Assessing non-normal effects in thermoacoustic systems with mean flow

Abstract

1In this paper, non-normal interactions in a thermoacoustic system are studied, using a low-order expansion of the state variables in terms of eigenmodes. The thermoa- The expansion in terms of eigenmodes is computationally efficient, making the approach potentially applicable to complex, 3D configurations including non-trivial boundary conditions and spatio-temporal distributions of heat release fluctuations.In the present paper, the method is applied to a 1D configuration that consists of a duct including a 1D heat source, followed by a choked isentropic nozzle. It is shown that for such a case it is essential to include the contribution of entropy perturbations in the calculation of the optimal initial perturbation and the maximum transient energy growth. Subsequently, the impact of increasing mean flow Mach number and increasing strength of flame/acoustic interaction on non normal effects is assessed in a parameter study.

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

“…Discussions about the appropriate energy norm are an ongoing matter of debate. 13,14,17,22,23 For example, Wieczorek et al 17 report that there can be differences in G max of several orders of magnitude when investigating the same non-isentropic thermoacoustic system subject to different energy norms. George and Sujith 22,23 attempt to shed light on the quality of different existing energy norms by qualifying their respective mathematical properties.…”

“…[13][14][15][16][17][18] Juniper 15,16,19 has shown that thermoacoustic triggering from smallamplitude perturbations may follow a path similar to that in bypass transition to turbulence. Non-normal effects can transiently amplify perturbations of low energy around an unstable oscillating limit cycle before the perturbations are repelled towards the undesired stable oscillating limit cycle by the action of nonlinearities.…”

“…To illustrate our arguments regarding the choice of output, we analyze two thermoacoustic setups with respect to the observed levels of non-normal transient growth: the above mentioned model studied by Wieczorek et al, 17 and a fluctuating heat source enclosed in a one-dimensional duct. In all cases, inspection of the net balance of flux and source terms associated with the definitions of output sheds light on the observed non-normal dynamics.…”

A Discrete-Time, State-Space Approach for the Investigation of Non-Normal Effects in Thermoacoustic Systems

“…Thus the 2-norm of the state vector can be related directly to acoustic energy. Note, however, that Wieczorek et al [11] have shown that acoustic energy density is not an adequate norm in thermo-acoustic systems with mean flow; total perturbation energy should be used instead in that case. The optimal definition of norm for the state space model is the subject of ongoing investigations, results and conclusions shall be reported elsewhere.…”

“…7 that the maximum amplification of perturbation energy (over all times and all initial conditions) G max ≈ 2 (see, e.g. [2,11,19,20] for pertinent definitions).…”