| 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 the noise radiated by a randomly excited two-dimensional mixing layer, which serves as near-nozzle model of a jet. The initial-vorticity-thickness Reynolds number is 500, and the free stream Mach numbers are 0.9 and 0.2. Controls are implemented into flow equations as general source terms (body forces, mass sources, and internal energy sources) with compact support near the "splitter plate." The noise to be reduced is defined by a space-time integral of the mean square 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 modeling approximations. The objective is to study the mechanics of the noise generation and its control. All controls effectively reduce noise requiring very little input power, with the most effective (internal energy control) reducing the noise intensity by 11 dB. Numerical tests show that the control is not by a simple acoustic cancellation ("anti-sound") mechanism, but results from a genuine change of the noise source. The comparison of otherwise identical flows without and with control applied shows little change of the flow's gross features: the evolution 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 nonperiodic streamwise direction, shows that the turbulence structures advect downstream more uniformly. This change appears to be the key to minimizing their acoustic efficiency. This perspective is clarified by contrasting it with a harmonically excited mixing layer. The noise in this case is not reduced by our control procedure, but is already at the low level achieved by the controller applied to the randomly excited mixing layers. The underlying empirical eigenfunctions in this case show a similar regular structure and behavior as achieved by the control in the randomly excited case. Unfortunately, from the perspective of any practical implementation with actuators, the control identified has a complex spatial and temporal structure. Two empirical eigenmodes were required to represent it sufficiently to reduce the noise about 50%, and their form was complex. Optimization of a single-degree-of-freedom control yielded only a 44% reduction by x-direction body-force control and less than 20% by others. |
