Turbulence intensity and the friction factor for smooth and roughwall pipe flow
Abstract
Turbulence intensity profiles are compared for smooth and roughwall 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.
keywords:
Turbulence intensity, Princeton Superpipe measurements, Flow in smooth and roughwall pipes, Friction factor1 Introduction
Measurements of streamwise turbulence in smooth and rough pipes have been carried out in the Princeton Superpipe hultmark_a () hultmark_b () smits_a (). We have treated the smooth pipe measurements as a part of russo_a (). In this paper, we add the rough pipe measurements to our previous analysis. The smooth (rough) pipe had a radius of 64.68 (64.92) mm and an RMS roughness of 0.15 (5) m, respectively. The corresponding sandgrain roughness is 0.45 (8) m langelandsvik_a ().
The smooth pipe is hydraulically smooth for all Reynolds numbers covered. The rough pipe evolves from hydraulically smooth through transitionally rough to fully rough with increasing . Throughout this paper, means the bulk defined using the pipe diameter .
We define the turbulence intensity (TI) as:
(1) 
where is the mean flow velocity, is the RMS of the turbulent velocity fluctuations and is the radius ( is the pipe axis, is the pipe wall).
The aim of this paper is to provide the fluid mechanics community with a scaling of the TI with , both for smooth and roughwall 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 is provided as Eq. (6.62) in ansys_a (). 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 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.
2 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 , 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 values where the rough pipe measurements are done. Further, we use a normalized pipe radius to account for the difference in smooth and rough pipe radii. The result is a comparison of the TI profiles at four , see Fig. 2. As 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):
(2) 
The TIR is shown in Fig. 3. The lefthand plot shows all radii; prominent features are:

The TIR on the axis is roughly one except for the highest where it exceeds 1.1.

In the intermediate region between the axis and the wall, an increase is already visible for the secondlowest , 1.98 10.
The events close to the wall are most clearly seen in the righthand plot of Fig. 3. A local peak of TIR is observed for all ; the magnitude of the peak increases with . 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 vs. , see Fig. 4. From this plot we find that the magnitude of the peak close to the wall () increases linearly with :
(3) 
Information on fits of the TI profiles to analytical expressions can be found in A.
3 Turbulence intensity scaling
We define the TI averaged over the pipe area as:
(4) 
In russo_a (), 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 B.
Scaling of the TI with for smooth and roughwall pipe flow is shown in Fig. 5.
For , the smooth and rough pipe values are almost the same. However, when 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.
4 Friction factor
The fits shown in Fig. 5 are:
(5) 
The Blasius smooth pipe (Darcy) friction factor blasius_a () is also expressed as an powerlaw:
(6) 
The Blasius friction factor matches measurements best for ; the friction factor by e.g. Gersten (Eq. (1.77) in gersten_a ()) is preferable for larger . The Blasius and Gersten friction factors are compared in Fig. 6. The deviation between the smooth and rough pipe Gersten friction factors above 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) to relate the pipe area TI to the Blasius friction factor:
(7) 
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:
(8) 
Now we are in a position to define an average velocity of the turbulent fluctuations:
(9) 
The friction velocity is:
(10) 
where is the wall shear stress and is the fluid density.
The relationship between and is illustrated in Fig. 8. From the fit, we have:
(11) 
which we approximate as:
(12) 
Eqs. (11) and (12) above correspond to the usage of the friction velocity as a proxy for the velocity of the turbulent fluctuations schlichting_a (). 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 mckeon_a ():
(13) 
where is the pressure loss, is the pipe length and is the pipe diameter. This can be reformulated as:
(14) 
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 (per pipe volume ) as:
(15) 
with so we have:
(16) 
where 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:
(17) 
5 Discussion
An overview of the general properties of turbulent velocity fluctuations can be found in marusic_a ().
5.1 The attached eddy hypothesis
Our quantification of the ratio as a constant can be placed in the context of the attached eddy hypothesis by Townsend townsend_a () marusic_b (). 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 mckeon_b ()) between the inner and outer scaling millikan_a (), it can be deduced that is a constant in this overlap region perry_a (); perry_b (). Such an overlap region has been shown to exist in perry_a (); hultmark_a (). The attached eddy hypothesis has provided the basis for theoretical work on e.g. the streamwise turbulent velocity fluctuations in flatplate marusic_c () and pipe flow hultmark_c () boundary layers. Work on the law of the wake in wall turbulence also makes use of the attached eddy hypothesis krug_a ().
As a consistency check for our results, we can compare the constant in Eq. (12) to the prediction by Townsend:
(18) 
where fits have provided the constants and . Here, is a universal constant whereas is not expected to be a constant for different wallbounded flows marusic_d (). The constants are averages of fits presented in hultmark_b () to the smooth and roughwall Princeton Superpipe measurements. The TownsendPerry constant was found to be 1.26 in marusic_d (). Performing the area averaging yields:
(19) 
Our finding is:
(20) 
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 pullin_a ().
5.2 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 yakhot_a () for . This corresponds to our Eq. (16).
A correspondence between the wallnormal Reynolds stress and the friction factor has been shown in orlandi_a (). 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 wallnormal and a streamwise component should exist which connects the two observations.
5.3 The turbulence intensity and the diagnostic plot
Other related work can be found beginning with alfredsson_a () 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 alfredsson_b (); alfredsson_c (); castro_a (). Eq. (3) in alfredsson_c () corresponds to our , see Eq. (21) in A.
6 Conclusions
We have compared TI profiles for smooth and roughwall pipe flow measurements made in the Princeton Superpipe.
The change of the TI profile with increasing from hydraulically smooth to fully rough flow exhibits propagation from the pipe wall to the pipe axis. The TIR at scales linearly with .
The scaling of TI with  on the pipe axis and averaged over the pipe area  shows that the smooth and roughwall level deviates with increasing Reynolds number.
We find that . This relationship can be useful to calculate the TI given a known , both for smooth and rough pipes. It follows that given a pressure loss in a pipe, the turbulent kinetic energy increase can be estimated.
Acknowledgement
We thank Professor A.J.Smits for making the Superpipe data publicly available smits_a ().
Appendix A Fits to the turbulence intensity profile
As we have done for the smooth pipe measurements in russo_a (), we can also fit the rough pipe measurements to this function:
(21) 
where , , , and are fit parameters. A comparison of fit parameters found for the smooth and roughpipe measurements is shown in Fig. 10. Overall, we can state that the fit parameters for the smooth and rough pipes are in a similar range for .
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. 11.
The core and wall fits for the smooth and rough pipe fits are compared in Fig. 12. Both the core and wall TI increase for the largest .
The position where the core and wall TI levels are equal is shown in Fig. 13. This position does not change significantly for the rough pipe; however, the position does increase with 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 russo_a (), we used the arithmetic mean (AM) instead:
(22) 
The AM leads to a somewhat different pipe area scaling for the smooth pipe measurements which is illustrated in Fig. 14. Compare to Fig. 5.
The scaling found in russo_a () using this definition is:
(23) 
The AM scaling also has implications for the relationship with the Blasius friction factor scaling (Eq. (7)):
(24) 
We can now define the AM version of the average velocity of the turbulent fluctuations:
(25) 
The AM definition can be considered as a first order moment equation for , 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. (11), see Fig. 15. The AM case can be fitted as:
(26) 
which we approximate as:
(27) 
As we did in Section 5, we can perform the AM averaging of Eq. (18) (also done in pullin_a ()):
(28) 
where we find:
(29) 
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