Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper ([email protected]) Engineering Department,

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



B. Re = 1000 to 10 000




B. Re = 1000 to 10 000




B. Re = 1000 to 10 000



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


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.


Combustion instability is still one of the biggest challenges facing gas turbine and rocket engine manufacturers.

SR71 engine test, with afterburner


Some combustion systems are described as ‘linearly stable but nonlinearly unstable’, which is a sign of a subcritical bifurcation.

oscillationamplitude

a systemparameter


But some systems seem able to trigger spontaneously from just the background noise.


But some systems seem able to trigger spontaneously from just the background noise.





behave?Can I find a

linear optimal?


How dothey differ?



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


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


Non-dimensional discretized governing equations


u p

The governing equations are discretized by considering the fundamental ‘open organ pipe’ mode and its harmonics. This is a Galerkin discretization.



Non-dimensional discretized governing equationsuj

pj




behave?Can I find a

linear optimal?


How dothey differ?



A continuation method is used to find stable and unstable periodic solutions.

Bifurcation diagrams for a 10 mode system stable periodic solutionunstable periodic solution


A continuation method is used to find stable and unstable periodic solutions.

Bifurcation diagrams for a 10 mode system stable periodic solutionunstable periodic solution


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



A surface separates the points that evolve to the stable fixed point from the points that evolve to the stable periodic solution.


boundary of the basinsof attraction of the two

stable solutions




unstable periodic solution


stable solutions

The unstable periodic solution sits on the basin boundary.






stable solutions

We want to find the lowest energy point on this boundary.






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


If the basin boundary looks like a potato, is it ...

Desiree



Desiree Pink eye



Desiree Pink eye Pink fur apple





behave?Can I find a

linear optimal?


How dothey differ?





stable fixed point


We start by examining the unstable periodic solution.


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.






First eigenvectorμ = 1.0422




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






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.






behave?Can I find a

linear optimal?


How dothey differ?



u p

We need to find the optimal initial state of the nonlinear governing equations



Non-dimensional discretized governing equations


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


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


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.


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.


A global optimization procedure finds the point with lowest energy on the basin boundary, called the ‘most dangerous’ initial state.


most dangerous initialstate


This has similar characteristics to a combination of the lowest energy point on the unstable periodic solution plus the first singular value.



first singularvalue





behave?Can I find a

linear optimal?


How dothey differ?





stable fixed point


So far we found the optimal initial state, which exploits transient growth around the unstable periodic solution ...


... 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.



first singularvalue

optimal state aroundstable fixed point

t


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





behave?Can I find a

linear optimal?


How dothey differ?



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


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)


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.


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



The most dangerous states can be represented on the bifurcation diagram to show the safe operating region.


‘linearly stable’





‘linearly stable but nonlinearly unstable’





safe





behave?Can I find a

linear optimal?


How dothey differ?



Experiments on a Rijke tube at IIT Madras, 2010

Pressure Heat release



behave?Can I find a

linear optimal?


How dothey differ?



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


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


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*

*


Some combustion systems are described as ‘linearly stable but nonlinearly unstable’, which is a sign of a subcritical bifurcation.

oscillationamplitude

a systemparameter


Documents

Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper ([email protected]) Engineering Department,