Gas Dynamics of Variable Area Ducts

P M V Subbarao

Professor

Mechanical Engineering Department

I I T Delhi

Development of Efficient and Compact Passive Devices….

0A

dA

V

dV

ρ

dρ

0V

dVM

2p

dp

D

fdxM

2 2

22

h

2 γγ

0

V

dV

2

M1

T

dT

TC

q2

22

p

'''

γδ

0T

dT

ρ

dρ

p

dp

0V

dV

M

dM

T

dT2

2

2

2

System of Equaions

Combine conservation, state equations– can algebraically show

2

0p

'''

h

22

2

2

2

M1TC

q

A

dA2

D

fdxM

M1

M2

11

M

dMγ

δγ

γ

So we have three ways to change Mach number of a flow

– area change (dA):

– friction: f > 0, same effect as –dA

– heat transfer: heating, q’’’ > 0, like –dA cooling, q’’’ < 0, like +dA

Effect of Shape of duct on Flow

2

0p

'''

h

22

2

2

2

M1TC

q

A

dA2

D

fdxM

M1

M2

11

M

dMγ

δγ

γ

Pure shape effects :

A

dA2

M1

M2

11

M

dM2

2

2

2γ

A

dA

1M

M12

M

dM2

2

2

2 γ

A

dA

1M2

M12

M

dM2

2γ

1M2

M12M

dM

A

dA

2

2

γ

Pure Shape Effects …..

p

dp1

u

duM2

γ

p

dp1

A

dA

M1

M2

2

γ

ρ

dρ

p

dpγ

Subsonic Nozzle Subsonic Diffuser

dA < 0 & M <1So, du > 0 & dp <0

dA > 0 & M <1So, du < 0 & dp>0

A

dA

1M

1

u

du2

T

dT

A

dA

M1

M12

2

γ

Supersonic Diffuser Supersonic Nozzle

dA < 0 & M >1So, du < 0 & dp >0

dA > 0 & M >1So, du >0 & dp<0

p

dp1

u

duM2

γ

p

dp1

A

dA

1M

M2

2

γ

A

dA

1M

1

u

du2

ρ

dρ

p

dpγ

ρ

dρ

A

dA

1M

M2

2

T

dT

A

dA

1M

M12

2

γ

Generation of High Pressure from Supersonic velocity

p

dpM

u

du 2

γ

p

dp1

M

M1

A

dA2

2

γ

p

dp1

ρ

dρ

γ

p

dp1

T

dT

γ

γ

A

dA

1M

1

u

du2

Generation of Supersonic Velocity from Rest

p

dpM

u

du 2

γ

p

dp1

M

M1

A

dA2

2

γ

p

dp1

ρ

dρ

γ

p

dp1

T

dT

γ

γ Rocket Nozzle

X – 15 : Hypersonic Research at the Edge of Space

The X-15 was air launched from a B-52 aircraft at about 45,000 ft and speeds upward of 500 mph. The airplane first set speed records in the Mach 4-6 range with Mach 4.43 on March 7, 1961; Mach 5.27 on June 23, 1961; Mach 6.04 on November 9, 1961; and Mach 6.7 on October 3, 1967. It also set an altitude record of 354,200 feet (67 miles) on August 22, 1963, and provided an enormous wealth of data on hypersonic air flow.Aerodynamic heating, control and stability at hypersonic speeds, reaction controls for flight above the atmosphere, piloting techniques for reentry, human factors, and flight instrumentation.

Intake at Design Altitude

p1

p2pthroat

p

p2

p*

p1

0p

dp1

M

M1

A

dA2

2

γAt M=1

Minimum Area = A* : Also called throat

*** AcρρuA

u

RT

ρ

ρ

u

c

ρ

ρ

A

A **

γ

Frictional Adiabatic Flow in A Variable Area Duct

2

0p

'''

h

22

2

2

2

M1TC

q

A

dA2

D

fdxM

M1

M2

11

M

dMγ

δγ

γ0

A

dA2

D

fdxM

M1

M2

11

M

dM

h

22

2

2

2

γ

γ

Constant Mach number frictional flow

h

2

2D

fMA

dx

dA γ