完全气体
内能和焓
热力学复习 热力学第一定律
熵及热力学第二定律
等熵关系式
压缩性定义
无粘可压缩流动的控制方程
总条件的定义
有激波的超音速流动的定性了解
第
七
章
路
线
图
7.3,DEFINITION OF COMPRESSIBILITY
(压缩性定义 )
All real substances are compressible to some
greater or lesser extend,When you squeeze or
press on them,their density will change,This is
particularly true of gases,
(所有的真实物质都是可压缩的,当我们压挤
它们时,它们的密度会发生变化,对于气体尤
其是这样,)
The amount by which a substance can be compressed
is given by a specific property of the substance called
the compressibilty,defined below,
物质可被压缩的大小程度称为物质的压缩性,
Consider a small element of fluid of volume,
The pressure exerted on the sides of the element is p,
If the pressure is increased by an infinitesimal
amount dp,the volume will change by a
negative amount,
v
dv
By definition,the compressibility is given by,
(7.33)
as
(7.36)
Physically,the compressibility is a fractional change in volume of
the fluid element per unit change in pressure.(从物理上讲,压缩
性就是每单位压强变化引起的流体微元单位体积内的体积
变化 )
dp
dv
v
1???
dp
d ?
?
? 1??
1?v
If the temperature of the fluid element is held
constant,then is identified as the
isothermal compressibility (等温压缩性 )
(7.34)
If the process takes place isentropically,then
(等熵压缩性 )
(7.35)
?
T
T p
v
v ?
?
?
?
??
?
?
?
?
??
1
?
s
s p
v
v ?
?
?
?
??
?
?
?
?
??
1
?
If the fluid is a gas,where compressibility is large,
then for a given pressure change from one point
to another in the flow,Eq.(7.37) states that can
be large,(如果流体为气体,则 值大,对于一个给定压强
变化,方程,(7.37)指出,也会大,)
Thus,is not constant; the flow of a gas is a
compressible flow,
The exception is the low-speed flow of a gas,Where is
the limit? If the Mach number
,the flow should be considered
compressible,
dpd ??? ?
?d
?
3.0/ ?? aVM
?
(7.37)
dp
?
dp ?d
7.4 GOVERNING EQUATIONS FOR INVISCID,
COMPRESSIBLE FLOW (无粘、可压缩流控制方程 )
For inviscid,incompressible flow,the primary
dependent variables are the pressure p and the
velocity, Hence,we need only two basic
equations,namely the continuity and the momentum
equations,
对于无粘,不可压缩流动,基本自变量是
压强 p和速度 。因此我们只需要 两个基
本方程,即 连续方程 和 动量方程。
Indeed,the basic equations are combined to obtain
Laplace’s equation and Bernoulli’s equation,which
are the primarily tools the applications discussed
in Chaps,3 to 6,Note that both and T are
assumed to be constant through out such inviscid,
incompressible flows,
连续方程与动量方程相结合可以得到 Laplace 方
程和 Bernoulli 方程，这是我们讨论第三章至第
六章内容用到的基本工具．对于无粘不可压缩
流动,我们假定密度和温度保持不变,
?
Basically,incompressible flows obey purely
mechanical laws and do not need thermodynamic
considerations,
In contrast,for compressible flow,is variable
and becomes an unknown,Hence we need an
additional equation – the energy equation – which in turn
introduces internal energy e as an unknown,
对于可压缩流，相反的是 是一个变量,
并且是一个未知数,因此，我们需要一个附加
方程－ 能量方程 －进而引入未知数 内能 e。
Internal energy e is related to temperature,then T
also becomes an important variable,
Therefore,the 5 primary dependent variables are,
To solve for these five variables,we need five
governing equations
?
TandeVp,,,,??
?
复习第二章知识,
AVm n???
nVA
m ??? ?fl u x M a s s
1,Continuity(连续方程）
Physical principle,mass can be neither created nor
destroyed
Net mass flow out of time rate of decrease of
control volume = mass inside control volume V
through surface S
通过控制体表面 S流出控制体的净质量流量＝控制体内的质量减少率
????? ????
?
S
dSVdv
t
0
?
??
?
(7.39)
or in the form of a partial differential equation (偏微分方程）
0)( ?????? Vt ???
(7.40)
where is the divergence of the vector field
? in Cartesian coordinates（在指角坐标系下）
)( V???? V??
? ? ? ? ? ?
z
w
y
v
x
uV
?
??
?
??
?
???? ???? )( ?
2,Momentum（动量方程）
Physical principle,
Force = time rate of change of momentum
(7.41)
where are the body forces,such as gravity,or electromagnetic
forces
? ?VmdtdF ?? ?
? ? ?? ???????? ?????
?
?
s vsv
dvfdSpVdSVvdV
t
????
???
f?
In terms of substantial derivative,
(7.42a)
the y and z directions of the vector can be easily found by substitution
(7.42b)
(7.42c)
xfx
p
Dt
Du ?? ?
?
???
yfy
p
Dt
Dv ?? ?
?
???
zfz
p
Dt
Dw ?? ?
?
???
写成矢量形式,
fpDt VD
??
?? ????
where is the substantial derivative which can be
written in Cartesian coordinates as,
)( ???
?
?
? V
tDt
D ?
zwyvxutDt
D
?
??
?
??
?
??
?
??
Dt
D
3,Energy
Physical principle,
Energy can be neither created nor destroyed; it can only
change in form
dsVVedvVet
sv
???
?
?
???
? ??
???
?
???
? ?
?
? ????? ?
22
22
??
? ? dvVfdsVpdvq
s vv
?? ?????? ????? ??? ??
(7.43)
Equation of energy can also be written as,
Assume that the flow is adiabatic and that body
forces are negligible, For such a flow
? ? ? ?VfVpq
Dt
VeD ???? ??????? ??? 2/2
? ?0?q?
? ?0?f?
? ? Vp
Dt
VeD ?????? 2/2?
(7.44)
(7.45)
4,Equation of state for a perfect gas,
5,Internal energy for a calorically perfect gas,
We have now 5 equations for 5 unknowns,
TRp ??
Tce v?
7.5 DEFINITION OF TOTAL CONDITIONS
(总条件的定义）
Consider a fluid element passing through a given point in a
flow where the local pressure,temperature,density,Mach number,and
velocity (local conditions)
are and,respectively,
假设流体微团通过一个给定点，对应的当地压强、温度、
密度、马赫数、速度分别为 。
Here,are static quantities,i.e.,static
pressure,static temperature,static density,respectively,
这里,是分别静变量（静参数），即静压、静
温、静密度。
,,,,MTp ? V
?
VMTp ?,,,,?
?,,a n dTp
?,,Tp
Now imagine that you grab hold of the fluid element and
adiabatically slow it down to zero velocity,Clearly,you would
expect (correctly) that the values of would change
as the element is brought to rest,?,,a n dTp
In particular,the value of the temperature of the fluid
element after it has been brought to rest adiabatically is
defined as the total temperature,denoted by,
特别地，假想流体微团被 绝热地 减速为静止所对应
的温度，定义此时流体微团对应的温度为 总温,
0T
The corresponding value of enthalpy is defined as total
enthalpy h0,where h0 = cpT0 for a calorically perfect gas,
*如何确定总温？
The energy equation,Eq,(7.44),provides some important
information about total enthalpy and hence total temperature,
(由能量方程可以的到总焓、因而总温的重要信息。 )
Assume that the flow is adiabatic and that body forces are negligible,
then the equation of energy can be written as,
(7.45) ? ? Vp
Dt
VeD ?????? 2/2?
注意 (7.45)式的前提条件：无粘、绝热、忽略体积力,
Expanding by using the following vector identity
And noting that
Vp ???
pVVpVp ???????? ???
? ?
Dt
Dp
Dt
DpDtpDDtDp
Dt
pD ?
??
??
?
?
? ??
?
?
2
///
Substituting the continuity equation
0???? VDtD ???
(7.47)
? ?
VppV
t
p
Vp
Dt
Dp
Dt
Dp
Dt
Dp
Dt
pD
??
?
??????
?
?
?
??????
/ ?
?
?
?
(7.48)
? ?
t
p
Dt
VhD
?
??? 2/2?
? ? Vp
Dt
VeD ?????? 2/2?
? ? VppV
t
p
Dt
pD ?? ??????
?
???? /
pVVpVp ???????? ???
(7.46)
(7.45)
(7.48)
(7.45)+(7.48),note,hpe ?? ?
(7.51)
If the flow is steady,(如果流动是定常的）
From the definition of the substantial derivative
Then the time rate of change of h+V2/2 following a
moving fluid element is zero,
0???tp
? ? 02/2 ??
Dt
VhD?
? ??????? VtDtD
c o n s tVh ??
2
2
(7.53)
Recall that the assumptions which led to Eq,(7.53) are that
the flow is steady,adiabatic,and inviscid,
(7.52)
Since h0 is defined as that enthalpy which would exist
at a point if the fluid element were brought to rest
adiabatically,where V = 0 and hence h = h0,then the
value of the constant is h0,
因为我们定义总焓 h0为 流体微元被绝热地减速为静止 时
对应的焓值,因此有能量方程我们可以得到总焓的值,即上式
(7.53)中的常数。
因此有,
0
2
2
hVh ??
(7.54)
Equation (7.54) is important; it states that at any point in a flow,
the total enthalpy is given by the sum of the static enthalpy plus
the kinetic energy,all per unit of mass,
方程 (7.54)很重要,它表明 在流动中任一点,总焓 由 每单位体积
的静焓和动能之和组成。
有了总焓的定义，能量方程可以用总焓来表示：对于定
常、绝热、无粘流动，方程（ 7.52）可以写成,
or
i.e,the total enthalpy is constant along a streamline,
即总焓沿流线为常数。
If all the streamlines of the flow originate from a common
uniform freestream (as the usually the case),then the h0 is the
same for each line,
如果像通常的情况那样，所有的流线都来自均匀自由来流，
那么 h0在不同流线也是相等的。
h0 =const,throughout the entire flow,and h0 is equal to its
freestream value,总焓在整个流场中为常数，等于自由来流
对应的总焓。
00 ?
Dt
Dh? c o n s th ?0
（ 7.55)
For a calorically perfect gas,h0 = cpT0, Thus,the above
results also state that the total temperature is constant
throughout the steady,inviscid,adiabatic flow of a
calorically perfect gas; i.e,
对于量热完全气体,h0 = cpT0 。因此，上面的结果
也表明了对于定常、无粘、绝热的量热完全气体，
总温保持不变，即
c o n s tT ?0
（ 7.56）
Keep in mind that the above discussion marbled two trains of
thought,On the one hand,we dealt with the general concept of
an adiabatic flow field [which led to Eqs,(7.51) to (7.53)],and
on the other hand,we dealt with the definition of total
enthalpy[which led to Eq,(7.54)],
要牢记在心的是：上面的讨论是沿着两条思路进行的，
一方面，我们讨论了绝热流场的一般概念 [导出了能量方程
(7.51) 至 (7.53)]；另一方面，我们讨论了总焓的定义 [给出了
（ 7.54)式 ]。
? ?
t
p
Dt
VhD
?
??? 2/2?
? ? 02/2 ??
Dt
VhD?
c o n s tVh ?? 2
2
(7.51)
(7.52)
(7.53)
0
2
2 h
Vh ?? (7.54)
总压与总密度的定义,
回到本节的开头，我们考虑流体微团通过一个给定点，
对应的当地压强、温度、密度、马赫数、速度分别
为 。
Once again,imagine that you grab hold of the fluid element and
slow it down to zero velocity,but this time,let us slow it down
both adiabatically and reversibly,That is,let us slow the fluid element
down to zero velocity isentropically,
When the fluid element is brought to rest isentropically,the resulting pressure
and density are defined as the total pressure p0 and total density,
定义, 当流体微元被等熵地减速至静止时对应的压强和密度
被定义为其总压和总密度。
VMTp ?,,,,?
0?
Since an isentropic process is also adiabatic,the
definition of total temperature remains unchanged,
As before,keep in mind that we do not have to actually
bring the flow to rest in real life in order to talk about
total pressure and total density; rather,the are defined
quantities that would exist at a point in a flow if (in our
imagination) the fluid element passing through that point
were brought to rest isentropically,Therefore,at a given
point in a flow,where the static pressure and static density
are p and ρ,respectively,we can also assign a value of
total pressure p0,and total density ρ0 defined as above,
6,SUMMARY
Total temperature T0 and total enthalpy h0 are
defined as the properties that would exist if the flow is
slowed to zero velocity adiabatically,
Total pressure p0 and total density ρ0 are defined as
the properties that would exist if the flow is slowed to
zero velocity isentropically,
If the general flow field is adiabatic,h0 is constant
throughout the flow,
If the general flow field is isentropic,p0 and ρ0 are constant throughout the flow,
7.6 Some Aspects of Supersonic Flow,Shock Waves
超音速流的一些特征：激波
51页图 1.30
An essential ingredient of a supersonic flow is the calculation of the
shape and strength of shock waves,This is the main thrust of chaps,
8 and 9,
超音速流动研究的一个重要内容就是计算激波的形状和强度。
这是第 8章和第 9章的主题。
A shock wave is an extremely thin region,typically on the order of
10-5cm,across which the flow properties can change drastically,
激波是一个极其薄的区域，厚度大约只有 10-5cm的量级，通过
激波流动特性发生剧烈变化。
7.7 Summary(小结）
1、热力学关系式,
状态方程,
对于量热完全气体,
( 7, 1 0 )
1-
R
( 9, 9 )
1-
( 7, 6 b )
( 7, 6 a )
( 7, 1 )
?
?
?
?
?
?
?
?
?
v
p
p
v
c
R
c
Tch
Tce
RTp
热力学第一定律的各种表达形式,
熵的定义,
热力学第二定律,
或对于绝热过程,
?
?
?
?
??
??
( 7, 2 0 )
( 7, 1 8 )
( 7, 1 1 )
d h - v d pT d s
pdvdeT d s
dewq ??
可逆过程,
( 7, 1 4 )
( 7, 1 3 )
i r r e v
r e v
ds
T
δq
ds
T
q
ds
??
?
及
?
( 7, 1 7 ) 0
( 7, 1 6 )
ds
T
q
ds
?
?
?
量热完全气体的熵增计算公式,
（ 7.25)
(7.26)
等熵流动的等熵关系式,
(7.32)
1
2
1
2
12 lnln p
pR
T
Tcss
p ???
1
2
1
2
12 lnln ?
?
? RT
Tcss ???
? ?1/
1
2
1
2
1
2
?
???
?
???
??
???
?
???
?? ???
?
?
T
T
p
p
2、压缩性
压缩性的一般定义,
（ 7.33）
对于等温过程,
(7.34)
对于等熵过程,
(7.35)
dp
dv
v
1???
T
T p
v
v ?
?
?
?
???
?
?
??? 1?
s
s p
v
v ?
?
?
?
???
?
?
??? 1?
3、无粘、可压缩流的控制方程,
连续方程,
????? ?????
S
dSVdvt 0???
?
0)( ?????? Vt ???
? ? ?? ???????? ???????
s vsv
dvfdSpVdSVvdVt ???? ???
xfx
p
Dt
Du ?? ?
?
???
yfy
p
Dt
Dv ?? ?
?
???
zfz
p
Dt
Dw ?? ?
?
???
动量方程,
（ 7.39)
(7.40)
(7.41)
(7.41a)
(7.41b)
(7.41c)
dsVVedvVet
sv
???
?
?
???
? ??
???
?
???
? ?
?
? ????? ?
22
22
??
? ? dvVfdsVpdvq
s vv
?? ?????? ????? ??? ??
? ? ? ?VfVpq
Dt
VeD ???? ??????? ??? 2/2
TRp ??
Tce v?
能量方程,
（ 7.43)
(7.44)
对于定常、绝热、无粘流，（ 7.44)和 (7.43)可以写成,
2 0
2
c o n s thc o n s tVh ??? 或
（ 7.53),(7.55)
完全气体状态方程,
量热完全气体内能,
（ 7.1)
(7.6a)
4、总温、总焓、总压、总密度的定义及概念,
总温和总焓定义为把流体微元（在我们的想象中） 绝热地
减速为静止时流体微元所对应的温度和焓值。
类似地，总压和总密度定义为把流体微元（在我们的想象
中） 等熵地 减速为静止时流体微元所对应的压强和密度。
在均匀自由来流的绝热流场中，总焓 h0在全流场中为常数，
相反，在非绝热流场中,h0随流场点的不同而不同。
类似地，在等熵流场中，总压和总密度在整个流场中为常
数，相反，在非等熵流场中，总压和总密度随流场点的不
同而不同。
5、激波
激波为超音速流中很薄的一层，通过激波压强、密度、温
度和熵增加；马赫数、流动速度、总压降低；总焓、总温
不变。
内能和焓
热力学复习 热力学第一定律
熵及热力学第二定律
等熵关系式
压缩性定义
无粘可压缩流动的控制方程
总条件的定义
有激波的超音速流动的定性了解
第
七
章
路
线
图
7.3,DEFINITION OF COMPRESSIBILITY
(压缩性定义 )
All real substances are compressible to some
greater or lesser extend,When you squeeze or
press on them,their density will change,This is
particularly true of gases,
(所有的真实物质都是可压缩的,当我们压挤
它们时,它们的密度会发生变化,对于气体尤
其是这样,)
The amount by which a substance can be compressed
is given by a specific property of the substance called
the compressibilty,defined below,
物质可被压缩的大小程度称为物质的压缩性,
Consider a small element of fluid of volume,
The pressure exerted on the sides of the element is p,
If the pressure is increased by an infinitesimal
amount dp,the volume will change by a
negative amount,
v
dv
By definition,the compressibility is given by,
(7.33)
as
(7.36)
Physically,the compressibility is a fractional change in volume of
the fluid element per unit change in pressure.(从物理上讲,压缩
性就是每单位压强变化引起的流体微元单位体积内的体积
变化 )
dp
dv
v
1???
dp
d ?
?
? 1??
1?v
If the temperature of the fluid element is held
constant,then is identified as the
isothermal compressibility (等温压缩性 )
(7.34)
If the process takes place isentropically,then
(等熵压缩性 )
(7.35)
?
T
T p
v
v ?
?
?
?
??
?
?
?
?
??
1
?
s
s p
v
v ?
?
?
?
??
?
?
?
?
??
1
?
If the fluid is a gas,where compressibility is large,
then for a given pressure change from one point
to another in the flow,Eq.(7.37) states that can
be large,(如果流体为气体,则 值大,对于一个给定压强
变化,方程,(7.37)指出,也会大,)
Thus,is not constant; the flow of a gas is a
compressible flow,
The exception is the low-speed flow of a gas,Where is
the limit? If the Mach number
,the flow should be considered
compressible,
dpd ??? ?
?d
?
3.0/ ?? aVM
?
(7.37)
dp
?
dp ?d
7.4 GOVERNING EQUATIONS FOR INVISCID,
COMPRESSIBLE FLOW (无粘、可压缩流控制方程 )
For inviscid,incompressible flow,the primary
dependent variables are the pressure p and the
velocity, Hence,we need only two basic
equations,namely the continuity and the momentum
equations,
对于无粘,不可压缩流动,基本自变量是
压强 p和速度 。因此我们只需要 两个基
本方程,即 连续方程 和 动量方程。
Indeed,the basic equations are combined to obtain
Laplace’s equation and Bernoulli’s equation,which
are the primarily tools the applications discussed
in Chaps,3 to 6,Note that both and T are
assumed to be constant through out such inviscid,
incompressible flows,
连续方程与动量方程相结合可以得到 Laplace 方
程和 Bernoulli 方程，这是我们讨论第三章至第
六章内容用到的基本工具．对于无粘不可压缩
流动,我们假定密度和温度保持不变,
?
Basically,incompressible flows obey purely
mechanical laws and do not need thermodynamic
considerations,
In contrast,for compressible flow,is variable
and becomes an unknown,Hence we need an
additional equation – the energy equation – which in turn
introduces internal energy e as an unknown,
对于可压缩流，相反的是 是一个变量,
并且是一个未知数,因此，我们需要一个附加
方程－ 能量方程 －进而引入未知数 内能 e。
Internal energy e is related to temperature,then T
also becomes an important variable,
Therefore,the 5 primary dependent variables are,
To solve for these five variables,we need five
governing equations
?
TandeVp,,,,??
?
复习第二章知识,
AVm n???
nVA
m ??? ?fl u x M a s s
1,Continuity(连续方程）
Physical principle,mass can be neither created nor
destroyed
Net mass flow out of time rate of decrease of
control volume = mass inside control volume V
through surface S
通过控制体表面 S流出控制体的净质量流量＝控制体内的质量减少率
????? ????
?
S
dSVdv
t
0
?
??
?
(7.39)
or in the form of a partial differential equation (偏微分方程）
0)( ?????? Vt ???
(7.40)
where is the divergence of the vector field
? in Cartesian coordinates（在指角坐标系下）
)( V???? V??
? ? ? ? ? ?
z
w
y
v
x
uV
?
??
?
??
?
???? ???? )( ?
2,Momentum（动量方程）
Physical principle,
Force = time rate of change of momentum
(7.41)
where are the body forces,such as gravity,or electromagnetic
forces
? ?VmdtdF ?? ?
? ? ?? ???????? ?????
?
?
s vsv
dvfdSpVdSVvdV
t
????
???
f?
In terms of substantial derivative,
(7.42a)
the y and z directions of the vector can be easily found by substitution
(7.42b)
(7.42c)
xfx
p
Dt
Du ?? ?
?
???
yfy
p
Dt
Dv ?? ?
?
???
zfz
p
Dt
Dw ?? ?
?
???
写成矢量形式,
fpDt VD
??
?? ????
where is the substantial derivative which can be
written in Cartesian coordinates as,
)( ???
?
?
? V
tDt
D ?
zwyvxutDt
D
?
??
?
??
?
??
?
??
Dt
D
3,Energy
Physical principle,
Energy can be neither created nor destroyed; it can only
change in form
dsVVedvVet
sv
???
?
?
???
? ??
???
?
???
? ?
?
? ????? ?
22
22
??
? ? dvVfdsVpdvq
s vv
?? ?????? ????? ??? ??
(7.43)
Equation of energy can also be written as,
Assume that the flow is adiabatic and that body
forces are negligible, For such a flow
? ? ? ?VfVpq
Dt
VeD ???? ??????? ??? 2/2
? ?0?q?
? ?0?f?
? ? Vp
Dt
VeD ?????? 2/2?
(7.44)
(7.45)
4,Equation of state for a perfect gas,
5,Internal energy for a calorically perfect gas,
We have now 5 equations for 5 unknowns,
TRp ??
Tce v?
7.5 DEFINITION OF TOTAL CONDITIONS
(总条件的定义）
Consider a fluid element passing through a given point in a
flow where the local pressure,temperature,density,Mach number,and
velocity (local conditions)
are and,respectively,
假设流体微团通过一个给定点，对应的当地压强、温度、
密度、马赫数、速度分别为 。
Here,are static quantities,i.e.,static
pressure,static temperature,static density,respectively,
这里,是分别静变量（静参数），即静压、静
温、静密度。
,,,,MTp ? V
?
VMTp ?,,,,?
?,,a n dTp
?,,Tp
Now imagine that you grab hold of the fluid element and
adiabatically slow it down to zero velocity,Clearly,you would
expect (correctly) that the values of would change
as the element is brought to rest,?,,a n dTp
In particular,the value of the temperature of the fluid
element after it has been brought to rest adiabatically is
defined as the total temperature,denoted by,
特别地，假想流体微团被 绝热地 减速为静止所对应
的温度，定义此时流体微团对应的温度为 总温,
0T
The corresponding value of enthalpy is defined as total
enthalpy h0,where h0 = cpT0 for a calorically perfect gas,
*如何确定总温？
The energy equation,Eq,(7.44),provides some important
information about total enthalpy and hence total temperature,
(由能量方程可以的到总焓、因而总温的重要信息。 )
Assume that the flow is adiabatic and that body forces are negligible,
then the equation of energy can be written as,
(7.45) ? ? Vp
Dt
VeD ?????? 2/2?
注意 (7.45)式的前提条件：无粘、绝热、忽略体积力,
Expanding by using the following vector identity
And noting that
Vp ???
pVVpVp ???????? ???
? ?
Dt
Dp
Dt
DpDtpDDtDp
Dt
pD ?
??
??
?
?
? ??
?
?
2
///
Substituting the continuity equation
0???? VDtD ???
(7.47)
? ?
VppV
t
p
Vp
Dt
Dp
Dt
Dp
Dt
Dp
Dt
pD
??
?
??????
?
?
?
??????
/ ?
?
?
?
(7.48)
? ?
t
p
Dt
VhD
?
??? 2/2?
? ? Vp
Dt
VeD ?????? 2/2?
? ? VppV
t
p
Dt
pD ?? ??????
?
???? /
pVVpVp ???????? ???
(7.46)
(7.45)
(7.48)
(7.45)+(7.48),note,hpe ?? ?
(7.51)
If the flow is steady,(如果流动是定常的）
From the definition of the substantial derivative
Then the time rate of change of h+V2/2 following a
moving fluid element is zero,
0???tp
? ? 02/2 ??
Dt
VhD?
? ??????? VtDtD
c o n s tVh ??
2
2
(7.53)
Recall that the assumptions which led to Eq,(7.53) are that
the flow is steady,adiabatic,and inviscid,
(7.52)
Since h0 is defined as that enthalpy which would exist
at a point if the fluid element were brought to rest
adiabatically,where V = 0 and hence h = h0,then the
value of the constant is h0,
因为我们定义总焓 h0为 流体微元被绝热地减速为静止 时
对应的焓值,因此有能量方程我们可以得到总焓的值,即上式
(7.53)中的常数。
因此有,
0
2
2
hVh ??
(7.54)
Equation (7.54) is important; it states that at any point in a flow,
the total enthalpy is given by the sum of the static enthalpy plus
the kinetic energy,all per unit of mass,
方程 (7.54)很重要,它表明 在流动中任一点,总焓 由 每单位体积
的静焓和动能之和组成。
有了总焓的定义，能量方程可以用总焓来表示：对于定
常、绝热、无粘流动，方程（ 7.52）可以写成,
or
i.e,the total enthalpy is constant along a streamline,
即总焓沿流线为常数。
If all the streamlines of the flow originate from a common
uniform freestream (as the usually the case),then the h0 is the
same for each line,
如果像通常的情况那样，所有的流线都来自均匀自由来流，
那么 h0在不同流线也是相等的。
h0 =const,throughout the entire flow,and h0 is equal to its
freestream value,总焓在整个流场中为常数，等于自由来流
对应的总焓。
00 ?
Dt
Dh? c o n s th ?0
（ 7.55)
For a calorically perfect gas,h0 = cpT0, Thus,the above
results also state that the total temperature is constant
throughout the steady,inviscid,adiabatic flow of a
calorically perfect gas; i.e,
对于量热完全气体,h0 = cpT0 。因此，上面的结果
也表明了对于定常、无粘、绝热的量热完全气体，
总温保持不变，即
c o n s tT ?0
（ 7.56）
Keep in mind that the above discussion marbled two trains of
thought,On the one hand,we dealt with the general concept of
an adiabatic flow field [which led to Eqs,(7.51) to (7.53)],and
on the other hand,we dealt with the definition of total
enthalpy[which led to Eq,(7.54)],
要牢记在心的是：上面的讨论是沿着两条思路进行的，
一方面，我们讨论了绝热流场的一般概念 [导出了能量方程
(7.51) 至 (7.53)]；另一方面，我们讨论了总焓的定义 [给出了
（ 7.54)式 ]。
? ?
t
p
Dt
VhD
?
??? 2/2?
? ? 02/2 ??
Dt
VhD?
c o n s tVh ?? 2
2
(7.51)
(7.52)
(7.53)
0
2
2 h
Vh ?? (7.54)
总压与总密度的定义,
回到本节的开头，我们考虑流体微团通过一个给定点，
对应的当地压强、温度、密度、马赫数、速度分别
为 。
Once again,imagine that you grab hold of the fluid element and
slow it down to zero velocity,but this time,let us slow it down
both adiabatically and reversibly,That is,let us slow the fluid element
down to zero velocity isentropically,
When the fluid element is brought to rest isentropically,the resulting pressure
and density are defined as the total pressure p0 and total density,
定义, 当流体微元被等熵地减速至静止时对应的压强和密度
被定义为其总压和总密度。
VMTp ?,,,,?
0?
Since an isentropic process is also adiabatic,the
definition of total temperature remains unchanged,
As before,keep in mind that we do not have to actually
bring the flow to rest in real life in order to talk about
total pressure and total density; rather,the are defined
quantities that would exist at a point in a flow if (in our
imagination) the fluid element passing through that point
were brought to rest isentropically,Therefore,at a given
point in a flow,where the static pressure and static density
are p and ρ,respectively,we can also assign a value of
total pressure p0,and total density ρ0 defined as above,
6,SUMMARY
Total temperature T0 and total enthalpy h0 are
defined as the properties that would exist if the flow is
slowed to zero velocity adiabatically,
Total pressure p0 and total density ρ0 are defined as
the properties that would exist if the flow is slowed to
zero velocity isentropically,
If the general flow field is adiabatic,h0 is constant
throughout the flow,
If the general flow field is isentropic,p0 and ρ0 are constant throughout the flow,
7.6 Some Aspects of Supersonic Flow,Shock Waves
超音速流的一些特征：激波
51页图 1.30
An essential ingredient of a supersonic flow is the calculation of the
shape and strength of shock waves,This is the main thrust of chaps,
8 and 9,
超音速流动研究的一个重要内容就是计算激波的形状和强度。
这是第 8章和第 9章的主题。
A shock wave is an extremely thin region,typically on the order of
10-5cm,across which the flow properties can change drastically,
激波是一个极其薄的区域，厚度大约只有 10-5cm的量级，通过
激波流动特性发生剧烈变化。
7.7 Summary(小结）
1、热力学关系式,
状态方程,
对于量热完全气体,
( 7, 1 0 )
1-
R
( 9, 9 )
1-
( 7, 6 b )
( 7, 6 a )
( 7, 1 )
?
?
?
?
?
?
?
?
?
v
p
p
v
c
R
c
Tch
Tce
RTp
热力学第一定律的各种表达形式,
熵的定义,
热力学第二定律,
或对于绝热过程,
?
?
?
?
??
??
( 7, 2 0 )
( 7, 1 8 )
( 7, 1 1 )
d h - v d pT d s
pdvdeT d s
dewq ??
可逆过程,
( 7, 1 4 )
( 7, 1 3 )
i r r e v
r e v
ds
T
δq
ds
T
q
ds
??
?
及
?
( 7, 1 7 ) 0
( 7, 1 6 )
ds
T
q
ds
?
?
?
量热完全气体的熵增计算公式,
（ 7.25)
(7.26)
等熵流动的等熵关系式,
(7.32)
1
2
1
2
12 lnln p
pR
T
Tcss
p ???
1
2
1
2
12 lnln ?
?
? RT
Tcss ???
? ?1/
1
2
1
2
1
2
?
???
?
???
??
???
?
???
?? ???
?
?
T
T
p
p
2、压缩性
压缩性的一般定义,
（ 7.33）
对于等温过程,
(7.34)
对于等熵过程,
(7.35)
dp
dv
v
1???
T
T p
v
v ?
?
?
?
???
?
?
??? 1?
s
s p
v
v ?
?
?
?
???
?
?
??? 1?
3、无粘、可压缩流的控制方程,
连续方程,
????? ?????
S
dSVdvt 0???
?
0)( ?????? Vt ???
? ? ?? ???????? ???????
s vsv
dvfdSpVdSVvdVt ???? ???
xfx
p
Dt
Du ?? ?
?
???
yfy
p
Dt
Dv ?? ?
?
???
zfz
p
Dt
Dw ?? ?
?
???
动量方程,
（ 7.39)
(7.40)
(7.41)
(7.41a)
(7.41b)
(7.41c)
dsVVedvVet
sv
???
?
?
???
? ??
???
?
???
? ?
?
? ????? ?
22
22
??
? ? dvVfdsVpdvq
s vv
?? ?????? ????? ??? ??
? ? ? ?VfVpq
Dt
VeD ???? ??????? ??? 2/2
TRp ??
Tce v?
能量方程,
（ 7.43)
(7.44)
对于定常、绝热、无粘流，（ 7.44)和 (7.43)可以写成,
2 0
2
c o n s thc o n s tVh ??? 或
（ 7.53),(7.55)
完全气体状态方程,
量热完全气体内能,
（ 7.1)
(7.6a)
4、总温、总焓、总压、总密度的定义及概念,
总温和总焓定义为把流体微元（在我们的想象中） 绝热地
减速为静止时流体微元所对应的温度和焓值。
类似地，总压和总密度定义为把流体微元（在我们的想象
中） 等熵地 减速为静止时流体微元所对应的压强和密度。
在均匀自由来流的绝热流场中，总焓 h0在全流场中为常数，
相反，在非绝热流场中,h0随流场点的不同而不同。
类似地，在等熵流场中，总压和总密度在整个流场中为常
数，相反，在非等熵流场中，总压和总密度随流场点的不
同而不同。
5、激波
激波为超音速流中很薄的一层，通过激波压强、密度、温
度和熵增加；马赫数、流动速度、总压降低；总焓、总温
不变。
