所需-气动空气动力学课件chapter

完全气体内能和焓热力学复习热力学第一定律熵及热力学第二定律等熵关系式压缩性定义无粘可压缩流动的控制方程总条件的定义有激波的超音速流动的定性了解第七章路线图7.3.DEFINITIONOFCOMPRESSIBILITY(压缩性定义)

Allrealsubstancesarecompressible

tosomegreaterorlesserextend. Whenyousqueezeorpressonthem,theirdensitywillchange.Thisisparticularlytrueofgases.(所有的真实物质都是可压缩的,当我们压挤它们时,它们的密度会发生变化,对于气体尤其是这样.)Theamountbywhichasubstancecanbecompressedisgivenbyaspecificpropertyofthesubstancecalledthecompressibilty,definedbelow.物质可被压缩的大小程度称为物质的压缩性.Considerasmallelementoffluidofvolume.Thepressureexertedonthesidesoftheelementisp.Ifthepressureisincreasedbyaninfinitesimalamountdp,thevolumewillchangebyanegativeamount.

Bydefinition,thecompressibilityisgivenby:

(7.33)as

(7.36)

Physically,thecompressibilityisafractionalchangeinvolumeofthefluidelementperunitchangeinpressure.(从物理上讲,压缩性就是每单位压强变化引起的流体微元单位体积内的体积变化)

Ifthetemperatureofthefluidelementisheldconstant,thenisidentifiedastheisothermalcompressibility(等温压缩性)

(7.34) Iftheprocesstakesplaceisentropically,then(等熵压缩性)(7.35)

Ifthefluidisagas,wherecompressibilityislarge,thenforagivenpressurechangefromonepointtoanotherintheflow,Eq.(7.37)statesthat

canbelarge.(如果流体为气体,则值大,对于一个给定压强变化,方程.(7.37)指出,也会大.)

Thus,isnotconstant;theflowofagasisacompressibleflow. Theexceptionisthelow-speedflowofagas.Whereisthelimit?IftheMachnumber ,theflowshouldbeconsideredcompressible.(7.37)7.4

GOVERNINGEQUATIONSFORINVISCID,COMPRESSIBLEFLOW(无粘、可压缩流控制方程) Forinviscid,pressibleflow,theprimarydependentvariablesarethepressurepandthevelocity.Hence,weneedonlytwobasicequations,namelythecontinuityandthemomentumequations.

对于无粘、不可压缩流动，基本自变量是压强p和速度。因此我们只需要两个基本方程，即连续方程和动量方程。Indeed,thebasicequationsarecombinedtoobtainLaplace’sequationandBernoulli’sequation,whicharetheprimarilytoolstheapplicationsdiscussedinChaps.3to6.NotethatbothandTareassumedtobeconstantthroughoutsuchinviscid,pressibleflows.连续方程与动量方程相结合可以得到Laplace方程和Bernoulli方程，这是我们讨论第三章至第六章内容用到的基本工具．对于无粘不可压缩流动，我们假定密度和温度保持不变．Basically,pressibleflowsobeypurelymechanicallawsanddonotneedthermodynamicconsiderations. Incontrast,forcompressibleflow,isvariableandesanunknown.Henceweneedanadditionalequation–theenergyequation–whichinturnintroducesinternalenergyeasanunknown.对于可压缩流，相反的是是一个变量，并且是一个未知数．因此，我们需要一个附加方程－能量方程－进而引入未知数内能e。Internalenergyeisrelatedtotemperature,thenTalsoesanimportantvariable. Therefore,the5primarydependentvariablesare:Tosolveforthesefivevariables,weneedfivegoverningequations复习第二章知识：

Continuity(连续方程） Physicalprinciple:masscanbeneithercreatednordestroyed

Netmassflowoutof timerateofdecreaseof controlvolume = massinsidecontrolvolumeV throughsurfaceS

通过控制体表面S流出控制体的净质量流量＝控制体内的质量减少率

(7.39)orintheformofapartialdifferentialequation

(偏微分方程）

(7.40)

whereisthedivergenceofthevectorfieldinCartesiancoordinates（在指角坐标系下）2.Momentum（动量方程）

Physicalprinciple:

Force=timerateofchangeofmomentum

(7.41)wherearethebodyforces,suchasgravity,orelectromagneticforcesIntermsofsubstantialderivative:(7.42a)theyandzdirectionsofthevectorcanbeeasilyfoundbysubstitution

(7.42b)(7.42c)

写成矢量形式：whereisthesubstantialderivativewhichcanbewritteninCartesiancoordinatesas:

3.Energy Physicalprinciple: Energycanbeneithercreatednordestroyed;itcanonlychangeinform(7.43)Equationofenergycanalsobewrittenas:

Assumethattheflowisadiabaticandthatbodyforcesarenegligible.Forsuchaflow

(7.44)(7.45)4.Equationofstateforaperfectgas:5.Internalenergyforacaloricallyperfectgas:Wehavenow5equationsfor5unknowns.7.5DEFINITIONOFTOTALCONDITIONS

(总条件的定义）

Considerafluidelementpassingthroughagivenpointinaflowwherethelocalpressure,temperature,density,Machnumber,andvelocity(localconditions)

are

and,

respectively.

假设流体微团通过一个给定点，对应的当地压强、温度、密度、马赫数、速度分别为。

Here,arestaticquantities,i.e.,staticpressure,statictemperature,staticdensity,respectively.

这里，是分别静变量（静参数），即静压、静温、静密度。Nowimaginethatyougrabholdofthefluidelementandadiabaticallyslowitdowntozerovelocity.Clearly,youwouldexpect(correctly)thatthevaluesofwouldchangeastheelementisbroughttorest.Inparticular,thevalueofthetemperatureofthefluidelementafterithasbeenbroughttorestadiabaticallyisdefinedasthetotaltemperature,denotedby.特别地，假想流体微团被绝热地减速为静止所对应的温度，定义此时流体微团对应的温度为总温．

Thecorrespondingvalueofenthalpyisdefinedastotalenthalpyh0,whereh0=cpT0

foracaloricallyperfectgas.

*如何确定总温？

Theenergyequation,Eq.(7.44),providessomeimportantinformationabouttotalenthalpyandhencetotaltemperature.

(由能量方程可以的到总焓、因而总温的重要信息。)Assumethattheflowisadiabaticandthatbodyforcesarenegligible,thentheequationofenergycanbewrittenas:

(7.45)注意(7.45)式的前提条件：无粘、绝热、忽略体积力．ExpandingbyusingthefollowingvectoridentityAndnotingthatSubstitutingthecontinuityequation(7.47)(7.48)

(7.46)(7.45)(7.48)(7.45)+(7.48),note:(7.51)

Iftheflowissteady,(如果流动是定常的）

Fromthedefinitionofthesubstantialderivative Thenthetimerateofchangeofh+V2/2followingamovingfluidelementiszero:(7.53)RecallthattheassumptionswhichledtoEq.(7.53)arethattheflowissteady,adiabatic,andinviscid.(7.52)Sinceh0isdefinedasthatenthalpywhichwouldexistatapointifthefluidelementwerebroughttorestadiabatically,whereV=0andhenceh=h0,thenthevalueoftheconstantish0.

因为我们定义总焓h0为流体微元被绝热地减速为静止时对应的焓值,因此有能量方程我们可以得到总焓的值,即上式(7.53)中的常数。因此有:(7.54)Equation(7.54)isimportant;itstatesthatatanypointinaflow,thetotalenthalpyisgivenbythesumofthestaticenthalpyplusthekineticenergy,allperunitofmass.方程(7.54)很重要,它表明在流动中任一点,总焓由每单位体积的静焓和动能之和组成。

有了总焓的定义，能量方程可以用总焓来表示：对于定常、绝热、无粘流动，方程（7.52）可以写成: ori.e.thetotalenthalpyisconstantalongastreamline.

即总焓沿流线为常数。

Ifallthestreamlinesofthefloworiginatefromacommonuniformfreestream(astheusuallythecase),thentheh0isthesameforeachline.

如果像通常的情况那样，所有的流线都来自均匀自由来流，那么h0在不同流线也是相等的。

h0=const,throughouttheentireflow,andh0isequaltoitsfreestreamvalue.总焓在整个流场中为常数，等于自由来流对应的总焓。（7.55)Foracaloricallyperfectgas,h0=cpT0

.Thus,theaboveresultsalsostatethatthetotaltemperature

isconstantthroughoutthesteady,inviscid,adiabaticflowofacaloricallyperfectgas;i.e.对于量热完全气体，h0=cpT0

。因此，上面的结果也表明了对于定常、无粘、绝热的量热完全气体，总温保持不变，即（7.56）Keepinmindthattheabovediscussionmarbledtwotrainsofthought:Ontheonehand,wedealtwiththegeneralconceptofanadiabaticflowfield[whichledtoEqs.(7.51)to(7.53)],andontheotherhand,wedealtwiththedefinitionoftotalenthalpy[whichledtoEq.(7.54)].

要牢记在心的是：上面的讨论是沿着两条思路进行的，一方面，我们讨论了绝热流场的一般概念[导出了能量方程(7.51)至(7.53)]；另一方面，我们讨论了总焓的定义[给出了（7.54)式]。(7.51)(7.52)(7.53)(7.54)

总压与总密度的定义：

回到本节的开头，我们考虑流体微团通过一个给定点，对应的当地压强、温度、密度、马赫数、速度分别为。

Onceagain,imaginethatyougrabholdofthefluidelementandslowitdowntozerovelocity,butthistime,letusslowitdownbothadiabaticallyandreversibly.Thatis,letusslowthefluidelementdowntozerovelocityisentropically.Whenthefluidelementisbroughttorestisentropically,theresultingpressureanddensityaredefinedasthetotalpressurep0

andtotaldensity.

定义：当流体微元被等熵地减速至静止时对应的压强和密度被定义为其总压和总密度。Sinceanisentropicprocessisalsoadiabatic,thedefinitionoftotaltemperatureremainsunchanged.Asbefore,keepinmindthatwedonothavetoactuallybringtheflowtorestinreallifeinordertotalkabouttotalpressureandtotaldensity;rather,thearedefinedquantitiesthatwouldexistatapointinaflowif(inourimagination)thefluidelementpassingthroughthatpointwerebroughttorestisentropically.Therefore,atagivenpointinaflow,wherethestaticpressureandstaticdensityarepandρ,respectively,wecanalsoassignavalueoftotalpressurep0,andtotaldensityρ0definedasabove.6.SUMMARY

TotaltemperatureT0andtotalenthalpyh0aredefinedasthepropertiesthatwouldexistiftheflowisslowedtozerovelocityadiabatically. Totalpressurep0

andtotaldensity

ρ0aredefinedasthepropertiesthatwouldexistiftheflowisslowedtozerovelocityisentropically. Ifthegeneralflowfieldisadiabatic,h0isconstantthroughouttheflow.

Ifthegeneralflowfieldisisentropic,p0andρ0areconstantthroughouttheflow.7.6SomeAspectsofSupersonicFlow:ShockWaves

超音速流的一些特征：激波51页图1.30Anessentialingredientofasupersonicflowisthecalculationoftheshapeandstrengthofshockwaves.Thisisthemainthrustofchaps.8and9.

超音速流动研究的一个重要内容就是计算激波的形状和强度。这是第8章和第9章的主题。Ashockwaveisanextremelythinregion,typicallyontheorderof10-5cm,acrosswhichtheflowpropertiescanchangedrastically.激波是一个极其薄的区域，厚度大约只有10-5cm的量级，通过激波流动特性发生剧烈变化。7.7Summary(小结