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90 3. Turbulence Models

3.5 Initial C o n d i t i o n s

Initial conditions needed in the solution of the transport equations for turbulent
flow are often obtained from empirical expressions when experimental data are
not available. The choice of these expressions can influence predictions of the
turbulence models and in some cases can even lead to the breakdown of calcula-
tions. This section presents useful and convenient expressions to use for external
boundary-layer flows. Before considering them, however, it is useful to review
the composite nature of a wall turbulent boundary layer. As was discussed in
[2, 3], for example, the behavior of turbulent flow in the inner region is different
from that in the outer region of the flow. Experiments and dimensional analysis
show that the length and velocity scales in the inner region, roughly y/6 < 0.2,
1 2
are v/u T and u r , where u T denotes the friction velocity defined by (r w/g) ' -
The velocity profile in the inner region of a smooth wall in the absence of surface
roughness and mass transfer can be expressed in the form
7/
+
u + = - = h(y ) (3.5.1)
u T
which is known as the "law of the wall". For y > 30u/u T approximately, but
y/6 < 0.20 approximately, Eq. (3.5.1) can be written as

u+ = -lny + + c (3.5.2)

where c is a constant found experimentally to be about 5.0-5.2.
Several expressions have been developed to express the relationship in Eq.
(3.5.1) from y = 0 to the limit of the validity of the logarithmic law. One con-
venient and useful expression due to Van Driest [19] is

u + / dy + (3.5.3)
2 y V J
Jo 1 + Vl + 4a
where
+
+
a = Ky [1 - exp(-y+/A )} (3.5.4)
Figure 3.2 shows the resulting profile from Eq. (3.5.3). Note that for large
+
y , it reduces to that given by Eq. (3.5.2). The exponential factor is often called
the "damping function."
In the outer region of a boundary layer in zero pressure gradient, the only
length and velocity scales are 6 and u T) and the velocity profile can be expressed
as

which is known as the velocity-defect law (see Fig. 3.3), valid for yu T/v > 30.
For zero pressure gradient flows, the function f\ has been found to be inde-
pendent of Reynolds number (except at Reynolds numbers RQ = u e6/v < 5000,