Chapter-7-Inviscid--Compressible-Flow-空气动力学英文课件.ppt

Chapter 7Inviscid,Compressible FlowWith the realization of airplane and missile speeds equal to or even surpassing the many times the speed of sound,thermodynamics has entered the scene and will never again leave our consideration.7.1 Introductionn nGiving the main differences between the Giving the main differences between the incompressible and compressible flows with respect incompressible and compressible flows with respect to aerodynamic properties.to aerodynamic properties.What flow properties should be introduced for the analysis of compressible flow problems.This chapter relates interesting,historical events dating back to the birth of modern aerodynamics;I advice all students to read this chapter carefully.Definition of compressible flows;The position of analysis of compressible flows in our textbook;The change between kinetic energy and internal energy should be taken into account;A High Speed Flow is a High Energy Flow;The change between kinetic energy and internal energy should be considered under the science of thermodynamics;Road map of this chapter7.2 A BRIEF REVIEW OF THERMODYNAMICS 7.2.1 PERFECT GAS A gas is a collection of particles that are in more or less random motion.If these particles are far enough apart,the influence of intermolecular forces can be neglected;this gas is defined as a perfect gas for which p,and T are related through the equation of state:7.2.2 INTERNAL ENERGY AND ENTHALPY Consider a molecule:its velocity and its rotational motion create kinetic energy its vibration creates vibrational energy the motion of electrons around the nuclei creates electronic energy The energy of a given molecule is the sum of these energies.Consider a finite volume of gas consisting of a large number of molecules.The sum of the energies of all the molecules in this volume is defined as the internal energy of the gas.Per unit of mass,it is denoted as .A related quantity is the specific enthalpy,denoted by and defined as:For a perfect gas,both and are functions of temperature only:If and represent differentials of and ,respectively;then,for a perfect gas:and are the specific heats at constant volume and constant pressure,respectively e and h are thermodynamic state variables,they depend only on the state of the gas and are independent of any process.Now,we have the thermodynamic variables as follows:P:pressure :density T:temperature e:internal energy h:enthalpyFor a specific gas,we have the following equations:Define .For air at standard conditions,.Then we get particularly useful equations:oror7.2.3 FIRST LAW OF THERMODYNAMICS Consider a fixed mass of gas called system.The region outside the system is called surroundings,the interface is called boundary.Hence is an exact differential,its value depend only on the initial and final states of the system,is a state variable.In contrast,and depend on the process in going from the initial to the final states.We consider 3 types of processes:1.Adiabatic process:no heat is added or taken away from the system.2.Reversible process:no dissipative phenomena occur;effects of viscosity,thermal conductivity,and mass diffusion are absent.3.Isentropic process:this process is both adiabatic and reversible For a reversible process,where is an incremental change in the volume due to a displacement of the boundary;thus Eq.(7.11)becomes How to add the heat to the system,reversibly?Reversible process:no dissipative phenomena occur;effects of viscosity,thermal conductivity,and mass diffusion are absent.More detailed definition of Reversible processEquilibrium state and quasi equilibrium process平衡状态 与 准平衡过程Entropy is a state variable used in conjunction with any type of process,reversible or irreversible.The quantity in Eq.(7.13)is just an artifice assigned to relate the initial and end points of an irreversible process,where the actual amount of heat added is ;an alternative relation is:In this equation,is the actual amount of heat added to the system during an actual irreversible process,and is the generation of entropy due to the irreversible,dissipative phenomena of viscosity,thermal conductivity and mass diffusion occurring within the system.These dissipative phenomena always increase the entropy:the equals sign denotes a reversible process.Combining Eqs.(7.14)and(7.15),we have Furthermore,if the process is adiabatic,and Eq.(7.16)becomes:The above equationa are forms of the second law of thermodynamics which predicts in what direction a process will take place,that the entropy of the system always increases or,at best,stays the same.Combining Eqs.(7.18)and(7.19)we obtain For a perfect gas,recall Eqs.(7.5a and b)and substituting into Eqs.(7.18)and(7.20),we obtain:Working with Eq.(7.22)and equation of state or into the last term:In a similar fashion,Eq.(7.21)leads to:Note from these equqtions that entropy is a function of two thermodynamic variables.or For such an isentropic process,Eq.(7.25)is written asor However,from Eq.(7.9),and hence Eq.(7.27)is written as In a similar fashion,from Eq.(7.26),we can get:However,from Eq.(7.10),and hence Eq.(7.29)is written as We can summarize the isentropic relations as:In many processes,dissipative effects of viscosity,etc.are very small and can be neglected or large regions of the flow can be assumed isentropic;this shows the importance of Equation above.7.3 DEFINITION OF COMPRESSIBILITYAll real substances are compressible to some greater or lesser extend.When you squeeze or press on them,their density will change.This is particularly true of gases.Qualitative definition of compressibilty:The amount by which a substance can be compressed is given by a specific property of the substance called the compressibilty Quantitative definition of compressibilty:Consider a small element of fluid of volumeThe pressure exerted on the side of the element is .If the pressure is increased by an infinitesimal amount ,the volume will change by a negative amount .With the quantitative definition of compressibilty,which is named as .It is the fractional change in volume of fluid element per unit change in pressure.If the temperature of the fluid element is held constant,then is identified as the isothermal compressibility If the process takes place isentropically,then asthenThus,whenever the fluid experiences a change in pressure,the corresponding change in density is,or to sayComparison of liquid and gas respect to the compressibility liquid:very small;constant;neglectable incompressiblegas :large;not constant;large compressibleException:Low-Speed flowFor low-speed flows,can be assumed to be constant;the low-speed gas flows can be analysed as incompressible flows.This can be considered when the Mach number is smaller than 0.3The speed of sound in a gas is related to the isentropic compressibility,that is 7.4 Governing Equations for Inviscid Compressible FlowFor inviscid,incompressible flow,the primary dependent variables are the pressure p and the velocity .Hence,we need only two basic equations,namely the continuity and the momentum equations.For inviscid and incompressible flows,there exists a stream function and velocity potential.And both stream function and velocity potential satisfy the Laplaces equation,that means the stream function and velocity potential can be superimposed to synthesize more complex inviscid incompressible flows.With the application of Bernoullis equation,the pressure distribution in the flow field or on the solid boy surface can be predicted.Note that both and T are assumed to be constant through out such inviscid and incompressible flow.As a result,no additional governing equations are reqired;in particular,there is no need for the energy equation or energy concept in general.Basically,incompressible flow obeys purely mechanical laws and does not require thermodynamic considerations.In contrast,for compressible flow,is variable and becomes an unknown.Hence we need an additional equation the energy equation which in turn introduces internal energy e as an unknown.Internal energy e is related to temperature,then T also becomes an important variable.Therefore,the 5 primary dependent variables are:To solve for the five variables,we need five governing equations.Review of the basic equations derived in Chapter 2.1.Continuity2.Momentum3.Energy The above continuity,momentum and energy equations are three equations in terms of the five unknowns assume a calorically perfect gas,two additional equations needed to complete the system are obtained4.Equation of state for a perfect gas:5.Internal energy for a calorically perfect gas:We have now 5 equations for 5 unknowns.Final note:we will use both the integral and differential forms of the above equations in our subsequent discussions.Make sure that you all have a fully understanding of the equations.7.5 Definition of Total(or Stagnation)Conditions Static quantities static pressure pStagnation quantities total pressure p0 Static pressure is a measure of the purely random motion of molecules in a gas.The total pressure was defined in Sec.3.4 as the pressure existing at a point(or points)in the flow where V=0.Consider a fluid element passing through a given point in a flow where the local pressure,temperature,density,Mach number,and velocity(local conditions)are p,T,M,and V,respectively.Here,p,T,and are static quantities,i.e.,static pressure,static temperature,static density,respectively.They are measured when you ride on along the with the gas at the local flow velocity.Now imagine that you grab hold of the fluid element and adiabatically slow it down to zero velocity.Clearly,you would expect(correctly)that the values of p,T,and would change as the element is brought to rest.In particular,the value of the temperature of the fluid element after it has been brought to rest adiabatically is defined as the total temperature,denoted by T0 The corresponding value of enthalpy is defined as total enthalpy h0,where h0=cpT0 for a calorically perfect gas.How to define the total temperature?The energy equation,Eq.(7.44),provides some important information about total enthalpy and hence total temperature.Assume that the flow is adiabatic and that body forces are negligible,then the equation of energy can be written as:Expanding by using the following vector identityAnd noting thatSubstituting the continuity equationnoteIf the flow is steady,that is thenFrom the definition of substantial derivative,then the time rate change(h+V2/2)following a moving fluid element is zero:that meansNote:the assumption led to the results above are that,the flow is steady,adiabatic,and inviscid.After the class,please try to locate where these assumptions had been used in the derivation of the equation bellowSince h0 is defined as that enthalpy which would exist at a point if the fluid element were brought to rest adiabatically,where V=0 and hence h=h0,then the value of the constant is h0.so(7.54)Equation(7.54)is important;it states that at any point in a flow,the total enthalpy is given by the sum of the static enthalpy plus the kinetic energy,all per unit of mass.With the definition of total enthalpy,the energy equation can be expressed with total enthalpy.For steady,adiabatic and inviscid flows,Eq.(7.52)can be written as(7.54)(7.48)and(2.99)Keep in mind that the above discussion followed two trains of thought:On the one hand,we dealt with the general concept of an adiabatic flow field which led to Eqs.(7.51)to(7.53)and on the other hand,we dealt with the definition of total enthalpy which led to Eq.(7.54).But,these two trains of thought are really separate and should not be confused.The above equations do not hold for a general non-adiabatic flow,such as a viscous boundary layer with heat transfer.However,the equation bellow hold locally at each point in the flow,because the assumption of an adiabatic flow contained in equation above is through the definition of h0 and has nothing to do with the general overall flow.Now,the definition of total enthalpy and total temperature have been give given.And whats the definition for the total pressure and total density?Return back to the beginning of this section,we consider a fluid element passing through a point in a flow where the local properties are p,T,M,and V.Once again,imagine that you grab hold of the fluid element and slow it down to zero velocity,but this time,let us slow it down both adiabatically and reversibly.That is,let us slow the fluid element down to zero velocity isentropically.When the fluid element is brought to rest isentropically,the resulting pressure and density are defined as the total pressure p0 and total density 0 Since an isentropic process is also adiabatic,the definition of total temperature remains unchanged.As before,keep in mind that we do not have to actually bring the flow to rest in real life in order to talk about total pressure and total density;rather,the are defined quantities that would exist at a point in a flow if(in our imagination)the fluid element passing through that point were brought to rest isentropically.Therefore,at a given point in a flow,where the static pressure and static density are p and,respectively,we can also assign a value of total pressure p0,and total density 0 defined as above.SUMMARYTotal temperature T0 and total enthalpy h0 are defined as the properties that would exist if the flow is slowed to zero velocity adiabatically.Total pressure p0 and total density 0 are defined as the properties that would exist if the flow is slowed to zero velocity isentropically.If the general flow field is adiabatic,h0 is constant throughout the flow.If the general flow field is isentropic,p0 and 0 are constant throughout the flow.7.6 Some Aspects of Supersonic Flow:Shock Waves 超音速流的一些特征：激波 An essential ingredient of a supersonic flow is the calculation of the shape and strength of shock waves.This is the main thrust of chaps.8 and 9.超音速流动研究的一个重要内容就是计算激波的形状和强度。这是第8章和第9章的主题。A shock wave is an extremely thin region,typically on the order of 10-5cm,across which the flow properties can change drastically.激波是一个极其薄的区域，厚度大约只有10-5cm的量级，通过激波流动特性发生剧烈变化。7.7 Summary(小结）1、热力学关系式：状态方程：对于量热完全气体：热力学第一定律的各种表达形式：熵的定义：热力学第二定律：或对于绝热过程，可逆过程：量热完全气体的熵增计算公式：（7.25)(7.26)等熵流动的等熵关系式：(7.32)2、压缩性压缩性的一般定义：（7.33）对于等温过程：(7.34)对于等熵过程：(7.35)3、无粘、可压缩流的控制方程：连续方程：动量方程：（7.39)(7.40)(7.41)(7.41a)(7.41b)(7.41c)能量方程：（7.43)(7.44)对于定常、绝热、无粘流，（7.44)和(7.43)可以写成：（7.53),(7.55)完全气体状态方程：量热完全气体内能：（7.1)(7.6a)4、总温、总焓、总压、总密度的定义及概念：总温和总焓定义为把流体微元（在我们的想象中）绝热地减速为静止时流体微元所对应的温度和焓值。类似地，总压和总密度定义为把流体微元（在我们的想象中）等熵地减速为静止时流体微元所对应的压强和密度。在均匀自由来流的绝热流场中，总焓h0在全流场中为常数，相反，在非绝热流场中，h0随流场点的不同而不同。类似地，在等熵流场中，总压和总密度在整个流场中为常数，相反，在非等熵流场中，总压和总密度随流场点的不同而不同。5、激波激波为超音速流中很薄的一层，通过激波压强、密度、温度和熵增加；马赫数、流动速度、总压降低；总焓、总温不变。