1. REYNOLDS PIPE FLOW
1.1 BACKGROUND
1.
OBJECTIVES:
At the completion of this experiment you will be able to:
1. Distinguish between laminar, transitional, and turbulent pipe flow.
2. Calculate pressure loss in a pipe using friction factors.
3. Explore the affect of streamwise pressure gradients on transition.
2.
Specific Applications:
Liquid propellant feed systems in satellites and boosters, turbine engine oil systems for
lubrication and cooling of critical engine seals, hydraulic systems in aircraft controls, lifesupport systems in spacecraft, duct flows in aerodynamics, HVXC flows for aircraft,
other vehicles, and buildings, etc.
3.
Reynolds Pipe Flow
In pipe flow, viscous effects become important, and the behavior of the boundary
layer must be taken into account. At the entrance of a pipe the boundary layer is
generally so thin that the flow in this region can be considered non-viscous, except near
the pipe walls. However, as the flow progresses down the pipe, there is a thickening of
the boundary layer until it encompasses the whole pipe cross-section, and can no longer
be considered a boundary layer. Fully developed flow is a flow that no longer changes in
the streamwise direction (see Figure 1).
Core
Fully developed
flow
Figure 1 Boundary Layer Development
In laminar flow fluid layers slide smoothly over one another in a well-ordered
pattern. In the pipe, after the flow becomes fully developed, laminar flow is purely axial,
and the velocity profile is independent of the coordinate along the direction of flow.
Laminar instability can readily be seen in the classic Reynolds experiment on viscous
flow, in which dye is injected into the flow of water through a glass pipe. The thread-like
form of the moving dye indicates the laminar behavior. When the velocity of the water is
increased a fluctuating motion appears in the dye, indicating a transition to an unsteady
flow. At higher velocities the thread of dye becomes mixed with the fluid; irregular radial
velocity fluctuations are superimposed upon the axial motion, and the flow is said to be
turbulent (see Figure 3).
Another method of determining the flow regime involves observation of the fluid
as it exits the pipe. For laminar flow, the stream of fluid leaving the pipe appears smooth
and glassy. The stream will also have a steady well-defined shape. A turbulent stream,
on the other hand, appears to be rough and dull in luster. A turbulent stream as it exits the
pipe is also characterized by fluctuations and small disturbances. For any given pressure
difference, a turbulent stream will have less momentum (and thus speed) as it exits the
pipe, compared to a laminar stream.
Oswald Reynolds determined that the transition from laminar to turbulent flow
depends on the Reynolds number (symbolized by Re or R), defined as
Re 
Vl Vl
 ,
 
V  Q/A
where  is the density, 1 is the pipe diameter,  is the viscosity, V is the mean velocity,
Q is the volumetric flow rate, and A is the pipe cross-sectional area.
The critical Reynolds number is the Reynolds number at which transition from
laminar to turbulent flow occurs. This critical Reynolds number (Rcrit) depends upon the
initial disturbances in the fluid entering the pipe. A critical Reynolds number of 40,000
or greater can be reached if great care is taken to reduce disturbances. However, the
critical Reynolds number below which even strong disturbances are damped by viscosity
and do not cause turbulence is approximately Rcrit = 2300.
Consider the first law of thermodynamics for pipe flow. This is similar to the
Bernoulli equation except for the "head loss" term h1, which represents the loss of
mechanical energy per unit of mass flowing through the pipe, due to friction:
p1 V 12
p2 V 22
  gz1    gz2  gh l
 2
 2
For steady incompressible flow with horizontal streamlines, and assuming fully
developed flow, this becomes
p
 hl
g
p   gh l
.
For details, see for example Sabersky et al., Fluid Flow, MacMillan, 1971, sec. 5.4.
Therefore we can consider the head loss as the loss in pressure head, p/g, due to
friction. This head loss is dependent upon the velocity profile, the type of fluid, and
sometimes the roughness of the pipe.
Intuition suggests that the pressure change is proportional to the length of the pipe.
The nondimensional pressure change can be written as equal to the nondimensional pipe
length times a function of Reynolds number and non-dimensional pipe roughness:
p
 VD  
 L fo 
, 
2
D

D

V
where  = average roughness height, and g is a function of Reynolds number and nondimensional pipe roughness. Letting 2f O = f and assuming horizontal streamlines and to
fully developed flow, this becomes:
h
2
 VD  
 V L f
, ,
l
2g D  
D
or
V2 L
p
hl 
f=
2g D
g
f=
p
 V 2   L  .

 
 2  D 
This relation is known as the Darcy-Weisbach formula. For details, see Schlichting,
Boundary-Layer Theory, McGraw-Hill, seventh ed., sections V.a. and XX.a.)
Many relations have been developed to predict the friction factor in certain flow
regimes and pipe conditions. Some of the more useful ones are:
For laminar flow, an exact theoretical solution shows that
f =
64
.
Re
For turbulent flow in SMOOTH pipes, empirical data is correlated to
f=
0.316
Re 0.25
.
These two equations are plotted in Figure 2. Another important tool in evaluating
the friction factor for a pipe is the Moody diagram. It can be used for many types of
pipes and flow ranges. Experiments determined the friction factor for a range of
Reynolds numbers for turbulent flow (see Figure 2). Values of f can be obtained for the
case of the plexiglas pipe by following the smooth pipe curve.
Figure 2 Moody Diagram (pdf version on web site )
Figure 3. Dye Streaks showing Flow Stability. Typical Profiles for Laminar and Turbulent
Flow (time averaged). Note that your hardware is oriented for right to left flow but we
generally illustrate flows as being from left to right.
Note from the graph that in the laminar region, f is independent of the roughness,
as is expected; roughness has no effect on the head loss in laminar flow.
For fully developed laminar flow, the drag on a pipe is given by the HagenPoiseuille equation:
Dp 
8LQ
R2
.
This is developed from the Navier-Stokes equations and Newton's law of friction,
which also show that the velocity profile in laminar flow is parabolic (see Figure 3, and
Schlichting (1979)). It can be shown for laminar flow that the drag is proportional to the
first power of the mean velocity:
Dp = 8LV
As noted, this theory is based upon the assumption of fully developed pipe flow.
Boussinesq, the first to make theoretical investigations in laminar pipe flow development,
found the length of transition from entrance conditions to fully developed flow to be
z  ( 0.02  0.08 Re)  D .
In the turbulent region, the pressure drop and therefore the drag becomes
approximately proportional to V2. Turbulent mixing dissipates a large amount of energy,
causing a substantial increase in resistance to flow. In a time-averaged sense, the speed
profile is nearly constant across the entire diameter of the pipe, and falls off very rapidly
to the no-slip condition near the wall (see Figure 3). There is a vigorous exchange of
momentum between the main flow and the slow fluid near the wall, for which reason the
wall is often considered a "momentum sink." Turbulent flow is unsteady -- at a given
position the velocity is not constant in time but exhibits irregular fluctuations, which are
often averaged to determine a mean flow.
Using a control volume and applying Newton's second law of motion results in:
Dp = pR2
where Dp is the drag on the walls of pipe, p is the pressure change over which drag acts,
and R is the pipe radius.
Recall the derivation for hl . Assuming horizontal fully developed flow results in
 V
D 
p
2
2
 V
D 
p
8
L R 2 f
D
2
LD f
where Dp is the drag on the walls of a pipe, V is the mean velocity,  is the fluid density,
L is the length of pipe over which drag acts, D is the pipe diameter, R is the pipe radius,
and f is the friction factor.
4.
Introduction to the Apparatus
In this experiment, the water height in a reservoir will be set, and water will flow
through a pipe. The velocity of the flow will depend on the height in the reservoir. Two
different pipe diameters and two different reservoir levels will be used to obtain data in
the three regimes of flow. A beaker will be used to catch the fluid flowing from the end
of the pipe over a certain period of time. From this, the volumetric flow rate can be
calculated.
Dye will be injected into the pipe flow so that the nature of the flow (laminar,
transition, turbulent) may be observed. The flow regime will also be observed by
studying the condition of the flow as it exits the pipe.
IMPORTANT NOTE:
A diagram of the apparatus is shown in Figure 4.
Water surface (1)
h
Reservoir
(3
)
(2)
Figure 4. Application of the Bernoulli Equation to Reynolds
Pipe Flow
Pressure at (1) and pressure at (2) is equal to atmospheric pressure, so the pressure drop
in the pipe is
p   gh