fluid_mechanics

9.4 Lift 513
Denotes p > p 0
Denotes p < p 0
U, p 0
F I G U R E 9.31
an automobile.
Pressure distribution on the surface of
E XAMPLE 9.14
Lift from Pressure and Shear Stress Distributions
GIVEN When a uniform wind of velocity U blows past the
semicircular building shown in Fig. E9.14a,b, the wall shear
stress and pressure distributions on the outside of the building are
as given previously in Figs. E9.8b and E9.9a, respectively.
FIND If the pressure in the building is atmospheric 1i.e., the
value, p 0 , far from the building2, determine the lift coefficient and
the lift on the roof.
SOLUTION
From Eq. 9.2 we obtain the lift as
As is indicated in Fig. E9.14b, we assume that on the inside of the
building the pressure is uniform, p p 0 , and that there is no
shear stress. Thus, Eq. 1 can be written as
or
where b and D are the length and diameter of the building,
respectively, and dA b1D22du. Equation 2 can be put into
dimensionless form by using the dynamic pressure, rU 2 2, planform
area, A bD, and dimensionless shear stress
to give
l bD 2
l p sin u dA t w cos u dA
p
l 1 p p 0 2 sin u b a D 2 b du
c p
p
1 p p 0 2 sin u du t w cos u du d
0
0
p
t w cos u b a D 2 b du
0
F1u2 t w 1Re2 1 2
1rU 2 22
l 1 2 rU 2 A c 1 2 p 1 p p 0 2
sin u du
1
2 rU 2
1
21Re
p
F1u2 cos u du d
0
From the data in Figs. E9.8b and E9.9a, the values of the two integrals
in Eq. 3 can be obtained by determining the area under the
0
0
(1)
(2)
(3)
curves of 31p p 0 21rU 2 224 sin u versus u and F1u2 cos u
versus u plotted in Figs. E9.14c and E9.14d. The results are
and
Thus, the lift is
or
and
p
0
l 1 2 rU 2 A ca 1 2 b 11.762 1
21Re 13.922d
C L
l
1
2 rU 2 A
(Ans)
(4) (Ans)
COMMENTS Consider a typical situation with D 20 ft,
U 30 fts, b 50 ft, and standard atmospheric conditions
1r 2.38 10 3 slugsft 3 and n 1.57 10 4 ft 2 s2, which
gives a Reynolds number of
Re UD
n 130 ft s2120 ft2
3.82
1.57 10 4 ft 2 106
s
Hence, the lift coefficient is
1 p p 0 2
sin u du 1.76
1
2 rU 2
p
F1u2 cos u du 3.92
0
l a0.88 1.96
1Re b a1 2 rU 2 Ab
0.88
1.96
1Re
1.96
C L 0.88
0.88 0.001 0.881
13.82 10 6 2 1 2