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A fully discrete formalism is introduced to perform stability analysis of aturbulent compressible flow whom dynamics is modeled with the Reynolds-AveragedNavier-Stokes (RANS) equations. The discrete equations are linearized usingfinite differences and the Jacobian is computed using repeated evaluation ofthe residuals. Stability of the flow is assessed solving an eigenvalue problem.The sensitivity gradients which indicate regions of the flow where a passivecontrol device could stabilize the unstable eigenvalues are defined within thisfully discrete framework. Second order finite differences are applied to thediscrete residual to compute the gradients. In particular, the sensitivitygradients are shown to be linked to the Hessian of the RANS equations. Theintroduced formalism and linearization method are generic: the code used toevaluate the residual of the RANS equations can be used in a black box manner,and the complex linearization of the Hessian is avoided. The method is testedon a two dimensional deep cavity case, the flow is turbulent with a Reynoldsnumber equal to 860 000 and compressible with a Mach number of 0.8. Severalturbulence models and numerical schemes are used to validate the method.Physical features of the flow are recovered, such as the fundamental frequencyof the natural flow as well as acoustic mechanisms, suggesting the validity ofthe method. The sensitivity gradients are then computed and validated, theerror in predicting the eigenvalue variation being found less than 3\%. Controlmaps using a small steady control device are finally obtained, indicating thatthe control area should be chosen in the vicinity of the leading edge of thecavity.

Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: Application to open-loop control