Application of modern tools for the thermo-acoustic study of annular combustion chambers

Transcription

1 Application of modern tools for the thermo-acoustic study of annular combustion chambers Franck Nicoud University Montpellier II I3M CNRS UMR 549 and CERFACS

2 Introduction A swirler (one per sector) November, 200 VKI Lecture 2

3 Introduction 0-24 sectors One swirler per sector November, 200 VKI Lecture 3

4 Introduction Y. Sommerer & M. Boileau CERFACS November, 200 VKI Lecture 4

5 Introduction November, 200 VKI Lecture 5

6 SOME KEY INGREDIENTS Flow physics turbulence, partial mixing, chemistry, two-phase flow, combustion modeling, heat loss, wall treatment, radiative transfer, Acoustics complex impedance, mean flow effects, acoustics/flame coupling, non-linearity, limit cycle, non-normality, mode interactions, Numerics Low dispersive low dissipative schemes, non linear stability, scalability, non-linear eigen value problems, November, 200 VKI Lecture 6

7 PARALLEL COMPUTING june 200 # Site Computer Oak Ridge National Laboratory USA 2 National Supercomputing Centre in Shenzhen (NSCS) China 3 DOE/NNSA/LANL USA 4 National Institute for Computational Sciences USA 5 Forschungszentrum Juelich (FZJ) Germany 6 NASA/Ames Research Center/NAS USA 7 National SuperComputer Center in Tianjin/NUDT China 8 DOE/NNSA/LLNL USA 9 Argonne National Laboratory USA 0 Sandia National Laboratories USA Cray XT5-HE Opteron Six Core 2.6 GHz Dawning TC3600 Blade, Intel X5650, NVidia Tesla C2050 GPU BladeCenter QS22/LS2 Cluster, PowerXCell 8i 3.2 Ghz / Opteron DC.8 GHz, Voltaire Infiniband Cray XT5-HE Opteron Six Core 2.6 GHz Blue Gene/P Solution SGI Altix ICE 8200EX/8400EX, Xeon HT QC 3.0/Xeon Westmere 2.93 Ghz, Infiniband NUDT TH- Cluster, Xeon E5540/E5450, ATI Radeon HD , Infiniband eserver Blue Gene Solution Blue Gene/P Solution Cray XT5-HE Opteron Six Core 2.6 GHz processors or more 458 Tflops or more November, 200 VKI Lecture 7

8 PARALLEL COMPUTING june 2007 (3 years ago ) # Site Computer DOE/NNSA/LLNL United States Oak Ridge National Laboratory United States NNSA/Sandia National Laboratories United States IBM Thomas J. Watson Research Center United States Stony Brook/BNL, New York Center for Computional United States DOE/NNSA/LLNL United States Rensselaer Polytechnic Institute, Computional Center United States NCSA United States Barcelona Supercomputing Center Spain BlueGene/L - eserver Blue Gene Solution IBM Jaguar - Cray XT4/XT3 Cray Inc. Red Storm - Sandia/ Cray Red Storm, Cray Inc. BGW - eserver Blue Gene Solution IBM New York Blue - eserver Blue Gene Solution IBM ASC Purple - eserver pseries p GHz IBM eserver Blue Gene Solution IBM Abe - PowerEdge 955, 2.33 GHz, Infiniband Dell MareNostrum - BladeCenter JS2 Cluster, IBM processors or more (factor 7) 56 Tflops or more (factor 8) 0 Leibniz Rechenzentrum Germany HLRB-II - Altix GHz SGI November, 200 VKI Lecture 8

9 PARALLEL COMPUTING Large scale unsteady computations require huge computing resources, an efficient codes Speed up processors November, 200 VKI Lecture 9

10 PARALLEL COMPUTING November, 200 VKI Lecture 0

11 Thermo-acoustic instabilities Premixed gas The Berkeley backward facing step experiment. November, 200 VKI Lecture

12 Thermo-acoustic instabilities Self-sustained oscillations arising from the coupling between a source of heat and the acoustic waves of the system Known since a very long time (Rijke, 859; Rayleigh, 878) Not fully understood yet but surely not desirable November, 200 VKI Lecture 2

13 Better avoid them AIR LPP SNECMA LPP injector (SNECMA) FUEL November, 200 VKI Lecture 3

14 Flame/acoustics coupling COMBUSTION Modeling problem Wave equation ACOUSTICS Rayleigh criterion: Flame/acoustics coupling promotes instability if pressure and heat release fluctuations are in phase November, 200 VKI Lecture 4

15 A tractable D problem IMPOSED VELOCITY FRESH GAS FLAME BURNT GAS IMPOSED PRESSURE p ' ( L, t ) = n, τ u ' (0, t) = 0 T T 2 = 4T 0 n τ model 0 L/2 L Kaufmann, Nicoud & Poinsot, Comb. Flame, 2002 γp0 q' ( x, t) = n u'( L / 2 γ for the L, t τ ) δ x 2 unsteady heat release n :interaction index τ : time lag November, 200 VKI Lecture 5

16 Equations u ' (0, t) = 0 T p ' ( L, t ) = 0 T 2 = 4T 0 L L 0 < x < : p' 2 p' c 2 = 2 t x CLASSICAL ACOUSTICS 2 wave amplitudes 0 L < x < L : p' 2 p' c2 = t x CLASSICAL ACOUSTICS 2 wave amplitudes TWO JUMP RELATIONS L / 2 L / 2 [ p' ] = 0 and [ u' ] = n u' ( L / 2, t ) L / 2 L / 2 τ November, 200 VKI Lecture 6

17 Dispersion relation Solve the 4x4 homogeneous linear system to find out the 4 wave amplitudes Consider Fourier modes p'( x, t) ( jωt pˆ( x e ) = R ) I I ( ω) ( ω) > < 0 : 0 : damped mode amplified mode Condition for non-trivial (zero) solutions to exist ωl cos 4c cos 2 ωl 4c ne ne jωτ jωτ + 3 = 0 Uncoupled modes Coupled modes November, 200 VKI Lecture 7

18 Stability of the coupled modes Eigen frequencies cos 2 ωl 4c ne ne jωτ jωτ + 3 = 0 Steady flame n=0: 4c 2 ω m 0 = ± arccos ± + 2, 3 m π m L, = 0,,2,... Asymptotic development for n<<: 4c ω = ωm 0 n cos ( ) ( ωm,0τ ) + j sin( ωm,0τ ) 9Lsin ωm,0l / 2c Complex [ ] o( ) m, + n pulsation November, 200 VKI Lecture 8 shift Kaufmann, Nicoud & Poinsot, Comb. Flame, 2002

19 Time lag effect The imaginary part of the frequency is 4c sin( ωm,0τ ) n 9Lsin ω L / 2c ( ) m,0 Steady flame modes such that ( L / 2c ) 0 sin ω m 0, < The unsteady HR destabilizes the flame if sin m,0 ( ω τ ) > 0 0 < ω τ < π[2π ] 0 < τ < [ T ] m,0 m,0 unstable unstable unstable 0 Tm,0 / 2 Tm,0 3T m, 0 / 2 2 m, 0 5T m, 0 / 2 T 2 m,0 T 3T m, 0 unstable τ November, 200 VKI Lecture 9

20 Time lag effect The imaginary part of the frequency is 4c sin n 9Lsin ( ωm,0τ ) ( ω L / 2c ) m,0 Steady flame modes such that ( L / 2c ) 0 sin ω m 0, > The unsteady HR destabilizes the flame if sin m,0 ( ω τ ) < 0 π < ω τ < 2π [2π ] < τ < T [ T ] m,0 m,0 unstable unstable unstable 0 Tm,0 / 2 Tm,0 3T m, 0 / 2 2 m, 0 5T m, 0 / 2 T 2 m,0 T 3T m, 0 m,0 τ November, 200 VKI Lecture 20

21 Numerical example T L = 300 = 0.5 K m Steady flame Unteady flame n=0.0 τ=0. ms Imaginary frequency (Hz) Uncoupled modes STABLE UNSTABLE Real frequency (Hz) November, 200 VKI Lecture 2

22 OUTLINE. Computing the whole flow 2. Computing the fluctuations 3. Boundary conditions 4. Analysis of an annular combustor November, 200 VKI Lecture 22

23 OUTLINE. Computing the whole flow 2. Computing the fluctuations 3. Boundary conditions 4. Analysis of an annular combustor November, 200 VKI Lecture 23

24 BASIC EQUATIONS reacting, multi-species gaseous mixture November, 200 VKI Lecture 24

25 BASIC EQUATIONS energy / enthalpy forms Sensible enthalpy of species k Specific enthalpy of species k Sensible enthalpy of the mixture Specific enthalpy of the mixture Total enthalpy of the mixture E = H p/ρ Total non chemical enthalpy of the mixture Total non chemical energy of the mixture November, 200 VKI Lecture 25

26 BASIC EQUATIONS Diffusion velocity / mass flux Exact form Practical model Satisfies mass conservation November, 200 VKI Lecture 26

27 BASIC EQUATIONS Stress and heat flux λ = µc p Pr November, 200 VKI Lecture 27

28 Turbulence. Turbulence is contained in the NS equations 2. The flow regime (laminar vs turbulent) depends on the Reynolds number : laminar Velocity Re = Ud ν Length scale viscosity turbulent November, 200 VKI Lecture 28 Re

29 RANS LES - DNS Streamwise velocity DNS LES RANS time November, 200 VKI Lecture 29

30 About the RANS approach Averages are not always enough (instabilities, growth rate, vortex shedding) Averages are even not always meaningful Oscillating flame T T2 T T2>T T time Prob(T=T mean )=0!! November, 200 VKI Lecture 30

31 The basic idea of LES E(k) Resolved scales Modeled scales k c k November, 200 VKI Lecture 3

32 LES equations Assumes small commutation errors Filtered version of the flow equations : November, 200 VKI Lecture 32

33 Laminar contributions Assumes negligible cross correlation between gradient and diffusion coefficients: November, 200 VKI Lecture 33

34 Sub-grid scale contributions Sub-grid scale stress tensor to be modeled τ sgs ij = ρ u~ u~ ρ u i j i u j Sub-grid scale mass flux to be modeled J sgs k, j ~ = ρ Y u~ + k j ρ Y k u j Sub-grid scale heat flux to be modeled q sgs j ~ = ρ E u~ + j ρ E u j November, 200 VKI Lecture 34

35 The Smagorinsky model From dimensional consideration, simply assume: u~ sgs sgs ~ ~ ~ u~ i τ ij τ kk δij = 2µ sgs Sij Skkδij, with Sij = x j x µ sgs = ρ ~ ~ 2 ( Cs ) 2Sij Sij i j The Smagorinsky constant is fixed so that the proper dissipation rate is produced, C s = 0.8 November, 200 VKI Lecture 35

36 The Smagorinsky model = τ The sgs dissipation is always sgs ij ij sgs ij ij positive Very simple to implement, no extra CPU time ε Any mean gradient induces sub-grid scale activity and dissipation, even in 2D!! sgs ~ S V 2µ = W = 0 ~ S because U ~ S but µ sgs 0 = U ( y) and S 2 0 Solid wall No laminar-to-turbulent transition possible Strong limitation due to its lack of universality. Eg.: in a channel flow, Cs=0. should be used November, 200 VKI Lecture 36

37 The Dynamic procedure (constant ρ) NS By performing, the following sgs contribution appears u ρ t i ui u + ρ x j j P = x i + sgs ( τ + τ ) ij x j ij τ sgs ij = ρu u ρu u i j i j Let s apply another filter to these equations sgs ij ij u u i i u τ τ j P + ρ + ρ = + t x x x j NS i j By performing, one obtains the following equations sgs ij Tij u u sgs i u τ i j P + ρ + ρ = + Tij = ρui u j ρuiu t x x x j i j τ sgs ij = ρu u ρu u i j i j j A B From A and B one obtains T sgs ij = τ ρu u + ρu sgs ij i j i u j November, 200 VKI Lecture 37

38 The dynamic Smagorinsky model Assume the Smagorinsky model is applied twice τ sgs ij 3 sgs τ kk δij = 2ρ 2 2 ( Cs ) 2Sij Sij Sij ( ) ij sgs sgs Tij Tkk δij = 2ρ Cs 2Sij Sij S 3 Assume the same constant can be used and write the Germano identity T sgs ij = τ ρu u + ρu sgs ij i j i u j ( ) 2 sgs 2 sgs Cs 2Sij Sij Sij + Tkk δij = 2ρ( Cs ) 2Sij Sij Sij + τ kk δij ρui u j ρui u j 2ρ C M = 2 s ij L C dynamically obtained s ij from the solution itself November, 200 VKI Lecture 38

39 The dynamic procedure The dynamic procedure can be applied locally : the constant depends on both space and time good for complex geometries, but requires clipping (no warranty that the constant is positive) 2 C M L ij ij L M ij ij M M ij ij depends on the SGS model can be computed from the resolved scales November, 200 VKI Lecture 39

40 How often should we accept to clip?. Example of a simple turbulent channel flow % clipping Y + 2. Not very satisfactory, and may degrade the results U + Local Dynamic Smagorinsky Plane-wise Dynamic Smagorinsky DNS Y + November, 200 VKI Lecture 40

41 The global dynamic procedure The dynamic procedure can also be applied globally: The constant depends only on time no clipping required, just as good as the static model it is based on Requires an improved time scale estimate 2 C M L ij ij L M ij ij M M ij ij depends on the SGS model can be computed from the resolved scales November, 200 VKI Lecture 4

42 Sub-grid scale model. Practically, eddy-viscosity models are often preferred 2. The gold standard today is the Dynamic Smagorinsky model Dynamically Computed From the grid or filter width strain rate : 2S ij S ij µ sgs = ρ C 2 2 ( time scale) 3. Looking for an improved model for the time scale November, 200 VKI Lecture 42

43 Description of the σ - model Eddy-viscosity based: Start to compute the singular values of the velocity gradient tensor (neither difficult nor expensive) µ sgs = ρ C ( ) 2 ( time scale) Null for axi-symmetic flows Null for isotropic ( time scale) = σ 3 ( σ σ )( σ σ ) σ AND near-wall behavior is O(y 3 )!! Null for 2D or 2C flows Null only if g ij = 0 November, 200 VKI Lecture 43

44 Sub-grid scale contributions Sub-grid scale stress tensor τ sgs ij = ρ u ~ u~ ρ u u µ i j i j sgs based model J Sub-grid scale mass flux of species k and heat flux q sgs k, j sgs j ~ = ρ Yku ~ j + ρ Yku j ρ E ~ u~ + ρ E u = j j In practice, constant SGS Schmidt and Prandtl numbers D sgs k µ µ sgsc sgs p ; Sck = 0.7 λsgs = ; Pr = Pr sgs = sgs sgs ρ Sck sgs 0.5 November, 200 VKI Lecture 44

45 Sub-grid scale heat release The chemical source terms are highly non-linear (Arrhenius type of terms) The flame thickness is usually very small (0. - mm), smaller than the typical grid size Poinsot & Veynante, 200 November, 200 VKI Lecture 45

46 The G-equation approach The flame is identified as a given surface of a G field The G-field is smooth and computed from Poinsot & Veynante, 200 Universal turbulent flame speed model not available November, 200 VKI Lecture 46

47 The thickened flame approach From laminar premixed flames theory: Multiplying a and dividing A by the same thickening factor F Poinsot & Veynante, 200 November, 200 VKI Lecture 47

48 The thickened flame approach The thickened flame propagates at the proper laminar speed but it is less wrinkled than the original flame: Total reaction rate R Total reaction rate R 2 R R < E R = R This leads to a decrease of the total consumption: 2 November, 200 VKI Lecture 48

49 The thickened flame approach An efficiency function is used to represent the sub-grid scale wrinkling of the thickened flame Diffusivity : Preexponential constant : a A Fa A/ F EFa EA/ F thickening SGS wrinckling This leads to a thickened flame (resolvable) with increased velocity (SGS wrinkling) with the proper total rate of consumption The efficiency function is a function of characteristic velocity and length scales ratios u S sgs Fδ 0 0 L L November, 200 VKI Lecture 49

50 The thickened flame approach Advantages. finite rate chemistry (ignition/ extinction) 2. fully resolved flame front avoiding numerical problems 3. easily implemented and validated 4. degenerates towards DNS: does laminar flames But the mixing process is not computed accurately outside the reaction zone. Extension required for diffusion or partially premixed flames 2. Introduction of a sensorto detect the flame zone and switch the F and E terms off in the non-reacting zones November, 200 VKI Lecture 50

51 About solid walls In the near wall region, the total shear stress is constant. Thus the proper velocity and length scales are based on the wall shear stress τ w : τ w uτ = l = ρ In the case of attached boundary layers, there is an inertial zone where the following universal velocity law is followed ν u τ y u u + C, κ : Von Karman constant, C + = ln y κ + u :"Universal constant", + = u u τ, κ 0.4 C 5. 2 y + = yu ν τ November, 200 VKI Lecture 5

52 Wall modeling A specific wall treatment is required to avoid huge mesh refinement or large errors, y u Velocity gradient at wall assessed from a coarse grid Exact velocity gradient at wall Use a coarse grid and the log law to impose the proper fluxes at the wall τ 32 model v u τ 2 model November, 200 VKI Lecture 52

53 About solid walls Close to solid walls, the largest scales are small Boundary layer along the x-direction: Vorticity ω x y z x - steep velocity profile, L t ~ κy - Resolution requirement: y + = O(), z + and x + = O(0)!! - Number of grid points: O(R τ2 ) for wall resolved LES u November, 200 VKI Lecture 53

54 Wall modeling in LES Not even the most energetic scales are resolved when the first off-wall point is in the log layer No reliable model available yet E(k) Resolved scales Modeled scales k c k November, 200 VKI Lecture 54

55 OUTLINE. Computing the whole flow 2. Computing the fluctuations 3. Boundary conditions 4. Analysis of an annular combustor November, 200 VKI Lecture 55

56 Considering only perturbations November, 200 VKI Lecture 56

57 Linearized Euler Equations assume homogeneous mixture neglect viscosity decompose each variable into its mean and fluctuation f ( x, t) = f0( x) + f ( x, t) f f 0 u c 0 assume small amplitude fluctuations ε << ; ε << ; November, 200 VKI Lecture 57 f c = ρ, p, T, s = 0 = γ p ρ 0 0 C v p ln ρ γ

58 Linearized Euler Equations the unknown are the small amplitude fluctuations, the mean flow quantities must be provided requires a model for the heat release fluctuation q contain all what is needed, and more : acoustics + vorticity + entropy November, 200 VKI Lecture 58

59 Zero Mach number assumption No mean flow or Zero-Mach number assumption f p f (, t ) = f ( ) + f (, t ); x ε << f = ρ p T s = Cv γ f ;,,, ln 0 x x 0 ρ u u( x, t ) = u c = 0 ( x) + u( x, t); ε << ; c0 0 γ p ρ 0 0 L a : acoustic wavelength Lf : flame thickness Probably well justified below 0.0 November, 200 VKI Lecture 59

60 Linear equations ρ Mass : + ρ0div 0 t Energy : ρ ( u ) + u ρ 0 = T t 0C v + u T0 = p0 u + Momentum : q State : u ρ0 t p p 0 ρ = ρ 0 = p T + T 0 - The unknowns are the fluctuating quantities ρ u, T,, p - The mean density, temperature, fields must be provided - A model for the unsteady HR q is required to close the system November, 200 VKI Lecture 60

61 Flame Transfer Functions Q =, t) dω ( t) q( x Ω Relate the global HR to upstream velocity fluctuations General form in the frequency space Qˆ ( ω) = F( ω) uˆ( x ref, ω) n-τ model (Crocco, 956), low-pass filter/saturation (Dowling, 997), laminar conic or V-flames (Schuller et al, 2003), entropy waves (Dowling, 995; Polifke, 200), Justified for acoustically compact flames November, 200 VKI Lecture 6

62 Local FTF model Flame not necessarily compact Local FTF model qˆ( x, ω) jωτ (, ) ˆ( ref, ) l ω u x ω n = nl ( x, ω) e x q U l mean n, τ : l Two scalar fields The scalar fields must be defined in order to match the actual flame response LES is the most appropriate tool assessing these fields bulk ref November, 200 VKI Lecture 62

63 Local FTF model ACOUSTIC WAVE + FLUCTUATING HEAT RELEASE ACOUSTIC VELOCITY AT THE REFERENCE POSITION n ( x, ω) e l ˆ ω ( x, ω) uˆ( x, ω) n jωτ ( x, ω) T l ref ref November, 200 Giauque et al., AIAA paper VKI Lecture 63

64 Back to the linear equations Let us suppose that we have a reasonable model for the flame response The set of linear equations still needs to be solved ρ Mass : + ρ0div 0 t Momentum : ( u ) + u ρ 0 = u ρ0 t = p State : p p 0 ρ = ρ 0 + T T 0 Energy : ρ T t 0C v + u T0 = p0 u + q November, 200 VKI Lecture 64

65 Time domain integration Use a finite element mesh of the geometry Prescribe boundary conditions Initialize with random fields Compute its evolution over time This is the usual approach in LES/CFD! November, 200 VKI Lecture 65

66 A simple annular combustor (TUM) Unsteady flame model with characteristic time delay τ Pankiewitz and Sattelmayer, J. Eng. Gas Turbines and Power, 2003 November, 200 VKI Lecture 66

67 Example of time domain integration TIME DELAY LEADS TO STABLE CONDITIONS TIME DELAY LEADS TO UNSTABLE CONDITIONS ORGANIZED ACOUSTIC FIELD Pankiewitz and Sattelmayer, J. Eng. Gas Turbines and Power, 2003 November, 200 VKI Lecture 67

68 The Helmholtz equation Since periodic fluctuations are expected, let s work in the frequency space p ( x, t) = R pˆ ( x) ( ) ( ) ( jωt ) ( ) ( jωt e u x t = R uˆ ( ) e ), x ( jωt q x, t = R qˆ x e ) From the set of linear equations for ρ, u, p, T, the following wave equation can be derived γ p r pˆ 2 pˆ 0 + ω = ρ0 jω( γ )qˆ November, 200 VKI Lecture 68

69 3D acoustic codes Let us first consider the simple steady flame case γ p r pˆ + ω pˆ ρ0 2 0 = 0 With simple boundary conditions pˆ = 0 or ρ ˆ ˆ 0 ω u n = p n = 0 Use the Finite Element framework to handle complex geometries November, 200 VKI Lecture 69

70 Discrete problem If m is the number of nodes in the mesh, the unknown is now P = [ p, pˆ,..., ˆ ] T ˆ 2 p m Applying the FE method, one obtains Linear Eigenvalue Problem of size N AP+ ω 2 P= discrete November, 200 VKI Lecture 70 0 version A : square of : γ p 0 ρ0 matrix

71 Solving the eigenvalue problem The QR algorithm is the method of choice for small/medium scale problems Shur decomposition: AQ=QT, Q unitary, T upper triangular Krylov-based algorithms are more appropriate when only a few modes needs to be computed Partial Shur decomposition: AQ n =Q n H n +E n with n << m A possible choice: the Arnoldi method implemented in the P- ARPACK library (Lehoucq et al., 996) November, 200 VKI Lecture 7

72 Solving the eigenvalue problem November, 200 VKI Lecture 72

73 Computing the TUM annular combustor PLENUM SWIRLED INJECTORS Experimentally, the first 2 modes are : 50 Hz (L) 300 Hz (C plenum) COMBUSTOR FE mesh of the plenum + 2 injectors + swirlers + combustor + 2 nozzles Mean temperature field prescribed from experimental observations November, 200 VKI Lecture 73

74 TUM combustor: first seven modes f (Hz) type 43 L 286 C plenum 503 2C plenum 594 2L 73 3C plenum 754 C combustor 769 3L November, 200 VKI Lecture 74

75 An industrial gas turbine burner Industrial burner (Siemens) mounted on a square cross section combustion chamber Studied experimentally (Schildmacher et al., 2000), by LES (Selle et al., 2004), and acoustically (Selle et al., 2005) November, 200 VKI Lecture 75

76 Instability in the LES From the LES, an instability develops at 98 Hz November, 200 VKI Lecture 76

77 Acoustic modes f (Hz) type 59 L 750 2L 92 LT 92 LT 350 3L 448 2LT 448 2LT November, 200 VKI Lecture 77

78 Turning mode The turning mode observed in LES can be recovered by adding the two 92 Hz modes with a 90 phase shift PRESSURE FLUCTUATIONS FOR 8 PHASES LES SOLVER: 98 Hz ACOUSTIC SOLVER: 92 Hz Selle et al., 2005 November, 200 VKI Lecture 78

79 Realistic boundary conditions Complex, reduced boundary impedance Z = pˆ ρcuˆ n A out A in n e. g.: Z Z = 0 pˆ = 0 = uˆ n = 0 Reflection coefficient R = A A in out = Z Z + Using the momentum equation, the most general BC is jω pˆ n pˆ = cz 0 November, 200 VKI Lecture 79

80 Discrete problem with realistic BCs In general, the reduced impedance depends on ω and the discrete EV problem becomes non-linear: AP + Boundary Terms jω pˆ n= pˆ cz ( ω) + ω 2 P = 0 Assuming Z( ω) ω ω AP+ BP+ CP= A, B, C = : Z 0 + C ω + square November, 200 VKI Lecture 80 C2 ω 2 ; Z 0, C 0 matrices, C 2 parameters Quadratic Eigenvalue Problem of size N

81 From quadratic to linear EVP Given a quadratic EVP of size N: 2 AP+ ωbp+ ω CP= 0 Add the variable: R = ω P Rewrite: IR + ω IP = 0 AP + BR + ω CR = 0 0 I P A B R I + ω 0 0 P C R = 0 Obtain an equivalent Linear EVP of size 2N and use your favorite method!! November, 200 VKI Lecture 8

82 Academic validation 0.5 m Z = / 2 + j Z = Z = / 2 + j Z = 0 T T 2 = 4T ACOUSTIC SOLVER EXACT STABLE UNSTABLE November, 200 VKI Lecture 82

83 Multiperforated liners Multiperforated plate (????) Dilution hole (resolved) Solid plate (Neumann) Example of a TURBOMECA burner November, 200 VKI Lecture 83

84 Multiperforated liners Designed for cooling purpose Burnt gas (combustion chamber) Cold gas (casing) but has also an acoustic effect. November, 200 VKI Lecture 84

85 Multiperforated liners 400 Hz 250 Hz Absorption coefficient November,

86 Multiperforated liners The Rayleigh conductivity (Howe 979) St ω a = U Under the M=0 assumption: (impedance-like condition) November, 200 VKI Lecture 86

87 Academic validation f = 382 8i Hz f = i Hz November, 200 VKI Lecture 87

88 Question There are many modes in the low-frequency regime They can be predicted in complex geometries Boundary conditions and multiperforated liners have first order effect and they can be accounted for properly All these modes are potentially dangerous Which of these modes are made unstable by the flame? November, 200 VKI Lecture 88

89 Accounting for the unsteady flame Need to solve the thermo-acoustic problem γ p r pˆ 2 pˆ 0 + ω = ρ0 jω( γ )qˆ e.g. : ( ) ( γ-) qmean ( ) jωγ qˆ = ρ 0,xref U bulk n l x, ω e jr ( ω ) τ l ( x, ω ) pˆ xref n ref " local FTF model" In discrete form AP + ω BP+ ω CP= D( ω) P A, B, C, D 2 :square matrices Non-Linear Eigenvalue Problem of size N November, 200 VKI Lecture 89

90 . Solve the Quadratic EVP Iterative method AP ω BP+ ω CP= At iteration k, solve the Quadratic EVP [ A D( ω )] P ω BP ω 2 CP 0 k + + = k k 3. Iterate until convergence < tol ω k ω k ω k,p is solution of the thermo-acoustic problem November, 200 VKI Lecture 90

91 Comparison with analytic results INLET n=5, τ=0. ms OUTLET exact modes Iteration 0 Iteration Iteration 2 Iteration 3 November, 200 VKI Lecture 9

92 About the iterative method No general prove of convergence can be given except for academic cases If it converges, the procedure gives the exact solution of the discrete thermo-acoustic problem The number of iterations must be kept small for efficiency (one Quadratic EVP at each step and for each mode) Following our experience, the method does converge in a few iterations in most cases!! November, 200 VKI Lecture 92

93 OUTLINE. Computing the whole flow 2. Computing the fluctuations 3. Boundary conditions 4. Analysis of an annular combustor November, 200 VKI Lecture 93

94 BC essential for thermo-acoustics p =0 u =0 Acoustic analysis of a Turbomeca combustor including the swirler, the casing and the combustion chamber C. Sensiau (CERFACS/UM2) AVSP code November, 200 VKI Lecture 94

95 Acoustic boundary conditions Boundary Conditions LES or Helmholtz solver November, 200 VKI Lecture 95

96 Analytical approach Assumption of quasi-steady flow (low frequency) M Nozzle Inlet M 2 Nozzle Outlet Theories (e.g.: Marble & Candle, 977; Cumpsty & Marble, 977; Stow et al., 2002; ) Conservation of Total temperature, Mass flow, Entropy fluctuations November, 200 VKI Lecture 96

97 Analytical approach 2 Mass Momentum Energy November, 200 VKI Lecture 97

98 Analytical approach Acoustic wave Acoustic wave Entropy wave Compact systems November, 200 VKI Lecture 98

99 Example: Compact nozzle. Compact choked nozzle: 2. Compact unchoked nozzle: ( ) ( ) 2 / 2 / M M + = + γ γ ω ω ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ] 2 / 2 / 2 / 2 / M M M M M M M M M M = + γ γ γ γ ω ω M 2 M November, VKI Lecture

100 Non compact elements M M2 The proper equations in the acoustic element are the quasi D LEE: Due to compressor or turbine November, 200 VKI Lecture 00

101 Principle of the method. The boundary condition is well known at x out (e.g.: p =0) 2. Impose a non zero incoming wave at x in 3. Solve the LEE in the frequency space 4. Compute the outgoing wave at x in 5. Deduce the effective reflection coefficient at x in November, 200 VKI Lecture 0

102 Example of a ideal compressor Imposed boundary condition p = 0 November, 200 VKI Lecture 02

103 Example of a ideal compressor Compressor Imposed boundary condition p = 0 November, 200 VKI Lecture 03

104 Example of a ideal compressor π c = p p t 2 = t 4 Compact theory November, 200 VKI Lecture 04

105 Realistic air intake Intlet B.C Outlet Choked B.C. ( taken as an acoustic wall ) Acoustic modes of the combustion chamber Need to be computed November, 200 VKI Lecture 05

106 Realistic air intake Air-Intake duct Intake BP Compressor HP Compressor Intlet B.C Outlet Choked B.C. ( taken as an acoustic wall ) November, 200 VKI Lecture 06

107 Realistic air intake Intake BP Compressor HP Compressor November, 200 VKI Lecture 07

108 Realistic air intake Intake BP Compressor HP Compressor November, 200 VKI Lecture 08

109 OUTLINE. Computing the whole flow 2. Computing the fluctuations 3. Boundary conditions 4. Analysis of an annular combustor November, 200 VKI Lecture 09

110 Helicopter engine 5 burners Overview of the configuration From experiment: A mode may run unstable November, 200 VKI Lecture 0

111 About the computational domain When dealing with actual geometries, defining the computational domain may be an issue Turbomachinery are present upstream/downstream The combustor involves many details : combustion chamber, swirler, casing, primary holes, multi-perforated liner Dassé et al., AIAA , Combustion Chamber Casing CC + casing + swirler CC + casing + swirler + primary holes 700 Hz 500 Hz 575 Hz 609 Hz Frequency of the first azimuthal mode A November, 200 VKI Lecture

112 Is this mode stable? The acoustic mode found at 609 Hz has a strong azimuthal component, like the experimentally observed instability Its stability can be assessed by solving the thermo-acoustic problem which includes the flame response In this annular combustor, there are 5 turbulent flames Do they share the same response? November, 200 VKI Lecture 2

113 Using the brute force Large Eddy Simulation of the full annular combustion chamber Staffelbach et al., 2008 The first azimuthal mode is found unstable from LES, at 740 Hz Same mode found unstable experimentally November, 200 VKI Lecture 3

114 In annular geometries, two modes may share the same frequency These two modes may be of two types Consider the pressure as a function of the angular position Mode structure A+ A- From simple acoustics: November, 200 VKI Lecture 4

115 Turning mode. A+ = & A- = 0 Companion mode turns counter clockwise November, 200 VKI Lecture 5

116 Standing mode. A+ = A- = Companion mode has its nodes/antinodes at opposite locations November, 200 VKI Lecture 6

117 LES mode. A+ = A- = 0.3 A simple analytical model can explain the global mode shape obtained from LES Can we predict the stability by using the Helmholtz solver? November, 200 VKI Lecture 7

118 Responses of the burners Because of the self-sustained instability, the pressure in front of each burner oscillates This causes flow rate oscillations Because the unstable mode is turning, the flow rate through the different burners are not in phase November, 200 VKI Lecture 8

119 Global Response of the flames From the full LES, the global flame transfer of the 5 burners can be computed from the global HR and the flow rate signals For all the 5 sectors, the amplitude of the response is close to 0.8 and the time lag is close to 0.6 ms This result support the ISSAC assumption November, 200 VKI Lecture 9

120 ISAAC Assumption As far as the flame response is concerned 5 Independent Sector Assumption for Annular Combustor Allows performing a single sector LES with better resolution to obtain more accurate FTF Of course thea mode cannot appear is such computation FTF deduced from single sector LES pulsated at 600 Hz November, 200 VKI Lecture 20

121 Local flame transfer function November, 200 Typical field of interaction index VKI Lecture 2

122 Extended the local n-τ model Define one point of reference upstream of each of the 5 burners Use the ISAAC Assumption n l ( x, ω) τ ( x, ω) the interaction index, and the time delay l are the same in all sectors ˆ ω ( x, ω) T n n M n n l l l l ( x, ω) e ( x, ω) e ( x, ω) e ( x, ω) e jωτ ( x, ω) l jωτ ( x, ω) l jωτ ( x, ω) l jωτ ( x, ω) l uˆ( x uˆ( x uˆ( x uˆ( x ref 2 ref 4 ref 5 ref, ω) n, ω) n, ω) n, ω) n ref 2 ref 4 ref 5 ref if if if if x x x x in sector in sector 2 in sector 4 in sector 5 November, 200 VKI Lecture 22

123 Extended the local n-τ model Assume ISAAC and duplicate the flame transfer function over the whole domain In the same way, duplicate the field of speed of sound November, 200 VKI Lecture 23

124 Stability of the A mode The A mode from the Helmholtz solver looks like the A from the full LES Helmholtz LES BUT. Its frequency remains close to 600 Hz instead of 740 Hz 2. It is found stable! November, 200 VKI Lecture 24

125 Shape of the A mode The A modes from the Helmholtz and LES solvers resemble But a closer look reveals important differences angle (rad) Helmholtz solver angle (rad) LES solver What is observed in the Helmholtz solver. Pressure amplitude (slightly) larger upstream of the burners 2. No phase shift between upstream and downstream November, 200 VKI Lecture 25

126 What s wrong? The Helmholtz solver is not as CPU demanding and allows parametric studies The A mode is found stable for time delays smaller than T/2 At 600 Hz, this corresponds to τ < 0.83 ms The time delay of the FTF (0.65ms) is below this critical value November, 200 VKI Lecture 26

127 What s wrong? Remember the full LES oscillates at 740 Hz This corresponds to a critical value for the time delay of T/2 = 0.67 ms, very close to the prescribed time delay From the LES, the flow rate and HR are indeed in phase opposition November, 200 VKI Lecture 27

128 Back to the stability of the A mode Instead of imposing the time delay in the FTF, let s impose the phase shift between flow rate and HR The A mode is now found unstable by the Helmholtz solver, consistently with LES and experiment The mode shape is also in better agreement Helmholtz solver LES solver November, 200 VKI Lecture 28

129 THANK YOU More details, slides, papers, November, 200 VKI Lecture 29