空气动力学基本公式集合

2
2,1
− ( − 1)
,1 = 1 sin
2 =
,2
sin( − )
2
( + 1),1
2
=
2
1 2 + ( − 1),1
2
2
2
=1+
(,1
− 1)
1
+1

2
2
2
2 + ( − 1),1
2
= [1 +
(,1
− 1)]





1
+
= −
+



流体表面应力张量


= [





] =

2
2
− ∙ −
3
…
[
1
= (
+
)
2

+ )

2

2
1
+ ( ) − × ( × ) = −

2

由热力学关系的矢量形式改写上述方程
1
= ℎ −

→

2
+ ( ) − × ( × ) = − ℎ

2
由滞止焓改写上述方程
2
ℎ0 = ℎ + ( )
2
→

− × ( × ) = − ℎ0
2 −1
=(
)
0
+1
20
a∗ = √
+1
定义速度系数
λ=
( + 1)2

√
=
∗
2 + ( − 1)2
2λ2
= √
( + 1) − ( − 1)λ2
定义气体动力学函数

−1 2
τ() = = (1 −
)
0
+1


− 1 2 −1
π() =
= (1 −
2
2 + ( − 1)12
2
(12 − 1)]
(12 − 1)]
} − ln [1 +
2
( + 1)1
+1
+1
斜激波关系式
tan = 2 cot
12 sin2 − 1
12 ( + cos 2) + 2
2
2 + ( − 1),1
,2 = √

2
不考虑粘性力
∭

2
[ ( + )] = ∭[ + () + ∙ − ∙ ()]

2

2
( + ) = + () + ∙ − ∙ ()

2









∙ () = ∙ ( ∙ ) = ∙ () + ∙ ( ) = ( − ) + [ ( ) − ( )] = ( ) −
)] −






3







2
( ∙ )

= −
+ [ ( +
)] +
)] + 2 (
)−
[ ( +
{






3

对于通常情况即不考虑 μ 随温度的变化，上述方程可化为
2
( + 1),1
1
+1
2 − 1 = ln {[1 +
02
=1
01
02
=
01
2
−1
( + 1),1
[
]
2
2 + ( − 1),1
1
−1
2
2
[1 + + 1 (,1
− 1)]
2
2
2 + ( − 1),1
2
2
2
(,1
− 1)]
1
d = d + d = dℎ − d = dℎ − d

2 − 1 = ln
2
2
2
1
− ln = ln + ln
1
1
1
2
等熵关系式

2
2
2 −1
=( ) =( )
1
1
1
滞止参数
ℎ0 = ℎ +
2
2
2
2
2 0


+ ∙ () =
+ = 0
Baidu Nhomakorabea

() () ()
+
+
+
=0




定常不可压

+
+
=0

动量方程




∭ = ∭ + ∯ ( ∙ ) = ∭ [
+
+ ∙ () + ( ∙ )]


= (2 − 1)


−1
2
1 + 2
=

1 − 2

+1
+1
2(−1)
2

1
2
−1
=
) (1 +
2 )]
[(
∗

+ 1
2

( ∗) =

2 −1
( − 1) (
+ 1)
+1


2 [1 − ( )
0
2

] ( )
0
无粘流基本方程
)
0
+1
1

− 1 2 −1
ε() =
= (1 −
)
0
+1
激波与膨胀波
正激波关系式
2 + ( − 1)12
2 = √
212 − ( − 1)
( + 1)12
2 1
=
=
1 2 2 + ( − 1)12
2
2
(12 − 1)
=1+
1









2
( ∙ )
= −
+ 2 ( ) +
)] −
[ ( + )] + [ ( +







3







2
( ∙ )

= −
+ [ ( + )] + 2 ( ) + [ ( +






1
=
+
+
+
= + (
+
+
)







{
N—S 方程








2
=
+
= −
+
[ (
+
)] −
(
)





3
+1

2
2
2 + ( − 1)12
(12 − 1)]
= [1 +
( + 1)12
1
+1
2 − 1 = ln {[1 +
02
=1
01
02
=
01
( + 1)12 −1
[
]
2 + ( − 1)12
1
−1
2
[1 + + 1 (12 − 1)]

+ ∙ s =
≥


粘性流体基本方程
连续方程


+ ∙ () =
+ ∙ = 0


定常不可压
() () ()
+
+
+
=0





+
+
=0

动量方程

1
=
+ ( ∙ ) = − +
雷诺输运定理及随体导数


∭ = ∭ + ∯ ( ∙ )




=
+ ( ∙ ) =
+ ∙



连续方程




=
∭ = ∭ + ∯ ( ∙ ) = ∭ [ + ∙ ()] = 0




≥




()

=
∭ = ∭ + ∯ ( ∙ ) = ∭ [
+ ∙ ()] ≥ ∭




()

+ ∙ () ≥


→




+
+ s ∙ () + ∙ s ≥



→


定常状态
× ( × ) = ℎ0 −
均能流（滞止焓均匀分布）
、均熵流及均能均熵流
× ( × ) = −
× ( × ) = ℎ0
× ( × ) =
能量方程

=
+




2
∭ ( + ) = ∭ + ∯ ( ∙ ) + ∭ ∙ − ∯ ( ∙ ) +

2
设质量力有势且在固定点处不随时间变化
=


∙ = ∙ = (
− )=


→


2

(ℎ +
− ) = + ∙ () +

2

绝热无机械功输入输出的定常流动
ℎ+
2
− = const
2
熵方程
≥


̇

1
+
= +







1
=
+
+
+
= + (
+
+
)











1
=
+
+
+
= + (
+
+
)





利用矢量恒等式改写欧拉方程
2
( ∙ ) = ( ) − × ( × )
2
→

2
1
+ ( ) − × ( × ) = − +

2

2
1
+ ( ) + 2( − ) = −
+
2

2
1











2


( + ) = + ∙ () + ∙ − [ ( ) − ]

2



2


2
( + + ) = + ∙ () + ∙ +
= (ℎ + )

2







2
=
+
= −
+ ∆ +




3



=
+ ( ∙ ) = − + ∆ + ( ∙ )


3
能量方程
} − ln [1 +
(,1
− 1)]
2
( + 1),1
+1
+1
膨胀波关系式
马赫角 μ
1 = arcsin
1
1
2 = arcsin
1
2
普朗特—迈耶函数
+1
−1
(2 − 1) − arc tan √2 − 1
ν() = √
arc tan √
−1
+1
2
− ∙ −
3
…
= 2 −

=
+ ( ∙ ) = + ∙


(
…
2
∙
3

( +
)


( +
)

2
2
− ∙ − ]
3
= −

2 2 /02 2 + ( − 1)12
=
=
1 1 /01 2 + ( − 1)22
= ν(2 ) − ν(1 )

2 2 /02
2 + ( − 1)12 −1
=
=(
)
1 1 /01
2 + ( − 1)22
准一维流动与喷管流动
面积-速度关系式
空气动力学基本公式集合
热力学参数及关系
δ + δ = d
ℎ=+


=
R = 287J/(kg ∙ K)
ℎ = + +
− =
= /
=
=
ℎ =

−1
=

−1
可逆过程（不一定绝热，等熵过程为可逆且绝热的过程）
+
( ) + 2( − ) = −
+
2

2
1
+ ( ) + 2( − ) = −
+

{ 2
克罗克运动方程（在葛罗米柯运动方程基础上吧焓梯度和熵梯度与旋涡量建立联系）
对于理想气体，忽略质量力后的葛罗米柯运动微分方程为




1
+
+
+
= −









1
+
+
+
= −









1
+
+
+
= −
{




葛罗米柯运动微分方程（把涉及运动旋涡部分的项分离出来而使研究无旋运动时方程简化）





= ∭ { [ + ( ∙ )]} = ∭ − ∯ + = ∭( − ) +

不考虑粘性力则为欧拉方程

1
=
+ ( ∙ ) = −





1
+
= −



02
∗ 2
∗ 2
= +
=
+
=
+
=
=
=
+
2
2
−1 2
−1 2
−1 −1 −1
2
0
−1
=1+
2

2

1
−1
0
−1
= (1 +
2 )

2
−1
0
−1
= (1 +
2 )

2
临界参数
∗
∗ 2
2
=( ) =
0
0
+1

∗
2 −1
=(
)
0
+1
1
∗