Dynamical Systems Approaches to Combus6on Instability

Transcription

1 Dynamical Systems Approaches to Combus6on Instability Prof. R. I. Sujith Indian Institute of Technology Madras India Acknowledgements: 1. P. Subramanian, L. Kabiraj, V. Jagadesan, V. Nair, G. Thampi,.. 2. P. Wahi, W. PoliGe, M. Juniper, P. Schmid, R. Govindarajan 3. Department of Science and Technology India

2 What is dynamical systems theory?

3 Dynamical systems theory describes changes in systems that evolve in 6me

4 Image from Wikipidea en.wikipedia.org/wiki/buckling

5 What is a bifurca6on? A small smooth change in parameter values causes a sudden 'qualita6ve' change in behaviour

6 Acknowledgement: Prof. Zinn s notes, with permissi

7 A system is nonlinearly unstable if some finite amplitude disturbance grows with 6me From Prof. Zinn s notes, with permission For triggering instability, the initial amplitude should be greater than a threshold amplitude

8 Two dis6nct kinds of instability can be iden6fied called Hopf bifurca6on in dynamical systems theory Super-critical bifurcation Sub-critical bifurcation Unstable Stable Unstable Stable Regions of linear & nonlinear instability are different in sub-critical bifurcation / triggered instability

9 Even if triggering does not occur, a sub- cri6cal bifurca6on is more dangerous for our system

10 We discuss the application of tools from dynamical systems theory in thermoacoustics Slow flow equations du dt = ( B 1 ε2 + ib 2 )U + ( B 3 + ib 4 )U 2 U Numerical continuation to obtain bifurcation plots Nonlinear time series analysis

11 Slow flow equa6ons

12 A horizontal Rijke tube is modeled Heating Element Air flow x f L Matveev & Culick (2003) Balasubramanian & Sujith (2008) We sidestep the effects of natural convection on the mean flow

13 The acous6c field within the thermoacous6c system evolves as γm u t + p x = 0 p t + ς p + γm u x = γ 1 ( ) q f ( t)δ( x x ) f N j=1 u = U j cos(jπx) and p = N j=1 γm jπ P j sin(jπx) Zinn & students (late 60s, early seven\es)

14 Method of mul6ple scales can be used to obtain an amplitude equa6on U +a 0 U +a 1 U +a 2p 1 + 3cos( πx f )U t τ ( ) 1 = 0 U ( t) = εy ( t) = A 1 t ( ( )) ( )sin t + φ 1 t Define multiple scales: y 2 ( ) ( ) ( ) ( ) ( 3 t = Y t, t, t + εy t, t, t + ε Y t, t, t + O ε )

15 Evolu6on equa6on for the slow flow is of the Stuart- Landau form Priya Subramanian dw dt = ( B 1 ε 2 + ib )( 3 σ σ H )W + ( B 2 + ib 4 )W 2 W JFM 2012 Slow flow equations for amplitude & phase: da dt = B ( σ σ 1 H )A + B 2 A 3 ; dϕ dt = B 3 ( σ σ H ) + B 4 ϕ 2 Triggering amplitude: A ( σ σ ) 1/2 H

16 Numerical Con6nua6on Jahanke & Culick (1994)

17 Numerical con6nua6on tracks the solu6on of a set of parameterized nonlinear equa6ons du dt + Fu (, λ) = 0 Simula\ons in \me domain are replaced by itera\ve root finding of the corresponding constrained set of equa\ons

18 The flow giving rise to a limit cycle can be also viewed as a map Flow: Map: du + Fu (, λ) = 0 dt u( n+ 1) T =ΦunT Poincare section Φ = State Transition Matrix Limit cycle Floquet multipliers are the eigenvalues of the state transition matrix

19 Increase in power destabilizes the system through a sub-critical Hopf bifurcation Heater power Subramanian et al. (IJSCD, 2010)

20 Stability varia6on with heater loca6on is not monotonic Heater loca\on 20 Subramanian et al. (IJSCD, 2010)

21 Nonlinear 6me series analysis

22 d χ dt = f χ ( ) χ = [ χ, χ, χ,...χ ] n

23 χ = [ χ, χ, χ,...χ ] n In a CFD simula6on, we calculate all the state variables; In an experiment, we have one pressure transducer!

24 The phase space is reconstructed using embedding theorem P (t) \me series False nearest neighbor Average Mutual Informa\on P (t + 2τ) d E d E Op\mum embedding dimension P (t) P (t+τ) P (t+2τ) P (t + τ) τ Op\mum \me delay P (t) Reconstructed phase space

25 The onset of instability is classified into soft and hard excitation Soft Excitation Hard Excitation (triggering)

26 Experimental data in a combustor with a bluff body flame holder - Nair & Sujith (2012)

27 We inves6gate the role of noise in a ducted non- premixed flame system Oxidizer Fuel Oxidizer

28 Quartz tube Traverse mechanism Oxygen plenum Flush Nitrogen Oxygen supply Brass tube Methane- Nitrogen mixture Sub woofer Fuel plenum Jagadesan & Sujith (2012 symposium)

29 System undergoes transi6on via subcri6cal Hopf bifurca6on.

30 Triggering instability is observed when the system is in hysteretic region Triggered Not triggered Scenario proposed by Balasubramanian & Sujith (2008) Juniper (2011)

31 Bifurcation diagram is separated in to globally stable, globally unstable and bistable regions Globally unstable zone Unstable zone Globally stable zone Hopf point Stable zone Fold point 31

32 Spurts in amplitude are observed 32

33 System undergoes transition analogous to bypass transition to turbulence Jagadesan & Sujith (Combus\on Symposium 2012)

34 In thermoacoustics, the final state is believed to be a limit cycle, when driving balanced damping

35 We observed intermittency in a turbulent swirl stabilized burner Thampi & Sujith (2012)

36 Experimental setup: Ducted laminar premixed flame Lipika Kabiraj Kabiraj & Sujith (JFM 2012)

37 Limit cycle breaks leading to a state of flame detachment and reattachment

38 We see a subcritical Hopf bifurcation

39 Bifurca6on: limit cycle, quasi- periodic and intermiuent oscilla6on x f = 56.5 cm A B C x f = 62cm x f = 64cm

40 During limit cycle, wrinkles originate at the base of the flame and propagate downstream

41 Flame response during quasi-periodic oscillations shows elongation, neck formation, cusping & pinch off

42 Dynamical behavior before flame blowout: Intermittent oscillations A B C A B C x f = 64 cm to x f = 68.5 cm

43 Two types of bursts are observed along with sections of laminar state A B C L- laminar state, B1 & B2- two types of burst, F-fixed point (no oscillation) x f = 64 cm

44 Recurrence plot is a graphical representation for visualizing system dynamics from short time series Embedding theorem A square binary recurrence matrix p(t) p(t) p(t + τ) - Time series p(t + 2τ) Reconstructed phase space

45 Recurrence plot: limit cycle and quasi-periodic oscillation

46 Features of RP for type-ii intermittency

47 Flame attachment and reattachment leads to intermittent oscillations

48 Simultaneous to these bursts, flame oscillates violently, lifts off & then reattaches to the burner rim a-c: Laminar state (L), d-o: Burst state (B2), p-r: Reattachment.

49 What does dynamical systems theory tell us about instabili6es in a turbulent combustor?

50 Thermoacous6c oscilla6ons have a micro- structure which can be revealed using nonlinear dynamics calcula6on

51 Can be used to detect the onset* of an impending instability before the instability occurs *Patent pending

52 In summary, dynamical systems theory provides us with tools that can be used to gain new insights Analytical & numerical tools to study thermoacoustics using bifurcation theory Time series analysis can be used for studying thermoacoustic systems experimentally Thermoacoustic systems have much richer behavior than just limit cycles

53 53

54 International journal of spray and combustion dynamics Editor-in-Chief: Dr. R. I Sujith ISSN Multi-Science Publishing Co. Ltd serving science & technology since Wates Way, Brentwood, Essex CM15 9TB, UK Tel: +44(0) Fax: +44(0) info@multi-science.co.uk Website: Recognized by Web o f Science (isiknowledge.com)