Computational methods applied to wind tunnel optimization

This report describes computational methods developed for optimizing the nozzle of a three-dimensional subsonic wind tunnel. This requires determination of a shape that delivers flow to the test section, typically with a speed increase of 7 or more and a velocity uniformity of.25% or better, in a compact length without introducing boundary layer separation. The need for high precision, smooth solutions, and three-dimensional modeling required the development of special computational techniques. These include: (1) alternative formulations to Neumann and Dirichlet boundary conditions, to deal with overspecified, ill-posed, or cyclic problems, and to reduce the discrepancy between numerical solutions and boundary conditions; (2) modification of the Finite Element Method to obtain solutions with numerically exact conservation properties; (3) a Matlab implementation of general degree Finite Element solvers for various element designs in two and three dimensions, exploiting vector indexing to obtain optimal efficiency; (4) derivation of optimal quadrature formulas for integration over simplexes in two and three dimensions, and development of a program for semi-automated generation of formulas for any degree and dimension; (5) a modification of a two-dimensional boundary layer formulation to provide accurate flow conservation in three dimensions, and modification of the algorithm to improve stability; (6) development of multi-dimensional spline functions to achieve smoother solutions in three dimensions by post-processing, new three-dimensional elements for $C1$ basis functions, and a program to assist in the design of elements with higher continuity; and (7) a development of ellipsoidal harmonics and Lamé's equation, with generalization to any dimension and a demonstration that Cartesian, cylindrical, spherical, spheroidal, and sphero-conical harmonics are all limiting cases.; The report includes a description of the Finite Difference, Finite Volume, and domain remapping methods, coordinate transformation theorems and techniques including the Method of Jacobians, and a derivation of the fluid flow fundamentals required for the model. It applies the methods to study the effect of cross-section and fillet variation, and to obtain a sample design of a high-uniformity nozzle.