所需-氣動(dòng)空氣動(dòng)力學(xué)課件chapter

完全氣體內(nèi)能和焓熱力學(xué)復(fù)習(xí)熱力學(xué)第一定律熵及熱力學(xué)第二定律等熵關(guān)系式壓縮性定義無(wú)粘可壓縮流動(dòng)的控制方程總條件的定義有激波的超音速流動(dòng)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(shí)物質(zhì)都是可壓縮的,當(dāng)我們壓擠它們時(shí),它們的密度會(huì)發(fā)生變化,對(duì)于氣體尤其是這樣.)Theamountbywhichasubstancecanbecompressedisgivenbyaspecificpropertyofthesubstancecalledthecompressibilty,definedbelow.物質(zhì)可被壓縮的大小程度稱為物質(zhì)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(qiáng)變化引起的流體微元單位體積內(nèi)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(duì)于一個(gè)給定壓強(qiáng)變化,方程.(7.37)指出,也會(huì)大.)

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

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

對(duì)于無(wú)粘、不可壓縮流動(dòng)，基本自變量是壓強(qiáng)p和速度。因此我們只需要兩個(gè)基本方程，即連續(xù)方程和動(dòng)量方程。Indeed,thebasicequationsarecombinedtoobtainLaplace’sequationandBernoulli’sequation,whicharetheprimarilytoolstheapplicationsdiscussedinChaps.3to6.NotethatbothandTareassumedtobeconstantthroughoutsuchinviscid,pressibleflows.連續(xù)方程與動(dòng)量方程相結(jié)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(nèi)容用到的基本工具．對(duì)于無(wú)粘不可壓縮流動(dòng)，我們假定密度和溫度保持不變．Basically,pressibleflowsobeypurelymechanicallawsanddonotneedthermodynamicconsiderations. Incontrast,forcompressibleflow,isvariableandesanunknown.Henceweneedanadditionalequation–theenergyequation–whichinturnintroducesinternalenergyeasanunknown.對(duì)于可壓縮流，相反的是是一個(gè)變量，并且是一個(gè)未知數(shù)．因此，我們需要一個(gè)附加方程－能量方程－進(jìn)而引入未知數(shù)內(nèi)能e。Internalenergyeisrelatedtotemperature,thenTalsoesanimportantvariable. Therefore,the5primarydependentvariablesare:Tosolveforthesefivevariables,weneedfivegoverningequations復(fù)習(xí)第二章知識(shí)：

Continuity(連續(xù)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(zhì)量流量＝控制體內(nèi)的質(zhì)量減少率

(7.39)orintheformofapartialdifferentialequation

(偏微分方程）

(7.40)

whereisthedivergenceofthevectorfieldinCartesiancoordinates（在指角坐標(biāo)系下）2.Momentum（動(dòng)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(shè)流體微團(tuán)通過一個(gè)給定點(diǎn)，對(duì)應(yīng)的當(dāng)?shù)貕簭?qiáng)、溫度、密度、馬赫數(shù)、速度分別為。

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

這里，是分別靜變量（靜參數(shù)），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(tuán)被絕熱地減速為靜止所對(duì)應(yīng)的溫度，定義此時(shí)流體微團(tuán)對(duì)應(yīng)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(wú)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(dòng)是定常的）

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

因?yàn)槲覀兌x總焓h0為流體微元被絕熱地減速為靜止時(shí)對(duì)應(yīng)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(shù)。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(dòng)中任一點(diǎn),總焓由每單位體積的靜焓和動(dòng)能之和組成。

有了總焓的定義，能量方程可以用總焓來(lái)表示：對(duì)于定常、絕熱、無(wú)粘流動(dòng)，方程（7.52）可以寫成: ori.e.thetotalenthalpyisconstantalongastreamline.

即總焓沿流線為常數(shù)。

Ifallthestreamlinesofthefloworiginatefromacommonuniformfreestream(astheusuallythecase),thentheh0isthesameforeachline.

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

h0=const,throughouttheentireflow,andh0isequaltoitsfreestreamvalue.總焓在整個(gè)流場(chǎng)中為常數(shù)，等于自由來(lái)流對(duì)應(yīng)的總焓。（7.55)Foracaloricallyperfectgas,h0=cpT0

.Thus,theaboveresultsalsostatethatthetotaltemperature

isconstantthroughoutthesteady,inviscid,adiabaticflowofacaloricallyperfectgas;i.e.對(duì)于量熱完全氣體，h0=cpT0

。因此，上面的結(jié)果也表明了對(duì)于定常、無(wú)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(jìn)行的，一方面，我們討論了絕熱流場(chǎng)的一般概念[導(dǎo)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(jié)的開頭，我們考慮流體微團(tuán)通過一個(gè)給定點(diǎn)，對(duì)應(yīng)的當(dāng)?shù)貕簭?qiáng)、溫度、密度、馬赫數(shù)、速度分別為。

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

andtotaldensity.

定義：當(dāng)流體微元被等熵地減速至靜止時(shí)對(duì)應(yīng)的壓強(qiáng)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(yè)圖1.30Anessentialingredientofasupersonicflowisthecalculationoftheshapeandstrengthofshockwaves.Thisisthemainthrustofchaps.8and9.

超音速流動(dòng)研究的一個(gè)重要內(nèi)容就是計(jì)算激波的形狀和強(qiáng)度。這是第8章和第9章的主題。Ashockwaveisanextremelythinregion,typicallyontheorderof10-5cm,acrosswhichtheflowpropertiescanchangedrastically.激波是一個(gè)極其薄的區(qū)域，厚度大約只有10-5cm的量級(jí)，通過激波流動(dòng)特性發(fā)生劇烈變化。7.7Summary(小結(jié)