Coupling Linearized Far-Field Boundary Conditions with Nonlinear Near-Field Solutions in Transonic Flow

摘要

Research has been conducted to evaluate the feasibility of coupling linearized far-field solutions with near-field finite difference equations to reduce the number of unknowns and thus the computer resources required in transonic flow calculations. For two dimensional flow, changes to an existing finite difference program involved distributing sources on the grid boundary in order to obtain the proper far field outgoing wave boundary conditions on a reduced grid. Validation of the matching procedure was made for zero thickness airfoils by comparison of the results with those of the kernel function method. For airfoils with finite thickness, a criterion based on the gradient of the flowfield Mach number was developed for establishing the minimum size finite difference region necessary for accurate unsteady calculations. This approach could not be applied directly to three-dimensional flow because of the large number of variables required in the exterior solution. However, it is shown that the number of unknowns can be reduced to a practical number by using both source and doublet distributions on the boundaries, describing these distributions with low-order polynomials, and using a least squares procedure to satisfy the matching conditions across the boundaries. Solutions for the flow over a wing of vanishing thickness were in very good agreement with results from original finite difference technique and with the kernel function method. Solutions for a wing thickness were in good agreement with results from the original finite difference technique. The pilot program used for this study is limited to rectangular wings.