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