High Reynolds number trends of centerline mean velocity and normal stress in pipe flow

TL;DR

The paper tackles inconsistencies in centerline mean velocity and turbulence data for fully developed pipe flow at high $Re_\tau$ across facilities. Using the high-resolution CICLoPE facility, the authors establish when high-$Re_\tau$ conditions emerge and quantify centerline metrics, including $U_{CL}^+$ scaling and turbulence intensities, while cross-validating with Superpipe and DNS data. They confirm $κ_{CL}=0.44$, show that $u'/U_{CL}$ saturates near $2.9\%$ and $(u'/u_\tau)^2$ near $0.85$ around $Re_\tau \approx 2.5\times 10^4$, and observe a clear $-5/3$ inertial range in centerline spectra; skewness and kurtosis of $U$ are Reynolds-number independent down to $Re_\tau=5\times 10^3$. The work reconciles prior conflicting trends, highlights the need for meticulous calibration and noise assessment in NSTAP data, and provides robust high-$Re_\tau$ pipe-flow benchmarks for turbulence modeling.

Abstract

The CICLoPE facility at the University of Bologna in Forli, Italy, is a unique facility that provides fully developed pipe flow up to Reynolds numbers of about $Re_τ$ of 50,000 with exceptional spatial resolution and stable operating conditions. Measurements obtained over the last two years, on the centerline of the pipe in the fully developed test section, with Pitot probes for the mean velocity in the stream ($\pm 0.2\%$ precision) and hot wires for the intensity of turbulence in the streamwise and the normal stress ( $\pm~5\%$ precision) are reported here and compared to other experimental results and some recent direct numerical simulation (DNS) data. The comparisons reveal that high Reynolds number conditions are only reached when $Re_τ$ is at least larger than $10,000$. The centerline von Kàrmàn constant, $κ_{CL}$ for pipe flow is confirmed with high confidence to be $0.44$. Unlike previously reported measurements on the pipe centerline of the streamwise turbulence intensity and normal stress reported in the literature with various conflicting trends, both quantities are found in CICLoPE to reach a level trend beyond $Re_τ$ around $25,000$ with values of approximately $2.9\%$ and $0.85$, respectively. Normalized streamwise velocity spectra on the centerline, with high resolution over seven decades of energy, exhibit a clear and extended $-5/3$ inertial range. Skewness and kurtosis of streamwise velocity are found to be independent of Reynolds number even down to $Re_τ= 5,000$ and equal to $-0.5$ and $3.5$, respectively.

Figures (10)

• Figure 1: Engineering drawing of CICLoPE facility and photograph of hot-wire probe in test section.
• Figure 2: Normalized centerline mean velocity, $U_{CL}^+= U_{CL}/u_\tau$, from CICLoPE by Mascotelli Lucia and Lazzarini Lorenzo, compared to other data in Superpipe by McKeon et al. McKeon, and to DNS results of Pirozzoli Pirozzoli.
• Figure 3: Centerline normalized streamwise turbulence intensity, $u'/U_{CL}$, as function of Reynolds number, $Re_\tau$, from CICLoPE by Fiorini Fiorini and Lazzarini Lorenzo, and comparisons with DNS data of Pirozzoli Pirozzoli, and experimental results by Zanoun et al. Zanoun24, and Fu et al. Fu.
• Figure 4: Centerline streamwise normal stress of turbulence, $(u'/u_\tau)^2$, as function of Reynolds number, $Re_\tau$, from CICLoPE by Lazzarini Lorenzo, compared to DNS data from Pirozzoli Pirozzoli, and experimental results of Ono et al. Ono and Hultmark et al. Hultmark; see Figure 10 regarding question mark near top right-hand side of this figure.
• Figure 5: Skewness and kurtosis of streamwise velocity from CICLoPE as function of Reynolds number, $Re_\tau$, by Fiorini Fiorini and Lazzarini Lorenzo.