Mean-velocity profile of smooth channel flow explained by a cospectral budget model with wall-blockage

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.