Sensitivity analysis to unsteady perturbations of complex flows: a discrete approach

A discrete framework for computing the global stability and sensitivity analysis to external perturbations for any set of partial differential equations is presented. In particular, a complex‐step approximation is used to achieve near analytical accuracy for the evaluation of the Jacobian matrix. Sensitivity maps for the sensitivity to base flow modifications and to a steady force are computed to identify regions of the flow field where an input could have a stabilising effect. Four test cases are presented: (1) an analytical test case to prove the theory of the discrete framework, (2) a lid‐driven cavity at low Reynolds case to show the improved accuracy in the calculation of the eigenvalues when using the complex‐step approximation, (3) the 2D flow past a circular cylinder at just below the critical Reynolds number used to validate the methodology, and finally, (4) the flow past an open cavity presented to give an example of the discrete method applied to a convectively unstable case. The latter three (2–4) of the aforementioned cases were solved with the 2D compressible Navier–Stokes equations using a Discontinuous Galerkin Spectral Element Method. Good agreement was obtained for the validation test case, (3), with appropriate results in the literature. Furthermore, it is shown that for the calculation of the direct and adjoint eigenmodes and their sensitivity maps to external perturbations, the use of complex variables is paramount for obtaining an accurate prediction. Copyright © 2014 John Wiley & Sons, Ltd.