aE  dAe
aW  dAw
a N  dAn
aS  dAs
aP  aE  aW  a N  aS
                     
bI , J  u ^ A w  u ^ A e  v ^ A s  v ^ A n    
ue  ue
      *
               d P  P 
                e
                    '
                    p    E
                          '


vn  vn
      *
               d P  P 
               n    P
                     '   '
                         N
  4/11/2012                   Arvind Deshpande(VJTI)        16
SIMPLER algorithm
1)      Initial guess P*,u*,v*,φ*
2)      Calculate pseudo velocities u^, v^

            ue^ 
                    a    u  be
                         nb nb

                          ae

            vn 
             ^      a    v  be
                         nb nb

                          an
3)     Solve pressure equation and calculate Pressure at all points.
                         aP PP  aW P  aE PE  aS PS  aN PN  bP
                                     W

4)     Set new value of P.

               P  PP
                    *
                    P
4/11/2012                               Arvind Deshpande(VJTI)         17
SIMPLER algorithm
5)      Solve discretized momentum equations and calculate u*,v*
                 *         *     *
                                     *
                                            
              aeue   anbunb  PP  PE Ae  be
              an vn   anbvnb
                  *         *
                                  P
                                    *
                                    P    P A
                                            *
                                            N   n    bn

6)      Solve pressure correction equation and calculate P’
                 aP P'P  aW P'W aE P'E aS P'S aN P' N b'P
7)     Correct velocities

                            
            ue  ue  d e PP'  PE'
                  *
                                     
            vn  vn  d n
                  *
                            P
                             P
                              '
                                  P   '
                                        N



4/11/2012                               Arvind Deshpande(VJTI)     18
SIMPLER algorithm
8) Solve all other discretized transport equations
                aPP  aW W  aEE  aSS  aNN  b

9) Check for convergence. If converged, stop. Otherwise set
    P*  P, u*  u, v*  v,  *  

10) Goto step 2




  4/11/2012                       Arvind Deshpande(VJTI)      19
SIMPLEC (SIMPLE Consistent)
Algorithm
 Van Doormal and
  Raithby
                                                         
                                                ue  d e PP'  PE'
                                                 '
                                                                     
                                                             Ae
 Momentum      equations                       de 
  are manipulated so that                              ae   anb
  velocity      correction                       '
                                                         
                                                vn  d n PP'  PN
                                                                '
                                                                     
  equations omit terms
                                                             An
  that are less significant                     dn 
  than those omitted in                                an   anb
  SIMPLE.



4/11/2012              Arvind Deshpande(VJTI)                            20
PISO (Pressure Implicit with Spliting
of Operators) - Issa
 Developed originally for non-iterative computation of
     unsteady compressible flows.
    Adapted for iterative solution of steady state
     problems.
    Involves one predictor and two corrector steps.
     Pressure correction equation is solved twice.
    Though the method implies considerable increase in
     computational efforts it has found to be efficient and
     fast.
    Extension of SIMPLE with a further correction step to
     enhance it.
4/11/2012                 Arvind Deshpande(VJTI)          21
PISO
   P  P  P'
       **       *


   u **  u *  u '
   v  v  v'
      **    *


                      
   ue*  ue  d e PP'  PE'
    *     *
                                    
   v  v  dn
      **
      n
            *
            n         P
                       P
                        '
                            P  '
                                N   

4/11/2012                    Arvind Deshpande(VJTI)   22
PISO
                      
aeue*   anbunb  PP**  PE** Ae  be
   *          *


a v   a v  P  P A  b
      **
    n n
                             *
                          nb nb
                                           **
                                           P
                                                      **
                                                      N         n                 n

a u   a u  P  P A  b
      ***
    e e
                             **
                          nb nb
                                            ***
                                            P
                                                          ***
                                                          E             e             e

a v   a v  P  P A  b
      ***
    n n
                             **
                          nb nb
                                            ***
                                            P
                                                          ***
                                                          N             n             n

a u  u    a u  u   P  P   P
    e
          ***
          e
                     **
                     e                nb
                                                **
                                                nb
                                                           *
                                                           nb
                                                                                   ***
                                                                                   P
                                                                                                    **
                                                                                                    P
                                                                                                         ***
                                                                                                         E      PE * Ae
                                                                                                                  *
                                                                                                                        
a v  v    a v  v   P  P   P
    n
          ***
          n
                     **
                     n                nb
                                                **
                                                nb
                                                          *
                                                          nb
                                                                                  ***
                                                                                  P
                                                                                                    **
                                                                                                    P
                                                                                                         ***
                                                                                                         N      PN*
                                                                                                                  *
                                                                                                                       A
                                                                                                                         n


u u 
    ***   a u  u   d P  P 
                **            nb
                                       **
                                       nb
                                                     *
                                                     nb                       ''           ''
    e           e                                                   e        P            E
                                      ae

v   ***
          v   **               
                        anb vnb  vnb
                             **    *
                                                        d P               ''
                                                                                    PN'
                                                                                       '
                                                                                                
    n           n                                               n           P
                                   an
4/11/2012                                                  Arvind Deshpande(VJTI)                                            23
PISO
  aP PP''  aE PE''  aW PW'  a N PN'  aS PS''  bP'
                           '         '              '


  aE  dAe
  aW  dAw
  a N  dAn
  aS  dAs
  aP  aE  aW  a N  aS
        A                      A                    
        a              
                 anb unb  unb  
                       **    *
                                       
                                         anb unb  unb  
                                                 **    *
                                                                     
  bP 
   ''      w                     a e                  
        A                     A                     
       
                      **
                          
               anb vnb  vnb  
                            *
                                      
                                        anb vnb  vnb 
                                                **   *
                                                                 
        a  s                   a n                    
4/11/2012                            Arvind Deshpande(VJTI)               24
PISO algorithm
1)      Initial guess P*,u*,v*,φ*
2)      Solve discretized momentum equations and calculate u*,v*
                   *         *     *
                                       *
                                              
                aeue   anbunb  PP  PE Ae  be
                an vn   anbvnb
                    *         *
                                    P
                                      *
                                      P    P A
                                              *
                                              N     n    bn
3)      Solve pressure correction equation and calculate P’
                 aP P'P  aW P'W aE P'E aS P'S aN P' N b'P
4)     Correct Pressure and velocities
            PP *  PP  PP'
             *      *


                            
            ue*  ue  d e PP'  PE'
             *     *
                                      
            vn*  vn
             *     *
                        d P
                          n   P
                               '
                                   P    '
                                          N

4/11/2012                                 Arvind Deshpande(VJTI)   25
PISO algorithm
5) Solve second pressure correction equation and calculate P’’
                       aP P' 'P  aW P' 'W aE P' 'E aS P' 'S aN P' ' N b' 'P
6)     Correct Pressure and velocities again.

            PP***  PP  PP'  PP''
                     *



            u   ***
                                  
                       u  de P  P 
                          *            '    '
                                                         **
                                                                
                                                      anb unb  unb
                                                                 *
                                                                                   d P ''
                                                                                                PE''   
                e         e           P    E                                         e   P
                                                                ae

            v   ***      *
                                 
                       v  d e P  PN '   '
                                                  a v  nb
                                                                    **
                                                                    nb    vnb
                                                                            *
                                                                                   d P ''
                                                                                                PN'
                                                                                                   '
                                                                                                        
                n        n            P                                              n   P
                                                                an

7) Set P = P***, u = u***, v = v***
4/11/2012                                       Arvind Deshpande(VJTI)                                      26
PISO algorithm
8) Solve all other discretized transport equations
    aI , J  'I , J  aI 1, J  'I 1, J aI 1, J  'I 1, J aI , J 1 'I , J 1 aI , J 1 'I , J 1 b'I , J

9) Check for convergence. If converged, stop. Otherwise set
        P*  P, u*  u, v*  v,  *  

 10) Goto step 2




     4/11/2012                                            Arvind Deshpande(VJTI)                                      27
General Comments
 Performance of each algorithm depends on flow conditions, the
     degree of coupling between the momentum equation and scalar
     equations, amount of under relaxation and sometimes even on the
     details of the numerical techniques used for solving the algebraic
     equations.
    SIMPLE algorithm is straightforward and has been successfully
     implemented in numerous CFD procedures.
    In SIMPLE, pressure correction P’ is satisfactory for correcting
     velocities, but not so good for correcting pressure.
    SIMPLER uses pressure correction for calculating velocity
     correction only. A separate pressure equation is solved to calculate
     the pressure field.
    Since no terms are omitted to derive the discretised pressure
     equation in SIMPLER, the resulting pressure field corresponds to
     velocity field.
    The method is effective in calculating the pressure field correctly.
     This has significant advantages when solving the momentum
     equations.
4/11/2012                      Arvind Deshpande(VJTI)                  28
General Comments
 Although calculations are more in SIMPLER, convergence is
     faster and effectively computer time reduces.
    SIMPLEC and PISO have proved to be as efficient as SIMPLER
     in certain types of flows.
    When momentum equations are not coupled to a scalar
     variable, PISO algorithm showed robust convergence and
     required less computational efforts than SIMPLER and
     SIMPLEC.
    When scalar variables were closely linked to velocities, PISO
     had no significant advantage over other methods.
    Iterative methods using SIMPLER and SIMPLEC have robust
     convergence behavior in strongly coupled problems. It is still
     unclear which of the SIMPLE variant is the best for general
     purpose computation.

4/11/2012                    Arvind Deshpande(VJTI)               29

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Pressure velocity coupling

  • 1. Pressure Velocity Coupling Arvind Deshpande
  • 2. Semi-Implicit Method for Pressure Linked Equations  Patankar and Spalding - Guess and Correct procedure for calculation of pressure on staggered grid arrangement 1. Initial guess for velocity and pressure field. 2. Convective mass flux per unit area F is evaluated from guessed velocity components. 3. Guessed pressure field is used to solve momentum equations to get velocity components. 4. Values of velocity components are substituted in continuity equation to get a pressure correction equation. 5. Values of pressure and velocity are updated. 6. The process is iterated until convergence of pressure and velocity fields. 4/11/2012 Arvind Deshpande(VJTI) 2
  • 3. SIMPLE   aeue   anbunb  PP  PE Ae  be a v   a v  P  P A  b n n nb nb P N n n a u   a u  P  P A  b * e e * nb nb * P * E e e a v   a v  P  P A  b * n n * nb nb * P * N n n a u  u    a u  u   P  P   P  P A e e * e nb nb * nb P * P E * E e a v  v    a v  v   P  P   P  P A n n * n nb nb * nb P * P N * N n a u   a u  P  P A ' e e ' nb nb P ' E ' e a v   a v  P  P A ' n n ' nb nb P ' ' N n 4/11/2012 Arvind Deshpande(VJTI) 3
  • 4. Omit a ' u & a v nb nb ' nb nb Ae An de  & dn  ae an   ue  PP'  PE' d e ' ' vn  P  P d P ' ' N n ue  u  d P  P  * e e P ' E ' vn  v  d P  P  * n n P ' ' N uw  u  d P  P  * w w W ' P ' vs  v  d P  P  * s s S ' P ' Aw As dw  & ds  aw as 4/11/2012 Arvind Deshpande(VJTI) 4
  • 5. Continuity equation uA i 1, J   uAi , J   vAI , j 1  vAI , j  0   A u e e * e     d e PP'  PE'   w Aw u w  d w PW  PP' * '    A v n n * n  d P n P '  P    A v ' N s s * s  ds P  P   0 S ' P ' dAe  dAw  dAn  dAs PP'  dAe PE'  dAw PW'  dAn PN'  dAs PS'  u A  u A  v A  v A  * w * e * s * n aP P  aE P  aW P  a N P  aS P  b P ' E ' W ' ' N S ' ' P 4/11/2012 Arvind Deshpande(VJTI) 5
  • 6. aE  dAe aW  dAw a N  dAn aS  dAs aP  aE  aW  a N  aS      bP  u * A w  u * A e  v* A s  v* A n '    PP  PP*  PP'  ue  ue  d e PP'  PE' *  vn  vn  d n * P P ' P  ' N 4/11/2012 Arvind Deshpande(VJTI) 6
  • 7. Discussion of Pressure Correction Equation 1. Omission of  a u &  a v ' ' nb nb nb nb 2. Semi-Implicit 3. Justification of omission 4. Mass source is useful indicator of convergence 5. Pressure correction equation is intermediate step to get correct pressure field 4/11/2012 Arvind Deshpande(VJTI) 7
  • 8. Under-relaxation  Pressure correction equation is susceptible to divergence unless some under-relaxation factor is used during iterative process.  αp, αu, αv,are under relaxation factors for pressure, u-velocity and v-velocity. u and v are corrected values without under relaxation and un-1 and vn-1 are values at the end of previous iteration. P new  P*   p P' u new   u u  (1   u )u n 1 v new   v v  (1   v )v n 1 4/11/2012 Arvind Deshpande(VJTI) 8
  • 9. Under-relaxation  A correct choice of these factors is important for cost effective simulation. Large value of α leads to oscillatory behavior or even divergence and small value cause extremely slow convergence.  There are no general rules for choosing the best value for α. Optimum values depends on nature of the problem, the number of grid points, grid spacing, and iterative procedures used.  Suitable value of α can be found by experience and from exploratory computations for the given problem.  Suggested values are 0.5 for α and 0.8 for αp  X-momentum and Y-momentum equations are modified considering under-relaxation factors instead of applying under- relaxing velocity correction as velocity values are continuity satisfying. 4/11/2012 Arvind Deshpande(VJTI) 9
  • 10. Under-relaxation  aeue   anbunb  PP*  PE* Ae  be *  ue    anbunb  PP*  PE* Ae  be *  ae ue  u e   * *    anbunb  PP*  PE Ae  be * *   ue     ae   ue  u e   u  * * *    anbunb  PP*  PE Ae  be *   ue     ae   ae     a  * ue   anbunb  PP*  PE Ae  be  (1   ) e u e * *    an  * an  nb nb  P *   N  vn   a v  P  P An  bn  (1   )  v e * *    4/11/2012 Arvind Deshpande(VJTI) 10
  • 11. SIMPLE algorithm 1) Initial guess P*,u*,v*,φ* 2) Solve discretized momentum equations and calculate u*,v* 1    * ue   anbunb  PP*  PE Ae  be   ae * * *      aeue 1    * vn   anbvnb  PP*  PN An  bn   an * * *   an u n    3) Solve pressure correction equation and calculate P’ aP P'P  aW P'W aE P'E aS P'S aN P' N b'P 4) Correct Pressure and velocities PP  PP   P PP' *  ue  u *  d e PP'  PE' e  vn  vn  d n * P P ' P  ' N 4/11/2012 Arvind Deshpande(VJTI) 11
  • 12. SIMPLE algorithm 5) Solve all other discretized transport equations aPP  aW W  aEE  aSS  aNN  b 6) Check for convergence. If converged, stop. Otherwise set P*  P, u*  u, v*  v,  *   7) Goto step 2 4/11/2012 Arvind Deshpande(VJTI) 12
  • 13. SIMPLER (SIMPLE Revised) - Patankar  Discretised continuity equation is used to derive discretised equation for pressure, instead of pressure correction equation as in simple.  Pressure field is obtained without correction.  Velocities are obtained through velocity corrections as in SIMPLE. 4/11/2012 Arvind Deshpande(VJTI) 13
  • 14. SIMPLER Algorithm ue  a u  be nb nb  Ae PP  PE   ae ae vn  a v  bn nb nb  An  PP  PN  an an u ^  a v  be nb nb e ae v ^  a v  bn nb nb n an  ue  ue^  d e PP  PE  vn  vn ^  A Pn P  PN  4/11/2012 Arvind Deshpande(VJTI) 14
  • 15. Continuity equation uAe  uAw   vAn  vAs   0  A u e e ^     d e PP  PE   w Aw u w  d w PW  PP ^    A v P  P   0 e n n ^ n  dn P P  P    A v N s s ^ s  ds S P dAe  dAw  dAn  dAs PP  dAe PE  dAw PW  dAn PN  dAs PS  u A  u A  v A  v A  ^ w ^ e ^ s ^ n aP PP  aW P  aE PE  a N PN  aS PS  bP W 4/11/2012 Arvind Deshpande(VJTI) 15
  • 16. SIMPLER Algorithm aE  dAe aW  dAw a N  dAn aS  dAs aP  aE  aW  a N  aS      bI , J  u ^ A w  u ^ A e  v ^ A s  v ^ A n   ue  ue *  d P  P  e ' p E ' vn  vn *  d P  P  n P ' ' N 4/11/2012 Arvind Deshpande(VJTI) 16
  • 17. SIMPLER algorithm 1) Initial guess P*,u*,v*,φ* 2) Calculate pseudo velocities u^, v^ ue^  a u  be nb nb ae vn  ^ a v  be nb nb an 3) Solve pressure equation and calculate Pressure at all points. aP PP  aW P  aE PE  aS PS  aN PN  bP W 4) Set new value of P. P  PP * P 4/11/2012 Arvind Deshpande(VJTI) 17
  • 18. SIMPLER algorithm 5) Solve discretized momentum equations and calculate u*,v* * * *  *  aeue   anbunb  PP  PE Ae  be an vn   anbvnb * *  P * P  P A * N n  bn 6) Solve pressure correction equation and calculate P’ aP P'P  aW P'W aE P'E aS P'S aN P' N b'P 7) Correct velocities  ue  ue  d e PP'  PE' *  vn  vn  d n * P P ' P  ' N 4/11/2012 Arvind Deshpande(VJTI) 18
  • 19. SIMPLER algorithm 8) Solve all other discretized transport equations aPP  aW W  aEE  aSS  aNN  b 9) Check for convergence. If converged, stop. Otherwise set P*  P, u*  u, v*  v,  *   10) Goto step 2 4/11/2012 Arvind Deshpande(VJTI) 19
  • 20. SIMPLEC (SIMPLE Consistent) Algorithm  Van Doormal and Raithby  ue  d e PP'  PE' '  Ae  Momentum equations de  are manipulated so that ae   anb velocity correction '  vn  d n PP'  PN '  equations omit terms An that are less significant dn  than those omitted in an   anb SIMPLE. 4/11/2012 Arvind Deshpande(VJTI) 20
  • 21. PISO (Pressure Implicit with Spliting of Operators) - Issa  Developed originally for non-iterative computation of unsteady compressible flows.  Adapted for iterative solution of steady state problems.  Involves one predictor and two corrector steps. Pressure correction equation is solved twice.  Though the method implies considerable increase in computational efforts it has found to be efficient and fast.  Extension of SIMPLE with a further correction step to enhance it. 4/11/2012 Arvind Deshpande(VJTI) 21
  • 22. PISO P  P  P' ** * u **  u *  u ' v  v  v' ** *  ue*  ue  d e PP'  PE' * *  v  v  dn ** n * n P P ' P ' N  4/11/2012 Arvind Deshpande(VJTI) 22
  • 23. PISO   aeue*   anbunb  PP**  PE** Ae  be * * a v   a v  P  P A  b ** n n * nb nb ** P ** N n n a u   a u  P  P A  b *** e e ** nb nb *** P *** E e e a v   a v  P  P A  b *** n n ** nb nb *** P *** N n n a u  u    a u  u   P  P   P e *** e ** e nb ** nb * nb *** P ** P *** E  PE * Ae *  a v  v    a v  v   P  P   P n *** n ** n nb ** nb * nb *** P ** P *** N  PN* * A n u u  ***  a u  u   d P  P  ** nb ** nb * nb '' '' e e e P E ae v *** v  **   anb vnb  vnb ** *   d P ''  PN' '  n n n P an 4/11/2012 Arvind Deshpande(VJTI) 23
  • 24. PISO aP PP''  aE PE''  aW PW'  a N PN'  aS PS''  bP' ' ' ' aE  dAe aW  dAw a N  dAn aS  dAs aP  aE  aW  a N  aS  A   A    a    anb unb  unb   ** *    anb unb  unb   ** *   bP  ''  w  a e   A   A    **    anb vnb  vnb   *    anb vnb  vnb  ** *    a  s  a n  4/11/2012 Arvind Deshpande(VJTI) 24
  • 25. PISO algorithm 1) Initial guess P*,u*,v*,φ* 2) Solve discretized momentum equations and calculate u*,v* * * *  *  aeue   anbunb  PP  PE Ae  be an vn   anbvnb * *  P * P  P A * N n  bn 3) Solve pressure correction equation and calculate P’ aP P'P  aW P'W aE P'E aS P'S aN P' N b'P 4) Correct Pressure and velocities PP *  PP  PP' * *  ue*  ue  d e PP'  PE' * *  vn*  vn * *  d P n P ' P  ' N 4/11/2012 Arvind Deshpande(VJTI) 25
  • 26. PISO algorithm 5) Solve second pressure correction equation and calculate P’’ aP P' 'P  aW P' 'W aE P' 'E aS P' 'S aN P' ' N b' 'P 6) Correct Pressure and velocities again. PP***  PP  PP'  PP'' * u ***   u  de P  P  * ' '   **  anb unb  unb *   d P ''  PE''  e e P E e P ae v *** *   v  d e P  PN ' '   a v nb ** nb  vnb *   d P ''  PN' '  n n P n P an 7) Set P = P***, u = u***, v = v*** 4/11/2012 Arvind Deshpande(VJTI) 26
  • 27. PISO algorithm 8) Solve all other discretized transport equations aI , J  'I , J  aI 1, J  'I 1, J aI 1, J  'I 1, J aI , J 1 'I , J 1 aI , J 1 'I , J 1 b'I , J 9) Check for convergence. If converged, stop. Otherwise set P*  P, u*  u, v*  v,  *   10) Goto step 2 4/11/2012 Arvind Deshpande(VJTI) 27
  • 28. General Comments  Performance of each algorithm depends on flow conditions, the degree of coupling between the momentum equation and scalar equations, amount of under relaxation and sometimes even on the details of the numerical techniques used for solving the algebraic equations.  SIMPLE algorithm is straightforward and has been successfully implemented in numerous CFD procedures.  In SIMPLE, pressure correction P’ is satisfactory for correcting velocities, but not so good for correcting pressure.  SIMPLER uses pressure correction for calculating velocity correction only. A separate pressure equation is solved to calculate the pressure field.  Since no terms are omitted to derive the discretised pressure equation in SIMPLER, the resulting pressure field corresponds to velocity field.  The method is effective in calculating the pressure field correctly. This has significant advantages when solving the momentum equations. 4/11/2012 Arvind Deshpande(VJTI) 28
  • 29. General Comments  Although calculations are more in SIMPLER, convergence is faster and effectively computer time reduces.  SIMPLEC and PISO have proved to be as efficient as SIMPLER in certain types of flows.  When momentum equations are not coupled to a scalar variable, PISO algorithm showed robust convergence and required less computational efforts than SIMPLER and SIMPLEC.  When scalar variables were closely linked to velocities, PISO had no significant advantage over other methods.  Iterative methods using SIMPLER and SIMPLEC have robust convergence behavior in strongly coupled problems. It is still unclear which of the SIMPLE variant is the best for general purpose computation. 4/11/2012 Arvind Deshpande(VJTI) 29