Adjoint analysis for noise control in a two-dimensional compressible mixing layer

Abstract

The adjoint of the compressible Navier–Stokes equations are solved on a two-dimensional mixing layer using Direct Numerical Simulation. The goal is to study the sensitivity of aeroacoustic noise radiated by this flow to various local perturbations of mass and energy, which is the first step to possible optimal control applications. The adjoint field shows the forcings that can be applied in order to achieve a given effect. Two test cases are presented. The first sensitivity analysis is based on a localized sinusoidal source in the dual space, corresponding to a localized sinusoidal source in the real space. The second one obtained from optimal control theory allows any frequency signature of the forcing. The adjoint perturbation, once it has reached the shear layer, propagates upstream until the inflow. The pressure perturbation exhibits a high sensitivity at the inflow as was found by Cervifio et al. [Cervifio L, Bewley TR, Freund JB, Lele SK. Perturbation and adjoint analyses of flow-acoustic interactions in an unsteady 2D jet. In: CTR Proceedings of the summer program, 2002. p. 27–40], and in the location of high shear in the mixing layer. Interactions between the direct field and the adjoint field lead to a large broadening of the adjoint variables spectra, with a strong low-frequency component, indicating a possibility for low-frequency control of higher frequency sound. Comparisons of sensitivities at the inflow and near the dual source show a particular amplification of the adjoint pressure near the inflow, indicating a high sensitivity to a forcing of the continuity equation. This gives a good basis for noise control by mass suction/blowing at the inflow.