Abstract

It was recently observed that finite difference schemes optimized to perform well with few points per wavelength (commonly referred to as DRP schemes) currently perform poorly when applied to waves of non-constant amplitude. In this paper, attempts are described at optimizing explicit symmetric finite difference schemes to require relatively few points per wavelength for waves with a range of growth-rates and decay-rates. Several optimization criteria are proposed, and some advantage may be gained from using such optimized schemes if the growth-and decay-rates present are known a priori. In particular, for the test case considered here, the usual 7 point 4th order DRP schemes are found to be overly ambitious in their optimization, and better performing schemes are derived which require almost half the number of points per wavelength to achieve the same accuracy. Without a priori knowledge, however, the best choice of finite difference schemes for waves of varying amplitude remain the classical maximal order schemes.

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 flows by including more information about the underlying flow in its predictions.

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 (OS) 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) that corresponds to the quasilinearized Navier–Stokes equations, we develop an extended OS analysis that also incorporates information about the fluctuation field. A more practical version of this – minimally extended OS analysis – 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.

have recently identified the ability of the streamwise-averaged streak fields U (𝑦, 𝑧, 𝑡) x in minimal channels to produce short-term transient growth as the key linear mechanism needed to sustain turbulence at 𝑅𝑒 𝜏 = 180. Here, we model this streak transient growth as a two-stage process by first selecting the dominant streak structure expected to emerge over the eddy turnover time on the turbulent mean profile 𝑈 (𝑦) x, and then examining the secondary growth on this (frozen) streak field U (𝑦, 𝑧) x. Choosing the mean streak amplitude and eddy turnover time consistent with simulations recovers the growth thresholds found by Lozano-Duran et al. (2021) for sustained turbulence. Extending this analysis to 𝑅𝑒 𝜏 = 360 and 720 suggests a Reynolds number independence of this threshold. This then indicates that the short-term growth properties of the turbulent mean profile U (𝑦, 𝑧) x are a much more plausible criterion of turbulence existence than its linear stability characteristics as originally suggested by Malkus (