7. Solution of the Navier-Stokes Equations
7.1 Special Features of the Navier-Stokes Equations
In Chaps. 3, 4 and 6 we dealt with the discretization of a generic conservation equation. The discretization principles described there apply t o the
momentum and continuity equations (which we shall collectively call the
Navier-Stokes equations). In this chapter, we shall describe how the terms in
the momentum equations which differ from those in the generic conservation
equation are treated.
The unsteady and advection terms in the momentum equations have the
same form as in the generic conservation equation. The diffusive (viscous)
terms are similar to their counterparts in the generic equation but, because
the momentum equations are vector equations, these contributions become
a bit more complex and their treatment needs to be considered in more
detail. The momentum equations also contain a contribution from the pressure, which has no analog in the generic equation. It may be regarded either as a source term (treating the pressure gradient as body force - nonconservatively) or as a surface force (conservative treatment) but, due to the
close connection of the pressure and the continuity equation, it requires special attention. Finally, the fact that the principal variable is a vector allows
more freedom in the choice of a grid.
7.1.1 Discretization of Convective and Viscous Terms
The convective term in the momentum equation is non-linear; its differential
and integral forms read:
The treatment of the convective term in the momentum equations follows that
of the convective term in the generic equation; any of the methods described
in Chaps. 3 and 4 can be used.
The viscous terms in the momentum equations correspond to the diffusive
term in the generic equation; their differential and integral forms are:
7.1 Special Features of the Navier-Stokes Equations
In Chaps. 3, 4 and 6 we dealt with the discretization of a generic conservation equation. The discretization principles described there apply t o the
momentum and continuity equations (which we shall collectively call the
Navier-Stokes equations). In this chapter, we shall describe how the terms in
the momentum equations which differ from those in the generic conservation
equation are treated.
The unsteady and advection terms in the momentum equations have the
same form as in the generic conservation equation. The diffusive (viscous)
terms are similar to their counterparts in the generic equation but, because
the momentum equations are vector equations, these contributions become
a bit more complex and their treatment needs to be considered in more
detail. The momentum equations also contain a contribution from the pressure, which has no analog in the generic equation. It may be regarded either as a source term (treating the pressure gradient as body force - nonconservatively) or as a surface force (conservative treatment) but, due to the
close connection of the pressure and the continuity equation, it requires special attention. Finally, the fact that the principal variable is a vector allows
more freedom in the choice of a grid.
7.1.1 Discretization of Convective and Viscous Terms
The convective term in the momentum equation is non-linear; its differential
and integral forms read:
The treatment of the convective term in the momentum equations follows that
of the convective term in the generic equation; any of the methods described
in Chaps. 3 and 4 can be used.
The viscous terms in the momentum equations correspond to the diffusive
term in the generic equation; their differential and integral forms are: