Department of Mechanical Engineering, Nottingham University

The following report is an experimental study of Boundary layer data obtained using hot-wire anemometry. In particular, the report presents and analyses mean velocity and turbulence intensity profiles as well as turbulence statistics. The aim is to discover the best methods of presenting and analysing data in order to determine and explain some of the turbulence phenomena by discussion of the data and its implications.

RESULTS

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Firstly the hot wire anemometer output voltage is calibrated using King’s Law against the velocities that are initially measured with a pitot tube or vane anemometer.

Hot-wire Calibration Data

Mean

Mean

Hot-wire

Velocity

Voltage

E

E2

U

U1/2

Volts

m/s

1.4227

2.02407529

1.02

1.00995

1.4466

2.09265156

1.222

1.105441

1.4687

2.15707969

1.441

1.200417

1.5002

2.25060004

1.785

1.336039

1.5204

2.31161616

2.029

1.42443

1.534

2.353156

2.198

1.482565

1.5507

2.40467049

2.434

1.560128

1.5684

2.45987856

2.689

1.639817

1.5837

2.50810569

2.928

1.71114

1.5979

2.55328441

3.16

1.777639

1.6115

2.59693225

3.407

1.845806

1.6252

2.64127504

3.656

1.912067

1.6366

2.67845956

3.886

1.971294

1.6488

2.71854144

4.132

2.032732

1.6599

2.75526801

4.378

2.092367

1.6712

2.79290944

4.629

2.151511

1.6814

2.82710596

4.869

2.206581

KINGS LAW

E2 = A + B(U)1/2

Where, B = slope =

0.6718895

from chart overleaf

and A = y – intercept =

1.35315794

Therefore,

E2 = 1.353 + 0.672(U)1/2

This calibration then allows for the mean and fluctuating velocities (turbulence intensity) to be determined. Results are shown below of the mean velocities and intensities along with distance from the wall.

Boundary Layer Traverse Data

Turbulence Intensity u’ = e’.dU/dE

y

y

E

e’

u

dU/dE

u’

Height

Height

Mean voltage

RMS voltage

Mean Velocity

Turbulence Intensity

mm

mm

volts, V

volts*40

m/s

m/sV

m/s

0.3

0.0003

1.2904

0.5209

0.215596503

3.56703479

0.046451711

0.4

0.0004

1.3003

1.1416

0.252502777

3.88990304

0.111017833

0.5

0.0005

1.3128

1.4394

0.303723675

4.30724946

0.154996372

0.6

0.0006

1.3295

1.5939

0.380425473

4.88185944

0.194529894

0.7

0.0007

1.3423

1.8144

0.44580487

5.33561062

0.242023298

0.8

0.0008

1.3554

1.7671

0.518808964

5.81210366

0.256764209

1

0.001

1.3796

1.8099

0.670421492

6.72495347

0.304287332

1.2

0.0012

1.3997

1.8563

0.813490755

7.51576872

0.348788037

1.5

0.0015

1.4147

1.8406

0.930779191

8.12549487

0.373894646

2

0.002

1.4381

1.6249

1.132359656

9.11052521

0.37009231

2.5

0.0025

1.4523

1.4689

1.266099436

9.72864551

0.357260185

3

0.003

1.4647

1.3288

1.39014779

10.2811407

0.341539494

3.5

0.0035

1.4721

1.1963

1.467466623

10.6165538

0.317514581

4

0.004

1.4758

1.1424

1.507060769

10.785866

0.308044334

4.5

0.0045

1.4762

1.0692

1.511378789

10.8042344

0.288797185

5

0.005

1.4802

1.0499

1.554964056

10.9886088

0.288423509

6

0.006

1.4854

0.9173

1.612731979

11.2301783

0.257536064

7

0.007

1.4883

0.8831

1.645496026

11.3658268

0.250929042

8

0.008

1.49

0.8748

1.664885752

11.4456542

0.250316457

9

0.009

1.4909

0.8219

1.675205896

11.4880084

0.236049853

10

0.01

1.494

0.8353

1.711045411

11.6343868

0.242955083

12

0.012

1.4975

0.7615

1.752056302

11.8005702

0.224653356

15

0.015

1.5024

0.7763

1.810452339

12.0348659

0.233566661

20

0.02

1.5106

0.6999

1.910759717

12.4312458

0.217515723

25

0.025

1.5175

0.6502

1.997698249

12.7689674

0.207559565

30

0.03

1.524

0.5861

2.081740071

13.0906241

0.19181037

35

0.035

1.5296

0.5488

2.155829997

13.3704882

0.183443098

40

0.04

1.5353

0.4805

2.232859842

13.6579688

0.16406635

45

0.045

1.5399

0.4233

2.296223886

13.8919038

0.147011072

50

0.05

1.5432

0.3681

2.342345592

14.0607936

0.129394453

55

0.055

1.5463

0.3083

2.386181023

14.2202615

0.109602666

60

0.06

1.549

0.2069

2.424763967

14.3597965

0.074276048

65

0.065

1.5501

0.1355

2.440591095

14.4168162

0.048836965

70

0.07

1.5502

0.1001

2.442033036

14.4220047

0.036091067

75

0.075

1.5508

0.0646

2.450695582

14.4531533

0.023341843

80

0.08

1.5508

0.0515

2.450695582

14.4531533

0.018608435

90

0.09

1.5506

0.0383

2.44780599

14.4427672

0.01382895

100

0.1

1.5506

0.0382

2.44780599

14.4427672

0.013792843

From such data the boundary layer integral parameters can be determined using:

Displacement thickness,

Momentum Thickness ,

Shape Factor,

The boundary layer thickness has been approximated here using the definition provided by F.M White. That is:

Boundary Layer thickness, ??? y???where u = 0.99U0

U/U0

dy

1-U/U0

Trap Areas

U/U0*(1-U/U0)

Trap Areas

y??

u’/U0

(y/?)1/7

mm

0.088077447

0.3

0.91192

0.13678838

0.080319811

0.01204797

0.005

0.018977

0.46933

0.103154735

0.1

0.89685

0.09043839

0.092513835

0.00864168

0.0067

0.045354

0.48902

0.124079962

0.1

0.87592

0.08863827

0.108684125

0.0100599

0.0084

0.063321

0.50486

0.15541488

0.1

0.84459

0.08602526

0.131261095

0.01199726

0.01

0.079471

0.51818

0.182124266

0.1

0.81788

0.08312304

0.148955018

0.01401081

0.0117

0.098874

0.52972

0.211948564

0.1

0.78805

0.08029636

0.16702637

0.01579907

0.0134

0.104896

0.53992

0.273886695

0.2

0.72611

0.15141647

0.198872773

0.03658991

0.0167

0.12431

0.55741

0.332334653

0.2

0.66767

0.13937787

0.221888332

0.04207611

0.0201

0.14249

0.57211

0.380250394

0.3

0.61975

0.19311224

0.235660032

0.06863225

0.0251

0.152747

0.59065

0.462601881

0.5

0.5374

0.28928693

0.248601381

0.12106535

0.0334

0.151193

0.61543

0.517238474

0.5

0.48276

0.25503991

0.249702835

0.12457605

0.0418

0.145951

0.63536

0.567915838

0.5

0.43208

0.22871142

0.245387439

0.12377257

0.0502

0.139529

0.65213

0.599502832

0.5

0.4005

0.20814533

0.240099186

0.12137166

0.0585

0.129714

0.66665

0.615678193

0.5

0.38432

0.19620474

0.236618556

0.11917944

0.0669

0.125845

0.67948

0.61744223

0.5

0.38256

0.19171989

0.236207323

0.11820647

0.0752

0.117982

0.69101

0.63524808

0.5

0.36475

0.18682742

0.231707957

0.11697882

0.0836

0.117829

0.70149

0.658847958

1

0.34115

0.35295198

0.224767326

0.22823764

0.1003

0.105211

0.72001

0.672233025

1

0.32777

0.33445951

0.220335785

0.22255156

0.117

0.102512

0.73604

0.680154293

1

0.31985

0.32380634

0.217544431

0.21894011

0.1337

0.102262

0.75021

0.684370372

1

0.31563

0.31773767

0.216007566

0.216776

0.1505

0.096433

0.76294

0.699011857

1

0.30099

0.30830889

0.210394281

0.21320092

0.1672

0.099254

0.77451

0.715766

2

0.28423

0.58522214

0.203445033

0.41383931

0.2006

0.091777

0.79495

0.739622481

3

0.26038

0.81691728

0.192581067

0.59403915

0.2508

0.095419

0.8207

0.780600965

5

0.2194

1.19944139

0.171263099

0.90961041

0.3344

0.088862

0.85513

0.816117886

5

0.18388

1.00820287

0.150069482

0.80333145

0.418

0.084794

0.88283

0.850451416

5

0.14955

0.83357674

0.127183805

0.69313322

0.5016

0.07836

0.90613

0.880719308

5

0.11928

0.67207319

0.105052808

0.58059153

0.5851

0.074942

0.9263

0.912188242

5

0.08781

0.51773112

0.080100853

0.46288415

0.6687

0.067026

0.94414

0.938074298

5

0.06193

0.37434365

0.058090909

0.34547941

0.7523

0.060058

0.96016

0.956916358

5

0.04308

0.26252336

0.041227442

0.24829588

0.8359

0.052861

0.97472

0.974824407

5

0.02518

0.17064809

0.024541782

0.16442306

0.9195

0.044776

0.98809

0.990586663

5

0.00941

0.08647232

0.009324726

0.08466627

1.0031

0.030344

1.00044

0.997052505

5

0.00295

0.03090208

0.002938807

0.03065883

1.0867

0.019951

1.01195

0.99764158

5

0.00236

0.01326479

0.002352858

0.01322916

1.1703

0.014744

1.02272

1.001180482

5

-0.0012

0.00294484

-0.00118188

0.00292745

1.2539

0.009536

1.03285

1.001180482

5

-0.0012

-0.0059024

-0.00118188

-0.0059094

1.3375

0.007602

1.04242

1

10

0

-0.0059024

0

-0.0059094

1.5047

0.00565

1.0601

1

10

0

0

0

0

1.6719

0.005635

1.07618

Bulk Velocity, U0 =

2.44780599

0.99U0 =

2.42332793

Boundary Layer Thickness =

59.81390261

Displacement Thickness

?* =

10.80487536

mm

Momentum Thickness

???

7.500002088

mm

Shape Factor

H =

1.440649647

To find the correct logarithmic velocity profile the Clauser plot technique was employed varying the friction velocity, u* to ‘match’ the logarithmic region of the profile to the known log law. This provided a graphical means of determining u*. Also on the following is the linear viscous sublayer plot of u+ = y+. This allowed the logarithmic region to be more precisely defined and hence a high-resolution graph of the logarithmic region could be used to match the profile to the Log Law.

LOG LAW u+ = 5.5 log y+ + 5.45

u+ = u/u*

log y+

y+

u+

1.959968212

0.342423

2.200000

7.333325

2.295479793

0.467361

2.933333

8.020488

2.761124316

0.564271

3.666667

8.553493

3.458413395

0.643453

4.400000

8.98899

4.052771545

0.710399

5.133333

9.357197

4.716445129

0.768391

5.866667

9.676153

6.094740837

0.865301

7.333333

10.20916

7.3953705

0.944483

8.800000

10.64465

8.46162901

1.041393

11.000000

11.17766

10.29417869

1.166331

14.666667

11.86482

11.50999487

1.263241

18.333333

12.39783

12.63770718

1.342423

22.000000

12.83332

13.34060567

1.409369

25.666667

13.20153

13.70055245

1.467361

29.333333

13.52049

13.73980717

1.518514

33.000000

13.80183

14.13603687

1.564271

36.666667

14.05349

14.66119981

1.643453

44.000000

14.48899

14.95905478

1.710399

51.333333

14.8572

15.13532502

1.768391

58.666667

15.17615

15.2291445

1.819544

66.000000

15.45749

15.55495828

1.865301

73.333333

15.70916

15.92778456

1.944483

88.000000

16.14465

16.45865762

2.041393

110.000000

16.67766

17.37054288

2.166331

146.666667

17.36482

18.16089317

2.263241

183.333333

17.89783

18.92490973

2.342423

220.000000

18.33332

19.59845452

2.409369

256.666667

18.70153

20.29872584

2.467361

293.333333

19.02049

20.8747626

2.518514

330.000000

19.30183

21.29405083

2.564271

366.666667

19.55349

21.69255476

2.605664

403.333333

19.78115

22.04330879

2.643453

440.000000

19.98899

22.18719177

2.678215

476.666667

20.18018

22.20030032

2.710399

513.333333

20.3572

22.27905074

2.740363

550.000000

20.52199

22.27905074

2.768391

586.666667

20.67615

22.25278172

2.819544

660.000000

20.95749

22.25278172

2.865301

733.333333

21.20916

Kinematic viscosity, ? =

1.50E-05

Viscosity, ???

1.80E-05

Friction velocity, u* (Clauser method)=

0.11

Another approach to obtaining the friction velocity, u* is to consider the linear viscous sub-layer. This region due to its linearity describes;

This is relationship between the shear stress close to the wall and the velocity gradient, which can be determined from the mean velocity profile.

The shear stress can then be applied to find the friction velocity using;

Friction Velocity, Using linear viscous sublayer

y+ = yu*??

588.3394

Take y+ = 5

?w = ??du/dy =

1.06E-02

y =

0.000681818

u* = (?w/?)1/2 =

0.093942

therefore use first 5 points

u*1 (clauser) =

0.11

u*2 (viscous sublayer) =

0.093942

% difference =

14.59822

%

Using the friction velocity, a calculation of the skin-friction coefficient can be determined using;

Skin-friction coefficient,

Therefore using both of the friction velocities computed by the two different methods;

cf = 2u*12/U02 =

0.00403888

cf = 2u*22/U02 =

0.00294575

% diff =

27.06536

%

Probability Density Function, PDF

Sample of Results

Time series of the velocity fluctuation

Max velocity

0.785281

at y+ = 2 in the turbulent boundary layer

Min velocity

-0.20274

Difference

0.988025

Sampling frequency = 1000 Hz

No.of intervals

1000

Total number of data points = 16,000

Interval

0.000988

Velocity (m/s)

Interval Height

Cumulative freq

Frequency

Kurtosis

Skewness

-0.0449941

-0.202744

1

1

-0.13662

1.102448773

-0.0389198

-0.201755975

3

2

-0.0299365

-0.20076795

9

6

-0.0227584

-0.199779925

12

3

Discussion

Due to the random, fluctuating flow process involved in turbulent flows it is important that precise experimental procedures are in place to enable us to analyse the flow and investigate its structures. This report has looked at the mean streamwise velocity and turbulence intensity profiles under different criterion and in different formats to investigate the differences between them and possibly find the ‘best’ method. At the end it has looked at turbulence statistics considering the probability density function of fluctuating streamwise velocity.

All the measurements taken in this report were conducted using a hot-wire anemometer that measures the velocity change in terms of the change of voltage across the wire. Firstly this was calibrated to using a static pitot tube. Using King’s Law a straight line curve is plotted clearly showing a very good linear relationship.

The mean velocity profile (u vs. y) shows the expected boundary layer curved profile due to the growth of a shear layer and the no-slip condition at the wall retarding the flow, and shows that the curve indeed, becomes asymptotic to the free-stream velocity, U0. In more detail there is evidence of a linear viscous sub-layer 0 ; y ;0.001m, a log region 0.0035 ; y ; 0.01m and also a wake region y;0.01m. The determination of U0 is in theory the asymptotic value to which the mean velocity profile tends to. A good approximation of U0 is important to reducing error throughout the rest of the data as when using non-dimensional analyses it is used. In this report and similar to Kline et al(1967) we have taken the furthest point way from the wall to be the bulk velocity as opposed to the maximum velocity obtained. This discrepancy is mainly due to random eddy motion and interference between the outer wake region and free flow. It could also suggest bursting or effects of bursting from the outer wake region into the free stream.

The turbulence intensity was found using the relationship; . To obtain consistent units the fluctuating voltage needed to be divided by the amplification gain. This amplification will obviously have amplified any error that was induced by the hot wire anemometer. However because the hot wire calibration is more than reasonable the error in U should be minimal. The intensity plot shows how there is high amounts of turbulence action within the buffer layer and the viscous wall region suggesting action due to the bursting of the collisions between high speed sweeps and low speed ejections transporting kinetic energy from the inner layer to the outer layer.

Once the mean velocity profile has been determined the integral parameters of the boundary layer can be determined. The boundary layer thickness has been approximated using the definition provided by F.M.White 2000 that the boundary layer thickness is the distance from the wall where the velocity is 99% of the bulk velocity. This is a somewhat approximate definition in that there is no fundamental reason as to why it is at 0.99U0 and not say, 0.98U0. The definition also has the disadvantage when using discrete data values as in this report because the value of 0.99U0 will not have an exactly corresponding value of distance from the wall. Hence, linear interpolation between the closest two points has been employed. The definition probably comes second to that provided by Pope, 2000 who states;

Other more empirical based formulae relating the integral parameters to the Reynolds Number can be equated to give relationships such as 0.35? = ?* etc. This method is seems naturally the least accurate of the three methods.

The other remaining values such as displacement thickness, momentum thickness and the shape factor have precise definitions but involve integration. As the integrals cannot be performed analytically due to the data being discrete we must employ a numerical method. This report has used the trapezium rule. Although it is an approximation the small increments in distance from the wall in relation to the overall boundary layer are small enough for a valid approximation.

Plotting the non-dimensional mean velocity and turbulence intensity profiles shows the same relationships as with u vs. y and u’ vs. y. On the mean velocity profile here the 1/7th law proposed by Prandtl for relatively low Reynolds number flows has been plotted. There is an obvious correlation between the two plots however there does exist a shift reduction in the velocity. Hence the log law predicts a greater velocity for the same distances away from the wall. There is good fit closer to the wall. This implies that there may be a retardation of the velocity. This could be due to the inaccuracy in predicting the boundary layer thickness.

It is evident that, close to the wall, the viscosity, and the wall shear stress are important parameters. From these we define viscous scales that are the appropriate velocity scales and length scales in the near-wall region. These are termed friction velocities. One such velocity is define as follows:

Obtaining the friction velocity allows for the calculation of the skin-friction coefficient.

In order to find the friction velocity two different techniques have been employed. The first technique involves using the termed ‘Clauser’ plotting technique, which involves fitting the logarithmic velocity profile u+ vs. log y+ to a known log law by matching the logarithmic region. In order to determine the friction velocity a visual comparison of the logarithmic velocity and the Log law and applying trial and error to find the ‘best’ match. This adequate for this study and is in fact the same technique employed by numerous researchers including Kline (1967). Before this however there was a requirement to identify the logarithmic region of the profile. This was done by considering previous data presented by Kline (1967), Wei and Willmarth (1989) and F.M.White (2000). As all these have presented data in non-dimensional format allowing for easy comparison and finding the logarithmic region. Using a plot of u+ vs. y+ also helps determine the inner layer region. Once the region had been identified a zoomed plot was conducted which allowed for better visual comparison.

In the second method, the friction velocity was determined by focusing in on the linear viscous sub-layer on the mean velocity profile u vs. y and computing the gradient. This can then be used in the known formula:

This can then be used to find the friction velocity in the formula provided above.

Definition of the viscous sublayer is provided by Laufer (1954) who states that the velocity profile should be linear from the wall out to at least y+ = 5. From this y was calculated and then the number of points identified. Five points was obtained and is enough for a valid straight line trend.

The two different methods provide friction velocities of u*1= 0.11 and u*2 = 0.093

This provides a % difference of 14%. This difference could be due to the errors involved in determining the best fits and trendlines, which for the first method was obtained visually. More importantly however, there is difficulty in defining both the logarithmic region and the viscous sub-layer in both cases. It seems intuitive to say that the second method is more accurate as the region is defined mathematically and there is no trial and error method involved plus there is enough points in that region to give a valid trend.

In any turbulent flows there is, unavoidably randomness. This is because the velocity field U(x,t) is random. For laminar flows, we can use theory (i.e. the Navier Stokes equations) to calculate U, a particular velocity at a particular location y from the wall. We also have the experimental methods to measure U directly using like earlier a hot-wire anemometer etc. The difficulty lies with the fact that the velocity predicted from the theory matches experiment for laminar flow but for turbulent flows the comparison is not so simple. Although the Navier Stokes equations still apply for turbulent flows, the aim of the theory must be different. Since, U is a random variable, its value is unpredictable, hence any theory that predicts a particular value for U is almost certain to be incorrect. To overcome this difficulty we adopt a method aimed at determining the probability of events i.e. velocity fluctuations. Specifically we U, for turbulent flows can be completely characterise by its probability density function, PDF.

With reference to the PDF, for 16000 velocity measurements recorded at 1000Hz, it is immediately apparent that there is a significant skewness and a deviation in shape from the nominal gaussian distribution. However before discussing the features of the plot it is worth noting the method employed for calculating the frequencies. It was chosen to use 1000 increments between the maximum and minimum velocity readings to safely suggest that the discrete random variables in the data form a continuous random variable. By computing the number of velocities within each increment we can gather the frequency. As each point was measured at an equal space in time i.e. 1000Hz the time increments do not need to be included.

From the PDF a skewness of 1.102 and kurtosis of -0.137. These are analogous to the asymmetry and flatness of the plot respectively. It is already known that in free shear flows the PDF is not Gaussian. The skewness is to the left, hence there is higher frequency of negative streamwise velocities at the position y+ = 2 therefore suggesting maybe a high influence of spanwise, vortical structures in the linear viscous sublayer. As we are concerned with the region close to the wall the spanwise vortices are in fact induced by the zero velocity retardation at the wall and the shear layer formation.