An analytical asymptotic solution method of the Euler equations for efficient flow analysis and aerodynamic design.

摘要

An efficient flow analysis and aerodynamic design tool was developed based on analytical asymptotic solutions of the Euler equations. The solution algorithm uses an analytical asymptotic formulation in the streamline coordinate system, wherein the governing equations are transformed into a non-homogeneous Cauchy-Riemann system. A sequence of coordinate transformations, mappings, and asymptotic expansions places the governing equations in a form suitable for classical mathematical techniques. The homogeneous solution to the governing system is obtained by the conformal mapping and Fourier analysis, and provides an exact solution for incompressible flows and an accurate approximate solution for compressible flows. The non-homogeneous solution is obtained by Green's function formulation and accounts for higher order compressibility effects. The focus of efforts has been on the evaluation of components of the analytical solution algorithm in terms of solution quality and computational efficiency, as compared to the conventional CFD method. The formulation was extended to cascade flow problems by a new mapping procedure, which preserves the Cauchy-Riemann form of the governing equations and therefore enables us to use the same analytical solution procedure. For transonic flow problems, the mass flux formulation has been derived and used with the Rankine-Hugoniot equations for locating shock waves and corresponding entropy jumps. Transonic flow solutions around 2D airfoils were obtained by an analytical shock modeling from the homogeneous solution, as a first-order approximation. Finally, in order to demonstrate the computational efficiency of the method, an analytic-based aerodynamic design tool was developed by combining the analytical flow solution with numerical optimization methods which include a genetic algorithm. Results from the current study show that the computational cost for flow analysis and aerodynamic design can be reduced significantly by using the analytical asymptotic solution method.