7.7 Boundary Conditions for the Navier-Stokes Equations
205
= O =+ I-,, = 2 p
= 0 . (7.144)
wall
Therefore, the diffusive flux in the v equation at the south boundary is:
This should be implemented directly, rather than using only the condition
that v = 0 at the wall. Since u p # 0, we would obtain a non-zero derivative
in the discretized flux expression if this were not done; v = 0 is used as
a boundary condition in the continuity equation. The shear stress can be
calculated by using a one-sided approximation of the derivative d u l d y ; one
possible approximation is (for the u equation and the situation from Fig.
7.7):
ear-boundary CV
symmetry plane
Fig. 7.7. On the boundary conditions at a wall and a symmetry plane
At a symmetry plane we have the opposite situation: the shear stress is
zero, but the normal stress is not, since (for the situation from Fig. 7.7):
The diffusive flux in the u equation is zero, and the diffusive flux in the v
equation can be approximated as:
where vs = 0 applies.
In a FV method using a staggered grid, the pressure is not required at
boundaries (except if the pressure at a boundary is specified; this is handled
in Chap. 10). This is due to the fact that the nearest CV for the velocity
component normal to the boundary extends only up to the center of the scalar
205
= O =+ I-,, = 2 p
= 0 . (7.144)
wall
Therefore, the diffusive flux in the v equation at the south boundary is:
This should be implemented directly, rather than using only the condition
that v = 0 at the wall. Since u p # 0, we would obtain a non-zero derivative
in the discretized flux expression if this were not done; v = 0 is used as
a boundary condition in the continuity equation. The shear stress can be
calculated by using a one-sided approximation of the derivative d u l d y ; one
possible approximation is (for the u equation and the situation from Fig.
7.7):
ear-boundary CV
symmetry plane
Fig. 7.7. On the boundary conditions at a wall and a symmetry plane
At a symmetry plane we have the opposite situation: the shear stress is
zero, but the normal stress is not, since (for the situation from Fig. 7.7):
The diffusive flux in the u equation is zero, and the diffusive flux in the v
equation can be approximated as:
where vs = 0 applies.
In a FV method using a staggered grid, the pressure is not required at
boundaries (except if the pressure at a boundary is specified; this is handled
in Chap. 10). This is due to the fact that the nearest CV for the velocity
component normal to the boundary extends only up to the center of the scalar