Unsteady Boundary Control for Optimal Control of Trailing Edge Noise

Abstract

The objective of the present work is to couple computational fluid dynamics analysis with optimal control theory, which has the potential to advance the strategies for active control of complex flows including aeroacoustics. Such an objective requires a highly accurate flow solver that accommodates unstructured grids to handle complex geometries and an efficient numerical algorithm to compute cost function gradients. To this end, we develop an optimal control framework for unsteady flows, based on the adjoint formulation. We pose the trailing-edge noise control problem within this framework. Specifically, we consider the two-dimensional unsteady flow past the Blake airfoil with unsteady blowing/suction on the trailing edge surface as the control, and find an optimal temporal/spatial control to reduce the trailing edge noise. The disparate range between the convective scales and the acoustic scales presents a computational challenge. It is usually handled by using a hybrid approach, which employs the so-called acoustic analogy or integral equation methodology to deal with the fast and large scales of noise propagation while solving the partial differential equations governing the flow by a sufficiently accurate numerical method. The present approach is based on the direct computation of noise as part of the flow solution. The compressible Navier-Stokes equations and their adjoint are solved numerically using a discontinuous spectral element method for spatial discretization and fourth-order RungeKutta for temporal discretization. The control is updated using a nonlinear conjugate gradient method. It is known that the discontinuous spectral element method is particularly suited for acoustics problems, as it requires, for the resolution of waves with a given accuracy, the least amount of grid points compared to the traditional finite-element or finite-difference methods. In addition to high order accuracy (spectral accuracy), the discontinuous spectral element method has the advantage of the local hp refinement. Furthermore, it is trivially parallelizable. The current work focuses on the tonal or narrowband noise associated with laminarflow airfoils. It is found that aeroacoustic control by trailing-edge blowing/suction directly affects the transient drag and lift histories, which is not surprising, as the trailing-edge noise is of dipole type (associated with the unsteady lift and drag). Such a noise-control strategy is found to obtain significant noise reduction with a direct impact on the thickness (drag) dipole with no drastic effect on the shedding region. Physical mechanisms for noise reduction are discussed in the end.