Revisiting the Lie-group symmetry method for turbulent channel flow with wall transpiration

Abstract

The Lie-group-based symmetry analysis, as first proposed in Avsarkisov et al. (2014) and then later modified in Oberlack et al. (2015), to generate invariant solutions in order to predict the scaling behavior of a channel flow with uniform wall transpiration, is revisited. By focusing first on the results obtained in Avsarkisov et al. (2014), we failed to reproduce two key results: (i) For different transpiration rates at a constant Reynolds number, the mean velocity profiles (in deficit form) do not universally collapse onto a single curve as claimed.(ii) The universally proposed logarithmic scaling law in the center of the channel does not match the direct numerical simulation (DNS) data for the presented parameter values. In fact, no universal scaling behavior in the center of the channel can be detected from their DNS data, as it is misleadingly claimed in Avsarkisov et al. (2014). Moreover, we will demonstrate that the assumption of a Reynolds-number independent symmetry analysis is not justified for the flow conditions considered therein. Only when including also the viscous terms, an overall consistent symmetry analysis can be provided. This has been attempted in their subsequent study Oberlack et al. (2015).But, also the (viscous) Lie-group-based scaling theory proposed therein is inconsistent, apart from the additional fact that this study of Oberlack et al. (2015) is also technically flawed. The reason for this permanent inconsistency is that their symmetry analysis constantly involves several unphysical statistical symmetries that are incompatible to the underlying deterministic description of Navier-Stokes turbulence, in that they violate the classical principle of cause and effect. In particular, as we consequently will show, the matching to the DNS data of the scalar dissipation, being a critical indicator to judge the prediction quality of any theoretically derived scaling law, fails exceedingly.

On the physical inconsistency of a new statistical scaling symmetry in incompressible Navier-Stokes turbulence

“…Thus, generating a useful statistical scaling law within Avsarkisov et al (2014) for any higher order velocity moment, which goes beyond the mean velocity profile, is predetermined to fail when considering all facts discussed herein. Indeed, this failure is confirmed in Khujadze & Frewer (2016).…”

A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach

“…The reason for this failure is that the proposed scalings in [1] are not solutions to the statistical Navier-Stokes equations, as incorrectly claimed, simply because these scalings are based on two nonphysical invariances (Eqs. [8][9] that violate the classical principle of cause and effect between the fluctuations and the mean fields [3][4][5][6][7][8]. This violation is suppressed and therefore not visible when analyzing the symmetry-based scaling of the full-field correlations, but becomes measurable and clearly visible when analyzing the corresponding fluctuation correlations.…”

“…To note is that this symmetry-based scaling failure is not specific to channel flow. It can also be clearly seen in other flow configurations when turbulence-relevant moments are explicitly matched to data [3,[9][10][11], and simply stems from the fact that this failure is methodologically rooted in the symmetry-based scaling approach as developed by Oberlack et al over the last two decades.…”

“…That is, also when generalizing the three streamwise scaling symmetries Eqs. (6)(7)9) to symmetries of the governing equation (Eq. 4) for arbitrary directions i, their scaling exponents will always remain independent of the coordinate direction.…”

Is the log-law a first principle result from Lie-group invariance analysis?

“…The even later studies, e.g., in Oberlack & Rosteck (2010); Oberlack & Zieleniewicz (2013); Wac lawczyk et al (2014) and Avsarkisov et al (2014) also suffer from the additional problem that new unphysical symmetries are generated, which in turn violate the classical principle of cause and effect. For more details, we refer to our other comments and reviews, Frewer et al (2014); Frewer (2015a); Frewer et al (2015a; Frewer & Khujadze (2016a); Khujadze & Frewer (2016), and to our reactions in Frewer (2015b); Frewer et al (2015bFrewer et al ( , 2016c and .…”