Simulation of Condensing Compressible Flows Maximilian Wendenburg
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 2
Schlieren Photography: Visualizing y ρ x subsonic supersonic x lamp mirror sensitive! wind tunnel mirror camera 3
Supersonic Wind Tunnel valve vacuum tank 10 mbar 34 m³ window with Laval nozzle atmosphere A* 4
Variation of Static Properties in Isentropic Flows T/T 0 p/p 0 ρ/ρ 0 M = r u a 5
Condensation of Dissolved Water Vapor in Air low cooling rate dt 3 10 dt K s dt 5 10 dt K s dt 6 10 dt high cooling rate K s 6
Partial Pressure of Vapor p v in Isentropic Expansion p v [Pa] 2000 Φ 0 = 65 % 1500 1000 S = pv p, ( T ) s i s e n t r o p i c e x p a n s i o n Φ 0 = pv p, ( T s ) T 0 = 295 K p 0 = 10 5 Pa 100% 500 p s, (T) (( )) 0 0 240 260 280 295 T [K] 7
Types of Non-Equilibrium Condensation Homogeneous Nucleation cluster nucleus droplet Droplet growth Dominates at high cooling rates Heterogeneous Particles serve as seeds for droplets Dominates at low cooling rates 8
Practical Effects Aircraft Influence on lift and drag Loss of thrust Steam Turbines Erosion Oscillation 9
Physical Aspects Isentropic expansion: T, ρ, p Transonic flows dt 6 K High cooling rates ( 10 ) Supersaturation non-equilibrium dt Homogeneous/heterogeneous condensation Influence of condensation on flow s 10
FLM 11 Euler Equations 0 ) ( ) ( 0 ) ( ) ( 0 ) ( = + + = + + = + pu Eu t E pi u u t u u t r r r r r r r ρ ρ ρ ρ ρ ρ RT a T p p e T T u E e κ ρ = = = = ), ( ) ( 2 2 1 r Dt D u t φ ρ φ ρ ρφ = + ) ( ) ( Thermodynamics, EOS:
Equations for Homogeneous Condensation ( ρn t ( ρg t hom hom ) ) + ( ρn + ( ρg hom hom r u) r u) = J hom 4π *3 = ρl 3 r J hom 14243 nucleation n kg -1 number density of droplets g - condensate mass fraction J m -3 s -1 nucleation rate + 2 drhom ρl 4πrhomρnhom 1444 2444dt 3 droplet growth J r hom hom σ ( T ) 3 m l g n l dr dt *2 r m l v σ R 2 = v exp hom = 3 2 π 3 4π ρ v hom ρ ρ 4π 3 hom = α ρ v p ( T ) T v p 2π R s, r v T g = ml ma + mv + ml 14243 4 total masses Hertz-Knudsen Law 12
Multiphase-Multicomponent Thermodynamics e T p κ a = = = E = T ( e, g p( ρ, T, g = κ( g κ u r 1 2 Air as carrier gas with 2 vapor: max, g) both ideal gases max p ρ max, g, T), g) Pure water vapor: real gas behavior 13
Solver CATUM Condensation 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 14
Finite Volume Method L FLUX R 15
Reconstruction at Cell Faces Average values stored in cell center Reconstruct the required value at the cell face 4 adjacent cells for 2 nd order accuracy Limiter functions: high order smoothness where continuous, low order sharpness at shocks 16
1-D Riemann Problem L R Q t F ( Q) = 0 + x Q ρ = ρu ρe F ( Q) = 1 0 ρu u + p E up expansion wave contact wave shock p L > p R 17
Boundary Conditions: Ghost Cells General: 2 ghost cells for 2 nd order accuracy Inlet Pressure extrapolated from outermost cell, other values calculated from stagnation conditions (p 0, T 0, ) by isentropic relationships, e.g. with p p 1 2 1+ = 1+ κ 1 2 κ 1 2 M M 2 2 2 1 κ κ 1 Wall Velocity normal to wall is zero Outlet Subsonic: set outlet pressure, extrapolate other values Supersonic: extrapolate all 18
Modeling and Simulation Euler equations Additional Eqns. for Condensation Influence on EOS Reconstruction Riemann problem BCs via ghost cells 19
Validation of the Implementation A 1 * M<1 M>1 20
Subcritical Heat Addition in Nozzle S2 M<1 M>1-0.05 0 0.05 x [m] log 10 J hom 25 20 15 10 p/p 01 log 10 J hom g/g max diabactic diabatic adiabatic 1 0.8 0.6 0.4 0.2 p/p 01 g/g max 5-0.05 0 0.05 x [m] 0 21
Supercritical Heat Addition in Nozzle S2 Nozzle S2 T 0 = 295 K M<1 M>1 M>1 shock: increase of T, ρ, p p 0 = 10 5 Pa Φ 0 = 65 % -0.05 0 0.05 x [m] M<1 log 10 J hom 25 20 15 10 p/p 01 onset earlier log 10 J hom shock g/g max diabactic diabatic adiabatic 1 0.8 0.6 0.4 0.2 p/p 01 g/g max 5-0.05 0 0.05 x [m] 0 22
Condensing Flows in Nozzle S2 p v [Pa] Φ 0 2000 1500 Φ 0 = 65 % Φ 0 = 50 % 1000 M=1 500 p s, (T) Nozzle S2 T 0 = 295 K p 0 = 10 5 Pa 0 (( )) 240 260 280 T [K] 23
Unsteady Effects at High Relative Humidity t 24
Validation of the Implementation T 0 = 291.7 K p 0 = 1.006 x 10 5 Pa Φ 0 = 91.1 % Mode 1 Result f = 948 Hz ½ grid: 241x21 cells 25
Symmetric Oscillation in Nozzle S2 T 0 = 291.7 K p 0 = 1.006 x 10 5 Pa Φ 0 = 91.1 % Mode 1 f = 948 Hz ½ grid: 241x21 cells 26
Hysteresis in Nozzle A1 28
Flow in Nozzle A1 T 0 = 305 K p 0 = 10 5 Pa Φ 0 = 77 % f = 1082 Hz full grid: 220 x 41 29
Asymmetric Oscillation in Nozzle S2 T 0 = 305 K p 0 = 10 5 Pa Φ 0 = 95 % enforced disturbance f = 3073 Hz full grid: 241x41 cells 30
Turbine Stage VKI with viscous effects 31
Rotor Stator Interaction Homogeneous Heterogeneously dominated g r p 01 = 0.417 bar T 01 = 357.5 K β 1,1 = 120 Re rotor = 1.13 10 6 M 2,2,is = 1.13 u = 175 m/s n het,0 = 10 16 m -3 r het = 10-8 m f rs = 2.46 khz f vs,stator = 10.8 khz f vs,rotor = 18.5 khz 33
Supersonic Axial Compressor NORD-1500 Griffon II (1957) M max =2.2, H max =16400m, 34.32 kn Alpha Jet (1973) M max =0.85, H max =14630m, 14.12 kn 34
First Stage of a Supersonic Axial Compressor Developed view of a cylindrical cut 35
Section of Axial Compressor Stage 36
Cascade Element 37
Experimental Setup 38
Cascade Element of an Axial Transonic Compressor π=p 2 /p 1 =1.764 adiabatic flow M 1 =1.3 π π diabatic = adiabatic 0.816 reduction of ca. 18% π=p 2 /p 1 =1.440 T 01 =291 K p 01 =1.008 bar φ 0 =80 % x=10 g/kg diabatic flow 39
Cascade Element of an Axial Transonic Compressor π=p 2 /p 1 =1.630 adiabatic flow M 1 =1.3 π π diabatic = adiabatic 0.833 reduction of ca. 17% π=p 2 /p 1 =1.358 T 01 =292 K p 01 =1.006 bar φ 0 =69.8 % x=9.6 g/kg diabatic flow 40
Cascade Element of an Axial Transonic Compressor π=p 2 /p 1 =1.055 adiabatic flow M 1 =1.3 π π diabatic = adiabatic 0.844 reduction of ca. 15% π=p 2 /p 1 =0.890 T 01 =294 K p 01 =1.005 bar φ 0 =80.3 % x=9.6 g/kg diabatic flow 41
Condensation Effects in Transonic Compressors Reduction of inlet Mach number stage compression ratio Loss of thrust efficiency 42
Results Steady flows Subcritical Supercritical Unsteady flows Symmetric/asymmetric oscillations Different modes Effects in Turbomachinery 43
Cпасибо Thank you for your attention!
Discussion
2 σ ( T ) * Critical Radius r = hom ρ ( T ) R T ln( S) l v p v [Pa] Φ 0 = 90 % 200 surface creation 2000 1500 i s e n t r o p i c e x p a n s i o n Φ 0 = 65 % Φ 0 = 50 % G / k B / T 150 100 50 0-50 expansion work r* 5.5 10-9 m M = 1.0 (( p 0 = 10 5 Pa 0 )) 240 260 280 T [K] -100 0 5 10 5E-10 1E-09 10-9 1.5 10 1.5E-09-9 r [m] 1000 500 p s, (T) T 0 = 295 K p 0 = 10 5 Pa Φ 0 = 65 % Φ 0 = 35 % Nozzle S2 T 0 = 295 K 46
Critical Radius 3E-19 r* 9 10-9 m M = 0.8 2E-19 G [J] 1E-19 r* 5.5 10-9 m M = 1.0 0 r* 4 10-9 m M = 1.2-1 10-1E-19-19 0 5 10 5E-10 10 1E-09-9 1.5 10 1.5E-09-9 r [m] T 0 = 295 K p 0 = 10 5 Pa Φ 0 = 65 % 47
Transition in Nozzle A1 p( t) 0-1 = p o ( t) p 01 p u ( t) log 10 ( p +ε) -2-3 -4-5 -6 0 10 20 30 40 t [ms] Nozzle A1 T 0 = 305 K p 0 = 10 5 Pa Φ 0 = 77 % 48
Transition in Nozzle S2 49
Appendix 1 50
Validation with Subcritical Flow in Nozzle S2 51
Heat Addition by Condensation latent heat L [kj/kg] of gases in air: H 2 O: 2260 Increase of static specific heat capacity c p [kj/kg/k]: air: 1.004 52