A Karhunen-Loeve Galerkin Technique with Shock Fitting for Optimization of a Blunt Body Geometry

A novel Karhunen-Loève (KL) Galerkin model for the supersonic, inviscid flow of a calorically perfect ideal gas about an axisymmetric, blunt body employing shock fitting is developed. The motivation for developing the KL Galerkin model is the need for an accurate and computationally efficient model for use in the optimal design of hypersonic vehicles. In constructing a KL Galerkin model, a set of flow field solutions representative of the design space are required. For this, a global polynomial pseudospectral method for the generalized coordinate, nonconservative form of the Euler equations is implemented. The variables in the equations are collocated via Lagrange interpolating polynomials defined at the zeroes of the Chebyshev polynomials, i.e. ChebyshevGauss-Lobatto nodes. Code verification, validation and grid convergence results are shown for the pseudospectral code, and an optimal geometry is identified from a single degree of freedom family of geometries. In conclusion the KL modes derived from pseudospectral solutions at Mach 3.5 from a uniform sampling of the design space are presented. This will be used in forthcoming work to develop a KL Galerkin model for the blunt body optimization.