1d steady compressible flow - virginia techdevenpor/aoe3114/4 - 1d steady flow.pdf · u 1d steady...
TRANSCRIPT
1D Steady Compressible Flow1D Steady Compressible Flow
1D Steady Flowu 1D Steady FlowTp
ix
,etcA
• Applies to flows in ducts (such as nozzles, engines, wind tunnels• Applies to external flows in streamtubesApplies to external flows in streamtubes
Conservation of MassConservation of Mass
ix
,
0.
CV CS
dSnUdVt
Conservation of MassConservation of Mass
0.. dSnUdSnU
ix
,21
Conservation of EnergyConservation of Energy
ix
, Q
CSCSCV
dSnUptQdSnUUedVUe
t .).()()( 2
212
21
Conservation of EnergyConservation of Energy0).()( 2
21 dSnUUh
ix
,
CS
ExampleExampleEstimate the temperature on the of the space shuttle nose during
re‐entryy
Other forms of the Energy Equation.2/2 constuTCp In adiabatic
flow
Other forms of the Energy Equation
Stagnation flowadiabaticconstMT .2
11 2
PropertiesflowisentropicconstMp .
211
2
12
flowisentropicconstM .2
111
1
2
2
0 211
MTT
So we define…
120 2
11
2
Mpp flowadiabaticconstT .0
11
20 2
11
M flowisentropicconstp .0
And thus the energy equation becomes…
flowisentropicconst.0
Flow Property 12
0 211
M
pp
Variations 11
2
0 211
M
1
12
0 211
M
TT
0.8
p/p0
/0T/T0
0.4
0.6
0
0.2
0 1 2 3 40
Mach number
TablesTablesNACA 1135 Equations, Tables and Charts for Compressible Flow
(on course website)
12
211
M
pp
(on course website)
11
2
0
211
2
M
p
12
0
211
2
MTT
0 2 T
TABLESTABLES
12
211
M
TT
11
2
0
11
2
M
T
12
0
11
2
Mp
0 21
M
p
1
1
12
0 211
M
TT
11
2
0 211
M
12
0 211
Mpp
ExampleTABLES
ExampleA Mach 0.8 air flow enters a nozzle with a pressure and temperature of 75kPa and 250K and leaves it at Mach 2 Find the exitand leaves it at Mach 2. Find the exit temperature and pressure for isentropic flow?
1 2
ExampleTABLES
ExampleA Mach 0.8 air flow enters a nozzle with a pressure and temperature of 75kPa and 250K and leaves it at Mach 2 Find the exit ISENTROPICand leaves it at Mach 2. Find the exit temperature and pressure for isentropic flow?
25075
1
1
KTkPap
?6.14
2
2
TkPap
ISENTROPIC
8.01
1
M 0.22
2
M
ExampleTABLES
ExampleA Mach 0.8 air flow enters a nozzle with a pressure and temperature of 75kPa and 250K and leaves it at Mach 2 Find the exit ISENTROPICand leaves it at Mach 2. Find the exit temperature and pressure for isentropic flow?
250751
KTkPap
1576.142
KTkPap
ISENTROPIC
8.0250
1
1
MKT
0.2157
2
2
MKT
Variations
1. Asked for 2 ?
2. Given u2 instead of M2 ?
3. Given u1 and u2 instead of M1 and M2 ?
4. What ifM1 had been given as 0 ?
5 Flow was adiabatic but not isentropic (e g shock present)?5. Flow was adiabatic, but not isentropic (e.g. shock present)?
6. Flow not adiabatic?
Conservation of MomentumConservation of Momentum
ix
,
CSCV CS
dSnpdSnUUdVUt
).(
CSCV CS
The Momentum EquationThe Momentum Equationn
Area A A+dA
x dSinpdSnUU ).().(
nn
dpp pu d
CSCS
ix
,dAAduudAu )())(( 22
u,
dpp 1
duu d
sAdppdAAdpppA sin)())(( 2
1
0 ududp EULER’S EQUATIONdpp 2As
0 ududp EULER S EQUATION
Isentropic Flow Solution ududp / Isentropic Flow Solution
constuTC
ududp
p
2/
/2
Using constp / gives (Redundant)constuTCp 2/Using constp / gives (Redundant)
The Momentum EquationThe Momentum Equationn
Area A A+dA
x dSinpdSnUU ).().(
nn
dpp pu d
CSCS
ix
,dAAduudAu )())(( 22
u,
dpp 1
duu d
sAdppdAAdpppA sin)())(( 2
1
0 ududp EULER’S EQUATIONdpp 2As
0 ududp EULER S EQUATION
Constant Area Flow Solution ududp Constant Area Flow Solution
constup
duudp
p
2
Useful in analyzing constup y gshock waves. Not Bernoulli!)