7.5 Solution Methods for the Navier-Stokes Equations
191
fluxes. This is because the matrices may not be diagonally dominant; these
equations are best solved using deferred correction approach described in
Sect. 5.6. In this method, the flux is expressed as:
CDS - U~,DS)m-l
FC w = me~y:S + k e ( u i , ,
1
(7.87)
where superscripts CDS and UDS denote approximation by central and upwind differences, respectively (see Sect. 4.4). The term in brackets is evaluated
using values from the previous iteration while the matrix is computed using
the UDS-approximation. At convergence, the UDS contributions cancel out,
leaving a CDS solution. This procedure usually converges a t approximately
the rate obtained for a pure upwind approximation.
The two schemes may also be blended; this is achieved by multiplying
the explicit part (the term in brackets in Eq. (7.87)) by a factor 0 5 / ? 5 1.
This practice can remove the oscillations obtained with central differences on
coarse grids. However, it improves the esthetics of the results a t the cost of decreasing the accuracy. Blending may be used locally, e.g. to allow calculation
of flows with shocks with CDS; this is preferable to applying it everywhere.
Calculation of the diffusive fluxes requires evaluation of the stresses T,,
and T ~ ,
a t the CV face 'e'. Since the outward unit normal vector a t this CV
face is i, we have:
where Se = y j - yj-1 = Ay for the u-CV and Se = ; ( ~ . j + ~
- yj-1) for the vCV. The stresses a t CV face require approximation of the derivatives; central
difference approximations lead to:
Note that T,, is evaluated a t the 'e' face of the u-CV, and T ~ ,
at the 'e' face
of the V-CV, so the indices refer t o locations on the appropriate CVs, see Fig.
7.4. Thus, une and use on the V-CV are actually nodal values of the u-velocity
and no interpolation is necessary.
At the other CV faces we obtain similar expressions. For u-CVs, we need
to approximate T,, at the faces 'e' and 'w', and T , ~ at the faces 'n' and 's'.
For v-CVs, T ~ ,
is needed at the faces 'e' and 'w' and T,, at the faces 'n' and
's'.
The pressure terms are approximated by:
191
fluxes. This is because the matrices may not be diagonally dominant; these
equations are best solved using deferred correction approach described in
Sect. 5.6. In this method, the flux is expressed as:
CDS - U~,DS)m-l
FC w = me~y:S + k e ( u i , ,
1
(7.87)
where superscripts CDS and UDS denote approximation by central and upwind differences, respectively (see Sect. 4.4). The term in brackets is evaluated
using values from the previous iteration while the matrix is computed using
the UDS-approximation. At convergence, the UDS contributions cancel out,
leaving a CDS solution. This procedure usually converges a t approximately
the rate obtained for a pure upwind approximation.
The two schemes may also be blended; this is achieved by multiplying
the explicit part (the term in brackets in Eq. (7.87)) by a factor 0 5 / ? 5 1.
This practice can remove the oscillations obtained with central differences on
coarse grids. However, it improves the esthetics of the results a t the cost of decreasing the accuracy. Blending may be used locally, e.g. to allow calculation
of flows with shocks with CDS; this is preferable to applying it everywhere.
Calculation of the diffusive fluxes requires evaluation of the stresses T,,
and T ~ ,
a t the CV face 'e'. Since the outward unit normal vector a t this CV
face is i, we have:
where Se = y j - yj-1 = Ay for the u-CV and Se = ; ( ~ . j + ~
- yj-1) for the vCV. The stresses a t CV face require approximation of the derivatives; central
difference approximations lead to:
Note that T,, is evaluated a t the 'e' face of the u-CV, and T ~ ,
at the 'e' face
of the V-CV, so the indices refer t o locations on the appropriate CVs, see Fig.
7.4. Thus, une and use on the V-CV are actually nodal values of the u-velocity
and no interpolation is necessary.
At the other CV faces we obtain similar expressions. For u-CVs, we need
to approximate T,, at the faces 'e' and 'w', and T , ~ at the faces 'n' and 's'.
For v-CVs, T ~ ,
is needed at the faces 'e' and 'w' and T,, at the faces 'n' and
's'.
The pressure terms are approximated by: