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7 . Solution of the Navier-Stokes Equations
7.5 Solution Met hods for the Navier-Stokes Equations
We have described discretization methods for the various terms in the transport equations. The linkage of the pressure and the velocity components in
incompressible flows was demonstrated and a few solution methods have been
given. Many other methods of solving the Navier-Stokes equations can be devised. It is impossible t o describe all of them here. However, many of them
have elements in common with the methods already described. Familiarity
with these methods should allow the reader to understand the others.
We describe below in some detail two methods that are representative
of a larger group of methods. First an implicit method using the pressurecorrection equation and staggered grids is described in enough detail to allow
straightforward translation into a computer code. The corresponding code is
available via Internet; see Appendix for details.
7.5.1 Implicit Scheme Using Pressure-Correction and a Staggered
Grid
In this section we present an implicit finite volume scheme that uses the
pressure-correction method on a staggered two-dimensional Cartesian grid.
Solution in complicated geometries is described in the next chapter.
The Navier-Stokes equations in integral form read:
For convenience, it is assumed that the only body force is buoyancy. The
macroscopic momentum flux vector ti, see Eq. 1.18, is split into a viscous
contribution rijij and a pressure contribution pii. We assume the density
constant except in the buoyancy term, i.e. we use the Boussinesq approximation. The mean gravitational force is incorporated into the pressure term, as
shown in Sect. 1.4.
Typical staggered control volumes are shown in Fig. 7.4. The control
volumes for u, and U , are displaced with respect to the control volume for
the continuity equation. For non-uniform grids, the velocity nodes are not a t
the centers of their control volumes. Cell faces 'e' and 'w' for u, and 'n' and
's' for u, lie midway between the nodes. For convenience, we shall sometimes
use u instead of u, and v instead of u,.
7 . Solution of the Navier-Stokes Equations
7.5 Solution Met hods for the Navier-Stokes Equations
We have described discretization methods for the various terms in the transport equations. The linkage of the pressure and the velocity components in
incompressible flows was demonstrated and a few solution methods have been
given. Many other methods of solving the Navier-Stokes equations can be devised. It is impossible t o describe all of them here. However, many of them
have elements in common with the methods already described. Familiarity
with these methods should allow the reader to understand the others.
We describe below in some detail two methods that are representative
of a larger group of methods. First an implicit method using the pressurecorrection equation and staggered grids is described in enough detail to allow
straightforward translation into a computer code. The corresponding code is
available via Internet; see Appendix for details.
7.5.1 Implicit Scheme Using Pressure-Correction and a Staggered
Grid
In this section we present an implicit finite volume scheme that uses the
pressure-correction method on a staggered two-dimensional Cartesian grid.
Solution in complicated geometries is described in the next chapter.
The Navier-Stokes equations in integral form read:
For convenience, it is assumed that the only body force is buoyancy. The
macroscopic momentum flux vector ti, see Eq. 1.18, is split into a viscous
contribution rijij and a pressure contribution pii. We assume the density
constant except in the buoyancy term, i.e. we use the Boussinesq approximation. The mean gravitational force is incorporated into the pressure term, as
shown in Sect. 1.4.
Typical staggered control volumes are shown in Fig. 7.4. The control
volumes for u, and U , are displaced with respect to the control volume for
the continuity equation. For non-uniform grids, the velocity nodes are not a t
the centers of their control volumes. Cell faces 'e' and 'w' for u, and 'n' and
's' for u, lie midway between the nodes. For convenience, we shall sometimes
use u instead of u, and v instead of u,.