fluid_mechanics

8.3 Fully Developed Turbulent Flow 401
laminar rather than turbulent. As shown in Chapter 9, turbulence can also aid in delaying flow
separation.
F l u i d s i n t h e N e w s
Smaller heat exchangers Automobile radiators, air conditioners,
and refrigerators contain heat exchangers that transfer energy
from (to) the hot (cold) fluid within the heat exchanger
tubes to (from) the colder (hotter) surrounding fluid. These
units can be made smaller and more efficient by increasing the
heat transfer rate across the tubes’ surfaces. If the flow through
the tubes is laminar, the heat transfer rate is relatively small.
Significantly larger heat transfer rates are obtained if the flow
within the tubes is turbulent. Even greater heat transfer rates
can be obtained by the use of turbulence promoters, sometimes
termed “turbulators,” which provide additional turbulent mixing
motion than would normally occur. Such enhancement
mechanisms include internal fins, spiral wire or ribbon inserts,
and ribs or grooves on the inner surface of the tube. While these
mechanisms can increase the heat transfer rate by 1.5 to 3 times
over that for a bare tube at the same flowrate, they also increase
the pressure drop (and therefore the power) needed to produce
the flow within the tube. Thus, a compromise involving increased
heat transfer rate and increased power consumption is
often needed.
V8.6 Stirring cream
into coffee
Turbulent flow parameters
can be described
in terms of
mean and fluctuating
portions.
Turbulence is also of importance in the mixing of fluids. Smoke from a stack would continue
for miles as a ribbon of pollutant without rapid dispersion within the surrounding air if the
flow were laminar rather than turbulent. Under certain atmospheric conditions this is observed to
occur. Although there is mixing on a molecular scale 1laminar flow2, it is several orders of magnitude
slower and less effective than the mixing on a macroscopic scale 1turbulent flow2. It is considerably
easier to mix cream into a cup of coffee 1turbulent flow2 than to thoroughly mix two colors
of a viscous paint 1laminar flow2.
In other situations laminar 1rather than turbulent2 flow is desirable. The pressure drop in pipes
1hence, the power requirements for pumping2 can be considerably lower if the flow is laminar rather
than turbulent. Fortunately, the blood flow through a person’s arteries is normally laminar, except
in the largest arteries with high blood flowrates. The aerodynamic drag on an airplane wing can
be considerably smaller with laminar flow past it than with turbulent flow.
8.3.2 Turbulent Shear Stress
The fundamental difference between laminar and turbulent flow lies in the chaotic, random behavior
of the various fluid parameters. Such variations occur in the three components of velocity, the
pressure, the shear stress, the temperature, and any other variable that has a field description. Turbulent
flow is characterized by random, three-dimensional vorticity 1i.e., fluid particle rotation or
spin; see Section 6.1.32. As is indicated in Fig. 8.12, such flows can be described in terms of their
mean values 1denoted with an overbar2 on which are superimposed the fluctuations 1denoted with
a prime2. Thus, if u u1x, y, z, t2 is the x component of instantaneous velocity, then its time mean
1or time-average2 value, u, is
u 1 T t 0T
(8.24)
where the time interval, T, is considerably longer than the period of the longest fluctuations, but considerably
shorter than any unsteadiness of the average velocity. This is illustrated in Fig. 8.12.
The fluctuating part of the velocity, u¿ , is that time-varying portion that differs from the average
value
u u u¿ or u¿ u u
Clearly, the time average of the fluctuations is zero, since
t 0
u1x, y, z, t2 dt
u¿ 1 T t 0T
1u u 2 dt 1 T a t 0T
t 0 T
u dt u dtb
(8.25)
t 0
t 0
t 0
1 1T u T u2 0
T