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- p. 9

Starting off with some initial values for the pressure field, the above equations can be solvedto obtain U and V.

However, these will not, in general, satisfy the continuity equation.

Now suppose we add corrections U, V, P to the velocities and pressure so that thecorrected variables

U = U+ U V = V + V P = P+ P (7)

satisfy both the momentum and continuity equations.

By substituting these expressions into the discretized momentum and continuity equations,we now attempt to solve for the corrections U, V, and P.

Substituting these new values into the discretized momentum equations gives:

(Ue + U

e) =X aui

aue(Ui + U

i) + Du(PP PE) + Du(P

P P

E) + Su (8a)

(Vn + V

n) =X avi

avn(Vi + V

i ) + Dv(PP PN) + Dv(P

P P

N) + Sv (8b)

- p. 10

Subtracting equations (6) from these, gives

Ue =X aui Ui

aue+ Du(P

P P

E) (9a)

Vn =X avi Vi

avn+ Dv(P

P P

N) (9b)

which provides relations between the corrections to the pressure and velocity that ensure

the momentum equations are still satisfied.

To simplify the analysis, we now split the velocity and pressure corrections into two parts:U = U

1+ U

2etc. Then we can write

Ue1 + U

e2 =X aui Ui

aue+ Du(P

P1 P

E1) + Du(P

P2 P

E2) (10a)

Vn1 + V

n2 =X avi Vi

avn+ Dv(P

P1 P

N1) + Dv(P

P2 P

N2) (10b)

- p. 11

In the SIMPLE scheme, the two parts of the corrections are defined such that

Ue1 = Du(P

P1 P

E1) U

e2 =X aui Ui

aue+ Du(P

P2 P

E2) (11a)

Vn1 = Dv(P

P1 P

N1) V

n2 = X avi V

i

avi+ Dv(P

P2 P

N2) (11b)

Note that the choice of this split ensures a very simple linkage between U1

, V1

and P1

values. The more complicated term involving the summation of velocity corrections overneighbouring nodes is put into the second part of the correction.

Since we want the corrected velocities to satisfy continuity, we now examine the discretizedcontinuity equation. Integrating the continuity equation over the pressure control volumeleads to

(eU

e wU

w)y + (nV

n sV

s )x = 0 (12)

or

(eU

e wU

w)y + (nV

n sV

s)x = Sm (13)

where Sm = (eUe wUw)y + (nVn sVs)x is simply the mass imbalance arisingfrom the original U, V field.

- p. 12

As a first approximation, we now assume that the second part of the corrections (U2

, V2

, P2

)

may be neglected. Substituting equations (11) into the discretized continuity equation thenresults in

y

[Du]e (P

P1 P

E1) [Du]w (P

W1 P

P1)

+ x

[Dv ]n (P

P1 P

N1) [Dv]s (P

S1 P

P1)

= Sm (14)

The above equation can then simply be written in the form

apP

P1 = aeP

E1 + awP

W1 + anP

N1 + asP

S1 + Su (15)

where

ae = y[Du]e aw = y[Du]w an = x[Dv]n as = x[Dv ]s

ap = ae + aw + an + as Su = Sm

This set of linear equations for the pressure correction P1

can be solved, using methods

considered in earlier lectures.

The corresponding corrections to the velocities, U1

and V1

, are then obtained from

equations (11), and the pressure and velocities are thus all updated.

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Since the SIMPLE scheme is typically embedded within an iterative flow solver, the wholealgorithm is:

1. Start with initial values

2. Solve momentum equations to get U, V

3. Calculate coefficients and source terms for the pressure correction equation (14)

4. Solve for the pressure corrections P1

5. Calculate velocity corrections U1

, V1

6. Update P, U, V

7. Repeat from step 2 until solution has converged

- p. 14

The PISO Scheme

One problem with the SIMPLE scheme as described is that it can be rather slow to converge.

One reason for this is the neglect of the corrections U2

, V2

and P2

.

There are a number of variants on the SIMPLE scheme, designed to improve convergencespeeds.

One such variant, which is very similar in structure, and may sometimes lead to better

convergence, is the PISO (Pressure Implicit solution by Split Operator method) schemeproposed by Issa (1982).

In the PISO scheme, the same decomposition of corrections to the velocity and pressure ismade as in the SIMPLE scheme, and U

1, V

1and P

1are computed as described earlier.

However, a second corrector stage is now added, in an attempt to account for the neglectedU2

, V2

and P2

in the first stage.

In this second corrector stage the second parts of equations ( 11) are approximated by

U

e2 =X aui Ui1

aue + Du(P

P2P

E2) (16a)

Vn2 =X avi Vi1

avi+ Dv(P

P2 P

N2) (16b)

- p. 15

These expressions are then substituted into the discretized continuity equation, which nowbecomes

(eU

e2 wU

w2)y + (nV

n2 sV

s2)x = 0 (17)

We thus get

y

[Du]e (P

P2P

E2)

[Du]w (P

W2P

P2)

+ x

[Dv ]n (P

P2 P

N2) [Dv ]s (P

S2 P

P2)

=

y

"X aui Ui1

aup

#w

y

"X aui Ui1

aup

#e

+ x

"X avi Vi1

avp

#n

x

"X avi Vi1

avp

#s

(18)

This can again be written in the generic form

apPP2 = aePE2 + awPW2 + anPN2 + asPS2 + Su (19)

where the ai coefficients are the same as those in the equation for the P1 correction, and the

source term Su is now given by the right hand side of equation (18) above.

In the PISO scheme, therefore, between steps 6 and 7 of the SIMPLE algorithm, the sourceterms Su of equation (19) are calculated, the equation for P2 is solved and the result used as

a second correction to the pressure.