  Adjoint-based sensitivity analysis methods are powerful tools for engineers
who use flow simulations for design. However, the conventional adjoint method
breaks down for scale-resolving simulations like large-eddy simulation (LES) or
direct numerical simulation (DNS), which exhibit the chaotic dynamics inherent
in turbulent flows. Sensitivity analysis based on least-squares shadowing (LSS)
avoids the issues encountered by conventional methods, but has a high
computational cost. The following report outlines a new, more computationally
efficient formulation of LSS, non-intrusive LSS, and estimates its cost for
several canonical flows using Lyapunov analysis.
