compress ible flow review
TRANSCRIPT
8/22/2019 Compress Ible Flow Review
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AE6050 Compressivle Flow -1
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• In continuum fluid mechanics, often start byconsidering conservation or transport equations,
e.g.,
– mass – momentum
• In compressible flow, must include energy equation
– kinetic energy of flow can not be neglected
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AE6050 Compressivle Flow -2
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Mass/continuity
• Momentum
• Energy
• Species conservation
0=∂
∂
ρ+
ρ
j
j
x
u
Dt
D
ρ= ii w
Dt
DY net mass production
rate of species i
j
j
j F
x
p
Dt
Duρ+
∂
∂−=ρ
body force
qu F t
p
Dt
Dh j j
o+ρ+
∂
∂=ρ
volumetric heating,
e.g., radiation
stagnation enthalpy,2
2
1
uhmix +
j j
xu
t Dt
D
∂
∂+
∂
∂=
???
substantial derivative
All molecular diffusion terms neglected in these equations
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AE6050 Compressivle Flow -3
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Thermal eq. state
• Caloric eq. state
– thermally perfect gas
– if reacting, R≠ const.
RT p =ρ
( )
( ) ( )
( )T ph
T T p RT e
T RY eY hY h
mix
mixmix
iiiiiimix
,
,
=
+=
+== ååå
he,
( ) ;T ee =
ρ+=+= pe pvehwith
thermally perfect (ideal) gas - TPG
( ) RT ba p =−ρρ+ 12 Van der Waals state eq: a,b constants
( ) RT T eh +=
dT decv = v p cc≡ Rcc v p +=dT dhc
p=
calorically
perfect gas - CPG
c p, cv, γ =constants
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Compressivle Flow -4
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Properties that would be achieved if flow brought to restadiabatically, reversibly and with no external work
• Stagnation Temperature – from energy conservation:
no work but flow work and adiabatic
R
c
T cT
T p
p
o
12
u11
2
−γ γ
+=Þ
2o M2
11
T
T −+=
12o
M2
1
1 p
p −γ γ
÷ ø
öçè
æ −γ
+=
( )
2
u2
=−T T c o p
( )1−γ γ
= T T p p oo
ÞTo (and ho) constant foradiabatic flow
• Stagnation Pressure – for rev. + adiabatic
=isentropic process (∆∆∆∆s=0)
Þpo (and so) constant if
also reversible
Expressions for TPG, CPG
Bulk KE
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Compressivle Flow -5
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Steady, nonreacting,
no body forces
02
12
22
=ρ
++τ
u
du
p
u
p
dpdx
A
L
p
p x
dx
p+dp
T+dT
ρ+dρu+du
A+dA
h+dh
p
T
ρu
A
h
τx
δq0
2
12
2
=++ρ
ρ
A
dA
u
dud
02
1q2
22
=−−δ
RT
dh
u
du
RT
u
RT
Mass
Momentum
Energy
Valid only for
dA/dx small
shear stress normalstress
momentum
change
heat addition KE change thermal energy change
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Compressivle Flow -6
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• In addition, limit to nonreacting
TPG (nonreact.ÞR=const.)
02 2
22 =
γ ++
τ
u
du M
p
dpdx
A
L
p
p x
dx
p+dp
T+dT
ρ+dρu+du
A+dA
γ +dγ
M+dM
p
T
ρu
A
γ
M
τx
δq0
2
12
2
=++ρ
ρ
A
dA
u
dud
( )0
2
1q2
2
2 =−−γ
−δ
T
dT
u
du M
T c p
Mass
Momentum
Energy
0=−
ρ
ρ−
T
dT d
p
dpIdeal Gas
Eq. State
02
2
2
2
=
γ
γ ++−
d
T
dT
u
du
M
dM
Mach
Number
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AE6050 Compressivle Flow -7
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Combine conservation/state equations
– can algebraically show if also CPG
( )ïþ
ïýü
ïî
ïíì
−γ +γ +δ
−
−+
= A
dA
D
dx f M M
T c M
M
M
dM
o p
21q
1
2
11
22
2
2
2
2
• So we have three ways to change M of flow
– area change (dA): e.g., conv.-div. nozzles – friction: f > 0, same effect as – dA
– heat transfer:heating, δδδδq>0, like – dA
cooling, δδδδq<0, like +dA
friction factor-from τx
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AE6050 Compressivle Flow -8
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Subsonic flow ( M <1): 1– M 2 > 0
– friction, heating, converging areaÞ increase M (dM >0)
– cooling, diverging areaÞ decrease M (dM <0)
• Supersonic flow ( M >1): 1– M 2
< 0 – friction, heating, converging areaÞ decrease M (d M <0)
– cooling, diverging areaÞ increase M (d M >0)
( ) ïþ
ïý
ü
ïî
ïí
ì
−γ +γ +δ
−
−+
= A
dA
D
dx f M M T c M
M
M
dM
o p21
q
1 2
11
222
2
2
2
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AE6050 Compressivle Flow -9
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Effect on transition point: sub⇔supersonic flow
• As M →1, 1– M 2→0, need { } term to approach 0
• For isentropic flow,
– sonic condition is dA=0, throat
( )ïþ
ïýü
ïî
ïíì
−γ +γ +δ
−
−+
= A
dA
D
dx f M M
T c M
M
M
dM
o p
21q
1
2
11
22
2
2
2
2
• For friction or heating, need dA>0 – sonic point in diverging section
• For cooling, need dA<0
– sonic point in converging section
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AE6050 Compressivle Flow -10
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• For TPG/CPG + steady, get equations for each TD property
change as function of Mach number change
• Analytic solutions if ONLY area change (Isentropic Nozzle)
OR friction (Fanno flow) OR heat xfer (Rayleigh flow)
– usually tabulated for given γ
2
2
2
2
2
11
2
1
M
dM
M
M
T
dT
T
dT
o
o
−γ +
−
−=
2
2
22
2
2
11
1
M
dM
M T
dT
u
du
o
o
−γ +
+=
A
dA
u
dud −−=
ρ
ρ2
2
2
1÷÷ ø
öççè
æ +γ
−= D
dx f
u
du M
p
dp2
22
2o po
o
T cT
dT qδ=
2
2
2
2
2
11
2
M
dM
M
M
p
dp
p
dp
o
o
−γ ++=
o
o
o
o
p
dp
T
dT
R
ds−
−γ =
1
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AE6050 Compressivle Flow -11
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Speed of sound and Mach waves
• Shocks – normal shocks
– oblique shocks
• act like normal shock in direction normal to wave – detached (bow) shocks
• Prandtl Meyer expansions and compressions
• Wave “reflections” – impose/“transmit” some boundary condition
(pressure or velocity) to flow
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AE6050 Compressivle Flow -12
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• For supersonic flow, can define region where
disturbance has had an effect (been “heard”)
• Conical region delineated by tangents tosound wave spheres
• Waves coalesce at edge of cone,
produce largest disturbance
– Mach wave (Mach line)
• Angle between Mach line
and body motion, Mach angle
0 -1 -2 -3
at
µ
Zone of
ActionZone of
Silence÷
ø
öç
è
æ =÷
ø
öç
è
æ =µ −−
v
asin
vt
atsin 11
vt
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AE6050 Compressivle Flow -13
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
1
1lim
1
2
1 p p 12 −γ
+=
ρ
ρ
>>
• p increase acrossnormal shock isgreatest staticproperty change
• Density ratio andvelocity ratioapproach limit
γ =1.4
0
5
10
15
20
25
30
35
40
1 2 3 4 5 6 7
M1
T 2 / T 1 , p 2 / p 1
, ρ ρρ ρ
2 / ρ ρρ ρ
1
p2/p1
T2/T1
ρ2/ρ1,v1/v2
v1
ρ1
T1
p1
M1
v2
ρ2
T2
p2
M2
• T, p and ρ increase,v decreases
Shock: compression wave with steep gradient
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AE6050 Compressivle Flow -14
School of Aerospace Engineering
Copyright © 2001,2003 by Jerry M. Seitzman. All rights reserved.
• Each turn produced by
infinitessimal flow change
• Prandtl Meyer function
1
11 M
1sin−=µ
212 M1sin−=µ
M1
M2
δÞÞÞÞMach waves
d νµM
M+dMM
dM
M2
11
1Md
2
2
−γ +
−= ν