CHAPTER 8 NORMAL SHOCK WAVES AND RELATED TOPICS2 空气动力学英文课件

u1
p2
2u2
u2
p1
1u1
p2
2u2
u2
u1
a12
u1
a22
u2
的标准大气条件）
• 临界参数的定义与计算公式
临界参数的定义：
Consider a point in a general flow where the velocity is exactly
sonic, i.e. where M=1. Denote the static temperature ,pressure, an
1 2 1 2 1
(8.34)
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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*.
8.5 WHEN IS A FLOW COMPRESSIBLE? 什么条件下流动是可压缩的？
We have stated several times in the preceding chapters the rule of thumb that a flow can be reasonably assumed to be incompressible when M<0.3, whereas it should be considered compressible when M>0.3. Why?
0
T0
T
(1)
（8.41)
p0 (1 1 M 2 ) ( 1)
p
2
（8.42)
0 (1 1 M 2 )1 ( 1)
2
（8.43)
方比定程20200（/，6/187 .4只2p)0由和Mp（、8和.430）表决明只定：依。总赖因压于此静马，压赫对比数于p。给0 定p 气、体总，密即度给静密度
cpT1u212cpT2u222cpT0cons(t8.39)
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For a calorically perfect gas, the ratio of total temperature to
sfotalltoicwtse:m（pe对rat于ure量, 热T完0 T全气is体a，fu总nc温tio和n 静of 温Ma的ch比nuTm0 bTe是r o马nly赫, a数s
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
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图8.2 第八章路线图
以温度表示：
cpT1u212
cpT2
u22 2
RT1 u12 RT2 u22 1 2 1 2
density at this sonic condition as T*,p*, and ρ*,respectively.
考虑流场中速度为准确的音速这一点，即M=1 的点。我们定
义这一点（音速条件）的静温、静压、静密度定义为临界参
数，用T*、p*和ρ*表示。
T* 2 T0 1
(8.44)
p * ( 2 ) ( 1)
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。
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• 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，p2
我们有：
p1
12
TT12
(1)
p0 p
p0
1
(8.45)
* ( 2 )1 ( 1)
0
1
(8.46)
T*0 .833 p*0 .528*0 .634
T 2020/6/17 0
p 0
0
• 特征马赫数（速度系数）M*的定义及计算公式
In the theory of supersonic flow, it is sometimes convenient to introduce a “characteristic” Mach number, M*, defined as:
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对于沿一条流线上的任意两点,有:
a 121u2 12a 221u2 22(2( 1 )a1* )2const(8.36)
1 a*2 a02 const 2(1) 1
(8.37)
Clearly, these defined quantities, a0 and a* , are both constants along a given in a steady, adiabatic, inviscid flow. If all the
defined T0,p0, and0 can be calculated from the actual conditions of
M ,T ,p and at a given point in general flow field (assuming
calorically perfect gas). They are so important that values of T0 T, p0 p
的唯一函数，证明如下：）
T T 0 12 u cp 2 T 12 Ru /2 T ( 1 ) 12 1(u a)2
T0 11M2
T
2
(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
a2 u2 (1)a*2
1 2 2(1)
（8.35）
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(a/u)21(1)a (*/u)2 1 2 2(1)
(1/M)2
1
2((11))(M 1*)21 2
M2(1)/M2*2(1)
M*2 2((1)1M)M2 2
（8.47） （8.48）
M*1
if M 1
M*1
if M 1
M*1
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。
Байду номын сангаас
if M 1
M* 1 1
if M
There, M* acts qualitatively in the same fashion as M except M* approaches a finite value when the actual Mach number approaches infinity.
and 0 obtained from Eqs. (8.40),(8.42), and (8.43), respectively ,
are tabulated as functions of M in App.A for 1.4 (which
corresponds to air at standard conditions).
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小结： In summary, a number of equations have been derived in this section, all of which stem in one fashion or another from the basic energy equation for steady, inviscid, adiabatic flow.
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• 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：
u2 cpT 2 cpT0
(8.38)
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。
M* u a*
Where a* is the value of the speed of sound at sonic conditions, not the actual local value. a* 是音速条件（流动速度u=a*时)的音速值。
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下面利用能量方程(8.35)得到M与M*的关系：
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0 (11M2)1(1)
2
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dpVdV
(3.12)
dpV2 dV
p pV
dp ( p)0
0V2
p
dV V
dp/ p (dp/ p)0 0
0
Hence, the degree by which deviates from unity as shown in Fig.8.5 is related to the same degree by which the fractional
a RT
(8.30) (8.31)
以音速表示：
a12 u12 a22 u22
1 2 1 2
(8.32)
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Definition of stagnation speed of sound：驻点音速的定义
a2 u2 a02
1 2 1
(8.33)
a12 u12 a22 u22 a02
方程(8.40),(8.42)和(8.43) 非常重要；应牢记于心。他们给出了对于
量热完全气体的任意流场，由某一给定点实际的M ,T ,p 和 的
值来计算定义的量 T0, p0和 0的公式。正因为其重要性，附录A 列表给出了 T 0 T,p0 p,0 随马赫数变化的函数关系。（对
1.4 应 2020/6/17
T0 11M2
T
2
p0 (1 1 M 2 ) ( 1)
p
2
0 (1 1 M 2 )1 ( 1)
2
Equation (8.40),(8.42)and (8.43) are very important; they should be
branded on your mind. They provided formulas from which the
pr2e02s0s/6u/17re change for a given dV/V.
8.6 CALCULATION OF NORMAL SHOCK-WAVE PROPERTIES
1u12u2
p11u12p22u22
h1
u12 2
h2
u22 2
h2 cpT2
p2 RT2
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p1
1u1