9.4 RANS Models
299
where E = eKB. The control volume nearest the wall has one face that lies
on the wall. In the equation for the momentum parallel to the wall for that
control volume, the shear stress a t the wall is required. It may be taken from
Eq. (9.49) i.e. this boundary condition allows us to obtain a closed set of
equations.
When these 'law of the wall' type boundary conditions are used, the diffusive flux of k through the wall is usually taken to be zero, yielding the
boundary condition that the normal derivative of k is zero.
The dissipation boundary condition is derived by assuming equilibrium i.e.
balance of production and dissipation in the near wall region. The production
in wall region is computed from:
da,
Pk M 7,,
dn
which is an approximation to the dominant term of Eq. (9.40) that is valid
near the wall; it is valid because the shear stress is nearly constant in this
region. We need the dissipation (= production) at the midpoint of the control
volume closest to the wall. The velocity derivative required can be derived
from the logarithmic velocity profile (9.46):
which, together with Eq. (9.49), provides a second equation relating the wall
shear stress and the velocity at the first grid point. From these two equations,
both quantities may be computed.
When the above approximations are used, the equation for E is not applied
in the control volume next to the wall; instead, E is at the CV center set equal
to:
This expression is derived from Eq. (9.41) using the approximation for the
length scale
which is valid near wall under the conditions used to derive the 'law of the
wall' model.
It should be noted that the above boundary conditions are valid when the
first grid point is within the logarithmic region, i.e. when n: > 30. Problems
299
where E = eKB. The control volume nearest the wall has one face that lies
on the wall. In the equation for the momentum parallel to the wall for that
control volume, the shear stress a t the wall is required. It may be taken from
Eq. (9.49) i.e. this boundary condition allows us to obtain a closed set of
equations.
When these 'law of the wall' type boundary conditions are used, the diffusive flux of k through the wall is usually taken to be zero, yielding the
boundary condition that the normal derivative of k is zero.
The dissipation boundary condition is derived by assuming equilibrium i.e.
balance of production and dissipation in the near wall region. The production
in wall region is computed from:
da,
Pk M 7,,
dn
which is an approximation to the dominant term of Eq. (9.40) that is valid
near the wall; it is valid because the shear stress is nearly constant in this
region. We need the dissipation (= production) at the midpoint of the control
volume closest to the wall. The velocity derivative required can be derived
from the logarithmic velocity profile (9.46):
which, together with Eq. (9.49), provides a second equation relating the wall
shear stress and the velocity at the first grid point. From these two equations,
both quantities may be computed.
When the above approximations are used, the equation for E is not applied
in the control volume next to the wall; instead, E is at the CV center set equal
to:
This expression is derived from Eq. (9.41) using the approximation for the
length scale
which is valid near wall under the conditions used to derive the 'law of the
wall' model.
It should be noted that the above boundary conditions are valid when the
first grid point is within the logarithmic region, i.e. when n: > 30. Problems