Abstract
This article studies how the proper orthogonal decomposition eigenvectors and eigenvalues of the two-point velocity correlation tensor scale with Reτ in turbulent channel flows. To this effect, the two-point correlation tensor is computed from velocity fields extracted from the Direct Numerical Simulations (DNS) of plane channel flows at Reτ = 547 and Reτ = 934. The analysis reveals that the eigenvalues exhibit a high degree of scaling with Reτ, across a very wide range of streamwise and spanwise wavenumbers. The eigenvectors also show near complete independence from Reτ, as long as the wall-normal lengthscales larger than the channel height are removed. The poor Reτ scaling of turbulent structures larger than the channel height is well documented in the literature, and thus one would not expect eigenvectors corresponding to these scales to exhibit favorable Reτ scaling. Two-point velocity correlations and their eigenvectors are also computed using Large Eddy Simulations (LES) at Reτ = 1000 and compared to the results of the DNS at Reτ = 934. Both the correlations and eigenvectors matched very well between LES and DNS.
Highlights
The search for modes of motion that could represent turbulent flows in a synthetic way can be traced back to the work of Townsend1 and Grant.2 Both authors were computing twopoint velocity correlations in canonical shear-flows, and in order to explain the shape of the correlations, they theorized the existence of large-scale, organized turbulent motion, which they called coherent structures
The two-point correlation tensor is computed from velocity fields extracted from the Direct Numerical Simulations (DNS) of plane channel flows at Reτ = 547 and Reτ = 934
The poor Reτ scaling of turbulent structures larger than the channel height is well documented in the literature, and one would not expect eigenvectors corresponding to these scales to exhibit favorable Reτ scaling
Summary
The search for modes of motion that could represent turbulent flows in a synthetic way can be traced back to the work of Townsend and Grant. Both authors were computing twopoint velocity correlations in canonical shear-flows, and in order to explain the shape of the correlations, they theorized the existence of large-scale, organized turbulent motion, which they called coherent structures. POD was applied to the time-averaged two-point velocity correlation tensor, allowing the authors to identify three dimensional modes of turbulent structures. They were able to confirm that the most energetic POD mode corresponded to a spanwise hairpin vortex, very similar in form to that observed in the measurements of Kline et al.. POD in this case do not directly correlate to turbulent structures because they are only one-dimensional and lack lengthscales in the streamwise and spanwise directions Another shortcoming of this analysis was that it lacked the resolution necessary to resolve the flow very close to the wall, i.e., the inner scales. An indepth analysis is presented of the Reτ scaling of the eigenvalues and eigenvectors of the two-point velocity correlation
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