Simulation of Condensing Compressible Flows · 2009. 7. 9. · Compressible Flows Maximilian Wendenburg. ... –Transonic Flows and Experiments –Condensation Fundamentals –Practical

Technische Universität MünchenLehrstuhl für Fluidmechanik – Fachgebiet Gasdynamik

Univ.-Prof. Dr.-Ing. habil. G. H. SchnerrFLM

Simulation of Condensing Compressible FlowsMaximilian Wendenburg



2

Outline• Physical Aspects

– Transonic Flows and Experiments– Condensation Fundamentals– Practical Effects

• Modeling and Simulation– Equations, Solver, Boundary Conditions

• Results– Validation– Effects in Laval Nozzles, Turbines and Compressors



3

Schlieren Photography: Visualizingx∂

∂ρ

lamp

mirror

mirror

camera

wind tunnel

subsonic

y

xsupersonic

sensitive!



4

Supersonic Wind Tunnel

A*

vacuumtank10 mbar34 m³

atmospherewindow with Laval nozzle

valve



5

T/T0ρ/ρ0

p/p0

Variation of Static Properties in Isentropic Flows

au

Mr

=



6high cooling rate

Condensation of Dissolved Water Vapor in Air

sK

dtdT 610−≈

low cooling rate

sK

dtdT 510−≈

sK

dtdT 310−−≈



7

Partial Pressure of Vapor pv in Isentropic Expansion

240 260 2800

500

1000

1500

2000 Φ0 = 65 %T0 = 295 Kp0 = 105 Pa

ps,∞(T)

i s e n t r o p ic e x p a n s

i o n

T [K]

pv [Pa]

(())0 295

%100)(,0⋅=Φ

∞ Tpp

s

v

)(,

TppS

s

v

∞

=



8

Types of Non-Equilibrium CondensationHomogeneous• Nucleation

cluster → nucleus → droplet• Droplet growth• Dominates at high

cooling rates

Heterogeneous• Particles serve as

seeds for droplets• Dominates at low

cooling rates



9

• Aircraft– Influence on

lift and drag– Loss of thrust

• Steam Turbines– Erosion– Oscillation

Practical Effects



10

Physical Aspects• Isentropic expansion: T↓, ρ↓, p↓• Transonic flows

→High cooling rates ( )→Supersaturation → non-equilibrium→Homogeneous/heterogeneous condensation→Influence of condensation on flow

sK

dtdT 610−≈



11

Euler Equations

0)()(0)()(

0)(

=+⋅∇+∂

∂

=+⊗⋅∇+∂

∂

=⋅∇+∂∂

upuEtE

Ipuutu

ut

rr

r

rr

r

r

ρρ

ρρ

ρρ

RTaTpp

eTTuEe

κ

ρ

=

=

=

−=

),()(

221 r

DtDu

tφρφρρφ =⋅∇+

∂∂ )()(

Thermodynamics, EOS:



12

Equations for Homogeneous Condensation

+=⋅∇+∂

∂

=⋅∇+∂∂

444 3444 2143421

r

r

growthdroplet

homhom

2hom

nucleationhom

3*3

4hom

hom

homhomhom

4)()(

)()(

dtrdnrJrugt

g

Juntn

ll ρπρρρρ

ρρ

π

TRpp

dtrd

ngr

TRmTr

mTJ

v

rsv

ll

vvl

v

v

⋅⋅−⋅=⋅⋅=

⋅⋅⋅⋅−⋅⋅⋅= ∞∞

πρα

ρπ

σπρρσ

π

243

)(34exp)(2

,hom3hom

hom

2*hom

2

3hom4434421

masses totallva

l

mmmmg

++=

Hertz-Knudsen Law

n kg-1 number density of dropletsg - condensate mass fractionJ m-3 s-1 nucleation rate



13

Multiphase-Multicomponent Thermodynamics

ρκ

κκρ

paTgg

ggTppggeTT

uEe

=

=

=

=

−=

),,(),,,(

),,(

max

max

max

221 r • Air as carrier gas with

vapor:both ideal gases

• Pure water vapor:real gas behavior



14

Solver• CATUM

– Condensation Technische Universität München– density-based FVM solver

• Cell-averaged values• Considering fluxes over cell boundaries: conservative

– 1-D, 2-D, 3-D– on structured multiblock grids– approach: solve local Riemann problems



15

Finite Volume Method

L RFLUX



16

Reconstruction at Cell Faces• Average values stored in

cell center• Reconstruct the required

value at the cell face• 4 adjacent cells for 2nd

order accuracy• Limiter functions: high

order smoothness where continuous, low order sharpness at shocks



17

1-D Riemann Problem

L R

=

EuQ

ρρρ

( )

+

=

upp

EuuQF

01ρ( ) 0=+ xt QFQ

shockcontact waveexpansion wave

pL > pR



18

Boundary Conditions: “Ghost Cells”• General: 2 “ghost cells”

for 2nd order accuracy• Inlet

– Pressure extrapolated from outermost cell, other values calculated from stagnation conditions (p0, T0, …) by isentropic relationships, e.g. with

• Wall– Velocity normal to wall is zero

• Outlet– Subsonic: set outlet pressure, extrapolate other values

– Supersonic: extrapolate all

1

212

1

222

1

2

1

11 −

++=

−

−κκ

κ

κ

MM

pp



19

Modeling and Simulation• Euler equations• Additional Eqns. for Condensation

– Influence on EOS• Reconstruction• Riemann problem• BCs via ghost cells



20

Validation of the Implementation

A1*

M1



21

Subcritical Heat Addition in Nozzle S2

x [m]

log10J ho

m

p/p01

g/gmax

-0.05 0 0.055

10

15

20

25

0

0.2

0.4

0.6

0.8

1log10 Jhom g/gmaxp/p01

diabactic

adiabatic

x [m]-0.05 0 0.05

diabatic

M1



22

Supercritical Heat Addition in Nozzle S2

x [m]

log10J ho

m

p/p01

g/gmax

-0.05 0 0.055

10

15

20

25

0

0.2

0.4

0.6

0.8

1log10 Jhom g/gmaxp/p01

diabactic

adiabatic

x [m]-0.05 0 0.05

Nozzle S2T0 = 295 Kp0 = 105 PaΦ0 = 65 %

diabatic

M1 M>1

M



23

Condensing Flows in Nozzle S2

240 260 2800

500

1000

1500

2000Φ0 = 65 %

Φ0 = 50 %

Nozzle S2T0 = 295 Kp0 = 105 Pa

ps,∞(T)(())

T [K]

pv [Pa]

M=1

Φ0



24

Unsteady Effects at High Relative Humidity

t



25

Validation of the Implementation

• T0 = 291.7 K• p0 = 1.006 x 105 Pa• Φ0 = 91.1 %• Mode 1

• f = 948 Hz• ½ grid: 241x21 cells

Result



26

Symmetric Oscillation in Nozzle S2• T0 = 291.7 K• p0 = 1.006 x 105 Pa• Φ0 = 91.1 %• Mode 1• f = 948 Hz• ½ grid: 241x21 cells



28

Hysteresis in Nozzle A1



29

Flow in Nozzle A1

• T0 = 305 K• p0 = 105 Pa• Φ0 = 77 %

• f = 1082 Hz• full grid: 220 x 41



30

Asymmetric Oscillation in Nozzle S2

• T0 = 305 K• p0 = 105 Pa• Φ0 = 95 %

• f = 3073 Hz• full grid: 241x41 cells

enforced disturbance



31

Turbine Stage VKIwith viscous effects



33

Rotor Stator InteractionHomogeneous Heterogeneously dominated

p01 = 0.417 bar T01 = 357.5 K β1,1 = 120°Rerotor = 1.13 · 106M2,2,is = 1.13 |u| = 175 m/s nhet,0 = 1016m-3rhet = 10-8 m frs = 2.46 kHz fvs,stator = 10.8 kHz fvs,rotor = 18.5 kHz

g

r



34

NORD-1500 Griffon II (1957)Mmax=2.2, Hmax=16400m, 34.32 kN

Alpha Jet (1973)Mmax=0.85, Hmax=14630m, 14.12 kN

Supersonic Axial Compressor



35

First Stage of a Supersonic Axial Compressor

Developed view ofa cylindrical cut



36

Section of Axial Compressor Stage



37

Cascade Element



38

Experimental Setup



39

Cascade Element of an Axial Transonic Compressor

π=p2/p1=1.764

diabatic flow

M1=1.3

T01=291 K p01=1.008 barφ0=80 %x=10 g/kg

reduction of ca. 18%

adiabatic flow

π=p2/p1=1.440

816.0ππadiabatic

diabatic =



40


π=p2/p1=1.630

diabatic flow

M1=1.3

T01=292 K p01=1.006 barφ0=69.8 %x=9.6 g/kg


adiabatic flow

π=p2/p1=1.358

833.0ππadiabatic

diabatic =



41


π=p2/p1=1.055

diabatic flow

M1=1.3

T01=294 K p01=1.005 barφ0=80.3 %x=9.6 g/kg


adiabatic flow

π=p2/p1=0.890

844.0ππadiabatic

diabatic =



42

Condensation Effects in Transonic Compressors

• Reduction of– inlet Mach number– stage compression ratio

• Loss of– thrust– efficiency



43

Results• Steady flows

– Subcritical– Supercritical

• Unsteady flows– Symmetric/asymmetric oscillations– Different modes

• Effects in Turbomachinery



Cпасибо

Thank you for your attention!



Discussion



46

Critical Radius )ln()( )(2*hom STRT Tr vl ⋅⋅⋅⋅

= ∞

ρσ

r [m]

∆G/k

B/T

0 5E-10 1E-09 1.5E-09-100

-50

0

50

100

150

200

T0 = 295 Kp0 = 105 PaΦ0 = 65 %

r* ≈ 5.5·10-9 m

M = 1.0

surfacecreation

expansionwork

5·10-10 10-9 1.5·10-9

240 260 2800

500

1000

1500

2000

Φ0 = 90 %

Φ0 = 65 %

Φ0 = 50 %

Φ0 = 35 %

Nozzle S2T0 = 295 Kp0 = 105 Pa

ps,∞(T)

i s e n t ro p i c e

x p a n si o n

(()) T [K]

pv [Pa]



47

Critical Radius

r [m]

∆G[J]

0 5E-10 1E-09 1.5E-09-1E-19

0

1E-19

2E-19

3E-19

T0 = 295 Kp0 = 105 PaΦ0 = 65 %

M = 1.2

M = 0.8

M = 1.0

r* ≈ 5.5·10-9 m

r* ≈ 9·10-9 m

r* ≈ 4·10-9 m

5·10-10 10-9 1.5·10-9-1·10-19



48

Transition in Nozzle A1

t [ms]

log10(|∆

p|+ε)

0 10 20 30 40-6-5-4-3-2-10

Nozzle A1T0 = 305 Kp0 = 105 PaΦ0 = 77 %

01

)()()( ptptptp uo −=∆



49

Transition in Nozzle S2



50

Appendix 1



51

Validation with Subcritical Flow in Nozzle S2



52

Heat Addition by Condensation• latent heat L [kJ/kg] of

gases in air:– H2O: 2260

• specific heat capacity cp [kJ/kg/K]:– air: 1.004

• Increase of static

Simulation of Condensing Compressible Flows · 2009. 7. 9. · Compressible Flows Maximilian Wendenburg. ... –Transonic Flows and Experiments –Condensation Fundamentals –Practical

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