ChaPteR 8 NORMAL SHOCK WAVES AND RELATED TOPIcs 第八章正激波及有关问题 Shock wave: A large-amplitude compression wave, such as that produced by an explosion, caused by supersonic motion of a body in a medium.激波是一个大振幅波,如由爆炸产生的波及物体在个质中 超音速运动而引起的波 8.1 INTRODUCTION The purpose of this chapter and Chap 9 is to develop shock-wave theory, thus giving us the means to calculate the changes in the flow properties across a wave 本章和第九章的目的是推导激波理论因而得出计算通过激波的 流动特性变化量的公式
CHAPTER 8 NORMAL SHOCK WAVES AND RELATED TOPICS 第八章 正激波及有关问题 Shock wave: A large-amplitude compression wave, such as that produced by an explosion, caused by supersonic motion of a body in a medium. 激波是一个大振幅波,如由爆炸产生的波及物体在介质中 超音速运动而引起的波. 8.1 INTRODUCTION The purpose of this chapter and Chap.9 is to develop shock-wave theory, thus giving us the means to calculate the changes in the flow properties across a wave. 本章和第九章的目的是推导激波理论,因而得出计算通过激波的 流动特性变化量的公式
The focus of this chapter is on normal shock waves, as sketched in Fig 8. 1. The supersonic flow over a blunt body and the supersonic flow established inside a nozzle are shown in Fig. 8.1 This portion of the bow shock is normal to the ∥ow Normal shock inside the nozzle M>1M Flow over a Overexpanded flow blunt bo through a nozzle FIGURE 8.1 Two examples where normal shock waves are of interes
The focus of this chapter is on normal shock waves, as sketched in Fig. 8.1. The supersonic flow over a blunt body and the supersonic flow established inside a nozzle are shown in Fig. 8.1
第八章的路线图: Derivation oi the basi mal shock equations def sau ot s ound S nal torms ot the energy equation when is a flow compressible? Derivation of de tailed equatons for the calculation of changes ucross d normal shock wave: discussion of physical trends Compressible airspeed measurements by means of a Pitot tube FIGURE 8.2 Road map for Chap. 8
第八章的路线图:
正激波基本控制方程的推导 速 能量方程的特殊形式 什么情况下流动是可压缩的? 用于计算通过正激波气体特性变化的方 程的详细推导;物理特性变化趋势的讨论 用皮托管测量可压缩流的流动速度 图82第八章路线图
正激波基本控制方程的推导 音速 能量方程的特殊形式 什么情况下流动是可压缩的? 用于计算通过正激波气体特性变化的方 程的详细推导; 物理特性变化趋势的讨论 用皮托管测量可压缩流的流动速度 图8.2 第八章路线图
8.2 THE BASIC NORMAL SHOCK EQUATIONS 正激波的基本方程 P p M,1◎‖③!M3 2 x direction u Unknown conditions Po.J P0,2 behind the 0, 2 wave 70. 02 The shock wave is a thin region of highly viscous flow. The tiow hrough the shock is adiabatic FIGURE 8.3 but nonisentropic sketch of a normal wave 激波是很薄的、具有强粘性的区域。通 过激波流动是绝热的但不等熵
8.2 THE BASIC NORMAL SHOCK EQUATIONS 正激波的基本方程 激波是很薄的、具有强粘性的区域。通 过激波流动是绝热的但不等熵
Consider the rectangular control volume abcd given by the dashed line in Fig8.3. The shock wave is inside the control volume 考虑矩形控制体abcd如图8.3虚线所示,激波在控制体内 We apply the integral form of conservation equations to this control volume.我们对这个控制体应用积分形式动量方程。 n the process, we observe four important physical facts about the flow given in Fig8.3:在进行过程中,注意图8.3给出的四个 重要物理事实: The flow is stead,ie.o/ar=0.流动是定常的。 2.The flow is adiabatic: no heat is added or taken away from the control volume.流动是绝热的,没有加入和带出控制体的热 里 3.There are no viscous effects on the sides of the control volume 控制体的边界上没有粘性的作用 4 There are no body forces;f=0。没有体积力
Consider the rectangular control volume abcd given by the dashed line in Fig.8.3. The shock wave is inside the control volume. 考虑矩形控制体abcd如图8.3虚线所示，激波在控制体内。 We apply the integral form of conservation equations to this control volume. 我们对这个控制体应用积分形式动量方程。 In the process, we observe four important physical facts about the flow given in Fig. 8.3: 在进行过程中，注意图8.3给出的四个 重要物理事实： 1.The flow is steady, i.e. ∂/∂t = 0. 流动是定常的。 2.The flow is adiabatic: no heat is added or taken away from the control volume. 流动是绝热的，没有加入和带出控制体的热 量。 3.There are no viscous effects on the sides of the control volume. 控制体的边界上没有粘性的作用。 4.There are no body forces; f=0。没有体积力
连续方程: pV ds=0 (8.1) (8.2) 动量方程: f(pF△)=-jp (8.3) ⅹ方向分量: (四dS)=,(ps),(84 P1+1l1=p2+P2l2 (86)
连续方程： = S V dS 0 (8.1) 1 u 1 = 2 u 2 (8.2) 动量方程： ( ) = − s s V dS V p dS (8.3) x方向分量： ( ) ( ) = − s x s V dS u p dS (8.4) 2 2 2 2 2 p 1 + 1 u 1 = p + u (8.6)
能量方程: ple+ ds=-Hpv ds (8.7) 2 2 h2+ (8.10)
= − + s S V ds pV dS V e 22 能量方程： (8.7) 2 2 2 2 2 2 1 1 u h u h + = + (8.10)
Repeating the above results for clarity the basic normal shock equations are:为了清晰起见,我们重复写出正激波基本方程: 连续 11=p2 (8.2) 动量 p1+P141=p2+2l2 (86 能量:h1+1=h2 ×3 (8.10) 2 2 焓 T 状态方程:P2=P2RT2
Repeating the above results for clarity, the basic normal shock equations are: 为了清晰起见，我们重复写出正激波基本方程： 1 u1 = 2 u2 2 2 2 2 2 p1 + 1 u1 = p + u 2 2 2 2 2 2 1 1 u h u h + = + 连续： 动量： 能量： 焓： 状态方程： (8.2) (8.6) (8.10) p2 = 2 RT2 h2 = c p T
Discussion Finally, we note that eqs. (8.2 ), (8.6),( 8.10)are not limited to normal shock waves they describe the changes that take place in any steady, adiabatic, inviscid flow where only one direction is involved That is in Fig. 8.3 the flow is in the x direction only. This type of flow, where the flow-field variables are functions of x only, Ip=plx), u=ulr),etc. is defined as one dimensional flow. Thus, Eqs. (8.2),(8.6), (8.10)are governing equations for one-dimensional, steady adiabatic, inviscid flow 讨论:最后,我们应注意方程(8.2)(86(8.10)并不只适用 于正激波,他们描述了只包含一个方向的定常、绝热、无粘流 动。在图8.3中,流动只沿x方向进行。这种类型的流动被定 义为一维流动,其流场变量只是x的函数[p=p(x),u=l(x,等 等]。因此,方程(8.2)、(86)、8,10)是一维、定常、绝热、无 粘流动的控制方程
Discussion: Finally, we note that Eqs. (8.2),(8.6),(8.10) are not limited to normal shock waves; they describe the changes that take place in any steady, adiabatic, inviscid flow where only one direction is involved. That is, in Fig. 8.3 the flow is in the x direction only . This type of flow, where the flow-field variables are functions of x only, [p= p(x), u=u(x),etc.], is defined as onedimensional flow. Thus, Eqs. (8.2),(8.6),(8.10) are governing equations for one-dimensional, steady, adiabatic, inviscid flow. 讨论： 最后，我们应注意,方程(8.2),(8.6),(8.10) 并不只适用 于正激波,他们描述了只包含一个方向的定常、绝热、无粘流 动。在图8.3中，流动只沿x方向进行。 这种类型的流动被定 义为一维流动，其流场变量只是x的函数[p= p(x), u=u(x),等 等]。因此，方程(8.2),(8.6),(8.10) 是一维、定常、绝热、无 粘流动的控制方程