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FULLY DEVELOPED PIPE AND
CHANNEL FLOWS
KUMAR DINKAR ANAND
3rd YEAR, MECHANICAL ENGG.
IIT-KHARAGPUR
GUIDANCE : PROF. S CHAKRABORTY
INDO-GERMAN WINTER ACADEMY-DECEMBER 2006

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: THE OUTLINE :
Hydraulically developing flow through pipes and channels and
evaluation of hydraulic entrance length. Hydraulically fully developed flows through pipes and channels .
Hydraulically fully developed flow through non-circular ducts.
Definition of Thermally fully developed flow and analysis of thermally
fully developed flow through pipe and channels.
Analysis of the problem of Thermal Entrance: The Graetz Problem.

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Fully Developed Flows
There are two types of fully developed flows :
1.) Hydraulically Fully Developed Flow
2.) Thermally Fully Developed Flow
Contd

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Hydraulically Fully Developed Flow
Definition: As fluid enters any pipe or channel , boundary layers keep on growing
till they meet after some distance downstream from the entrance region. After this
distance velocity profile doesn't change, flow is said to be Fully Developed.
Analysis of fluid flow before it is fully developed:
Velocity in the core of the flow outside the boundary layer increases with
increasing distance from entrance. This is due to the fact that through any cross
section same amount of fluid flows, and boundary layer is growing.
This means
hence
0>dxdU
0

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Schematic picture of internal flow through a pipe :
Velocity Profile ,Using the boundary conditions :
1.) At
2.) At
3.) At
We get the velocity profile as :
Contd
2
)( cybyayu ++=
=y Uu =
0=u
=y 0=dydu2
)()(2
)(
yyU
yu
=
0=y

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Where Free stream velocity of entering fluid
Free stream core velocity inside the tube
Core velocity of fully developed flow
Radius of pipe
Now from the principle of conservation of mass :
Hence ,
Contd
=U=U
+=
R
R
R
UrdrurdrRU
0
2
22*
rRy ==R=eU
2)/(6/1)/(3/21
1
RRU
U
+=
2
2
)/(6/1)/(3/21
)/()/(2
RR
yy
U
u
+
=

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+=
0 0
2 })/1()/1(/{ dyUudx
dUUdyUuUuUdxd
w
Boundary Layer momentum integral equation:
Where, Shear stress at wall,
From Bernoulli's Equation for free stream flow through core:
Using Navier-Stokes equation at the wall
Contd
0==
y
wy
u
x
p
dx
dUU
=
1
0
2
2
=
=
yy
u
x
p

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Solving for boundary layer thickness :
Integrate momentum Integral Equation
Using the boundary condition at
For determination of Entrance Length :
putting at
We get the expression for Entrance Length as:
Contd
)(
0= 0=x
)( eL
R=eLx =
)( eL
De
D
LRe03.0=

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Analytical expression for Entrance Length :
Hence it can be observed that our expression for Entrance Length differsfrom the analytical expression due to the following reasons:
1.) We have assumed parabolic velocity profile in the boundary layer
2.)We have not used the Navier-Stokes boundary equation at wall for velocity
profile determination
3.) We are doing boundary layer analysis which gives approximate results
Contd
)( eL
De
DL Re06.0=
2)/()/(2 yyUu =
0
2
2
==
yy
u
x
p
2)/(6/1)/(3/211
RRUU
+=

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Schematic picture of internal flow through a channel:
Velocity Profile
Using the boundary conditions :
1.) At
2.) At
3.) At
We get the velocity profile as :
Contd
2)( cybyayu ++=
0=y 0=u
=y Uu ==y 0=dydu
2)()(2)( yyU
yu =

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Here , Distance between the parallel plates of channel
Width of the Channel
Free stream velocity of entering fluid
Free stream velocity inside channel
Core velocity of fully developed flow
Entrance Length
Hydraulic Diameter
Contd
=D
=W
=U
=U
=eU
=eL
=HD DW
WD
P
AH 22
44 ===

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From the principle of conservation of mass:
Hence when flow is fully developed
Contd
+=
)2/(
00
22*
D
UdyudyDU
)/(3/21 1 DUU =
)/(3/21)/()/(2 2
D
yy
U
u
=
)2/( D=
= UUe 5.1

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From Boundary layer momentum integral equation :
Where, Shear stress at wall,
From Bernoulli's Equation for free stream flow through core:
Using Navier-Stokes equation at the wall
Contd
+=
0 0
2 })/1()/1(/{ dyUudx
dUUdyUuUuUdxd
w
0==
y
wy
u
x
p
dx
dUU
=
1
0
2
2
=
=
yy
u
x
p

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Solving for boundary layer thickness :
Integrate momentum Integral Equation
Using the boundary condition at
For determination of Entrance Length :
putting at
We get the expression for Entrance Length as:
OR
Contd
)(
0= 0=x
)(e
L
eLx = R=
)( eL
D
e
D
L
Re025.0=HD
H
e
D
L
Re00625.0=

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Analytical expression for Entrance Length :
Hence it can be observed that our expression for Entrance Length differs
from the analytical expression due to the following reasons:
1.) We have assumed parabolic velocity profile in the boundary layer
2.) We have not used the Navier-Stokes boundary equation at wall for
velocity profile determination
3.) We are doing boundary layer analysis which gives approximate results.
Contd
)( eL
De
D
LRe05.0=
2
)/()/(2 yyU
u
=
0
2
2
=
=
yy
u
x
p
)/(3/21
1
DU
U
=

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Analysis of fully developed fluid flow:
Fully Developed Flow Through a Pipe:
From Equation of continuity in cylindrical coordinates:
for an incompressible fluid flowing through a pipe
Contd
0)(1 =+
xuru
rrr

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Here, radial velocity
axial velocity
radius of pipe
No fluid property varies with ,
,at wall of the pipe
hence it is zero everywhere.
Hence Equation of continuity reduces to :
Momentum Equation in radial coordinate:
Contd
=ru=u
=a
0=r
u
,0=xu )(ruu =
,0=
r
p )(xpp =

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Momentum Equation in axial direction :
)( dr
du
rdr
d
rdx
dp
=
Solving above differential equation in (r) using the boundary conditions:
1.) Axial velocity (u) is zero at wall of pipe (r =R)
2.) Velocity is finite at the pipe centerline (r=0).
We get the fully developed velocity profile:
Contd
=
22
14 a
rxpau

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Shear Stress Distribution :
Shear stress ,
Maximum shear stress at wall ,
=
=
x
pr
dr
durx
2
=
x
pa
20
Contd
Hence it can be observed that
Shear stress decreases from
maximum to zero at pipe
centerline and then increases
to maximum again at wall.

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Volume Flow Rate :
volume flow rate ,
== xpaurdrQ
a
0
4
82
Now in a fully developed flow pressure gradient is constant ,
Hence ,( )
L
p
L
pp
x
p entexit
=
=
LpaQ 84
=
Contd

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Average Velocity :
Average velocity ,
=== x
pa
a
Q
A
QV 8
2
2
Maximum Velocity :
At the point of maximum velocity , 0=dr
du
This corresponds to core of pipe , 0=r
Hence VxpaUuu r 2
4
2
0max =
=== =
Contd

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Fully Developed Flow through Channel :
From equation of continuity within the entrance length : 0=
+
y
v
x
u
In entrance length boundary layers growing , 0xu 0v
It means flow is not parallel to walls in entrance region
Contd
)(
a

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Equation of Continuity for an incompressible fluid in fully developed region :
0=
x
u
)(yuu =
Momentum equation in y-direction (transverse direction) :
0=yp )(xpp =
Momentum equati

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