## 5.11Boundary ﬂuxes

The equations in Sec. 5.10 are used to compute the velocity , ﬂux and pressure based on conservation of mass and momentum. At boundaries, the ﬂux corrector Eq. (5.18 ) must calculate in a manner consistent with and and their respective boundary conditions.

For example, at an impermeable stationary wall the calculated ﬂux must be , consistent with the no-slip condition . At boundary faces, and must be compatible to evaluate the correct according to Eq. (5.18 ). The ﬁgure shows the fundamental boundary conditions from Sec. 4.3 and corresponding ﬂux evaluations. At the inlet and wall boundaries, is directly assigned from the boundary velocity by within the ﬂux corrector Eq. (5.18 ).

In the absence of body forces, the boundary condition is commonly applied, as discussed in Sec. 4.4 . The ﬂux is then equivalent to assigning in Eq. (5.18 ).

At the outlet, is not prescribed since is not a ﬁxed value condition. Instead, is evaluated from Eq. (5.18 ) using taken from cells adjacent to the boundary and calculated on the boundary.

### Fluxes with a body force

When a body force is present in the momentum equation, the gradient condition for , e.g. at an inlet or wall, in principle becomes , as discussed in Sec. 4.4 . The precise details of the boundary condition in fact depend on how is incorporated within the coupling algorithm, discussed below. One approach, illustrated by the algorithm in Sec. 5.10 , is to include the body force within in Eq. (5.16 ). In that case, the assignment cannot be valid, so is adopted instead.

With established, the condition is calculated based on the known by inverting Eq. (5.18 ). This approach causes to include a contribution from viscous stresses, as in Eq. (4.5 ), which may cause instability.

To avoid this problem, is omitted from in Eq. (5.16 ), appearing instead as an extra term in the other equations in Sec. 5.10 , e.g. the ﬂux corrector Eq. (5.18 ) which becomes (5.20)
Assigning at boundaries where is ﬁxed satisﬁes Eq. (5.20 ) when . This is the condition ignoring viscous stresses described in Sec. 4.4 .
Notes on CFD: General Principles - 5.11 Boundary ﬂuxes 