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Bjo¨rn Birnir
The Kolmogorov-Obukhov
Theory of Turbulence
A Mathematical Theory of Turbulence
123
Bjo¨rnBirnir
UniversityofCalifornia
DepartmentofMathematics
SantaBarbara,CA93106
USA
ISSN2191-8198 ISSN2191-8201(electronic)
ISBN978-1-4614-6261-3 ISBN978-1-4614-6262-0(ebook)
DOI10.1007/978-1-4614-6262-0
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012956056
MathematicsSubjectClassification(2010):60H15,35R60,76B79,35Q35,35Q30,76D05,37L55
(cid:2)c Bjo¨rnBirnir2013
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To Inga,Addaand Einar.
Preface
Inthisbookwepresenttherecentlydevelopedstatistical theoryofturbulenceina
formthatcanbeappreciatedbyphysicists,mathematicians,andengineers.Thethe-
oryisgroundedinprobabilitytheoryandwedevelopfromprobabilityalltheresults
that are necessary to understand the turbulence theory, without proofs. However,
referencesaregiventostandardtextswheretheproofsandmorebackgrounddetails
can be found.Thegoalsare to find estimates forstructurefunctionsof turbulence
thatare realizedin simulationand experiments;to derivethe invariantmeasureof
turbulencebothforthe one-pointstatistics andthe two-pointstatistics; andfinally
to derive the probability density function (PDF) for the statistics that are used in
practice.Wedonotassumeanymathematicalbackgroundbutfamiliaritywithbasic
probabilitytheoryandpartialdifferentialequationsobviouslyhelps.
We willseethattheNavier–Stokesequationforallbutthelargestscalesintur-
bulent flow can be expressed as a stochastic Navier–Stokes equation (1.65). The
stochasticforcingresultsfrominstabilitiesoftheflowthatmagnifiessmallambient
noise and saturates its growth into large stochastic forcing. This has been mod-
eledbeforebyaReynoldsdecompositionandbyacoarsegrainingoftheflow.The
stochasticforceisgenericandisdeterminedbythegeneralprinciplesofprobability
with a minimumof physicalinputs. It consists of two components:additivenoise
and multiplicative noise and the additive component is determined by the central
limittheoremandthelargedeviationprinciple.Thephysicalinputisthatthesetwo
termsmustproducesimilarscalingsbecausetheyarethedetaileddescriptionofthe
samedissipativeprocesses.Thisdeterminestherateinthelargedeviationprinciple.
The multiplicative noise multiplies the fluid velocity and models jumps (vorticity
concentrations)inthevelocitygradient.ItisexpressedbyagenericPoissonprocess
whereonlytherateneedstobegiven.Thisrateisdeterminedbythespectralanaly-
sisofthe(linearized)Navier–Stokesoperatorandtherequirement,following[64],
thatthe dimensionofthemostsingularvorticitystructure(filaments)is one.Thus
thestochasticforcingisgenericanddeterminedwithtwomildphysicalinputs.
ThestochasticNavier–Stokesequationcanbeexpressedasanintegralequation
(2.17) and the log-Poissonian processes found by She and Leveque and explored
by She and Waymire and Dubrulle are produced from the multiplicative noise by
vii
viii Preface
the Feynman–Kacformula. This givesa satisfying mathematicalderivationof the
intermittency phenomena that had earlier been derived from impirical considera-
tions.Moreover,theintegralequationsshowhowtheNavier–Stokesevolutionand
thelog-Poissonianintermittencyprocessesactonthe dissipationprocessestopro-
duce the intermittencyin the dissipation. This is a mathematical derivation of the
experimental observation that intermittent dissipation processes accompany inter-
mittent velocity variations. Using the integral equation, we get an estimate on all
thestructurefunctionsofthevelocitydifferencesinturbulence.Theevidencefrom
simulations and experimentsis that this upper bound is reached in turbulentflow.
Why the inertial cascade achieves this maximal efficiency in the energy transfer
remainstobeexplained.
WethenbuildonHopf’s[29]ideastocomputetheinvariantmeasureofturbulent
flow.Thismeasurecanbecomputedbecauseitsolvesalinearfunctionaldifferential
equation, the Kolmogorov–Hopfequation; see [56]. It turns out to be an infinite-
dimensionalGaussianmultipliedbya(discrete)Poissondistribution.ThisPoisson
distributioncorrespondstotheintermittencyandthelog-Poissonprocesses.Thenby
takingthetraceoftheinvariantmeasurewegetthePDFofthevelocitydifferences.
Wefirstderivethefunctionaldifferentialequation(PDE)forthePDFandthenshow
that there are infinitely many PDFs, each corresponding to a particular moment
becauseoftheintermittencycorrections.ThePDE(3.15)forthesequenceofPDFs
can also be solved and the PDFs turn out to be the normalized inverse Gaussian
(NIG)distributionsofBarndorff–Nielsen[7].Theirparametersareeaslycomputed
andweseehowtodothisforbothsimulationsandexperiments.
ItisinterestingtonoticethatalthoughthesolutionoftheNavier–Stokesequation
maynotbeuniqueorsmooththeinvariantmeasureofthevelocitydifferences(3.12)
isstillwelldefinedbyLeray’s[42]existencetheory.Moreover,differentvelocities
produceequivalentmeasures,sothestatisticalobservablesofturbulenceareunique
althoughtheturbulentvelocitymaynotbe.
Thetheorypresentedinthisbookmustbecomplementedbyadynamicalsystems
theoryforthelarge-scalestructuresinfluidflowandeventuallyonewantstowork
out how the small-scale flow presented here influences the large-scale dynamics.
Thisisamaterialforfutureresearch,buthopefullythetoolspresentedinthisbook
willalsobehelpfulinthatendeavor.
SantaBarbara,California, Bjo¨rnBirnir
UnitedStates
Acknowledgements Theauthor wouldliketoacknowledge a largenumber ofcolleagues with
whomtheresultsinthisbookhavebeendiscussedandwhohaveprovidedvaluableinsights.They
includeOleBarndorff-NilsenandJurgenSchmiegelinAarhus,HenryMcKean,K.R.Sreenivasan,
andR.VaradhaninNewYork,Z.-S.ZheinBeijing,EdWaymireinOregon,E.BodenschatzandH.
XuinGottingen,MichaelWilczekinMunsterandM.SørenseninCopenhagen.Healsobenefitted
fromconversationswithJ.Peinke,M.Oberlack,E.Meiburg,B.Eckhardt,S.Childress,L.Biferale,
L-S. Yang, A. Lanotto, K. Demosthenes, M. Nelkin, A. Gylfason, V. L’vov, and many others.
Thisresearch wassupported inpart bytheTheCourant InstituteandtheProject ofKnowledge
InnovationProgram(PKIP)ofChineseAcademyofSciences,GrantNo.KJCX2.YW.W10,whose
supportisgratefullyacknowledgedandbyagrantfromtheUCSantaBarbaraAcademicSenate.
Contents
1 TheMathematicalFormulationofFullyDevelopedTurbulence...... 1
1.1 IntroductiontoTurbulence................................... 1
1.2 TheNavier–StokesEquationforFluidFlow .................... 4
1.2.1 EnergyandDissipation ............................... 5
1.3 LaminarVersusTurbulentFlow............................... 6
1.4 TwoExamplesofFluidInstabilityCreatingLargeNoise.......... 8
1.4.1 Stability............................................ 10
1.5 TheCentralLimitTheoremandtheLargeDeviationPrinciple,
inProbabilityTheory ....................................... 15
1.5.1 Crame´r’sTheorem ................................... 17
1.5.2 StochasticProcessesandTimeChange .................. 18
1.6 PoissonProcessesandBrownianMotion ....................... 20
1.6.1 Finite-DimensionalBrownianMotion ................... 24
1.6.2 TheWienerProcess .................................. 26
1.7 TheNoiseinFullyDevelopedTurbulence...................... 28
1.7.1 TheGenericNoise ................................... 31
1.8 TheStochasticNavier–StokesEquationforFullyDeveloped
Turbulence ................................................ 32
2 ProbabilityandtheStatisticalTheoryofTurbulence ............... 35
2.1 ItoProcessesandIto’sCalculus .............................. 35
2.2 TheGeneratorofanItoDiffusionandKolmogorov’sEquation .... 37
2.2.1 TheFeynman–KacFormula ........................... 39
2.2.2 Girsanov’sTheoremandCameron–Martin............... 39
2.3 JumpsandLe´vyProcesses................................... 40
2.4 SpectralTheoryfortheOperatorK............................ 42
2.5 TheFeynman–KacFormulaandtheLog-PoissonianProcesses .... 46
2.6 TheKolmogorov–Obukhov–She–LevequeTheory............... 48
2.7 EstimatesoftheStructureFunctions........................... 49
2.8 TheSolutionoftheStochasticLinearizedNavier–Stokes
Equation.................................................. 53
ix
