fluid_mechanics

302 Chapter 6 ■ Differential Analysis of Fluid Flow
y
dθ
p s
F y
a
F x
θ
x
F I G U R E 6.28
on a circular cylinder.
The notation for determining lift and drag
V6.9 Potential and
viscous flow
Potential theory incorrectly
predicts
that the drag on a
cylinder is zero.
and
F x
2p
F y p s sin u a du
0
(6.118)
where is the drag 1force parallel to direction of the uniform flow2 and is the lift 1force perpendicular
to the direction of the uniform flow2. Substitution for p s from Eq. 6.116 into these two equations, and
subsequent integration, reveals that F x 0 and F y 0 1Problem 6.732. These results indicate that both
the drag and lift as predicted by potential theory for a fixed cylinder in a uniform stream are zero.
Since the pressure distribution is symmetrical around the cylinder, this is not really a surprising result.
However, we know from experience that there is a significant drag developed on a cylinder when it is
placed in a moving fluid. This discrepancy is known as d’Alembert’s paradox. The paradox is named
after Jean le Rond d’Alembert 11717–17832, a French mathematician and philosopher, who first showed
that the drag on bodies immersed in inviscid fluids is zero. It was not until the latter part of the nineteenth
century and the early part of the twentieth century that the role viscosity plays in the steady fluid
motion was understood and d’Alembert’s paradox explained 1see Section 9.12.
F y
E XAMPLE 6.8
Potential Flow—Cylinder
GIVEN When a circular cylinder is placed in a uniform
stream, a stagnation point is created on the cylinder as is shown in
Fig. E6.8a. If a small hole is located at this point, the stagnation
pressure, p stag , can be measured and used to determine the approach
velocity, U.
FIND
(a) Show how p stag and U are related.
(b) If the cylinder is misaligned by an angle (Figure E6.8b),
but the measured pressure is still interpreted as the stagnation
pressure, determine an expression for the ratio of the true
velocity, U, to the predicted velocity, U. Plot this ratio as a function
of for the range 20 20.
U
U
U
p 0
y
a
r
θ
x
α
a
β
β
Stagnation
point
(a)
(b)
(d)
U__
U'
1.5
1.4
1.3
1.2
1.1
1.0
–20° –10° 0° 10° 20°
α
(c)
F I G U R E E6.8