Proper Orthogonal Decomposition Extensions and Their Applications in Steady Aerodynamics

The development and application of model reduction techniques have been widely investigated for problems in unsteady aerodynamic systems, driven mainly by the need to develop effective economical computational models capturing as much flow physics as possible. These techniques which have been based on a number of approaches such as balanced truncation, Arnoldi, proper orthogonal decomposition etc are designed to obtain a reduced-order model for both linear and nonlinear systems. However there has been a paucity in the application of these techniques in the area of steady aerodynamics. The work reported in this thesis demonstrates how the proper orthogonal decomposition (POD) technique can be extended to a number of applications in steady aerodynamics. In the first instance the POD approach is coupled with a cubic spline interpolation procedure to develop reliable fast, low-order models for accurately predicting steady aerodynamic flowfields for arbitrary values or variation in flow parameters such as the angle of attack or inflow Mach number. Results on the prediction of steady transonic aerodynamic flowfield solution past an airfoil at arbitrary values of angle of attack or Mach number show that accurate flow-field predictions can be obtained, including cases that were not sampled in the ensemble of snapshots. The second extension concerns a " gappy " POD technique for the reconstruction of incomplete or inaccurate aerodynamic flowfield data. The first case corresponds to the complete reconstruction of pressure field around an airfoil from the knowledge of pressure data defined only at the airfoil surfaces. The second case corresponds to the one in which POD snapshots are reconstructed from an incomplete set of aerodynamic data. Gappy POD is shown to be an effective technique for reconstruction of complete aerodynamic flow field data from limited measurements or incomplete data. This approach demonstrates an effective way in which experimental and computational aerodynamic data can be combined to predicate accurate aerodynamic flow fields cheaply. Finally it is shown how an extension of the gappy POD can be used to formulate and cast inverse airfoil shape design and flowfield prediction for an arbitrary airfoil problems, in which steady aerodynamic plays a role, as gappy data problems. The extension of the methodology to constrained airfoil shape design problem is also discussed. The gappy design methodology has been shown to be a simple, effective and an efficient airfoil shape design technique.