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

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 $550<Re_τ<16,000$, 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 $Re_τ<2,000$, the NSP did not contain either of the proposed trends. As $Re_τ$ 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 $Re_τ$ 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''.