Bypass transition in thermoacoustics (Triggering)

1. Bypass transition in thermoacoustics(Triggering) IIIT Pune & Idea Research, 3rd Jan 2011 Matthew Juniper (mpj1001@cam.ac.uk) Engineering Department, University of Cambridge with thanks to Peter Schmid, R. I. Sujith and Iain Waugh Bypass transition in thermoacoustics

2. In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable? A. Re = 100 to 1000 B. Re = 1000 to 10 000 C. Re = 10 000 to 100 000 D. It never becomes unstable Phone a friend 50/50 Ask the audience

3. In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable? B. Re = 1000 to 10 000 D. It never becomes unstable Phone a friend 50/50 Ask the audience

4. In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable? B. Re = 1000 to 10 000 D. It never becomes unstable Phone a friend 50/50 Ask the audience

5. In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable? B. Re = 1000 to 10 000 D. It never becomes unstable Phone a friend 50/50 Ask the audience

6. In fluid mechanics, at what Reynolds number does the flow within a pipe become unstable? B. Re = 1000 to 10 000 Phone a friend 50/50 Ask the audience

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

8. What is the model? How does it behave? Can I find a linear optimal? What do I mean? How does this compare with experiments? Can I find a non- linear optimal? How do they differ? How does triggering occur?

9. 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

10. 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

11. 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

12. But some systems seem able to trigger spontaneously from just the background noise. Bypass transition in thermoacoustics

13. But some systems seem able to trigger spontaneously from just the background noise. Bypass transition in thermoacoustics

14. Bypass transition in thermoacoustics

15. What is the model? How does it behave? Can I find a linear optimal? What do I mean? How does this compare with experiments? Can I find a non- linear optimal? How do they differ? How does triggering occur?

16. air flow 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. Diagram of the Rijke tube hot wire Non-dimensional governing equations (note the time delay in the heat release term) acoustics damping heat release at the hot wire Definition of the non-dimensional acoustic energy Bypass transition in thermoacoustics

17. 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 p u Non-dimensional discretized governing equations Definition of the non-dimensional acoustic energy Bypass transition in thermoacoustics

18. 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 p u Non-dimensional discretized governing equations uj pj Definition of the non-dimensional acoustic energy Bypass transition in thermoacoustics

19. What is the model? How does it behave? Can I find a linear optimal? What do I mean? How does this compare with experiments? Can I find a non- linear optimal? How do they differ? How does triggering occur?

20. 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

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

22. 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 periodic solution stable fixed point Bypass transition in thermoacoustics

23. 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 stable periodic solution boundary of the basins of attraction of the two stable solutions Bypass transition in thermoacoustics

24. The unstable periodic solution sits on the basin boundary. 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 Bypass transition in thermoacoustics

25. We want to find the lowest energy point on this boundary. 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 Bypass transition in thermoacoustics

26. The lowest energy point on the unstable periodic solution is a good starting point but can a better point be found? 3-D cartoon of 20-D state space stable periodic solution unstable periodic solution lowest energy point on the unstable periodic solution boundary of the basins of attraction of the two stable solutions Bypass transition in thermoacoustics

27. If the basin boundary looks like a potato, is it ... Desiree Bypass transition in thermoacoustics

28. If the basin boundary looks like a potato, is it ... Desiree Pink eye Bypass transition in thermoacoustics

29. If the basin boundary looks like a potato, is it ... Desiree Pink eye Pink fur apple Bypass transition in thermoacoustics

30. Bypass transition in thermoacoustics

31. What is the model? How does it behave? Can I find a linear optimal? What do I mean? How does this compare with experiments? Can I find a non- linear optimal? How do they differ? How does triggering occur?

32. We start by examining the unstable periodic solution. 3-D cartoon of 20-D state space stable periodic solution unstable periodic solution stable fixed point Bypass transition in thermoacoustics

33. We evaluate the monodromy matrix around the unstable periodic solution and find its eigenvalues and eigenvectors. Floquet multipliers of the unstable periodic solution (eigenvalues of monodromy matrix) Bypass transition in thermoacoustics

34. We evaluate the monodromy matrix around the unstable periodic solution and find its eigenvalues and eigenvectors. Floquet multipliers of the unstable periodic solution (eigenvalues of monodromy matrix) Bypass transition in thermoacoustics

35. We evaluate the monodromy matrix around the unstable periodic solution and find its eigenvalues and eigenvectors. Floquet multipliers of the unstable periodic solution (eigenvalues of monodromy matrix) First eigenvector μ = 1.0422 Bypass transition in thermoacoustics

36. The first singular value exceeds the first eigenvalue, which means that transient growth is possible around the unstable periodic solution. Floquet multipliers of the unstable periodic solution (eigenvalues of monodromy matrix) First eigenvector μ = 1.0422 First singular vector σ = 1.6058 Bypass transition in thermoacoustics

37. 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. 3-D cartoon of 20-D state space stable periodic solution unstable periodic solution lowest energy point on the unstable periodic solution boundary of the basins of attraction of the two stable solutions Bypass transition in thermoacoustics

38. Bypass transition in thermoacoustics

39. What is the model? How does it behave? Can I find a linear optimal? What do I mean? How does this compare with experiments? Can I find a non- linear optimal? How do they differ? How does triggering occur?

40. We need to find the optimal initial state of the nonlinear governing equations Discretization into basis functions p u Non-dimensional discretized governing equations Definition of the non-dimensional acoustic energy Bypass transition in thermoacoustics

41. We find a non-linear optimal initial state by defining an appropriate cost functional, J, and expressing the governing equations as constraints.  Lagrange optimization Costfunctional: Constraints: Define a Lagrangian functional: Bypass transition in thermoacoustics

42. We re-arrange the Lagrangian functional to obtain the adjoint equations of the non-linear governing equations Re-arrange: The optimal value of J is found when: Bypass transition in thermoacoustics

43. p1 u1 The (local) optimal initial state is found by adjoint looping of the governing equations, nested within a conjugate gradient algorithm. Linear governing equations, constrained E0 Non-linear governing equations, unconstrained E0 p1 SVD solution u1 contours: cost functional, J arrows: gradient information returned from adjoint looping of non-linear governing equations dots: path taken by conjugate gradient algorithm Bypass transition in thermoacoustics

44. The (local) optimal initial state is found by adjoint looping of the governing equations, nested within a conjugate gradient algorithm. Linear governing equations, constrained E0 Non-linear governing equations, unconstrained E0 p1 p1 SVD solution u1 u1 contours: cost functional, J arrows: gradient information returned from adjoint looping of non-linear governing equations dots: path taken by conjugate gradient algorithm Bypass transition in thermoacoustics

45. 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

46. 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 first singular value most dangerous initial state Bypass transition in thermoacoustics

47. Bypass transition in thermoacoustics

48. What is the model? How does it behave? Can I find a linear optimal? What do I mean? How does this compare with experiments? Can I find a non- linear optimal? How do they differ? How does triggering occur?

49. So far we found the optimal initial state, which exploits transient growth around the unstable periodic solution ... 3-D cartoon of 20-D state space stable periodic solution unstable periodic solution stable fixed point Bypass transition in thermoacoustics

50. t ... 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 first singular value optimal state around stable fixed point most dangerous initial state Bypass transition in thermoacoustics