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
8.6 CALCULATION OF NORMAL SHOCK-WAVE PROPERTIES正激波性质的计算 本节要点只有一个 计算通过正激波的流动特性变化 ?P=?2=?2-1=?102=22=? 0.1 问题:已知激波前区域1 227M 的条件,计算激波后区域 2 x direction 2的条件。 Unknown i Po2 conditions ho 2 wave The shock wave is a thin region of highly viscous flow. The flow through the shock is adiabatic FIGURE 8.3 but nonisentro sketch of a normal wave
8.6 CALCULATION OF NORMAL SHOCK-WAVE PROPERTIES 正激波性质的计算 本节要点只有一个: • 计算通过正激波的流动特性变化 即 问题: 已知激波前区域1 的条件，计算激波后区域 2的条件。 ? ? T T ? ? ? ? ? 0,1 0,2 0,1 0,2 2 1 1 2 1 2 1 2 2 = = = = − = = = p p s s T T p p M
复习我们在82节中推导出的正激波基本方程: 11=P2l (8.2) P1+P1=p2+2l (86) 2 h1+=h2+ (8.10) 2 2 h=c t (8.49) p-2 2R7 (8.50) Examining the five equations given above, we see that they involve five unknowns, namely, p2, u2, p2, h, and T2. Hence, Eqs. (8.2 ),(8.6),(8.10) (8.49), and(8.50)are sufficient for determining the properties behind a normal shock wave in a calorically perfect gas. Let us proceed
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 + = + 2 T2 h c = p p2 = RT2 复习我们在8.2节中推导出的正激波基本方程： (8.2) (8.6) (8.10) (8.49) (8.50) Examining the five equations given above, we see that they involve five unknowns, namely, ρ2 ,u2 ,p2 ,h2 ,and T2 . Hence, Eqs.(8.2) , (8.6), (8.10), (8.49),and (8.50) are sufficient for determining the properties behind a normal shock wave in a calorically perfect gas. Let us proceed
2 (86)式除以(8.2)式: +u 1 l1(8.51 P11P2 因为a=√p/: (8.52) 41n 由公式(8.35) (y+1)a* 可得: 122(y-1) 2y+1,*2y-1,2(853) a,2=2y+l C (8.54 2 2
2 2 2 2 1 1 1 1 u u p u u p + = + 2 1 2 2 2 1 1 1 u u u p u p − = − 2 1 2 2 2 1 2 1 u u u a u a − = − 2( 1) ( 1) * 1 2 2 2 2 − + + = − a u a 2 1 2 2 1 2 1 * 2 1 a a u − − + = 2 2 2 2 2 2 1 * 2 1 a a u − − + = (8.6)式除以(8.2)式： (8.51) 因为 ： (8.52) a = p 由公式(8.35): 可得： (8.53) (8.54)
将(853)8.54)式代入(8.52)式 y+1a*2 y+1 2 n y 3 y y+1 2 整理为 1 y r+ 两边同除以2l1 at+ 2 2 2 y Q兴? uu (8.55)
2 2 1 2 2 1 1 2 2 * 1 2 1 2 * 1 2 1 u u u u a u u a = − − + + − − − + 2 1 2 1 2 2 1 1 2 ( ) 2 1 ( ) * 2 1 u u a u u u u u u − = − − − + + 1 2 1 * 2 1 2 1 2 = − + + a u u 1 2 2 a* = u u 将(8.53),(8.54)式代入(8.52)式： 整理为： 两边同除以u2 -u1 : (8.55)
C 米=1L1L (855) Equation( 8.55)is called the Prandtl relation and is a useful intermediate relation for normal shock waves.方程(855)被称为Pzm关系式, 是一个很有用的正激波中间关系式 (855)式还可写成:112 (8.56 C 来刀米 由特征马赫数的定义:M* 可得: 1=M1*M,*
1 2 2 a* = u u (8.55) Equation (8.55) is called the Prandtl relation and is a useful intermediate relation for normal shock waves. 方程(8.55)被称为Prandtl 关系式， 是一个很有用的正激波中间关系式． * * 1 1 2 a u a u (8.55)式还可写成： = (8.56) 由特征马赫数的定义： 可得： * * a u M = 1= M1 *M2 *
(8.57 应用(848)式: M*2 (y+1)M 2+(y-1)M (+1)M2 (y+1)M1 2+(y-1)M2[2+(y-1)M (8.58) 1+(y-1)/2]M1 M2-(y-)/2(859 Equation(8.59)is our first major result for a normal shock wave Examine Eq. (8.59)closely; it states that the Mach number behind the wave, M2, is a function only of the Mach number ahead of the wave, MI 方程(8.59)是我们得到的第一个主要正激波关系式,表明波后马赫 数M2是波前马赫数M1的唯一函数
* 1 * 1 2 M M = 2 2 2 2 ( 1) ( 1) * M M M + − + = 1 2 1 2 1 2 2 2 2 2 ( 1) ( 1) 2 ( 1) ( 1) − + − + = + − + M M M M ( 1)/ 2 1 [( 1)/ 2] 2 1 2 2 1 2 − − + − = M M M (8.57) 应用（8.48)式： (8.58) (8.59) Equation (8.59) is our first major result for a normal shock wave. Examine Eq. (8.59) closely; it states that the Mach number behind the wave, M2 , is a function only of the Mach number ahead of the wave, M1 . 方程（8.59)是我们得到的第一个主要正激波关系式，表明波后马赫 数M2是波前马赫数M1的唯一函数．
Moreover, if M=l, then M, -l. This is the case of an infinitely weak normal shock wave. defined as a mach wave 如果M1=1,则M2=1。这种情况对应无限弱的正激波,我们定义 为马赫波。 Furthermore. ifM>l then m1,则M2<1;也就是:正激波后的流动是亚音速的。 As M, increases above l, the normal shock wave becomes stronger and M, becoming progressively less than 1 当M1由1逐渐增大时,正激波越来越强,激波后马赫数M越来越 小(在小于1的范围内) However, in the limit as m1→→∞o,M2 approaches a finite minimum value,M2→√-)/2y, which for air is0.378 然而,当M趋于无穷大,M2趋于一有限的最小值M2→√y-1)/2y, 对于空气其值为0.378
Moreover, if M1=1, then M2=1. This is the case of an infinitely weak normal shock wave, defined as a Mach wave. 如果M1=1，则M2=1。这种情况对应无限弱的正激波，我们定义 为马赫波。 Furthermore, if M1>1, then M21, 则 M2<1；也就是： 正激波后的流动是亚音速的。 As M1 increases above 1, the normal shock wave becomes stronger, and M2 becoming progressively less than 1. 当 M1 由1逐渐增大时，正激波越来越强，激波后马赫数M2越来越 小（在小于1的范围内）。 However, in the limit as M1→∞,M2 approaches a finite minimum value，M2→ , which for air is 0.378. 然而，当M1趋于无穷大,M2趋于一有限的最小值M2→ ， 对于空气其值为0.378。 ( −1) 2 ( −1) 2
下面我们来推导通过正激波的热力学特性即P2/、P2/P1、T/T 的表达式 2l1 M,*2 +1)M1 (8.61) Pl22+(y-1)M12 p2-P1=p11-P2l2=P11(1-2) (862) 2 PB=2(1-42)=22(1-)=m2(1-)(863)
2 2 1 2 1 1 2 2 1 2 1 1 2 * * M a u u u u u u = = = = 2 1 2 1 2 1 1 2 2 ( 1) ( 1) M M u u + − + = = ( ) (1 ) 1 2 2 1 1 1 2 1 1 2 2 2 2 2 1 1 1 u u p − p = u − u = u u −u = u − (1 ) (1 ) (1 ) 1 2 2 1 1 2 2 1 2 1 1 2 1 2 1 1 1 2 1 u u M u u a u u u p u p p p = − = − = − − 下面我们来推导通过正激波的热力学特性,即 、 、 的表达式： 2 1 p2 p1 T2 T1 （8.61） (8.62) (8.63)
P2-Pi=MM?|1 2+(y-1)M2 y+1)M3 (8.64) pI 2=112y (M2-1)(865) p1 y+1 P2‖2 (866) 7(P1八P2 (M12-1) 2+(y-1)M (867) h ,r+ y+1)M
+ + − = − − 2 1 2 2 1 1 1 2 1 ( 1) 2 ( 1) 1 M M M p p p ( 1) 1 2 1 2 1 1 2 − + = + M p p = 2 1 1 2 1 2 p p T T 2 1 2 2 1 1 1 2 1 2 ( 1) 2 ( 1) ( 1) 1 2 1 M M M h h T T + + − − + = = + (8.65) (8.64) (8.66) (8.67)