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CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 1
By
Dr. Ajit Pratap Singh,
Civil Engineering Department,
BITS, Pilani-333031
Turbulent Flow through PipesObjective
� Theoretical discussion on Turbulent flow includingturbulent shear stress and Prandtl’s mixing length theory.
� To explain the development of velocity boundary layer inpipe and to explain how to get length required to establish afully developed flow.
� To study Velocity Distribution in a pipe for turbulent flowand to obtain velocity profile.
� To classify hydrodynamically smooth and rough pipes.
� To measure the pressure drop in the straight section ofsmooth, rough, and packed pipes as a function of flow rate.
� To correlate this in terms of the friction factor and Reynoldsnumber.
� To compare results with available theories and correlations.
� To determine the influence of pipe fittings on pressure drop.
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�When 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”
�In 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.
�For this case almost all terms in the Navier-
Stokes equations are important and there is no
simple solution
∆∆∆∆P = ∆∆∆∆P (D, µµµµ, ρρρρ, L, U,)
uz
úz
Uz
average
ur
úr
Ur
average
p
P’
p
average
Time
Laminar vs Turbulent Flow
• Laminar • Turbulent
Turbulent flow
Laminar Flow
Turbulent Flow
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 2
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”
Reynolds Experiment
• Reynolds Number
• Laminar flow: Fluid moves in
smooth streamlines
• Turbulent flow: Violent mixing,
fluid velocity at a point varies
randomly with time
∝>
−
∝<
=2lowfTurbulent4000
flowTransition40002000
flowLaminar2000
Re
Vh
VhVD
f
f
µ
ρ
Laminar Turbulent
Boundary layer buildup in a pipe
Pipe
Entrance
v vv
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• Developing flow
– Includes boundary layer andcore,
– viscous effects grow inwardfrom the wall
• Fully developed flow
– In between the entrancesection and section AA, wherethe boundary layer thicknessequal to the radius of the pipe,the velocity of pipe will varyfrom section to section due tovariation in thickness of BL
– Shape of velocity profile issame at all points along pipeafter a section AA. The flow inpipe will be then truly uniformand the flow is said to beestablished.
eL
Entrance length LeFully developed
flow region
Region of linear
pressure drop
Entrance
pressure drop
Pressure
x
≈
flowTurbulent for 50
or 4.4Re
flowLaminar for R 0.07
D
L 1/6
e
e
Problem
• A 15-mm-diameter water pipe is 20 m long
and delivers water at 0.0005 m3/sec at 200C. 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.
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 3
Fully Developed Turbulent Flow: Overview
One see fluctuation or randomness on the macroscopic scale.
One of the few ways we can describe turbulent flow is by describing it in terms of time-averaged means and fluctuating parts.
mean fluctuating
Turbulent Flow – Shear stressesThere 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.
� We estimate shear stress by using experimental data,
semiempirical formulas and dimensional analysis
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, for turbulent flow.
Laminar Flow:
Shear relates to random motion as particles glide smoothly past
each other.
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.
Turbulent Flow:
Fully Developed Turbulent Flow: Overview
The turbulent shear components are known as Reynolds Stresses.
Shear Stress in Turbulent Flows: Turbulent Velocity Profile:
In the outer layer: τtirb > τlaminar 100 to 1000 time greater.
In viscous sublayer: τlaminar > τturb 100 to 1000 times 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
– Propose an apparent viscosity
– In general µT>µ , so ordinary viscosity can be
neglected
dy
udµτ TT =
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 4
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.
uuu ′+= ppp ′+= and so on
(T and ρ 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-continuityContinuity - what goes in must come out!
In laminar flow: 0z
w
y
v
x
u=
∂
∂+
∂
∂+
∂
∂
( ) ( ) ( )0
z
ww
y
vv
x
uu=
∂
′+∂+
∂
′+∂+
∂
′+∂In turbulent flow:
Separating: 0z
w
y
v
x
u
z
w
y
v
x
u=
∂
′∂+
∂
′∂+
∂
′∂+
∂
∂+
∂
∂+
∂
∂
Taking a time average: 0z
w
y
v
x
u=
∂
∂+
∂
∂+
∂
∂
0z
w
y
v
x
u=
∂
′∂+
∂
′∂+
∂
′∂Therefore, the fluctuating part
also satisfies the continuity equation
Reynolds stresses - Navier Stokes
Similarly, the N-S equations become (Schlichting, Ch 18)
∂
∂+
∂
∂+
∂
∂=
∂
′′∂+
∂
′′∂+
∂
′∂−
∂
∂+
∂
∂+
∂
∂+
∂
∂−
z
uw
y
uv
x
uuρ
z
wu
y
vu
x
uρ
z
u
y
u
x
uµ
x
pρg
2
2
2
2
2
2
2
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)
s
uu
t
u
dt
du:(remember
∂
∂+
∂
∂=
Reynolds stresses in 2D
x
yu
∂
∂+
∂
∂+
∂
∂=
∂
′′∂+
∂
′′∂+
∂
′∂−
∂
∂+
∂
∂+
∂
∂+
∂
∂−
z
uw
y
uv
x
uuρ
z
wu
y
vu
x
uρ
z
u
y
u
x
uµ
x
pρg
2
2
2
2
2
2
2
x
•No z,w terms
•Steady, turbulent flow in x
direction
•Ignore gravity
Reynolds stresses in 2D
y
uρv
y
vuρ
y
uµ
x
p2
2
∂
∂=
∂
′′∂−
∂
∂+
∂
∂−
In 2D, the turbulent N-S equation therefore reduces to:
Note that there are now two shear stress terms.
Re-writing:y
uρvvuρ
y
uµ
yx
p
∂
∂=
′′−
∂
∂
∂
∂+
∂
∂−
vuρy
uµτ ′′−
∂
∂=
In turbulent flow, therefore
the shear stress is given by
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 5
Reynolds and Boussinesq
Boussinesq proposed an additive turbulent shear stress:
y
uµ
y
uµτ T
∂
∂+
∂
∂=
So the additive term is equivalent to the Reynolds’ stress.
However, we need to know values for vu ′′
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
Prandtl‘s Mixing Length Theory
)y(u
lump of
turbulence
x,u
mean
velocity
y,v
mixing length l
defined as that downstream distance, which is
needed for the lump of turbulence to be
completely mixed with the surrounding fluid
turbulent shear
flow along
solid wall(not valid close
to the wall)
lll
y
ulu
y
ulv
==
∂
∂⋅±′⇒
∂
∂⋅+′
21
21 ~~
lump of
turbulence
(mixed)
v~ ′′′′
u~ ′′′′
y
u
∂∂∂∂
∂∂∂∂
Prandtl’s Mixing Length…
•Analogous to the kinetic theory of gases
•Used because ‘it works’
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
y
y1
y2l
(y)u
Prandtl’s Mixing Length
y
uu
∂
∂∝′ l
Turbulence is even in all directions (homogeneous)
y
uuv
∂
∂∝′∝′ l
2
2
y
uvu
∂
∂∝′′ l
So the Reynolds shear stress must be proportional to the
square of the mixing length times the velocity gradient:
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Prandtl’s Mixing Length…
vuρy
uµτ ′′−
∂
∂=
Returning to the equation for the shear stress:
2
2
Ty
uρvu
y
uµ
∂
∂∝′′−=
∂
∂lρ
y
uµ
y
uµτ T
∂
∂+
∂
∂=
2
2
Ty
uρµ
∂
∂= l
This gives a direct relationship between turbulent ‘viscosity’
and velocity gradient in the flow
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 6
Total Shear stress at any point
• Total Shear stress at any point is the sum of the viscousshear stress and turbulent shear stress and may beexpressed as
• The significance of Prandtl’s turbulent shear stressequation is that it is possible to make suitable assumptionsregarding the variation of the mixing length
2
2
dy
vdρl
dy
vdµτ
+=
CE F312: Hydraulic Engineering by Dr. A. P. Singh
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)
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Mixing length measurement in pipes
CE F312: Hydraulic Engineering by Dr. A. P. Singh
The Universal Law of The Wall
First define the friction velocity, V*, which is characteristic
of the fluctuating flow:
y
uvuV
*
∂
∂=′′= l
Assuming that the shear stress remains constant throughout,
then V* = const (typically V*~4% u)
y
ukyV*
∂
∂=
Using the relation from above, l=ky, gives the differential
equation
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• In general a boundary with irregularities of largeaverage height k, on its surface is considered to berough boundary and the one with smaller k valuesis 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
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Hydrodynamically Smooth and
Rough Pipe Boundaries
A systematic study is complicated by different types:
1. Shape
2. Height
3. Density
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 7
• We should also consider flow and fluid characteristics forproper classification of smooth and rough boundaries.
• If k is the average height of rough projections on the surfaceof the plate and δ is the thickness of the boundary layer,then the relative roughness (k/ δ) is a significant parameterindicating the behavior the boundary surface of a plate.
• If the boundary layer is turbulent from the leading edge ofthe plate, the front portion of the plate will act ashydrodynamically rough followed by transition region andthe downstream portion of the plate will behydrodynamically smooth if the plate is sufficiently long.
Hydrodynamically Smooth and
Rough Pipe Boundaries
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Hydrodynamically Smooth and
Rough Pipe Boundaries
δ
Turbulent
Laminar
• 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<δ, then the law of friction
for smooth pipes will apply
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• With the increase in Reynolds number, the thickness ofthe laminar sub-layer decreases, and it can evenbecome much smaller than the average height k, ofsurface irregularities. The irregularities will thenproject through the laminar sub-layer and laminar sub-layer is completely destroyed. The eddies will thuscome in contact with the surface irregularities and largeamount of energy loss will take place and thus theboundary act as a rough boundary.
Hydrodynamically Smooth and
Rough Pipe Boundaries
CE F312: Hydraulic Engineering by Dr. A. P. Singh
From Nikuradse’s experiment
• Hydrodynamically smooth pipe
• Transition region in a pipe
• Hydrodynamically Rough Pipe
50.2δk
'<
0.6δk
25.0'<<
0.6δ
k'
>
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• Hydrodynamically smooth Plate
• Plate in Transition region
• Hydrodynamically Rough
5υ
kV s* <
70υkV
5 s* <<
70υkV s* >
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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
From Nikuradse’s experiment
• Hydrodynamically smooth pipe
• Transition region in a pipe
• Hydrodynamically Rough Pipe
5V
or 50.2δ
k *
'≤
≤
υ
k
70V
3or 0.6δk
25.0 *
'<
<<<
υ
k
70V
or 0.6δk *
'≥
≥
υ
k
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 8
The Universal Law of The Wall
+=
+=
=
R
yLog2.5V v v
CyLogK
V v
dy
dvyρkτ
e*max
e*
2
220
=
−
y
RLog2.5V
V
vv e*
*
max
CE F312: Hydraulic Engineering by Dr. A. P. SinghCE F312: Hydraulic Engineering
by Dr. A. P. Singh
CE F312: Hydraulic Engineering by Dr. A. P. Singh
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
107
δy
V
0.108υyy y for 0.108
υyV
V
11.6υδδ y for 11.6
υyV
''
*
'''
*
*
''*
=⇒
=⇒==
=⇒==
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Velocity Distribution for turbulent
flow
• Velocity Distribution in a hydrodynamically smooth pipe
• Velocity Distribution in a hydrodynamically Rough Pipes
5.5υ
yV log 5.75
V
v *10
*
+=
8.5k
y log 5.75
V
v10
*
+
=
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Velocity Distribution for turbulent
flow in terms of Mean Velocity (V)
• Velocity Distribution in a hydrodynamically smooth pipe
• Velocity Distribution in a hydrodynamically Rough Pipes
75.1υRV
log 5.75V
V *10
*
+=
4.75k
R log 5.75
V
V10
*
+
=
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 9
Law of the Wall
*V
u
υyV
ln*
Turbulent
layer
Laminar
sub-layer
Buffer
zone
5.5
CE F312: Hydraulic Engineering by Dr. A. P. Singh
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. n/1
R
r1
V
u
−=
where V is the velocity
at the centerline
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Losses due to Friction/The Friction
FactorFor 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.
D
V 2L
hf ∝
By introducing the friction factor, f:
D
V f
2L
hf = where
)2/)(/(2
gVDL
hf
f=
where L=length of the pipe,
D=diameter of the pipe, V=velocity,
CE F312: Hydraulic Engineering by Dr. A. P. Singh
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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Nikuradse’s Experiments• In general, friction factor
• Function of Re and
roughness
• Laminar region
– Independent of
roughness
• Turbulent region
– Smooth pipe curve
• All curves
coincide @
~Re=2300
– Rough pipe zone
• All rough pipe
curves flatten out
and become
independent of Re
Re
64=f
( )Blausius
Re4/1
kf =
Rough
Smooth
Laminar Transition Turbulent
Blausius OK for smooth pipe
)(Re,D
eFf =
Re
64=f
2
9.010Re
74.5
7.3log
25.0
+
=
D
e
f
CE F312: Hydraulic Engineering by Dr. A. P. Singh
The Friction FactorThe mechanical energy equation can be written:
Knowledge of the friction factor allows us to estimate the
loss term in the energy equation
−=−+−+
ρ−
ρ 24)()
22()(
2
12
21
2212 V
D
Lf
m
Wzzg
VVPP shaft
&
Or in terms of heads:
−=−+−+
ρ−
ρ g
V
D
Lf
gm
Wzz
g
V
g
V
g
P
g
P shaft
24)()
22()(
2
12
21
2212
&
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 10
Friction factor: The Moody ChartThe Moody Chart (Figure 14.10 textbook) provides a convenient
representation of the functional dependence f = f(Re, κ/D)
� For laminar flow:
f = 64 / Re
� For turbulent flow:
+−=
f
k
f Re
51.2
3.7
/D log 2
1Colebrook formula
� For turbulent flow, with Re<105 and for hydraulically smooth surfaces:
4/1Re
316.0=f Blasius formula
+
ε+⋅=
3/16
Re
10
D000,201001375.0f
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering by Dr. A. P. Singh
( )
( )
( ) 237.0
10
10
745
Re
0.221 0.0032 f
relation emperical sNikurdse'
8.0fRelog 2.0f
1
found Nikurdse results alexperiment fromBut
91.0fRelog 2.03f
1
pipessmooth for )104 to105 from Refor (Valid 10 ReFor
+=
−=
−=
××>
CE F312: Hydraulic Engineering by Dr. A. P. Singh
( )
( ) 74.1Re/klog 2.0f
1
found Nikurdse results alexperiment fromBut
68.1Re/klog 2.03f
1
pipesRough in
flowTurbulent for )104 Refor (Valid 10 ReFor
10
10
35
+=
+=
×>>
Surface Roughness
Additional dimensionless group κκκκ/D need
to be characterize
Thus more than one curve on friction factor-Reynolds 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: Hydraulic Engineering by Dr. A. P. Singh
Variation of friction factor for
Commercial pipes
White Equation
Colebrook formula
+=
−
fRe
R/K18.71log 2.0-1.74
K
Rlog 2.0
f
11010
+−=
fRe
51.2
3.7
k/Dlog 2.0
f
110
CE F312: Hydraulic 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.
+≅
11.1
7.3
/
Re
6.9log 8.1
1 Dk
f
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 11
Moody Diagram
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Moody Diagram
0.010
0.100
1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08
Re
0.05
0.04
0.03
0.02
0.015
0.010.008
0.006
0.004
0.002
0.0010.0008
0.0004
0.0002
0.0001
0.00005
0
laminar flow
ε/D
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Fanning Diagram
f =16/Re
1
f= 4.0 * log
D
ε+ 2.281
f= 4.0 * log
D
ε+ 2.28− 4.0 * log 4.67
D /ε
Re f+1
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Following observations from the Moody
chart:
• For laminar flow, the friction factordecreases with increasing Reynolds number,and it is independent of surface roughness.
• The friction factor is a minimum for asmooth pipe (but still not zero because ofthe no-slip condition) and increases withroughness. The Colebrook equation in thiscase (k=0) reduces to the Prandtl equationexpressed as
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• The transition region from the laminar to turbulentregime (2300 < Re < 4000) is indicated by theshaded area in the Moody chart. The flow in thisregion may be laminar or turbulent, depending onflow disturbances, or it may alternate betweenlaminar and turbulent, and thus the friction factormay also alternate between the values for laminarand turbulent flow. The data in this range are theleast reliable. At small relative rougnesses, thefriction factor increases in the transition regionand approaches the value for smooth pipes.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Equivalent roughness values for new
commercial pipesMaterial 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
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 12
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 head-
loss 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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• 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 from
– Using Coolebrook formula and the above equation, Obtain f
– Obtain Re from the Moody’s chart and hence Q
1/2
2
f
LV
D2gh
υVD
fRe
=
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• 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
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Swamee and Jain in 1986 proposed the following
explicit relations that are accurate to within 2% of
the Moody chart
2000Re hgD
L3.17
3.7D
kln
L
hgD0.965Q
0.5
L
3
20.5
L
5
>
+
−=
υ
8
260.04
5.2
9.4
4.75
L
21.25
103Re5000
10k/D10
gh
LυQ
gh
LQk0.66D
×<<
<<
+
=
−−
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE C371: Hydraulics & Fluid Mechanics by Dr. A. P. Singh
APPARATUS
Pipe Network
Rotameters
Manometers
Problem
• 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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 13
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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• 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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Problem
• After 15 years of service a steel water 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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Problem• 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 .
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering by Dr. A. P. Singh
• 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?
CE F312: Hydraulic Engineering by Dr. A. P. Singh
CE F312: Hydraulic Engineering By Prof. Ajit Pratap Singh 14
• 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.
CE F312: Hydraulic Engineering by Dr. A. P. Singh CE F312: Hydraulic Engineering by Dr. A. P. Singh
Pipe Flow Summary
�The statement of conservation of mass, momentum and
energy becomes the Bernoulli equation for steady state
constant density of flows.
� Dimensional analysis gives the relation between flow rate and
pressure drop.
�Turbulent flow losses and velocity distributions require
experimental results.
�Experiments give the relationship between the fraction factor
and the Reynolds number.
� Head loss becomes minor when fluid flows at high flow rate
(fraction factor is constant at high Reynolds numbers).
CE F312: Hydraulic Engineering by Dr. A. P. Singh
Images - Laminar/Turbulent Flows
Laser - induced florescence image of an
incompressible turbulent boundary layer
Simulation of turbulent flow coming out of a
tailpipe
Laminar flow (Blood Flow)
Laminar flowTurbulent flow
CE F312: Hydraulic Engineering by Dr. A. P. Singh
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