Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew...
-
Upload
clifford-lawson -
Category
Documents
-
view
220 -
download
0
Transcript of Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew...
![Page 1: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/1.jpg)
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
![Page 2: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/2.jpg)
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
![Page 3: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/3.jpg)
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?
![Page 4: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/4.jpg)
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?
![Page 5: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/5.jpg)
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?
![Page 6: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/6.jpg)
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?
![Page 7: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/7.jpg)
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
![Page 8: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/8.jpg)
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?
![Page 9: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/9.jpg)
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
![Page 10: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/10.jpg)
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
![Page 11: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/11.jpg)
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
![Page 12: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/12.jpg)
But some systems seem able to trigger spontaneously from just the background noise.
Bypass transition in thermoacoustics
![Page 13: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/13.jpg)
But some systems seem able to trigger spontaneously from just the background noise.
Bypass transition in thermoacoustics
![Page 14: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/14.jpg)
Bypass transition in thermoacoustics
![Page 15: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/15.jpg)
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?
![Page 16: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/16.jpg)
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
![Page 17: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/17.jpg)
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
![Page 18: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/18.jpg)
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
![Page 19: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/19.jpg)
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?
![Page 20: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/20.jpg)
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
![Page 21: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/21.jpg)
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
![Page 22: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/22.jpg)
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
![Page 23: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/23.jpg)
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
![Page 24: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/24.jpg)
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
![Page 25: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/25.jpg)
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
![Page 26: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/26.jpg)
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
![Page 27: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/27.jpg)
If the basin boundary looks like a potato, is it ...
Desiree
Bypass transition in thermoacoustics
![Page 28: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/28.jpg)
If the basin boundary looks like a potato, is it ...
Desiree Pink eye
Bypass transition in thermoacoustics
![Page 29: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/29.jpg)
If the basin boundary looks like a potato, is it ...
Desiree Pink eye Pink fur apple
Bypass transition in thermoacoustics
![Page 30: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/30.jpg)
Bypass transition in thermoacoustics
![Page 31: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/31.jpg)
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?
![Page 32: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/32.jpg)
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
![Page 33: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/33.jpg)
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
![Page 34: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/34.jpg)
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
![Page 35: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/35.jpg)
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
![Page 36: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/36.jpg)
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
![Page 37: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/37.jpg)
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
![Page 38: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/38.jpg)
Bypass transition in thermoacoustics
![Page 39: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/39.jpg)
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?
![Page 40: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/40.jpg)
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
![Page 41: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/41.jpg)
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
![Page 42: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/42.jpg)
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
![Page 43: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/43.jpg)
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
![Page 44: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/44.jpg)
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
![Page 45: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/45.jpg)
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
![Page 46: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/46.jpg)
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
![Page 47: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/47.jpg)
Bypass transition in thermoacoustics
![Page 48: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/48.jpg)
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?
![Page 49: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/49.jpg)
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
![Page 50: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/50.jpg)
... 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
![Page 51: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/51.jpg)
The first paper about this was in 2008 paper by Balasubmrananian and Sujith at IIT Madras
Triggering in the Rijke tube
![Page 52: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/52.jpg)
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
![Page 53: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/53.jpg)
Bypass transition in thermoacoustics
![Page 54: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/54.jpg)
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?
![Page 55: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/55.jpg)
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
![Page 56: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/56.jpg)
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
![Page 57: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/57.jpg)
• 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
![Page 58: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/58.jpg)
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
![Page 59: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/59.jpg)
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
![Page 60: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/60.jpg)
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
![Page 61: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/61.jpg)
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
![Page 62: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/62.jpg)
![Page 63: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/63.jpg)
Bypass transition in thermoacoustics
![Page 64: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/64.jpg)
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?
![Page 65: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/65.jpg)
Experiments on a Rijke tube at IIT Madras, 2010
Pressure Heat release
![Page 66: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/66.jpg)
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?
![Page 67: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/67.jpg)
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
![Page 68: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/68.jpg)
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
![Page 69: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/69.jpg)
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
![Page 70: Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department,](https://reader036.fdocuments.in/reader036/viewer/2022062309/5697bff41a28abf838cbd493/html5/thumbnails/70.jpg)
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