Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper Engineering Department, University of Cambridge with thanks to Peter Schmid, R. I. Sujith and Iain Waugh Bypass transition in thermoacoustics

A. Re = 100 to 1000B. Re = 1000 to C. Re = to D. It never becomes unstable In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable? Phone a friend50/50Ask the audience

B. Re = 1000 to D. It never becomes unstable Phone a friend50/50Ask the audience In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable?

B. Re = 1000 to D. It never becomes unstable Phone a friend50/50Ask the audience In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable?

B. Re = 1000 to D. It never becomes unstable Phone a friend50/50Ask the audience In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable?

B. Re = 1000 to Phone a friend50/50Ask the audience In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable?

Bypass transition in thermoacoustics (Triggering) IIIT Pune & Idea Research, 3 rd Jan 2011 Matthew Juniper Engineering Department, University of Cambridge with thanks to Peter Schmid, R. I. Sujith and 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 do they differ? How does triggering occur? How does this compare with experiments?

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. oscillation amplitude a system parameter 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

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 do they differ? How does triggering occur? How does this compare with experiments?

Diagram of the Rijke tube Non-dimensional governing equations hot wire air flow acousticsdampingheat 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 equations ujuj pjpj 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 do they differ? How does triggering occur? How does this compare with experiments?

A continuation method is used to find stable and unstable periodic solutions. Bifurcation diagrams for a 10 mode system stable periodic solution unstable 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 solution unstable 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 basins of 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 basins of 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 basins of 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 basins of 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 the unstable 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... DesireePink eye Bypass transition in thermoacoustics

If the basin boundary looks like a potato, is it... DesireePink eyePink fur apple 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 do they differ? How does triggering occur? How does this compare with experiments?

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 eigenvector μ = 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 eigenvector μ = The first singular value exceeds the first eigenvalue, which means that transient growth is possible around the unstable periodic solution. First singular vector σ = Bypass transition in thermoacoustics

3-D cartoon of 20-D state space stable periodic solution unstable periodic solution boundary of the basins of 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 the unstable periodic solution 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 do they differ? How does triggering occur? How does this compare with experiments?

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

u1u1 p1p1 u1u1 p1p1 Linear governing equations, constrained E 0 contours: cost functional, J arrows: gradient information returned from adjoint looping of non-linear governing equations Non-linear governing equations, unconstrained E 0 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

u1u1 p1p1 u1u1 p1p1 Linear governing equations, constrained E 0 contours: cost functional, J arrows: gradient information returned from adjoint looping of non-linear governing equations Non-linear governing equations, unconstrained E 0 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 the unstable periodic solution most dangerous initial state 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 the unstable periodic solution most dangerous initial state first singular value 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 do they differ? How does triggering occur? How does this compare with experiments?

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 the unstable periodic solution most dangerous initial state first singular value optimal state around stable 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

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 do they differ? How does triggering occur? How does this compare with experiments?

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

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 solution unstable periodic solution most 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 solution unstable periodic solution most 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 solution unstable periodic solution most 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 solution unstable periodic solution most 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

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 do they differ? How does triggering occur? How does this compare with experiments?

Experiments on a Rijke tube at IIT Madras, 2010 PressureHeat 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 do they differ? How does triggering occur? How does this compare with experiments?

We force stochastically with a noise profile that has most energy at low frequencies (red noise). Most dangerous initial state Spectrum of forcing signalforcing 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 dangerous initial state Spectrum of forcing signalforcing 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 simulationsExperimental results* * Bypass transition in thermoacoustics

Some combustion systems are described as ‘linearly stable but nonlinearly unstable’, which is a sign of a subcritical bifurcation. oscillation amplitude a system parameter Bypass transition in thermoacoustics