紊流理論(基本理論)-課件

課程內(nèi)容（10-18周）一.緒論二.基本方程三.基本理論四.紊流模型五.明渠紊流六.紊流前沿成果（楊勝發(fā)）二、紊流基本理論1、層流穩(wěn)定性理論2、科莫戈羅夫理論3、紊流猝發(fā)現(xiàn)象層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-圓管流動(dòng)1883年，雷諾進(jìn)行圓管流動(dòng)的實(shí)驗(yàn)，觀察到層流向紊流的轉(zhuǎn)捩。Re較小時(shí)，流體質(zhì)點(diǎn)沿著與管道中心平行的直線勻速前進(jìn)，不同的流層互不干擾和摻混，為層流。Re增大到一定的數(shù)值時(shí)，不同流層中的質(zhì)點(diǎn)開始摻混，發(fā)生動(dòng)量交換，一點(diǎn)的流速和壓強(qiáng)呈隨機(jī)性的脈動(dòng)，但是時(shí)間平均值趨于均勻，為紊流。層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-圓管流動(dòng)當(dāng)Re處在臨界值附近的一個(gè)范圍內(nèi)時(shí)，流動(dòng)具有間歇性，時(shí)而為層流時(shí)而為紊流。羅塔于1956年觀察到這樣一個(gè)現(xiàn)象，當(dāng)雷諾數(shù)為2550時(shí)，圓管中的流動(dòng)呈現(xiàn)間歇性。間歇系數(shù)：層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)在雷諾數(shù)較大的流動(dòng)中，緊貼著物體表面，流動(dòng)受到粘性的顯著影響，流速沿壁面法向的變化非常急劇，摩擦切應(yīng)力不能略去不計(jì)的極薄的一層流體，稱為邊界層。層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)以經(jīng)典的圓柱繞流為例。可以看出，流體與圓柱之間存在滑移，流線是對(duì)稱的，流動(dòng)方向無阻力。層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)以經(jīng)典的圓柱繞流為例。粘性流動(dòng)壁面無滑移，產(chǎn)生邊界層，在背流面發(fā)生分離，形成一個(gè)由漩渦組成的尾流區(qū)。層流穩(wěn)定性理論層流到紊流的轉(zhuǎn)捩-壁面邊界層流動(dòng)臨界Re附近，邊界層流動(dòng)由層流向紊流轉(zhuǎn)捩，分離點(diǎn)下移，尾流區(qū)縮小，形狀阻力降低，產(chǎn)生了阻力危機(jī)。來流的紊動(dòng)度和壁面的粗糙程度都影響轉(zhuǎn)捩的發(fā)生，粗糙表面的邊界層更容易發(fā)展為紊流，因此會(huì)導(dǎo)致阻力危機(jī)的提前發(fā)生。因此，繞流阻力主要取決于物體背流面尾流中的負(fù)壓，與背流面的形狀關(guān)系密切。這一問題直到邊界層理論提出后才得到解決。層流穩(wěn)定性理論基本點(diǎn)1930年，普朗特建立了層流穩(wěn)定性理論。層流穩(wěn)定性的基本點(diǎn)是：層流流動(dòng)總會(huì)受到一些擾動(dòng)，可能是受進(jìn)口邊壁粗糙或者來流自身的紊動(dòng)，如果擾動(dòng)隨時(shí)間衰減，則層流穩(wěn)定，否則會(huì)逐漸過渡至穩(wěn)流。研究?jī)?nèi)容尋求各種流動(dòng)情況下，層流對(duì)微小擾動(dòng)失去抑制時(shí)的雷諾數(shù)，即臨界雷諾數(shù)。考慮一個(gè)二維的情況，將流動(dòng)分解為主流和加在上面的一個(gè)擾動(dòng)。層流穩(wěn)定性理論問題：對(duì)于這樣一個(gè)主流流動(dòng)，主流滿足N-S方程，疊加后的流動(dòng)也滿足，那么擾動(dòng)將隨時(shí)間放大還是衰減。層流穩(wěn)定性理論層流穩(wěn)定性理論層流穩(wěn)定性理論層流穩(wěn)定性理論層流穩(wěn)定性理論u方向?qū)取微分，v方向?qū)取微分，相減消去壓強(qiáng)擾動(dòng)項(xiàng)，則得到兩個(gè)方程式，含有兩個(gè)未知量u’、v’。邊界條件：邊壁處u’=0、v’=0；無窮遠(yuǎn)處同樣。科莫戈羅夫理論漩渦的產(chǎn)生：假設(shè)某一水流的分離面，由于擾動(dòng)，流線發(fā)生彎曲流線集中的地方流速大，壓力低，分散地方相反，加劇流線彎曲，最終產(chǎn)生漩渦?？颇炅_夫理論漩渦的產(chǎn)生：流速梯度大的地方，機(jī)理相似。漩渦抬升，擴(kuò)散至全流區(qū)?？颇炅_夫理論漩渦的結(jié)構(gòu)和組成：漩渦抬升過程中逐漸增大。大尺度漩渦的尺寸與容器尺寸（管徑、水深等）屬于同一量級(jí)。大尺度漩渦的分布及方向取決于形成條件，不是各向均勻同性的，由于強(qiáng)烈的摻混作用，大漩渦不穩(wěn)定，會(huì)崩解稱為次一級(jí)的小漩渦。大漩渦分解，把能量傳遞給次一級(jí)漩渦，次一級(jí)漩渦仍然不穩(wěn)定，會(huì)進(jìn)一步分解。分解的過程中，漩渦的幾何方向性逐漸喪失，形成條件的影響越來越弱，越來越接近各向同性。一直到與水團(tuán)相關(guān)的雷諾數(shù)低到不能再產(chǎn)生更小的漩渦為止。這些最低級(jí)漩渦的能量會(huì)通過粘性轉(zhuǎn)化為熱能。科莫戈羅夫理論L.F.Richardson(“WeatherPredictionbyNumericalProcess.”CambridgeUniversityPress,1922)summarizedthisinthefollowingoftencitedverse:BigwhirlshavelittlewhirlsWhichfeedontheirvelocity;Andlittlewhirlshavelesserwhirls,Andsoontoviscosity inthemolecularsense.科莫戈羅夫理論漩渦能量分布：漩渦抬升過程中逐漸增大?？颇炅_夫理論Kolmogorov’stheorydescribeshowenergyistransferredfromlargertosmallereddieshowmuchenergyiscontainedbyeddiesofagivensizehowmuchenergyisdissipatedbyeddiesofeachsizethreemainturbulentlengthscalestheintegralscale,theTaylorscale,andtheKolmogorovscale;correspondingReynoldsnumberstheconceptofenergyanddissipationspectra科莫戈羅夫理論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:科莫戈羅夫理論科莫戈羅夫理論EnergytransferandDissipationThelargeeddiesareunstableandbreakup,transferringtheirenergytosomewhatsmallereddies.Thesesmallereddiesundergoasimilarbreak-upprocessandtransfertheirenergytoyetsmallereddies.Thisenergycascade–inwhichenergyistransferredtosuccessivelysmallerandsmallereddies–continuesuntiltheReynoldsnumberRe(l)u(l)l/issufficientlysmallthattheeddymotionisstable,andmolecularviscosityiseffectiveindissipatingthekineticenergy.Atthesesmallscales,thekineticenergyofturbulenceisconvertedintoheat.科莫戈羅夫理論EnergytransferandDissipationNotethatdissipationtakesplaceattheendofthesequenceofprocesses.Therateofdissipationisdetermined,thereforebythefirstprocessinthesequence,whichisthetransferofenergyfromthelargesteddies.Theseeddieshaveenergyoforderu02andtimescalet0=l0/u0sotherateoftransferofenergycanbesupposedtoscaleasu02/t0=u03/l0Consequently,consistentwithexperimentalobservationsinfreeshearflows,thispictureoftheenergycascadeindicatesthatisproportionaltou03/l0independentof(athighReynoldsnumbers).科莫戈羅夫理論Manyquestionsremainunanswered.Whatisthesizeofthesmallesteddiesthatareresponsiblefordissipatingtheenergy?Asldecreases,dothecharacteristicvelocityandtimescalesu(l)and(l)increase,decrease,orstaythesame?TheassumeddecreaseoftheReynoldsnumberu0l0/byitselfisnotsufficienttodeterminethesetrends.TheseandothersareansweredbyKolmogorov’stheoryofturbulence.Kolmogorov’stheoryisbasedonthreeimportanthypothesescombinedwithdimensionalargumentsandexperimentalobservations.科莫戈羅夫理論Kolmogorov’shypothesisoflocalisotropyForhomogenousturbulence,theturbulentkineticenergykisthesameeverywhere.Forisotropicturbulencetheeddiesalsobehavethesameinalldirections.Kolmogorov’shypothesisoflocalisotropystatesthatatsufficientlyhighReynoldsnumbers,thesmall-scaleturbulentmotions(l<<l0)arestatisticallyisotropic.Here,thetermlocalisotropymeansisotropyatsmallscales.Largescaleturbulencemaystillbeanisotropic.lEIisthelengthscalethatformsthedemarcationbetweenthelargescaleanisotropiceddies(l>lEI)

andthesmallscaleisotropiceddies(l<lEI).FormanyhighReynoldsnumberflowslEIcanbeestimatedaslEI

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

>lDIThislengthscalesplitstheuniversalequilibriumrangeintotwosubranges:Theinertialsubrange(lEI>l

>lDI)wheremotionsaredeterminedbyinertialeffectsandviscouseffectsarenegligible.Thedissipationrange(l

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

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

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