Shape optimization for aerodynamic noise control

Noise generated by turbulent boundary layers near the trailing edge of lifting surfaces continues to pose a challenge for many applications. Much of the previous work on this topic has focused on development of accurate computational methods for the prediction of trailing edge noise. For instance, aeroacoustic calculations of the flow over an airfoil using large-eddy simulation (LES) and aeroacoustic theory have been presented in Wang &Moin (2000). These results compare favorably with the experiments of Blake (1975) and Blake and Gershfeld (1988). To make the simulations more cost-effective, Wang & Moin (2001) successfully employed wall models in the trailing-edge flow LES; this resulted in a drastic reduction in computational cost with minimal degradation of the flow solutions. In this study, we extend the earlier work to noise control, using shape optimization and control theory in conjunction with the simulation tools developed previously. For trailing-edge noise control, a shape design method based on control theory for partial differential equations and a gradient-based minimization algorithm is employed to optimize the trailing-edge shape. The main difficulty in gradient-based optimization methods is the calculation of the gradient of the cost function with respect to the control parameters. The most widely-used method is to solve an adjoint equation in addition to the flow equations. While the adjoint method has been successful in many aerodynamic calculations, (e.g. Jameson et al. 1998; Pironneau 1984), it is expensive and not well suited to unsteady flow problems. In addition, the adjoint equation is dependent on the flow solver and thus is not portable. Because of these issues, we have employed the method of incomplete sensitivities for the gradient evaluation. This method, developed by Mohammadi & Pironneau (2001), offers the advantage that effects of geometric changes on the flow field can be neglected when computing the gradient of the cost function. This makes it far more cost effective than solving the full adjoint problem. In LES-based aeroacoustic shape design, the efficiency of the optimization routine is crucial. The use of incomplete gradients allows us to perform the optimization with only a small additional cost to the flow computation. In order to validate and gain experience with the method of incomplete sensitivities, we have applied it to a model problem consisting of the two-dimensional unsteady laminar flow over an airfoil. This problem allows us to define the cost function based on aeroacoustic theory. In this article, we present results for the model problem which show a significant reduction in the cost function. We also discuss the addition of aerodynamic constraints. Section 2 outlines the problem formulation and the definition of the cost function. Sections 3 and 4 discuss the gradient evaluation and the optimization procedure. Results are presented in Section 5.