Eigenvalue Sensitivity Computations for Linear Stability Theory

To realize the drag reduction benefit of boundary-layer transition control strategies, it is crucial to integrate transition prediction into the vehicle design through an optimization process. The integration of transition prediction based on linear stability analysis into adjoint-based design optimization requires coupling an adjoint-enabled computational fluid dynamics (CFD) solver with an adjoint-enabled linear stability code. In particular, the boundary-layer transition location is often predicted using the [Formula: see text]-factor method based on linear stability theory (LST). Thus, the sensitivity of the linear-stability eigenvalues constitutes an essential building block for optimizing the laminar flow performance. The present paper describes an implementation of LST eigenvalue sensitivity analysis that can be easily coupled with a CFD solver. Specifically, we describe a discrete adjoint formulation for the transition location prediction based on the [Formula: see text]-factor method. The verification of this formulation is carried out by comparing the adjoint-based sensitivity of the local growth rate of a given instability mode with respect to the disturbance frequency and the adjoint-based sensitivity of the transition location with respect to the spanwise wavenumber with those sensitivities computed using a finite-difference approximation. Finally, the adjoint LST formulation is applied to flat-plate boundary-layer flows at transonic, supersonic, and hypersonic conditions to determine the behavior and sensitivities of the transition location with respect to a range of disturbance spanwise wavenumbers.