7.5 Solution Methods for the Navier-Stokes Equations
189
Fig. 7.4. Control volumes for a staggered grid: for mass conservation and scalar
quantities (left), for x-momentum (center) and for y-momentum (right)
We will use the second order implicit three time level scheme described in
Sect. 6.2.4 for integration in time. This leads to the following approximation
of the unsteady term:
where
3 p A 0
A R
Ab =
and Qki = !!--- ( 4 4 - uY-') .
2At
We shall drop the superscript n + 1; all terms are evaluated at t,+' unless
stated otherwise. Because the scheme is implicit, the equations require iterative solution. If the time steps are as small as those used in explicit schemes,
one or two iterations per time step will suffice. For flows with slow transients,
we may use larger time steps and more iterations will be necessary. As noted
earlier, these iterations are called outer iterations to distinguish them from
the inner iterations used to solve linear equations such as the pressure correction equation. We assume that one of the solvers described in Chap. 5 is
used for the latter and shall concentrate on the outer iterations.
We now consider the approximation of the convective and diffusive fluxes
and the source terms. The surface integrals may be split into four CV face
integrals. Let us concentrate attention on CV face 'e'; the other faces are
treated in the same way, and the results can be obtained by index substitution. We shall adopt the second order central difference approximations
presented in Chap. 4 . Fluxes are approximated by assuming that the value of
a quantity at a CV face center represents the mean value over the face (midpoint rule approximation). On the mth outer iteration, all nonlinear terms
are approximated by a product of an 'old' (from the preceding outer iteration) and a 'new' value. Thus, in discretizing the momentum equations, the
189
Fig. 7.4. Control volumes for a staggered grid: for mass conservation and scalar
quantities (left), for x-momentum (center) and for y-momentum (right)
We will use the second order implicit three time level scheme described in
Sect. 6.2.4 for integration in time. This leads to the following approximation
of the unsteady term:
where
3 p A 0
A R
Ab =
and Qki = !!--- ( 4 4 - uY-') .
2At
We shall drop the superscript n + 1; all terms are evaluated at t,+' unless
stated otherwise. Because the scheme is implicit, the equations require iterative solution. If the time steps are as small as those used in explicit schemes,
one or two iterations per time step will suffice. For flows with slow transients,
we may use larger time steps and more iterations will be necessary. As noted
earlier, these iterations are called outer iterations to distinguish them from
the inner iterations used to solve linear equations such as the pressure correction equation. We assume that one of the solvers described in Chap. 5 is
used for the latter and shall concentrate on the outer iterations.
We now consider the approximation of the convective and diffusive fluxes
and the source terms. The surface integrals may be split into four CV face
integrals. Let us concentrate attention on CV face 'e'; the other faces are
treated in the same way, and the results can be obtained by index substitution. We shall adopt the second order central difference approximations
presented in Chap. 4 . Fluxes are approximated by assuming that the value of
a quantity at a CV face center represents the mean value over the face (midpoint rule approximation). On the mth outer iteration, all nonlinear terms
are approximated by a product of an 'old' (from the preceding outer iteration) and a 'new' value. Thus, in discretizing the momentum equations, the