Mechanics of Fluids

308 Boundary layers, wakes and other shear layers
which, from eqn 8.17, is given by
CF = 2
� �1/2 2ABµ
= 2(2AB)
ϱuml
1/2 Re −1/2
l
where Re l = ϱuml/µ.
The local skin friction coefficient C f is defined as
which using eqn 8.16 yields
C f = τ0
1
2 ϱu2 m
c f = (2AB) 1/2 Re −1/2
x
(8.18)
Equation 8.15 shows that the thickness of a laminar boundary layer on a
flat plate with zero pressure gradient is proportional to the square root of
the distance from the leading edge, and inversely proportional to the square
root of the velocity of the main flow relative to the plate.
In order to evaluate δ from eqn 8.15, we must calculate A and B, and we
therefore require the form of the function f (η). Referring to Fig. 8.5, the
function f (η) must satisfy the following conditions:
(I)
(II)
(III)
(IV)
u
um
u
um
= 0 when η = 0
= 1 when η = 1
d(u/um)
= 0
dη
when η = 1
d(u/um)
�= 0
dη
when η = 0
The fourth condition follows because τ0 is finite and
� �
∂u
τ0 = µ
∂y
y=0
= µum
δ
� �
d(u/um)
dη
The above four conditions must be satisfied by any equation describing the
velocity profile within the laminar boundary layer on a flat plate with zero
pressure gradient. A number of simple but approximate relations have been
proposed. Some of these are considered in examples 8.2 to 8.4. The German
engineer P. R. H. Blasius (1883–1970) developed an exact solution of the
laminar boundary layer equations for the flow on a flat plate with zero
pressure gradient. It is therefore possible to compare results obtained by the
approximate methods with the exact results of Blasius (see Table 8.1).
y=0