Chapter7CompressibleFlow:SomePreliminaryAspects第七章可壓縮流動(dòng):相關(guān)的預(yù)備知識(shí)7.1引言空氣動(dòng)力學(xué)

A.連續(xù)流動(dòng)B.低密度和自由分子流

ContinuumFlowLow-densityandfree-moleculeflows

C.粘性流動(dòng)D.無(wú)粘流動(dòng)

ViscousFlowInviscidFlow

E.不可壓縮流動(dòng)IncompressibleFlow

F.可壓縮流動(dòng)CompressibleFlowG.亞音速流H.跨音速流I.超音速流J.高超音速流SubsonicTransonicSupersonicHypersonic可壓縮流動(dòng)的基本特征:＊Thepivotalaspectofhigh-speedflowisthatthedensityisavariable－－密度是變量．＊Anotherpivotalaspectofhigh-speedcompressibleflowisenergy.Ahigh-speedflowisahighenergyflow.－－是一個(gè)高能量的流動(dòng)．＊Energytransformationandtemperaturechangesareimportantconsiderations.－－必須考慮能量轉(zhuǎn)換與溫度變化．

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７.2ABRIEFREVIEWOFTHERMODYNAMICS

(熱力學(xué)簡(jiǎn)要復(fù)習(xí)）7.2.1Perfectgas(完全氣體)定義：Agasinwhichtheintermolecularforcesareneglectedisdefinedasaperfectgas.

(忽略分子間作用力的氣體定義為完全氣體）完全氣體滿足狀態(tài)方程：（equationofstate)

(7.1)(7.2)R為氣體常數(shù)（specificgasconstant)R=287J/(kg·K)7.2.2InternalEnergyandEnthalpy(內(nèi)能和焓）Theenergyofagivenmoleculeisthesumofitstranslational,rotational,vibrational,andelectronicenergies.

（一個(gè)給定分子的能量是其平動(dòng)動(dòng)能、轉(zhuǎn)動(dòng)動(dòng)能、振動(dòng)動(dòng)能和電子能的總和。）對(duì)于由大量分子組成的給定體積的氣體，所有分子所具有的能量的總和稱為氣體的內(nèi)能。單位質(zhì)量氣體的內(nèi)能稱為氣體的比內(nèi)能。（Theinternalenergyperunitmassofgasisdefinedasthespecificinternalenergy.)用e表示．與比內(nèi)能e相聯(lián)系的另一個(gè)量為比焓h，定義為：(7.3)對(duì)于完全氣體，e和h都只是溫度的函數(shù)：－－定容比熱－－定壓比熱Specificheatsatconstant

volumeSpecificheatatconstantpressureT<1000Ｋ，氣體為量熱完全氣體（caloricallyperfectgas),

、為常數(shù)注意：e和h均為熱狀態(tài)變量（thermodynamicstatevariables),它們只依賴與氣體的狀態(tài)而與過程無(wú)關(guān)．（theydependonlyonthestateofthegasandareindependentofanyprocess)(7.6b)-(7.6a):(7.7)定義比熱比(7.９)(7.１０)7.2.3FirstLawofThermodynamics（熱力學(xué)第一定律）系統(tǒng)－system環(huán)境-surroundings邊界-boundary：外界通過邊界加于系統(tǒng)的熱增量：外界對(duì)系統(tǒng)的做的功（7.11)要使系統(tǒng)產(chǎn)生內(nèi)能增量de，有無(wú)數(shù)多種對(duì)系統(tǒng)做功和給系統(tǒng)加熱的方式（過程）．--Foragivende,thereareingeneralaninfinitenumberofdifferentways(processes)bywhichheatcanbeaddedandworkdoneonthesystem.Assumethissystemisstationary:(假定系統(tǒng)為靜止的)我們主要考慮三個(gè)常見過程：１．Adiabaticprocess

(絕熱過程）

Aprocessduringwhichthereisnoheattransfer.

在過程中沒有熱傳導(dǎo)。２．Reversibleprocess(可逆過程）

Nodissipativephenomenaoccur.沒有耗散現(xiàn)象發(fā)生.

３．Isentropicprocess(等熵過程）絕熱、可逆。對(duì)于可逆過程，有：(7.12)熱力學(xué)第一定律小結(jié):熱力學(xué)第一定律就是能量守恒原理的一種表達(dá)，說明能量既不能產(chǎn)生也不能消失，只能從一種形式轉(zhuǎn)換為另一種形式。

系統(tǒng)與外界的能量交換可以通過加熱(heat)、做功（work)、質(zhì)量流（massflow)實(shí)現(xiàn)。對(duì)于我們研究的封閉系統(tǒng)（closedsystem)我們不考慮質(zhì)量流。Heatisdefinedastheformofenergythatistransferredbetweenasystemanditssurroundingsbyvirtueofatemperaturedifference.(熱量是一種能量，它是由于系統(tǒng)與外界的溫度差而引起。）heattransfer：Temperatureisthedrivingforceforheattransfer.Thelargerthetemperaturedifference,thehigheristherateofheattransfer.(溫度差是熱傳導(dǎo)的驅(qū)動(dòng)力)熱傳遞的形式：Conduction(傳導(dǎo)）,Convection（對(duì)流）,Radiation（輻射）.Work:Workistheenergytransferassociatedwithaforceactingthroughadistance.（功是力與力的作用距離相聯(lián)系的能量形式）加熱和做功的共同特征：1、都通過系統(tǒng)邊界作用于系統(tǒng)。2、系統(tǒng)擁有能量，而不是熱和功。3、熱和功都和過程相聯(lián)系，而不是狀態(tài)。4、它們都是路徑的函數(shù)。（即它們的大小依賴于過程和狀態(tài)）。*可逆過程的理解：可逆過程的定義：對(duì)于一個(gè)給定的系統(tǒng)和它的外界，如果它們完全可以由終止?fàn)顟B(tài)回到他們的初始狀態(tài)，這樣的過程就是可逆過程。（Ifaprocesscanbereversedsothatthesystemofparticlesanditssurroundingsarerestoredsothatthesysteminallrespectstotheirinitialconditions,theexchangeprocessissaidtobereversible.)熱力學(xué)研究的是平衡的系統(tǒng)。處于平衡狀態(tài)的系統(tǒng)內(nèi)沒有力的不平衡，沒有溫度的梯度，沒有質(zhì)量的擴(kuò)散?？赡孢^程舉例：1、無(wú)摩擦的擺錘2、氣體的準(zhǔn)平衡膨脹與壓縮*不可逆過程：舉例:不可逆現(xiàn)象7.2.4EntropyandtheSecondLawofThermodynamics(熵及熱力學(xué)第二定律)熱力學(xué)第一定律解決了能量在一個(gè)過程中的守恒問題。熱力學(xué)第二定律則要解決過程會(huì)向哪個(gè)方向進(jìn)行的問題，表明能量既有“質(zhì)”（quality)也有“量”（quantity)。

Aprocesscannottakeplaceunlessitsatisfiesboththefirstandsecondlaw.（一個(gè)過程只能向同時(shí)滿足熱力學(xué)第一定律和第二定律的方向進(jìn)行。）

PROCESS1stlaw2ndlawThesecondlawleadstothedefinitionofanewpropertycalledentropy.

（熱力學(xué)第二定律引入了一個(gè)新性質(zhì)，熵）

Letusdefineanewstatevariable,theentropys,asfollows:

（7.13）

whereisanincrementalamountofheataddedreversiblytothesystemandT

isthesystemtemperature.(為可逆地加于系統(tǒng)的熱增量，T

為系統(tǒng)的溫度。)

Thequantityisjustanartifice(人為的量）whichcanalwaysbeassignedtorelatetheinitialandendpointsofanirreversibleprocess,wheretheactualamountofheataddedis

（實(shí)際的加熱量）

Hence:

(7.14)

isthegenerationofentropyduetotheirreversible,dissipativephenomena

Thesedissipativephenomenaalwaysincreasetheentropys:

(7.15)Theequalssigndenotesareversibleprocess

（等號(hào)表示可逆過程）Therefore:(7.16)Iftheprocessisadiabatic,then

(7.17)

Thisisthesecondlawofthermodynamicsthattellsinwhatdirectionaprocesswilltakeplace:

（熱力學(xué)第二定律指明過程進(jìn)行的方向）

Theentropyofthesystemanditssurroundingsalwaysincreasesor,atbest,staysthesame.

(系統(tǒng)和其環(huán)境的熵總是增加的或不變的）

Insummary,theconceptofentropyincombinationwiththesecondlawallowsustopredictthedirectionthatnaturetakes.(熵與熱力學(xué)第二定律相結(jié)合，使我們能預(yù)計(jì)過程進(jìn)行的方向）Practicalcalculationofentropy(熵的實(shí)際計(jì)算）

Ifheadisaddedreversibly,then:and（7.18)Fromthedefinitionofenthalpy(7.19)and,bycombiningtheseequations:

(7.20)Foraperfectgas,andSubstitutinginprecedentrelations,oneobtain(7.21)(7.22)Withtheequationofstateor

wegetaconvenientformforintegration

Considerathermodynamicprocesswithinitialstate1andendstate2,theintegrationleadsto:

Foracaloricallyperfectgas,bothRandcp

areconstant;hence:

(7.25)

Inasimilarfashion,weobtain:(7.26)Notethatsisafunctionoftwothermodynamicstatevariables;（熵是兩個(gè)熱力學(xué)狀態(tài)參數(shù)的函數(shù)）e.g.

or

7.2.5Isentropicrelations(等熵關(guān)系式）Foranadiabaticprocess,Andforanisentropicprocess,Hence:or(7.28)(7.27)Inasimilarfashion(7.29)(7.30)(7.31)Equation(7.32)isveryimportant;itrelatespressure,density,andtemperatureforanisentropicprocess.方程（7.32)非常重要，它將等熵過程中的壓強(qiáng)、密度、溫度聯(lián)系起來Whyistheseequationsoimportant?Anisentropicprocessseemssorestrictive,requiringbothadiabaticand

reversibleconditions?

Becausealargenumberofpracticalcompressibleflowscanbeassumedtobeisentropic（絕大多數(shù)實(shí)際流動(dòng)問題可以被假設(shè)為等熵的）:validforacaloricallyperfectgas,(量熱完全氣體)andtheflowoutsidetheboundarylayerwhichissmallcomparedtotheentireflowfield(附面層之外的絕大多數(shù)流動(dòng))(7.32)Conceptofbarotropic:正壓流的概念滿足密度是壓力的唯一函數(shù)的流動(dòng),稱為正壓流.即(3.12)1.不可壓流2.等熵流完全氣體內(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.)Theamountbywhichasubstancecanbecompressedisgivenbyaspecificpropertyofthesubstancecalledthe

compressibilty,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) Iftheprocesstakesplace

isentropically,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,incompressibleflow,theprimarydependentvariablesarethepressurepand

thevelocity.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,incompressibleflows.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,incompressibleflowsobeypurelymechanicallawsanddonotneedthermodynamicconsiderations.

Incontrast,forcompressibleflow,isvariableandbecomesanunknown.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,thenTalsobecomesanimportantvariable. 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.Internalenergyfora

caloricallyperfectgas: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。Nowimaginethatyougrabholdofthefluidelementand

adiabaticallyslowitdowntozerovelocity.Clearly,youwouldexpect(correctly)thatthevaluesofwouldchangeastheelementisbroughttorest.Inparticular,thevalueofthetemperatureofthefluidelementafterithasbeenbroughttorest

adiabaticallyisdefinedasthetotaltemperature,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(gè)流場(chǎng)中為常數(shù)，等于自由來流對(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,letusslowitdownboth

adiabaticallyandreversibly.Thatis,letusslowthefluidelementdowntozerovelocity

isentropically.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

TotaltemperatureT0andtotalenthalpyh0aredefinedasthepropertiesthatwouldexistiftheflowisslowedtozerovelocityadiaba