紊流理论(基本理论).ppt

紊流理论Turbulence 主讲 李文杰杨胜发河海学院303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 Andsoontoviscosityinthemolecularsense 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 Eddiesinthelargestsizerangearecharacterizedbythelengthscalel0comparabletotheflowlengthscaleL Theircharacteristicvelocityu0 u l0 isontheorderofther m s Turbulenceintensityu 2k 3 1 2whichiscomparabletoU Turbulentkineticenergyisdefinedas TheReynoldsnumberoftheseeddiesRe0 u0l0 islarge comparabletoRe andthedirecteffectsofviscosityontheseeddiesarenegligiblysmall IntegralscaleWecanderiveanestimateofthelengthscalel0ofthelargereddiesbasedonthefollowing Eddiesofsizel0haveacharacteristicvelocityu0andtimescalet0 l0 u0Theircharacteristicvelocityu0 u l0 isontheorderofther m s turbulenceintensityu 2k 3 1 2Assumethatenergyofeddywithvelocityscaleu0isdissipatedintimet0Wecanthenderivethefollowingequationforthislengthscale 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 Therateofdissipation isdetermined thereforebythefirstprocessinthesequence whichisthetransferofenergyfromthelargesteddies Theseeddieshaveenergyoforderu02andtimescalet0 l0 u0sotherateoftransferofenergycanbesupposedtoscaleasu02 t0 u03 l0Consequently consistentwithexperimentalobservationsinfreeshearflows thispictureoftheenergycascadeindicatesthat isproportionaltou03 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 llEI 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 haveauniversalformthatisuniquelydeterminedby and 科莫戈罗夫理论 Giventhetwoparameters and wecanformthefollowinguniquelength velocity andtimescales Kolmogorovscaleisindicativeofthesmallesteddiespresentintheflow thescaleatwhichtheenergyisdissipated NotethefactthattheKolmogorovReynoldsnumberRe ofthesmalleddiesis1 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 haveauniversalformthatisuniquelydeterminedby independentof WeintroduceanewlengthscalelDI withlDI 60 formanyturbulenthighReynoldsnumberflows sothatthisrangecanbewrittenaslEI l lDIThislengthscalesplitstheuniversalequilibriumrangeintotwosubranges Theinertialsubrange lEI l lDI wheremotionsaredeterminedbyinertialeffectsandviscouseffectsarenegligible Thedissipationrange l lDI wheremotionsexperienceviscouseffects 科莫戈罗夫理论 Foreddiesintheinertialsubrangeofsizel using andthepreviouslyshownrelationshipsbetweentheturbulentReynoldsnumberandvariousscales velocityscalesandtimescalescanbeformedfrom andl Aconsequence then ofthesecondsimilarityhypothesisisthatintheinertialsubrangethevelocityscalesandtimescalesu l and l decreaseasldecreases 科莫戈罗夫理论 TaylormicroscaleThedissipationratedependsontheviscosityandvelocitygradients shear intheturbulenteddies Forisotropicturbulence mainlybookkeepingforalltheterms WecannowdefinetheTaylormicroscale asfollows 科莫戈罗夫理论 ThisthenresultsinthefollowingrelationshipfortheTaylormicroscale Fromk 1 2 u 2 v 2 w 2 wecanderivek 3 2 u 2 and TheTaylormicroscalefallsinbetweenthelargescaleeddiesandthesmallscaleeddies whichcanbeseenbycalculatingtheratiosbetween andl0and 科莫戈罗夫理论 ThebulkoftheenergyiscontainedinthelargereddiesinthesizerangelEI l0 6 l 6l0 whichisthereforecalledtheenergy containingrange EIandDIindicatethatlEIisthedemarcationlinebetweenenergy E andinertial I ranges aslDIisthatbetweenthedissipation D andinertial I ranges Kolmogorovlengthscale Taylormicroscale Integrallengthscale 科莫戈罗夫理论 TherateatwhichenergyistransferredfromthelargerscalestothesmallerscalesisT l Undertheequilibriumconditionsintheinertialsubrangethisisequaltothedissipationrate andisproportionaltou l 2 科莫戈罗夫理论 EnergyspectrumTheturbulentkineticenergykisgivenby Itremainstobedeterminedhowtheturbulentkineticenergyisdistributedamongtheeddiesofdifferentsizes ThisisusuallydonebyconsideringtheenergyspectrumE HereE istheenergycontainedineddiesofsizelandwavenumber definedas 2 l BydefinitionkistheintegralofE overallwavenumbers Theenergycontainedineddieswithwavenumbersbetween Aand Bisthen 科莫戈罗夫理论 EnergyspectrumWewilldevelopanequationforE intheinertialsubrange AccordingtothesecondsimilarityhypothesisE willsolelydependon and Wecanthenperformthefollowingdimensionalanalysis ThelastequationdescribesthefamousKolmogorov 5 3spectrum CistheuniversalKolmogorovconstant whichexperimentallywasdeterminedtobeC 1 5 科莫戈罗夫理论 FullenergyspectrumModelequationsforE intheproductionrangeanddissipationrangehavebeendeveloped Wewillnotdiscussthetheorybehindthemhere Thefullspectrumisgivenby logE log Dissipationrange Inertialsubrange Energycontainingrange slope 5 3 mostoftheenergy 80 iscontainedineddiesoflengthscalelEI l0 6 l 6l0 科莫戈罗夫理论 Forgivenvaluesof andk thefullspectrumcannowbecalculatedbasedontheseequations Itis howevercommontonormalizethespectruminoneoftwoways basedontheKolmogorovscalesorbasedontheintegrallengthscale BasedonKolmogorovscale Measureoflengthscalebecomes E ismadedimensionlessasE u 2 Basedonintegralscale Measureoflengthscalebecomes l0 E ismadedimensionlessasE kl0 Insteadofhavingthreeadjustableparameters k thenormalizedspectrumthenhasonlyoneadjustableparameter R 科莫戈罗夫理论 TheenergyspectrumasafunctionofR R 30 100 300 1000 科莫戈罗夫理论 TheenergyspectrumasafunctionofR R 30 100 300 1000 科莫戈罗夫理论 Measurementsofspectra Thefigureshowsexperimentallymeasuredonedimensionalspectra onevelocitycomponentwasmeasuredonly asindicatedbythe 1 and 11 subscripts ThenumberattheendofthereferencedenotesthevalueofR forwhichthemeasurementsweredone Source Pope page235 Determinationofthespectrumrequiressimultaneousmeasurementsofallvelocitycomponentsatmultiplepoints whichisusuallynotpossible Itiscommontomeasureonevelocitycomponentatonepointoveracertainperiodoftimeandconvertthetimesignaltoaspatialsignalusingx UtwithUbeingthetimeaveragedvelocity ThisiscommonlyreferredtoasTaylor shypothesisoffrozenturbulence Itisonlyvalidforu U 1 whichisnotalwaysthecase Spectrummeasurementsremainachallengingfieldofresearch 科莫戈罗夫理论 Summary ReynoldsnumbersThefollowingReynoldsnumbershavebeendefined FlowReynoldsnumber TurbulenceReynoldsnumber TaylorReynoldsnumber KolmogorovReynoldsnumber TheflowReynoldsnumber