Explorations into the inertial and integral scales of homogeneous axisymmetric turbulence

9 MB·English

by  Kelken Chang

EXPLORATIONS INTO THE INERTIAL AND
INTEGRAL SCALES OF HOMOGENEOUS
AXISYMMETRIC TURBULENCE
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DoctorofPhilosophy
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KelkenChang
January2012
©2012KelkenChang
ALLRIGHTSRESERVED
EXPLORATIONSINTOTHEINERTIALANDINTEGRALSCALESOF
HOMOGENEOUSAXISYMMETRICTURBULENCE
KelkenChang,Ph.D.
CornellUniversity2012
A flow generator is described in which homogeneous axisymmetric turbu-
lent air flows with varying and fully controllable degrees of anisotropy, includ-
ing the much studied isotropic case, are generated by the combined agitations
produced by 32 acoustic mixers focusing at the center of the system. The ax-
isymmetric turbulence in a central volume of the size of the inertial scale is
showntohavenegligiblemeanandshear. TheTaylorReynoldsnumberisabout
480.
The influence of large scale anisotropy on the turbulence is examined from
three aspects, namely the velocity structure functions, the velocity correlation
functions,andtheintegrallengths. Thedirectionaldependenceoftwodifferent
second order transverse structure functions, in which one of them has separa-
tions stretched along the axis of symmetry of the turbulence and the other one
normal to it, is studied. It is shown that the inertial range scaling exponents,
determined using the extended-self-similarity procedure, and the Kolmogorov
constantsofthetwostructurefunctionsareunaffectedbythedirectioninwhich
thestructurefunctionsaremeasured.
Asanextension,becauseofitsrelevancetothestudyofintermittency,thedi-
rectionaldependenceoftransversestructurefunctionsofthefourthtothe sixth
order is studied. Despite some issues with measurement noise and statistical
convergence, some indications are found that anisotropy in the velocity field
intensifies the asymmetry of the probability density of the velocity increments.
In addition, some evidence is found that the inertial range scaling exponents of
thefourth,fifth,andsixthorderareindependentoftheanisotropy.
Finally, it is found that, except in the isotropic case, the second order trans-
verse velocity correlation functions deviate from each other at the large scale
with increasing anisotropy. A self-similarity argument similar to one found in
thestudyofcriticalphenomenaisproposed. Itisshownthattheargumentleads
toapower-lawrelationshipbetweenthelargescalevelocityfluctuationandthe
correlation length, with an exponent that depends on the inertial range scaling
exponent of the turbulence. The data collapse predicted by the self-similarity
hypothesis is verified. It is demonstrated that the value of the power-law expo-
nentisconsistentwiththevalueoftheinertialrangescalingexponent.
BIOGRAPHICALSKETCH
Kelken Chang was born in Muar, Malaysia in September, 1978, the fourth child
of five of Chia-Ming and Kim-Leu Chang. Chia-Ming Chang was a laboratory
technician who was working in the Lee Rubber Company in Medan, Indonesia
at the time his second son was born. In fact the original family name had been
Teo,thefamilybeingethnicChinesefromFujian,China,butChia-MingTeohad
changed his name to Chia-Ming Chang in 1965 when he took up Singaporean
citizenship. Kelken’s mother was the fifth child of eight of a well-to-do chinese
trader and she was ten years younger than her husband. Kelken was educated
at Muar High School, a former Government English School established in 1902
by the British government during the colonization period. He went on to ter-
tiary education at the National University of Singapore (NUS) and earned a
Bachelor of Science degree in Physics with first class honors in 2003. He spent
the first half of his graduate career at Cornell in Ithaca, New York, from 2003 to
2006, and afterwards at the Max-Planck-Institut fu¨r Dynamik und Selbstorgan-
isation in Go¨ttingen, Germany. In his spare time, he enjoys swimming in the
lakes,readingpoetry,andifhefindsagoodvictim,playingpracticaljokes.
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Tomyfamilyandfriends.
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ACKNOWLEDGEMENTS
Many outstanding individuals have made my graduate experience one of the
highlights of my life so far. I thank my advisor Eberhard Bodenschatz, the in-
trepid explorer whose undiminishable energy and contagious enthusiasm for
science has provided much of the impetus for this dissertation. Eberhard pro-
vided me the freedom to grow as an independent thinker. He is also very kind
andunderstandingwhenexperimentsdidnotgoaswellasplanned.
Iambeholdentomyco-workerGregBewleyforsharingwithmehistechni-
cal expertise and numerous invaluable ideas. His work attitude is very instruc-
tive and decisive every time in pushing the project a step forward. I appreciate
the patience and rigor shown by both Greg and Eberhard in weeding out the
waffleofmythoughts.
I am grateful to Erich Mueller and Itai Cohen for agreeing to sit on my
special committee and for providing the many levels of administrative sup-
port throughout the entire stage of my studies. Outside of classroom, Erich
has taught me the value of scientific collaboration, a value that has also been
reprised by many Cornell professors. I thank David Cassel, Veit Elser, Henry
Tye, and Michelle Wang for the advice they have given me at various stages of
mygraduatestudies.
I am indebted to Sned for teaching me the rudiments of machining and for
running a superb student machine shop; Andreas Kopp, Dr. Artur Kubitzek,
Ortwin Kurre, Gerhard Nolte, and Andreas Renner for their invaluable tech-
nical assistance; Angela Meister for her meticulous handling of administrative
issues at the institute; and Katharina Schneider for understandingly hearing
out my many grouses – our daily banter at the lunch table has been a bulwark
againstthemonotonyoflaboratorywork.
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I owe a debt of gratitude to my inspiring office mates from Cornell: Albert,
Amgad,Dario,andHaitao;andfromtheMaxPlanckInstituteforDynamicsand
Self-Organization: Azam, Christian, Eva, Ewe Wei, Fabio, Gabriel A., Gabriel
S., Haitao, Hengdong, Holger, Jens, Marco, Mathieu, Matthias, Mireia, Noriko,
Robert, Shinji, Stephan, Toni, Vladimir, and Walter. Foremost among them, I
wishtoexpressmygratitudetoMathieuGibertforgraciouslymakinghisapart-
ment available at critical stages of the writing of this thesis. I enjoyed the gen-
uine camaraderie of these wonderful individuals, with whom I have had some
of the most engaging conversations, sometimes discussing far into the night in
some pub in Go¨ttingen and finally thinking that we had solved our respective
problemswithalcoholicallyinducedelation.
I have received financial support from the National Science Foundation,
through grants PHY-9988755 and PHY-0216406, for my studies at Cornell and
from the Max Planck Society and Deutsche Forschungsgemeinschaft (German
Science Foundation), through the grant XU91/3-1, for my studies in Go¨ttingen,
Germany.
I thank my family for their unfaltering support that has carried me through
the ups and downs of graduate school, and my friends for riding with me
throughthickandthinwood.
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TABLEOFCONTENTS
BiographicalSketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
TableofContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Introduction 1
1.1 Kolmogorovtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 VerificationofKolmogorovtheory . . . . . . . . . . . . . . . . . . 12
1.3 Relationtootherinertial-rangeconstants . . . . . . . . . . . . . . 19
1.4 Higher-orderstructurefunctions . . . . . . . . . . . . . . . . . . . 23
1.5 Anomaloustransversescalingexponents . . . . . . . . . . . . . . 26
1.6 Anisotropicturbulence . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.7 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Theflowapparatusandthemeasurementtechnique 38
2.1 Theflowapparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.1 Thechamber . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.2 Theloudspeaker . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.3 Thedrivingalgorithm . . . . . . . . . . . . . . . . . . . . . 44
2.1.4 Thedistributionofamplitudes . . . . . . . . . . . . . . . . 48
2.1.5 Theamplifier . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2 LaserDopplerVelocimetry . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.1 TheDopplermodel . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.2 Thefringemodel . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.3 TheBraggcell . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2.4 Dataprocessing . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2.5 LimitationsoftheLDVtechnique . . . . . . . . . . . . . . 65
3 Theexperimentalprocedure 69
3.1 Thecoordinatesystem . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Themeasurementprotocol . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Theflows 74
4.1 Anisotropyofthefluctuations . . . . . . . . . . . . . . . . . . . . . 74
4.2 Axisymmetryofthefluctuations . . . . . . . . . . . . . . . . . . . 75
4.3 Homogeneityofthefluctuations . . . . . . . . . . . . . . . . . . . 79
4.4 Anisotropyofthemeanflow . . . . . . . . . . . . . . . . . . . . . 81
4.5 Reflectionalsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 Reynoldsshearstresses . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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5 Universalityinthestructurefunctions 89
5.1 Isotropyofthestructurefunctions . . . . . . . . . . . . . . . . . . 89
5.2 EqualityoftransverseESSscalingexponents . . . . . . . . . . . . 91
5.3 Scalar-valuednessoftheKolmogorovconstant . . . . . . . . . . . 97
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 Higherorderstatistics 105
6.1 Higherorderstructurefunctions . . . . . . . . . . . . . . . . . . . 106
6.2 Kurtosisandhyperkurtosisofstructurefunctions . . . . . . . . . 113
6.3 Higherorderscalingexponents . . . . . . . . . . . . . . . . . . . . 117
6.4 Anomalousscaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Theintegralscalesofturbulence 123
7.1 Theintegralscaleandcorrelationfunction . . . . . . . . . . . . . . 124
7.2 Scalingandself-similarity . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Integrallengthscaling . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.4 Correlationfunctionscollapse . . . . . . . . . . . . . . . . . . . . . 137
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8 SummaryandOutlook 139
A EvaluatingthedimensionlessintegralA 142
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B Statisticaltools 145
C Generatingexponentiallycorrelatedcolorednoise 147
D Calculatingstatisticswithinter-arrivaltimeweighting 149
E ResamplingLDVvelocitysignals 151
F Calculatingvelocityautocorrelation 153
G Calculatingleast-squaresfitcoefficients 155
Bibliography 158
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