Turbulence in wall-bounded flows is characterized by stable statistics for the mean flow and the fluctuations both for the case of the ensemble and the time mean. Although in a substantial set of turbulent systems, this stable statistical state corresponds to a stable fixed point of an associated statistical state dynamics (SSD) closed at second order, referred to as S3T, this is not the case for wall turbulence. In wall turbulence the trajectory of the statistical state evolves on a transient chaotic attractor in the S3T statistical state phase space and the time-mean statistical state is neither a stable fixed point of this SSD nor, if the time-mean statistical state is maintained as an equilibrium state, is it stable. Nevertheless, sufficiently small perturbations from the ensemble or time-mean state of wall turbulence are expected to relax back to the mean statistical state following an effective linear dynamics. In this work the dynamics of spanwise uniform perturbations to the time-mean flow are studied using a linear inverse model to identify the linear operator governing the ensemble stability of the ensemble or time-mean state by obtaining the time mean stability properties over the transient attractor of the turbulence identified by the S3T SSD. The ensemble or time-mean stability of an unstable equilibrium can be understood by noting that, even when every member of an ensemble is unstable, the ensemble mean may be stable with perturbations following an identifiable stable dynamics. While simplifying insight into turbulent flows has commonly been obtained by identifying and studying ensemble mean statistical states, less attention has been accorded to identifying and studying the ensemble mean dynamics. We show that, in the case of wall turbulence, even though stable fixed-point SSD equilibria are not available to allow application of traditional perturbation analysis methods to identify the perturbation stability of the mean state, an effective linear stability analysis can be obtained to identify the perturbation dynamics of the ensemble or time-mean statistical state.

• Received 12 April 2023
• Accepted 12 January 2024

DOI:https://doi.org/10.1103/PhysRevFluids.9.024605