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We have characterized a transition of turbulence intensity (TI) scaling for friction Reynolds numbers $Re_{\tau} \sim 10^4$ in the companion papers [Basse, N.T. Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 065127}] and [Basse, N.T. Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 125109}]. Here, we build on those results to extrapolate TI scaling for $Re_{\tau} \gg 10^5$, under the assumption that no further transitions exist. Scaling of the core, area-averaged and global peak TI demonstrates that they all scale inversely with the logarithm of $Re_{\tau}$, but with different multipliers. Finally, we confirm the prediction that the TI squared is proportional to the friction factor for $Re_{\tau} \gg 10^5$.

Presentation on turbulence intensity and how it can be applied to e.g. computational fluid dynamics boundary conditions.

We extend the procedure outlined in Basse [“Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region,” Phys. Fluids 33, 065127 (2021)] to study global, i.e., radially averaged, scaling of streamwise velocity fluctuations. A viscous term is added to the log-law scaling, which leads to the existence of a mathematical abstraction, which we call the “global peak.” The position and amplitude of this global peak are characterized and compared to the inner and outer peaks. A transition at a friction Reynolds number of order 10 000 is identified. Consequences for the global peak scaling, length scales, non-zero asymptotic viscosity, turbulent energy production/dissipation, and turbulence intensity scaling are appraised along with the impact of including an additional wake term.

We extend the procedure outlined in [Basse, "Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region," Phys. Fluids Vol. 33, 065127 (2021)] to study global, i.e. radially averaged, scaling of streamwise velocity fluctuations. A viscous term is added to the log-law scaling which leads to the existence of a mathematical abstraction which we call the "global peak". The position and amplitude of this global peak are characterized and compared to the inner and outer peaks. A transition at a friction Reynolds number of order 10000 is identified. Consequences for the global peak scaling, length scales, non-zero asymptotic viscosity, turbulent energy production/dissipation and turbulence intensity scaling are appraised along with the impact of including an additional wake term.

We study the global, i.e. radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as I^2 = overbar(u^2)/U^2 , where u and U are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark M, Vallikivi M, Bailey SCC and Smits AJ. Logarithmic scaling of turbulence in smooth-and rough-wall pipe flow. J. Fluid Mech. Vol. 728, 376-395 (2013)]. Scaling expressions are derived both for log-law and power-law functions of radius. A transition to asymptotic scaling is found at a friction Reynolds number Re_τ ∼ 11000.

An asymptotic scaling law for drag in flat-plate turbulent boundary layers has been proposed [Dixit SA, Gupta A, Choudhary H, Singh AK and Prabhakaran T. Asymptotic scaling of drag in flat-plate turbulent boundary layers. Phys. Fluids Vol. 32, 041702 (2020)]. In this paper we suggest to amend the scaling law by using a correction term derived from the logarithmic law for the mean velocity in the streamwise direction.

We explore the analogy between asymptotic scaling of two canonical wall-bounded turbulent flows, i.e. zero-pressure-gradient and pipe flows; we find that these flows can be characterised using similar scaling laws which relate streamwise turbulence intensity and friction.

Python code to calculate turbulence intensity based on Reynolds number and surface roughness.

We study streamwise turbulence intensity definitions using smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. Scaling of turbulence intensity with the bulk
(and friction) Reynolds number is provided for the definitions. The turbulence intensity scales with the friction factor for both smooth- and rough-wall pipe flow. Turbulence intensity definitions
providing the best description of the measurements are identified. A procedure to calculate the turbulence intensity based on the bulk Reynolds number (and the sand-grain roughness for
rough-wall pipe flow) is outlined.

Turbulence intensity profiles are compared for smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. The profile development in the transition from hydraulically smooth to fully rough flow displays a propagating sequence from the pipe wall towards the pipe axis. The scaling of turbulence intensity with Reynolds number shows that the smooth- and rough wall level deviates with increasing Reynolds number. We quantify the correspondence between turbulence intensity and the friction factor.

Turbulence intensity profiles are compared for smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. The profile development in the transition from hydraulically smooth to fully rough flow displays a propagating sequence from the pipe wall towards the pipe axis. The scaling of turbulence intensity with Reynolds number shows that the smooth- and rough wall level deviates with increasing Reynolds number. We propose a correspondence between turbulence intensity and the friction factor.