fluid_mechanics

498 Chapter 9 ■ Flow over Immersed Bodies
2.5
•
Flat plate
normal to flow
C D
2.0
1.5
U
D
Re = UD
v = 10 5
b = length
1.0
C D =
1
ρU 2 bD
2
Flat plate
0.5
parallel to flow
C D =
1
ρU 2 b
2
0 0 1 2 3 4 5 6
D
F I G U R E 9.19 Drag coefficient
for an ellipse with the characteristic area either
the frontal area, A bD, or the planform area,
A b / (Ref. 5).
Re < 1
U
= f(U, , )
After all, it is the planform area on which the shear stress acts, rather than the much smaller 1for thin
bodies2 frontal area. The ellipse drag coefficient based on the planform area, C D d1rU 2 b/22, is
also shown in Fig. 9.19. Clearly the drag obtained by using either of these drag coefficients would be
the same. They merely represent two different ways to package the same information.
The amount of streamlining can have a considerable effect on the drag. Incredibly, the drag
on the two two-dimensional objects drawn to scale in Fig. 9.20 is the same. The width of the wake
for the streamlined strut is very thin, on the order of that for the much smaller diameter circular
cylinder.
Reynolds Number Dependence. Another parameter on which the drag coefficient can
be very dependent is the Reynolds number. The main categories of Reynolds number dependence
are 112 very low Reynolds number flow, 122 moderate Reynolds number flow 1laminar boundary
layer2, and 132 very large Reynolds number flow 1turbulent boundary layer2. Examples of these three
situations are discussed below.
Low Reynolds number flows 1Re 6 12 are governed by a balance between viscous and pressure
forces. Inertia effects are negligibly small. In such instances the drag on a threedimensional
body is expected to be a function of the upstream velocity, U, the body size, /, and
the viscosity, m. Thus, for a small grain of sand settling in a lake 1see margin figure2
d f 1U, /, m2
From dimensional considerations 1see Section 7.7.12
d Cm/U
(9.38)
where the value of the constant C depends on the shape of the body. If we put Eq. 9.38 into dimensionless
form using the standard definition of the drag coefficient, we obtain
C D
d
1
2rU 2 / 2Cm/U
2 rU 2 / 2
2C
Re
a = b
U, ρ
U, ρ
10 D
Diameter = D
(a)
(b)
F I G U R E 9.20 Two objects of considerably different size that have the same drag force:
(a) circular cylinder C D 1.2; (b) streamlined strut C D 0.12.