By
Dr. Ajit Pratap Singh,
Civil Engineering Department,
BITS, Pilani-333031
Turbulent Flow through Pipes
Objective
➤ Theoretical discussion on Turbulent flow including
turbulent shear stress and Prandtl’s mixing length theory.
➤ To explain the development of velocity boundary layer in
pipe and to explain how to get length required to establish a
fully developed flow.
➤ To study Velocity Distribution in a pipe for turbulent flow
and to obtain velocity profile.
➤ To classify hydrodynamically smooth and rough pipes.
➤ To measure the pressure drop in the straight section of
smooth, rough, and packed pipes as a function of flow rate.
➤ To correlate this in terms of the friction factor and Reynolds
number.
➤ To compare results with available theories and correlations.
➤ To determine the influence of pipe fittings on pressure drop.
1
Theoretical Discussion
Fluid flow in pipes is of considerable importance in
process.
•Animals and Plants circulation systems.
•In our homes.
•City water.
•Irrigation system.
•Sewer water system
Turbulent flow
ØWhen fluid flow at higher flowrates, the
streamlines are not steady and straight and the
flow is not laminar. Generally, the flow field
will vary in both space and time with
fluctuations that comprise "turbulence”
ØIn turbulent flow the fluid particles are in
extreme state of disorder, their movement is
haphazard and large scale eddies are developed
which results in complete mixing of the fluid.
ØFor this case almost all terms in the NavierStokes equations are important and there is no
simple solution
DP = DP (D, µ, r, L, U,)
uz
úz
Uz
average
ur
úr
Ur
average
p
p
average
P’
Time
2
Laminar vs Turbulent Flow
• Laminar
• Turbulent
Turbulent flow
Laminar Flow
Turbulent Flow
3
Definition of turbulent
flow(Hinze)
“Turbulent fluid motion is an irregular
condition of flow in which the various
quantities show a random variation in time
and space, so that statistically distinct
average values can be discerned”
4
Reynolds Experiment
ì
•
•
•
< 2000
Laminar flow
hf µV
rVD ï
Re =
Reynolds Number
í2000 - 4000 Transition flow
µ ï
Turbulent f low h f µ V 2
Laminar flow: Fluid moves in î > 4000
smooth streamlines
Turbulent flow: Violent mixing,
fluid velocity at a point varies
randomly with time
Laminar
Turbulent
Boundary layer buildup in a pipe
Because of the shear force near the pipe wall, a boundary layer forms on the
inside surface and occupies a large portion of the flow area as the distance
downstream from the pipe entrance increase. At some value of this distance the
boundary layer fills the flow area. The velocity profile becomes independent of
the axis in the direction of flow, and the flow is said to be fully developed.
Pipe
Entrance
v
v
v
5
• Developing flow
– Includes boundary layer and
core,
– viscous effects grow inward
from the wall
• Fully developed flow
– In between the entrance
section and section AA, where
Pressure
the boundary layer thickness
equal to the radius of the pipe, Entrance
the velocity of pipe will vary pressure drop
from section to section due to
variation in thickness of BL
– Shape of velocity profile is
same at all points along pipe
after a section AA. The flow in
Le
pipe will be then truly uniform
D
and the flow is said to be
established.
Pipe Entrance
Fully developed
flow region
Entrance length Le
Region of linear
pressure drop
ì0.07 R e
ï
» í4.4Re 1/6 or
ï50
î
Le
for Laminar flow
x
ü
ýfor Turbulent flow
þ
Problem
• A 15-mm-diameter water pipe is 20 m long
and delivers water at 0.0005 m3/sec at 20
0C. What fraction of this pipe is taken up by
the entrance region so that after this region
fluid flow becomes fully developed? Take ν
= 1.01x10-6 m2/sec.
6
Fully Developed Turbulent Flow: Overview
One see fluctuation or randomness on the macroscopic scale.
mean
fluctuating
One of the few ways we can describe turbulent flow is by describing it in
terms of time-averaged means and fluctuating parts.
Turbulent Flow – Shear stresses
There are several theoretical models available for the
prediction of shear stresses in turbulent flow. However,
there is no general, useful model that can accurately
predict the shear stress for turbulent flow.
Ø We estimate shear stress by using experimental data,
semiempirical formulas and dimensional analysis
7
Modelling turbulent flow
• Why not solve the Navier-Stokes equations?
– No analytical solution possible
– In a computer, every small whirl would need to
be modelled. Even a 10cm3 volume would
require ~ 100,000,000 nodes
• Need to simplify
– Crossing of streamlines transfers momentum
between parts of the flow
Fully Developed Turbulent Flow: Overview
Now, shear stress:
However,
Laminar Flow:
Shear relates to random motion
as particles glide smoothly past
each other.
for turbulent flow.
Turbulent Flow:
Shear comes from eddy motion
which have a more random motion
and transfer momentum.
For turbulent flow:
Is the combination of laminar and turbulent shear. If there are no fluctuations,
the result goes back to the laminar case. The turbulent shear stresses are
positive, thus turbulent flows have more shear stress.
8
Fully Developed Turbulent Flow: Overview
The turbulent shear components are known as Reynolds Stresses.
Shear Stress in Turbulent Flows:
Turbulent Velocity Profile:
In viscous sublayer: tlaminar > tturb 100 to 1000 times greater.
In the outer layer: ttirb > tlaminar 100 to 1000 time greater.
The viscous sublayer is extremely small.
Apparent shear stress
• Apparent shear stress - Boussinesq(1877)
– Turbulence provides a shear in the flow in
addition to viscous shear
– Even in low viscosity fluids, there will be a
shear
du
τ
=
µ
T
T
– Propose an apparent viscosity
dy
– In general µT>µ , so ordinary viscosity can be
neglected
9
Reynolds stresses
Object: to include the random fluctuations in the
Navier-Stokes equations for the mean flow.
Method: represent all quantities by the mean plus fluctuation.
u = u + u¢
p = p + p¢
and so on
(T and r must also be considered for compressible flow)
Putting these into the Navier-Stokes equations and separating
out the time averaged and variable terms leads to a
modified set of equations
Reynolds stresses-continuity
Continuity - what goes in must come out!
¶u ¶v ¶w
+ +
=0
¶x ¶y ¶z
¶ (u + u ¢) ¶ (v + v¢) ¶ (w + w ¢)
In turbulent flow:
+
+
=0
¶x
¶y
¶z
¶u ¶v ¶w ¶u ¢ ¶v¢ ¶w ¢
Separating:
+ +
+
+
+
=0
¶x ¶y ¶z ¶x ¶y ¶z
¶u ¶v ¶w
Taking a time average:
+ +
=0
¶x ¶y ¶z
¶u¢ ¶v¢ ¶w¢
Therefore, the fluctuating part
+
+
=0
also satisfies the continuity equation ¶x ¶y
¶z
In laminar flow:
10
Reynolds stresses - Navier Stokes
Similarly, the N-S equations become (Schlichting, Ch 18)
ρg x -
æ ¶ 2 u ¶ 2 u ¶ 2 u ö æ ¶ u ¢2 ¶ u ¢v¢ ¶ u ¢w ¢ ö÷
¶p
+ µçç 2 + 2 + 2 ÷÷ - ρç
+
+
ç
¶x
¶
x
¶
y
¶
z
¶
x
¶
y
¶z ÷ø
è
ø è
æ ¶u
¶u
¶u ö
= ρçç u
+ v + w ÷÷
¶y
¶z ø
è ¶x
Shear stresses
Direct stress
Reynolds stresses
• Compared to the laminar Navier-Stokes
equation, one new term has been added.
The other terms have been averaged to
remove the time dependency.
• The terms on the left are the forcing terms,
gravity, pressure, viscosity and turbulence
• The terms on the right are the response
terms
du ¶u
¶u
(remember :
dt
=
¶t
+u
¶s
)
11
Reynolds stresses in 2D
•No z,w terms
y
u
•Steady, turbulent flow in x
direction
•Ignore gravity
x
¶p æ ¶ 2 u ¶ 2 u ¶ 2 u ö æç ¶ u¢2 ¶ u¢v¢ ¶ u¢w ¢ ö÷
ρg x - + µçç 2 + 2 + 2 ÷÷ - ρ
+
+
¶x è ¶x ¶y ¶z ø çè ¶x
¶y
¶z ÷ø
æ ¶u
¶u
¶u ö
= ρçç u + v + w ÷÷
¶y
¶z ø
è ¶x
Reynolds stresses in 2D
In 2D, the turbulent N-S equation therefore reduces to:
¶p
¶2u
¶ u¢v¢
¶u
+µ 2 -ρ
= ρv
¶x
¶y
¶y
¶y
Note that there are now two shear stress terms.
Re-writing: -
ö
¶p ¶ æ ¶u
¶u
+ çç µ - ρ u ¢v¢ ÷÷ = ρv
¶x ¶y è ¶y
¶y
ø
In turbulent flow, therefore
the shear stress is given by
τ=µ
¶u
- ρ u ¢v¢
¶y
12
Reynolds and Boussinesq
Boussinesq proposed an additive turbulent shear stress:
τ=µ
¶u
¶u
+ µT
¶y
¶y
So the additive term is equivalent to the Reynolds’ stress.
However, we need to know values for
u¢v¢
in order to use this
Are the Reynolds’ stresses related to the flow velocity?
Prandtl‘s Mixing Length Theory
• That distance in the transverse direction which
must be covered by a lump of fluid particles
traveling with its original mean velocity in order
to make the difference between its velocity and the
velocity of new layer equal to the mean transverse
fluctuation in turbulent flow
13
Prandtl‘s Mixing Length Theory
u( y)
mean
velocity
lump of
turbulence
lump of
turbulence
(mixed)
~ v¢
~ u¢
y,v
¶u
¶y
x,u
turbulent shear
flow along
solid wall
(not valid close
to the wall)
mixing length l
v¢ ~ + l1 ×
¶u
¶y
Þ u ¢ ~ ± l2 ×
¶u
¶y
defined as that downstream distance, which is
needed for the lump of turbulence to be
completely mixed with the surrounding fluid
l1 = l2 = l
Prandtl’s Mixing Length…
•Analogous to the kinetic theory of gases
•Used because ‘it works’
u(y)
y
y2
y1
l
Suppose ‘lumps’ of fluid move
randomly from one shear layer
to another, a distance l apart.
This carries momentum and the
velocity difference must
therefore be related to the
turbulence
14
Prandtl’s Mixing Length
u¢ µ l
¶u
¶y
Turbulence is even in all directions (homogeneous)
v¢ µ u ¢ µ l
¶u
¶y
So the Reynolds shear stress must be proportional to the
square of the mixing length times the velocity gradient:
æ ¶u ö
u ¢v¢ µ l çç ÷÷
è ¶y ø
2
2
Prandtl’s Mixing Length…
Returning to the equation for the shear stress:
τ=µ
¶u
- ρ u ¢v¢
¶y
τ=µ
¶u
¶u
+ µT
¶y
¶y
æ ¶u ö
¶u
µT
= - r u ¢v¢ µ ρl 2 çç ÷÷
¶y
è ¶y ø
2
This gives a direct relationship between turbulent ‘viscosity’
and velocity gradient in the flow
¶u
µ T = ρl
¶y
2
2
15
Total Shear stress at any point
• Total Shear stress at any point is the sum of the viscous
shear stress and turbulent shear stress and may be
expressed as
æ dv ö
dv
τ=µ
+ ρl 2 çç ÷÷
dy
è dy ø
2
• The significance of Prandtl’s turbulent shear stress
equation is that it is possible to make suitable assumptions
regarding the variation of the mixing length
Prantdl’s Mixing Length
• We still need a value for the mixing length, l.
• In free turbulence, l will be constant.
• In wall generated turbulence, l will vary as
the distance from the wall. (l=ky)
• For a smooth wall y=0, l=0
• For a rough wall y=0, l=k (the surface
roughness)
16
Mixing length measurement in pipes
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
The Universal Law of The Wall
CE CF312: Hydraulics by Dr. A. P. Singh
17
The Universal Law of The Wall
The Universal Law of The Wall
æ dv ö
τ 0 = ρk 2 y 2 çç ÷÷
è dy ø
2
æyö
v = v max + 2.5V* Log e ç ÷
èRø
v max - v
æRö
= 2.5V* Log e çç ÷÷
V*
èyø
CE CF312: Hydraulics by Dr. A. P.
Singh
18
Hydrodynamically Smooth and
Rough Pipe Boundaries
• In general a boundary with irregularities of large
average height k, on its surface is considered to be
rough boundary and the one with smaller k values
is considered as smooth boundary.
• Hopf found two types of roughness:
– Coarse, dense roughness where f is a function
of roughness ratio, k/D, and is independent of
the Reynolds number
– Gentle, less dense roughness, where f is a
function of both Re and roughness ratio
– The significant factor is the roughness height
compared to the laminar sub-layer.
Hydrodynamically Smooth and
Rough Pipe Boundaries
A systematic study is complicated by different types:
1. Shape
2. Height
3. Density
19
Hydrodynamically Smooth and
Rough Pipe Boundaries
• We should also consider flow and fluid characteristics for
proper classification of smooth and rough boundaries.
• If k is the average height of rough projections on the surface
of the plate and δ is the thickness of the boundary layer,
then the relative roughness (k/ δ) is a significant parameter
indicating the behavior the boundary surface of a plate.
• If the boundary layer is turbulent from the leading edge of
the plate, the front portion of the plate will act as
hydrodynamically rough followed by transition region and
the downstream portion of the plate will be
hydrodynamically smooth if the plate is sufficiently long.
Hydrodynamically Smooth and
Rough Pipe Boundaries
Turbulent
Laminar
d
• As the flow outside the laminar sub-layer is turbulent, eddies
of various sizes are present which try to penetrate through the
laminar sub-layer. But due to greater thickness of the laminar
sub-layer the eddies cannot reach the surface irregularities
and thus the boundary act as a smooth boundary.
• In the laminar sub-layer, any vortices generated by the
roughness is damped out, so if k<d, then the law of friction
for smooth pipes will apply
20
Hydrodynamically Smooth and
Rough Pipe Boundaries
• With the increase in Reynolds number, the thickness of
the laminar sub-layer decreases, and it can even
become much smaller than the average height k, of
surface irregularities. The irregularities will then
project through the laminar sub-layer and laminar sublayer is completely destroyed. The eddies will thus
come in contact with the surface irregularities and large
amount of energy loss will take place and thus the
boundary act as a rough boundary.
From Nikuradse’s experiment
• Hydrodynamically
smooth pipe
• Transition region in a
pipe
• Hydrodynamically
Rough Pipe
k
< 0.25
δ'
0.25 <
k
< 6.0
δ'
k
> 6.0
δ'
21
• Hydrodynamically smooth Plate
V*k s
<5
υ
• Plate in Transition region
5<
V*k s
< 70
υ
• Hydrodynamically Rough
V*k s
> 70
υ
Where ks is equivalent
sand grains roughness
defined as that value of
the roughness which
would offer the same
resistance to the flow
past the plate as that of
due to the actual
roughness on the surface
of the plate.
From Nikuradse’s experiment
• Hydrodynamically
smooth pipe
k
æ V*k ö
£
0.2
5
or
ç
÷£3
δ'
u
è
ø
• Transition region in a
k
æVkö
0.25 < ' < 6.0 or 3 < ç * ÷ < 70
pipe
δ
è u ø
• Hydrodynamically
Rough Pipe
k
æVkö
³ 6.0 or ç * ÷ ³ 70
'
δ
è u ø
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
22
Nikuradse’s experimental studies of turbulent
flow in smooth pipes have also shown that
• In smooth pipes of
any size the value of
the parameter
V* y
11.6υ
= 11.6 for y = δ ' Þ δ ' =
υ
V*
V* y '
0.108υ
= 0.108 for y = y ' Þ y ' =
υ
V*
Þ y' =
δ'
107
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
Velocity Distribution in Smooth Pipes
•
•
In the vicinity of a smooth boundary there exists a laminar
sublayer. The flow in the laminar sublayer being laminar has a
parabolic velocity distribution.
As such the velocity which is zero at the pipe boundary
increases parabolically in the zone of laminar motion, which
extends up to certain distance from the boundary. Above the
zone of laminar motion there exists a transition zone where the
flow changes from laminar to turbulent Beyond the transition
zone the flow is turbulent having logarithmic velocity
distribution. Since the change from parabolic to logarithm
i.e. distribution is gradual, the zone of laminar motion will
extend well beyond the distance y', as shown in Fig.. However,
in the absence of any specific line of demarcation between the
different zones of flow near the boundary, the intersection of
the parabolic and the logarithmic velocity distribution curves,
as shown in Fig. is arbitrarily chosen as nominal border line
between the two types of flow.
23
Velocity Distribution for turbulent
flow
• Velocity Distribution
v
= 5.75 log
in a hydrodynamically
V*
smooth pipe
10
• Velocity Distribution
v
= 5.75 log
in a hydrodynamically
V*
Rough Pipes
10
V* y
+ 5.5
υ
æyö
ç ÷ + 8.5
èkø
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
Velocity Distribution for turbulent
flow in terms of Mean Velocity (V)
• Velocity Distribution V
= 5.75 log
in a hydrodynamically
V
smooth pipe
*
10
• Velocity Distribution V
= 5.75 log
in a hydrodynamically
V
Rough Pipes
*
æRö
÷ + 4.75
10 ç
èkø
V*R
+ 1.75
υ
24
Law of the Wall
u
V*
5.5
Laminar
sub-layer
Buffer
zone
Turbulent
layer
yV*
ln
υ
Turbulent Flow – Velocity Profile
For turbulent flow in tubes the time-averaged velocity profile can be
expressed in terms of the power law equation. n =7 is usually a good
approximation.
1/ n
u æ
rö
= ç1 - ÷
V è Rø
where V is the velocity
at the centerline
25
Losses due to Friction/The Friction
Factor
For turbulent flow there is no rigorous theoretical treatment available. In
order to determine an expression for the losses due to friction we must
resort to experimentation.
L V2
hf µ
D
where L=length of the pipe,
D=diameter of the pipe, V=velocity,
By introducing the friction factor, f:
L V2
hf = f
D
where
f =
hf
( L / D )(V 2 / 2 g )
Flow in Pipes
Hopf found two types of roughness:
•Coarse, dense roughness where f is a function of
roughness ratio, k/D, and is independent of the Reynolds
number
•Gentle, less dense roughness, where f is a function of
both Re and roughness ratio
The significant factor is the roughness height compared to
the laminar sub-layer.
26
Nikuradse’s Experiments
•
In general, friction factor
•
Function of Re and
roughness f = F (Re,
•
•
e
)
D
Laminar region
– Independent of
64
roughness
f =
Re
Turbulent
region
– Smooth pipe curve
• All curves
coincide @
~Re=2300
– Rough pipe zone
• All rough pipe
curves flatten out
and become
independent of Re
f =
0.25
é
5.74 öù
æ k
êlog 10 ç 3.7 D + Re 0.9 ÷ú
è
øû
ë
2
f =
f =
k
(Re )1 / 4
Rough
Blausius
64
Re
Blausius OK for smooth pipe
Laminar
Transition
Smooth
Turbulent
CE F312: Hydraulics Engineering by Dr. A. P. Singh
The Friction Factor
The mechanical energy equation can be written:
(
Wshaft æ
P2 P1
V2 V2
L V 2 ö÷
- ) + ( 2 - 1 ) + g ( z 2 - z1 ) =
-ç4 f
ç
r r
2
2
m!
D 2 ÷ø
è
Or in terms of heads:
(
Wshaft æ
P2 P1
V2 V2
L V 2 ö÷
- ) + ( 2 - 1 ) + ( z 2 - z1 ) =
-ç4 f
ç
rg rg
2g 2g
m! g
D 2 g ÷ø
è
Knowledge of the friction factor allows us to estimate the
loss term in the energy equation
27
Friction factor: The Moody Chart
The Moody Chart (Figure 14.10 textbook) provides a convenient
representation of the functional dependence f = f(Re, k/D)
Ø For laminar flow:
f = 64 / Re
Ø For turbulent flow:
æ k/D
1
2.51
= -2 log ç
+
ç 3.7 Re f
f
è
ö
÷
÷
ø
Colebrook formula
1/ 3
é æ
k 106 ö ù
÷ ú
f = 0.001375 × ê1 + çç 20,000 +
D Re ÷ø ú
êë è
û
Ø For turbulent flow, with Re<105 and for hydraulically smooth surfaces:
f =
0.316
Re 1/ 4
Blasius formula
Surface Roughness
Additional dimensionless group k/D need
to be characterize
Thus more than one curve on friction factorReynolds number plot
Fanning diagram or Moody diagram
Depending on the laminar region.
If, at the lowest Reynolds numbers, the laminar portion
corresponds to f =16/Re Fanning Chart
or f = 64/Re Moody chart
CE F312: Hydraulics Engineering by Dr. A. P. Singh
28
Variation of friction factor for
Commercial pipes
White Equation
R/K ö
æ
æRö
- 2.0 log 10 ç ÷ = 1.74 - 2.0 log 10 ç1 + 18.7
÷
f
Re f ø
èKø
è
1
Colebrook formula
1
æ k/D 2.51 ö
= -2.0 log 10 ç
+
÷
3.7
f
Re
f
è
ø
CE F312: Hydraulics Engineering by Dr. A. P. Singh
• The Colebrook equation is
implicit in f, and thus the
determination
of
the
friction factor requires
some
iteration.
An
approximate
explicit
relation for was given by
S.E. Haaland in 1983.
• The results obtained from
this relation are within 2
percent of those obtained
from
the
Colebrook
equation.
é 6.9 æ k / D ö1.11 ù
1
@ 1.8 log ê
+ç
÷ ú
Re
f
è 3.7 ø ûú
ëê
CE F312: Hydraulics Engineering by Dr. A. P. Singh
29
Moody Diagram
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
Fanning Diagram
é
ù
1
D
D/e
= 4.0 * log + 2.28 - 4.0 * logê4.67
+ 1ú
e
f
Re f
ë
û
1
D
= 4.0 * log + 2.28
e
f
f =16/Re
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
30
Following observations from the Moody
chart:
• For laminar flow, the friction factor
decreases with increasing Reynolds number,
and it is independent of surface roughness.
• The friction factor is a minimum for a
smooth pipe (but still not zero because of
the no-slip condition) and increases with
roughness. The Colebrook equation in this
case (k=0) reduces to the Prandtl equation
• The transition region from the laminar to turbulent
regime (2300 < Re < 4000) is indicated by the
shaded area in the Moody chart. The flow in this
region may be laminar or turbulent, depending on
flow disturbances, or it may alternate between
laminar and turbulent, and thus the friction factor
may also alternate between the values for laminar
and turbulent flow. The data in this range are the
least reliable. At small relative rougnesses, the
friction factor increases in the transition region
and approaches the value for smooth pipes.
31
Equivalent roughness values for new
commercial pipes
Material
Roughness, k (mm)
Glass, plastic
0 (smooth)
Concrete
0.9 to 9
Wood stave
0.5
Rubber, smoothed
0.01
Copper or brass tubing
0.0015
Cast iron
0.26
Galvanized iron
0.15
Wrought iron
0.046
Stainless steel
0.002
Commercial steel
0.045
Type of Problems
• Determining the head-loss or pressure drop from
the given values of Q, L, D, pipe roughness κ,
kinematic viscosity ν.
• Determining the Q from the given values of headloss or pressure drop due to friction, L, D, pipe
roughness κ, kinematic viscosity ν.
• Determining the dia of pipe from the given values
of head-loss or pressure drop due to friction, Q, L,
pipe roughness κ, kinematic viscosity ν.
32
• Type 1
– Calculate Re and k/D from the given data
– Obtain f from the Moody’s chart
• Type 2
– Calculate k/D from the given data and Re√f
æ 2gh D ö
from Re f = VD
ç
÷
υ è V L ø
– Using Coolebrook formula and the above
equation, Obtain f
– Obtain Re from the Moody’s chart and hence Q
1/2
f
2
• Type 3: Dia is unknown
– Assume a suitable value of f and calculate Dia
from Darcy-Weisbatch equation
– With this trial value of D, calculate k/D and Re
– With this k/D and Re, calculate f from Moody’s
diagram
– Repeat the process till f becomes same
33
Swamee and Jain in 1986 proposed the following
explicit relations that are accurate to within 2% of
the Moody chart
0.5
0.5
é k
æ 3.17u 2 L ö ù
æ gD 5 h L ö
÷÷ ú
÷÷ ln ê
Q = -0.965çç
+ çç
3
êë 3.7D è gD h L ø úû
è L ø
4.75
5.2
é
æ LQ 2 ö
æLö ù
÷÷ + υQ 9.4 çç ÷÷ ú
D = 0.66 êk1.25 çç
êë
è gh ø úû
è gh L ø
0.04
Re > 2000
10 -6 < k/D < 10 - 2
5000 < Re < 3 ´108
APPARATUS
Pipe Network
Rotameters
Manometers
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
34
Problem 1
• Water at 15 0C is flowing steadily in a 5
cm-diameter horizontal pipe made of
stainless steel at a rate of 0.34 m3/min.
Determine the pressure drop, the head loss,
and the required pumping power input for
flow over a 61m-long section of the pipe.
Dynamic viscosity = 1.138x10-3 kg/m-sec
Problem 2
• A person with no experience in fluid mechanics wants to
estimate the friction factor for 2.5-cm-diameter
commercial galvanized iron pipe (k=0.150 mm) at a
Reynolds number of 8000. They stumble across the simple
equation of f = 64/Re and use this to calculate the friction
factor. Explain the problem with this approach and
determine percentage error in evaluating friction factor by
the person. (Answer: 80%)
35
Problem 3
• After 15 years of service a steel pipe main
0.6 m in diameter is found to require 40%
more power to deliver the 300 liters/second
for which it was originally designed.
Determine the corresponding magnitude of
the rate of roughness increase α. Take
kinematic viscosity of water ν = 0.015
Stokes roughness protrusions height for new
steel pipe = 0.045 mm.
Problem 4
• A 15-mm-diameter water pipe is 20 m long
and delivers water at 0.0005 m3/sec at 20
0C. What fraction of this pipe is taken up by
the entrance region so that after this region
fluid flow becomes fully developed? Take ν
= 1.01x10-6 m2/sec.
36
Problem 5
• A new reservoir will use gravity to supply drinking water
to a water treatment plant serving several surrounding
towns as shown in Figure. The required flow rate is 0.315
m3/sec. The surface of the reservoir is 61 m above the plain
where the water treatment plant is located, and the supply
pipe is commercial steel, 914.4 mm in diameter. If the
minimum pressure required at the water treatment plant is
347.7 kpa (gage), how far away can the reservoir be
located with this size pipe? Assume that minor losses are
negligible and that the water is at 283.1 K. The average
height of the pipe wall roughness protrusions may be taken
as 0.0458 mm. Take kinematic viscosity of water ν =
0.13x10-5 m2/sec .
37
Problem
• For flow in open channels assume turbulent
shear to the constant т = тo and the mixing
length variation with y is given by l = 0.40 y
for y ≤ 0.20 D, and l = 0.08D for y ≥ 0.20D
where D is the depth of flow. Obtain the
velocity distribution law which will satisfy
the boundary condition, v = V at y = D.
• A commercial new galvanized iron service
pipe from a water main is required to
deliver 200 L/s of water during a fire. If the
length of the service pipe is 35 m, the
allowable head loss in the pipe is 50 m and
kinematic viscosity of water at 20 0C is 1.00
x 10-6 m2/sec, what will the pipe diameter to
be used for this purpose?
38
• Water at 200C is to be pumped from a reservoir (ZA = 2 m)
to another reservoir at a higher elevation (ZB = 9 m)
through two 25-m long plastic pipes connected in parallel.
The diameters of the two pipes are 3 cm and 5 cm. Water
is to be pumped by a 68 percent efficient motor-pump unit
that draws 7 kW of electric power during operation. The
minor losses and the head loss in the smaller single pipes
that connect both the parallel pipes to the two reservoirs
are considered to be negligible. Determine the total flow
rate between the reservoirs and the flow rates through each
of the parallel pipes.
39
Pipe Flow Summary
ØThe statement of conservation of mass, momentum and
energy becomes the Bernoulli equation for steady state
constant density of flows.
Ø Dimensional analysis gives the relation between flow rate and
pressure drop.
ØTurbulent flow losses and velocity distributions require
experimental results.
ØExperiments give the relationship between the fraction factor
and the Reynolds number.
Ø Head loss becomes minor when fluid flows at high flow rate
(fraction factor is constant at high Reynolds numbers).
Images - Laminar/Turbulent Flows
Laser - induced florescence image of an
incompressible turbulent boundary layer
Laminar flow (Blood Flow)
Simulation of turbulent flow coming out of a
tailpipe
Turbulent flow
CE C371: Hydraulics & Fluid
Mechanics by Dr. A. P. Singh
Laminar flow
40