Michael Bauerheim

Progress in analytical methods to predict and control azimuthal combustion instability modes in annular chambers

Longitudinal low-frequency thermoacoustic unstable modes in combustion chambers have been intensively studied experimentally, numerically, and theoretically, leading to significant progress in both understanding and controlling these acoustic modes. However, modern annular gas turbines may also exhibit azimuthal modes, which are much less studied and feature specific mode structures and dynamic behaviors, leading to more complex situations. Moreover, dealing with 10–20 burners mounted in the same chamber limits the use of high fidelity simulations or annular experiments to investigate these modes because of their complexity and costs. Consequently, for such circumferential acoustic modes, theoretical tools have been developed to uncover underlying phenomena controlling their stability, nature, and dynamics. This review presents recent progress in this field. First, Galerkin and network models are described with their pros and cons in both the temporal and frequency framework. Then, key features of such acoustic modes are unveiled, focusing on their specificities such as symmetry breaking, non-linear modal coupling, forcing by turbulence. Finally, recent works on uncertainty quantifications, guided by theoretical studies and applied to annular combustors, are presented. The objective is to provide a global view of theoretical research on azimuthal modes to highlight their complexities and potential.

Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification

Monte Carlo and Active Subspace Identification methods are combined with first- and second-order adjoint sensitivities to perform (forward) uncertainty quantification analysis of the thermo-acoustic stability of two annular combustor configurations. This method is applied to evaluate the risk factor, i.e., the probability for the system to be unstable. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculations by as much as the Monte Carlo samples.

Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part I. Sensitivity

We present an adjoint-based method for the calculation of eigenvalue perturbations in nonlinear, degenerate and non-self-adjoint eigenproblems. This method is applied to a thermo-acoustic annular combustor network, the stability of which is governed by a nonlinear eigenproblem. We calculate the first- and second-order sensitivities of the growth rate and frequency to geometric, flow and flame parameters. Three different configurations are analysed. The benchmark sensitivities are obtained by finite difference, which involves solving the nonlinear eigenproblem at least as many times as the number of parameters. By solving only one adjoint eigenproblem, we obtain the sensitivities to any thermo-acoustic parameter, which match the finite-difference solutions at much lower computational cost.

Numerical investigation of combustion noise: The Entropy Wave Generator

A Large Eddy Simulation of a subsonic operating point (test case number two from Bake et al. [9]) of the EWG is presented in this paper. This study has three goals. The first is to provide a larger database on the generation of indirect noise than the one provided by RANS simulations and analytical models. The second is to determine the dominant noise in the subsonic case (direct or indirect). The third is to understand the reason of the overestimation of the pressure peak noise when the nozzle Mach number increases (i.e. Figure 1). As an explanation to the third phenomenon, Howe [20] suggested firstly that in cases with flow separation in the nozzle diffuser, vortex sound is strongly correlated with entropy noise and can dramatically reduce the overall sound level. Secondly, the strong deformation of the hot slug in the nozzle throat reduces the entropy gradients within the front and rear interfaces of the slug generating a decrease in acoustic pressure.

Three-Dimensional Modeling of Annular Cascade Trailing-Edge Noise

The paper addresses various aspects of an analytical methodology for the modeling of the sound transmission through the outlet guide vanes of an axial-flow fan architecture, in view of predicting the trailing-edge noise with a proper account of the cascade effect. The first part extends a previous two-dimensional model by including the stagger and the curvature of the vanes. This is achieved by iteratively solving matching equations at the leadingedge and trailing-edge interfaces of the stator, with a multiple-scale analysis between two iterations, considering the inter-vane channels as bifurcated waveguides of slowly varying cross-section. The second part is aimed at extending a previous two-dimensional cascade trailing-edge noise model to an annular cascade described in cylindrical coordinates. For this the trailing-edge noise sources of a vane section are replaced by an equivalent lift dipole, the direct sound of which is expanded as a series of annular-duct modes. The scattering of each mode by the complete cascade is calculated by a three-dimensional mode-matching technique and the complete radiation of the trailing-edge source obtained by summing all modal contributions. The last part explains how the aforementioned two-dimensional model of curved-channel can be generalized in the three-dimensional annular geometry at the price of some approximations. Only preliminary results are given at each step, the paper being aimed at demonstrating the methodology but not yet at simulating a complete configuration. The objective is to formulate three-dimensional blade/vane row aeroacoustic problems without resorting to a strip-theory approach.

Saturation of a turbulent mixing layer over a cavity: response to harmonic forcing around mean flows

Turbulent mixing layers over cavities can couple with acoustic waves and lead to undesired oscillations. To understand the nonlinear aspects of this phenomenon, a turbulent mixing layer over a deep cavity at Reynolds number 150 000 is considered and its response to harmonic forcing is analysed with large-eddy simulations (LES) and linearised Navier-Stokes equations (LNSE). As a model of incoming acoustic perturbations, spatially uniform time-harmonic forcing is applied at the cavity end, with amplitudes in the wide range 0.045-8.9% of the bulk velocity. Compressible LES provide reference nonlinear responses of the shear layer, and the associated mean flows. Linear responses are calculated with the incompressible LNSE around the LES mean flows; they predict well the amplification (both measured with kinetic energy and with a proxy for vortex sound production) and capture the nonlinear saturation observed as the forcing amplitude increases and the mixing layer thickens. Perhaps surprisingly, LNSE calculations based on a monochromatic (single frequency) assumption yield a good agreement even though higher harmonics and their nonlinear interaction (Reynolds stresses) are not negligible. However, the leading Reynolds stresses do not force the mixing layer efficiently, as shown by a comparison with the optimal volume forcing obtained in a resolvent analysis. Thus, they cannot fully benefit from the potential for amplification available in the flow. Finally, the sensitivity of the optimal harmonic forcing at the cavity end is computed with an adjoint method. The sensitivities to mean flow modification and to a localised feedback (structural sensitivity) both identify the upstream cavity corner as the region where a small-amplitude modification has the strongest effect. This can guide in a systematic way the design of strategies for the control of amplification and saturation mechanisms.