Modelling rectangular-jet screech via dimensional reduction

Abstract

This work presents a linear stability analysis of the waves involved in the screech resonance loop generated within a rectangular supersonic jet. Linear stability analysis of rectangular jets has often been performed using a planar-jet approximation, due to the significant reduction in complexity and computational cost. However, the impact of this simplification on the predictive power of the stability model has not been considered in detail thus far. In this work, the predictions of both a simplified planar model and a more representative two-dimensional model are compared for two waves of particular relevance to jet noise. These waves are the downstream-propagating Kelvin-Helmholtz (KH) instability, and the upstream-propagating guided-jet mode (GJM). Disparity in the predicted wavenumber for both waves is considered, as well as disparities in the KH growth rate, and the GJM band of existence. A parametric sweep through nozzle-pressure ratio is performed for rectangular jets with aspect ratios ranging from a square jet with \(\textrm{AR}=1\) to an \(\textrm{AR}=8\) rectangular jet. It is demonstrated that the KH wavenumber can be well approximated by a planar model for rectangular geometries with an aspect ratio as low as 2. The growth rate is more sensitive, but for \(\textrm{AR} \ge 4\) the differences between the two models are negligible. In contrast to the KH waves, the band of existence of the GJM, defined as the frequency range between the mode’s branch and saddle points, shows significant dependence on the modelling approach. For higher aspect ratios, the planar model can reasonably predict the saddle point of the GJM, but the predicted branch point of the GJM differs significantly from that predicted by the two-dimensional model. Performing the analysis about an experimentally derived mean-flow profile demonstrates that the predictions for the GJM are sensitive to small changes in the flow profile, with both branch and saddle points showing a strong dependence on the thickness of the shear layer. In all cases tested, the two-dimensional model predicts a much narrower band of frequencies over which the GJM is supported by the flow, as compared to the planar model. To verify the predictions made from the two models, screech frequency predictions are made using the modified weakest-link model and compared to experimental data. The planar model, though it overpredicts the band of existence of the GJM, still produces correct predictions of screech frequency. The two-dimensional model, when linearized about the experimental mean flow, also correctly predicts the screech tones, though performs no better than the planar model.