84
4. Finite Volume Methods
Expression (4.30) is exact on any grid, since the velocity u,,, is constant along
the face. The flux approximation is then:
The linear interpolation coefficient A, is defined by Eq. (4.14). Analogous
expressions for the fluxes through the other CV faces produce the following
coefficients in the algebraic equation for the case of UDS:
Ak=min(me,O.);
Ah=min(m,,O.),
Ah = min(m,, 0.) ;
A: = min(ms, 0.) ,
A",
-(AA", + A h + Ah + A;) .
For the CDS case, the coefficients are:
The expression for ACp follows from the continuity condition:
which is satisfied by the velocity field. Note that m, and A, for the CV centered around node P are equal to -me and 1 - A, for the CV centered around
node W, respectively. In a computer code the mass fluxes and interpolation
factors are therefore calculated once and stored as me, m, and A,, A, for
each CV.
The diffusive flux integral is evaluated using the midpoint rule and CDS
approximation of the normal derivative; this is the simplest and most widely
used approximation:
Note that XE =
+xi) and xp = :(xi +xi-l), see Fig. 4.2. The diffusion
coefficient r is assumed constant; if not, it could be interpolated linearly
between the nodal values at P and E. The contribution of the diffusion term
to the coefficients of the algebraic equation are: