Extension of the law of the wall exploiting weak similarity of velocity fluctuations in turbulent channels

TL;DR

This paper investigates velocity fluctuation similarities in turbulent channel flows using POD, proposing an extended law of the wall based on weak similarity, which improves flow reconstruction accuracy in the wall layer.

Contribution

It introduces a novel extension to the law of the wall leveraging weak similarity of velocity fluctuations, enhancing flow modeling in turbulent channels.

Findings

Weak similarity exists in the viscous and outer layers.

The extended law of the wall improves flow reconstruction accuracy.

Strong similarity is observed in the viscous and wake regions.

Abstract

This paper explores the similarity of the streamwise velocity fluctuations in a channel. In the analysis, we employ a one-dimensional scalar variant of the proper orthogonal decomposition (POD). This approach naturally motivates the introduction of two different levels of similarity which we will refer to as strong and weak similarity. Strong similarity requires that the two-point correlation, and thus, all POD modes, show Reynolds number similarity, while weak similarity only requires that the first few POD modes show similarity. As POD concerns information at more than one location, these similarities are more general than various similarities found in the literature concerning single-point flow statistics. We examine flows at $Re_\tau=$180, 540, 1000, and 5200. Strong similarity is observed in the viscous layer and the wake region, and weak similarity is found in both the viscous wall region and the outer part of the logarithmic layer. The presence of weak similarity suggests the existence of an extension to the law of the wall (LoW). We propose such an extension based on the results from the one-dimensional POD analysis. The usefulness of the LoW extension is then assessed by comparing flow reconstructions according to the conventional equilibrium LoW and the extended LoW. We show that the extended LoW provides accurate flow reconstructions in the wall layer, capturing fine-scale motions that are entirely missed by the equilibrium LoW.