fluid_mechanics

506 Chapter 9 ■ Flow over Immersed Bodies
for the smooth golf ball
d 1 2 10.00238 slugs ft 3 21200 fts2 2 p 2
4 a1.69 12 ftb 10.512
0.378 lb
(Ans)
and for the table tennis ball
d 1 2 10.00238 slugs ft 3 2160 fts2 2 p 2
4 a1.50 12 ftb 10.502
0.0263 lb
(Ans)
The corresponding decelerations are a dm gdw,
where m is the mass of the ball. Thus, the deceleration relative to
the acceleration of gravity, ag 1i.e., the number of g’s deceleration2
is ag dw or
a 0.185 lb
1.86 for the standard golf ball
g 0.0992 lb
a 0.378 lb
3.81 for the smooth golf ball
g 0.0992 lb
a 0.0263 lb
4.77 for the table tennis ball
g 0.00551 lb
(Ans)
(Ans)
(Ans)
COMMENTS Note that there is a considerably smaller deceleration
for the rough golf ball than for the smooth one. Because of
its much larger drag-to-mass ratio, the table tennis ball slows down
relatively quickly and does not travel as far as the golf ball. Note
that with U 60 fts the standard golf ball has a drag of
d 0.0200 lb and a deceleration of ag 0.202, considerably
less than the ag 4.77 of the table tennis ball. Conversely, a
table tennis ball hit from a tee at 200 fts would decelerate at a
rate of a 1740 fts 2 , or ag 54.1. It would not travel nearly
as far as the golf ball.
By repeating the above calculations, the drag as a function of
speed for both a standard golf ball and a smooth golf ball is shown
in Fig. E9.12.
The Reynolds number range for which a rough golf ball has
smaller drag than a smooth one (i.e., 4 10 4 to 3.6 10 5 ) corresponds
to a flight velocity range of 45 U 400 ft/s. This is
comfortably within the range of most golfers. (The fastest tee
shot by top professional golfers is approximately 280 ft/s.) As
discussed in Section 9.4.2, the dimples (roughness) on a golf
ball also help produce a lift (due to the spin of the ball) that allows
the ball to travel farther than a smooth ball.
, lb
1.2
1
0.8
0.6
0.4
0.2
Smooth
golf ball
Standard
golf ball
0
0 100 200 300 400 500
U, ft/s
F I G U R E E9.12
F l u i d s i n t h e N e w s
Dimpled baseball bats For many years it has been known that
dimples on golf balls can create a turbulent boundary layer and reduce
the aerodynamic drag, allowing longer drives than with
smooth balls. Thus, why not put dimples on baseball bats so that tomorrow’s
baseball sluggers can swing the bat faster and, therefore,
hit the ball farther? MIT instructor Jeffery De Tullio pondered that
question, performed experiments with dimpled bats to determine
the answer, and received a patent for his dimpled bat invention.
The result is that a batter can swing a dimpled bat approximately
3 to 5% faster than a smooth bat. Theoretically, this extra speed
will translate to an extra 10 to 15 ft distance on a long hit. (See
Problem 9.89.)
V9.12 Jet ski
The drag coefficient
for surface ships is
a function of the
Froude number.
Froude Number Effects. Another parameter on which the drag coefficient may be strongly
dependent is the Froude number, Fr U 1g/. As is discussed in Chapter 10, the Froude number
is a ratio of the free-stream speed to a typical wave speed on the interface of two fluids, such as the
surface of the ocean. An object moving on the surface, such as a ship, often produces waves that
require a source of energy to generate. This energy comes from the ship and is manifest as a drag.
[Recall that the rate of energy production 1power2 equals speed times force.] The nature of the waves
produced often depends on the Froude number of the flow and the shape of the object—the waves
generated by a water skier “plowing” through the water at a low speed 1low Fr2 are different than
those generated by the skier “planing” along the surface at high speed 1large Fr2.
Thus, the drag coefficient for surface ships is a function of Reynolds number 1viscous effects2
and Froude number 1wave-making effects2; C D f1Re, Fr2. As was discussed in Chapter
7, it is often quite difficult to run model tests under conditions similar to those of the prototype
1i.e., same Re and Fr for surface ships2. Fortunately, the viscous and wave effects can often be
separated, with the total drag being the sum of the drag of these individual effects. A detailed account
of this important topic can be found in standard texts 1Ref. 112.
As is indicated in Fig. 9.26, the wave-making drag, d w , can be a complex function of the
Froude number and the body shape. The rather “wiggly” dependence of the wave drag coefficient,