Valida&on of the FDF methodology with a mul&dimensional Helmholtz solver

Transcription

1 Valida&on of the FDF methodology with a mul&dimensional Helmholtz solver Alexis Cuquel, Frédéric Boudy, Daniel Durox, Thierry Schuller and Sébas;en Candel Laboratoire EM2C CNRS Ecole Centrale Paris. KIAI & TECC- AE Public Workshop, Florence, Italy September, 17-18th 2012

2 Thermo- acous,c instability consequences. Many combustors are prone to thermo- acous;c instabili;es, poten;ally leading to unsteady flow parameter fluctua;ons. It can induce severe damages on the combustor itself, such as: Mul;ple swirled injector combustor. Damaged system with melted perforates. Noise. Increased heat fluxes. Vibra;ons. Flashback. Blow- out. Goy et al. Chapter 8 in: Combus&on instabili&es in gas turbine engines: opera&onal experience, fundamental mechanisms, and modeling. (2005) 2

3 Thermo- acous,c resonant coupling Flow Combus;on Acous;cs Acous&cs Culick et al. (2001) Proceedings RTO/VKI Combus&on dynamics Candel et al. (1996) In Unsteady combus;on Schlieren images of a conical flame subjected to acous;c modula;on 3

4 Linear and non- linear instability components. Many experiments during thermoacous;c instabili;es indicate that the main nonlinearity results from the flame oscilla;on. Linear acoustic" Linear treatment of acoustic with wave equation! 4

5 Linear and non- linear instability components. Many experiments during thermoacous;c instabili;es indicate that the main nonlinearity results from the flame oscilla;on. Linear acoustic" Nonlinear flame dynamic" Linear treatment of acoustic with wave equation! Need of nonlinear treatment of flame / flow interaction! FDF: 5

6 Objec,ves (1) Give modeling elements on current state of the art to assess thermo- acous;c instabili;es. (2) Develop a non- linear extension of the Flame Transfer Func;on: the Flame Describing Func&on (FDF). (3) Exhibit results from a non- linear stability analysis combining an acous;c network model and the FDF. (4) Transfer and validate the FDF methodology into an acous&c solver, in order to extend it towards prac;cal and complex configura;ons. 6

7 Outline 1. Experimental configura;on. 2. Stability analysis combining the FDF methodology with low- order acous;c network models. 3. Transfer of the FDF methodology into the AVSP code and its valida;on on a generic case. 7

8 Outline 1. Experimental configura;on. 2. Stability analysis combining the FDF methodology with low- order acous;c network models. 3. Transfer of the FDF methodology into the AVSP code and its valida;on on a generic case. 8

9 Experimental setup and Burner schema,c 9

10 Typical experimental result: unstable regime.

11 Typical experimental result: limit cycle oscilla,on During unstable opera;on, aher igni;on and transient growth, oscilla;ons of flow variables reach a limit cycle: Acous&c pressure in the plenum OH* chemiluminescence Flow velocity in the plenum 11

12 Outline 1. Experimental configura;on. 2. Stability analysis combining the FDF methodology with low- order acous;c network models. 3. Transfer of the FDF methodology into the AVSP code and its valida;on on a generic case. 12

13 Stability analysis: from linear to non- linear. Three different levels of modeling can be considered: Acoustic Analysis Linear Stability Analysis Nonlinear Stability Analysis No unsteady flame effects. Flame Transfer Function (FTF) Flame Describing Function Acoustic mode frequencies and modal structures. No informations on mode stability. Frequency and exponential growth rate for vanishingly small oscillations only. Limited to the prediction of linearly unstable modes. No information on the limit cycle or non-linearly unstable modes. Instability frequencies and growth rate evolution up to the limit cycle. Limit cycle amplitude, frequency shift, hysteresis, mode switching, triggering. 13

14 Stability analysis: from linear to non- linear. Three different levels of modeling can be considered: Acoustic Analysis Linear Stability Analysis Nonlinear Stability Analysis No unsteady flame effects. Flame Transfer Function (FTF) Flame Describing Function Acoustic mode frequencies and modal structures. No informations on mode stability. Frequency and exponential growth rate for vanishingly small oscillations only. Limited to the prediction of linearly unstable modes. No information on the limit cycle or non-linearly unstable modes. Instability frequencies and growth rate evolution up to the limit cycle. Limit cycle amplitude, frequency shift, hysteresis, mode switching, triggering. 14

15 Stability analysis: from linear to non- linear. Three different levels of modeling can be considered: Acoustic Analysis Linear Stability Analysis Nonlinear Stability Analysis No unsteady flame effects. Flame Transfer Function (FTF) Flame Describing Function Acoustic mode frequencies and modal structures. No informations on mode stability. Frequency and exponential growth rate for vanishingly small oscillations only. Limited to the prediction of linearly unstable modes. No information on the limit cycle or non-linearly unstable modes. Instability frequencies and growth rate evolution up to the limit cycle. Limit cycle amplitude, frequency shift, hysteresis, mode switching, triggering. 15

16 The Flame Describing Func,on (FDF). The FDF is determined by a set of FTF measured for increasing input levels: The FDF gain and phase are here both non- linear and depend on the frequency and the acous;c perturba;on amplitude. Combined to acous;c low- order network model, its use leads to frequencies and growth rates as a func;on of the perturba;on level. 16

17 The FDF methodology for stability analysis. Solu;on : 17

18 The FDF methodology for stability analysis. Solu;on : Growth rates and eigenfrequencies are calculated separetely for different modes using the following procedure:" These growth rates and frequencies depend on the fluctuation amplitude:" 18

19 Linearly unstable mode reaching limit cycle. Growth rates ω i =ω i ( u,l 1 ) are calculated for the first eigenmode of the combustor. Mode 1 Stable band for mode 1 Limit cycle is reached when ω i equals zero. The method yields the limit cycle oscilla;on level and oscilla;on frequency. Regions where ω i is nega;ve are linearly stable. Linearly unstable mode : Posi;ve growth rate for infinitesimally small amplitude. 19

20 Non- linearly unstable mode reaching limit cycle. Mode 1 Nonlinearly unstable mode : Nega;ve growth rate for small perturba;on amplitudes, but posi;ve values above a certain threshold. Limit cycle is reached when ; 20

21 Mode switching predic,on. FDF calculation prediction" Experiment" Switch 3 rd (769 Hz) " 2 nd (465 Hz)" Switch naturally to mode 2 during growth" 3 rd 3 rd 737 Hz 492 Hz 21

22 Current valida,on of the FDF methodology Unconfined laminar flames:! Confined laminar flames:! Noiray et al. (2008) JFM 615" Boudy et al. (2011) JEGTP 133" Boudy et al. (2011) PCI 33" Confined turbulent swirling flame:! Palies et al. (2011) C&F 158 " Possibility to anticipate: Limit cycle amplitudes Limit cycle frequencies (including frequency shift) Hysteresis Triggering Mode switching 22

23 Outline 1. Experimental configura;on. 2. Stability analysis combining the FDF methodology with low- order acous;c network models. 3. Transfer of the FDF methodology into the AVSP code and its valida;on on a generic case. 23

24 The AVSP code: a Helmholtz equa,on solver. Developed at CERFACS. Finite volume formula;on. On unstructured grids with tetrahedral elements. Solves the Helmholtz equa;on defined by: (c 2 0 p 1 )+ω 2. p 1 = iω(γ 1) q 1 By including a local Flame Transfer Func;on, it reads: (c 2 0 p 1 )+ω 2 p 1 = (γ 1) q 0. ρ ref v 0 (x ref ) F loc ref p 1 (x ref ) n ref e remaining issue is to link the local q Flame Transfer Func with F loc (ω r ) = 1 / q 0 v 1 (x ref ) n ref / v 0 (x ref ) The discre;za;on hile of the velo Helmholtz referenceequa;on point ison related a mesh to leads t to a non- linear Eigen- value problem: AP + ωb(ω)p + ω 2 CP = D(ω)P 24

25 Numerical configura,on: the confinement tube. L 2 25

26 Numerical configura,on: the plenum. L 2 L 1 26

27 Numerical configura,on: the mesh. Unstructured mesh refined in the flame zone, above the perforated plate: L 2 L 2 L 1 L 1 500k cells and 100k nodes. 27

28 Boundary condi,on and pressure jump. Unflanged open pipe impedance: Z(ω) = 1 4 (ωr c )2 i0.61 ωr c real part allows to take into Melling model for perforated plate: p + 1 (x) p 1 (x) = h 1+ l ν a (1 + i) Piston- head reflec;on coefficient determined experimentally: $ σ # p 1 (x) n b Confinement tube Plenum R!%" &'#!# &'# arg(r) (rad)!! "!!# $!!! Frequency (Hz) "!!# $!!! # Frequency (Hz) 28

29 Modal analysis in absence of combus,on. A first valida;on can be done on the configura;on without combus;on. The acous;c Eigen- frequencies were measured and numerically computed to make sure the geometry considered is valid: Combustor geometry Exp. AVSP LOM L 1 = 0.12 m 462 Hz 472 Hz 482 Hz l p = m 882 Hz 881 Hz 909 Hz L 1 = 0.35 m 225 Hz 219 Hz 228 Hz l p = m 604 Hz 608 Hz 609 Hz 792 Hz 814 Hz 844 Hz 1190 Hz 1194 Hz 1225 Hz Acoustic L 2 = 0.10m; eigenmodes T 1 = 300 K; T of 2 = 300 the K; Boudy setup, taking int , ; L 1 : 120mm Frequency (Hz) , ; L 1 : 350mm Frequency (Hz) 29

30 Effect of heat release rate. 1. Steady heat release rate effects: T 1 = 300 K; T 2 = 900 K 2. Unsteady heat release rate effects: T 2 = 900 K - Generated through the FDF. - The flame region is compact with respect to the acous;c wavelength. (2 mm- high, 36.1 mm- wide) - The flame zone is located 1 mm above the perforated plate. T 1 = 300 K 30

31 Linearly unstable mode: L 1 = 0.12 m. Experimental signal at limit cycle" Prediction with AVSP/FDF " " "and LOM/FDF" 31

32 Linearly unstable mode: L 1 = 0.12 m. Experimental signal at limit cycle" AVSP/FDF prediction" 1/4- wave mode of the plenum. 32

33 Current issues on instability predic,on. Some issues remains on both the acous;c and the flame dynamic components of the models: Acous;cs: How to determine accurately the damping rate α? In prac;cal combustor, the limit cycle is reached when ω i = α Flame dynamics: How to transpose FTF from one configura;on to another? Influence of the chamber geometry? Influence of the injector geometry? 33

34 Current issues on instability predic,on. Some issues remains on both the acous;c and the flame dynamic components of the models: Acous;cs: How to determine accurately the damping rate α? Flame dynamics: How to transpose FTF from one configura;on to another? Influence of the chamber geometry? Influence of the injector geometry? 34

35 Current issues on instability predic,on. Some issues remains on both the acous;c and the flame dynamic components of the models: Acous;cs: How to determine accurately the damping rate α? Flame dynamics: How to transpose FTF from one configura;on to another? Influence of the chamber geometry? Influence of the injector geometry? 35

36 Conclusion The predic;on of thermo- acous;c instability frequency and amplitude is achievable through the FDF framework. Coupled to a low- order network analysis, it allows to predict non- linear behavior such as hysteresis, mode triggering and mode switching. The methodology has been extended to the mul;- dimensionnal acous;c solver AVSP to expand its applicability towards prac;cal burner. The combina;on AVSP/FDF has been successfully validated on a mul;point injec;on generic configura;on. 36

37 The research leading to these results has received funding from the European Communitys Seventh Framework Programme (FP7/ ) under Grant Agreement #ACP8- GA We would also like to acknowledge F. Nicoud, C. Silva and E. Motheau for the great support they provided on the acous&c solver AVSP, and CERFACS for gran&ng us access to their computa&onal ressources. Thank you for your ahen&on KIAI & TECC- AE Public Workshop, Florence, Italy September, 17-18th 2012