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Towards Aerodynamic Shape Optimization of Unsteady Turbulent Flows

Optimization is an important phase of modern engineering design, and gradient-based optimization methods are especially effective for design problems involving a large number of design variables, such as aerodynamic shape optimization. However, if the system of interest exhibits chaotic dynamics and the quantity of interest is a long-time average, for example, drag in a turbulent flow, conventional sensitivity analysismethodswill fail to provide useful gradients and gradient-based optimization becomes infeasible. Progress has been made in the sensitivity analysis of chaotic systems, but most proposed methods are computationally expensive. We propose a newmethod of stabilizing the tangent-sensitivity equations by introducing a minimal perturbation to the linearized equations. The goal is to eliminate linear instabilities in the sensitivity equations, while ensuring useful gradients are obtained. We implement the tangent sensitivity method on a model chaotic problem, demonstrate the effectiveness of our stabilizations, and validate the gradient information obtained from the stabilized tangent sensitivity. Finally, we describe how the method of stabilization can easily be adapted to the sensitivity analysis of large-scale computational fluid dynamics problems.