A direct-adjoint framework for stability and sensitivity analyses of turbomachinery aeroacoustics

In order to meet new stringent emissions and noise reduction targets, the turbomachinery community is increasingly concerned with addressing inefficiencies in aeroengines, and of particular interest is the issue of sound generated through aeroacoustic processes within the components of the turbomachine. At the same time, precipitous increases in computing power over the last thirty years are driving larger and more detailed simulations. Increasing data output from these requires the development of new techniques that are able to identify the underlying processes and facilitate future efforts for optimisation. This thesis seeks to address this by creating a direct-adjoint framework that can be applied in two sub-categories of flow problems commonly found in turbomachinery aeroacoustics, and lays the groundwork for future efforts in the efficient analysis and optimisation of these flows. For this, the in-house, high-order, time-domain pseudo-wave-based aeroacoustics code, previously used for isolated airfoil self-noise studies, has been extensively modified and adapted for application in turbomachinery geometries. Using this code as a starting point, self-excited instabilities and the resulting acoustic fields are simulated within a representative compressor blade row, at off-design conditions. Prior experiments and numerical calculations of the flow at these conditions have confirmed the presence of an unsteady reattaching laminar separation bubble on the suction side of the blade, shedding vortices that are subsequently scattered by the blunt trailing edge. Nonlinear flow analyses are carried out first within a single blade passage, followed by a linear impulse response analysis. A detailed linear direct-adjoint mean flow global stability analysis is performed, and the direct spectrum obtained and correlated with the nonlinear pressure signals, showing excellent agreement. The direct global modes are categorised and corresponding adjoint modes leveraged to compute structural sensitivities with wavemaker regions. The sensitivity analysis confirms the presence of aeroacoustic feedback loops between the pressure and suction sides of the blade, and is consistent with earlier observations in similar studies of isolated airfoils. The theory of block-circulant matrices is then detailed and exploited to achieve computationally efficient decompositions of the governing system matrices, and the previous global stability analysis reproduced with ten passages per period window. An increase in linear growth rates is observed, new low-frequency structures identified and analysed in the context of new feedback loop pathways which propagate disturbances throughout the domain. The effects of aerodynamic mistuning are then briefly addressed. Rotor-stator interaction is then considered in the context of an adjoint sensitivity problem. For this, a new high-order, time-domain adjoint sliding-plane interface treatment, which is capable of modelling and propagating self-excited instabilities and tonal blade row interaction noise, is developed and extensively validated. This work concludes with the application of this theory to a simplified rotor-stator sensitivity analysis within the context of a time-domain nonlinear-adjoint looping algorithm. Sensitivities identified here are briefly discussed. Applicability of these methods to different codes and numerical treatments is highlighted throughout this thesis.