| (79) |
The discrete continuity equation is written:
|
| (79) |
The uncorrected face mass flow rate 
is computed after the discrete momentum equations have been solved. The mass flow correction 
is required to satisfy continuity. The following sections describe how the discretized pressure correction equation is obtained, both at interior faces and at boundaries.
The uncorrected mass flow rate at an interior face may be written in terms of the cell variables as follows:
| (80) |
where 
and 
are the cell velocities after the discrete momentum equations have been solved. 
is the grid flux. 
is the Rhie-and-Chow-type dissipation at the face, given by:
| (81) |
where:
| (82) |
and 
are the volumes for cell-0 and cell-1, respectively. 
and 
are the average of the momentum coefficients for all components of momentum for cells 0 and 1, respectively. 
and 
are the cell pressures from the previous iteration. 
is the volume-weighted average of the cell gradients of pressure, 
and 
. The vector 
is given by Eqn. 350.
For compressible flow, the density correction 
is introduced. Noting that:
| (83) |
where the subscript "fn" denotes the face-normal component, we define:
| (84) |
where 
and 
are the cell pressure corrections, and:
| (85) |
where:
| (86) |
Since we choose to neglect the second-order term 
, these equations can be combined to define the mass flow correction as:
| (87) |
The discrete pressure correction equation is obtained from Eqns. (79) and (87) and is written in coefficient form as:
| (88) |
The residual r is simply the net mass flow into the cell:
| (89) |
In the cases where the boundary velocity is specified, such as wall, symmetry and inlet boundaries, the value of 
is calculated directly from the known velocity 
on boundaries as:
| (90) |
For these boundaries, a Neumann condition is used for the pressure correction:
| (91) |
and the mass flux corrections are zero.
On a specified-pressure boundary (stagnation inlet and pressure outlet), the pressure corrections will not be zero. The uncorrected boundary mass flux is given by:
| (92) |
where 
is the boundary velocity. The dissipation 
is defined as:
| (93) |
where:
| (94) |
Similar to Eqn. 87, we can postulate a mass flux correction of the form:
| (95) |
For supersonic inflow, 
and 
, and
.
For supersonic outflow, 
and 
, and Eqn. 95 becomes:
| (96) |
For subsonic outflow, 
and 
, and Eqn. 95 becomes:
| (97) |
For subsonic inflow, 
. However, even though this simplifies Eqn. 95 somewhat, it is still necessary to eliminate 
from the equation.
Noting that:
| (98) |
this is combined with Eqn. 95 to obtain:
| (99) |
and:
| (100) |
A similar approach is followed for the free-stream boundary type, with slightly different results.