Non-normality, Transient Growth & Subcritical...
Transcript of Non-normality, Transient Growth & Subcritical...
![Page 1: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/1.jpg)
Non-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability
R. I. SujithDept. of Aerospace EngineeringIIT Madras
![Page 2: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/2.jpg)
Non-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability
R. I. SujithDept. of Aerospace EngineeringIIT Madras
Acknowledgements
Balasubramanian, Kulkarni, Kedia, Nagaraja, BharatPriya, Sathesh, Lipika, Subhas, Nishant, Sreerekha, Joseph, Vikrant, Vineeth,....
Funding from DST, ISRO; Wolfgang Polikfe & Rama Govindarajan
![Page 3: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/3.jpg)
We clarify the role of non-normality & nonlinearity in thermoacoustics
![Page 4: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/4.jpg)
Non-normality & its role in subcritical transition to instability
We clarify the role of non-normality & nonlinearity in thermoacoustics
![Page 5: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/5.jpg)
Non-normality & its role in subcritical transition to instability
Rijke tube
We clarify the role of non-normality & nonlinearity in thermoacoustics
![Page 6: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/6.jpg)
Non-normality & its role in subcritical transition to instability
How to analyse non-normal systems?
Rijke tube
We clarify the role of non-normality & nonlinearity in thermoacoustics
![Page 7: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/7.jpg)
Positive Feedback
Acoustics
Heat Release
Combustion instability is caused by the feedback between acoustics & combustion
![Page 8: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/8.jpg)
![Page 9: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/9.jpg)
Network model using normal modesPolifke (2004)
Combustion instability is investigated by testing for unstable eigenvalues of the linearized system
![Page 10: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/10.jpg)
Reasoning: At low amplitudes, linearised eqns are sufficient to model the evolution
Network model using normal modesPolifke (2004)
Combustion instability is investigated by testing for unstable eigenvalues of the linearized system
![Page 11: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/11.jpg)
Initial excitation at small identical amplitudes.
A linearly stable system can go unstable!
![Page 12: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/12.jpg)
Who is the villain?
![Page 13: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/13.jpg)
Who is the villain?
Nonlinearity
![Page 14: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/14.jpg)
Who is the villain?
Nonlinearity?
![Page 15: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/15.jpg)
Let us linearize the acoustics and the heat release.
Momentum
Energy
LinearizedHeat release
![Page 16: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/16.jpg)
A system is non-normal when its evolution operator does not commute with its adjoint
![Page 17: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/17.jpg)
A system is non-normal when its evolution operator does not commute with its adjoint
Thermoacoustic interaction is non-normal
Balasubramanian & Sujith: JFM (2008), POF (2008); Nicoud et al. (AIAA J 2007)
![Page 18: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/18.jpg)
![Page 19: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/19.jpg)
“Non-normality”
![Page 20: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/20.jpg)
A non-normal system can have transient growth even if the individual modes decay
Φ1
Φ2
Φ1
Φ2
Φ1
Φ2
Individual Eigenvalues: Wrong tool to analyse a non-normal system
![Page 21: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/21.jpg)
Superposition of decaying eigenvectors can produce growth in short term
Schmid (2007)
Non-normality: not considered before in thermoacoustic instability
![Page 22: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/22.jpg)
Transient growth can trigger nonlinearities when the amplitude reaches high enough values
Triggerring: Balasubramanian & Sujith: JFM (2008), POF (2008)
Linear
Nonlinear
![Page 23: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/23.jpg)
Transient growth can trigger nonlinearities when the amplitude reaches high enough values
Triggerring: Balasubramanian & Sujith: JFM (2008), POF (2008)
![Page 24: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/24.jpg)
![Page 25: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/25.jpg)
![Page 26: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/26.jpg)
![Page 27: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/27.jpg)
Combustion makes the acoustic modes non-normal
![Page 28: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/28.jpg)
Combustion makes the acoustic modes non-normal
Combustion modes are non-normal even in the absence of acoustic feedback
![Page 29: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/29.jpg)
![Page 30: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/30.jpg)
Rijke tube
POF (2008)AIAA Paper 2007-3428
![Page 31: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/31.jpg)
Horizontal Rijke tube is modeled
We sidestep the effects of natural convection on the mean flow
Air flow
![Page 32: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/32.jpg)
The acoustic field is solved in the time domain using the Galerkin technique
Momentum:
Energy: Compact Heat Source
Modal expansion:
![Page 33: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/33.jpg)
Heckl’s modified form of the King’s law is used to model the heat release
![Page 34: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/34.jpg)
Heckl’s modified form of the King’s law is used to model the heat release
![Page 35: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/35.jpg)
Heckl’s modified form of the King’s law is used to model the heat release
King’s law predicts nonlinearity only for velocity perturbations greater than the mean velocity
![Page 36: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/36.jpg)
BNN is non-normal
BNL is nonlinear
Evolution equations obtained from the Galerkin projection are non-normal & nonlinear
![Page 37: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/37.jpg)
No damping
![Page 38: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/38.jpg)
.
.
![Page 39: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/39.jpg)
.
.
![Page 40: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/40.jpg)
.
.
![Page 41: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/41.jpg)
Subcritical transition to instability is observed
.
.
![Page 42: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/42.jpg)
Subcritical transition to instability is observed
.
.
![Page 43: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/43.jpg)
Subcritical transition to instability is observed
.
.
![Page 44: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/44.jpg)
Subcritical transition to instability is observed
.
.
Limit cycle in the absence of damping
![Page 45: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/45.jpg)
Subcritical transition to instability is observed
.
.
Limit cycle in the absence of damping
Phase evolves to 90°
![Page 46: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/46.jpg)
Include damping
![Page 47: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/47.jpg)
“Triggering” occurs with damping as well
c1 = 0.05, c2 = 0.005
![Page 48: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/48.jpg)
Linear
Nonlinear
Identical linear & nonlinear simulations at low amplitudes.
Nonlinearity “picks up” after sufficient transient growth
![Page 49: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/49.jpg)
Bootstrapping in an initially decaying system that grows later
Lw = 2m and Tw = 900K, xf = 0.3
![Page 50: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/50.jpg)
Transient growth of oscillations is quantified by growth rate
![Page 51: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/51.jpg)
Transient growth of oscillations is quantified by growth rate
The growth rate is the envelope of energy evolution for different initial conditions
![Page 52: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/52.jpg)
Transient growth of oscillations is quantified by growth rate
The growth rate is the envelope of energy evolution for different initial conditions
![Page 53: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/53.jpg)
Transient growth of oscillations is quantified by growth rate
The growth rate is the envelope of energy evolution for different initial conditions
Gmax is the maximum growth rate within a time interval
![Page 54: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/54.jpg)
Transient growth of oscillations is quantified by growth rate
The growth rate is the envelope of energy evolution for different initial conditions
Gmax is the maximum growth rate within a time interval
Achieved through singular value decomposition.
![Page 55: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/55.jpg)
Gmax = Infinity Classical linear instability
![Page 56: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/56.jpg)
Gmax = Infinity Classical linear instability
Gmax = 1 Classical linear stability
![Page 57: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/57.jpg)
Gmax = Infinity Classical linear instability
Gmax = 1 Classical linear stability
Gmax > 1, finite Transient growth
![Page 58: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/58.jpg)
n - τ model of Crocco captures transient dynamics
Nagaraja, Kedia & Sujith (Montreal Symposium, 2008)
![Page 59: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/59.jpg)
There are regions of stability, instability and “potential” instability
Nagaraja, Kedia & Sujith (Montreal Symposium, 2008)
![Page 60: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/60.jpg)
Pseudospectra for studying non-normal systems
![Page 61: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/61.jpg)
Pseudospectra for studying non-normal systems
Z is an ε - pseudoeigenvalue of A if it satisfies ||(ZI-A)-1|| > ε -1
![Page 62: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/62.jpg)
Pseudospectra for studying non-normal systems
Set of all points on the complex plane whose minimum value of the singular value of (ZI-A) is less than ε
Z is an ε - pseudoeigenvalue of A if it satisfies ||(ZI-A)-1|| > ε -1
![Page 63: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/63.jpg)
log10ε
![Page 64: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/64.jpg)
log10ε
Contours should protrude into the right half plane for the system to exhibit transient growth
![Page 65: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/65.jpg)
log10ε
Contours should protrude into the right half plane for the system to exhibit transient growth
“Necessary condition” for transient growth
![Page 66: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/66.jpg)
![Page 67: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/67.jpg)
Lord Rayleigh
![Page 68: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/68.jpg)
Predicting transient growth using Rayleigh criteria requires precise knowledge of initial conditions
Lord Rayleigh
![Page 69: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/69.jpg)
Ambiguity of initial conditions due to noise makes the identification of transient growth using Rayleigh Criteria difficult.
Predicting transient growth using Rayleigh criteria requires precise knowledge of initial conditions
Lord Rayleigh
![Page 70: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/70.jpg)
The conditions are now on the evolution operator; hence do not depend on initial conditions
![Page 71: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/71.jpg)
Thermoacoustics involves multiphysics & multiscales
![Page 72: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/72.jpg)
Ducted Burke-Schumann Flame
JFM (2008)AIAA Paper 2007- 0567
![Page 73: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/73.jpg)
In summary, thermoacoustic interaction is non-normal and nonlinear
![Page 74: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/74.jpg)
In summary, thermoacoustic interaction is non-normal and nonlinear
Transient growth can lead to high enough amplitudes where nonlinearities become significant
![Page 75: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/75.jpg)
In summary, thermoacoustic interaction is non-normal and nonlinear
Transient growth can lead to high enough amplitudes where nonlinearities become significant
Non-normality & nonlinearity leads tosubcritical transition to instability
![Page 76: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/76.jpg)
In summary, thermoacoustic interaction is non-normal and nonlinear
Transient growth can lead to high enough amplitudes where nonlinearities become significant
Non-normality & nonlinearity leads tosubcritical transition to instability
Individual eigenvalues: wrong tools for analyzing a non-normal system.
![Page 77: Non-normality, Transient Growth & Subcritical …home.iitk.ac.in/~mkv/Conf/Talks_files/Sujith.pdfNon-normality, Transient Growth & Subcritical Transition to Thermoacoustic Instability](https://reader031.fdocuments.net/reader031/viewer/2022022005/5ab559b37f8b9a156d8cb86d/html5/thumbnails/77.jpg)
In summary, thermoacoustic interaction is non-normal and nonlinear
Transient growth can lead to high enough amplitudes where nonlinearities become significant
Non-normality & nonlinearity leads tosubcritical transition to instability
Individual eigenvalues: wrong tools for analyzing a non-normal system.
Need to adopt tools such as SVD and ε pseudospectra