Sensitivity analysis of a time-delayed thermo-acoustic system via an adjoint-based approach

Abstract

We apply adjoint-based sensitivity analysis to a time-delayed thermo-acoustic system: a Rijke tube containing a hot wire. We calculate how the growth rate and frequency of small oscillations about a base state are affected either by a generic passive control element in the system (the structural sensitivity analysis) or by a generic change to its base state (the base-state sensitivity analysis). We illustrate the structural sensitivity by calculating the effect of a second hot wire with a small heat release parameter. In a single calculation, this shows how the second hot wire changes the growth rate and frequency of the small oscillations, as a function of its position in the tube. We then examine the components of the structural sensitivity in order to determine the passive control mechanism that has the strongest influence on the growth rate. We find that a force applied to the acoustic momentum equation in the opposite direction to the instantaneous velocity is the most stabilizing feedback mechanism. We also find that its effect is maximized when it is placed at the downstream end of the tube. This feedback mechanism could be supplied, for example, by an adiabatic mesh. We illustrate the base-state sensitivity by calculating the effects of small variations in the damping factor, the heat-release timedelay coefficient, the heat-release parameter, and the hot wire location. The successful application of sensitivity analysis to thermo-acoustics opens up new possibilities for the passive control of thermo-acoustic oscillations by providing gradient information that can be combined with constrained optimization algorithms in order to reduce linear growth rates.

Experimental sensitivity analysis via a secondary heat source in an oscillating thermoacoustic system

“…3 This does not support the predictions of Magri and Juniper, 17 because they do not take into account base flow modifications caused by an increase in mean flow temperature.…”

“…17 Further information can be found in the recent reviews of literature. [18][19][20] Historically, sensitivity analysis is performed via the finite difference approach.…”

Multiple-scale thermo-acoustic stability analysis of a coaxial jet combustor

“…(15) was introduced in thermo-acoustics to calculate the effect of external feedback mechanisms, such as a hot wire, and design-parameter changes, such as the flame inlet geometric ratio, in the stability [7][8][9] via appropriate models for the thermo-acoustic perturbation δJ . We introduce the concept of intrinsic sensitivity , extending the structural sensitivity used in [7,18] , defined as the eigenvalue's sensitivity to intrinsic physical mechanisms. Here, the perturbation operator of the eigenvalue's sensitivity formula (15) is the Jacobian itself, i.e., δJ = εJ .…”

“…caam.rice.edu/software/ARPACK/ ). We use a Discrete Adjoint approach [7,25] to ensure that the adjoint spectra match the complex conjugate of the direct spectra to machine precision. The code was validated against the solutions of [18] and the spectra matched with an error smaller than 0.5%.…”

“…If at least one eigenvalue has positive growth rate, the system is unstable. The sensitivity of the stability to external feedback mechanisms or passive control has been calculated with adjoint-based methods [7][8][9] , which were recently experimentally validated [10] . However, these versatile methods were applied only to simplified thermo-acoustic models with uniform mean flows.…”