Turbulence intensity and the friction factor for smooth- and rough-wall pipe flow

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.


Introduction
Measurements of streamwise turbulence in smooth and rough pipes have been carried out in the Princeton Superpipe [1] [2] [3]. We have treated the smooth pipe measurements as a part of [4]. In this paper, we add the rough pipe measurements to our previous analysis. The smooth (rough) pipe had a radius R of 64.68 (64.92) mm and an RMS roughness of 0.15 (5) µm, respectively. The corresponding sand-grain roughness is 0.45 (8) µm [5].
The smooth pipe is hydraulically smooth for all Reynolds numbers Re covered. The rough pipe evolves from hydraulically smooth through transitionally rough to fully rough with increasing Re. Throughout this paper, Re means the bulk Re defined using the pipe diameter D.
We define the turbulence intensity (TI) I as: where v is the mean flow velocity, v RMS is the RMS of the turbulent velocity fluctuations and r is the radius (r = 0 is the pipe axis, r = R is the pipe wall). The aim of this paper is to provide the fluid mechanics community with a scaling of the TI with Re, both for smooth-and rough-wall pipe flow. An application example is computational fluid dynamics (CFD) simulations where the TI at an opening can be specified. A scaling expression of TI with Re is provided as Eq. (6.62) in [6]. However, this formula does not appear to be documented, i.e. no reference is provided.
Our paper is structured as follows: In Section 2, we study how the TI profiles change over the transition from smooth to rough pipe flow. Thereafter we present the resulting scaling of the TI with Re in Section 3. Quantification of the correspondence between the friction factor and the TI is contained in Section 4 and we discuss our findings in Section 5. Finally, we conclude in Section 6.

Turbulence intensity profiles
We have constructed the TI profiles for the measurements available, see Fig. 1. Nine profiles are available for the smooth pipe and four for the rough pipe. In terms of Re, the rough pipe measurements are a subset of the smooth pipe measurements.
To make a direct comparison of the smooth and rough pipe measurements, we interpolate the smooth pipe measurements to the four Re values where the rough pipe measurements are done. Further, we use a normalized pipe radius r n = r/R to account for the difference in smooth and rough pipe radii. The result is a comparison of the TI profiles at four Re, see Fig. 2. As Re increases, we observe that the rough pipe TI becomes larger than the smooth pipe TI.
To make the comparison more quantitative, we define the turbulence intensity ratio (TIR): The TIR is shown in Fig. 3  • The TIR on the axis is roughly one except for the highest Re where it exceeds 1.1.
• In the intermediate region between the axis and the wall, an increase is already visible for the second-lowest Re, 1.98 × 10 6 .
The events close to the wall are most clearly seen in the right-hand plot of Fig. 3. A local peak of TIR is observed for all Re; the magnitude of the peak increases with Re. Note that we only analyse data to 99.8% of the pipe radius. So the 0.13 mm closest to the wall are not considered.  The TIR information can also be represented by studying the TIR at fixed r n vs. Re, see Fig. 4. From this plot we find that the magnitude of the peak close to the wall (r n = 0.99) increases linearly with Re: Information on fits of the TI profiles to analytical expressions can be found in Appendix A.

Turbulence intensity scaling
We define the TI averaged over the pipe area as: In [4], another definition was used for the TI averaged over the pipe area. Analysis presented in Sections 3 and 4 is repeated using that definition in Appendix B.
Scaling of the TI with Re for smooth-and rough-wall pipe flow is shown in Fig. 5.
For Re = 10 6 , the smooth and rough pipe values are almost the same. However, when Re increases, the TI of the rough pipe increases compared to the smooth pipe; this increase is to a large extent caused by the TI increase in the intermediate region between the pipe axis and the pipe wall, see Figs. 3 and 4. We have not made fits to the rough wall pipe measurements because of the limited number of datapoints.

Friction factor
The fits shown in Fig. 5 are: The Blasius smooth pipe (Darcy) friction factor [7] is also expressed as an Re power-law: The Blasius friction factor matches measurements best for Re < 10 5 ; the friction factor by e.g. Gersten (Eq. (1.77) in [8]) is preferable for larger Re. The Blasius and Gersten friction factors are compared in Fig. 6. The deviation between the smooth and rough pipe Gersten friction factors above Re = 10 5 is qualitatively similar to the deviation between the smooth and rough pipe area TI in Fig. 5. For the Gersten friction factors, we have used the measured pipe roughnesses.
For the smooth pipe, we can combine Eqs. (5) and (6) The TI and Blasius friction factor scaling is shown in Fig. 7. For axisymmetric flow in the streamwise direction, the mean flow velocity averaged over the pipe area is: Now we are in a position to define an average velocity of the turbulent fluctuations: The friction velocity is: where τ w is the wall shear stress and ρ is the fluid density. The relationship between v RMS and v τ is illustrated in Fig. 8. From the fit, we have: which we approximate as: Eqs. (11) and (12) above correspond to the usage of the friction velocity as a proxy for the velocity of the turbulent fluctuations [9]. We note that the rough wall velocities are higher than for the smooth wall.
Eqs. (9) and (12) can be combined with Eq. (1.1) in [10]: where ∆P is the pressure loss, L is the pipe length and D is the pipe diameter. This can be reformulated as: We show how well this approximation works in Fig. 9. Overall, the agreement is within 15%.
We proceed to define the average kinetic energy of the turbulent velocity fluctuations E kin,RMS (per pipe volume V ) as:    with V = LπR 2 so we have: where m is the fluid mass. The pressure loss corresponds to an increase of the turbulent kinetic energy. The turbulent kinetic energy can also be expressed in terms of the mean flow velocity and the TI or the friction factor:

Discussion
An overview of the general properties of turbulent velocity fluctuations can be found in [11].

The attached eddy hypothesis
Our quantification of the ratio v RMS /v τ as a constant can be placed in the context of the attached eddy hypothesis by Townsend [12] [13]. Our results are for quantities averaged over the pipe radius whereas the attached eddy hypothesis provides a local scaling with distance from the wall. By proposing an overlap region (see Fig. 1 in [14]) between the inner and outer scaling [15], it can be deduced that v RMS /v τ is a constant in this overlap region [16,17]. Such an overlap region has been shown to exist in [16,1]. The attached eddy hypothesis has provided the basis for theoretical work on e.g. the streamwise turbulent velocity fluctuations in flat-plate [18] and pipe flow [19] boundary layers. Work on the law of the wake in wall turbulence also makes use of the attached eddy hypothesis [20].
As a consistency check for our results, we can compare the constant 9/5 in Eq. (12) to the prediction by Townsend: where fits have provided the constants B 1 = 1.5 and A 1 = 1.25. Here, A 1 is a universal constant whereas B 1 is not expected to be a constant for different wall-bounded flows [21]. The constants are averages of fits presented in [2] to the smooth-and rough-wall Princeton Superpipe measurements. The Townsend-Perry constant A 1 was found to be 1.26 in [21]. Performing the area averaging yields: v RMS,Townsend Our finding is: which is within 5% of the result in Eq. (19). The reason that our result is smaller is that Eq. (18) is overpredicting the turbulence level close to the wall and close to the pipe axis. Eq. (18) as an upper bound has also been discussed in [22].

The friction factor and turbulent velocity fluctuations
The proportionality between the average kinetic energy of the turbulent velocity fluctuations and the friction velocity squared has been identified in [23] for Re > 10 5 . This corresponds to our Eq. (16).
A correspondence between the wall-normal Reynolds stress and the friction factor has been shown in [24]. Those results were found using direct numerical simulations. The main difference between the cases is that we use the streamwise Reynolds stress. However, for an eddy rotating in the streamwise direction, both a wall-normal and a streamwise component should exist which connects the two observations.

The turbulence intensity and the diagnostic plot
Other related work can be found beginning with [25] where the diagnostic plot was introduced. In following publications a version of the diagnostic plot was brought forward where the local TI is plotted as a function of the local streamwise velocity normalised by the free stream velocity [26,27,28]. Eq.

Conclusions
We have compared TI profiles for smooth-and rough-wall pipe flow measurements made in the Princeton Superpipe.
The change of the TI profile with increasing Re from hydraulically smooth to fully rough flow exhibits propagation from the pipe wall to the pipe axis. The TIR at r n = 0.99 scales linearly with Re.
The scaling of TI with Re -on the pipe axis and averaged over the pipe area -shows that the smooth-and rough-wall level deviates with increasing Reynolds number.
We find that I Pipe area ∼ 9 10 √ 2 × √ λ. This relationship can be useful to calculate the TI given a known λ, both for smooth and rough pipes. It follows that given a pressure loss in a pipe, the turbulent kinetic energy increase can be estimated.

Appendix A. Fits to the turbulence intensity profile
As we have done for the smooth pipe measurements in [4], we can also fit the rough pipe measurements to this function: where α, β, γ, δ and ε are fit parameters. A comparison of fit parameters found for the smooth-and rough-pipe measurements is shown in Fig. A.10.
Overall, we can state that the fit parameters for the smooth and rough pipes are in a similar range for 10 6 < Re < 6 × 10 6 . The min/max deviation of the rough pipe fit from the measurements is below 10%; see the comparison to the smooth wall fit min/max deviation in Fig. A.11.
The core and wall fits for the smooth and rough pipe fits are compared in Fig. A.12. Both the core and wall TI increase for the largest Re.
The position where the core and wall TI levels are equal is shown in Fig.  A.13. This position does not change significantly for the rough pipe; however, the position does increase with Re for the smooth pipe: This indicates that the wall term becomes less important relative to the core term.    Appendix B. Arithmetic mean definition of turbulence intensity averaged over the pipe area In the main paper, we have defined the TI over the pipe area in Eq. (4). In [4], we used the arithmetic mean (AM) instead: The AM leads to a somewhat different pipe area scaling for the smooth pipe measurements which is illustrated in Fig. B  The scaling found in [4] using this definition is: The AM scaling also has implications for the relationship with the Blasius friction factor scaling (Eq. (7)): The AM definition can be considered as a first order moment equation for v RMS , whereas the definition in Eq. (9) is a second order moment equation.
Again, we find that the AM average turbulent velocity fluctuations are proportional to the friction velocity. However, the constant of proportionality is different than the one in Eq. which we approximate as: As we did in Section 5, we can perform the AM averaging of Eq. (18) (also done in [22]):