Aerodynamic shape optimization of unsteady, chaotic flows

Abstract

First, a novel ensemble Newton-Krylov method is presented, which successfully improves upon a chosen objective function for the chaotic Lorenz '63 system. Next, inspired by the accurate yet costly least-squares shadowing (LSS) adjoint method, a Petrov-Galerkin approximation of the LSS adjoint is developed and investigated on the Lorenz system. Finally, an energy-stabilization method is derived from an energy-stability analysis of the sensitivity equations, and promising results are shown on the Lorenz problem.

Two stabilization approaches are investigated and their results analyzed. The energy stabilization method is subsequently extended to the sensitivity analysis of computational fluid dynamics and applied to a two-dimensional inviscid airfoil test case. The energy stabilization method is shown to successfully bound the exponential growth of the tangent sensitivity. The sensitivities produced by the energy-stabilization method are usable in the sense that they are not exponentially large, but they contain some bias when compared with a least-squares fit of the data, though the error is small in absolute terms. As predicted, the method is computationally inexpensive compared to other leading methods for obtaining sensitivities under chaotic dynamics.

Improving the efficiency and/or mission capabilities of modern aircraft is becoming more costly, both in terms of delays and budget overruns. This cost can often be reduced using computational optimization, and one discipline that has benefited from optimization in particular is aerodynamics. Gradient-based computational aerodynamic shape optimization (ASO) is an attractive and efficient application of optimization to aircraft design, and its application to cruise conditions is becoming common. Extending the benefits of steady ASO to unsteady phenomena could potentially offer significant improvements to aircraft design. However, gradient-based optimization methods fail in the presence of chaotic dynamics, a characteristic of turbulent flows, because conventional sensitivity analysis methods (e.g. tangent and adjoint) are unstable for chaotic flows, resulting in unusable sensitivities. Resolution of turbulent flows for ASO is important because turbulence models such as RANS may not suffciently capture turbulent structures that are important for ASO. Several methods are investigated for performing optimization of, or obtaining usable sensitivity information for, chaotic dynamical systems.