A dynamically-consistent framework for mean flow stability and resolvent analysis

Mean flow stability and resolvent analysis remain popular reduced-order approaches for modeling coherent structures across a broad range of turbulent shear flows. These techniques project high-dimensional measured or simulated flow data onto the RANS equations and compute the linear dynamics that follow from a time average reference state, often further invoking a simplified or decoupled turbulence closure to simplify the formulation. However, both of these assumptions introduce potentially significant errors that cannot be quantified a priori. Building on related recent efforts by Yim, Meliga, & Gallaire (RSPA, 2019) and von Saldern et al. (JFM, 2024), this work develops a dynamically-consistent formulation of the mean flow analysis approach via a continuation and bifurcation analysis of the RANS equations. Such an approach ensures that the reference state is a bona fide solution of the complete RANS equations and that the linear dynamics are a fully consistent asymptotic expansion of the closure model. Consequently, errors associated with both the projection of the true dynamics onto the RANS system and the use of inconsistent or decoupled turbulence closures may be eliminated, allowing the role of each to be isolated and studied independently. Moreover, this method gives immediate access to state space parameter gradients, directly exposing the role of the various closure parameters on the linearized dynamics of the RANS system.