Resolvent Analysis of Shock-Laden Flows

We present a semi-analytic investigation of the resolvent operator and its associated forcing and response modes for quasi-one-dimensional shock-laden flows. Utilizing a Green's function approach, we derive resolvent solutions for sub- and supersonic isentropic and shock-laden transonic flows in convergent-divergent nozzles. Our analysis demonstrates that shocks induce heightened sensitivity in the resolvent across flow discontinuities, leading to significant discrepancies between numerically computed and analytical input and output modes if the shock is not appropriately accounted for. In particular, we find that the leading inviscid resolvent modes do not converge to the leading viscous resolvent modes as the viscosity $\mu \rightarrow 0^+$, which affects the accuracy of resolvent-based methods for analysis and flow control. Furthermore, we use the analytical solutions as verification guides for numerical schemes designed to compute adjoint and resolvent modes in shock-laden flows.