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

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.

37 pages, 14 figures

References in corpus (6)

• Model-based design of transverse wall oscillations for turbulent drag reduction
• Direct Statistical Simulation of Out-of-Equilibrium Jets
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• Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence
• Linear instability of turbulent channel flow
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Cited by in corpus (4)

• Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence
• Patterns in transitional shear turbulence. Part 2: Emergence and optimal wavelength
• Threshold transient growth as a criterion for turbulent mean profiles
• Non-equivalence of quasilinear dynamical systems and their statistical closures