Mech 392: Mechanics of Fluids II
Experiment I
Friction Coefficient and Velocity Distribution in Turbulent Pipe Flow
Objective
To determine the friction coefficient and velocity distribution in a turbulent flow and to
demonstrate the validity of theoretical and empirical relations.
Background
Knowledge of the pressure drop and velocity distribution is important if we want to know
the shear stress in a fluid flow, the vorticity distribution, the flow rate, and the shear force
on the inner surface of a duct, channel or pipe. This type of information is required for the
design of fluid flow machinery, pipeline and other fluid flow devices such as gas turbine
blades, aircraft wings, heat exchangers, etc...
For fully turbulent flow in a smooth pipe, the velocity distribution is given by the logarithmic law of the wall:
u+ = 2.5 ln y+ + 5.5
(1)
+
+
where
pu = u/uτ , y = uτ y/ν , y = D/2 − r , and the “friction velocity” is defined as
uτ = τ/ρ.
Figure 1: Geometry of Pipe Flow
The velocity profile outside the viscous sublayer should differ little from the law of the
wall. Note however that equation (1) is not valid close to the centerline. An alternative
empirical power-law relation is sometimes used to describe the velocity distribution:
u+ = 8.6 (y+ )1/7
(2)
This equation gives usually a good fit for 104 < Re < 5 × 104 .
Equations (1) and (2) can be used to derive an expression relating the local friction
coefficient, C f = τ/( 21 ρV 2 ), to the Reynolds number, Re = V D/ν:
(2)
(1)
⇒
⇒
1
p
C f /2
C f = 0.078 Re−0.25
q
= 2.46 ln(Re C f /2) + 0.30
1
(3)
(4)
Equation (4) is a little awkward to use, and a commonly employed empirical equation
giving a good fit over 3 × 104 < Re < 106 is
C f = 0.046 Re−0.2
(5)
Apparatus
The Air-line Apparatus will be used. This consists of a 6 m long, smooth aluminum alloy
circular pipe, with an inner diameter of 76mm. The pipe inlet is fitted with a smooth shaped
nozzle, and the flow is produced by a centrifugal fan connected to the outlet of the pipe by
a diffuser. The fan speed is variable, and a slide valve at the fan outlet controls the flow
rate.
Tests for this experiment will be performed with a 75mm diameter nozzle, for which the
applicable discharge equation is
s
∆pN T
(6)
Q = 0.1016
pb
where Q is the volumetric flow rate, ∆pN = p2 − p0 is the difference between nozzle and
atmospheric pressures, T is the absolute temperature of ambient air, and pb is the barometric pressure (all in S.I. units).
For fully developed flow, the pressure drop is related to the friction coefficient by
∆p = 4C f
L V2
ρ
D 2
(7)
This equation will be used to deduce C f from pressure drop measurements between taps
#14 and 17 (i.e. ∆p = p17 − p14 with L = 2.03m ) where the flow may be expected to be
fully developed.
A perspex section at the downstream end of the pipe is equipped with a total head
probe (tap #16) which may be traversed across the pipe using a micrometer screw. The
static pressure is measured at the wall by a pressure tap (# 17) located in the same plane
as the total head probe. The dynamic head, and hence the local velocity, is given by the
pressure difference.
Procedure
1. Frictional loss:
• Measure the ambient temperature and barometric pressure.
• Ensure that the throttle is wide open.
• Using equation (6), calculate the required ∆pN to achieve Re = 50, 000 and 100, 000.
• Measure and tabulate the pressure drop in the nozzle and the pipe for 10 settings
between these two values.
2
• Compute the flow rate from equation (6), the Reynolds number, and the friction
coefficient from equation (7).
• Plot C f vs. Re on a log-log scale comparing the experimental results with equations
(3) and (5).
2. Velocity profiles:
• Vary the motor RPM to achieve a height difference between tap 0 and 2 corresponding to Re = 52, 000 and determine C f as before.
• With the total head probe, perform a velocity traverse across the diameter of the
pipe. A total of approximately 15 to 20 data points should be adequate, with smaller
increments close to the wall where the velocity gradients are highest. Perform one
traverse in each direction and average the readings.
• Repeat the measurements for Re = 95, 000.
• Compute the experimental velocity distributions, and integrate (graphically or numerically) the profiles to determine the volumetric flow rate comparing the values
with those obtained from the pressure drop in the nozzle. It is convenient to plot the
velocity against the square of the corresponding radial position.
• Plot u+ vs. y+ on the same graph for both runs and compare to the predicted distributions using equations (1) and (2).
Sample Discussion Points
The following must be covered in the discussion of your results. They must not be treated
as questions, but rather as guiding points to ensure that your discussion is complete.
• Why is equation (1) not valid at the centerline?
• When performing velocity traverse measurements, the static pressure is measured
at the wall rather than at the location of the probe. Why is this acceptable?
• When the Reynolds number is increased, equation (1) is valid closer to the wall.
Why?
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