fluid_mechanics

9.3 Drag 503
The corresponding Reynolds number 1assuming v 1.57
10 4 ft 2 s2 is
Re UD
n
91.2 ft s 10.125 ft2
1.57 10 4 ft 2 s
7.26 104
For this value of Re we obtain from Fig. 9.21, C D 0.5. Thus,
our assumed value of C D 0.5 was correct. The corresponding
value of U is
U 91.2 fts 62.2 mph (Ans)
COMMENTS By repeating the calculations for various altitudes,
z, above sea level (using the properties of the U.S. Standard
Atmosphere given in Appendix C), the results shown in Fig.
E9.11b are obtained. Because of the decrease in density with altitude,
the hail falls even faster through the upper portions of the
storm than when it hits the ground.
Clearly, an airplane flying through such an updraft would feel
its effects 1even if it were able to dodge the hail2. As seen from
Eq. 2, the larger the hail, the stronger the necessary updraft.
Hailstones greater than 6 in. in diameter have been reported. In reality,
a hailstone is seldom spherical and often not smooth. However,
the calculated updraft velocities are in agreement with measured
values.
U, mph
140
120
100
80
60
40
20
(0, 62.2 mph)
0
0 10,000 20,000 30,000 40,000
z, ft
F I G U R E E9.11b
The drag coefficient
is usually independent
of Mach number
for Mach
numbers up to approximately
0.5.
Compressibility Effects. The above discussion is restricted to incompressible flows. If
the velocity of the object is sufficiently large, compressibility effects become important and the
drag coefficient becomes a function of the Mach number, Ma Uc, where c is the speed of sound
in the fluid. The introduction of Mach number effects complicates matters because the drag
coefficient for a given object is then a function of both Reynolds number and Mach number—
C D f1Re, Ma2. The Mach number and Reynolds number effects are often closely connected because
both are directly proportional to the upstream velocity. For example, both Re and Ma increase
with increasing flight speed of an airplane. The changes in C D due to a change in U are due
to changes in both Re and Ma.
The precise dependence of the drag coefficient on Re and Ma is generally quite complex
1Ref. 132. However, the following simplifications are often justified. For low Mach numbers, the
drag coefficient is essentially independent of Ma as is indicated in Fig. 9.23. For this situation, if
Ma 6 0.5 or so, compressibility effects are unimportant. On the other hand, for larger Mach number
flows, the drag coefficient can be strongly dependent on Ma, with only secondary Reynolds
number effects.
For most objects, values of C D increase dramatically in the vicinity of Ma 1 1i.e., sonic
flow2. This change in character, indicated by Fig. 9.24, is due to the existence of shock waves as
3.0
2.5
U
D
2.0
__________
U 2 bD
1__
2 ρ
1.5
C D =
1.0
0.5
0
0
U
b = length
4D
D
0.5 1.0
Ma
F I G U R E 9.23
Drag coefficient as a function
of Mach number for
two-dimensional objects in
subsonic flow (Ref. 5).