Mechanics of Fluids

312 Boundary layers, wakes and other shear layers
is therefore difficult, and an experimental check on theoretical results is
more easily obtained by measurements of the drag force exerted by the
fluid on the plate. Table 8.1 shows, however, that results do not vary
widely when different assumptions are used for the velocity distribution
within the boundary layer. The usefulness of the approximate method is
thus further demonstrated. Because of the asymptotic approach to the mainstream
velocity, Blasius’s solution does not give a finite value of the boundary
layer thickness, but the value of y at which the velocity is 0.99um is about
4.91x/ √ (Rex), andu = 0.999um at y ≏ 6.01x/ √ (Rex).
Example 8.4 Air of density 1.21 kg · m −3 and kinematic viscosity
1.5 × 10 −5 m 2 · s −1 passes over a thin flat plate, of dimensions 1.2 m
×1.2 m, parallel to the airstream. If transition takes place at the trailing
edge of the plate, determine
(a) the velocity of the airstream
(b) the frictional drag of the plate, DF.
Assume that transition takes place at Ret = 5 × 10 5 and the velocity
profile is given by
u
um
= 3
2
�
y
�
δ
− 1
2
�
y
�3 δ
Solution
At the trailing edge x = xt = x l; Rex = Ret = Re l; θ = θ l.
(a) Ret = umxt
ν
so
um = νRet
xt
(b) From Table 8.1
Hence
= (1.5 × 10−5 ) m 2 · s −1 × (5 × 10 −5 )
1.2 m
θ = 0.646x
√ Rex
θ √ Rex
x
= 0.646 × 1.2 m
�
= cf Rex = 0.646
� (5)(10 5 ) = 1.096 × 10 −3 m
= 6.25 m · s −1