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Learning Outcomes:
PIPE FLOW: Boundary Layer
(Lecture 2)
At the end of this session, learners are able to:
▪ Differentiate between laminar and turbulent pipe flows in
terms of the Reynold‘s number
▪ Sketch and calculate the typical velocity profile for
turbulent pipe flow and estimate the thickness of the
viscous sublayer and the transition zone between laminar
and turbulent flows
▪ Recognize the condition for fully smooth and fully rough
pipes in terms of Reynold‘s number and absolute surface
roughness
▪ Use the turbulent velocity profile to improve the estimate
for kinetic energy and momentum, which are usually based
on the average velocity
INSTRUCTOR:
Dr S Rwanga
Department of Civil Engineering Sciences
University of Johannesburg
Department of Civil Engineering Sciences
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Reynold’s Experiment
Reynolds designed an experiment in which a filament of dye was injected into a flow of
water (below). The discharge was carefully controlled and passed through a glass tube so
that observations could be made. Reynolds discovered
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Reynolds Number
In summary,
Reynolds” experiments revealed that the onset of turbulence was a function of
fluid velocity, viscosity and a typical dimension. This led to the formation of the
dimensionless Reynolds number (symbol Re):
It is also possible to show that the Reynolds number represents a ratio of forces
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Development of Boundary Layers
Figure right: Shows the development
of laminar flow in a pipe. At entry to
the pipe, a laminar boundary layer
begins to grow. However, the growth
of the boundary layer is halted when it
reaches the pipe centreline, and
thereafter the flow consists entirely of
a boundary layer of thickness r. The
resulting velocity distribution is as
shown in Figure a.
For the case of turbulent flow shown
in Figure b, the growth of the
boundary layer is not suppressed until
it becomes a turbulent boundary layer
with the accompanying laminar
sublayer. The resulting velocity profile
therefore differs considerably from the
laminar case. The existence of the
laminar sublayer is of prime
importance in explaining the difference
between smooth and rough pipes.
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Boundary layers and velocity
distributions:
(a) laminar flow and
(b) turbulent flow.
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A parabolic velocity distribution
The discharge (Q) may be determined as well
Momentum equation
Derived elsewhere
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Example
In practice, it is usual to express (V) in terms of frictional head
loss by making the substitution
Oil flows through a 25 mm diameter pipe
with a mean velocity of 0.3 m/s. Given that
μ = 4.8 × 10−2 kg/m s and ρ = 800 kg/m3,
calculate (a) the pressure drop in a 45 m
length and (b) the maximum velocity.
This is the Hagen–Poiseuille
equation, named after the two
people who first (independently)
carried out the experimental work
leading to it.
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Fully developed laminar flow in a circular pipe
Pressure Drop and Head Loss
This equation is known as Poiseuille’s law, and this flow is called Hagen–
Poiseuille flow in
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Examples Flow Rates in Horizontal and Inclined
Pipes
Inclined
Pipes
Oil at 20°C (density = 888 kg/m3 and m
0.800 kg/m · s) is flowing steadily through a
5-cm-diameter, 40-m-long pipe (Fig.
below). The pressure at the pipe inlet and
outlet are measured to be 745 and 97 kPa,
respectively.
Determine the flow rate of oil through the
pipe assuming the pipe is
(a) horizontal,
(b) inclined 15° upward,
(c) inclined 15° downward.
Also verify that the flow through the pipe
is laminar.
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The pressure drop across the pipe and the pipe cross-sectional
area are
Solution
Assumptions
i. The flow is steady and incompressible.
ii. The entrance effects are negligible, and thus the
flow is fully developed.
iii. The pipe involves no components such as bends,
valves, and connectors.
iv. The piping section involves no work devices such
as a pump or a turbine.
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The flow rate for all three cases can be determined
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Discussion
Note that the flow is driven by the combined effect of pressure difference and gravity. As
can be seen from the flow rates we calculated, gravity opposes uphill flow, but enhances
downhill flow. Gravity has no effect on the flow rate in the horizontal case. Downhill flow
can occur even in the absence of an applied pressure difference. For the case of P1 = P2 = 97
kPa (i.e., no applied pressure difference), the pressure throughout the entire pipe would
remain constant at 97 Pa, and the fluid would flow through the pipe at a rate of 0.00043
m3/s under the influence of gravity. The flow rate increases as the tilt angle of the pipe from
the horizontal is increased in the negative direction and would reach its maximum value
when the pipe is vertical.
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Power-law velocity profile – Turbulent
Flow
Power-law velocity profile
• The velocity profile in turbulent flow is flatter in
the central part of the pipe (i.e., in the turbulent
core) than in laminar flow. The flow velocity drops
rapidly, extremely close to the walls. This is due to
the diffusivity of the turbulent flow.
• In the case of turbulent pipe flow, there are many
empirical velocity profiles. The simplest and the
best known is the power-law velocity profile:
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where the exponent n is a constant whose value depends on
the Reynolds number. This dependency is empirical, and it is
shown in the picture. In short, the value n increases with
increasing Reynolds number. The one-seventh power-law
velocity profile approximates many industrial flows.
Example on turbulent velocity profile
In an experimental setup, a flow velocity of 1.36
m/s is measured 45 mm away from the wall of the
pipe with diameter of 250 mm. Calculate the
Vmax. Choose n = 7
Turbulent Velocity Profile
Turbulent flow – profiles
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Solution
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