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1、High Speed Flows At high speeds,the kinetic energy of a fluid flow At high speeds,the kinetic energy of a fluid flow becomes significant compared to the random becomes significant compared to the random kinetic energy of the particles themselves.kinetic energy of the particles themselves.As a result
2、,changes in flow velocity can produce As a result,changes in flow velocity can produce changes in the fluid internal energy and thus its changes in the fluid internal energy and thus its temperature and density.temperature and density.To account for this energy conversion between To account for this
3、 energy conversion between random and flow KE,we use THERMODYNAMICSrandom and flow KE,we use THERMODYNAMICSDirected FlowMotionRandom ParticleMotion11/5/202227AE 301 Aerodynamics IThermodynamics Thermodynamics is the study of the way energy Thermodynamics is the study of the way energy is stored and
4、exchanged.is stored and exchanged.A fundamental principle is the A fundamental principle is the First LawFirst Law:The internal energy,E,of a closed system can be The internal energy,E,of a closed system can be increased only by heat added across its boundaries,increased only by heat added across it
5、s boundaries,Q,or by work done on the system,Q,or by work done on the system,W.W.(per unit mass)11/5/202228AE 301 Aerodynamics IThermo.-First Law Internal energy,EInternal energy,E The energy stored in the particles themselves,I.e.The energy stored in the particles themselves,I.e.random KE,rotation,
6、vibration,chemical bonding,etc.random KE,rotation,vibration,chemical bonding,etc.e is specific internal energy,E/m,and de represents a e is specific internal energy,E/m,and de represents a small change in e.small change in e.Work done,Work done,W,and Heat added,W,and Heat added,Q Q The symbol The sy
7、mbol represents a small incremental process represents a small incremental process since both W and Q are methods of exchanging energy since both W and Q are methods of exchanging energy and not fluid properties themselves.and not fluid properties themselves.the work done and heat added per unit mas
8、s of fluid the work done and heat added per unit mass of fluid are then are then w and w and q.q.11/5/202229AE 301 Aerodynamics IThermo.-Work Only one work process is of interest in Only one work process is of interest in aerodynamics:the work done by squeezing a aerodynamics:the work done by squeez
9、ing a fluid element against the resistance of pressure.fluid element against the resistance of pressure.Consider squeezing a sphere:Consider squeezing a sphere:thusthus And the first law becomesAnd the first law becomesdrpA11/5/202230AE 301 Aerodynamics IThermo.-Enthalpy Before proceeding,we must in
10、troduce a new Before proceeding,we must introduce a new fluid property,Enthalpy,given by the symbol Hfluid property,Enthalpy,given by the symbol H Mathematically,Mathematically,Or,more commonly,the specific enthalpy is Or,more commonly,the specific enthalpy is defined by:defined by:Enthalpy is the s
11、um of the internal energy and Enthalpy is the sum of the internal energy and the energy associated with having brought all the the energy associated with having brought all the particles together into a given volume of space.particles together into a given volume of space.11/5/202231AE 301 Aerodynam
12、ics IThermo.-Enthalpy(continued)Small changes in specific internal energy and Small changes in specific internal energy and specific enthalpy are related by:specific enthalpy are related by:Therefore,the First Law can be rewritten in Therefore,the First Law can be rewritten in terms of specific enth
13、alpy:terms of specific enthalpy:Note:Enthalpy is much more useful in Note:Enthalpy is much more useful in aerodynamics than internal energy is.aerodynamics than internal energy is.Get Get comfortable with it!comfortable with it!11/5/202232AE 301 Aerodynamics IThermo.-Specific Heats The factors relat
14、ing changes in temperature to The factors relating changes in temperature to the amount of heat added are called the the amount of heat added are called the Specific HeatsSpecific Heats Because heat addition is a process,we must also Because heat addition is a process,we must also specify the condit
15、ions under which heat is addedspecify the conditions under which heat is added At constant volume,V=constant(At constant volume,V=constant(=constant)=constant)From the First Law,however:From the First Law,however:0thus11/5/202233AE 301 Aerodynamics IThermo.-Specific Heats(continued)At constant press
16、ure,p=constantAt constant pressure,p=constant From the First Law,however:From the First Law,however:A little note:A little note:We use the above relations as definitions of cWe use the above relations as definitions of c and c and cp p.Thus,they are valid Thus,they are valid all the timeall the time
17、,whether or not heat,whether or not heat is actually added!is actually added!This is important because heat addition is not the only This is important because heat addition is not the only way we can change the temperature and thus e and h!way we can change the temperature and thus e and h!0thus11/5
18、/202234AE 301 Aerodynamics IThermo.-Specific Heats(continued)Futher assumptionsFuther assumptions Already assumed that air is a perfect gas under most Already assumed that air is a perfect gas under most conditions,I.e.except for very high conditions,I.e.except for very high s s and low Ts.and low T
19、s.Now neglect high Ts where we have to worry about Now neglect high Ts where we have to worry about vibrational excitation and dissociation of Nvibrational excitation and dissociation of N2 2 and O and O2 2.With these restrictions,we can assume that air is With these restrictions,we can assume that
20、air is thermally perfect,or that cthermally perfect,or that cp p and c and cv v are constants are constants The previous relations can then be integrated to getThe previous relations can then be integrated to get For convenience,we have set e=h=0 when T=0.For convenience,we have set e=h=0 when T=0.1
21、1/5/202235AE 301 Aerodynamics IThermo.-Heat addition Now consider heat addition to our systemNow consider heat addition to our system Lets make this simple Lets make this simple q=0 or q=0 or adiabaticadiabatic A reasonable assumption;we dont often try to heat or A reasonable assumption;we dont ofte
22、n try to heat or cool the air around an airplane!cool the air around an airplane!This This does notdoes not mean the temperature remains mean the temperature remains constant-doing work can still change the energy of constant-doing work can still change the energy of the fluid and thus its temperatu
23、re.the fluid and thus its temperature.Also,lets assume the flow is Also,lets assume the flow is reversiblereversible Thus,no friction-a reasonable assumption Thus,no friction-a reasonable assumption everywhere but near the skin surface.everywhere but near the skin surface.And also that there are not
24、 abrupt property changes.And also that there are not abrupt property changes.Abrupt changes induce dissipative losses.Abrupt changes induce dissipative losses.11/5/202236AE 301 Aerodynamics IThermo.-Isentropic flow A flow which is adiabatic A flow which is adiabatic andand reversible is called rever
25、sible is called isentropic.isentropic.Literally,this means entropy,S,is a constant.But we Literally,this means entropy,S,is a constant.But we dont want to have to deal with entropy quite yet.dont want to have to deal with entropy quite yet.Practically,this means that some special relations exist Pra
26、ctically,this means that some special relations exist between our fluid properties.between our fluid properties.To see this,start with the energy equations:To see this,start with the energy equations:include the definitions of specific heats:include the definitions of specific heats:0011/5/202237AE
27、301 Aerodynamics IThermo.-Isentropic flow(continued)rearrange and dividerearrange and divide where where =c=cp p/c/c =ratio of specific heats=ratio of specific heats Now,integrate over the change from one condition,1,Now,integrate over the change from one condition,1,to another,2to another,2oror11/5
28、/202238AE 301 Aerodynamics IThermo.-Isentropic flow(continued)We can also use the perfect gas law to introduce TWe can also use the perfect gas law to introduce T To summarize,for isentropic flow,r,p and T are To summarize,for isentropic flow,r,p and T are related byrelated byor11/5/202239AE 301 Aer
29、odynamics IThermo.-Ratio of Specific HeatTypical values:Typical values:=5/3=1.67 for monatomic gases=5/3=1.67 for monatomic gases =7/5=1.4 for diatomic gases=7/5=1.4 for diatomic gases 1.1 for complex,polyatomic gases 1.1 for complex,polyatomic gasesSince:Since:oror From this we can get the relation
30、s:From this we can get the relations:11/5/202240AE 301 Aerodynamics IThermo.-Energy Equation Finally,lets use our thermodynamics to Finally,lets use our thermodynamics to formulate a conservation of energy equationformulate a conservation of energy equation Start with the First Law for adiabatic flo
31、w:Start with the First Law for adiabatic flow:Replace dp by using Eulers equation,dp=-Replace dp by using Eulers equation,dp=-VdVVdV Now,integrate to get:Now,integrate to get:11/5/202241AE 301 Aerodynamics IThermo.-Energy Equation(continued)Alternate forms of the energy equationAlternate forms of th
32、e energy equation Note the resemblance to Bernoullis Eqn.Note the resemblance to Bernoullis Eqn.11/5/202242AE 301 Aerodynamics ISpeed of Sound Derived as example in class.Derived as example in class.Final result for the speed of sound,a:Final result for the speed of sound,a:The Mach number is the ra
33、tio of flow velocity to The Mach number is the ratio of flow velocity to that of the speed of sound:M=V/athat of the speed of sound:M=V/a Note that both M and a will vary around a body Note that both M and a will vary around a body as velocity and temperature change!as velocity and temperature chang
34、e!11/5/202243AE 301 Aerodynamics ISpeed Regimes We define the flight speed regimes by:We define the flight speed regimes by:Incompressible Subsonic:Incompressible Subsonic:M M 0.3 0.3Insignificant density variationsInsignificant density variations Compressible Subsonic:Compressible Subsonic:0.3 M0.3
35、 M 0.7 0.7Significant density variations,but local flow is always Significant density variations,but local flow is always subsonicsubsonic Transonic:Transonic:0.7 M0.7 M 1.2 1.2Mixture of subsonic and supersonic flowMixture of subsonic and supersonic flow Supersonic:Supersonic:1.2 M1.2 M 5 5All supe
36、rsonic flow-aero heating is manageableAll supersonic flow-aero heating is manageable Hypersonic:Hypersonic:5 M 5 M Aero heating is excessive and/or real gas effectsAero heating is excessive and/or real gas effects11/5/202244AE 301 Aerodynamics IMach Relations The total temperature,TThe total tempera
37、ture,T0 0,is the temperature a,is the temperature a flow has if brought adiabatically to rest:flow has if brought adiabatically to rest:And introducing the Mach numberAnd introducing the Mach number T T0 0 is a measure of the total energy in a flow!is a measure of the total energy in a flow!011/5/20
38、2245AE 301 Aerodynamics IMach Relations(continued)Also,we can define the total pressure and total Also,we can define the total pressure and total density as those conditions we would have if the density as those conditions we would have if the flow was brought flow was brought isentropicallyisentrop
39、ically to rest:to rest:11/5/202246AE 301 Aerodynamics IMach Relations(continued)Special note:Special note:The dynamic pressure is used in compressible flow as The dynamic pressure is used in compressible flow as much as incompressible flow.much as incompressible flow.A simpler way to evaluate exists,however:A simpler way to evaluate exists,however:thusthus11/5/202247AE 301 Aerodynamics I