fluid_mechanics

8.3 Fully Developed Turbulent Flow 403
A
y
(2)
(1)
Velocity profile,
u = u(y)
A
u
u 1 < u 2
A
y
A
Average velocity profile,
u = u(y)
Turbulent
eddies
u
(a)
(b)
F I G U R E 8.14 (a) Laminar flow shear stress caused by random motion of molecules.
(b) Turbulent flow as a series of random, three-dimensional eddies.
Turbulent flow
shear stress is
larger than laminar
flow shear stress
because of the
irregular, random
motion.
with the motion of fluid particles, u. As the molecules dart across a given plane 1plane A– A, for example2,
the ones moving upward have come from an area of smaller average x component of velocity
than the ones moving downward, which have come from an area of larger velocity.
The momentum flux in the x direction across plane A– A gives rise to a drag 1to the left2 of
the lower fluid on the upper fluid and an equal but opposite effect of the upper fluid on the lower
fluid. The sluggish molecules moving upward across plane A– A must be accelerated by the fluid
above this plane. The rate of change of momentum in this process produces 1on the macroscopic
scale2 a shear force. Similarly, the more energetic molecules moving down across plane A– A must
be slowed down by the fluid below that plane. This shear force is present only if there is a gradient
in u u1y2, otherwise the average x component of velocity 1and momentum2 of the upward and
downward molecules is exactly the same. In addition, there are attractive forces between molecules.
By combining these effects we obtain the well-known Newton viscosity law: t m dudy, where
on a molecular basis m is related to the mass and speed 1temperature2 of the random motion of the
molecules.
Although the above random motion of the molecules is also present in turbulent flow, there
is another factor that is generally more important. A simplistic way of thinking about turbulent flow
is to consider it as consisting of a series of random, three-dimensional eddy type motions as is depicted
1in one dimension only2 in Fig. 8.14b. (See the photograph at the beginning of this chapter.)
These eddies range in size from very small diameter 1on the order of the size of a fluid particle2 to
fairly large diameter 1on the order of the size of the object or flow geometry considered2. They move
about randomly, conveying mass with an average velocity u u1y2. This eddy structure greatly promotes
mixing within the fluid. It also greatly increases the transport of x momentum across plane
A– A. That is, finite particles of fluid 1not merely individual molecules as in laminar flow2 are randomly
transported across this plane, resulting in a relatively large 1when compared with laminar
flow2 shear force. These particles vary in size but are much larger than molecules.
F l u i d s i n t h e N e w s
Listen to the flowrate Sonar systems are designed to listen to
transmitted and reflected sound waves in order to locate submerged
objects. They have been used successfully for many years
to detect and track underwater objects such as submarines and
aquatic animals. Recently, sonar techniques have been refined so
that they can be used to determine the flowrate in pipes. These
new flow meters work for turbulent, not laminar, pipe flows because
their operation depends strictly on the existence of turbulent
eddies within the flow. The flow meters contain a sonar-based
array that listens to and interprets pressure fields generated by the
turbulent motion in pipes. By listening to the pressure fields associated
with the movement of the turbulent eddies, the device can
determine the speed at which the eddies travel past an array of sensors.
The flowrate is determined by using a calibration procedure
which links the speed of the turbulent structures to the volumetric
flowrate.