A new approach for efficient construction of asymptotic solutions of the two-dimensional Euler equations

Abstract

A new method for analytical solution of the twodimensional, steady-state Euler equations is presented. The equations are written with flow angle and mass flux as dependent variables and streamline coordinates as independent variables. The solution procedure uses an asymptotic formulation wherein higher-order compressibility effects appear as non-homogeneous forcing terms. A sequence of transformations, mappings and asymptotic methods places the Euler equations in the form of a boundary value problem amenable to solution using classical mathematical techniques. Mass flux as a dependent variable permits extension of the new approach to transonic flow. Solution of the non-homogeneous system for full compressibility effects does not require a Green's function as in earlier formulations. For airfoil problems (and many other types), the non-homogeneous solution can be constructed from closed-form indefinite integrals. The approach is derived herein for airfoil-type flows. Extensive use can be made of symbolic manipulation software (e.g., MATHEMATICA, MACSYMA). The new procedure is computationally orders of magnitude more efficient than the conventional Computational Fluid Dynamics (CFD) approach because it requires only function evaluations; furthermore, field grid generation is not required. Aerodynamic sensitivity derivatives for design optimization can be evaluated analytically by differentiating and using the same closed-form solution techniques. Results are presented which demonstrate the potential efficiency improvements that can be realized by this approach. Introduction In today's aerospace environment, the emphasis is on product affordability, which translates into a need for more efficient numerical simulation tools for use in aerodynamic analysis and design. There are many CFD *Research Fellow, AIAA Associate Fellow t Project Engineer, AIAA member Copyright © 1998 by The Boeing Company. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. methods at the present time which provide accurate numerical predictions for subsonic, transonic, and supersonic airfoil flowfields. The cost of these methods in terms of present day computers is very reasonable. However, when used in design optimization procedures, the CFD code is often run hundreds of times, especially when many design variables are used. The resulting cost of such a process in terms of computer time (and elapsed time) can quite large. Extension of the CFD procedure to threedimensions to design a wing, for example, usually requires computer run times which are prohibitive, especially in an advanced design environment. To make the problem more tractable, the designer is often forced to limit the number of design variables, thereby compromising the size of the design space. Analytic-based design optimization methods offer an alternative (or complement) to the CFD approach. Preliminary results (References 1 and 2) have shown reductions in computational time by as much as two orders of magnitude. Furthermore, manual field grid generation is not required. An improved procedure for analytical solution of the Euler equations for airfoils is described herein. It can form the basis for an efficient design procedure because the analytical solutions can subsequently be differentiated with respect to any design variable to analytically determine aerodynamic design sensitivity. The steady-state Euler equations are transformed to a natural (s,ri) streamline coordinate system with mass flux and flow angle as dependent variables. For mild compressibility conditions, the flow is described by the familiar Cauchy-Riemann system. Higher-order compressibility and rotational effects appear explicitly as right-hand-side (RHS) forcing terms. The CauchyRiemann system is transformed to a semi-infinite strip using a velocity transformation from physical space and conformal mapping. The resulting boundary value problem is then solved efficiently using Fourier methods whereby discontinuities in surface flow angle (i.e., stagnation point and trailing edge) are treated explicitly. The solution procedure is iterative starting from some initial approximation for surface velocity.