gas dynamics equations
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7/29/2019 gas dynamics equations
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Isentropic Equations
T o
T = 1+
γ −1
2 M
2
(3.12)
Po
P = 1+γ
−1
2 M
2
γ
γ −1
(3.15)
ρ o
ρ = 1+
γ −1
2 M
2
1
γ −1(3.17)
A
A*
=1
M
2
γ +11+
γ − 1
2 M
2
γ +12 γ −1( )
(3.23)
Mass Flow Rate
&m = ρ AV = Po RT o
A γ M 1+ γ −1
2 M 2
−γ +1
2 γ −1( )(3.19)
Normal Shock Equations
M 2
2 = M
1
2 +2
γ −1
2γ
γ −1 M
1
2 −1
, M 1
2 = M
2
2 +2
γ −1
2γ
γ −1 M
2
2 −1
(4.9)
T 2T 1
=1+
γ − 1
2
M 1
2
1+γ − 1
2 M
2
2
(4.11)
P2
P1
=1+ γ M
1
2
1+ γ M 2
2(4.12)
ρ 2
ρ 1
=M
1
M 2
T 1
T 2
=M
1
M 2
1+γ −1
2 M
2
2
1+γ −1
2 M
1
2
(4.13)
Po2
Po1=A1
*
A2
*=
1+ γ −12
M 2
2
1+γ −1
2 M
1
2
γ
γ −1
2γ M 1
2
γ +1−γ −1
γ +1
=
γ +12
M 12
1+γ −1
2 M
1
2
γ
γ −1
1
2γ M 1
2
γ +1−γ − 1
γ + 1
1
γ −1
(4.15)
7/29/2019 gas dynamics equations
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ρ
1
ρ 2
= 1−V
S
(5.6)
p2
p1
= 1+γ SV
a1
2 (5.7)
p2
p1
=2γ
γ +1
S
a1
2
−γ −1
γ +1(5.8)
V
S =
2
γ +11−
a1
S
2
(5.9)
S =γ +1
4
V +
γ +1
4
V
2
+ a1
2 (5.10)
S =γ −1
2
p1
ρ 1
+γ +1
2
p1
ρ 1
p
2
p1
(5.11)
p2
p1
= 1+ γ γ +1
4
V
a1
2
+γ V
a1
1+γ +1
4
2
V
a1
2
(5.12)
T 2
T 1
=a2
2
a1
2= 1+
γ −1
a1
2SV −
V 2
2
(5.13)
Expansion waveRiemann invariants:
Left moving wave2
γ −1a +V = cons tan t , Right moving wave
2
γ −1a −V = cons tan t .
Isentropic relationship:
P = C 1 ρ γ = C
2T
γ
γ −1 = C 3a
2γ
γ −1
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Oblique Shock
P2
P1
=2γ M
1
2sin
2 θ ( )
γ +1
−γ −1
γ +1
(6.10)
ρ 2
ρ 1
=V n1
V n2
=V 1sin θ ( )
V 2sin θ − δ ( )
=γ +1( ) M
1
2sin
2 θ ( )
γ −1( ) M 1
2sin
2 θ ( )+ 2(6.11)
T 2
T 1
=a2
a1
2
=1+
γ −1
2 M
1
2sin
2 θ ( )
2γ
γ −1 M
1
2sin
2 θ ( )−1
γ +1( )22 γ −1( ) M
1
2sin
2 θ ( )
(6.12)
P02
P01
=
γ +1
2 M
1
2sin
2 θ ( )
1+γ −1
2 M
1
2sin
2 θ ( )
γ
γ −1
1
2γ
γ +1 M
1
2sin
2 θ ( )−γ − 1
γ + 1
1
γ −1
(6.13)
T 02
T 01
= 1
M 2
2 =1+
γ −1
2 M
1
2
γ M 1
2sin
2θ −γ −1
2
+M
1
2cos
2θ
1+γ −1
2 M
1
2sin
2θ (6.17)
tanδ = cotθ M
1
2sin
2θ −1
γ +1
2 M
1
2 − M 1
2sin
2θ −1( )(6.18)
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tan δ ( )= ±
P2
P1
−1
1+ γ M 1
2( )−P2
P1
2γ
γ +1 M
1
2 −γ −1
γ +1
−
P2
P1
γ −1
γ +1+P2
P1
(6.32)
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Prandtl-Meyer Expansion
ν =γ +1
γ −1tan
−1 γ −1
γ +1 M
2 −1( )
− tan
−1 M
2 −1( (7.10)
ν =γ +1
γ −1tan
−1 2
γ +1
po
p
γ −1( ) / γ
− tan−1 2
γ −1
po
p
γ −1( ) / γ
−γ +1
γ −1
(7.16)
Fanno Flow
fLmax
D=γ +12
ln
γ +12
1+γ −12
M 2
−1
γ 1−
1
M 2
−
γ +12γ
ln1
M 2
(9.16)
T
T *
=γ +1
2 + γ −1( ) M 2 (9.25)
P
P*
=1
M
γ +1
2 + γ −1( ) M 2
1/2
(9.26)
ρ
ρ *=V
*
V =
1
M
2 + γ −1( ) M 2
γ +1
1/2
(9.27)
Po
Po
*=
1
M
2 + γ −1( ) M 2
γ +1
γ +1
2 γ −1( )(9.28)
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Rayleigh Flow
q = c pT
o2−T
o1( ) (10.10)
P
P*
=1+ γ
1+ γ M 2 (10.11)
T
T *
=1+ γ ( )
2
M 2
1+ γ M 2( )
2 (10.12)
V
V *
=ρ *
ρ =
1+ γ ( ) M 2
1+ γ M 2 (10.13)
T o
T o
*=
1+ γ ( ) M 2
2 + γ −1( ) M 2
1+ γ M 2( )
2 (10.14)
Po
Po
*=
1+ γ
1+ γ M 2
2 + γ −1( ) M 2
1+ γ
γ
γ −1
(10.15)