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 m<l: ie. the mach number behind the normal shock wave is subsonic 如果M1>1,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 M2<1; i.e., the Mach number behind the normal shock wave is subsonic. 如果M1>1, 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)