Adjoints in Thermo-Fluids
Motivation
In response to strict emission regulations, lean, premixed combustion techniques have become prevalent in gas turbines. Unfortunately, engines operating at these conditions are more susceptible to thermoacoustic instabilities, a phenomenon manifesting itself as large-amplitude pressure oscillations. This may affect engine emissions, reduce the life-spans, or even cause structural damage to the engine. In consequence, thermoacoustic instabilities are a major concern in the development of modern gas turbine combustors and need to be incorporated into the design process.
For stability analysis, the governing equations of the thermoacoustic system are typically reduced to a non-linear eigenvalue problem. A system is rendered linearly stable if the growth rate, i.e. the real part of the eigenvalue, is negative, as oscillations at the corresponding frequency are damped. However, thermoacoustic systems are highly sensitive to a large number of input parameters which may cause the eigenvalue to become unstable for only small chages in the input parameters. Adjoint methods enable the calculation of the sensitivity of output parameters, i.e. the system eigenvalues, with regard to all input parameters requiring only one additional computation of the system, opposed to (at least) one additional computation for each input parameter required for the standard finite difference approach. This significantly reduces the computational costs for thermoacoustic stability analysis. Furthermore, the adjoint formalism is capable to explicitly quantify implicit dependencies, making adjoint solvers particularly efficient for the calculation of inverse and optimization problems. This allows us to cheaply identify parameter settings for which an unstable thermoacoustic system is stablilized.
However, standard stability analysis may not suffice as parameters are always subject to uncertainties, i.e. the operational conditions deviate from the nominal condition due to stochastic fluctuations. As a result, a system that has been predicted to be stable for nominal parameter settings may, in fact, become unstable as the parameters deviate from their nominal values. Quantifying the uncertainty and confidence intervals of predictions by taking parameter variations into account is, therefore, essential for reliable claims concerning system stability. Adjoint theory can be applied to cheaply determine these system uncertainties.
Objectives and Strategy
Objective #1 - Exceptional Points
Low-order network models are commonly used for preliminary studies of thermoacoustic systems. Our in-house network model taX reformulates the system equations into a state-space system, thus the resulting system is described by a linear eigenvalue problem, even for systems including a flame. To obtain the discrete adjoint system, the direct state-space model can be transformed by replacing the system matrix by its Hermitian. Based on this approach, the eigenvalue sensitivities can be calculated in a straight-forward manner for any type of thermoacoustic system that can be modeled using the original taX software.
We exploited this characteristic to detect and study so-called exceptional points, utilizing the adjoint derivation of eigenvalue sensitivities. We seek to assess the influence of exceptional points on the reliability of stability predictions, taking into account that model parameters are always uncertain. The study further investigated the reliability of surrogate models in the vicinity of exceptional points to quantify system uncertainties.
Objective #2 - Continuous Adjoints
We strive to gain new insights into the physics of thermoacoustic systems by analyzing the corresponding adjoint systems. Since adjoint eigenmodes obtained from the discrete adjoint approach exhibit faulty values at boundary and jump conditions, we aim to modify the original taX software to get automated solutions of the continuous adjoint system. The software architecture promises to be advantageous, since adjoint jump conditions may be formulated based on the direct jump conditions, i.e. the adjoint equations don't need to be derived from first principles.
Based on the analysis of adjoint eigenmodes gained from the continuous adjoint system we want to efficiently optimize the system parameters to stabilize the system and minimize risk factors.
Objective #3 - Uncertainty Quantification
Although there have been studies regarding adjoint methods used in uncertainty quantification, this topic still offers a lot of potential. Adjoint analysis can be coupled with perturbation theory to obtain reliable surrogate models to quantify uncertainty propagation. In addition, we aim to identify relevant model parameters by performing global sensitivity analysis (GSA). The results of standard GSA depend on the distribution of input uncertainties. However, not all uncertainty distributions of the model parameters can be derived from physical information. As a result, uncertainty quantification usually draws on strong assumptions concerning input uncertainties. We aim to improve uncertainty predictions by determining realistic uncertainty distributions for the input parameters.
References
[1] Silva et al. Uncertainty quantification of growth rates of thermoacoustic instability by an adjoint Helmholtz solver. In ASME/IGTI Turbo Expo 2016, GT2016-57659, Seoul, Korea, 2016. ASME.
[2] Magri et al. Stability analysis of thermoacoustic nonlinear eigenproblems in annular combustors. Part I. Sensitivity. Journal of Computational Physics, 325:395–410, November 2016.
[3] Magri et al. Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification. Computational Physics, 325:411–421, November 2016.