A new theory for the turbulent plane wall jet without external stream is proposed
based on a similarity analysis of the governing equations. The asymptotic invariance
principle (AIP) is used to require that properly scaled profiles reduce to similarity
solutions of the inner and outer equations separately in the limit of infinite Reynolds
number. Application to the inner equations shows that the appropriate velocity scale
is the friction velocity, u∗, and the length scale is
v/u∗. For finite Reynolds numbers, the profiles retain a dependence on the length-scale ratio,
y+1/2 = u∗y1/2/v, where
y1/2 is the distance from the wall at which the mean velocity has dropped to 1/2 its
maximum value. In the limit as y+1/2 → ∞,
the familiar law of the wall is obtained. Application of the AIP to the outer equations
shows the appropriate velocity scale to be Um, the velocity maximum, and the length scale
y1/2; but again the profiles retain a dependence on y+1/2
for finite values of it. The Reynolds shear stress in the outer layer
scales with u2*, while the normal stresses scale with
U2m. Also Um ∼
yn1/2 where n < −1/2 and must be
determined from the data. The theory cannot rule out the possibility that
the outer flow may retain a dependence on the source conditions, even asymptotically.
The fact that both these profiles describe the entire wall jet for finite values of
y+1/2, but reduce to inner and outer profiles in the limit, is used to determine their
functional forms in the ‘overlap’ region which both retain. The result from near
asymptotics is that the velocity profiles in the overlap region must be power laws, but
with parameters which depend on Reynolds number y+1/2 and are only asymptotically
constant. The theoretical friction law is also a power law depending on the velocity
parameters. As a consequence, the asymptotic plane wall jet cannot grow linearly,
although the difference from linear growth is small.
It is hypothesized that the inner part of the wall jet and the inner part of the
zero-pressure-gradient boundary layer are the same. It follows immediately that all of
the wall jet and boundary layer parameters should be the same, except for two in the
outer flow which can differ only by a constant scale factor. The theory is shown to be
in excellent agreement with the experimental data which show that source conditions
may determine uniquely the asymptotic state achieved. Surprisingly, only a single parameter,
B1 = (Umv/Mo)/
(y+1/2Mo/v2)n
= constant where n ≈ −0.528, appears to
be required to determine the entire flow for a given source.