Whistling of turbulent cavity flows: self-consistent predictions with the linearised compressible Navier–Stokes equations

A self-consistent saturation model for the prediction of aeroacoustic limit cycles emerging in turbulent low-Mach cavity flows (Reynolds number upper R e equals script upper O left parenthesis 10 Superscript 5 Baseline right parenthesis R e = O ( 10 5 ) Re=O(10⁵) , Mach number upper M asymptotically equals 0.2 M ≃ 0.2 M 0.2 ) is proposed. It predicts the nonlinear interactions between the acoustic modes of a deep rectangular cavity and the hydrodynamic instabilities of the turbulent shear layer that forms over its opening due to the presence of a grazing flow. The model is based on the triple decomposition of the flow variables and the compressible Navier–Stokes equations. At each step of the iterative process, the nonlinear eigenvalue problem associated with perturbations around the mean flow is updated with the steady component of the forcing from the unstable eigenmode’s Reynolds stress. The iterations are performed until the dominant eigenmode becomes marginally stable, i.e. its growth rate vanishes. The evolution of the coherent velocity fluctuations as a function of the oscillation amplitude is in good qualitative agreement with previously published compressible large eddy simulations. Furthermore, the predictions of the frequency and amplitude of the aeroacoustic limit-cycle oscillations are validated against the ones obtained from a low-order model, whose parameters were adjusted to reproduce the experimental measurements of the deep cavity whistling.