Abstract
A series of recent studies has shown that a model of the turbulent vertical velocity variance spectrum (Fvv) combined with a simplified cospectral budget can reproduce many macroscopic flow properties of turbulent wall-bounded flows, including various features of the mean-velocity profile (MVP), i.e., the “law of the wall”. While the approach reasonably models the MVP’s logarithmic layer, the buffer layer displays insufficient curvature compared to measurements. The assumptions are re-examined here using a direct numerical simulation (DNS) dataset at moderate Reynolds number that includes all the requisite spectral and co-spectral information. Starting with several hypotheses for the cause of the “missing” curvature in the buffer layer, it is shown that the curvature deficit is mainly due to mismatches between (i) the modelled and DNS-observed pressure-strain terms in the cospectral budget and (ii) the DNS-observed Fvv and the idealized form used in previous models. By replacing the current parameterization for the pressure-strain term with an expansive version that directly accounts for wall-blocking effects, the modelled and DNS reported pressure-strain profiles match each other in the buffer and logarithmic layers. Forcing the new model with DNS-reported Fvv rather than the idealized form previously used reproduces the missing buffer layer curvature to high fidelity thereby confirming the “spectral link” between Fvv and the MVP across the full profile. A broad implication of this work is that much of the macroscopic properties of the flow (such as the MVP) may be derived from the energy distribution in turbulent eddies (i.e., Fvv) representing the microstate of the flow, provided the link between them accounts for wall-blocking.
Highlights
Sufficiently high Reynolds number (Re), of a logarithmic region in the mean-velocity profile (MVP) away from the wall[3] such that
A cospectral budget model is shown to be capable of reproducing the full lawof-the-wall MVP with no tunable parameters, when forced with the right vertical-velocity variance spectra Fvv+ and T K E dissipation rates
Parameterizations used for Fvv+ used in previous studies overestimate the true spectra, at least at the relatively low Reynolds number considered in this study
Summary
Gioia et al.[11] first proposed a mechanism for such a link, relying on a heuristic argument in which turbulent stresses were generated by the product of a longitudinal velocity excursion set in size by “dominant eddies” of radius y+, and a vertical velocity excursion determined by the turbulent kinetic energy T K E( y+) This phenomenological model has been shown to yield a range of known macroscopic properties of turbulent flows. The new physics learned from the cospectral budget model is that the distribution of turbulent vertical velocity fluctuations (the “microstate” of the flow, represented by Fvv(k)) contains sufficient information to generate the MVP (the “macrostate” of the flow) This establishes a link between two previously unrelated areas of the turbulence literature: (1) Kolmogorov’s theory of homogeneous, isotropic turbulence, including the k−5/3 scaling of the energy spectrum and (2) the “law of the wall”. The specific value of this study is in testing and refining assumptions from previous studies by using DNS data, and making modifications to the theory where conflicts arise with the DNS data
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