完全氣體內能和焓熱力學復習熱力學第一定律熵及熱力學第二定律等熵關系式壓縮性定義無粘可壓縮流動的控制方程總條件的定義有激波的超音速流動的定性了解第七章路線圖7.3.DEFINITIONOFCOMPRESSIBILITY(壓縮性定義)

Allrealsubstancesarecompressible

tosomegreaterorlesserextend. Whenyousqueezeorpressonthem,theirdensitywillchange.Thisisparticularlytrueofgases.(所有的真實物質都是可壓縮的,當我們壓擠它們時,它們的密度會發生變化,對于氣體尤其是這樣.)Theamountbywhichasubstancecanbecompressedisgivenbyaspecificpropertyofthesubstancecalledthecompressibilty,definedbelow.物質可被壓縮的大小程度稱為物質的壓縮性.Considerasmallelementoffluidofvolume.Thepressureexertedonthesidesoftheelementisp.Ifthepressureisincreasedbyaninfinitesimalamountdp,thevolumewillchangebyanegativeamount.

Bydefinition,thecompressibilityisgivenby:

(7.33)as

(7.36)

Physically,thecompressibilityisafractionalchangeinvolumeofthefluidelementperunitchangeinpressure.(從物理上講,壓縮性就是每單位壓強變化引起的流體微元單位體積內的體積變化)

Ifthetemperatureofthefluidelementisheldconstant,thenisidentifiedastheisothermalcompressibility(等溫壓縮性)

(7.34) Iftheprocesstakesplaceisentropically,then(等熵壓縮性)(7.35)

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

canbelarge.(如果流體為氣體,則值大,對于一個給定壓強變化,方程.(7.37)指出,也會大.)

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

GOVERNINGEQUATIONSFORINVISCID,COMPRESSIBLEFLOW(無粘、可壓縮流控制方程) Forinviscid,pressibleflow,theprimarydependentvariablesarethepressurepandthevelocity.Hence,weneedonlytwobasicequations,namelythecontinuityandthemomentumequations.

對于無粘、不可壓縮流動，基本自變量是壓強p和速度。因此我們只需要兩個基本方程，即連續方程和動量方程。Indeed,thebasicequationsarecombinedtoobtainLaplace’sequationandBernoulli’sequation,whicharetheprimarilytoolstheapplicationsdiscussedinChaps.3to6.NotethatbothandTareassumedtobeconstantthroughoutsuchinviscid,pressibleflows.連續方程與動量方程相結合可以得到Laplace方程和Bernoulli方程，這是我們討論第三章至第六章內容用到的基本工具．對于無粘不可壓縮流動，我們假定密度和溫度保持不變．Basically,pressibleflowsobeypurelymechanicallawsanddonotneedthermodynamicconsiderations. Incontrast,forcompressibleflow,isvariableandesanunknown.Henceweneedanadditionalequation–theenergyequation–whichinturnintroducesinternalenergyeasanunknown.對于可壓縮流，相反的是是一個變量，并且是一個未知數．因此，我們需要一個附加方程－能量方程－進而引入未知數內能e。Internalenergyeisrelatedtotemperature,thenTalsoesanimportantvariable. Therefore,the5primarydependentvariablesare:Tosolveforthesefivevariables,weneedfivegoverningequations復習第二章知識：

Continuity(連續方程） Physicalprinciple:masscanbeneithercreatednordestroyed

Netmassflowoutof timerateofdecreaseof controlvolume = massinsidecontrolvolumeV throughsurfaceS

通過控制體表面S流出控制體的凈質量流量＝控制體內的質量減少率

(7.39)orintheformofapartialdifferentialequation

(偏微分方程）

(7.40)

whereisthedivergenceofthevectorfieldinCartesiancoordinates（在指角坐標系下）2.Momentum（動量方程）

Physicalprinciple:

Force=timerateofchangeofmomentum

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

(7.42b)(7.42c)

寫成矢量形式：whereisthesubstantialderivativewhichcanbewritteninCartesiancoordinatesas:

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

Assumethattheflowisadiabaticandthatbodyforcesarenegligible.Forsuchaflow

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

(總條件的定義）

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

are

and,

respectively.

假設流體微團通過一個給定點，對應的當地壓強、溫度、密度、馬赫數、速度分別為。

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

這里，是分別靜變量（靜參數），即靜壓、靜溫、靜密度。Nowimaginethatyougrabholdofthefluidelementandadiabaticallyslowitdowntozerovelocity.Clearly,youwouldexpect(correctly)thatthevaluesofwouldchangeastheelementisbroughttorest.Inparticular,thevalueofthetemperatureofthefluidelementafterithasbeenbroughttorestadiabaticallyisdefinedasthetotaltemperature,denotedby.特別地，假想流體微團被絕熱地減速為靜止所對應的溫度，定義此時流體微團對應的溫度為總溫．

Thecorrespondingvalueofenthalpyisdefinedastotalenthalpyh0,whereh0=cpT0

foracaloricallyperfectgas.

*如何確定總溫？

Theenergyequation,Eq.(7.44),providessomeimportantinformationabouttotalenthalpyandhencetotaltemperature.

(由能量方程可以的到總焓、因而總溫的重要信息。)Assumethattheflowisadiabaticandthatbodyforcesarenegligible,thentheequationofenergycanbewrittenas:

(7.45)注意(7.45)式的前提條件：無粘、絕熱、忽略體積力．ExpandingbyusingthefollowingvectoridentityAndnotingthatSubstitutingthecontinuityequation(7.47)(7.48)

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

Iftheflowissteady,(如果流動是定常的）

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

因為我們定義總焓h0為流體微元被絕熱地減速為靜止時對應的焓值,因此有能量方程我們可以得到總焓的值,即上式(7.53)中的常數。因此有:(7.54)Equation(7.54)isimportant;itstatesthatatanypointinaflow,thetotalenthalpyisgivenbythesumofthestaticenthalpyplusthekineticenergy,allperunitofmass.方程(7.54)很重要,它表明在流動中任一點,總焓由每單位體積的靜焓和動能之和組成。

有了總焓的定義，能量方程可以用總焓來表示：對于定常、絕熱、無粘流動，方程（7.52）可以寫成: ori.e.thetotalenthalpyisconstantalongastreamline.

即總焓沿流線為常數。

Ifallthestreamlinesofthefloworiginatefromacommonuniformfreestream(astheusuallythecase),thentheh0isthesameforeachline.

如果像通常的情況那樣，所有的流線都來自均勻自由來流，那么h0在不同流線也是相等的。

h0=const,throughouttheentireflow,andh0isequaltoitsfreestreamvalue.總焓在整個流場中為常數，等于自由來流對應的總焓。（7.55)Foracaloricallyperfectgas,h0=cpT0

.Thus,theaboveresultsalsostatethatthetotaltemperature

isconstantthroughoutthesteady,inviscid,adiabaticflowofacaloricallyperfectgas;i.e.對于量熱完全氣體，h0=cpT0

。因此，上面的結果也表明了對于定常、無粘、絕熱的量熱完全氣體，總溫保持不變，即（7.56）Keepinmindthattheabovediscussionmarbledtwotrainsofthought:Ontheonehand,wedealtwiththegeneralconceptofanadiabaticflowfield[whichledtoEqs.(7.51)to(7.53)],andontheotherhand,wedealtwiththedefinitionoftotalenthalpy[whichledtoEq.(7.54)].

要牢記在心的是：上面的討論是沿著兩條思路進行的，一方面，我們討論了絕熱流場的一般概念[導出了能量方程(7.51)至(7.53)]；另一方面，我們討論了總焓的定義[給出了（7.54)式]。(7.51)(7.52)(7.53)(7.54)

總壓與總密度的定義：

回到本節的開頭，我們考慮流體微團通過一個給定點，對應的當地壓強、溫度、密度、馬赫數、速度分別為。

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

andtotaldensity.

定義：當流體微元被等熵地減速至靜止時對應的壓強和密度被定義為其總壓和總密度。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頁圖1.30Anessentialingredientofasupersonicflowisthecalculationoftheshapeandstrengthofshockwaves.Thisisthemainthrustofchaps.8and9.

超音速流動研究的一個重要內容就是計算激波的形狀和強度。這是第8章和第9章的主題。Ashockwaveisanextremelythinregion,typicallyontheorderof10-5cm,acrosswhichtheflowpropertiescanchangedrastically.激波是一個極其薄的區域，厚度大約只有10-5cm的量級，通過激波流動特性發生劇烈變化。7.7Summary(小結