紊流理论(基本理论)-课件

课程内容（10-18周）一.绪论二.基本方程三.基本理论四.紊流模型五.明渠紊流六.紊流前沿成果（杨胜发）二、紊流基本理论1、层流稳定性理论2、科莫戈罗夫理论3、紊流猝发现象层流稳定性理论层流到紊流的转捩-圆管流动1883年，雷诺进行圆管流动的实验，观察到层流向紊流的转捩。Re较小时，流体质点沿着与管道中心平行的直线匀速前进，不同的流层互不干扰和掺混，为层流。Re增大到一定的数值时，不同流层中的质点开始掺混，发生动量交换，一点的流速和压强呈随机性的脉动，但是时间平均值趋于均匀，为紊流。层流稳定性理论层流到紊流的转捩-圆管流动当Re处在临界值附近的一个范围内时，流动具有间歇性，时而为层流时而为紊流。罗塔于1956年观察到这样一个现象，当雷诺数为2550时，圆管中的流动呈现间歇性。间歇系数：层流稳定性理论层流到紊流的转捩-壁面边界层流动层流稳定性理论层流到紊流的转捩-壁面边界层流动层流稳定性理论层流到紊流的转捩-壁面边界层流动在雷诺数较大的流动中，紧贴着物体表面，流动受到粘性的显著影响，流速沿壁面法向的变化非常急剧，摩擦切应力不能略去不计的极薄的一层流体，称为边界层。层流稳定性理论层流到紊流的转捩-壁面边界层流动层流稳定性理论层流到紊流的转捩-壁面边界层流动层流稳定性理论层流到紊流的转捩-壁面边界层流动层流稳定性理论层流到紊流的转捩-壁面边界层流动层流稳定性理论层流到紊流的转捩-壁面边界层流动以经典的圆柱绕流为例。可以看出，流体与圆柱之间存在滑移，流线是对称的，流动方向无阻力。层流稳定性理论层流到紊流的转捩-壁面边界层流动以经典的圆柱绕流为例。粘性流动壁面无滑移，产生边界层，在背流面发生分离，形成一个由漩涡组成的尾流区。层流稳定性理论层流到紊流的转捩-壁面边界层流动临界Re附近，边界层流动由层流向紊流转捩，分离点下移，尾流区缩小，形状阻力降低，产生了阻力危机。来流的紊动度和壁面的粗糙程度都影响转捩的发生，粗糙表面的边界层更容易发展为紊流，因此会导致阻力危机的提前发生。因此，绕流阻力主要取决于物体背流面尾流中的负压，与背流面的形状关系密切。这一问题直到边界层理论提出后才得到解决。层流稳定性理论基本点1930年，普朗特建立了层流稳定性理论。层流稳定性的基本点是：层流流动总会受到一些扰动，可能是受进口边壁粗糙或者来流自身的紊动，如果扰动随时间衰减，则层流稳定，否则会逐渐过渡至稳流。研究内容寻求各种流动情况下，层流对微小扰动失去抑制时的雷诺数，即临界雷诺数。考虑一个二维的情况，将流动分解为主流和加在上面的一个扰动。层流稳定性理论问题：对于这样一个主流流动，主流满足N-S方程，叠加后的流动也满足，那么扰动将随时间放大还是衰减。层流稳定性理论层流稳定性理论层流稳定性理论层流稳定性理论层流稳定性理论u方向对y取微分，v方向对x取微分，相减消去压强扰动项，则得到两个方程式，含有两个未知量u’、v’。边界条件：边壁处u’=0、v’=0；无穷远处同样。科莫戈罗夫理论漩涡的产生：假设某一水流的分离面，由于扰动，流线发生弯曲流线集中的地方流速大，压力低，分散地方相反，加剧流线弯曲，最终产生漩涡。科莫戈罗夫理论漩涡的产生：流速梯度大的地方，机理相似。漩涡抬升，扩散至全流区。科莫戈罗夫理论漩涡的结构和组成：漩涡抬升过程中逐渐增大。大尺度漩涡的尺寸与容器尺寸（管径、水深等）属于同一量级。大尺度漩涡的分布及方向取决于形成条件，不是各向均匀同性的，由于强烈的掺混作用，大漩涡不稳定，会崩解称为次一级的小漩涡。大漩涡分解，把能量传递给次一级漩涡，次一级漩涡仍然不稳定，会进一步分解。分解的过程中，漩涡的几何方向性逐渐丧失，形成条件的影响越来越弱，越来越接近各向同性。一直到与水团相关的雷诺数低到不能再产生更小的漩涡为止。这些最低级漩涡的能量会通过粘性转化为热能。科莫戈罗夫理论L.F.Richardson(“WeatherPredictionbyNumericalProcess.”CambridgeUniversityPress,1922)summarizedthisinthefollowingoftencitedverse:BigwhirlshavelittlewhirlsWhichfeedontheirvelocity;Andlittlewhirlshavelesserwhirls,Andsoontoviscosity inthemolecularsense.科莫戈罗夫理论漩涡能量分布：漩涡抬升过程中逐渐增大。科莫戈罗夫理论Kolmogorov’stheorydescribeshowenergyistransferredfromlargertosmallereddieshowmuchenergyiscontainedbyeddiesofagivensizehowmuchenergyisdissipatedbyeddiesofeachsizethreemainturbulentlengthscalestheintegralscale,theTaylorscale,andtheKolmogorovscale;correspondingReynoldsnumberstheconceptofenergyanddissipationspectra科莫戈罗夫理论ConsiderfullyturbulentflowathighReynoldsnumberRe=UL/.Eddiesofsizelhaveacharacteristicvelocityu(l)andtimescalet(l)l/u(l).Eddiesinthelargestsizerangearecharacterizedbythelengthscalel0

comparabletotheflowlengthscaleL.Theircharacteristicvelocityu0u(l0)isontheorderofther.m.s.Turbulenceintensityu’(2k/3)1/2whichiscomparabletoU.Turbulentkineticenergyisdefinedas:TheReynoldsnumberoftheseeddiesRe0

u0l0/islarge(comparabletoRe)andthedirecteffectsofviscosityontheseeddiesarenegligiblysmall.IntegralscaleWecanderiveanestimateofthelengthscalel0ofthelargereddiesbasedonthefollowing:Eddiesofsizel0haveacharacteristicvelocityu0andtimescalet0

l0/u0Theircharacteristicvelocityu0u(l0)isontheorderofther.m.s.turbulenceintensityu’(2k/3)1/2

Assumethatenergyofeddywithvelocityscale

u0

isdissipatedintimet0

Wecanthenderivethefollowingequationforthislengthscale:Here,(m2/s3)istheenergydissipationrate.Theproportionalityconstantisoftheorderone.Thislengthscaleisusuallyreferredtoastheintegralscaleofturbulence.TheReynoldsnumberassociatedwiththeselargeeddiesisreferredtoastheturbulenceReynoldsnumberReL,whichisdefinedas:科莫戈罗夫理论科莫戈罗夫理论EnergytransferandDissipationThelargeeddiesareunstableandbreakup,transferringtheirenergytosomewhatsmallereddies.Thesesmallereddiesundergoasimilarbreak-upprocessandtransfertheirenergytoyetsmallereddies.Thisenergycascade–inwhichenergyistransferredtosuccessivelysmallerandsmallereddies–continuesuntiltheReynoldsnumberRe(l)u(l)l/issufficientlysmallthattheeddymotionisstable,andmolecularviscosityiseffectiveindissipatingthekineticenergy.Atthesesmallscales,thekineticenergyofturbulenceisconvertedintoheat.科莫戈罗夫理论EnergytransferandDissipationNotethatdissipationtakesplaceattheendofthesequenceofprocesses.Therateofdissipationisdetermined,thereforebythefirstprocessinthesequence,whichisthetransferofenergyfromthelargesteddies.Theseeddieshaveenergyoforderu02andtimescalet0=l0/u0sotherateoftransferofenergycanbesupposedtoscaleasu02/t0=u03/l0Consequently,consistentwithexperimentalobservationsinfreeshearflows,thispictureoftheenergycascadeindicatesthatisproportionaltou03/l0independentof(athighReynoldsnumbers).科莫戈罗夫理论Manyquestionsremainunanswered.Whatisthesizeofthesmallesteddiesthatareresponsiblefordissipatingtheenergy?Asldecreases,dothecharacteristicvelocityandtimescalesu(l)and(l)increase,decrease,orstaythesame?TheassumeddecreaseoftheReynoldsnumberu0l0/byitselfisnotsufficienttodeterminethesetrends.TheseandothersareansweredbyKolmogorov’stheoryofturbulence.Kolmogorov’stheoryisbasedonthreeimportanthypothesescombinedwithdimensionalargumentsandexperimentalobservations.科莫戈罗夫理论Kolmogorov’shypothesisoflocalisotropyForhomogenousturbulence,theturbulentkineticenergykisthesameeverywhere.Forisotropicturbulencetheeddiesalsobehavethesameinalldirections.Kolmogorov’shypothesisoflocalisotropystatesthatatsufficientlyhighReynoldsnumbers,thesmall-scaleturbulentmotions(l<<l0)arestatisticallyisotropic.Here,thetermlocalisotropymeansisotropyatsmallscales.Largescaleturbulencemaystillbeanisotropic.lEIisthelengthscalethatformsthedemarcationbetweenthelargescaleanisotropiceddies(l>lEI)

andthesmallscaleisotropiceddies(l<lEI).FormanyhighReynoldsnumberflowslEIcanbeestimatedaslEI

l0/6.科莫戈罗夫理论Kolmogorov’sfirstsimilarityhypothesisKolmogorovalsoarguedthatnotonlydoesthedirectionalinformationgetlostastheenergypassesdownthecascade,butthatallinformationaboutthegeometryoftheeddiesgetslostalso.Asaresult,thestatisticsofthesmall-scalemotionsareuniversal:theyaresimilarineveryhighReynoldsnumberturbulentflow,independentofthemeanflowfieldandtheboundaryconditions.ThesesmallscaleeddiesdependontherateTEIatwhichtheyreceiveenergyfromthelargerscales(whichisapproximatelyequaltothedissipationrate)andtheviscousdissipation,whichisrelatedtothekinematicviscosity.Kolmogorov’sfirstsimilarityhypothesisstatesthatineveryturbulentflowatsufficientlyhighReynoldsnumber,thestatisticsofthesmallscalemotions(l<lEI)haveauniversalformthatisuniquelydeterminedbyand.科莫戈罗夫理论Giventhetwoparametersandwecanformthefollowinguniquelength,velocity,andtimescales:Kolmogorovscaleisindicativeofthesmallesteddiespresentintheflow,thescaleatwhichtheenergyisdissipated.NotethefactthattheKolmogorovReynoldsnumberReofthesmalleddiesis1,isconsistentwiththenotionthatthecascadeproceedstosmallerandsmallerscalesuntiltheReynoldsnumberissmallenoughfordissipationtobeeffective.科莫戈罗夫理论Whenweusetherelationshipl0~k3/2/andsubstituteitintheequationsfortheKolmogorovscales,wecancalculatetheratiosbetweenthesmallscaleandlargescaleeddies.Asexpected,athighReynoldsnumbers,thevelocityandtimescalesofthesmallesteddiesaresmallcomparedtothoseofthelargesteddies.Since/l0decreaseswithincreasingReynoldsnumber,athighReynoldsnumbertherewillbearangeofintermediatescaleslwhichissmallcomparedtol0andlargecomparedwith.科莫戈罗夫理论BecausetheReynoldsnumberoftheintermediatescaleslisrelativelylarge,theywillnotbeaffectedbytheviscosity.Basedonthat,Kolmogorov’ssecondsimilarityhypothesisstatesthatineveryturbulentflowatsufficientlyhighReynoldsnumber,thestatisticsofthemotionsofscalelintherangel0>>l>>haveauniversalformthatisuniquelydeterminedbyindependentof.WeintroduceanewlengthscalelDI,(withlDI60formanyturbulenthighReynoldsnumberflows)sothatthisrangecanbewrittenaslEI>l

>lDIThislengthscalesplitstheuniversalequilibriumrangeintotwosubranges:Theinertialsubrange(lEI>l

>lDI)wheremotionsaredeterminedbyinertialeffectsandviscouseffectsarenegligible.Thedissipationrange(l

<lDI)wheremotionsexperienceviscouseffects.科莫戈罗夫理论Foreddiesintheinertialsubrangeofsizel,using: andthepreviouslyshownrelationshipsbetweentheturbulentReynoldsnumberandvariousscales,velocityscalesandtimescalescanbeformedfromandl:Aconsequence,then,ofthesecondsimilarityhypothesisisthatintheinertialsubrangethevelocityscalesandtimescalesu(l)and(l)decreaseasldecreases.科莫戈罗夫理论TaylormicroscaleThedissipationratedependsontheviscosityandvelocitygradients(“shear”)intheturbulenteddies.Forisotropicturbulence(mainlybookkeepingforalltheterms):WecannowdefinetheTaylormicroscaleasfollows:科莫戈罗夫理论ThisthenresultsinthefollowingrelationshipfortheTaylormicroscale:Fromk=(1/2)(u’2+v’2+w’2)wecanderivek=(3/2)u’2,and:TheTaylormicroscalefallsinbetweenthelargescaleeddiesandthesmallscaleeddies,whichcanbeseenbycalculatingtheratiosbetweenandl0and:科莫戈罗夫理论ThebulkoftheenergyiscontainedinthelargereddiesinthesizerangelEI=l0/6<l<6l0,whichisthereforecalledtheenergy-containingrange.EIandDIindicatethatlEIisthedemarcationlinebetweenenergy(E)andinertial(I)ranges,aslDIisthatbetweenthedissipation(D)andinertial(I)ranges.InertialsubrangeDissipationrangeEnergycontainingrangeUniversalequilibriumrangelDIlEIl0LKolmogorovlengthscaleTaylormicroscaleIntegrallengthscale科莫戈罗夫理论TherateatwhichenergyistransferredfromthelargerscalestothesmallerscalesisT(l).Undertheequilibriumconditionsintheinertialsubrangethisisequaltothedissipationrate,andisproportionaltou(l)2/.InertialsubrangeDissipationEnergycontainingrangeDissipationrangelDIlEIl0LProductionPT(l)Transferofenergytosuccessivelysmallerscales科莫戈罗夫理论EnergyspectrumTheturbulentkineticenergykisgivenby:Itremainstobedeterminedhowtheturbulentkineticenergyisdistributedamongtheeddiesofdifferentsizes.ThisisusuallydonebyconsideringtheenergyspectrumE().HereE()istheenergycontainedineddiesofsizelandwavenumber,definedas=2/l.BydefinitionkistheintegralofE()overallwavenumbers:TheenergycontainedineddieswithwavenumbersbetweenAandBisthen:科莫戈罗夫理论EnergyspectrumWewilldevelopanequationforE()intheinertialsubrange.AccordingtothesecondsimilarityhypothesisE()willsolelydependonand.Wecanthenperformthefollowingdimensionalanalysis:ThelastequationdescribesthefamousKolmogorov–5/3spectrum.CistheuniversalKolmogorovconstant,whichexperimentallywasdeterminedtobeC=1.5.科莫戈罗夫理论FullenergyspectrumModelequationsforE()intheproductionrangeanddissipationrangehavebeendeveloped.Wewillnotdiscussthetheorybehindthemhere.Thefullspectrumisgivenby:logE()logDissipationrangeInertialsubrangeEnergycontainingrangeslope–5/3mostoftheenergy(80%)iscontainedineddiesoflengthscale

lEI=l0/6<l<6l0.科莫戈罗夫理论Forgivenvaluesof,,andk，thefullspectrumcannowbecalculatedbasedontheseequations.Itis,howevercommontonormalizethespectruminoneoftwoways:basedontheKolmogorovscalesorbasedontheintegrallengthscale.BasedonKolmogorovscale:Measureoflengthscalebecomes().E()ismadedimensionlessasE()/(u2)Basedonintegralscale:Measureoflengthscalebecomes(l0).E()ismadedimensionlessasE()/(k

l0)Insteadofhavingthreeadjustableparameters(,,k),thenormalizedspectrumthenhasonlyoneadjustableparameter:R.科莫戈罗夫理论TheenergyspectrumasafunctionofRR=301003001000科莫戈罗夫理论TheenergyspectrumasafunctionofRR=301003001000科莫戈罗夫理论MeasurementsofspectraThefigureshowsexperimentallymeasuredonedimensionalspectra(onevelocitycomponentwasmeasuredonly,asindicatedbythe“1”and“11”subscripts).ThenumberattheendofthereferencedenotesthevalueofRforwhichthemeasurementsweredone.Source:Pope,page235.Determinationofthespectrumrequiressimultaneousmeasurementsofallvelocitycomponentsatmultiplepoints,whichisusuallynotpossible.Itiscommontomeasureonevelocitycomponentatonepointoveracertainperiodoftimeandconvertthetimesignaltoaspatialsignalusingx=UtwithUbeingthetimeaveragedvelocity.ThisiscommonlyreferredtoasTaylor’shypothesisoffrozenturbulence.Itisonlyvalidforu’/U<<1,whichisnotalwaysthecase.Spectrummeasurementsremainachallengingfieldofresearch.科莫戈罗夫理论Summary–ReynoldsnumbersThefollowingReynoldsnumbershave