compress ible flow review

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






• 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






• 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



• 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






• 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






• 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






• 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






• 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






• 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






• 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






• 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






• 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






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






• 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

compress ible flow review

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