A simplified resolvent model for connecting linear amplifications to linear mechanisms in channel flow

The linearised Navier–Stokes (LNS) equations are employed in channel flow to study linear amplification mechanisms from an input–output point of view. We consider two models: the full LNS system and a simplified model. In the simplified model, we retain only the forcing pathways that pertain to the shear-driven lift-up mechanism. This approach enables individual analysis of the Orr–Sommerfeld and Squire operators as subsystems, revealing the relationship between linear amplification and the linear mechanisms from which it arises. We examine wavenumber regions corresponding to streamwise streaks, oblique waves and Tollmien–Schlichting (TS) waves, linking the underlying mechanisms back to the LNS equations. Analysis is performed for laminar Poiseuille flow, laminar Couette flow and turbulent Poiseuille flow using an eddy-viscosity model. Results indicate that the Orr–Sommerfeld system amplifies regions for streamwise streaks, oblique waves and TS waves, whereas the Squire system only amplifies streamwise streaks and oblique waves. Leading modal structures between the full and simplified models are also compared.