Page 170 - Bird R.B. Transport phenomena

154 Chapter 5 Velocity Distributions in Turbulent Flow

§5.1 COMPARISONS OF LAMINAR AND TURBULENT FLOWS
Before discussing any theoretical ideas about turbulence, it is important to summarize
the differences between laminar and turbulent flows in several simple systems. Specifi-
cally we consider the flow in conduits of circular and triangular cross section, flow along
a flat plate, and flows in jets. The first three of these were considered for laminar flow in
§2.3, Problem 3B.2, and §4.4.

Circular Tubes

For the steady, fully developed, laminar flow in a circular tube of radius R we know that
the velocity distribution and the average velocity are given by

iv )
• = 1 - К- and z (Re < 2100) (5.1-1, 2)
-, К / r ma x
and that the pressure drop and mass flow rate w are linearly related:

(Re < 2100) (5.1-3)

For turbulent flow, on the other hand, the velocity is fluctuating with time chaotically at
each point in the tube. We can measure a "time-smoothed velocity" at each point with,
say, a Pitot tube. This type of instrument is not sensitive to rapid velocity fluctuations,
but senses the velocity averaged over several seconds. The time-smoothed velocity
(which is defined in the next section) will have a z-component represented by v , z and its
shape and average value will be given very roughly by 1

1/7 (v ) A
5
4
and z (10 < Re < 10 ) (5.1-4,5)
5
This \-power expression for the velocity distribution is too crude to give a realistic veloc-
ity derivative at the wall. The laminar and turbulent velocity profiles are compared in
Fig. 5.1-1.

Tube center

Fig. 5.1-1. Qualitative comparison of lami-
nar and turbulent velocity profiles. For a
0 more detailed description of the turbulent
1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 velocity distribution near the wall, see
r/R—+- Fig. 5.5-3.

1
H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 7th edition (1979), Chapter XX
(tube flow), Chapters VII and XXI (flat plate flow), Chapters IX and XXIIV (jet flows).