Laplacian Equivalents to Subsonic Potential Flows

Abstract

This paper presents a mapping technique for the solution of Laplacian equivalents to steady subsonic potential flows in the plane. The method employs a coordinate transformation to map a given closed profile and domain in a compressible flow to a related closed profile and domain in a Laplacian flow, with differences in the geometries stemming only from compressibility effects. Unlike certain past methods, the mapping is valid for nonlinear flows with arbitrary density–velocity relations. The theory is implemented in an iterative numerical method in which the Laplacian flow is solved with a boundary element technique, and the compressible flow is then found from a finite difference solution to the mapping. Validation is provided by comparing the resulting subsonic flow solutions to knownresultsintheliteratureforbothacircularcylinderandaNACA0012profile.Themethodisexactinthesense that the inverse mapping of the equivalent Laplacian field recovers the subsonic full-potential solution over the originalprofile,uptonumericalaccuracy.Becauseincompressiblepotential flowsareLaplacian,themappingyields insights into the phenomenon of compressibility itself in the subsonic setting, and it may have utility in certain aerodynamic test applications.