7 - normal shock waves
TRANSCRIPT
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 1/16
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 2/16
• –
propagating supersonically, producing almost instantaneous
• deceleration, • compression,• increase in pressure, density and temperature, of the flow
• Normal Shocks – – perpendicular to the flow
– always decelerate flow to subsonic speed
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 3/16
• Shock is so thin that we only
in properties across them
•
1 2
isentropic
• In 1D flow onl normal shocks Isentropic flow Isentropic flow
Adiabatic flow
are possible
• Need… )()()( 13
212
211
2 M F T
M F M F u
=== ρ
)()()( 161215214
1
2 M F ss M F M M F p
p=−==
Rankine
)()()( 19*1
2
1801
02
1701
02
M F A M F p
p
M F ===
ρ
ρ Hugoniot
Relations
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 4/16
•
1 2
• Momentum Isentropic flow Isentropic flow
Adiabatic flow
• Energy
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 5/16
u2/u
1 = F
1 (M
1)
1 2From Energy
• 2121 −=… 22 p p
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 6/16
u2/u
1 = F
1 (M
1)
1 2From Mass & Momentum
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 7/16
u2/u
1 = F
1 (M
1)
1 2Solution
Energy Mass & Momentum
2
1
2
221
212
1
2
2
21
211
uu M M
aa −−−+= γ γ
2
1
2
221
1
221
1
2
uu M
uu M
uu γ γ −+=
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 8/16
2
1.5
1
1 u 2
/ u
0.5
0 2 4 6 8 100
M1
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 9/16
Density, Temp. and )1(2 2
12
F
M u
=
−+
=
γ
Pressure Ratios 11u +γ
=
2
2
22
1
2
12
2
2 11
1
u
M M
a −
−
−
+=
γ γ
Energy
T 2/T 1 = F 3 (M1)
p2/ p1 = F 4 (M1)
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 10/16
20
ρ2/ρ1
152 1
p2/p1
10
5
0 2 4 6 8 10
M1
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 11/16
1 2 M2
Entropy Gain
⎥⎤
⎢⎡
⎟⎟ ⎞
⎜⎜⎛
=−
−1
1212 log
γ
ρ
T
T
C
sse
v
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 12/16
For Air
1.8
2
1.2
1.4
.
0.6
0.8
1 M 2
0.2
0.4
0 2 4 6 8 100
M1
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 13/16
For Air
2
2.5
1.5
C v
0.5
1
( s 2 - s 1
)
0
0 2 4 6 8 10-0.5
M
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 14/16
Stagnation Pressure and 1 2Density Ratio
p01 , ρ 01, T 01Ima ine the entro e uation a lied between h othetical 02 , 02, 02
isentropic stagnation points on either side of the wave…
⎤⎡ ⎞⎛ − −1
010212
γ
ρ T ss
⎥⎦⎢⎣ ⎠⎝ 0201
16 ρ T C
e
v
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 15/16
Stagnation Pressure and Density Ratio
For Air
1.5
1
0 2 / ρ 0 1
0.5 p 0 2
/ p 0 1 , ρ
0 2 4 6 8 10
M1
8/13/2019 7 - Normal Shock Waves
http://slidepdf.com/reader/full/7-normal-shock-waves 16/16
T u
2
13
1
12
1
11
1
pT u
=−==
=== ρ
1 2
)()()( 19*
*
218
0217
02
161215214
1
M F A M F p M F
p
=== ρ
10101 p ρ Will meet later
TABULATED along with p1 /p02 in NACA 1135
Don’t need tables for u u or since 12 ρ u0202 p ρ
and 21 ρ u
0101 p ρ