Asymptotics of stream-wise Reynolds stress in wall turbulence

TL;DR

This paper develops a new asymptotic expansion for stream-wise Reynolds stress in wall turbulence, accurately fitting DNS and experimental data across a wide range of Reynolds numbers, and clarifies the scaling behavior of stress peaks.

Contribution

It introduces a novel two-term inner asymptotic expansion for $ angle uu angle^+$, matched with an outer expansion, improving understanding of stress scaling in high Reynolds number flows.

Findings

Inner peak saturates at 11.3 and decreases with $Re_\tau^{-1/4}$

Wall loglaw slope becomes positive at $Re_\tau \approx 20,000$

Outer logarithmic overlap with negative slope approaches zero as $Re_\tau \to \infty$

Abstract

The scaling of different features of stream-wise normal stress profiles $\langle uu\rangle^+(y^+)$ in turbulent wall-bounded flows, in particular in truly parallel flows, such as channel and pipe flows, is the subject of a long running debate. Particular points of contention are the scaling of the "inner" and "outer" peaks of $\langle uu\rangle^+$ at $y^+\approxeq 15$ and $y^+ =\mathcal{O}(10^3)$, respectively, their infinite Reynolds number limit, and the rate of logarithmic decay in the outer part of the flow. Inspired by the landmark paper of Chen and Sreenivasan (2021), two terms of the inner asymptotic expansion of $\langle uu\rangle^+$ in the small parameter $Re_\tau^{-1/4}$ are extracted for the first time from a set of direct numerical simulations (DNS) of channel flow. This inner expansion is completed by a matching outer expansion, which not only fits the same set of channel DNS within 1.5\% of the peak stress, but also provides a good match of laboratory data in pipes and the near-wall part of boundary layers, up to the highest $Re_\tau$'s of order $10^5$. The salient features of the new composite expansion are first, an inner $\langle uu\rangle^+$ peak, which saturates at 11.3 and decreases as $Re_\tau^{-1/4}$, followed by a short "wall loglaw" with a slope that becomes positive for $Re_\tau \gtrapprox 20'000$, leading up to an outer peak, and an outer logarithmic overlap with a negative slope continuously going to zero for $Re_\tau \to\infty$.