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Compressible Flow: Through Nozzles and Diffusers
Aerospace Engineering, International School of Engineering (ISE) Academic year : 2013-2014 (January – May, 2014)
Jeerasak Pitakarnnop , Ph.D.
[email protected] [email protected]
March 22, 2014 Aerodynamics II 1
Exhaust Nozzle
Shock Diamond
Mach diamonds from an F-‐16 taking off with a7erburner
CFD simula>on of a shock wave inside an Exhaust Nozzle
March 22, 2014 Aerodynamics II 2
Separated Flow
Viscous Model
Inviscid Model
Supersonic Diffuser a) External Compression b) Internal Compression c) Combine Compression
March 22, 2014 Aerodynamics II 3
Scramjet Engine
Quasi-‐One-‐Dimensional Flow
ρ1u1A1 = ρ2u2A2
p1A1 + ρ1u12A1 + pdA
A1
A2∫ = p2 + ρ2u22
h1 +u12
2
= h2 +u22
2
h = cpT p = ρRT
ConKnuity: Momentum: Energy: Enthalpy & Eq. of state:
March 22, 2014 Aerodynamics II 4
Governing EquaKon for Quasi-‐One-‐Dimensional Flow
dAA+dρρ+duu= 0
ρudu = −dp
dht = dh+udu = 0dpp=dρρ+dTT
p = ρa2
γMarch 22, 2014 Aerodynamics II 5
Compressible Flow in Converging and Diverging Duct
• 0 ≤ M < 1 (subsonic flow): Area decrease à Vel. increase
• M > 1 (supersonic flow): Area decrease à Vel. decrease
• M = 1 (sonic flow), dA = 0: local maximum or minimum area distribuKon.
dAA= M 2 −1( ) duu
March 22, 2014 Aerodynamics II 6
Supersonic Nozzle and Diffuser
March 22, 2014 Aerodynamics II 7
Nozzle Flows: Area-‐Mach RelaKon • Mach n o . a t a n y locaKon is a funcKon of the local to the sonic throat area.
• A < A* is impossible in an isentropic flow.
AA*!
"#
$
%&2
=1M 2
2γ +1
1+ γ −12
M 2!
"#
$
%&
(
)*
+
,-
γ+1( )γ−1( )
March 22, 2014 Aerodynamics II 8
Isentropic Supersonic Nozzle Flow
AA*!
"#
$
%&2
=1M 2
2γ +1
1+ γ −12
M 2!
"#
$
%&
(
)*
+
,-
γ+1( )γ−1( )
March 22, 2014 Aerodynamics II 9
Calculated from equation below
Isentropic Subsonic Nozzle Flow 1) Low Speed Subsonic Flow:
pe is slightly less than p0 Athroat is more than A*
2) Moderate Speed Subsonic Flow: sKll subsonic flow at throat
3) Supersonic Flow at Throat
March 22, 2014 Aerodynamics II 10
Infinite no. of possible isentropic subsonic soluKons where
p0 ≥ pe ≥ pe,3
Choked Flow
March 22, 2014 Aerodynamics II 11
Once the flow becomes sonic at the throat, the upstream is frozen.
Downstream disturbance can no longer communicate with upstream,
the change in exit pressure could not be detected.
Supersonic Nozzle with Internal Shock
March 22, 2014 Aerodynamics II 12
Normal Shock At The Exit
March 22, 2014 Aerodynamics II 13
Supersonic Nozzle with External Shock
Overexpanded Nozzle
Back Pressure = Exit Pressure
Underexpanded Nozzle
March 22, 2014 Aerodynamics II 14
Flow through Exhaust Nozzle 1. pe = pa: Subsonic Flow
Th rough ou t t he nozzle.
2. pe = pa: Subsonic Flow Th rough ou t t he nozzle but Mthroat =1.
3. pe = pa: Subsonic Flow in the converging secKon & Supersonic Flow in the diverging secKon. – MAXIMUM THRUST – Design CondiKon for
the ideal case
March 22, 2014 Aerodynamics II 15
pi pe pa
Flow through Exhaust Nozzle 4. pe < pa: Overexpand
5. pe > pa: Underexpand
March 22, 2014 Aerodynamics II 16
pi pe pa
Flow through Exhaust Nozzle 6. pe < pa: Overexpand +
No rma l Shock @ diverging secKon.
7. pe < pa: Overexpand + Normal Shock @ exit plane.
March 22, 2014 Aerodynamics II 17
pi pe pa
Supersonic Diffuser
March 22, 2014 Aerodynamics II 18
EX: Flow through Nozzle • Consider the isentropic flow through a convergent-‐divergent nozzle with an exit-‐to-‐throat area raKo of 2. The reservoir pressure and temperature are 1 atm and 288 K, respecKvely. Calculate the Mach no., pressure, and temperature at both the throat and the exit for the cases where: a) The flow is supersonic at the exit. b) The flow is subsonic throughout the enKre nozzle
except at throat, where M = 1 c) If the exit pressure is 0.973 atm (Determine only
Mach no.)
March 22, 2014 Aerodynamics II 19