Overview of pressure correction

Chapter 7 Overview of pressure correction

methods

7.1 SIMPLE algorithm

The linearized Navier Stokes equation written in operator form is

(7.1) Within the SIMPLE algorithm the solution is seeking at each time step in

u D Au C Bp C C

form of the loop: Calculation of the auxiliary velocity

D C u .m 1/ Au Bp C C (7.2)

Calculation of the pressure correction

(7.3) Calculation of the velocity correction

7.2 P ISO algorithm

In the SIMPLE algorithm we neglected the term rAu 0 (see 6.25). In PISO algorithm this term is taken into account. Actually the term rAu 0 can not

be calculated before the velocity correction is computed. Therefore, the term is taken into account in an iterative way.

7.2.1 First iteration

The term Au 0 is neglected, i.e. Au 0 D 0 . The pressure correction is found from the Poisson equation

r Bp 0 D r u

The velocity correction is then u 0 D Bp 0 (7.7)

7.2.2 Second iteration

The pressure correction within the second iteration is found from the Poisson equation

r Bp 00 D r Au 0 (7.8) The velocity correction within the second iteration is then

u 00 D Au 0 C Bp 00 (7.9)

7.2.3 Correction

Corrected velocities and pressure are u .m/ D u C u 0 C u 00 ;p .m/ D p .m 1/ C p 0 C p 00 (7.10)

Using formula derived above

r Bp 0 D r u ;u 0 D Bp 0 ;u 00 D Au 0 C Bp 00 ; rBp 00 D r Au 0

it is easy to prove that the velocity u .m/ satisfies the continuity equation.

Now we prove the equation u .m/ D Au .m/ C Bp .m/ C C :

(7.11) Since

C 0 C 00 D C 0 C 00 u .m 1/ u u A.u u u / C B.p C p 0 C p 00 /CC

0 0 00 0 u 00 D Au C Bp C C; u D Bp ;u D Au C Bp the equation (7.11) is not satisfied. The residual is Au 00 . The residual can be

.m 1/

reduced within next iterations. However, usually, PISO algorithm uses only two iterations.

7.2.4 Summary

The PISO algorithm can be summarized as follows: Calculation of the auxiliary velocity

u .m 1/ D Au C Bp CC (7.12)

Calculation of the pressure correction p 0 :

(7.13) Calculation of the velocity correction u 0

r 0 Bp D r u

(7.14) Calculation of the pressure correction p 00 :

0 u 0 D Bp

(7.15) Calculation of the velocity correction u 00

(7.17) Both algorithms PISO and SIMPLE are widely used in CFD codes.

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7.3 SIMPLEC algorithm

Another way to hold the term rAu 0 (see 6.25) is implemented in the SIM- PLEC algorithm. The velocity correction at ˛ th control volume u 0 ˛ can

be calculated using the interpolation over N adjacent control volumes:

where ˇ is the number of adjacent control volumes around the control volume with the number ˛.

The equation for the velocity correction is u 0 D Au 0 C Bp 0 (7.20)

Substitution of (7.19) into (7.21) yields

The pressure correction equation

r Bp 0 D r u

r Au 0

takes the form

The computational steps are the same as these in SIMPLE algorithm with only difference that the equation (7.24) is solved instead of (7.3).