flat plate boundary layer 7over10 complete
TRANSCRIPT
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
1/23
Abstract
The following experiment will give the proper procedure to determining and measuring the
boundary layer using the momentum integral equation. During this experiment, Airflow
bench, Test apparatus, Micrometer scale and Velocity measurement were used as apparatus to
study the boundary layer behaviour when using smooth and rough plate. By measuring the
boundary layer thickness, () displacement thickness, ( ) momentum thickness,
(
)thereforevelocity profile can be measured.
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
2/23
TABLE OF CONTENTS
CONTENTS PAGE
1. Introduction 1
2. Objectives 5
3. Apparatus 6
4. Experimental Procedures 8
5. Data Collected 12
6. Sample of calculations 16
7. Results
8.
Discussions 17
9. Conclusions 28
10.References 39
Appendices and Raw Data -
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
3/23
List of figures, graphs and tables
Figure 1.1: Boundary layer on a flat plate 1
Figure 1.2: Boundary layer thickness 3
Figure 4.1: 6
Figure 4.2: 6
Figure 4.3: 7
Figure 5.1: 8
Graph 6.1: Graph y of vs u/U for test 1.
Graph 6.2: Graph of y vs (u/U)(1-u/U) for test 1
Graph 6.3: Graph y of vs u/U for test 2
Graph 6.4: Graph of y vs (u/U)(1-u/U) for test 2
Graph 6.5: Graph y of vs u/U for test 3.
Graph 6.6: Graph of y vs (u/U)(1-u/U) for test 3
Graph 6.7: Graph y of vs u/U for test 4
Graph 6.8: Graph of y vs (u/U)(1-u/U) for test 4
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
4/23
1. INTRODUCTION
As for flow in a duct, flow in boundary layer may be laminar or turbulent.
There is no unique value of the Reynold number at which transition from laminar to
turbulent flow occur in a boundary layer. Among the factors that affect boundary
layer transition are pressure, gradient, surface, roughness, heat transfer, body forces
and free stream disturbances.
In many real life flow situations, a boundary layer develops over a long,
essentially flat surface. A qualitative picture of the boundary layer growth over a flat
plate is shown in Figure 1.1 below.
Figure 1.1: Boundary layer on a flat plate.
Some measures of boundary layers are describe in Figure 1.2 below.
Figure 1.2: Boundary layer thickness.
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
5/23
The boundary layer thickness,
, is used for a thickness beyond which the velocity is
essentially the free-stream velocity, U. This is customarily defined as the distance from the
wall to the point where
u|y= =0.99
U
The displacement thickness, *, is the distance by the solid boundary would have to be
displaced in a frictionless flow the same mass deficit exist in the boundary layer. The
mathematcal definiton of the displacement thickness for incompressible flow is given by
*= ( )
The momentum thickness, , is defined as the thickness of the layer fluid of velocity, U(free stream velocity), for which the momentum flux is equal to the deficit of momentum flux
through the boundary layer. Mathematically it is defines as
= ( )
The equation for velocity measured by pitot tube is given as
u =
The Blasiuss exact solution to the laminar boundary layer yields the following equations for
the above properties.
=
* =
=
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
6/23
Due to the complexity of the flow, there is no exact solution to the turbulent boundary layer.
The properties of the boundary layer are approximated using the momentum integral equation
which results in the following expression
=
* =
=
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
7/23
2. OBJECTIVES
1. To measured the boundary layer velocity layer and observed the growth of the
boundary layer for the flat plate with smooth and rough surface.
2. To measured the boundary layer properties for the measured velocity profile.
3. To studied the effect of surface roughness on the development of the boundary
layer.
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
8/23
3. APPARATUS
1. Airflow benchprovie adjustable air stream which enables a series of
experiment to be connected
2. Test apparatusconsists of rectangular duct with flat plate. One side of the
plate is smooth and other rough. Pitot tube is set in zero plane of scale. By
moving plate up and down, the leading edge can be set to given
distance from pitot tube tip.
3. Micrometer scaleto measure the displacement of pitot tube from wall.
4. Velocity measurementvelocity is measured using total and static probes
which connected to multi-tube manometer.
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
9/23
4. EXPERIMENTAL PROCEDURES
1. The apparatus had been set up on the bench. Smooth surface of the plat was
used for the first part of experiment.
2. To set the measurement playe at the required distance from leading edge, the
position of the central plate was adjusted to 50mm.
3. The fan and the air flow speeds was switched on to set the air stream velocity
at medium speed. The total pressure of the pitot tube was read for a range of
several points as the tube traverse toward the plate at an interval of 0.25mm.
4. The increment of the advanced reduced as the pressure begins to fall to clearly
define the velocity profile. The pressure reading was not fall to zero as the
pitot tube has a finite thickness.
5. Step 2 to 4 were repeated for set up of measurement plate at 200mm.
6. The entire experiments was repeated for the rough surface plate.
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
10/23
5. DATA COLLECTED
Test 1: Smooth Surface Plate
Distance from leading edge, x: 0.05 m
Free stream velocity, U: 22.37 m/sRe: 0.738 X 105
Micrometer
reading, y
(mm)
Static
pressure
manometer,
(mbar)
Total
pressure
Manometer
(mbar)
Differential
manometer
height
(mbar)
Differential
manometer
height
h, (mm)
u
(m/s)
0 10.4 12.4 2.0 25.60 18.08 0.81 0.15
0.25 10.4 12.8 2.4 30.72 19.81 0.89 0.10
0.50 10.4 13.0 2.6 33.28 20.62 0.92 0.07
0.75 10.4 13.2 2.8 35.84 21.40 0.96 0.04
1.00 10.4 13.4 3.0 38.40 22.15 0.99 0.01
1.25 10.4 13.4 3.0 38.40 22.15 0.99 0.01
1.50 10.4 13.4 3.0 38.40 22.15 0.99 0.01
1.75 10.4 13.4 3.0 38.40 22.15 0.99 0.01
2.00 10.4 13.4 3.0 38.40 22.15 0.99 0.01
2.25 10.4 13.4 3.0 38.40 22.15 0.99 0.01
2.50 10.4 13.4 3.0 38.40 22.15 0.99 0.01
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
11/23
Test 2: Smooth Surface Plate
Distance from leading edge, x: 0.2 m
Free stream velocity, U: 23.82 m/s
Re: 3.14 X 105
Micrometer
reading, y
(mm)
Static
pressure
manometer,
(mbar)
Total
pressure
Manometer
(mbar)
Differential
manometer
height
(mbar)
Differential
manometer
height
h, (mm)
u
(m/s)
0 10.2 12.4 2.2 28.16 18.97 0.80 0.16
0.25 10.2 12.8 2.6 33.28 20.62 0.87 0.11
0.50 10.2 13.0 2.6 33.28 20.62 0.87 0.11
0.75 10.2 13.2 3.0 38.40 22.15 0.93 0.07
1.00 10.2 13.2 3.0 38.40 22.15 0.93 0.07
1.25 10.2 13.4 3.2 40.96 22.88 0.96 0.04
1.50 10.2 13.4 3.2 40.96 22.88 0.96 0.04
1.75 10.2 13.4 3.2 40.96 22.88 0.96 0.04
2.00 10.2 13.6 3.4 43.52 23.58 0.99 0.01
2.25 10.2 13.6 3.4 43.52 23.58 0.99 0.01
2.50 10.2 13.6 3.4 43.52 23.58 0.99 0.01
2.75 10.2 13.6 3.4 43.52 23.58 0.99 0.01
3.0 10.2 18.6 3.4 43.52 23.58 0.99 0.01
3.25 10.2 13.6 3.4 43.52 23.58 0.99 0.01
3.5 10.2 13.6 3.4 43.52 23.58 0.99 0.01
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
12/23
Test 3: Rough Surface Plate
Distance from leading edge, x: 0.05 m
Free stream velocity, U: 23.11 m/s
Re: 0.762 X 105
Micrometer
reading, y
(mm)
Static
pressure
manometer,
(mbar)
Total
pressure
Manometer
(mbar)
Differential
manometer
height
(mbar)
Differential
manometer
height
h, (mm)
u
(m/s)
0 10.4 12.4 2.0 25.60 18.08 0.78 0.17
0.25 10.4 12.8 2.4 30.72 19.81 0.86 0.12
0.50 10.4 13.0 2.6 33.28 20.62 0.89 0.10
0.75 10.4 13.2 2.8 35.84 21.40 0.93 0.07
1.00 10.4 13.4 3.0 38.40 22.15 0.96 0.04
1.25 10.4 13.4 3.0 38.40 22.15 0.96 0.04
1.50 10.4 13.4 3.0 38.40 22.15 0.96 0.04
1.75 10.4 13.4 3.0 38.40 22.15 0.96 0.04
2.00 10.4 13.6 3.2 40.96 22.88 0.99 0.01
2.25 10.4 13.6 3.2 40.96 22.88 0.99 0.01
2.50 10.4 13.6 3.2 40.96 22.88 0.99 0.01
2.75 10.4 13.6 3.2 40.96 22.88 0.99 0.01
3.0 10.4 18.6 3.2 40.96 22.88 0.99 0.01
3.25 10.4 13.6 3.2 40.96 22.88 0.99 0.01
3.5 10.4 13.6 3.2 40.96 22.88 0.99 0.01
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
13/23
Test 4: Rough Surface Plate
Distance from leading edge, x: 0.2 m
Free stream velocity, U: 24.51 m/s
Re: 3.23 X 105
Micrometer
reading, y
(mm)
Staticpressure
manometer,
(mbar)
Totalpressure
Manometer
(mbar)
Differentialmanometer
height
(mbar)
Differentialmanometer
height
h, (mm)
u
(m/s)
0 10.2 12.2 2.0 25.60 18.08 0.74 0.19
0.25 10.2 12.4 2.2 28.16 18.97 0.77 0.18
0.50 10.2 12.8 2.6 33.28 20.62 0.84 0.13
0.75 10.2 13.0 2.8 35.84 21.40 0.87 0.11
1.00 10.2 13.0 2.8 35.84 21.40 0.87 0.11
1.25 10.2 13.2 3.0 38.40 22.15 0.90 0.09
1.50 10.2 13.4 3.2 40.96 22.88 0.93 0.07
1.75 10.2 13.4 3.2 40.96 22.88 0.93 0.07
2.00 10.2 13.6 3.4 43.52 23.58 0.93 0.07
2.25 10.2 13.6 3.4 43.52 23.58 0.93 0.07
2.50 10.2 13.6 3.4 43.52 23.58 0.93 0.07
2.75 10.2 13.8 3.6 46.08 24.26 0.99 0.01
3.0 10.2 13.8 3.6 46.08 24.26 0.99 0.01
3.25 10.2 13.8 3.6 46.08 24.26 0.99 0.01
3.5 10.2 13.8 3.6 46.08 24.26 0.99 0.01
3.75 10.2 13.8 3.6 46.08 24.26 0.99 0.01
4.0 10.2 13.8 3.6 46.08 24.26 0.99 0.01
4.25 10.2 13.8 3.6 46.08 24.26 0.99 0.01
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
14/23
6. SAMPLE OF CALCULATIONS
Smooth surface with distance from the leading edge, x = 0.05m
air = 1.204 kg/m3
air= 1.825 x 10-5
m2
/soil = 784 kg/m
3
velocity, u =a
o hg
2m/s
=204.1
106.2581.97842 3
m/s
= 18.08 m/s
Free stream velocity, U = highest value of u / 0.99
= 22.15 = 22.37
Reynolds number, Rex = airUXair
=5
10825.1
05.037.22204.1
= 0.735 X 105< 500000 (laminar)
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
15/23
Sample calculation for boundary layer thickness,(
)displacement thickness, (
) momentum
thickness, (
)by usingtheoretical
For smooth surface with x = 50mm
x
x
Re
0.5
=
= 0.9203mm
x
x
Re
72.1
=
= 0.3166mm
x
x
Re
664.0
=
= 0.1222mm
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
16/23
Sample calculation for boundary layer thickness,(
)displacement thickness, (
) momentum
thickness, (
) for experimental values
Boundary layer thickness, = 1.0 mm
Displacement thickness, * =- u/U)dy= - y/ 1/7)dy=/
= 1/8(1.0)
= 0.125 mm
Momentum thickness, = u/U - u/U)
= y/ 1/7)(1- y/ 1/7)=/
= 7/72(1.0)
= 0.0972 mm
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
17/23
7. RESULTS
Graph 6.1: Graph of y vs u/U for test 1.
Graph 6.2: Graph of y vs (u/U)(1-u/U) for test 1.
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1 1.2
m
icrometerreading,y
(mm)
u/U
test1 graph y versus u/U
0
0.5
1
1.5
2
2.5
3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
heigh
t,y(mm)
(u/U)(1-u/U)
test1 graph y vs (u/U)(1-u/U)
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
18/23
Graph 6.3: Graph of y vs u/U for test 2.
Graph 6.4: Graph of y vs (u/U)(1-u/U) for test 2.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2
height,y(mm)
u/U
test2, y vs u/U
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
height,y(mm)
(u/U)(1-u/U)
test2 y vs (u/U)(1-u/U)
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
19/23
Graph 6.5: Graph of y vs u/U for test 3.
Graph 6.6: Graph of y vs (u/U)(1-u/U) for test 3.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2
height,y(mm)
u/U
test3 y vs u/U
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
height,y(mm)
(u/U)(1-u/U)
test3 y vs (u/U)(1-u/U)
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
20/23
Graph 6.7: Graph of y vs u/U for test 4.
Graph 6.8: Graph of y vs (u/U)(1-u/U) for test 4.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1 1.2
height,y(mm)
u/U
test4 y vs u/U
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
height,y(mm)
(u/U)(1-u/U)
test4 y vs (u/U)(1-u/U)
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
21/23
Boundary layer
thickness,() (mm)
Displacement
thickness, (
) (mm)
Momentum
thickness, ()
Test1 theoretical 0.9203 0.3166 0.1222
Test1 experimental 1.0 0.125 0.0972
Test2 theoretical 1.7845 0.6139 0.2370
Test2 experimental 2.0 0.2500 0.1944
Test3 theoretical 0.9057 0.3115 0.1202
Test3 experimental 2.0 0.2500 0.1944
Test4 theoretical 1.7595 0.6053 0.2337
Test4 experimental 2.75 0.3438 0.2674
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
22/23
9. REFERENCES
i.
P.M Gerhart, R.J. Gross and J.I. Hochstein, Fundamentals of fluid Mechanics, 2nd
Edition, 1992.
ii.
F.M White, Fluid Mechanics, McGraw-Hill, 4th Edition, 1999.
iii. Fundamental of Fluid Mechanics, B.r. Munson, D.F. Young and T.H. Okiishi, John
Wiley and Sons, 3rd Edition, 1998
iv.
Yunus A. engel and John M. Cimbala, 2010, Fluid Mechanics Fundamentals And
Applications Second Edition in SI Unit, Published By McGraw Hill International
Edition. In Singapore.
v. A.Cengel, Y., & John M. Cimbala. (N.D.), Fluid Mechanics Fundamental and
Applications, (2nd Edition).
TABLE OF CONTENTS
CONTENTS PAGE
11.OBJECTIVES 2
12.INTRODUCTION 2
13.
THEORY 3
14.APPARATUS 5
15.PROCEDURES 7
16.RESULTS 8
17.DISCUSSIONS 10
-
8/10/2019 Flat Plate Boundary Layer 7over10 Complete
23/23
18.CONCLUSIONS 11
19.
RECOMMENDATIONS 12
20.REFERENCES 39