On the use of applying Lie-group symmetry analysis to turbulent channel flow with streamwise rotation

TL;DR

This paper critically examines the application of Lie-group symmetry analysis to turbulent channel flow with streamwise rotation, highlighting the limitations due to turbulence closure problems and the necessity of empirical data for meaningful scaling laws.

Contribution

The paper revisits previous symmetry analysis, demonstrating the inherent arbitrariness in scaling laws due to turbulence closure issues and clarifying misconceptions in earlier work.

Findings

Invariant scaling laws can be arbitrarily generated for turbulence.

Turbulence closure problem prevents a priori predictions of scaling.

Symmetry analysis alone cannot resolve turbulence scaling without empirical input.

Abstract

The study by Oberlack et al. (2006) consists of two main parts: a direct numerical simulation (DNS) of a turbulent plane channel flow with streamwise rotation and a preceding Lie-group symmetry analysis on the two-point correlation equation (TPC) to analytically predict the scaling of the mean velocity profiles for different rotation rates. We will only comment on the latter part, since the DNS result obtained in the former part has already been commented on by Recktenwald et al. (2009), stating that the observed mismatch between DNS and their performed experiment is possibly due to the prescription of periodic boundary conditions on a too small computational domain in the spanwise direction. By revisiting the group analysis part in Oberlack et al. (2006), we will generate more natural scaling laws describing better the mean velocity profiles than the ones proposed. However, due to the statistical closure problem of turbulence, this improvement is illusive. As we will demonstrate, any arbitrary invariant scaling law for the mean velocity profiles can be generated consistent to any higher order in the velocity correlations. This problem of arbitrariness in invariant scaling persists even if we would formally consider the infinite statistical hierarchy of all multi-point correlation equations. The closure problem of turbulence simply cannot be circumvented by just employing the method of Lie-group symmetry analysis alone: as the statistical equations are unclosed, so are their symmetries! Hence, an a priori prediction as how turbulence scales is thus not possible. Only a posteriori by anticipating what to expect from numerical or experimental data the adequate invariant scaling law can be generated through an iterative trial-and-error process. Finally, apart from this issue, also several inconsistencies and incorrect statements to be found in Oberlack et al. (2006) will be pointed out.