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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
What is dynamical systems theory?
Dynamical systems theory describes changes in
systems that evolve in 6me
Image from Wikipidea en.wikipedia.org/wiki/Buckling
A small smooth change in parameter values causes a sudden 'qualita6ve' change in behaviour
What is a bifurca6on?
Acknowledgement: Prof. Zinn’s notes, with permission
A system is nonlinearly unstable if some finite amplitude disturbance grows with 6me
For triggering instability, the initial amplitude should be greater than a “threshold amplitude”
From Prof. Zinn’s notes, with permission
Two dis6nct kinds of instability can be iden6fied – called Hopf bifurca6on in dynamical systems theory
Stable
Unstable
Unstable
Stable
Super-critical bifurcation Sub-critical bifurcation
Regions of linear & nonlinear instability are different in sub-critical bifurcation / triggered instability
Even if triggering does not occur, a sub-‐cri6cal
bifurca6on is more dangerous for our system
We discuss the application of tools from dynamical systems theory in thermoacoustics
Nonlinear time series analysis
Slow flow equations
dUdt
= B1ε2 + iB
2( )U + B3
+ iB4( )U 2
U
Numerical continuation to obtain
bifurcation plots
Slow flow equa6ons
fx
L
Heating Element
Air flow
A horizontal Rijke tube is modeled
We sidestep the effects of natural convection on the mean flow
Matveev & Culick (2003) Balasubramanian & Sujith (2008)
The acous6c field within the thermoacous6c system evolves as
γM∂ ′u∂t
+∂ ′p∂x
= 0
∂ ′p∂t
+ ς ′p + γM∂ ′u∂x
= γ−1( ) ′qft( )δ x −x
f( )
′u = Ujcos(jπx) and ′p =
j=1
N
∑ −γMjπ
Pjsin(jπx)
j=1
N
∑
Zinn & students (late 60s, early seven\es)
Method of mul6ple scales can be used to obtain an amplitude equa6on
Define multiple scales:
U t( ) = εy t( ) = A
1t( )sin t +φ
1t( )( )
( ) ( ) ( ) ( ) ( ) ...,,,,,, 32102
221012100 ++++= εεε OtttYtttYtttYty
U +a
0U +a
1U +a
2p1+ 3cos πx
f( )U t− τ( ) −1⎡
⎣⎢⎢
⎤
⎦⎥⎥= 0
Evolu6on equa6on for the slow flow is of the Stuart-‐Landau form
Triggering amplitude:
dW
dt= B
1ε2 + iB
3( ) σ−σH
( )W + B2
+ iB4( )W 2
W
A∝ σ−σ
H( )1/2
Slow flow equations for amplitude & phase:
dA
dt= B
1σ−σ
H( )A + B
2A
3 ; dϕdt
= B3 σ −σ H( ) + B4ϕ 2
Priya Subramanian
JFM 2012
Numerical Con6nua6on
Jahanke & Culick (1994)
Numerical con6nua6on tracks the solu6on of a set of parameterized nonlinear equa6ons
( , ) 0du F udt
λ+ =
Simula\ons in \me domain are replaced by itera\ve root finding of the corresponding constrained set of equa\ons
( , ) 0du F udt
λ+ =
( 1)n T nTu u+ =Φ
Flow:
Map:
Limit cycle
Poincare section
The flow giving rise to a limit cycle can be also viewed as a map
Φ = State Transition Matrix
Floquet multipliers are the eigenvalues of the state transition matrix
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 Subramanian et al. (IJSCD, 2010)
Nonlinear 6me series analysis
d χdt
= f χ( )
χ = χ1,χ2,χ3,...χn[ ]
χ = χ1,χ2,χ3,...χn[ ]
In a CFD simula6on, we calculate all the state variables; In an experiment, we have one pressure transducer!
The phase space is reconstructed using embedding theorem
P’ (t) – \me series
Average Mutual Informa\on
P’ (t) P’ (t+τ) P’ (t+2τ)
False nearest neighbor
dE
τ – Op\mum \me delay
dE – Op\mum embedding dimension
P’(t)
P’ (t + 2τ)
P’(t + τ)
Reconstructed phase space
The onset of instability is classified into soft and hard excitation
Soft Excitation
Hard Excitation
(triggering)
Experimental data in a combustor with a bluff body flame holder -‐ Nair & Sujith (2012)
Fuel
Oxidizer
Oxidizer
We inves6gate the role of noise in a ducted non-‐premixed flame system
Oxygen plenum
Flush Nitrogen Oxygen supply
Quartz tube
Traverse mechanism
Brass tube
Sub woofer
Fuel plenum
Methane-‐Nitrogen mixture
Jagadesan & Sujith (2012 symposium)
System undergoes transi6on via subcri6cal Hopf bifurca6on.
Triggering instability is observed when the system is in hysteretic region
Triggered Not triggered
Scenario proposed by Balasubramanian & Sujith (2008) Juniper (2011)
Globally stable zone
Globally unstable
zone Unstable zone
Stable zone
Fold point Hopf point
Bifurcation diagram is separated in to globally stable, globally unstable and bistable regions
31
Spurts in amplitude are observed
32
System undergoes transition analogous to bypass transition to turbulence
Jagadesan & Sujith (Combus\on Symposium 2012)
In thermoacoustics, the final state is “believed” to be a limit cycle, when driving balanced damping
We observed intermittency in a turbulent swirl stabilized burner
Thampi & Sujith (2012)
Experimental setup: Ducted laminar premixed flame
Lipika Kabiraj
Kabiraj & Sujith (JFM 2012)
Limit cycle breaks leading to a state of flame detachment and reattachment
We see a subcritical Hopf bifurcation
Bifurca6on: limit cycle, quasi-‐periodic and intermiUent oscilla6on
ABC
xf = 56.5 cm
xf = 62cm
xf = 64cm
During limit cycle, wrinkles originate at the base of the flame and propagate downstream
Flame response during quasi-periodic oscillations shows elongation, neck formation, cusping & pinch off
ABC
Dynamical behavior before flame blowout: Intermittent oscillations
ABC
xf = 64 cm to xf = 68.5 cm
Two types of bursts are observed along with sections of laminar state
L- laminar state, B1 & B2- two types of burst, F-fixed point (no oscillation)
AB
xf = 64 cm
C
Recurrence plot is a graphical representation for visualizing system dynamics from short time series
Embedding theorem
- Time series
Reconstructed phase space
p(t)
p(t) p(t + τ)
p(t + 2τ) A square binary recurrence matrix
Recurrence plot: limit cycle and quasi-periodic oscillation
Features of RP for type-II intermittency
Flame attachment and reattachment leads to intermittent oscillations
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.
What does dynamical systems theory tell us about instabili6es in a turbulent combustor?
Thermoacous6c oscilla6ons have a “micro-‐structure” which can be revealed using nonlinear dynamics calcula6on
Can be used to detect the onset* of an impending instability before the instability occurs
*Patent pending
In summary, dynamical systems theory provides us with tools that can be used to gain new insights
Time series analysis can be used for studying thermoacoustic systems experimentally
Analytical & numerical tools to study thermoacoustics using bifurcation theory
Thermoacoustic systems have much richer behavior than just limit cycles
53
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