Extended Orr-Sommerfeld Stability Analysis

The idea of statistical stability of turbulent states was first introduced by Malkus in 1956 (J. Fluid Mech. 1(5), 521-539). Despite the debate about the validity of Malkus's theory, his proposed approach of solving a linear Orr-Sommerfeld equation around a turbulent mean velocity profile remains a popular tool in turbulence analysis (see for example Beneddine et al 2016, J. Fluid Mech. 798 485-504). Inspired by the statistical nature of turbulence and Malkus's ideas, we use statistical state equations truncated at second order (see Farrell & Ioannou 2017, Phys. Rev. Fluids 2, 084608) to derive the Extended Orr-Sommerfeld linear stability analysis model in primitive variables space. By incorporating two-point correlations of the flow, this model takes into account the fluctuation-fluctuation interactions between the base and the perturbations fields. We apply the new stability model to a range of statistically steady states obtained via direct numerical simulations of 2D channel flow, discovering an improvement in statistical stability classification for some of the test cases. We carefully test the assumptions made in the model to understand when the improvement can be expected.