Adjoint-based sensitivity analysis of low-order thermoacoustic networks using a wave-based approach

Abstract

Strict pollutant emission regulations are pushing gas turbine manufacturers to develop devices that operate in lean conditions, with the downside that combustion instabilities are more likely to occur. Methods to predict and control unstable modes inside combustion chambers have been developed in the last decades but, in some cases, they are computationally expensive. Sensitivity analysis aided by adjoint methods provides valuable sensitivity information at a low computational cost. This paper introduces adjoint methods and their application in wave-based low order network models, which are used as industrial tools, to predict and control thermoacoustic oscillations. Two thermoacoustic models of interest are analysed. First, in the zero Mach number limit, a nonlinear eigenvalue problem is derived, and continuous and discrete adjoint methods are used to obtain the sensitivities of the system to small modifications. Sensitivities to base-state modification and feedback devices are presented. Second, a more general case with non-zero Mach number, a moving flame front and choked outlet, is presented. The influence of the entropy waves on the computed sensitivities is shown.

Adjoint Methods as Design Tools in Thermoacoustics

“…The adjoint boundary conditions are defined such that the boundary terms are zerô ρ + * ūρ +ρ + * ρû +û + * ρūû +û + * p +p + * ūp +p + * γpû = 0. (105) As shown by Aguilar et al [162], the set of adjoint partial differential equations is hyperbolic: Two adjoint acoustic waves propagate at speedsc +ū andc −ū and an adjoint entropy wave convects at the mean speedū. The flame and frequency-dependent boundary conditions make the system non-self-adjoint.…”

“…The individual influence of the hydrodynamic and acoustic fields on the thermoacoustic stability were identified and calculated by an adjoint method. The adjoint sensitivity framework was applied in [162] to thermoacoustic networks, which are used in the preliminary design of aero-engines and gas turbines for power generation. Adjoint methods were developed to accelerate uncertainty quantification of thermoacoustic stability in an annular-combustor network [163] and in a turbulent swirled combustor [164], where first-and second-order corrections for nonlinear degenerate eigenproblems were used [165].…”

“…Importantly, it is straightforward to check the important property that ∂([q + , q])/∂t = 0, which is used for debugging unsteady adjoint codes (see dot-product test in Appendix B.2). The authors of [153,162] presented a comparison between the numerical truncation errors between the CA and DA methods as applied to thermoacoustics. Although the two formulations should converge in principle, it has been shown that convergence is actually not guaranteed a priori [134,[254][255][256][257].…”

“…The adjoint variables for the governing equations (19) are defined asρ + (x),û + (x),p + (x), whereas the adjoint variables for the jump conditions (28) are defined asf + ,ĝ + ,ĥ + . More details can be found in [162]. Integration by parts of Green's identity (62), yields the continuous adjoint equations…”

“…As explained in [162], the receptivity to the mass equation, used to compute the corresponding feedback mechanisms, is given byρ + (x)+c 2p+ (x). In the same reference, the discrete adjoint equations can be found.…”