Shock-Free Transonic Airfoil Design by a Hodograph Method.
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH
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Refined mathematical methods are required for the analytical solution of the partial differential equation governing steady, two-dimensional, compressible, transonic, potential fluid flow. This equation is nonlinear in the physical plane and so does not lend itself to standard analytical solution methods. The Molenbroek-Chaplygin transformation, where the physical Cartesian coordinates as the independent variables are replaced by the velocity magnitude and direction as the independent variables, linearizes the governing equation which may then be analytically solved. The plane where the said velocity parameters are the independent variables is termed the hodograph plane. Likewise, the transformed differential equation is known as the hodograph equation and it is solved by hodograph methods. This mathematical study addresses the solution of transonic flow phenomena by an extension of Lighthills hodograph method. Lighthills method transforms a given solution of the Laplace equation into a solution of the hodograph equation for subsonic flows only. A new relation is developed in this study extending this transformation technique to include flows up to March 2.2735 in air. Requiring only numerical data concerning the velocity field, this hodograph method is computationally efficient and mathematically straightforward.
- Fluid Mechanics