bypass transition in thermoacoustics (triggering) iiit pune & idea research, 3 rd jan 2011...
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Bypass transition in thermoacoustics(Triggering)
IIIT Pune & Idea Research, 3rd Jan 2011
Matthew Juniper ([email protected])Engineering Department, University of Cambridge
with thanks to Peter Schmid, R. I. Sujithand Iain Waugh
Bypass transition in thermoacoustics
A. Re = 100 to 1000 B. Re = 1000 to 10 000
C. Re = 10 000 to 100 000 D. It never becomes unstable
In fluid mechanics, at what Reynolds number does the flow within a pipebecome unstable?
Phone a friend 50/50 Ask the audience
B. Re = 1000 to 10 000
D. It never becomes unstable
Phone a friend 50/50 Ask the audience
In fluid mechanics, at what Reynolds number does the flow within a pipebecome unstable?
B. Re = 1000 to 10 000
D. It never becomes unstable
Phone a friend 50/50 Ask the audience
In fluid mechanics, at what Reynolds number does the flow within a pipebecome unstable?
B. Re = 1000 to 10 000
D. It never becomes unstable
Phone a friend 50/50 Ask the audience
In fluid mechanics, at what Reynolds number does the flow within a pipebecome unstable?
B. Re = 1000 to 10 000
Phone a friend 50/50 Ask the audience
In fluid mechanics, at what Reynolds number does the flow within a pipebecome unstable?
Bypass transition in thermoacoustics(Triggering)
IIIT Pune & Idea Research, 3rd Jan 2011
Matthew JuniperEngineering Department, University of Cambridge
with thanks to Peter Schmid, R. I. Sujithand Iain Waugh
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
A flame in a pipe can be unstable and generate sustained acoustic oscillations. This occurs if heat release occurs at the same time as localized high pressure.
Bypass transition in thermoacoustics
Combustion instability is still one of the biggest challenges facing gas turbine and rocket engine manufacturers.
SR71 engine test, with afterburner
Bypass transition in thermoacoustics
Some combustion systems are described as ‘linearly stable but nonlinearly unstable’, which is a sign of a subcritical bifurcation.
oscillationamplitude
a systemparameter
Bypass transition in thermoacoustics
But some systems seem able to trigger spontaneously from just the background noise.
Bypass transition in thermoacoustics
But some systems seem able to trigger spontaneously from just the background noise.
Bypass transition in thermoacoustics
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
Diagram of the Rijke tube
Non-dimensional governing equations
hot wireair flow
acoustics damping heat release at the hot wire
(note the time delay in the heat release term)
We will consider a toy model of a horizontal Rijke tube. Heat release at the wire is a function of the velocity at the wire at a previous time.
Definition of the non-dimensional acoustic energy
Bypass transition in thermoacoustics
u p
The governing equations are discretized by considering the fundamental ‘open organ pipe’ mode and its harmonics. This is a Galerkin discretization.
Discretization into basis functions
Definition of the non-dimensional acoustic energy
Non-dimensional discretized governing equations
Bypass transition in thermoacoustics
u p
The governing equations are discretized by considering the fundamental ‘open organ pipe’ mode and its harmonics. This is a Galerkin discretization.
Discretization into basis functions
Definition of the non-dimensional acoustic energy
Non-dimensional discretized governing equationsuj
pj
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
A continuation method is used to find stable and unstable periodic solutions.
Bifurcation diagrams for a 10 mode system stable periodic solutionunstable periodic solution
Bypass transition in thermoacoustics
A continuation method is used to find stable and unstable periodic solutions.
Bifurcation diagrams for a 10 mode system stable periodic solutionunstable periodic solution
Bypass transition in thermoacoustics
Every point in state space is attracted to the stable fixed point or the stable periodic solution.
3-D cartoon of 20-D state space
stable fixed point
stable periodic solution
Bypass transition in thermoacoustics
stable periodic solution
A surface separates the points that evolve to the stable fixed point from the points that evolve to the stable periodic solution.
3-D cartoon of 20-D state space
boundary of the basinsof attraction of the two
stable solutions
Bypass transition in thermoacoustics
3-D cartoon of 20-D state space
stable periodic solution
unstable periodic solution
boundary of the basinsof attraction of the two
stable solutions
The unstable periodic solution sits on the basin boundary.
Bypass transition in thermoacoustics
3-D cartoon of 20-D state space
stable periodic solution
unstable periodic solution
boundary of the basinsof attraction of the two
stable solutions
We want to find the lowest energy point on this boundary.
Bypass transition in thermoacoustics
3-D cartoon of 20-D state space
stable periodic solution
unstable periodic solution
boundary of the basinsof attraction of the two
stable solutions
The lowest energy point on the unstable periodic solution is a good starting point but can a better point be found?
lowest energy point on theunstable periodic solution
Bypass transition in thermoacoustics
If the basin boundary looks like a potato, is it ...
Desiree
Bypass transition in thermoacoustics
If the basin boundary looks like a potato, is it ...
Desiree Pink eye
Bypass transition in thermoacoustics
If the basin boundary looks like a potato, is it ...
Desiree Pink eye Pink fur apple
Bypass transition in thermoacoustics
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
stable periodic solution
unstable periodic solution
stable fixed point
3-D cartoon of 20-D state space
We start by examining the unstable periodic solution.
Bypass transition in thermoacoustics
Floquet multipliers of the unstable periodic solution(eigenvalues of monodromy matrix)
We evaluate the monodromy matrix around the unstable periodic solution and find its eigenvalues and eigenvectors.
Bypass transition in thermoacoustics
Floquet multipliers of the unstable periodic solution(eigenvalues of monodromy matrix)
We evaluate the monodromy matrix around the unstable periodic solution and find its eigenvalues and eigenvectors.
Bypass transition in thermoacoustics
Floquet multipliers of the unstable periodic solution(eigenvalues of monodromy matrix)
First eigenvectorμ = 1.0422
We evaluate the monodromy matrix around the unstable periodic solution and find its eigenvalues and eigenvectors.
Bypass transition in thermoacoustics
Floquet multipliers of the unstable periodic solution(eigenvalues of monodromy matrix)
First eigenvectorμ = 1.0422
The first singular value exceeds the first eigenvalue, which means that transient growth is possible around the unstable periodic solution.
First singular vectorσ = 1.6058
Bypass transition in thermoacoustics
3-D cartoon of 20-D state space
stable periodic solution
unstable periodic solution
boundary of the basinsof attraction of the two
stable solutions
Close to the lowest energy point on the unstable periodic solution there must be a point with lower energy that is also on the basin boundary.
lowest energy point on theunstable periodic solution
Bypass transition in thermoacoustics
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
u p
We need to find the optimal initial state of the nonlinear governing equations
Discretization into basis functions
Definition of the non-dimensional acoustic energy
Non-dimensional discretized governing equations
Bypass transition in thermoacoustics
Cost functional:
Constraints:
Define a Lagrangian functional:
We find a non-linear optimal initial state by defining an appropriate cost functional, J, and expressing the governing equations as constraints. Lagrange optimization
Bypass transition in thermoacoustics
Re-arrange:
The optimal value of J is found when:
We re-arrange the Lagrangian functional to obtain the adjoint equations of the non-linear governing equations
Bypass transition in thermoacoustics
u1
p1
u1
p1
Linear governing equations,
constrained E0
contours: cost functional, J
arrows: gradient information returned from adjoint looping of non-linear governing equations
Non-linear governing equations,
unconstrained E0
dots: path taken by conjugate gradient algorithm
SVD solution
The (local) optimal initial state is found by adjoint looping of the governing equations, nested within a conjugate gradient algorithm.
Bypass transition in thermoacoustics
u1
p1
u1
p1
Linear governing equations,
constrained E0
contours: cost functional, J
arrows: gradient information returned from adjoint looping of non-linear governing equations
Non-linear governing equations,
unconstrained E0
dots: path taken by conjugate gradient algorithm
SVD solution
The (local) optimal initial state is found by adjoint looping of the governing equations, nested within a conjugate gradient algorithm.
Bypass transition in thermoacoustics
A global optimization procedure finds the point with lowest energy on the basin boundary, called the ‘most dangerous’ initial state.
lowest energy point on theunstable periodic solution
most dangerous initialstate
Bypass transition in thermoacoustics
This has similar characteristics to a combination of the lowest energy point on the unstable periodic solution plus the first singular value.
lowest energy point on theunstable periodic solution
most dangerous initialstate
first singularvalue
Bypass transition in thermoacoustics
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
stable periodic solution
unstable periodic solution
stable fixed point
3-D cartoon of 20-D state space
So far we found the optimal initial state, which exploits transient growth around the unstable periodic solution ...
Bypass transition in thermoacoustics
... but it is different from the optimal initial state around the stable fixed point, which is found with the SVD of the linearized stability operator.
lowest energy point on theunstable periodic solution
most dangerous initialstate
first singularvalue
optimal state aroundstable fixed point
t
Bypass transition in thermoacoustics
The first paper about this was in 2008 paper by Balasubmrananian and Sujith at IIT Madras
Triggering in the Rijke tube
G(T,E0) of the non-linear system can be found with adjoint looping over a wide range of optimization times and initial energies.
G(T,E0) for the non-linear system
~ linear
Gmax (lin)
local Gmax (nonlin)
triggering threshold
Bypass transition in thermoacoustics
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
1. transient growth
2. settles around unstable periodic solution
3. grows to stable periodic solution
With an infinitesimal amplification, the most dangerous state evolves to the stable periodic solution, after initial attraction towards the unstable periodic solution.
Evolution from the most dangerous initial state
Bypass transition in thermoacoustics
With an infinitesimal amplification, the most dangerous state evolves to the stable periodic solution, after initial attraction towards the unstable periodic solution.
Evolution from the most dangerous initial state (higher solution)
Bypass transition in thermoacoustics
0.7
1.330
• Triggering is like bypass transition to turbulence, but occurs due to transient growth towards the unstable periodic solution rather than transient growth away from the stable fixed point.
• Nonnormality describes the start of the journey, while nonlinearity describes the end.
• Triggering (in this model) is simpler than bypass transition to turbulence. There is only one unstable attractor.
• In thermoacoustics, the nonlinear terms contribute to transient growth as much as the nonnormal terms do.
Bypass transition in thermoacoustics
stable periodic solutionunstable periodic solutionmost dangerous states
The most dangerous states can be represented on the bifurcation diagram to show the ‘safe operating region’.
Bifurcation diagram for a 10 mode system
Bypass transition in thermoacoustics
stable periodic solutionunstable periodic solutionmost dangerous states
The most dangerous states can be represented on the bifurcation diagram to show the safe operating region.
Bifurcation diagram for a 10 mode system
‘linearly stable’
Bypass transition in thermoacoustics
stable periodic solutionunstable periodic solutionmost dangerous states
The most dangerous states can be represented on the bifurcation diagram to show the safe operating region.
Bifurcation diagram for a 10 mode system
‘linearly stable but nonlinearly unstable’
Bypass transition in thermoacoustics
stable periodic solutionunstable periodic solutionmost dangerous states
The most dangerous states can be represented on the bifurcation diagram to show the safe operating region.
Bifurcation diagram for a 10 mode system
safe
Bypass transition in thermoacoustics
Bypass transition in thermoacoustics
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
Experiments on a Rijke tube at IIT Madras, 2010
Pressure Heat release
What do I mean?What is
the model?How does it
behave?Can I find a
linear optimal?
Can I find a non-linear optimal?
How dothey differ?
How doestriggering occur?
How does thiscompare withexperiments?
We force stochastically with a noise profile that has most energy at low frequencies (red noise).
Most dangerousinitial state
Spectrum of forcing signal forcing signal in time domain
Bypass transition in thermoacoustics
We force stochastically with a noise profile that has most energy at low frequencies (red noise).
Most dangerousinitial state
Spectrum of forcing signal forcing signal in time domain
Bypass transition in thermoacoustics
When we add this noise to our toy model, we see the same double jump and, as expected, it coincides with the unstable periodic solution.
pressure
acoustic energy
Numerical simulations Experimental results*
*
Bypass transition in thermoacoustics
Some combustion systems are described as ‘linearly stable but nonlinearly unstable’, which is a sign of a subcritical bifurcation.
oscillationamplitude
a systemparameter
Bypass transition in thermoacoustics