Utilizing indicator functions with computational data to confirm nature of overlap in normal turbulent stresses: logarithmic or quarter-power

TL;DR

This study uses indicator functions derived from DNS data to analyze the presence of logarithmic or quarter-power trends in turbulent stresses across various Reynolds numbers, revealing a dominant quarter-power region at high Reynolds numbers.

Contribution

It provides new evidence that the quarter-power law dominates in high Reynolds number turbulent stresses, challenging previous assumptions of logarithmic trends.

Findings

Quarter-power region develops at Re_tau > 2000

Range of quarter-power NSP extends to y+ between 1000 and 10,000

Logarithmic trend not observed in the examined DNS data at high Re_tau

Abstract

Indicator functions of the streamwise normal-stress profiles (NSP), based on careful differentiation of some of the best direct numerical simulations (DNS) data from channel and pipe flows, over the range $550<Re_\tau<16,000$, are examined to establish the existence and range in wall distances of either a logarithmic-trend segment or a $1/4$-power region. For the nine out of fifteen cases of DNS data we examined where $Re_\tau<2,000$, the NSP did not contain either of the proposed trends. As $Re_\tau$ exceeds around $2,000$ a $1/4$-power, reflecting the ``bounded-dissipation'' predictions of Chen \& Sreenivasan and data analysis of Monkewitz , develops near $y^+=1,000$ and expands with Reynolds numbers extending to $1,000<y^+<10,000$ for $Re_\tau$ around $15,000$. This range of $1/4$-power NSP corresponds to a range of outer-scaled $Y$ between around $0.3$ and $0.7$. The computational database examined did not include the zero-pressure-gradient boundary layer experiments at higher Reynolds numbers where the logarithmic trend in the NSP has been previously reported around $y^+$ of $1,000$ by Marusic et al. according to a ``wall-scaled eddy model''.