ME408 Fluid Mechanics II
I. Viscous Flow in Circular Pipe
1. Reynolds Experiment
1.1. Flow Rate and Mean Velocity
The steady flow rate (Q) can be measured by collecting a volume
(Vol) of water within a period of time (t), and the mean velocity
(V) is found by dividing Q by the inner cross-section area (A) of pipe:
Q
Vol
t
,
V 
Q 4Q

A D 2
1.2. Reynolds Number (Re) is defined by the mean velocity (V), inner
diameter (D) of the pipe, density () and viscosity () or kinematic
viscosity () of fluid, and is the ratio of inertial force to viscous
force:
VD VD
Re 


Reynolds observed that the flow is:
(a) Laminar when Re < 2000
(b) Transitional when 2000 < Re < 4000

(c) Turbulent when Re > 4000
Reynold’s Experiment
Typical measured time-history of a local velocity u(t,r).
1.3. Entrance Flow
At the entrance region of the pipe, the velocity profile is varying in both the radial direction (r) and the axial direction (x),
until the growing viscous boundary layer along the inner wall meet at the centerline. Experimental observation shows that
this entrance length depends on the Reynolds number:
(a) For laminar flow (Re < 2300):
(b) For turbulent flow (Re > 4000):
Le
 0.06 Re
D
Le
 4.4 Re 1/ 6
D
2. Fully-Developed Laminar Flow (Re < 2000)
Downstream of the entrance length, the flow is called fully-developed because the
velocity profile u(r) no longer changes in the axial direction. For laminar flow, we
can derive the exact mathematical expression for u(r), and subsequently predict the
frictional loss along the pipe.
2.1. Velocity Profile
Consider a volume of the fluid in the form of a circular cylindrical in a circular
pipe inclined at a slope angle of  from the horizon. For steady flow (zero
acceleration), the net force in the direction of flow must be zero:
F
x
 p(r 2 )  ( p  p)(r 2 )   (r 2l ) g (sin  )  (2rl )  0
Recall the definition between laminar viscosity and shear
du
du
stress (y is the distance from the pipe wall, hence dy = - dr):    dy    dr
Simplifying the governing equation above and substituting this definition into it
results in the following simple differential equation:
 p  gl (sin  ) 
du
 
r
dr
2


l


Problem 1-1: Integrating with respect to r and applying the no-slip boundary condition at the inner wall u(R) = 0, show that
the velocity profile is parabolic:
2
 p  gl (sin  )  2
 p  z  2 
r 
2
u
 ( R  r )   4   l  R 1  R 2 
4


l



 

In which R is the radius of the inner wall, g is the specific weight of the fluid, and z=l sin is simply the change in
elevation.
2.2. Centerline Velocity, Flow Rate, and Mean Velocity
(a) The centerline velocity along the axis (r=0) is therefore:
 p  z  2
Vc  u (0)  
R
 4  l 
2


(b) The volume flow rate can be found by integrating dQ  udA  Vc 1  r (2rdr ) from r=0 to R.
2



Problem 1-2: Show that the volume flow rate through the pipe is:
(c) The mean velocity is therefore
Q
R 
R 2Vc
2
D 4 [p    l (sin  )]

128  l
Vc D 2 [p    l (sin  )]
Q
V


R 2 2
32  l
This velocity is also the uniform velocity at the very entrance of the pipe before the boundary layer starts.
2.3. Viscous Head Loss and Darcy’s Friction Factor
Since the velocity profile remains the same along the pipe (steady fully-developed flow), there is no change in linear
momentum. The momentum equation in the x-direction (net force = net change of linear momentum = 0) between two typical
cross-sections (1 & 2) becomes:
F
x
 p(R 2 )  ( p  p)(R 2 )   (R 2l ) g (sin  )  (2Rl ) w  0 where
p
Simplifying:

 z 
p1  p2

 z1  z 2 
 du 
is shear stress along wall.

 dr  r  R
 w   
2l w
 hf
R
The term on the RHS is called viscous (friction) head loss (hf) since it is proportional to the shear stress
2
along the wall and has the same dimension as z (elevation change). Furthermore, it is convenient to define h  f l V
f
a friction factor (f) with the following correlation between the head loss and the mean velocity head:
D 2g
Problem 1-3:
Show that for fully-developed laminar flow in a circular pipe, Darcy’s friction factor is:
f 
8 w
64

2
V
Re
2.4. Kinetic Energy Correction Factor
Since the velocity profile is parabolic the rate of kinetic
energy carried by the fluid at each cross-section of the pipe
can be found by:
2
2
Ru
Ru
R
u2
3

dQ


d
(
udA
)


u
(
2

rdr
)


2
0 2
0 2
0 u (rdr )
For convenience, a correction factor () is introduced to rewrite this kinetic energy in terms of
R
V2
3

(

Q
)


the mean velocity and the total volume flow rate:
0 u (rdr )
2
Problem 1-4:
Show that for fully-developed laminar flow in a circular pipe, the kinetic energy correction factor  is exactly = 2.
2.5. Energy Equation (Conservation of Energy/First Law of Thermodynamics)
Between two typical cross-sections (1 & 2) of a steady
2
2
 p1
 
 p2

V1
V2
flow in a pipe, the balance of rate of energy for can be
Q   
 gz1  uˆ1   Qin  Q   
 gz 2  uˆ 2 
simplified to:
2
2




Recall that Q is the mass flow rate, and the terms in the brackets are the flow work, kinetic energy, potential energy and
internal energy per unit mass, respectively; and the last term on the LHS is the rate of heat added from the surrounding.
For a pipe with constant diameter, the mean velocity remains constant. Hence the
kinetic energies at sections 1 and 2 are the same and are cancelled out. Dividing
through by Qg (=Q):
p1  p2

uˆ 2  uˆ1 Q in
 z1  z 2 

g
Q
The LHS of this equation is precisely equal to the friction head loss, when compared with the

momentum equation derived above. The physical meaning of this result is all but familiar to us: h  uˆ2  uˆ1  Qin
f
g
Q
mechanical energy loss due to friction is dissipated into the surrounding in the form of heat transfer, or
causes an increase in internal energy, or both:
Increase of internal energy is associated with a rise of temperature T, related through the specific heat (C): uˆ2  uˆ1  C (T2  T1 )
3. Turbulent Pipe Flow (Re>4,000)
When the flow in the pipe is turbulent, the simple relation between the shear stress and rate of strain through viscosity is no
longer valid. The velocity profile, head loss and friction factor, etc. cannot be derived analytically, but can be found
experimentally. These experiments show that these quantities are functions of both Reynolds number and the relative
roughness (/D) of the pipe wall.
3.1. Experimental Velocity Profile
It is observed experimentally that viscous turbulent flow in pipes can be subdivided
into three layers: a very thin wall layer where the flow is essentially still laminar (in
which the adjacent wall suppresses turbulent mixing), an overlap layer which serves
as a transition from the wall layer to the fully turbulent outer layer.
Although the velocity u(r,t) at each point (r-location) is always fluctuating in all
directions in time, the time-average of u is still steady if the volume flow rate steady.
The overall time-average velocity profile can be measured with tiny sensors (such as
a hotwire), and the following power law is generally observed:
u  Vc  (1 
r 1 / n In which Vc  u(0) is the centerline velocity, and
)
R
7 < n < 8, depending on Re.
3.2. Mean Velocity and Kinetic Energy Correction Factor
R
R
Since Q  R 2V  u (2rdr )  2Vc (1  r )1 / n (rdr )
0
0 R
One can evaluate the integration and conclude that the mean velocity is:
2n 2
V
Vc
(n  1)( 2n  1)
For n = 7, V = 0.817 U0 ; and for n = 8, V = 0.837 U0 .
Hence, the results are not much different in the range 7 < n < 8.
Problem 1-5:
From the definition

R
3
V2
 ( Q )    u rdr
0
2
(a) Show that the Kinetic Energy Correction
Factor for turbulent flow in circular pipe is:
(b) Evaluate  for n = 7 and n = 8. Conclude that
the difference is small, and is approximately = 1.

2n 2
(n  3)( 2n  3)


2n 2
 (n  1)( 2n  1) 


3
3.3. Experimental Head Loss and Friction Factor
Turbulent Pipe Flow: Experimental Friction Factor
By measuring the flow rate discharging from a pipe, and
the pressure and elevation at two stations (1 & 2) along the
pipe of a certain average /D, one can readily calculate the
Reynolds number, and apply the momentum equation (as
in Section 2.3) to calculate the friction head loss:
0.06
0.05
hf 
The friction factor is simple found from its definition (as
introduced in Section 2.3):
0.04
Friction Factor (f )
2l w p1  p2

 z1  z 2
R

 2g D 
f  hf  2 
V l 
0.03
By changing the flow rate with a valve, different Re can be
set, and the measurements and calculations are repeated to
obtain the corresponding f. A curve fitting through these
data points can be plotted for each /D. Repeat the
experiment with pipes of different /D would establish a
set of similar curves on the Re versus f plot.
0.02
e/D=0
e/D=0.005
e/D=0.01
e/D=0.015
e/D=0.02
0.01
The empirical Colebrook formula is found to fit these
experimental data closely:
  / D 2.51 
1
 2.0 log 

3
.
7
Re f 
f

0
0
5000
10000
15000
20000
Renolds No. ( Re )
25000
30000
35000
But it is inconvenient to apply this empirical formula since
f appears on both sides, and has to be found by iteration.
A more convenient empirical formula, called Haaland’s approximation, is also fairly accurate:

  / D 1.11 6.9  


f  1.8 log 
 

Re  

 3.7 


2
4. Moody’s Chart
The dependence of f on Re (and /D for turbulent flow) is summarized in this chart by plotting the curves calculated from
Colebrook’s formula. It is shown on log-log scale to emphasize the variation at low Re and de-emphasize the change at high
Re. One can conclude that:
(a) For laminar flow, f (=64/Re) strongly depends on Re.
(b) For turbulent flow with relatively low /D, f is also sensitive to Re.
(c) In the “wholly-turbulent flow” region (at high Re or high /D), f is only sensitive to /D, and almost independent of Re.
(d) In the transition range (2000 < Re < 4000), the friction factor f is uncertain. Hence, it is wise to avoid designing pipe flows
in this Re-range of uncertainty.
Moody’s Chart
5. Energy Equation (Conservation of Energy/First Law of Thermodynamics) for a Single Pipe Flow
Between two typical cross-sections (1 & 2) of a steady flow in a pipe, the balance of rate of energy can be simplified to:
2
2
 p1
 
 p2

V1
V2
Q   
 gz1  uˆ1   Qin  Q   
 gz 2  uˆ 2 
2
2




In which Q is the mass flow rate, and the terms in the brackets are the flow work, kinetic energy, potential energy and
internal energy per unit mass, respectively; and the last term on the LHS is the rate of heat added from the surrounding.
For a pipe with constant diameter, the mean velocity remains constant. Hence the kinetic energies at sections 1 and 2 are the
same and are cancelled out. Dividing through by Qg (=Q):
p1  p2

 z1  z 2 
uˆ 2  uˆ1 Q in

g
Q
The LHS of this equation is precisely equal to the friction head loss, when compared with the momentum equation derived
above. The physical meaning of this result is all but familiar to us: mechanical energy loss due to friction is dissipated into the
surrounding in the form of heat transfer, or causes an increase in internal energy, or both:
uˆ 2  uˆ1 Q in
hf 

g
Q
Increase of internal energy is associated with a rise of temperature T, related through the specific heat (C):
uˆ2  uˆ1  C(T2  T1 )
6. Pipe Systems
6.1. Minor Losses
The friction head loss is also known as the major loss in a single pipe of significant length. In addition to the pipe, a typical
piping system usually contains other components such as entrances and exits, elbows and bends, contractions and expansions,
fittings and valves, etc. Each of these components causes a minor head loss. When summed together, the total minor loss may
become as significant as the major loss. Each minor loss is evaluated in terms of an experimentally determined loss coefficient
(KL), defined so that the minor head loss is calculated in terms of the mean velocity head:
2
hL min or  K L
V
2g
6.2. Total Head Loss
The total head loss, which becomes heat dissipation and/or increase of internal energy
V 2 uˆ2  uˆ1 Q in
of the pipe system, is therefore the sum of the friction head loss in the pipe and all the
hL  h f   K L


minor losses:
2g
g
Q
6.3. Machine Heads
A complete piping system may also contain machines such as pumps and turbines. Pumps are needed to move the fluid through
the pipe by adding power to the fluid, and turbines may be inserted to extract mechanical power from the fluid. Pump head
(input) and turbine head (output) are defined by dividing their power by the weight flow rate (Q):
W P
hP 
Q
WT
hT 
Q
6.4. Energy Equation
A complex piping system may also contain pipes of different diameters in series, or reservoirs with free surfaces at which
velocity is negligible. Hence the kinetic energy terms at the inlet and outlet sections must be retained. For steady flow in a pipe
system with pumps, turbines and minor losses:
2
2
 p1
 
 p2

V1
V2

Q   1
 gz1  uˆ1   Qin  WP  Q    2
 gz 2  uˆ 2   WT
2
2




Dividing through by Q, and substituting in the total head loss for the
heat dissipation and internal energy terms:
p1
2
2
V
p
V
 1 1  z1  hP  2   2 2  z 2  hT  hL

2

2
7. Non-Circular Pipes
Analysis of viscous flow in a non-circular pipe or a partially-filled pipe is essentially the same as that for circular pipes simply
by replacing the pipe diameter with the “hydraulic diameter” defined as:
4A
Dh 
P
where A is the cross-section area of the fluid in the pipe and P is the wetted perimeter of the cross-section of the inner wall.
8. Measurement of Volume Flow Rate in Pipes
8.1 Flow Meters
Orifice, nozzle and venture meters are inexpensive devices to measure the volume flow rate in a pipe. By measuring the
pressure drop from the inlet to the throat of the device, and solving the steady-state continuity equation and the Bernoulli’s
equation (which neglects viscous head loss) simultaneously, the ideal flow rate can be calculated.
The actual flow rate is found by multiplying a discharge coefficient, which is determined experimentally for each device. The
discharge coefficient depends significantly on the throat-to-pipe diameter ratio of each device and slightly on the Reynold’s
number of the pipe flow.
8.2. Commercial Flow-Meters
These are more expensive devices that read the flow rate directly, and there are numerous types and designs.