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Experimental Techniques; Analysis of Boundary Layer Data.

Extracts from this document...

Introduction

Joseph Gransden        Case Study 3: Experimental Techniques        10/05/07

CASE STUDY 3: Experimental Techniques; Analysis of Boundary Layer Data

By Joseph Gransden

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

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.

...read more.

Middle

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

Boundary Layer thickness,        δ = ywhere u = 0.99U0

 U/U0 dy 1-U/U0 Trap Areas U/U0*(1-U/U0) Trap Areas y/δ u'/U0 (y/δ)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 δ* = 10.80487536 mm Momentum Thickness θ = 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+.

...read more.

Conclusion

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.

It is also worth noting that the negative sign on the kurtosis has no significance.

References

• KLINE et al (1967) “ The structure of turbulent boundary layers”. J of Fluid Mech. Vol 30 pp 741- 773
• CHOI K-S (2001) “ Boundary layers” Fluid Mechanics 2 notes.
• WHITE F.M (2000) “Fluid Mechanics”
• POPE S.B (2000) “Turbulent flows”

...read more.

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