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This paper explores the similarity of the streamwise velocity fluctuations ina channel. In the analysis, we employ a one-dimensional scalar variant of theproper orthogonal decomposition (POD). This approach naturally motivates theintroduction of two different levels of similarity which we will refer to asstrong and weak similarity. Strong similarity requires that the two-pointcorrelation, and thus, all POD modes, show Reynolds number similarity, whileweak similarity only requires that the first few POD modes show similarity. AsPOD concerns information at more than one location, these similarities are moregeneral than various similarities found in the literature concerningsingle-point flow statistics. We examine flows at $Re_\tau=$180, 540, 1000, and5200. 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 partof the logarithmic layer. The presence of weak similarity suggests theexistence of an extension to the law of the wall (LoW). We propose such anextension based on the results from the one-dimensional POD analysis. Theusefulness of the LoW extension is then assessed by comparing flowreconstructions according to the conventional equilibrium LoW and the extendedLoW. We show that the extended LoW provides accurate flow reconstructions inthe wall layer, capturing fine-scale motions that are entirely missed by theequilibrium LoW.

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