1D Steady Compressible Flow - Virginia Techdevenpor/aoe3114/4 - 1D Steady Flow.pdf · u 1D Steady...

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!)