Improved assessment of the statistical stability of turbulent flows using extended Orr-Sommerfeld stability analysis

TL;DR

This paper introduces an extended Orr-Sommerfeld stability analysis that incorporates fluctuation information via cumulant expansion, improving the assessment of statistical stability in turbulent shear flows.

Contribution

It develops an extended Orr-Sommerfeld analysis using CE2 closure to include fluctuation effects, enhancing stability predictions in turbulent flows.

Findings

Extended analysis improves stability assessment accuracy.

Tested on turbulent channel flows with positive results.

Incorporates fluctuation data for better flow predictions.

Abstract

The concept of statistical stability is central to Malkus's 1956 attempt to predict the mean profile in shear flow turbulence. Here we discuss how his original attempt to assess this - an Orr-Sommerfeld analysis on the mean profile - can be improved by considering a cumulant expansion of the Navier-Stokes equations. Focusing on the simplest non-trivial closure (commonly referred to as CE2) which corresponds to the quasilinearized Navier-Stokes equations, we develop an extended Orr-Sommerfeld analysis (EOS) which also incorporates information about the fluctuation field. A more practical version of this - minimally extended Orr-Sommerfeld analysis (mEOS) - is identified and tested on a number of statistically-steady and therefore statistically stable turbulent channel flows. Beyond the concept of statistical stability, this extended stability analysis should also improve the popular approach of mean-flow linear analysis in time-dependent shear flows by including more information about the underlying flow in its predictions as well as for other flows with additional physics such as convection.