University of Southampton

Faculty of Engineering and the Environment

Part II, FEEG2003: Fluid DynamicsLaboratory 1: Boundary Layer Flow

Vincent LamLaboratory date: 23/02/2015

IntroductionObjectiveThe aim of this experiment is to investigate the differences in the velocity distributions in laminar and turbulent boundary layer flows. Then evaluate the accuracy of the approximations in the analyse process.

BackgroundIn viscous flows, the fluid directly adjacent to a solid surface has the same velocity as the surface. This is known as the ‘no-slip’ condition which means that there is a boundary layer where the flow velocity varies from zero at the surface to the external flow velocity. After a certain height above the surface the flow has a freestream velocity.

For streamlined objects the boundary layers are thin, usually only 1% of the chord length. Boundary layer forms are dependent on the flow environment and the Reynolds number. For low Reynolds numbers and low background instabilities then the flow in the boundary layer over smooth surfaces is laminar, so smooth and steady. At high Reynolds numbers or if the flow has high background disturbances then the flow is turbulent, so unsteady and chaotic motions. . Figure 1 shows laminar, transitional and turbulent flows with corresponding Reynolds numbers.

Figure 1

For boundary layers with zero free stream pressure gradients, the approximate forms for the change of average velocity are:

.

.

.

Description of experimental procedure We consider the flow along a flat plate aligned with the oncoming flow to study boundary layers with zero freestream pressure gradient by using a wind tunnel. A small pitot tube, which can travel across the boundary layer, is used to measure the flow velocity. The pitot tube is connected to an inclined manometer, so a static pressure tapping from a hole in the plate to the other end. This means that the manometer reads:

p0− p=12ρ U2

Procedure steps:

1. Record the temperature and pressure of the laboratory room.2. Check that the plate leading edge is smooth and record the distance between this edge and

to the pitot probe. 3. Vary the wind tunnel speed so that the tunnel reference reading is 12 throughout the

experiment. 4. When the pitot is furthest from the plate edge, measure the pressure difference. Record the

inclination angle of the manometer and convert the pressure difference to N/m². Then calculate the velocity U eoutside the boundary layer.

5. Increase the traverse distance of the probe across the boundary layer and record this reading with the manometer reading. When the manometer reading reaches the same reading as the free –stream velocity then stop the traversing.

6. Turn off the tunnel. Stick a piece of sand (glass) paper to the leading edge of the flat plate. This should cause the boundary layer to become turbulent.

7. Repeat steps 3 to 5 for the new case and check that the two graphs match what is expected of laminar and turbulent flows.

Data Collection and AnalysisExperimental dataIn the laboratory, the temperature of air is 15°C (288K) and the pressure is 101325 Pa. The density of air under these conditions is 1.225 kg/m³. The distance from edge of plate to pitot is 0.251 m. All the manometer readings can be converted to velocity in 𝑚/𝑠 using:

p0− p=12ρ U2

Free stream velocity calculationRearranging formula (1) gives:

U e=√ 2 ( po−p )ρ

Input values for dynamic pressure and density at free stream velocity data point gives:

(1)

U e=√ 2×3701.225=24.578m /s

Turbulent caseTable 1 below shows the pitot probe height above the plate, manometer readings, corrected dynamic pressure and velocity using equation 1. A correction factor of 0.025 inches were added to the original values to compensate for the radius of the pitot. Then to convert the pitot probe height from inches to metres, the corrected distance was multiplied by 0.0254.

Distance from flat plate (inches)

Corrected distance (inch)

Corrected distance (m)

Dynamic pressure (kPa)

Corrected dynamic pressure (Pa)

Velocity (m/s)

0 0.025 0.000635 0.63 126 14.34274330.005 0.03 0.000762 0.72 144 15.33303760.01 0.035 0.000889 0.83 166 16.46269750.015 0.04 0.001016 0.89 178 17.0473530.02 0.045 0.001143 0.93 186 17.42622940.025 0.05 0.00127 0.97 194 17.79704170.03 0.055 0.001397 1 200 18.07015810.035 0.06 0.001524 1.03 206 18.33920750.04 0.065 0.001651 1.06 212 18.60436640.045 0.07 0.001778 1.09 218 18.86579890.05 0.075 0.001905 1.12 224 19.12365770.055 0.08 0.002032 1.15 230 19.37808570.06 0.085 0.002159 1.18 236 19.6292160.065 0.09 0.002286 1.2 240 19.79486640.07 0.095 0.002413 1.23 246 20.04077480.08 0.105 0.002667 1.28 256 20.44405010.09 0.115 0.002921 1.33 266 20.83952290.1 0.125 0.003175 1.38 276 21.22762930.11 0.135 0.003429 1.43 286 21.60876620.12 0.145 0.003683 1.47 294 21.90890230.13 0.155 0.003937 1.52 304 22.27838710.14 0.165 0.004191 1.56 312 22.56962020.15 0.175 0.004445 1.6 320 22.85714290.16 0.185 0.004699 1.64 328 23.14109340.17 0.195 0.004953 1.68 336 23.42160180.18 0.205 0.005207 1.72 344 23.69879010.195 0.22 0.005588 1.75 350 23.90457220.21 0.235 0.005969 1.8 360 24.24366110.225 0.25 0.00635 1.825 365 24.41143930.23 0.255 0.006477 1.83 366 24.44485670.24 0.265 0.006731 1.84 368 24.5115550.255 0.28 0.007112 1.85 370 24.5780722

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.0080

5

10

15

20

25

30Graph of velocity against distance from surface for turbulent

flow

Corrected distance from the surface (m)

Vel

ocity

(m/s

)

From graph above, the boundary layer thickness can be found where y (pitot tube height) matches with the velocity of 99% of the free stream velocity.

0.99×U e=24.332po−p=362.635 Pa

¿0.00600m

This graph shows that there is a steep increase in velocity of the flow when close to the wall. It also takes a larger thickness for the velocity to reach free stream velocity compared to the laminar case, which is shown later.

The next step is to normalise U and y as U /U e and y/Table 2 below shows the result of this normalisation. Theoretical values of U /U e can be calculated using equation 2 from the lab document.

UU e

=( y❑ )1n

Where n is the turbulent intensity and is assumed to be 5. y (m) U (m/s) U/Ue y / Theoretical U/Ue0.000635 14.34274331 0.583558515 0.105995785 0.6383483190.000762 15.33303756 0.623850294 0.127194942 0.6620548460.000889 16.46269748 0.669812398 0.148394099 0.6827839860.001016 17.04735302 0.693600087 0.169593256 0.7012642770.001143 17.42622935 0.709015305 0.190792413 0.7179797910.00127 17.79704172 0.724102427 0.211991571 0.7332696640.001397 18.07015806 0.735214622 0.233190728 0.7473813490.001524 18.33920747 0.746161348 0.254389885 0.7605013120.001651 18.6043664 0.756949782 0.275589042 0.772773799

0.001778 18.86579888 0.7675866 0.296788199 0.7843128410.001905 19.12365775 0.77807802 0.317987356 0.7952102520.002032 19.37808567 0.788429846 0.339186513 0.8055411210.002159 19.62921604 0.798647505 0.36038567 0.8153676990.002286 19.79486637 0.805387266 0.381584827 0.8247422050.002413 20.04077476 0.815392461 0.402783984 0.8337089020.002667 20.44405008 0.831800392 0.445182298 0.8505651360.002921 20.83952289 0.847890865 0.487580612 0.8661822620.003175 21.22762928 0.863681623 0.529978926 0.8807481110.003429 21.60876617 0.879188815 0.57237724 0.894409640.003683 21.9089023 0.891400356 0.614775555 0.9072841380.003937 22.27838707 0.906433462 0.657173869 0.919466810.004191 22.56962018 0.918282769 0.699572183 0.931036070.004445 22.85714286 0.92998111 0.741970497 0.9420573180.004699 23.14109341 0.941534113 0.784368811 0.9525857120.004953 23.42160175 0.952947065 0.826767125 0.9626682310.005207 23.69879012 0.964224937 0.869165439 0.9723452340.005588 23.90457219 0.972597525 0.93276291 0.9861756020.005969 24.24366107 0.986393924 0.996360381 0.9992710140.00635 24.41143927 0.993220261 1.059957853 1.0117139060.006477 24.44485673 0.994579906 1.08115701 1.015728770.006731 24.51155496 0.997293635 1.123555324 1.0235731670.007112 24.57807219 1 1.187152795 1.034906999

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

Experimental and theoretical graph of U/Ue against y/d for turbulent flow

Theoretical U/Ue Experimental turbulent U/Ue

y/

U/U

e

The graph on the above shows that there is a tiny difference between the theoretical and the experimental values since the shape of both the curves are consistently similar. The average percentage difference between the two curves is 2.2%. This is reasonable since there were many possible sources of errors in the experiment. It can be concluded that the equation is suitable for estimating the velocity profile as the difference is relatively small. Although the value of turbulent intensity n=5 may not be optimal and this should be taken into consideration.

y / Theoretical U/Ue Experimental U/Ue Difference % difference

0.105995785 0.638348319 0.583558515 0.054789804 9.3889135150.127194942 0.662054846 0.623850294 0.038204552 6.1239935450.148394099 0.682783986 0.669812398 0.012971588 1.9366001110.169593256 0.701264277 0.693600087 0.007664189 1.1049867820.190792413 0.717979791 0.709015305 0.008964486 1.2643572030.211991571 0.733269664 0.724102427 0.009167238 1.2660139480.233190728 0.747381349 0.735214622 0.012166726 1.6548537070.254389885 0.760501312 0.746161348 0.014339964 1.921831580.275589042 0.772773799 0.756949782 0.015824017 2.0904975710.296788199 0.784312841 0.7675866 0.016726241 2.1790688870.317987356 0.795210252 0.77807802 0.017132232 2.2018655130.339186513 0.805541121 0.788429846 0.017111275 2.1702977480.36038567 0.815367699 0.798647505 0.016720194 2.0935636130.381584827 0.824742205 0.805387266 0.019354939 2.4031840920.402783984 0.833708902 0.815392461 0.018316441 2.2463343830.445182298 0.850565136 0.831800392 0.018764744 2.2559191420.487580612 0.866182262 0.847890865 0.018291397 2.157281960.529978926 0.880748111 0.863681623 0.017066488 1.976016060.57237724 0.89440964 0.879188815 0.015220824 1.7312349940.614775555 0.907284138 0.891400356 0.015883782 1.7818908890.657173869 0.91946681 0.906433462 0.013033348 1.4378714420.699572183 0.93103607 0.918282769 0.012753301 1.3888206670.741970497 0.942057318 0.92998111 0.012076208 1.2985433170.784368811 0.952585712 0.941534113 0.011051599 1.1737863560.826767125 0.962668231 0.952947065 0.009721166 1.0201160780.869165439 0.972345234 0.964224937 0.008120297 0.8421579180.93276291 0.986175602 0.972597525 0.013578077 1.3960632460.996360381 0.999271014 0.986393924 0.01287709 1.3054713831.059957853 1.011713906 0.993220261 0.018493645 1.8619882961.08115701 1.01572877 0.994579906 0.021148864 2.1264117371.123555324 1.023573167 0.997293635 0.026279532 2.6350847061.187152795 1.034906999 1 0.034906999 3.490699854

Next step is to calculate:

Rex=U e x❑

Where x is the distance from the leading edge of the plate to the pitot tube and is kinematic viscosity with a value of 14.55×10−6m2/s

R ex=24.578×0.25114.55×10−6

=423991.615

Then calculate:

Re❑=U e

❑

Re❑=24.578×0.0060014.55×10−6 =10135.258

Then use expression (4) from lab document to find the boundary layer thickness, which is a rough approximation for turbulent flow:

❑x

=0.375ℜ x−0.2

¿0.375×423991.615−0.2×0.251=0.00705mPercentage difference for boundary layer thickness:

0.00705−0.006000.00600

×100=17.5%

Therefore equation (4) is barely suitable for calculating turbulent flow boundary layer thickness. This is due to the fairly large percentage difference of 17.5% between the experimental and theoretical values. Although the equation used is a rough approximation and not a theoretical result like for the laminar flow. Although it makes sense that the theoretical boundary layer thickness would be much greater for turbulent flow than laminar.

Laminar caseTable 4 below shows the pitot probe height above the plate, manometer readings, corrected dynamic pressure and velocity using equation 1.

Distance from flat plate (inches)

Corrected distance (inch)

Corrected distance (m)

Dynamic pressure (kPa)

Corrected dynamic pressure (Pa)

Velocity (m/s)

0 0.025 0.000635 0.49 9812.6491106

0.005 0.03 0.000762 0.57 11413.6426702

0.01 0.035 0.000889 0.71 14215.2261858

0.015 0.04 0.001016 0.85 17016.6598626

0.02 0.045 0.001143 0.98 19617.8885438

0.025 0.05 0.00127 1.11 22219.0380929

0.03 0.055 0.001397 1.22 24419.9591419

0.035 0.06 0.001524 1.31 26220.6822412

0.04 0.065 0.001651 1.4 28021.3808994

0.045 0.07 0.001778 1.49 29822.0574389

0.05 0.075 0.001905 1.55 31022.4971654

0.055 0.08 0.002032 1.62 32422.9995563

0.06 0.085 0.002159 1.67 334 23.351790

6

0.065 0.09 0.002286 1.71 34223.6297979

0.07 0.095 0.002413 1.74 34823.8361756

0.075 0.1 0.00254 1.76 35223.9727737

0.08 0.105 0.002667 1.78 35624.1085978

0.085 0.11 0.002794 1.79 35824.1762238

0.09 0.115 0.002921 1.81 36224.3109113

0.095 0.12 0.003048 1.815 36324.3444667

0.1 0.125 0.003175 1.82 364 24.377976

0.105 0.13 0.003302 1.83 36624.4448567

0.11 0.135 0.003429 1.835 36724.4782286

0.115 0.14 0.003556 1.84 368 24.511555

0.12 0.145 0.003683 1.845 36924.5448361

0.125 0.15 0.00381 1.85 37024.5780722

0.13 0.155 0.003937 1.85 37024.5780722

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.00450

5

10

15

20

25

30

Graph of velocity against distance from surface for laminar flow

Corrected distance from the surface (m)

Vel

ocity

(m/s

)

Free stream velocity calculation

Rearranging formula (1) gives:

U e=√ 2 ( po−p )ρ

Input values for dynamic pressure and density at free stream velocity data point gives:

U e=√ 2×3701.225=24.578m /s

From graph above, the boundary layer thickness can be found where y (pitot tube height) matches with the velocity of 99% of the free stream velocity.

0.99×U e=24.332po−p=362.635 Pa

¿0.00296m

Next step to normalise U and y as U/Ue and y/ and table 5 below shows these results. From the lab document, expression (1) can be used to compute the theoretical values of U/Ue:

UU e

=2 y❑−2( y❑ )3

+( y❑ )4

y (m) U (m/s) U/Ue y / Theoretical U/Ue0.000635 12.64911064 0.514650235 0.214438063 0.4112693210.000762 13.64267016 0.555074867 0.257325675 0.4849575630.000889 15.22618582 0.619502852 0.300213288 0.5544343780.001016 16.65986256 0.677834389 0.3431009 0.6192809220.001143 17.88854382 0.727825343 0.385988513 0.6791595480.00127 19.03809286 0.774596669 0.428876126 0.7338138070.001397 19.95914194 0.812071093 0.471763738 0.7830684440.001524 20.68224122 0.841491597 0.514651351 0.8268294040.001651 21.38089935 0.869917672 0.557538963 0.8650838260.001778 22.05743893 0.897443817 0.600426576 0.8979000470.001905 22.49716535 0.915334823 0.643314188 0.9254276020.002032 22.99955634 0.935775441 0.686201801 0.9478972210.002159 23.35179056 0.95010668 0.729089413 0.965620830.002286 23.62979786 0.961417872 0.771977026 0.9789915550.002413 23.83617556 0.969814694 0.814864638 0.9884837160.00254 23.97277367 0.975372417 0.857752251 0.994652830.002667 24.10859784 0.98089865 0.900639864 0.9981356120.002794 24.17622377 0.983650125 0.943527476 0.9996499720.002921 24.3109113 0.989130112 0.986415089 0.999995020.003048 24.34446675 0.990495372 1.029302701 1.0000510590.003175 24.37797601 0.991858752 1.072190314 1.000779590.003302 24.44485673 0.994579906 1.115077926 1.0032233130.003429 24.47822856 0.995937695 1.157965539 1.0085061210.003556 24.51155496 0.997293635 1.200853151 1.0178331070.003683 24.54483611 0.998647734 1.243740764 1.032490559

0.00381 24.57807219 1 1.286628377 1.0538459620.003937 24.57807219 1 1.329515989 1.083347998

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

Experimental and theoretical graph of U/Ue against y/d for laminar flow

Theoretical U/Ue Experimental laminar U/Ue

y/

U/Ue

The graph above shows a comparison between the experimental and theoretical values of U/Ue. The graph shows that the two curves have similar shapes apart from at the initial and end points. This is because when flow is lose to the wall there is a gradual increase in velocity until it reaches the freestream velocity.

From using table 6 below, it can be calculated that the average percentage difference is 1.19% which is low, despite the initial point having an initially percentage difference of -20.087%. Therefore this equation is consistent enough to be deemed suitable for this case.

Theoretical U/Ue Experimental U/Ue Difference % difference0.411269321 0.514650235 -0.1033809 -20.087606580.484957563 0.555074867 -0.0701173 -12.632044460.554434378 0.619502852 -0.0650685 -10.503337290.619280922 0.677834389 -0.0585535 -8.6383146490.679159548 0.727825343 -0.0486658 -6.6864660660.733813807 0.774596669 -0.0407829 -5.2650449110.783068444 0.812071093 -0.0290026 -3.5714420660.826829404 0.841491597 -0.0146622 -1.7424052280.865083826 0.869917672 -0.0048338 -0.555667146

0.897900047 0.8974438170.00045623 0.050836598

0.925427602 0.9153348230.01009278 1.102632514

0.947897221 0.9357754410.01212178 1.295372739

0.96562083 0.95010668 0.0155141 1.632885142

5

0.978991555 0.9614178720.01757368 1.827892308

0.988483716 0.9698146940.01866902 1.925009179

0.99465283 0.9753724170.01928041 1.976723226

0.998135612 0.980898650.01723696 1.757262219

0.999649972 0.9836501250.01599985 1.62657915

0.99999502 0.9891301120.01086491 1.098430609

1.000051059 0.9904953720.00955569 0.964738193

1.00077959 0.9918587520.00892084 0.899406125

1.003223313 0.9945799060.00864341 0.869051032

1.008506121 0.9959376950.01256843 1.261969143

1.017833107 0.9972936350.02053947 2.059521013

1.032490559 0.9986477340.03384282 3.388865105

1.053845962 10.05384596 5.384596199

1.083347998 1 0.083348 8.334799823

Next step is to calculate:

Rex=U e x❑

Where x is the distance from the leading edge of the plate to the pitot tube and is kinematic viscosity with a value of 14.55×10−6m2/s

Rex=24.578×0.25114.55×10−6

=423991.615

Then calculate:

Re❑=U e

❑

Re❑=24.578×0.0029614.55×10−6 =5000.060

Then use expression (3) from lab document to find the boundary layer thickness, which is theoretical result for laminar flow:

❑x

=5.83×ℜx−0.5

¿5.83×423991.615−0.5×0.251=0.00225mPercentage difference for boundary layer thickness:

0.00296−0.002250.00296

×100=31.5%

Therefore equation (3) is not suitable for calculating turbulent flow boundary layer thickness. This is due to the very large percentage difference between the experimental and theoretical values. Although it makes sense that the theoretical boundary layer thickness would be much less for laminar flow than turbulent.

Qualitative differences between the laminar and turbulent flow casesThe thickness of the boundary layer is the main distinction between the turbulent and laminar cases. The turbulent boundary layer had a thickness of 0.00600 m whereas the laminar had 0.00296 m. This means that there is a percentage difference of 102.7% which is very large but expected.

0.00600−0.002960.00296

×100=102.70%

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

Comparison of experiment U/Ue values for laminar and tubur-lent cases

Experimental laminar U/Ue Experimental turbulent U/Ue

y/

U/Ue

The graph above highlights the differences between laminar and turbulent cases. For the turbulent curve, the steep velocity gradient when adjacent and close to the wall caused a greater shear stress, which is due to the higher Reynolds number. This meant that at the beginning of the experiment, the velocity stayed at zero for a longer period of time due to the high friction in the fluid. Since turbulent flow is more chaotic, it has higher energy which results in the steeper increase in velocity.

Possible sources of error in the experimentThere are many possible errors in this experiment:

1. The wind tunnel machine used for the experiment is old and inaccurate so it unlikely that the tunnel reference reading was consistently at 12. This would have affected the accuracy and reliability of the initial dynamic pressure and hence velocity.

2. In the laboratory, the room pressure may fluctuate.3. There may be variance in temperature. 4. Throughout the experiment, there people walking across the intake of the wind tunnel

which would temporarily block it. Therefore this would affect the velocity of the flow in the experiment.

5. The pitot tube may be loose.6. The pitot tube did not lie exactly on the plate due to a difference of 0.025 inches due to the

radius of the tube. 7. The flat end of the leading edge had bits of residue sand paper left behind by the previous

group. This affected the results of the laminar case.