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
8.4 SPECIAL FORMS OF THE ENERGY EQUATIONS 能量方程的特殊形式 本节需要掌握的内容要点: 能量方程的各种特殊表达形式 总温的计算公式 总压、总密度的计算公式 临界参数的定义与计算公式 特征马赫数(速度系数)M的定义及计算公式
8.4 SPECIAL FORMS OF THE ENERGY EQUATIONS 能量方程的特殊形式 本节需要掌握的内容要点： • 能量方程的各种特殊表达形式 • 总温的计算公式 • 总压、总密度的计算公式 • 临界参数的定义与计算公式 • 特征马赫数（速度系数）M*的定义及计算公式
能量方程的各种特殊表达形式 在75节中我们得到了定常、绝热、无粘流动的能量方程: V=h2+2 2 (828) 其中V1、V2条三维流线上的任意两点的速度。对于 我们现在研究的一维流动,能量方程为: (8.29) 2 2 However, keep in mind that all the subsequent results in this section hold in general along a streamline and are by no means limited to just one- dimensional flows.然而,应当记住的是: 这一节中所有的结论对于一般的沿流线的问题都适用,并不 只是局限于一维流动
2 2 2 2 2 2 1 1 V h V h + = + 2 2 2 2 2 2 1 1 u h u h + = + • 能量方程的各种特殊表达形式 在7.5节中我们得到了定常、绝热、无粘流动的能量方程： 其中V1、V2一条三维流线上的任意两点的速度。 对于 我们现在研究的一维流动，能量方程为： （8.28) （8.29) However, keep in mind that all the subsequent results in this section hold in general along a streamline and are by no means limited to just one –dimensional flows. 然而，应当记住的是： 这一节中所有的结论对于一般的沿流线的问题都适用，并不 只是局限于一维流动
以温度表示: l C T+1=C,T2+2 (8.30) 2 T yRT, (8.31) 2y-12 AT 以音速表示: (8.32) 2 2
2 2 2 2 2 2 1 1 u c T u c p T + = p + 1 2 1 2 2 2 2 2 RT1 u1 RT u + − + = − 1 2 1 2 2 2 2 2 2 1 2 a1 u a u + − + = − (8.30) (8.31) a = RT (8.32) 以温度表示： 以音速表示：
Definition of stagnation speed of sound:驻点音速的定义 C (8.33) y 12 (8.34)
1 2 1 2 0 2 2 − + = − a u a 1 2 1 2 1 2 0 2 2 2 2 2 1 2 1 − + = − + = − a u a u a Definition of stagnation speed of sound：驻点音速的定义 (8.33) (8.34)
Definition of a*:a*的定义 7.5节最后一段: As a corollary to the above considerations,we need another defined temperature, denoted by 1, and defined as follows. Consider a point in a subsonic flow where the local static temperature is T. At this point, imagine that the fluid element is speeded up to sonic velocity, adiabatically. The Temperature it would have at such sonic conditions is denoted as T. Similarly, consider a point in a supersonic flow, where the local static temperature is T. At this point, imagine that the fluid element is slowed down to sonic velocity, adiabatically. Again, the Temperature it would have at such sonic conditions is denoted as Ta 用*号表示的变量被称为临界参数a*=√2RT*称为临界音速 分 (y+1)a (8.35) 12 y 22(y-1) In Equation(8.35), a and u are the speed of sound and velocity respectively, at any point of flow, and a* is a characteristic value associated with that same point
2( 1) ( 1) * 1 2 2 2 2 − + + = − a u a 2 * 1 * 1 2 2 2 2 2 a u a a + − + = − (8.35) Definition of a*: a*的定义 7.5节最后一段：As a corollary to the above considerations, we need another defined temperature, denoted by T*, and defined as follows. Consider a point in a subsonic flow where the local static temperature is T. At this point, imagine that the fluid element is speeded up to sonic velocity, adiabatically. The Temperature it would have at such sonic conditions is denoted as T*. Similarly, consider a point in a supersonic flow, where the local static temperature is T. At this point, imagine that the fluid element is slowed down to sonic velocity, adiabatically. Again, the Temperature it would have at such sonic conditions is denoted as T*. 用*号表示的变量被称为临界参数. a* = RT * 称为临界音速. In Equation (8.35), a and u are the speed of sound and velocity, respectively, at any point of flow, and a* is a characteristic value associated with that same point
对于沿一条流线上的任意两点,有 2 (y+1)a* const y-12y-122(y-1) 8.36) y+1 q兴 const 2(y-1) (8.37) y Clearly, these defined quantities, ao and a'*, are both constants along a given in a steady, adiabatic, inviscid flow. If all the streamlines emanate from the same uniform freestream conditions, then ao and a are constants throughout the entire flow field.很明显,a和a*为定义的量,沿定常、绝热、无粘 流动的给定流线为常数。如果所有流线都来自于均匀自由来 流,则ao和a*在整个流场为常数
const a u a u a = − + + = − + + − 2( 1) ( 1) * 1 2 1 2 2 2 2 2 2 2 1 2 1 对于沿一条流线上的任意两点,有: (8.36) const a a = − = − + 1 * 2( 1) 1 2 2 0 (8.37) Clearly, these defined quantities, a0 and a* , are both constants along a given in a steady, adiabatic, inviscid flow. If all the streamlines emanate from the same uniform freestream conditions, then a0 and a* are constants throughout the entire flow field. 很明显, a0 和 a*为定义的量, 沿定常、绝热、无粘 流动的给定流线为常数。如果所有流线都来自于均匀自由来 流，则a0 和 a*在整个流场为常数
总温的计算公式 回忆7.5节中总温7的定义,有方程(830可得: 2 =cT (8.38) 2 Equation(8.38 )provides a formula from which the defined total temperature To can be calculated from the given actual conditions of Tand u at any given points in a general flow field.方程(8.38)给出 了由流场中给定点处的实际温度7和速度计算总温T的计算公 式 1 2+ (8.39) 2 2 p const
0 2 2 c T u c p T + = p c T const u c T u c p T + = p + = p 0 = 2 2 2 2 1 1 2 2 (8.38) • 总温的计算公式 回忆7.5节中总温T0的定义，有方程(8.30)可得： (8.39) Equation (8.38) provides a formula from which the defined total temperature T0 can be calculated from the given actual conditions of T and u at any given points in a general flow field. 方程(8.38) 给出 了由流场中给定点处的实际温度T和速度u计算总温T0的计算公 式
For a calorically perfect gas, the ratio of total temperature to static temperature s a function of Mach number only, as follows:(对于量热完全气体,总温和静温的比是马赫数 的唯一函数,证明如下:) 1+ 2C T 2R(x-D1+ 1+ M (8.40) Equation( 8.40)is very important; it states that only M(and of course, the value of y) dictates the ratio of total temperature to static temperature 方程(840)非常重要;表明只有马赫数(及的值)决定总 温与静温的比
2 2 2 0 ( ) 2 1 1 2 /( 1) 1 2 1 a u RT u c T u T T p − = + − = + = + 0 2 2 1 1 M T T − = + (8.40) Equation (8.40) is very important; it states that only M (and ,of course, the value of ) dictates the ratio of total temperature to static temperature. 方程(8.40)非常重要；表明只有马赫数（及 的值）决定总 温与静温的比。 For a calorically perfect gas, the ratio of total temperature to static temperature, is a function of Mach number only, as follows: （对于量热完全气体，总温和静温的比 是马赫数 的唯一函数，证明如下：） T0 T T0 T
总压、总密度的计算公式: 回忆75节总压和总密度的定义,在定义中包含了将气流速度 等熵地缩为零速度。由(732)式,盘丫(Y 我们有: y/(-1) T (841) (y-1) (842) 2 2)1/(-1) (8.43) 2 方程(842)和(8.43)表明:总压静压比、总密度静密度 比只由M和决定。因此,对于给定气体,即给 定,、只依赖于马赫数
0 2 1 ( 1) 0 2 ( 1) ) 2 1 (1 ) 2 1 (1 − − − = + − = + M M p p ( 1) 0 0 0 − = = T T p p • 总压、总密度的计算公式： 回忆7.5节总压和总密度的定义， 在定义中包含了将气流速度 等熵地压缩为零速度。由（7.32）式， 我们有： ( 1) 1 2 1 2 1 2 − = = T T p p （8.41) （8.42) （8.43) 方程（8.42)和（8.43）表明：总压静压比 、总密度静密度 比 只由M 和 决定。因此，对于给定气体，即给 定 ， 、 只依赖于马赫数。 p0 p 0 p0 p 0