DATE:
TO:
April 10, 2008
Jeffrey Caniparoli, Mechanical Engineering Test Manager
FROM: V. Dakshina Murty, Test Engineer (Member of Group A)
SUBJECT: Boundary Layers (Experiment 12)
As per your request, an experiment was conducted on March 31, 2008 to study boundary layers on a flat plate and measure the displacement thickness and momentum thickness.
The equipment for this experiment is located in Shiley Hall 106 on the University of
Portland campus. The purpose of this letter is to give the details of the procedure and results.
The equipment consisted of the Air Flow bench which has a fan providing air at various speeds. By means of the choke at the exit of the fan, the speed could be adjusted. A flat plate which was smooth on one side and rough on the other side was placed in the flow.
By means of a fine Pitot tube, measurements of the stagnation pressure of the fluid were taken at various distances from the plate surface. Initially the Pitot tube was placed sufficiently far from the plate, to ensure that it was completely outside the boundary layer. Then it was slowly moved till it was on the edge of the boundary layer. This was determined when the manometer readings changed when the tube was moved. Once this was determined, readings were taken every 0.4 millimeters till the Pitot tube was in contact with the plate. This procedure was repeated for four different positions along the length of the plate. Local pressure and temperature were recorded to calculate air density.
For the laboratory conditions, density of air was measured to be 1.17 kg/m
3
. Inlet air speed was practically a constant and was equal to 34.8 m/s. The measured values of the displacement thickness ranged from 0.31 mm to 0.58 mm. When compared with the theoretical predictions the percent error varied between 2% and 33%. The momentum thickness ranged from 0.26 mm to 0.88 mm. The corresponding errors when compared with theoretical predictions were between 41% and 283%.
Considering the nature of the experiment which involved measurements in turbulent flow, the equipment performed reasonably well. One difficulty was the exact determination of the outer edge of the boundary layer due to the fluctuations of the readings. The maximum error of 33% associated with displacement thickness is acceptable. However, results for momentum thickness appear questionable in view of the large error between the measured and calculated values. It is recommended that this experiment be repeated several times and average trends should be taken, before confident predictions can be made about boundary layers. If you have any questions, please contact me at murty@up.edu
.

EXPERIMENT 12
BOUNDARY LAYERS
Department of Mechanical Engineering
University of Portland
April 10, 2008
Report Submitted by: V. Dakshina Murty
Experimental work performed on March 31, 2008 in
Room 106 Engineering Building by
V. Dakshina Murty
Allan Hanson
Craig Henry

EXECUTIVE SUMMARY
An experiment was conducted to measure the displacement thickness and momentum thickness of boundary layers on a flat plate. The equipment consisted of the Air Flow bench is located in 106 Shiley Hall in University of Portland. It consisted mainly of a fan providing air at various speeds to which various attachments could be added. One such attachment consisted of a channel into which a flat plate could be inserted. By means of a fine Pitot tube, measurements of the stagnation pressure of the fluid were taken at various distances from the plate surface. The edge of boundary layer was detected by initially p[lacing the Pitot tube sufficiently far from the plate, then moving it slowly till was on the edge of the boundary layer. This was determined when the manometer readings changed when the tube was moved. Once inside the boundary layer, readings were taken every 0.4 millimeters till the Pitot tube was in contact with the plate. This procedure was repeated for four different positions along the length of the plate. Local pressure and temperature were recorded to calculate air density. Details of the equipment and other test materials are given in the Equipment and Materials section of this report.
For the temperature and pressure measured in the laboratory, density of air was measured to be 1.17 kg/m
3
. Inlet air speed was equal to 34.8 m/s and was practically constant throughout the experiment. The displacement thickness ranged from 0.31 mm to 0.58 mm. When compared with the theoretical predictions the percent error varied between
2% and 33%. The momentum thickness ranged from 0.26mm to 0.88 mm with the corresponding errors between 41% and 283%.
A major difficulty was noticed in connection with the exact determination of the outer edge of the boundary layer. It was due to the fluctuations of the manometer readings indicating that the edge is unstable. Although the maximum error associated with displacement thickness is acceptable, results for momentum thickness appear questionable in view of the large error between the measured and calculated values. This has been the disappointing aspect of this experiment. To make confident predictions about boundary layers, it is recommended that this experiment be repeated several times and average trends should be taken.
INTRODUCTION
The purpose of this experiment is to determine how the boundary layer grows along the length on a smooth flat plate. This growth rate can be related to the shear stress on the wall from which skin friction drag can be calculated. To obtain an accurate estimate of the power required to overcome drag, it is necessary to understand the features of the boundary layer. Aeronautical engineers, naval architects, and automobile engineers are interested in minimizing the friction drag which is directly related to the resistance of motion. Too much friction drag may result in the vehicles to be uneconomical because of excessive costs of the propulsive mechanisms.
BACKGROUND

Any object that moves through a fluid experiences a force that resists motion. This is called drag and is due to two different mechanisms: pressure difference between the front and the rear of the body and friction on the surface of the body. The former is called pressure drag which is present for all blunt objects while the latter is called skin friction drag which is present on slender bodies like airfoils, ships, etc. Even for fluids which have very small viscosities the shear stress on the surface can be quite significant. This is due to the fact that the shear stress is the product of viscosity and strain rate. While the former can be vanishingly small, the latter can be quite significant and result in a reasonably large shear stress which is the product of the two terms. Thus for fluids with small viscosities, while most of the flow may behave in a very non viscous way, there will be regions near the solid surfaces where the velocity changes from zero on the wall
(due to the no slip condition) to the free stream speed in very small distance. This results in high velocity gradients. These regions where shear stresses are large near solid surfaces are called boundary layers. An effective understanding of boundary layers is important to predict drag forces on flat objects.
Consider the flow over a flat plate as shown in Fig. 1. The free stream speed is given by
U. Along the length of the plate, the velocity is zero and at a sufficient distance from the plate the fluid velocity is equal to the free stream speed, U. The distance required for the speed to increase from zero on the wall to the free stream speed is called the boundary layer. Its thickness is defined as the distance when the boundary layer speed is 99% of the free stream speed. The thickness of the boundary layer is zero near the leading edge of the plate. Along the length of the plate, its value increases till the boundary layer becomes turbulent. For flat plates the transition to turbulence occurs when the Reynolds number based on distance from leading edge is between 10
5
and 10
6
(see Crowe [1]). The thickness of the boundary layer δ for laminar and turbulent cases is given as follows (see
Crowe [1]):
 x

5
Re
1
2 x
(1)
 x

0 .
16
Re
1
7 x
(2)
Because of the reduction of velocity associated with the boundary layer, the streamlines outside the boundary are shifted away from the boundary. This amount of displacement is called the displacement thickness, denoted by δ *
. By applying the conservation of mass, it can be shown that the displacement thickness is given by the following formula (see
Schlichting [3]):


*

(

0

1 u
U
) dy (3)

Fig. 1: Boundary layer formation on a flat plate (from reference [2])
By applying the momentum equation in conjunction with conservation of mass, another useful measure of thickness of boundary layer called the momentum thickness is obtained
(see the ME 374 manual [2]). This can be written as:
 


0
U u
( 1
 u
U
) dy (4)
For laminar boundary layers, the displacement thickness

* and momentum thickness θ are given as follows (see the ME 374 manual [2]):

*

1 .
721 x
Re x
 
0 .
664 x
Re x
(5)
For turbulent boundary layers, these variables can be written as (see the ME 374 manual
[2]):

*

0 .
046 x
Re x
0 .
2
(6)
 
0 .
036 x
Re x
0 .
2
The shear stress on the wall is given in terms of the momentum thickness as (see the ME
374 manual [2])

0
 
U
2 d

(7) dx

Thus by measuring the velocity profiles inside the boundary layer at various lengths of a plate, it is possible to obtain values for the momentum thickness and estimate the shear stress on the plate and hence the skin-friction drag.
LABORATORY METHODOLOGY
Equipment and Materials
The equipment was a self contained air flow bench consisting of a fan, barometer, manometer, thermometer, etc. Given below are the details of the equipment
Multi-tube Manometer
Air Flow Bench
Model # AF 10a
Model # AF 10
Procedure
The arrangement of the test section is shown in Fig. 2. A flat plate was placed in the middle of the section with the sharp edge facing the oncoming flow. One side of the plate is smooth and the other side is rough. The experiment was conducted only for the smooth side of the flat plate. The fine Pitot tube was traversed through the boundary layer at a given section of the plate. The Pitot tube was set about 10 mm from the surface and moved towards the plate. This distance was enough to make sure that the Pitot tube was outside the boundary layer. It was then moved towards the plate in small increments and the pressures were noted. Once the instrument was inside the boundary layer, the readings began to change. It was noted that the readings do not go to zero even when the
Pitot tube touches the plate, the reason being the finite thickness of the Pitot tube. The following data refers to the physical dimensions of the Pitot tube.
Thickness of Pitot tube at tip, 2t = 0.4 mm
Length of plate from leading edge to traverse section = 0.265 mm

Fig. 2: Experimental setup (Reference [2])
RESULTS
The experimental data is given in Appendix I. Using the room temperature and pressure the density of air was calculated to be 1.17 kg/m 3 . Using the inlet and stagnation pressures in the channel, the velocity ratios are calculated (sample calculations are shown in Appendix II) and are shown in Table 1. The measurements were taken at four notch locations which were separated by 5.1 cm from each other. From the velocity profiles inside the boundary layer, the displacement thickness and momentum thickness are calculated. They are shown in Table 2. The highest value of the Reynolds number is about 630,000. The transition of the boundary layer from laminar to turbulent takes place around Reynolds numbers of 500,000 to 1 million. The calculated values of the displacement and momentum thicknesses (δ and θ) are given from Eqs. 5 - 6. These results are shown graphically in Figs. 3 and 4.

Table 1: Variation of velocity ratios inside the boundary layer along the length of the plate
Readings taken at notch 1 Readings taken at y+zero u/U 1-u/U
(1u/U)*u/U y+zero u/U 1-u/U
0.2 0.52475 0.47525 0.249387
0.6 0.761387 0.238613 0.181677
1 0.825324 0.174676 0.144164
1.4 0.908893 0.091107 0.082806
1.8 0.940244 0.059756 0.056186
2.2 0.978019 0.021981 0.021498
2.6 0.985401 0.014599 0.014386
3
3.4
1
1
0
0
0
0
Readings taken at notch 3 Readings taken at notch 2
(1u/U)*u/U
0.2 0.722315 0.277685 0.200576
0.6 0.816497 0.183503 0.14983
1 0.868115 0.131885 0.114492
1.4 0.916831 0.083169 0.076252
1.8 0.947919 0.052081 0.049368
2.2 0.970582 0.029418 0.028553
2.6 0.985401 0.014599 0.014386
3
3.4
3.8
0.992727 0.007273
1
1
0
0
0.00722
0
0 notch 4 y+zero u/U 1-u/U
(1u/U)*u/U y+zero u/U 1-u/U
(1u/U)*u/U
0.2 0.637022 0.362978 0.231225
0.7 0.780189 0.219811 0.171494
1.2 0.825324 0.174676 0.144164
1.7 0.892805 0.107195 0.095704
2.2 0.932505 0.067495 0.06294
2.7 0.963087 0.036913 0.035551
3.2 0.978019 0.021981 0.021498
3.7 0.985401 0.014599 0.014386
4.2 0.992727 0.007273 0.00722
4.7 1 0 0
5.2 1 0 0
0.2 0.57735 0.42265 0.244017
0.7 0.770846 0.229154 0.176643
1.2 0.825324 0.174676 0.144164
1.7 0.884652 0.115348 0.102043
2.2 0.924701 0.075299 0.069629
2.7 0.947919 0.052081 0.049368
3.2 0.963087 0.036913 0.035551
3.7 0.978019 0.021981 0.021498
4.2 0.985401 0.014599 0.014386
4.7 0.992727 0.007273 0.00722
5.2
5.7
0.99637
0.99637
0.00363
0.00363
0.003617
0.003617
6.2 0.99637 0.00363 0.003617
Table 2: Displacement and Momentum thickness along the length of the plate for various Reynolds numbers x Re
Displacement thickness
 expt)

Momentum
Thickness
 calc)
  expt)
  calc)

(m) (mm)
0.119 274206 0.4304
(mm)
0.3911
(mm)
0.3000
(mm)
0.1509
0.4688 0.2563 0.1809 0.171 394027 0.3118
0.222 511544 0.5065
0.273 629061 0.5804
0.5342
0.5924
0.5530
0.8754
0.2061
0.2286

Fig. 3: Velocity profile inside the boundary layer.
Fig. 4: Linear momentum profile inside the boundary layer.

DISCUSSION
The results of the experiment were marginally satisfactory. The percentage errors in the displacement thickness ranged from a low of 2 % to a high of 33 %. The corresponding values for momentum thickness were much worse, ranging from a 41% to 283%. Results for the former are acceptable since measurements in turbulent flows are very difficult in view of rapidly fluctuating velocities. Since the momentum thickness is derived from the velocity profiles, any error in the latter exacerbates the former. This could partially explain the large values of error for momentum thickness. Another significant source of error is the delicate nature of the Pitot tube itself. Due to repeated usage, its tip was bent and hence when it was just in contact with the plate, it was not totally aligned with the flow. This was the case even when readings were taken inside the boundary layer. When the readings are not aligned with the flow, the pressure readings would not be stagnation readings. Also the edge of the boundary layer could not be determined exactly since the flow fluctuated quite rapidly and it was difficult to take accurate readings. It can also be seen that the boundary layer transitioned from laminar to turbulent beyond half the length of the plate. Theoretical predictions in such situations are difficult to make since the formulae are given for tripped boundary layers. This experiment would be suitable for qualitative study of boundary layers but cannot be recommended for making quantitative predictions.
REFERENCES
1.
Crowe, C. T., Elger, D. F., and Roberson, J. A., Engineering Fluid Mechanics ,
John Wiley, 8 th
Edition, 2005.
2.
ME 374 Fluids Laboratory Requirements and Experiments , School of
3.
Engineering, University of Portland, 2008-2009.
Schlichting, H., Boundary Layer Theory , McGraw-Hill, New York, 1979 .

APPENDIX I
Experimental Data

Room conditions:
Boundary Layer Experiment: Date March 31, 2008 p atm
(mm) T o C
762 29
Distance from leading edge =
Micrometer reading
0
400
800
1200
1600
2000
2400
2800
3200
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2 y
(mm)
0 y+zero
0.2
0.6
1
1.4
1.8
2.2
2.6
3
3.4 notch 1 p
(mbar)
13.2
15.3
16
17
17.4
17.9
18
18.2
18.2
Chamber Inlet
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
11.3
11.3
11.3
11.3
11.3
11.3
11.3
11.3
11.3
Micrometer reading
0
400
800
1200
1600
2000
2400
2800
3200
3600
Micrometer reading
0
500 y
(mm)
0
0.5
(mm)
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6 y
Distance from leading edge = y+zero
0.2
0.6
1
1.4
1.8
2.2
2.6
3
3.4
3.8
Distance from leading edge = y+zero
0.2
0.7 p
(mbar)
14.9
15.9
16.5
17.1
17.5
17.8
18
18.1
18.2
18.2 notch 2
Chamber Inlet
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4 p
(mbar)
14.1
15.5 notch 3
Chamber Inlet
18.4
18.4
11.4
11.4

1000
1500
2000
2500
3000
3500
4000
4500
5000
Micrometer reading
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6 y
(mm)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1.2
1.7
2.2
2.7
3.2
3.7
4.2
4.7
5.2
Distance from leading edge = y+zero
0.2
0.7
1.2
1.7
2.2
2.7
3.2
3.7
4.2
4.7
5.2
5.7
6.2
16
16.8
17.3
17.7
17.9
18
18.1
18.2
18.2
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4 p
(mbar)
13.6
15.4
16
16.7
17.2
17.5
17.7
17.9
18
18.1
18.15
18.15
18.15 notch 4
Chamber Inlet
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
18.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4
11.4

APPENDIX II
Calculations