At a wall the velocity is zero. However, it is more effective to calculate the stress at the wall directly. The mass conservation amounts to
![]() |
(556) |
For stationary flow (
) at the wall
(
) we arrive at
![]() |
(557) |
or (Figure 159)
![]() |
(558) |
![]() |
(559) |
Now, because
![]() |
(560) |
one obtaines , since
(just derived) and also the
turbulent kinetic energy at the wall is zero. For the tangential component
one obtains:
![]() |
(561) |
Since , one arrives at
The velocity at P (Figure 159) is now decomposed into a component normal and a component tangent to the wall:
![]() |
(563) |
and
![]() |
(564) |
where
and
are unit vectors in n- and
t-direction, respectively. The stress tensor amounts to:
![]() |
(565) |
and the normal vector orthogonal and external to the surface satisfies:
![]() |
(566) |
This leads to the following stress vector
:
![]() |
(567) |
or
Approximating by
![]() |
(569) |
one obtains by combining Equations (562) and (568):
![]() |
(570) |
Therefore, the integral at the wall can be approximated by:
![]() |
(571) |
where is the area of the wall face.
The first term contributes to the left hand side, the second term to the
right hand side of the system of equations.