Compressible Flow

Compressible Flow

Equations:

Incompressibla:

•Continuity

•Momentum

Unknowns:

Velocity, pressure

Kompressible:

•Continuity

•Momentum

•Energy

•Equation of state.

Unknowns:

Velocity, pressure,

density energy (enthalpy,

temperature)

Compressible FlowKompressibel strömning

Thermodynamis, a short review

Ideal gas: A gas that follows the equation of state RTp

Gas constantM

R

K kgJ 8314

M

Universal gas constant

Molecular weight

vp ccR

pc

vc

Specific heat at constant pressure

Specific heat at constant volume

v

p

c

ck Specific heat ratio

Compressible Flow

Internal energy

2

1

12 ˆˆ dTcuu v

2

1

12ˆˆ dTchh p

Enthalpy

If cp and cv are constant: 1212 ˆˆ TTcuu v

1212ˆˆ TTchh p

Compressible Flow

Isentropic process (adiabatic, reversibel)

0ˆ pdvuddqr

vdppdvudpvudhd ˆˆˆ

rdqpdvudvdphd ˆˆ

T

dqds r

pdvuddp

hdTds ˆˆ

Added heat

Enthalpy

Entropy

1v

Compressible Flow

Isentropic

pdvuddp

hdTds ˆˆ

v

dvRTdTc

p

dpRTdTcTds vp

2

1

2

1

2

1

2

1

2

1v

dvR

T

dTc

p

dpR

T

dTcds vp

21 ss

1

2

1

2 lnlnp

pR

T

Tcp

1

2

1

2 lnln

R

T

Tcv

kk

k

T

T

p

p

1

21

1

2

1

2

Speed of Sound

Consider a pressure wave

moving with the speed C

C

V

TT

p

pp

0V

T

p

Instead consider that the

gas is flowing through a

standing wave

VCV

TT

p

pp

CV

T

p

Speed of Sound

VCV

TT

p

pp

CV

T

p

Momentum inut VVmF

CVCACApppA

VCp

Continuity AVCAC

122 pCCp

VCp

CV

CV

Speed of Sound

VCV

TT

p

pp

CV

T

p

12 p

C

In a sound wave are small ,p

Låt220 a

pC

speed of sound

Adiabatic processkonst.konst.

Ts

pk

pa

For an ideal gas: kRTkp

a

Speed of Sound

When can one assume incompressible flow?

x

u

xu

x

u

x

u

x

u

xu

dvs.

Can be written asV

dVd

VV

dadp 2

Speed of sound

From Bernoulli: VdVdp

11 2

2

2

22 Ma

a

V

V

dp

a

dp

Mach numberNormally the limit is set to

3.0Ma

Compressible Flow

Ma < 0.3 Incompressible

0.3 < Ma < 0.8 Subsonic flow

0.8 < Ma < 1.2 Transonic flow

1.2 < Ma < 3.0 Supersonic flow

3.0 < Ma Hypersonic flow

Compressible Flow

Adiabatic och isentropic stationary flow

The energy equation along a stream linevwqgzVhgzVh 2

2221

211

2

1ˆ2

1ˆ

For gases one may neglect 12 zzg

For y larger than T

0

0

q

wv0

222

211

ˆkonstant2

1ˆ2

1ˆ hVhVh

Perfect gas: 0

2

2ˆ Tc

VTcTch ppp

Definition: Stagnation enthalpy/temperature = The enthalpy/temperature

the gas would get if brought to rest adiabatically

Stagnation enthalpy

20

2

11 Ma

k

T

T

Compressible Flow

If the flow is isentropic:

121

1

00

12100

2

11

2

11

k

k

k

k

k

k

k

Mak

T

T

Mak

T

T

p

p

Critical values, values at Ma=1

1

0

*

0

*

1

2

1

2

k

k

kp

p

kT

T

2

1

0

*

1

1

0

*

1

2

1

2

ka

a

k

k

Both the stagnation

values and thecritical

values are useful as

references

Note that stagnation pressure and

stagnation density are not

constant in adiabatic flow, only in

isentropic flow.

Nozzles

Isentropic flow with area change

x

y

xh

xR

yxV , Assume

1. very thin boundary layers

2. small increase in area

3. Large curvaturr

1dx

dh

xRxh

x

y

xh

xV

Nozzles

Continuitet constant mxAxVx

Take the diferential forms of continuity and

momentum equations

22

2

1

10

0

V

dp

MaA

dA

V

dV

dadp

VdVdp

A

dA

V

dVd

Nozzles

22 1

1

V

dp

MaA

dA

V

dV

0dA

0dA

1Ma 1Ma

0

0

dp

dV

0

0

dp

dV

0

0

dp

dV

0

0

dp

dV

1

012

Ma

dAMa otherwise unphysical!

in the smallest section

dV

Normal shocks

Normal shocks(adiabatic but not reversible)

1

1

1

1

1

1

1

ˆ

Ma

s

h

p

A

V

2

2

2

2

2

2

2

ˆ

Ma

s

h

p

A

V

shock

continuitet 222111 VAVA

Momentum2

1112

2222211 VAVAApAp

Energy0

222

211

ˆconstant2

1ˆ2

1ˆ hVhVh

0

22

2

21

122

TcV

TcV

Tc ppp

(3)

Normal shocks

1

1

1

1

1

1

1

ˆ

Ma

s

h

p

A

V

2

2

2

2

2

2

2

ˆ

Ma

s

h

p

A

V

21 AA2211 VV

211

22221 VVpp

Only compression shocks are possible, i.e. p2 > p1

1111

ˆ

k

pk

k

p

c

cc

pc

cc

pc

R

pcTch

p

vp

p

vp

ppp

(1)

(2)

Combine (1), (2) och (3) 21

1

21

1

211 kMa

kRT

kV

p

V

02

222

11ˆ

2

1ˆ2

1ˆ hVhVh (3)

Normal shocks

1

1

1

1

1

1

1

ˆ

Ma

s

h

p

A

V

2

2

2

2

2

2

2

ˆ

Ma

s

h

p

A

V

Combinera (1), (2) och (3)

21

1

21

1

211 kMa

kRT

kV

p

V

1

2

1

1

1

211

1

2 kp

V

kp

p

Use:

121

1 21

1

2

kkMakp

p(4)

11

2 p

pif 11 Ma

Normal shocks

1

1

1

1

1

1

1

ˆ

Ma

s

h

p

A

V

2

2

2

2

2

2

2

ˆ

Ma

s

h

p

A

V

(2) can be written as

22

21

1

2

1

1

kMa

kMa

p

p

Introduce (4)

12

2121

212

2

kkMa

MakMa

11 21 MaMa

01020102

*1

*2121212

121221 11

TTpp

AATTVV

ppssMaMa

Nozzles

AVm

maxmm då 1Ma**** VAm

Further decreasing pb will not change the

mass flow since 1max Ma

2

1

00*

112

1

2

1

2

1

0*1

1

0***

max

1

2

1

2

1

2

RTAk

k

RTk

Ak

VAm

kk

k

Nozzles

Convergent-divergent

Note that the flow is only supersonic

at the exit in cases G, H and I

Film

Schlieren visualisation

Kompressibel strömning

FILM