Boundary Layer
TITLE : FLAT PLATE BOUNDARY LAYER
OBJECTIVES The objective of this experiment are as follows:
•
To measure the boundary layer velocity profiles and observes the growth of the boundary layer for flat plate with smooth and rough surfaces.
•
To measure the boundary layer properties for the measured velocity profiles.
•
To study the effect of surface roughness on the development of boundary layer.
INTRODUCTION The conce concept pt of boun bounda dary ry laye layerr was was first first introd introduc uced ed by Ludw Ludwig ig Prand Prandtl, tl, a Germa German n aerodynamicist in !"#. The boundary layer concept provided the lin$ that had been missin missing g betw betwee een n theo theory ry and and practi practice ce.. %urth %urtherm ermore ore,, the the boun bounda dary ry layer layer conce concept pt permitted the solution of viscous flow problems that would have been impossible through application the &avier'(to$es e)uation to the complete flow field.
*s for flow in a duct, flow in a boundary layer may be laminar or turbulent. There is no uni)ue value of the +eynolds number at which transition from laminar to turbulent flow occur in a boundary layer. *mong the factors that affect boundary'layer transition are pressure pressure gradient gradient,, surface surface roughne roughness, ss, heat heat transfe transfer, r, body forces forces and free stream stream disturbances. n many real flow situations, a boundary layer develops over a long, essentially flat surface. * )ualitative picture of the boundary layer growth over a flat plate is shown in figure below.
%igure .: -oundary layer on a flat plate /ertical thic$ness exaggerate greatly0
Boundary Layer
THEORY
(ome measures of boundary layers are described in figure .1 below.
%igure .1 : -oundary Layer thic$ness definitions
The boundary layer thic$ness, δ, is defined as the distance from the surface to the point where the velocity is within percent of the stream velocity. The displacement thic$ness,
δ2, is the the dista distance nce by which which the solid solid bounda boundary ry would would have to be displa displaced ced in a frictionless flow to give the same mass deficit as exists in the boundary layer.
The momentum thic$ness, θ, is define as the thic$ness of a layer of fluid of velocity, 3 free stream velocity0, for which the momentum flux is e)ual to the deficit of momentum flux through the boundary layer. The -lasius4s exact solutions to the laminar boundary yield the following e)uations for the above properties. δ =
5.0 x Re x
∗
δ
1.72 x =
Re x
θ =
0.664 x Re x
Boundary Layer
5ue to the complexity of the flow, there is no exact solution to the turbulent boundary layer. The velocity profile within the boundary layer commonly approximated using the 67 power law. u U
1
y 7 = δ
The properties of boundary layer are approximated using the momentum integral e)uation, which result in the following expression.
δ =
0.370 x 1
Re x 5
δ ∗
=
0.0463 x 1
Re x 5
θ =
0.036 x 1
Re x 5 *nother measure of the boundary layer is the shape factor, 8, which is the ratio of the displacement thic$ness to the momentum thic$ness, 8 9 δ26θ. %or laminar flow, 8 increases from 1. to ;.< at separation. %or turbulent boundary layer, 8 increases from .; to approximately 1.< at separation.
EXPERIMENT APPARATUS
Boundary Layer
The experiment set up consists of: .
*irflow -ench
1.
Test *pparatus
;.
Total and static tube pressure probes and multi tube manometer.
Plenm otlet On-off switch
Damper control rod
PROCEDURES
Boundary Layer
. The apparatus has been set up on the bench as shown on figure # uses the flat plate with the smooth surface for the first part of the experiment. 1. (et the pitot tube about . +epeat the entire experiment for the rough surface.
RESULT AND CALCULATION
Boundary Layer
TABULATION OF DATA AND SAMPLE CALCULATION:
. (mooth surface with distance from the leading edge, x 9 #" +oom temperature: ;" ο?
ρair 9 .77 $g6m ; ν 9 .;" x " '< m16s ρoil 9 7># $g6m ; %ree stream velocity, 3 9
9
2 ρ o g ∆h
ρ a
m6s
2 × 7!4 × ".!1 × 22 × 10−3 1.177
m6s
9 .! m6s
+eynolds number, +e x 9 9
Ux ν
16."6 × 40 × 10 −3 1.30 × 10− 5
9 <1>#.
1. (mooth surface with distance from the leading edge, x 9 <"
Boundary Layer
+oom temperature: ;" ο?
ρair 9 .77 $g6m ; ν 9 .;" x " '< m16s ρoil 9 7># $g6m ;
2 ρ o g ∆h
%ree stream velocity 3 9
ρ a
m6s
2 × 7!4 × ".!1 × 22 × 10−3
9
1.177
m6s
9 .!m6s
+eynolds number, +e x9
Ux ν
9
16."6 × 150 × 10 −3 1.30 × 10
−5
9 !<!1.;
;. +ough surface with distance from the leading edge, x 9 #"
Boundary Layer
+oom temperature: ;" ο?
ρair 9 .77 $g6m ; ν 9 .;" x " '< m16s ρoil 9 7># $g6m ;
2 ρ o g ∆h
%ree stream velocity 3 9
ρ a
m6s
2 × 7!4 × ".!1 × 26 × 10 −3
9
1.177
m6s
9 >.#;m6s
+eynolds number, +e x 9
Ux ν
9
1!.43 × 40 × 10 −3 1.30 × 10
−5
9 <7>.1
#. +ough surface with distance from the leading edge, x 9 <"
Boundary Layer
+oom temperature: ;" ο?
ρair 9 .77 $g6m ; ν 9 .;" x " '< m16s ρoil 9 7># $g6m ;
%ree stream velocity 3 9
9
2 ρ o g ∆h
ρ a
m6s
2 × 7!4 × ".!1 × 22 × 10−3 1.177
9 .! m6s
+eynolds number, +e 9 9
Ux ν
16."6 × 150 × 10 −3 1.30 × 10
9 !<!1.;
TABLE:
−5
m6s
Boundary Layer
Table : Tabulation of data for smooth surface with x 9 #"mm 5ifferential %ree (tatic Total @icrometer pressure pressure manometer stream reading velocity, manometer,manometer, height y mm0 ∆h mm0 U m6s0 h mm0 h mm0
+eynolds number
U
1 − u U U
u
u
"
""
11
11
.!
<1>#.1
".77
".7!
".<
""
1
1
>.#;
<7"7.!
".7"<
".1">
."
""
1>
1>
!.;
<>>.<#
".>"
".1>
.<
""
1>
1>
!.;
<>>.<#
".>"
".1>
1."
""
1>
1>
!.;
<>>.<#
".>"
".1>
1.<
"1
;"
1>
!.;
<>>.<#
".>"
".1>
;."
"1
;"
1>
!.;
<>>.<#
".>"
".1>
;.<
"1
;"
1>
!.;
<>>.<#
".>"
".1>
#."
"1
;"
1>
!.;
<>>.<#
".>"
".1>
#.<
"1
;"
1>
!.;
<>>.<#
".>"
".1>
Table 1: Tabulation of data for smooth surface with x 9 <"mm
Boundary Layer
5ifferential (tatic Total %ree @icrometer pressure pressure manometer stream reading manometer,manometer, height velocity, y mm0 ∆h mm0 3 m6s0 h mm0 h mm0
u
+eynolds number
U
u 1 − U U u
"
""
11
11
.!
!<!1.;
".77
".7!
".<
""
1
1
>.#;
11!;.;
".7"<
".1">
."
""
1>
1>
!.;
11"7;".>
".>"
".1>
.<
""
;"
;"
!.>"
11>#!.;
".<7
".11<
1."
""
;"
;"
!.>"
11>#!.;
".<7
".11<
1.<
""
;1
;1
1".#<
1;
".;
".1;1
;."
"1
;1
;"
!.>"
11>#!.;
".<7
".11<
;.<
"1
;1
;"
!.>"
11>#!.;
".<7
".11<
#."
"1
;1
;"
!.>"
11>#!.;
".<7
".11<
#.<
"1
;1
;"
!.>"
11>#!.;
".<7
".11<
Table ;: Tabulation of data for rough surface with x 9 #"mm
Boundary Layer
5ifferential (tatic Total %ree @icrometer pressure pressure manometer stream reading manometer,manometer, height velocity, y mm0 ∆h mm0 3 m6s0 h mm0 h mm0
u
+eynolds number
U
1 − u U U u
"
""
1
1
>.#;
<7>.1
".7"<
".1">
".<
""
1>
1>
!.;
<>>.<
".>"
".1>
."
""
1>
1>
!.;
<>>.<
".>"
".1>
.<
""
;"
;"
!.>"
"!1<.
".<7
".11<
1."
"1
;"
;"
!.>"
"!1<.
".<7
".11<
1.<
"1
;"
;"
!.>"
"!1<.
".<7
".11<
;."
"1
;"
;"
!.>"
"!1<.
".<7
".11<
;.<
"1
;"
;"
!.>"
"!1<.
".<7
".11<
#."
"1
;"
;"
!.>"
"!1<.
".<7
".11<
#.<
"1
;"
;"
!.>"
"!1<.
".<7
".11<
Table #: Tabulation of data for rough surface with x 9 <"mm
Boundary Layer
5ifferential (tatic Total %ree @icrometer pressure pressure manometer stream reading manometer,manometer, height velocity, y mm0 ∆h mm0 3 m6s0 h mm0 h mm0
*.
u
+eynolds number
U
u U
u 1 − U
"
""
11
11
.!
!<!1.;
".77
".7!
".<
""
1#
1#
7.7
1"#;#!.
".7;#
".!<
."
""
1
1
>.#;
11!;.;
".7"<
".1">
.<
""
1>
1>
!.;
11"7;".>
".>"
".1>
1."
"1
;"
1>
!.;
11"7;".>
".>"
".1>
1.<
"1
;"
1>
!.;
11"7;".>
".>"
".1>
;."
"1
;"
1>
!.;
11"7;".>
".>"
".1>
;.<
"1
;"
1>
!.;
11"7;".>
".>"
".1>
#."
"1
;"
1>
!.;
11"7;".>
".>"
".1>
#.<
"1
;"
1>
!.;
11"7;".>
".>"
".1>
(ample calculation for boundary layer thic$ness, , displacement thic$ness, momentum thic$ness, and shape factor, H by using experimental.
,
Boundary Layer
For smooth surface with x = 40mm i. -oundary layer thic$ness, ii. 5isplacement thic$ness,
9
9 ". x < 9 .< mm iii. @omentum thic$ness, = *1 x δ 9 "." x < 9 ".!< mm iv. (hape factor, H 9 =
∗ δ
θ
1.65 0."15
= .>";
For smooth surface with x = 150mm i. -oundary layer thic$ness, ii. 5isplacement thic$ness,
9
9 "."!; x < 9 .##< mm iii. @omentum thic$ness, = *# x δ 9 "."7 x < 9 ."< mm iv. (hape factor, H 9 =
∗ δ
θ
1.445 1.05
= .;7
For rough surface with x = 40mm i. -oundary layer thic$ness,
9
Boundary Layer
ii. 5isplacement thic$ness,
9 *< x δ
9 ".">;< x < 9 .1< mm iii. @omentum thic$ness, = * x δ 9 "."< x < 9 ".!7< mm iv. (hape factor, H 9 = =
∗ δ
θ
1.25 0."75
1.2!2
For rough surface with x = 150mm i. -oundary layer thic$ness, ii. 5isplacement thic$ness,
9
9 "."!7! x < 9 .#! mm iii. @omentum thic$ness, = *> x δ 9 "."71 x < 9 .">! mm iv. (hape factor, H 9 =
∗ δ
θ
1.46" 1.0!"
= .;#!
-.
(ample calculation for boundary layer thic$ness, , displacement thic$ness, momentum thic$ness, and shape factor, H by using theoretical.
,
Boundary Layer
L *@&*+ -A3&5*+B L*BC+
For smooth surface with x = 40mm
i. δ
5.0 x
=
Re x 5.0 × 0.04
=
521!4.62
= !.755 × 10 − 4 m = 0.!76 mm
∗
ii. δ
1.72 x
=
Re x 1.72 × 0.04
=
521!4.62
= 3.012 × 10 − 4 m = 0.301 mm
iii.
θ
= =
0.664 x Re x 0.664 × 0.04 521!4.62
= 1.163 × 10− 4 m = 0.116 mm
i#.
H =
=
δ ∗ θ 0.301
0.116 = 2.5"5
For smooth surface with x = 150mm
i. δ
=
5.0 x Re x
Boundary Layer
5.0 × 0.15
=
1"56"2.3
= 1.6"5 × 10 − 3 m = 1.6"5 mm
∗
ii. δ
1.72 x
=
Re x 1.72 × 0.15
=
1"56"2.3
= 5.!32 × 10 − 4 m = 0.5!3 mm
iii.
θ
= =
0.664 x Re x 0.664 × 0.15 1"56"2.3
= 2.252 × 10 − 4 m = 0.225 mm
iv. H =
= =
δ ∗ θ 0.5!3 0.225 2.5"1
For rough surface with x = 40mm
i. δ
=
5.0 x Re x
Boundary Layer
5.0 × 0.04
=
5671!.2
= !.3"! × 10 − 4 m = 0.!40 mm
∗
ii. δ
1.72 x
=
Re x
= 1.72 × 0.04 5671!.2
= 2.!!" × 10 − 4 m = 0.2!" mm
iii.
θ
= =
0.664 x Re x 0.664 × 0.04 5671!.2
= 1.115 × 10 − 4 m = 0.112 mm
iv. H =
=
δ ∗ θ 0.2!"
0.112 = 2.5"1
For rough surface with x = 150mm
i. δ
=
5.0 x Re x
Boundary Layer
5.0 × 0.15
=
1"56"2.3
= 1.6"5 × 10 − 3 m = 1.6"5 mm
∗
ii. δ
1.72 x
=
Re x
=
1.72 × 0.15 1"56"2.3
= 5.!32 × 10 − 4 m = 0.5!3 mm
iii.
θ
= =
0.664 x Re x 0.664 × 0.15 1"56"2.3
= 2.252 × 10 − 4 m = 0.225 mm
i#. H =
= =
δ ∗ θ 0.5!3 0.225 2.5"1
T3+-3LC&T -A3&5*+B L*BC+
For smooth surface with x = 40mm
i. δ
=
0.370 x %Re $ x
1 5
Boundary Layer
=
0.370 × 0.04 %521!4.6$
1
5
= 1.6!6 × 10 − 3 m = 1.6!6 mm
ii. δ
∗
0.0463 x
=
%Re$ =
1
5
0.0463 × 0.04 %521!4.6$
1
5
=
2.10" × 10− 4 m
=
0.210" mm
iii. θ
0.036 x
=
%Re $
1
5
x
=
0.036 × 0.04 %521!4.6$
1 5
= 1.640 × 10− 4 m = 0.164mm
i#. H =
δ ∗ θ
= 0.210" 0.1640 = 1.2!6
For smooth surface with x = 150mm
i. δ
=
0.370 x %Re $ x
1 5
Boundary Layer
0.370 × 0.15
=
%1"56"2.3$
1
5
= 4.!53 × 10 − 3 m = 4.!53 mm
ii. δ
∗
0.0463 x
=
%Re$
1
5
0.0463 × 0.15
=
%1"56"2.3$
1
5
=
6.072 × 10− 4 m
=
0.6072 mm
iii. θ
0.036 x
=
%Re $
1
5
x
0.036 × 0.15
=
%1"56"2.3$
1
5
=
4.721 × 10 − 4 m
=
0.4721mm
i#. H =
δ ∗ θ
= 0.6072
0.4721 = 1.2!6
For rough surface with x = 40mm i. δ
=
0.370 x %Re $ x
1 5
Boundary Layer
0.370 × 0.04
=
%5671!.2$
1
5
= 1.65! × 10 − 3 m = 1.65! mm
ii. δ
∗
0.0463 x
=
%Re$ =
1
5
0.0463 × 0.04 %5671!.2$
iii. θ
1
5
=
2.074 × 10− 4 m
=
0.2074 mm
0.036 x
=
%Re $
1
5
x
=
0.036 × 0.04 %5671!.2$
1
5
= 1.613 × 10− 4 m = 0.1613 mm
iv. H =
=
δ ∗ θ 0.2074
0.1613 = 1.2!6
For rough surface with x = 150mm (refer to data D)
i. δ
=
0.370 x %Re $ x
1 5
Boundary Layer
0.370 × 0.15
=
%1"56"2.3$
1
5
= 4.!53 × 10 − 3 m = 4.!53 mm
ii. δ
∗
=
0.0463 x %Re$
=
1
5
0.0463 × 0.15 %1"56"2 .3$
iii. θ
1
5
=
6.072 × 10− 4 m
=
0.6072 mm
=
0.036 x %Re $
1
5
x
=
0.036 × 0.15 %1"56"2.3$
1
5
=
4.721 × 10 − 4 m
=
0.4721mm
iv. H =
=
δ ∗ θ 0.6072
0.4721 = 1.2!6
TABLE 5: Table of comparison for smooth and rough surface under experimental value and theoretical.
%or smooth surface: #"mm +ex 9<1>#.0 EXPERIMENT (m)
THEORY LAMINAR ( m ) TURBULENT ( m )
Boundary Layer
δ δ2 θ 8
< x "'; .< x " '; ".!< x " '; .>">
>."; x " '# ;.;;; x " '# .1>7 x " '# 1.
.1> x " '; 1.";> x " '# .<># x " '# .1>7
%or smooth surface: <"mm +e x 9!<!1.;0
δ δ2 θ 8
EXPERIMENT (m) < x "'; .##< x" '; ."< x " '; .;7
THEORY LAMINAR ( m ) TURBULENT ( m ) '; .<1> x " #.<< x " '; <.1<7 x " '# <.>1< x " '# 1."1! x " '# #.<1! x " '# 1.
%or rough surface: #"mm +e x 9<7>.10
δ δ2 θ 8
EXPERIMENT (m) < x "'; .1< x " '; ".!7< x " '; .1>1
THEORY LAMINAR ( m ) TURBULENT ( m ) '# >.17 x " .; x " '; 1.7! x " '# 1."#7 x " '# ."7! x " '# .
%or rough surface: <"mm +e x 9!<!1.;0
δ δ2 θ 8
EXPERIMENT (m) < x "'; .#! x " '; .">! x " '; .;#!
THEORY LAMINAR ( m ) TURBULENT ( m ) '; .<># x " #.711 x " '; <.##! x " '# <.!"! x " '# 1."; x " '# #.
DISCUSSION •
The micrometer reading y0 has to be started from ."mm as shown in the table , table 1, table ; and table #.
Boundary Layer
•
The value of displacement thic$ness δ20 is obtained by the graph of
•
The y δ
vs
value
of
momentum
thic$ness
θ0
is
obtained
by
y δ
vs
the
u U ∞
.
graph
of
u
u 1 − . U ∞ U ∞
CONCLUSION This experiment, we can say that the various boundary layer velocity profiles such as boundary layer thic$ness δ0, displacement thic$ness δ20, momentum thic$ness θ0 and shape factor 80 are depend on the distance from the leading edge and the surface condition. *ll the result is as state in table <. %rom the table, the boundary layer property is increasing between smooth and rough surface. *nother facts that we can conclude are the micrometer reading y0 for the smooth surface is lower than the rough surface. t is because the free stream at rough surface occurs faster than the smooth surface. *lso as expected δ is increasing with increasing distance from leading edge for both smooth and rough surface. %rom experiment note that shape factor decreasing as distance from leading edge increasing showing boundary layer is changing from laminar to turbulent.
REFERENCES 1! FLUID MECHANICS, "! F! Doug#as$ "! %! &asiore'$ "! ! Swaffie#d$ hird Editio*$ +o*gma* Scie*tific , ech*ica# -! INTRODUCTION TO FLUID MECHANICS, Ro.ert /! Fox$ #a* %cDo*a#d$ Seco*d Editio*$ "oh* /i#e , So*s!
