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. Using a Green’s function approach, we derive resolvent solutions for isentropic (subsonic and supersonic) and transonic flows with shocks in converging–diverging nozzles of arbitrary geometry. Our analysis demonstrates that shock-induced heightened sensitivity in the resolvent across flow discontinuities leads to significant discrepancies between numerically computed and the analytical input and output modes if shock effects are not properly accounted for. In particular, we find that the resolvent operator exhibits singular behaviour at the shock location. Specifically, the inviscid (where the shock is treated purely as a flow discontinuity) and viscous analytical leading resolvent modes do not converge as the viscosity parameter $\mu \rightarrow 0$ , which affects the accuracy of flow control and stability analyses that rely on resolvent-based methods. Furthermore, the derived solutions serve as benchmarks for verifying numerical schemes designed to compute adjoint and resolvent modes in shock-laden flows, ensuring that they capture the correct physical behaviour in the presence of shocks.