A systems perspective on non-normality in low-order thermoacoustic models: Full norms, semi-norms and transient growth

Blumenthal, Ralf; Tangirala, Arun K.; Sujith, R. I.; Polifke, Wolfgang

doi:10.1177/1756827716652474

Cited by 10 publications

(23 citation statements)

References 59 publications

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“…The above described shapes of the optimal initial condition for transient and long-time dynamics have also been investigated when the norm is such that only the instantaneous acoustic energy is considered. This was done to ensure that the results are independent of the choice of norm, as highlighted by Blumenthal et al (2017). The results showed no qualitative differences from those described above.…”

Section: Optimal Initial Conditionsmentioning

confidence: 98%

“…As the memory dimension is discretised on Gauss-Lobatto points, we use Clenshaw-Curtis quadrature weights for representing the integral in this coordinate direction (Trefethen & Embree 2005). Before proceeding, we wish to justify our choice of norm in light of the work of Blumenthal et al (2017). The above integral is taken over both the spatial and the memory direction.…”

Section: The Fully Discretised Time-domain Modelmentioning

confidence: 99%

“…Alternatively, it can also be based on the instantaneous acoustic energy. This can be done either by using the methodology proposed by Blumenthal et al (2017) or the technique used by Skene & Schmid (2019). For completeness, we have also investigated transient growth by maximising the instantaneous acoustic energy, using the methodology from Skene & Schmid (2019).…”

Section: ) Transient Behaviour At Intermediate Timesmentioning

confidence: 99%

“…In another study, Zhao (2012) considered the vertical Rijke tube and investigated the most dangerous flame location as well as the effect in non-orthogonality as mean heat is added. Recently, Blumenthal et al (2017) pointed out that non-normal analysis in thermoacoustic systems should be addressed using a carefully selected energy norm. By revisiting previous works, they show that transient growth results can only identify mechanisms linked to the state variables included in the selected norm and have to be interpreted accordingly.…”

Section: Introductionmentioning

confidence: 99%

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Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

2019

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This study analyses the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. The analysis is performed using a frequency-domain low-order network model as well as a time-domain spatially discretised model. Anti-correlated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. The time-domain analysis results in a delay-differential equation and allows the investigation of non-normal behaviour and its consequences. Pseudospectral analysis reveals that energy amplification is crucially linked to an interplay between acoustic and ITA modes. While higher non-orthogonality between two modes is correlated with peaks in modal sensitivity, transient energy growth does not necessarily involve the most sensitive modes. In particular, growth estimates based on the Kreiss constant demonstrate that transient amplification relies critically on the proximity of the non-normal modes to the imaginary axis. The time scale for transient amplification is identified as the flame time delay, which is further corroborated by determining the optimal initial conditions responsible for the bulk of the non-modal energy growth. The flame is identified as an active and dominant contributor to energy gain. The frequency of the optimal perturbation matches the acoustic time scale, once more confirming an interplay between acoustic and ITA structures. Flame-based amplification factors of two to five are found, which are significant when feeding into the acoustic dynamics and eventually triggering nonlinear limit-cycle behaviour.

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Section: Optimal Initial Conditionsmentioning

confidence: 98%

Section: The Fully Discretised Time-domain Modelmentioning

confidence: 99%

Section: ) Transient Behaviour At Intermediate Timesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

2019

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“…The calculations were performed both by singular value decomposition [38] and semi-norm Lagrangian optimization [49]. In ducted premixed flames, the level of nonnormality was shown to be small [50]. The small degree of non-normality justifies the use of eigenvalue analysis in thermoacoustic stability.…”

Section: Transient Growthmentioning

confidence: 99%

Adjoint Methods as Design Tools in Thermoacoustics

Magri

2019

Applied Mechanics Reviews

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In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor's geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to calculate the sensitivity to design parameters of limit cycles with adjoint Floquet analysis. To show the robustness and versatility of adjoint methods, examples of applications are provided for different acoustic and flame models, both in longitudinal and annular combustors, with deterministic and probabilistic approaches. The successful application of adjoint sensitivity analysis to thermoacoustics opens up new possibilities for physical understanding, control and optimization to design safer, quieter, and cleaner aero-engines. The versatile methods proposed can be applied to other multiphysical and multiscale problems, such as fluid–structure interaction, with virtually no conceptual modification.

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Optimisation of chaotically perturbed acoustic limit cycles

Huhn

Magri

2020

Nonlinear Dyn

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In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure. The objective of manufacturers is to design stable thermoacoustic configurations. In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation. In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down. First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles. Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time-averaged cost functionals exist and can be robustly calculated by a shadowing method. Third, we calculate the sensitivities of the time-averaged acoustic energy and Rayleigh index to small changes to the heat-source intensity and time delay. By embedding the sensitivities into a gradient-update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source. The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification.

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A systems perspective on non-normality in low-order thermoacoustic models: Full norms, semi-norms and transient growth

Cited by 10 publications

References 59 publications

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

Adjoint Methods as Design Tools in Thermoacoustics

Optimisation of chaotically perturbed acoustic limit cycles

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