Mechanics of Fluids

Table 8.1 Comparison of results for various approximate velocity distributions in
a laminar boundary layer with the Blasius’s exact solution
u
um
δ √ (Rex)
x
δ ∗√ (Rex)
x
θ √ (Rex)
x
√
= cf (Rex)
The laminar boundary layer on a flat plate 309
�
τ 0
ϱu 2 m
� √(Rex)
2η − η 2 5.48 1.826 0.730 0.365
3
2 η − 1 2 η3 4.64 1.740 0.646 0.323
2η − 2η 3 + η 4 5.84 1.751 0.685 0.343
sin(πη/2) 4.80 1.743 0.655 0.328
Blasius’s — 1.721 0.664 0.332
exact solution
Example 8.2 Which of the following expressions describes better the
velocity distribution for a laminar boundary layer on a flat plate in the
absence of a streamwise pressure gradient?
(a) u
=
um
3
�
y
�
−
2 δ
1
�
y
�3 2 δ
(b) u
=
um
3
�
y
�
−
2 δ
1
�
y
�2 2 δ
Give the reasons for your decision.
Solution
Write η = y/δ.
(a) u
um
u
um
u
um
= 3 1
η −
2 2 η3
= 0 when η = 0
= 1 when η = 1
d(u/um)
dη
d(u/um)
dη
d(u/um)
dη
= 3 3
−
2 2 η2
= 3
2
when η = 0
= 0 when η = 1
(b) u
um
u
um
u
um
= 3 1
η −
2 2 η2
= 0 when η = 0
= 1 when η = 1
d(u/um)
dη
d(u/um)
dη
d(u/um)
dη
= 3
− η
2
= 3
2
= 1
2
when η = 0
when η = 1
Velocity profile (a) is the better of the two, as it satisfies all four of the
required boundary conditions, whereas profile (b) fails condition III. ✷