Identification of cross-frequency interactions in compressible cavity flow using harmonic resolvent analysis

The resolvent analysis reveals the worst-case disturbances and the most amplified response in a fluid flow that can develop around a stationary base state. The recent work by Padovan et al. ( J. Fluid Mech. , vol. 900, 2020, A14) extended the classical resolvent analysis to the harmonic resolvent analysis framework by incorporating the time-varying nature of the base flow. The harmonic resolvent analysis can capture the triadic interactions between perturbations at two different frequencies through a base flow at a particular frequency. The singular values of the harmonic resolvent operator act as a gain between the spatiotemporal forcing and the response provided by the singular vectors. In the current study, we formulate the harmonic resolvent analysis framework for compressible flows based on the linearized Navier–Stokes equation (i.e. operator-based formulation). We validate our approach by applying the technique to the low-Mach-number flow past an airfoil. We further illustrate the application of this method to compressible cavity flows at Mach numbers of 0.6 and 0.8 with a length-to-depth ratio of $2$ . For the cavity flow at a Mach number of 0.6, the harmonic resolvent analysis reveals that the nonlinear cross-frequency interactions dominate the amplification of perturbations at frequencies that are harmonics of the leading Rossiter mode in the nonlinear flow. The findings demonstrate a physically consistent representation of an energy transfer from slow-evolving modes toward fast-evolving modes in the flow through cross-frequency interactions. For the cavity flow at a Mach number of 0.8, the analysis also sheds light on the nature of cross-frequency interaction in a cavity flow with two coexisting resonances.