5.11 Boundary fluxes

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

For example, at an impermeable stationary wall the calculated flux must be eqn, consistent with the no-slip condition eqn. At boundary faces, eqn and eqn must be compatible to evaluate the correct eqn according to Eq. (5.18 ).

PICT\relax \special {t4ht=

The figure shows the fundamental boundary conditions from Sec. 4.3 and corresponding flux evaluations. At the inlet and wall boundaries, eqn is directly assigned from the boundary velocity eqn by eqn within the flux corrector Eq. (5.18 ).

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

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

Fluxes with a body force

When a body force eqn is present in the momentum equation, the gradient condition for eqn, e.g. at an inlet or wall, in principle becomes eqn, as discussed in Sec. 4.4 . The precise details of the boundary condition in fact depend on how eqn is incorporated within the coupling algorithm, discussed below.

PICT\relax \special {t4ht=

One approach, illustrated by the algorithm in Sec. 5.10 , is to include the body force eqn within eqn in Eq. (5.16 ). In that case, the assignment eqn cannot be valid, so eqn is adopted instead.

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

To avoid this problem, eqn is omitted from eqn 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

 ℬ S H(u)- jSfj- r p + S b- : f f A f A f n f f A f \relax \special {t4ht=
Assigning eqn at boundaries where eqn is fixed satisfies Eq. (5.20 ) when eqn . This is the condition ignoring viscous stresses described in Sec. 4.4 .
Notes on CFD: General Principles - 5.11 Boundary fluxes