186
7. Solution of the Navier-Stokes Equations
Let us now see how the artificial compressibility approach may be used to
derive a pressure-correction equation similar to those described in preceding
sections.
We may recall from the previous section that from the implicitly discretized momentum equations, one can demonstrate the following relationship between pui and the pressure gradient (see Eq. (7.32)):
Instead of Eq. (7.70) we postulate the following expression for ( ~ u ~ ) ~ + l :
From Eq. (7.71) we find that:
If we further introduce the pressure correction
= pn+l - pn ,
we may finally rewrite the Eq. (7.72) as:
Obviously, if the densities in ( p ~ i ) ~ + '
and ( p ~ f ) ~ + '
are the same, this implies
a velocity correction of the same form as in the SIMPLE method, see Eq.
(7.37), i.e.:
By substituting expression (7.75) into Eq. (7.69), we obtain:
which can, with the help of Eq. (7.68), be further rearranged to give:
7. Solution of the Navier-Stokes Equations
Let us now see how the artificial compressibility approach may be used to
derive a pressure-correction equation similar to those described in preceding
sections.
We may recall from the previous section that from the implicitly discretized momentum equations, one can demonstrate the following relationship between pui and the pressure gradient (see Eq. (7.32)):
Instead of Eq. (7.70) we postulate the following expression for ( ~ u ~ ) ~ + l :
From Eq. (7.71) we find that:
If we further introduce the pressure correction
= pn+l - pn ,
we may finally rewrite the Eq. (7.72) as:
Obviously, if the densities in ( p ~ i ) ~ + '
and ( p ~ f ) ~ + '
are the same, this implies
a velocity correction of the same form as in the SIMPLE method, see Eq.
(7.37), i.e.:
By substituting expression (7.75) into Eq. (7.69), we obtain:
which can, with the help of Eq. (7.68), be further rearranged to give: