Aerodynamic Sensitivity Analysis Methods for the Compressible Euler Equations

Abstract

A mathematical formulation is developed for aerodynamic sensitivity coefficients based on a discretized form of the compressible, two-dimensional Euler equations. A brief motivating introduction to the aerodynamic sensitivity analysis and the reasons behind an integrated flow/sensitivity analysis for design algorithms are presented. Two approaches to determine the aerodynamic sensitivity coefficients, namely, the finite difference approach, and the quasi-analytical approach are discussed with regards to their relative accuracies and involved computational efforts. In the quasi-analytical approach, the direct and the adjoint variable methods are formulated and assessed. Also, several methods to solve the system of linear algebraic equations, that arises in the quasi-analytical approach, are investigated with regards to their accuracies, computational time and memory requirements. A new flow prediction concept, which is an outcome of the direct method in the quasi-analytical approach, is developed and illustrated with an example. Surface pressure coefficient distributions of a nozzle-afterbody configuration obtained from the predicted flow-field solution are compared successfully with their corresponding values obtained from a flowfield analysis code and the experimental data.

Discrete Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids

“…Conventional black-box finite difference methods [1] suffer from well-known step-size limitations and incur a computational expense that grows linearly with the number of design variables. Forward, or direct, differentiation methods [2] and techniques based on the use of complex variables [3] mitigate the step-size limitation but still suffer from excessive cost in the presence of many design variables, as is often the case with aerodynamic design applications.…”

Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation

“…These methods promise to allow computational fluid dynamics methods to become true aerodynamic design methods. References [1,2,3,5,4,7,8,6,32,29,16,35,30,28,34,45,31,36,37,47,15,33,14] , but was not adopted in that work because of the complexity of the resulting discrete adjoint system. The approach has been favored by Taylor et al [29,16,35,30,28] and Baysal et al [1,2,3,5,4,7,8,6,321. It seems that both alternatives have some advantages.…”

Multidisciplinary design optimization techniques - Implications and opportunities for fluid dynamics research

“…At a 1986 conference Sobieski 4 challenged the aerodynamics community to develop a general sensitivity analysis capability, and the first papers on exact aerodynamic sensitivity analysis appeared a few years afterwards. [5][6][7][8] Of course, there had been earlier related developments. Sensitivity analysis of a limited sort was implicit in the aerodynamic optimization methods utilizing adjoint equations 9,10 that originated in the late 1970s.…”