Efficient evaluation of the direct and adjoint linearized dynamics from compressible flow solvers

The direct and adjoint operators play an undeniably important role in a vast number of theoretical and practical studies that range from linear stability to flow control and nonlinear optimization. Based on an existing nonlinear flow solver, the design of efficient and straightforward procedures to access these operators is thus highly desirable. In the case of compressible solvers, the use of high-order numerical schemes combined with complicated governing equations makes the derivation of efficient procedures a challenging and often tedious undertaking. In this work, a novel technique for the evaluation of the direct and adjoint operators directly from compressible flow solvers is presented and extended to include nonlinear differentiation schemes and turbulence models. The application to the incompressible counterpart is also discussed. The presented method requires minimal additional programming effort and automatically takes into account subsequent modifications in the governing equations and boundary conditions. The introduced methodology is demonstrated on existing numerical codes, and direct and adjoint global modes are calculated for three typical flow configurations. Implementation issues and the performance measures are also discussed. The proposed algorithm presents an easy-to-implement and efficient technique to extract valuable information for the quantitative analysis of complex flows.