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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 ParticleMotion3/8/202327AE 301 Aerodynamics IThermodynamics Thermodynamics is the study of the way energy Thermodynamics is the study of the way energy is stored and e
4、xchanged.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 its
5、 boundaries,Q,or by work done on the system,Q,or by work done on the system,W.W.(per unit mass)3/8/202328AE 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,vi
6、bration,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,WW,and Heat added,and Heat added,Q Q The symbol The symbo
7、l 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 mass o
8、f fluid the work done and heat added per unit mass of fluid are then are then w w and and q q.3/8/202329AE 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 squeezing
9、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 becomesdrpA3/8/202330AE 301 Aerodynamics IThermo.-Enthalpy Before proceeding,we must introdu
10、ce 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 sum of
11、 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.3/8/202331AE 301 Aerodynamics IT
12、hermo.-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 enthalpy:t
13、erms 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!3/8/202332AE 301 Aerodynamics IThermo.-Specific Heats The factors relating cha
14、nges 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 conditions un
15、der 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:0thus3/8/202333AE 301 Aerodynamics IThermo.-Specific Heats(continued)At constant pressure,p=co
16、nstantAt 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,whether
17、 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!0thus3/8/202334AE
18、 301 Aerodynamics IThermo.-Specific Heats(continued)FutherFuther assumptions 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 Ts.Now neg
19、lect high Ts where we have to worry about Now neglect high Ts where we have to worry about vibrational vibrational excitation and dissociation of Nexcitation 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 air is th
20、ermally perfect,or that cthermally perfect,or that cp p and and c cv v are constantsare 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=0.3/8/202335A
21、E 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 q=0 or=0 or adiabaticadiabatic A reasonable assumption;we dont often try to heat or A reasonable assumption;we dont often try to he
22、at 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 temperature.the flui
23、d 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 abrupt pro
24、perty changes.And also that there are not abrupt property changes.Abrupt changes induce dissipative losses.Abrupt changes induce dissipative losses.3/8/202336AE 301 Aerodynamics IThermo.-Isentropic flow A flow which is adiabatic A flow which is adiabatic andand reversible is called reversible is cal
25、led 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 Practically,thi
26、s 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:003/8/202337AE 301 Aerodynam
27、ics 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,2oror3/8/202338AE 301
28、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 byor3/8/202339AE 301 Aerodynamics ITher
29、mo.-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 1.1 for complex,for complex,polyatomic polyatomic gases gasesSince:Since:oror From this we can get the relations:From this we
30、can get the relations:3/8/202340AE 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 flow:Start with the
31、 First Law for adiabatic flow:Replace Replace dp dp by using by using Eulers Eulers equation,equation,dpdp=-=-VdVVdV Now,integrate to get:Now,integrate to get:3/8/202341AE 301 Aerodynamics IThermo.-Energy Equation(continued)Alternate forms of the energy equationAlternate forms of the energy equation
32、 Note the resemblance to Bernoullis Note the resemblance to Bernoullis EqnEqn.3/8/202342AE 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 ratio of flow velocit
33、y 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 change!3/8/202343AE 301
34、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 M 0.7 0.7Significan
35、t 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 supersonic flow-aero hea
36、ting 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 effects3/8/202344AE 301 Aerodynamics IMach Relations The total temperature,TThe total temperature,T0 0,is the temp
37、erature a,is the temperature a flow has if brought flow has if brought adiabatically adiabatically to rest: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!03/8/202345AE 301 Aerodynamic
38、s 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 isentropicallyisentropically to rest:to rest
39、:3/8/202346AE 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:thusthus3/8/202347AE 301 Aerodynamics I