ay2014 ce2134 hydraulics 4p boundary layer flow over a flat plate ii
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8/10/2019 AY2014 CE2134 Hydraulics 4P Boundary layer flow over a flat plate II
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LAMINAR BOUNDARY LAYER OVER A SMOOTH FLAT
PLATE
At any x
" #w =d
dx0
$
%& u2 "Umu( )dz'(
)
*+
,
-".u
.z
/
01
2
34w
=
d
dx&u2dz
0
$
%/
01
2
34"Um
d
dx&udz
0
$
%/
01
2
34
For similar velocity profilesu
Um= f
z
$
/
01
2
34= f 5( )
-Um
$
df 5( )
d5
'(
)
*+
,5=0=&
.
.x
Um2$ 1" f 5( )[ ]f 5( )d5
0
1
%'(
)
*+
,
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LAMINAR BOUNDARY LAYER OVER A SMOOTH FLAT
PLATE
Let
"= 1# f $( )[ ]f $( )d$0
1
%
&=df $( )d$
'()
*+,$= 0
-Um&
.=/Um
2"0.
0x
Integrating
&x =1
2
/Um".2+ const
With .= 0 atx = 0
.=2&
"
x
Rex
'()
*+,
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LAMINAR BOUNDARY LAYER OVER A
SMOOTH FLAT PLATE
"W
=
d
dx#U
m$u( )udz
0
1
%&
'(
)
*+
=#Um
2,d-
dx
=#Um
2, 2.
#Um,
&
'(
)
*+1
2x
$1/ 2
=#Um
2 ,.
2Rex
F = "Wdx
0
L
#
=$Um
2%&0
L
= 2%'$Um
3L[ ]
The total friction force on one side of
the smooth flat plate between x=0
and x=L for a unit width of the plate
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EXAMPLE: LAMINAR BOUNDARY LAYER
Ftotaldragforce
! = 22 sides
! 2"#$Um3
L[ ] where "= 1% f&( )[ ]
0
1
' f&( )d& and #=
df&( )d&
(
)*
+
,-& =0
" = 1 # 2$# $2( ){ }0
1
% 2$# $2( )d$ = 2$# 5$2 + 4$3 #$4{ }0
1
% d$
= $2 #
5
3$
3+$
4 #1
5$
5&
'(
)
*+0
1
=1 #5
3+1 #
1
5=
2
15 ,
" =df#( )
d#
$
%
&'
(
)
# =0
= 2 * 2#[ ]# =0
= 2 +
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EXAMPLE : LAMINAR BOUNDARY LAYER
F = 2 2"#$U3L
[ ] = 2 2
2
15
%
&'
(
)*2( )$U3L
+
,-
.
/0=1.46 $U
3L
By definition,
F =CD
1
2$ 2L(
)U
2
1CD =1.46 $U3L
$LU2=
1.46
$UL
=
1.46
ReL
2
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TURBULENT BOUNDARY LAYER OVER A FLAT
PLATE
In the case of the turbulent boundary layer, the shearing stress atthe plate may be expressed by
"w= +#
t( ) $u
$z
%
&'
(
)*z= 0
"tthe eddy viscosity is a property of the flow; i.e. it depends onthe character of the flow. The eddy viscosity can be different
all over the flow and it is not readily determined
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BLASIUS
FORMULA FOR TURBULENT SHEAR
STRESS OVER SMOOTH SURFACE (Re
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(b) Blasius formula for f for smooth pipes with diameter D and
mean velocity U. This appears to fit the experimental results forReynolds number between 3000 and 105.
f = 0.32UD
"
#
$%
&
'(
)1/ 4
* f = 0.32 0.8UmD"
#
$% &
'(
)1/ 4
= 0.32 1.6UmR"
#
$% &
'(
)1/ 4
= 0.321.6Um+
"
#
$%
&
'(
)1/ 4
,w =
-fU2
8=
-
8 0.8Um( )
2
0.32( ) "
1.6Um+
#
$%
&
'(
1/ 4
= 0.0227-Um2 "
Um+
#
$%
&
'(
1/ 4
.
BLASIUS
FORMULA FOR TURBULENT SHEAR
STRESS OVER SMOOTH SURFACE (Re
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TURBULENT BOUNDARY LAYER OVER A FLAT PLATE
Use the momentum equation of von Karman for the mean timequantities, assume similarity in velocity profiles along the plate andaccount for the wall shear stress by using an experimentally derivedvalue.
If thenand Umis constant along the direction of x outside the boundarylayer.
"p*
"x=0
"Um
"x=0
Simple velocity profile suggested by Prandtl in the turbulent
boundary layer
u
Um
= z
"
#
$%
&
'(
n
, n =1
7
(c) Assume a velocity profile
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TURBULENT BOUNDARY LAYER OVER A FLAT PLATE
"
Um
#
$%
&
'(
1/ 4
x = 3.45)5 / 4
+ constant
*)
x= 0.376
Umx
"
#
$%
&
'(
+1/ 5
,
Turbulent boundary layer starts at the transition location and has some
thickness at this position. Location of the transition is difficult todetermine.
The length of the laminar boundary layer is generally small and one canapproximately imagine that the turbulent boundary layer commences atthe leading edge x = 0.
In the turbulent boundary layer, the thickness varies as
In the laminar boundary layer, the thickness varies as
The turbulent boundary layer grows faster along x than the laminar case.
5/4x
x1/ 2
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TURBULENT BOUNDARY LAYER DRAG OVER A FLAT PLATE
The drag for unit width on both sides of the smooth plate
Drag = 20
L
! w! dx= 0.072"Um2L #
UmL
"#$ %
&'
1/5
=0.072 "Um2LR
eL
(1/5
The drag coefficient CDof an object immersed in a fluid stream ofvelocity U and mass density #is defined by
CD =Drag force
1
2"Um
2Area( )
CD
=
0.072!Um
2L.1( )ReL
!1/5
0.5!Um
22L
two sides
!
= 0.072ReL
!1/5
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DRAG ON FLAT PLATE
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BLANK
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