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,
第八章的路线图,
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
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 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,
?? ??
S
dSV 0??
(8.1)
2211 uu ?? ?
(8.2)
动量方程,
? ? ???? ???
ss
dSpVdSV ???
(8.3)
x方向分量,
? ? ? ????? ???
s
x
s
dSpudSV ??
(8.4)
22222111 upup ?? ??? (8.6)
???? ???????????? ?
Ss
dSVpdsVVe
??
2
2
?
能量方程,
(8.7)
22
2
2
2
2
1
1
uhuh ??? (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,
2211 uu ?? ?
22222111 upup ?? ???
22
2
2
2
2
1
1
uhuh ???
连续,
动量,
能量,
焓,
状态方程,
(8.2)
(8.6)
(8.10)
222 TRp ??
Tch p?2
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 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),(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。
8.3 SPEED OF SOUND 音速 (声速）
? What is the physical mechanism of the propagation of sound
waves? 声波传播的物理机理是什么？
? How can we calculate the speed of sound? 我们怎样计算声音
的速度？
? What properties of the gas does it depend on? 声音由气体的什
么特性决定？
The purpose of this section is to address these questions,本节的
目的就是要回答、讨论这些问题。
? What is the physical mechanism of the propagation of sound waves?
声波传播的物理机理是什么？
The physical mechanism of sound propagation in a gas is based on
molecular motion,声音在气体中的传播机理基于分子的运动。
For example,firecracker goes off,例如：点燃爆竹
When the firecracker detonates,chemical energy (basically a form of
heat release) is transferred to the air molecules adjacent to the
firecracker,These energized molecules are moving about in a random
fashion,The eventually collide with their neighboring molecules and
transfer their high energy to these neighbors,In turn,these
neighboring molecules eventually collide with their neighbors and
transfer energy in the process,By means of this,domino” effect,the
energy released by the firecracker is propagated through the air by
molecular collisions,Moreover,because T,p,and ρfor a gas are
macroscopic average of the detailed microscopic molecular motion,
the regions of energized molecules are also regions of slight
variations in the local temperature,pressure,and density,Hence,as
this energy wave from the firecracker passes over our eardrum,we
“hear” the slight pressure changes in the wave,This is sound,and the
propagation of the energy wave is simply the propagation of a sound
wave through the gas,
当爆竹爆炸时,化学能 (基本上是热释放的形式 )被传递到紧邻爆
竹的空气分子上,这些接收到能量的分子无规则地向四周运动,他
们会与其相邻分子相碰撞,并将高能量传给其相邻分子,通过这种
,多米诺, 效应，爆竹释放出的能量通过分子间的碰撞传播出
去。更进一步，因为 T,p和 ρ作为气体的温度、压强和密度是分
子 微观 运动的 宏观 平均值，接收到能量的分子所占区域也是温
度、压强和密度发生微小变化的区域。因此，当能量波传递到
我们的耳膜时，我们“听”到声波中微小的压强变化。这就是
声音，能量波的传播实际上就是声波在气体中传播。
macroscopic,宏观的 microscopic,微观的
? How can we calculate the speed of sound? 我们怎样计算声音
的速度？
Although the propagation of sound is due to molecular collisions,
we do not use such a microscopic picture for our derivation,
Rather,we take advantage of the fact that the macroscopic
properties p,T,ρ,etc.,change across the wave,and we use our
macroscopic equations of continuity,momentum,and energy to
analyze these changes,
尽管声音的传播是由于分子碰撞引起的,我们在推导气体音速
的方程时并不采用这一微观物理画面,相反,我们利用气体的宏
观性 p,T,ρ,等 通过声波将发生变化这一事实,应用连续、动
量、能量宏观方程来分析这些变化。
Consider a sound wave propagating through a stagnant gas
with velocity a,as sketched in Fig,8.4a,Here,the sound
wave is moving from right to left into a stagnant gas
(region1),where the local pressure,temperature,and
density are p,T,and ρ,respectively,Behind the sound wave
(region 2),the gas properties are slightly different and given
by p + dp,T + dT,and ρ + dρ,respectively,
假设 声波在气体中以速度 a在静止气体中 传播，如图
8.4a所示。 这里，声波从右向左进入当地压强、温度和
密度分别为 p,T,and ρ的静止气体区（区域 1）。在声波
之后（区域 2），其气体的性质与区域 1的气体性质有微
小的不同，分别用 p + dp,T + dT,和 ρ + dρ来 表示 。
Now imagine that you hop on the wave and ride with it,When
you look upstream,into region 1,you see the gas moving toward
you with a relative velocity a,as sketched in Fig,8.4b,When you
look downstream,into region 2,you see the gas receding away
from you with a relative velocity a + da,as also shown in Fig,
8.4b,
想象你跳上声波并乘波运动,这时你将看到上游 (区域 1)气体
以相对速度 a向你运动,。向下游（区域 2）看时，你看到气
体以相对速度 a + da离你远去, 同样如图 8.4b所示。
The pictures in Fig.8.4a and Fig.8.4b are analogous; only the
perspective is different,For purposes of analysis,we use Fig.8.4b.
图 8.4a 和 b的图画是相同的，只是观察参照点不同。为便于分
析，我们采用图 8.4b,
Why?
采用图 8.4b进行分析的优点是：原来的 非定常 问题转化成了 定
常 问题，可以采用与图 8.3分析静止正激波类似的方法分析声波。
Please note that the sound wave in Fig,8.4b is nothing more than an
infinitely weak normal shock wave,
请注意，图 8.4b所示的声波就是无限弱的正激波。
声波与激波的不同之处在于：通过激波流动特性发生突变，是
一个间断（ discontinuities)，是一个绝热但不等熵过程；通过声
波流动特性发生无限小的微弱变化，流动特性变化是连续的，
是一个等熵过程。
Finally,the gradients within the wave are very small-the changes dp,
dT,dρ and da are infinitesimal。 Therefore,the influence of
dissipative phenomena (viscosity and thermal conduction) is
negligible,最后，我们还知道通过声波发生的气体特性变化非常
小 ——变化量可用无限小的微分 dp,dT,dρ and da 来表示 。 因此，
耗散现象（粘性与热传导）的影响可以忽略不计。
对图 8.4b应用连续方程,
))(( daada ??? ???
daddaadaa ????? ???? （ 8.11)
忽略微分的乘积 dadρ,
?? d
daa ?? （ 8.12) 或
对图 8.4b应用动量方程,
22 ))(()( daaddppap ?????? ???
（ 8.13)
22222 22 daddaaddadaa d aadppap ??????? ?????????
略去微分的乘积,
?? daadadp 22 ??? （ 8.14)
daad ?? ?? （ 8.12b)
?? daadadp 22 ??? （ 8.14)
将 (8.12b) 式 代入 (8.14) 式（ ） 得,daad ?? ??
????? dadadadadaadp 2222 2)(2 ??????
?d
dpa ?2即,
（ 8.17)
s
pa )(
??
??
（ 8.18)
Equation (8.18) is a fundamental expression for the speed of
sound in a gas,方程 (8.18) 是气体音速的一个基本表达式。
假设气体是量热完全的。对于这种情况，等熵关系式
（ 7.32)成立。
?
?
?
???
?
???
?
?
2
1
2
1
p
p （ 8.19)
cc o n s tp ????
??cp ?
（ 8.20)
?
?pa ? (8.23)
应用状态方程,
RTa ??
(8.25)
which is our final expression for the speed of sound; it clearly
states that the speed of sound in a calorically perfect gas is a
function of temperature only,
是我们得到的音速计算公式的表达式；它清楚
地表明,对于量热完全气体，音速是温度的唯一函数 。 RTa ??
? What properties of the gas does it depend on? 声音由气体的什
么特性决定？
?音速与压缩性的 关系,
s
s p
v
v ?
?
?
?
??
?
?
?
??? 1?
22
11
ap
s
s ?
?
?
?? ?
?
?
?
?
?
?
?
?
??
?
?
??
?
?
?
?
??? (8.26)
s
a
??
1? (8.27)
?马赫数的物理意义,
考虑一沿流线运动的流体微元。其单位质量的动能和内能分别
为 和 e。 动能和内能的比为,
2
2
2222
2
)1(
)1/(
)2/(
)1/(
222 M
a
V
RT
V
Tc
V
e
V
v
??
?????
??
?
?
?
Hence,we see that the square of the Mach number is proportional
to the ratio of kinetic energy of a gas flow,In other words,the
Mach number is a measure of the directed motion of the gas
compared with the random thermal motion of the molecules,
因此，我们看到马赫数的平方正比于气体动能内能之比。用
另一句话说：马赫数是气体的有序运动和分子无规则热运动
的程度对比的度量。
22V
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,
第八章的路线图,
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
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 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,
?? ??
S
dSV 0??
(8.1)
2211 uu ?? ?
(8.2)
动量方程,
? ? ???? ???
ss
dSpVdSV ???
(8.3)
x方向分量,
? ? ? ????? ???
s
x
s
dSpudSV ??
(8.4)
22222111 upup ?? ??? (8.6)
???? ???????????? ?
Ss
dSVpdsVVe
??
2
2
?
能量方程,
(8.7)
22
2
2
2
2
1
1
uhuh ??? (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,
2211 uu ?? ?
22222111 upup ?? ???
22
2
2
2
2
1
1
uhuh ???
连续,
动量,
能量,
焓,
状态方程,
(8.2)
(8.6)
(8.10)
222 TRp ??
Tch p?2
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 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),(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。
8.3 SPEED OF SOUND 音速 (声速）
? What is the physical mechanism of the propagation of sound
waves? 声波传播的物理机理是什么？
? How can we calculate the speed of sound? 我们怎样计算声音
的速度？
? What properties of the gas does it depend on? 声音由气体的什
么特性决定？
The purpose of this section is to address these questions,本节的
目的就是要回答、讨论这些问题。
? What is the physical mechanism of the propagation of sound waves?
声波传播的物理机理是什么？
The physical mechanism of sound propagation in a gas is based on
molecular motion,声音在气体中的传播机理基于分子的运动。
For example,firecracker goes off,例如：点燃爆竹
When the firecracker detonates,chemical energy (basically a form of
heat release) is transferred to the air molecules adjacent to the
firecracker,These energized molecules are moving about in a random
fashion,The eventually collide with their neighboring molecules and
transfer their high energy to these neighbors,In turn,these
neighboring molecules eventually collide with their neighbors and
transfer energy in the process,By means of this,domino” effect,the
energy released by the firecracker is propagated through the air by
molecular collisions,Moreover,because T,p,and ρfor a gas are
macroscopic average of the detailed microscopic molecular motion,
the regions of energized molecules are also regions of slight
variations in the local temperature,pressure,and density,Hence,as
this energy wave from the firecracker passes over our eardrum,we
“hear” the slight pressure changes in the wave,This is sound,and the
propagation of the energy wave is simply the propagation of a sound
wave through the gas,
当爆竹爆炸时,化学能 (基本上是热释放的形式 )被传递到紧邻爆
竹的空气分子上,这些接收到能量的分子无规则地向四周运动,他
们会与其相邻分子相碰撞,并将高能量传给其相邻分子,通过这种
,多米诺, 效应，爆竹释放出的能量通过分子间的碰撞传播出
去。更进一步，因为 T,p和 ρ作为气体的温度、压强和密度是分
子 微观 运动的 宏观 平均值，接收到能量的分子所占区域也是温
度、压强和密度发生微小变化的区域。因此，当能量波传递到
我们的耳膜时，我们“听”到声波中微小的压强变化。这就是
声音，能量波的传播实际上就是声波在气体中传播。
macroscopic,宏观的 microscopic,微观的
? How can we calculate the speed of sound? 我们怎样计算声音
的速度？
Although the propagation of sound is due to molecular collisions,
we do not use such a microscopic picture for our derivation,
Rather,we take advantage of the fact that the macroscopic
properties p,T,ρ,etc.,change across the wave,and we use our
macroscopic equations of continuity,momentum,and energy to
analyze these changes,
尽管声音的传播是由于分子碰撞引起的,我们在推导气体音速
的方程时并不采用这一微观物理画面,相反,我们利用气体的宏
观性 p,T,ρ,等 通过声波将发生变化这一事实,应用连续、动
量、能量宏观方程来分析这些变化。
Consider a sound wave propagating through a stagnant gas
with velocity a,as sketched in Fig,8.4a,Here,the sound
wave is moving from right to left into a stagnant gas
(region1),where the local pressure,temperature,and
density are p,T,and ρ,respectively,Behind the sound wave
(region 2),the gas properties are slightly different and given
by p + dp,T + dT,and ρ + dρ,respectively,
假设 声波在气体中以速度 a在静止气体中 传播，如图
8.4a所示。 这里，声波从右向左进入当地压强、温度和
密度分别为 p,T,and ρ的静止气体区（区域 1）。在声波
之后（区域 2），其气体的性质与区域 1的气体性质有微
小的不同，分别用 p + dp,T + dT,和 ρ + dρ来 表示 。
Now imagine that you hop on the wave and ride with it,When
you look upstream,into region 1,you see the gas moving toward
you with a relative velocity a,as sketched in Fig,8.4b,When you
look downstream,into region 2,you see the gas receding away
from you with a relative velocity a + da,as also shown in Fig,
8.4b,
想象你跳上声波并乘波运动,这时你将看到上游 (区域 1)气体
以相对速度 a向你运动,。向下游（区域 2）看时，你看到气
体以相对速度 a + da离你远去, 同样如图 8.4b所示。
The pictures in Fig.8.4a and Fig.8.4b are analogous; only the
perspective is different,For purposes of analysis,we use Fig.8.4b.
图 8.4a 和 b的图画是相同的，只是观察参照点不同。为便于分
析，我们采用图 8.4b,
Why?
采用图 8.4b进行分析的优点是：原来的 非定常 问题转化成了 定
常 问题，可以采用与图 8.3分析静止正激波类似的方法分析声波。
Please note that the sound wave in Fig,8.4b is nothing more than an
infinitely weak normal shock wave,
请注意，图 8.4b所示的声波就是无限弱的正激波。
声波与激波的不同之处在于：通过激波流动特性发生突变，是
一个间断（ discontinuities)，是一个绝热但不等熵过程；通过声
波流动特性发生无限小的微弱变化，流动特性变化是连续的，
是一个等熵过程。
Finally,the gradients within the wave are very small-the changes dp,
dT,dρ and da are infinitesimal。 Therefore,the influence of
dissipative phenomena (viscosity and thermal conduction) is
negligible,最后，我们还知道通过声波发生的气体特性变化非常
小 ——变化量可用无限小的微分 dp,dT,dρ and da 来表示 。 因此，
耗散现象（粘性与热传导）的影响可以忽略不计。
对图 8.4b应用连续方程,
))(( daada ??? ???
daddaadaa ????? ???? （ 8.11)
忽略微分的乘积 dadρ,
?? d
daa ?? （ 8.12) 或
对图 8.4b应用动量方程,
22 ))(()( daaddppap ?????? ???
（ 8.13)
22222 22 daddaaddadaa d aadppap ??????? ?????????
略去微分的乘积,
?? daadadp 22 ??? （ 8.14)
daad ?? ?? （ 8.12b)
?? daadadp 22 ??? （ 8.14)
将 (8.12b) 式 代入 (8.14) 式（ ） 得,daad ?? ??
????? dadadadadaadp 2222 2)(2 ??????
?d
dpa ?2即,
（ 8.17)
s
pa )(
??
??
（ 8.18)
Equation (8.18) is a fundamental expression for the speed of
sound in a gas,方程 (8.18) 是气体音速的一个基本表达式。
假设气体是量热完全的。对于这种情况，等熵关系式
（ 7.32)成立。
?
?
?
???
?
???
?
?
2
1
2
1
p
p （ 8.19)
cc o n s tp ????
??cp ?
（ 8.20)
?
?pa ? (8.23)
应用状态方程,
RTa ??
(8.25)
which is our final expression for the speed of sound; it clearly
states that the speed of sound in a calorically perfect gas is a
function of temperature only,
是我们得到的音速计算公式的表达式；它清楚
地表明,对于量热完全气体，音速是温度的唯一函数 。 RTa ??
? What properties of the gas does it depend on? 声音由气体的什
么特性决定？
?音速与压缩性的 关系,
s
s p
v
v ?
?
?
?
??
?
?
?
??? 1?
22
11
ap
s
s ?
?
?
?? ?
?
?
?
?
?
?
?
?
??
?
?
??
?
?
?
?
??? (8.26)
s
a
??
1? (8.27)
?马赫数的物理意义,
考虑一沿流线运动的流体微元。其单位质量的动能和内能分别
为 和 e。 动能和内能的比为,
2
2
2222
2
)1(
)1/(
)2/(
)1/(
222 M
a
V
RT
V
Tc
V
e
V
v
??
?????
??
?
?
?
Hence,we see that the square of the Mach number is proportional
to the ratio of kinetic energy of a gas flow,In other words,the
Mach number is a measure of the directed motion of the gas
compared with the random thermal motion of the molecules,
因此，我们看到马赫数的平方正比于气体动能内能之比。用
另一句话说：马赫数是气体的有序运动和分子无规则热运动
的程度对比的度量。
22V
