The adjoint of the perturbed and linearized compressible viscous flow equations is formulated in such a way that its solution can be used to optimize control actuation in order to reduce flow-generated sound. We apply it to a direct numerical simulation of a randomly excited two-dimensional mixing layer, with inflow vorticity-thickness Reynolds number 500 and free-stream Mach numbers 0.9 and 0.2. The control actuation is implemented as general source terms in the flow equations (body forces, mass sources, and internal energy sources) with compact support near the inflow boundary. The noise to be reduced is defined by a space-time integral of the meansquare pressure fluctuations on a line parallel to the mixing layer in the acoustic field of the low-speed stream. Both the adjoint and flow equations are solved numerically and without modelling approximations. The objective is to study the mechanics of the noise generation and its control. All controls reduce targeted noise with very little required input power, with the most effective (the internal energy control) reducing the noise intensity by 11 dB. Numerical tests confirm that the control is not by any simple acoustic cancellation mechanism but instead results from a genuine change of the flow as a source of sound. The comparison of otherwise identical flows with and without control applied shows little change of the flow's gross features: the evolution and pairings of the energetic structures, turbulence kinetic energy, spreading rate, and so on are superficially unchanged. However, decomposition of the flow into empirical eigenfunctions, as surrogates for Fourier modes in the non-periodic streamwise direction, shows that the turbulence structures advect downstream more uniformly. This change appears to be the key to reducing their acoustic efficiency, a perspective that is clarified by comparing the randomly excited mixing layer to a harmonically excited mixing layer, which is relatively quiet because it is highly ordered. Unfortunately, from the perspective of any practical implementation with actuators, the optimized control identified has a complex spatial and temporal structure, but it can be simplified. Two empirical eigenmodes were required to represent it sufficiently to reduce the targeted noise intensity by about 50 %. Optimization of a simple single-degree-of-freedom control with an ad hoc spatial structure is less effective.

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We develop low-dimensional models for the evolution of a free shear layer in a periodic domain. The goal is to obtain models simple enough to be analysed using standard tools from dynamical systems theory, yet including enough of the physics to model nonlinear saturation and energy transfer between modes (e.g. pairing). In the present paper, two-dimensional direct numerical simulations of a spatially periodic, temporally developing shear layer are performed. Low-dimensional models for the dynamics are obtained using a modified version of proper orthogonal decomposition (POD)/Galerkin projection, in which the basis functions can scale in space as the shear layer spreads. Equations are obtained for the rate of change of the shear-layer thickness. A model with two complex modes can describe certain single-wavenumber features of the system, such as vortex roll-up, nonlinear saturation, and viscous damping. A model with four complex modes can describe interactions between two wavenumbers (vortex pairing) as well. At least two POD modes are required for each wavenumber in space to sufficiently describe the dynamics, though, for each wavenumber, more than 90% energy is captured by the first POD mode in the scaled space. The comparison of POD modes to stability eigenfunction modes seems to give a plausible explanation. We have also observed a relation between the phase difference of the first and second POD modes of the same wavenumber and the sudden turning point for shear-layer dynamics in both direct numerical simulations and model computations.

Comparisons are made between direct numerical simulations ͑DNS͒ of uncontrolled and optimally noise-controlled two-dimensional mixing layers in order to identify the physical mechanism responsible for the noise reduction. The analysis is carried out in the time domain to identify events that are significant in sound generation and which are acted upon by the control. Results show that a triple vortex interaction in the uncontrolled mixing layer radiates high-amplitude pressure waves to the far acoustic field; the elimination of this triple merging accounts for 70% of the noise reduction accomplished by a body force control applied normal to the shear layer. The effect of this control is shown to comprise vertical acceleration of vortical structures; the acceleration, whose action on the structures is convected across the control volume, leads to changes in their relative convection velocities and a consequent regularization of their evolution, which prevents the triple merger. Analysis of a longer time series for the DNS of the uncontrolled mixing layer using a wavelet transform identifies several similar intermittent, noisy events. The sound production mechanism associated with such noisy events can be understood in terms of cancellation disruption in a noncompact source region, such as described by a retarded-potential formalism. This shows that acoustic analogies formulated from the perspective of quadrupole acoustic sources are, in principle, useful for the modeling of such events. However, this study also illustrates the extent to which time-averaged statistical analysis of sound producing flows can mask the most important source activity, suggesting that intermittency should be explicitly modeled in sound prediction methodologies.

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