Mean momentum structure and kinetic energy production in fully developed turbulent pipe flow
Abstract
In light of recent data from hot wire anemometry and laser Doppler velocimetry, this article explores experimentally the momentum layer structure and kinetic energy production in fully developed turbulent pipe flow for shear Reynolds numbers in the range . Difficulties in carrying out simultaneous measurements of both the streamwise and the wallnormal velocity fluctuations with adequate spatial and temporal resolutions are addressed. Specifically, the difference between the direct measure of the Reynolds shear transport, and that deduced from the mean flow data and the mean momentum balance are discussed. A significant dynamical viscous effect is observed in wall layer extending hundreds of wall units beyond the buffer layer depending on the Reynolds number. The present analysis also reinforces the universality of the viscous stress gradient to the Reynolds stress gradient within both the extended buffer layer and the pipe core region. Examining the premultiplied kinetic energy production indicates a dominance of the inertial sublayer, contributing most to the bulk turbulent energy production, in particular, at high Reynolds number. The present assessment of the existing high Reynolds number pipe flow data underlines the importance of measuring the Reynolds shear stress directly, and the wall friction velocity independent of each other to utilize the momentum balance to ascertain the quality of the data.
urbulent pipe flow, momentum structure, kinetic energy production
1 Introduction and Theoretical Considerations
In wallbounded shear flows, examining the momentum transport in various wall layers and its contribution to the turbulent kinetic energy is a key issue for better understanding of turbulent flow structures, e.g. the large scale and very large scale motions, i.e. LSM VLSM, see e.g. Refs. [1, 2]. The Reynoldsaveraged NavierStokes equations:
(1) 
with stands for the flow direction, and for component, when adapted to fully developed turbulent pipe flow, the following streamwise mean momentum equation results:
(2) 
representing gradients of the viscous shear stress (), and the Reynolds shear stress () in balance with the gradient of the mean pressure . Here is the streamwise distance, is the distance normal to the wall, is the streamwise mean velocity component, is the streamwise velocity fluctuation, is the wallnormal velocity fluctuation, is the dynamic viscosity, is the fluid density, and is the mean flow pressure. Thus equation (2) describes the mechanism of the momentum transport in fully developed turbulent pipe flow, see e.g. Ref. [3]. Integrating equation (2), and introducing the definition of the wall shear stress, , gives:
(3) 
Normalizing all terms in equation (3) with the socalled wall units, i.e. with the characteristic velocity , length , and time scales, and rearranging it results in the following normalized form for the streamwise mean momentum equation:
(4) 
Here denotes the friction Reynolds number or the socalled Kármán number and is defined as , with being the pipe radius. When the friction Reynolds number tends to infinity, i.e. , equation (4) is simplified, in terms of wall units, to represent flow in the wall layer as follows:
(5) 
where the mean pressure gradient has a minimal effect. In wall vicinity, i.e. , the viscous shear stress is dominant, see Fig. 1, however, it is still of vital importance even far from the wall to analyze the rest of the flow field, see Ref. [4, 6].
Far enough from the pipe wall, i.e. for , where , see Fig. 1, equation (4) in outer scaling reads as:
(6) 
expressing the balance between the Reynolds shear stress and the mean pressure gradient. Here is a notation for the normalized wall distance which is due to the mean pressure gradient.
Various studies attempted to match the inner [equation (5)] and the outer [equation (6)] regions of the flow; see, e.g., Refs. [6, 7, 8, 9]. An accurate description for the Reynolds shear stress is given in Ref. [8, 9] for twodimensional turbulent shear flows:
(7) 
where is a constant that was recommended by Panton [8] to have a value of 7.8 for a von Kármán constant () of 0.41. For the present pipe flow data in Fig. 2, a revised value of is adopted for , proposed by Ref. [10] for the pipe flow. Equation (7) was found to be valid for all values of and permits the prediction of the turbulent shear momentum transport for any Reynolds number as follows:
(8) 
Considering equation (2) to estimate the turbulent shear momentum transport, indirectly, it requires measurements of the streamwise mean pressure gradient along the pipe test section, in addition to the gradient of the crosssectional mean velocity distribution. The later reflects the dynamics of the momentum transport and their scaling with the Reynolds number. Wei et al. (2005) addressed, for instance, the primary importance associated with the mean velocity profile in its inherent description of the momentum/vorticity redistribution processes. On contrary, to obtain directly via measuring the streamwise and the wall normal velocity fluctuations, i.e. , in the region where it is the much larger term in equation (2) was claimed not to be the right approach since a small error in will yield a large error in distribution, see [4]. In spite of that, early attempts have been made to measure simultaneously the streamwise and the wallnormal fluctuations to obtain . For instance, [11] concluded that the directly measured turbulent shear stress is on average 10 smaller than the distribution deduced from the momentum balance and the mean flow data in agreement with [12, 13]. It was observed also by [14] that the total shearing stress obtained from the direct fluctuations measurements was approximately 20 lower than those values computed from the measurements of the gradients of the mean velocity and the mean pressure.
Such discrepancy observed between the direct and indirect measures of the shear momentum transport might be attributed to:

difficulties in conducting simultaneous measurements of due to uncertainties in the calibration of the measuring devices if, e.g., hot wire anemometry is to be used. In addition, difficulties in proper alignment of the hot wire probes might occur.

inadequate spatial and temporal resolutions, in particular, in the region close to the wall where flow exhibits strong gradient.

temperature drift, independencey and locality of the wall shear stress, and uncertainties in positioning the measuring probes with respect to the wall surface.
The above summary motivates the present work to review aspects on both the low and the high momentum transport distributions in fully developed turbulent pipe flows, and discusses them in light of recent hot wire anemometry and laser Doppler velocimetry measurements performed in CoLaPipe facility [15], in addition to hot wire data obtained from CICLoPE facility [16, 27] over a wide range of Reynolds number. In particular, the following two main points are to be highlighted:

which appropriate method would be to estimate the turbulent shear momentum transport and structure in terms of amplitudes and how do they scale with Reynolds number?

what is the contribution of the logarithmic region to the turbulent kinetic energy production in turbulent pipe flow?
To outline the present manuscript, the main sections are summarized as follows. Section 1 outlines briefly the governing equations and Panton’ stress function to treat the turbulent momentum transport. The structure of the momentum transport and kinetic energy production data are presented in Section 2. This section presents the experimental data over a wide range of Reynolds number from two different pipe facilities, namely, CoLaPipe and CICLoPE. Section 3 summarizes the outcome of the present study with some concluding remarks.
2 Discussion of Results
2.1 The mean momentum balance
Studying the explicit mean momentum balance described by equation (2), and its scaling is of considerable utility for better understanding of the dynamics of turbulent flow structures [3]. Hence, the main interest of the present section is being given to the distributions of the viscous (), the turbulent shear momentum transports , and turbulent kinetic energy production, , in fully developed turbulent pipe flows at high Reynolds number. Two measuring techniques, namely, hot wire anemometry and laser Doppler velocimetry, were utilized, and supported by some recent numerical data to yield reliable information about the characteristics of the viscous and turbulent momentum transports of fully developed turbulent pipe flows.
Referring to [3], a primary and relevant structure of the streamwise mean momentum in wallbounded flows is the fourlayer description:

laminar sublayer: ; viscous forcepressure force balance

buffer layer: ; viscous forceturbulent inertia balance

logarithmic layer: ; dominance of turbulent inertia

core region: ; turbulent inertiapressure force balance
The above fourlayer description of momentum transport is closely tied to properties of the mean profile, and thus a more subtle influence of the mean profile structure relates to the associated physical interpretations of wallflow physics [3]. However, to examine better the mean momentum balance, equation (4) ca be rewritten as follows:
(9) 
expressing balance among the most significant forces acting on the flow, i.e. viscous force, turbulent inertia and pressure force, respectively. Flow dynamics in each of the fourlayer description is characterized either by a balance among all three forces in equation (9) or by a balance of two forces while the third diminishes in effect [3]. Considering the ratio of the first term (a) to the second term (b) in equation (9), Fig. 3 is produced versus the wall normal location (). It is worth noting that Panton’ stress relation, i.e. Equation (7), was utilized to produce Fig. 3. The figure is, therefore, examines the ratio of the gradient of the viscous shear stress [] to the gradient of the Reynolds shear stress [] for low () and high () Reynolds numbers.
In agreement with [3], Figs. 1 and 3 support the fourlayer structures, and show, in particular, the behavior of the viscous and the Reynolds shear stresses in the various wall layers. The physical extent of each layer from such four layer structure presented in Fig. 3 is subjected to changes, depending on the Reynolds number except the inner viscouspressure balance layer I, see Fig. 3.
In the wall vicinity, i.e. , Fig. 1 shows a minimal effect for the Reynolds shear stress, resulting in the gradient ratio (a/b) as shown in Fig. 3 is becoming much greater than unity by approaching the pipe surface. Within this layer, however, pressure (c) and viscous (a) forces are in approximate balance. The next layer, i.e. layer II, is the stressgradient balance layer, where the stressgradient ratio shows a magnitude very close to unity. The viscous and the Reynolds stress gradients are of equal importance within this layer, but having opposite signs. The thickness of this layer, however, shows a strong Reynoldsnumber dependence as Fig 3 illustrates. For instance, this layer extends from to for in agreement with a value deduced from , a formula proposed by [3]. The extent of this layer, however, increases as the Reynolds number increases interfering the logarithmic region, e.g. extends to approximately at .
Far enough from the wall, i.e. along the log layer, the Reynolds shear stress dominates with a minimal effect for the viscous stress, however, still of importance Ref. [4]. Within layer III, gradients of the viscous stress and the mean pressure are in balance, see also Fig. 1. Along this layer, the gradient of the Reynolds shear stress shows a zero value due to its maximum behavior, therefore the stressgradient ratio tends to infinity as can seen in Fig. 3. In pipe core region (IV), i.e. , gradients of the mean pressure and the Reynolds shear stress are in balance.
Both Figs. 1 3 reveal a significant dynamical viscous effect from the wall extending hundreds of wall units, however, for high enough . This finding is online with earlier observations made, e.g.; by [17, 10] that the inner limit of the logarithmic layer lies between and for the pipe flow. This observation is also in relevance for such viscous effects observed by [18] outside the buffer layer. On contrary, this turns to be not in agreement with a belief that the viscous effects are limited to the outer edge of the buffer layer, i.e. , e.g. see [19, 11, 20].
2.2 Wall friction in both facilities
Based on measurements of the bulk flow velocity, and the mean pressure gradient along the pipe test sections in both facilities, the pipe wall friction () data were obtained independently of the mean profile measurements. Figure 4 shows, therefore, a comparison between present pipe friction data and some data extracted from the literature. Good agreement is to be observed in Fig. 4(a) when the present data were compared with experimental data of Zagarola and Smits [17], and also with PrandtlvonKármán logarithmic relation:
(10) 
which is known as a universal friction law for hydraulically smooth pipe flows. It is worth, however, noting that the above logarithmic friction law was deduced based on a complete similarity assumption of the mean velocity profile in both the inner and the outer flow regions [21]. In spite of the incorrect assumption made to derive the logarithmic friction law, it is often used to obtain the wall skin friction data with acceptable accuracy, however, for high enough Reynolds numbers, i.e. . The Reynolds number value proposed, i.e. , might be considered as a lower limit for the validity of the logarithmic law of the wall [17, 10].
On the other hand, for the variation of the smooth pipe wall friction versus the Reynolds number is well represented by the power skin friction relation proposed by [22]:
(11) 
Figure 4(b) illustrates a satisfactory agreement of the present experimental data for with the above power friction relation. Note also that this power formula is in line with the Reynolds numberdependent power law representation of the mean velocity profile proposed in the overlap region by [23], and recently by [24].
2.3 Direct measurements of the shear momentum transport
To obtain the shear momentum transport (), it is either to model or to measure simultaneously the streamwise (), and the wall normal () velocity fluctuations. Hence, the purpose of this section is to report about 2D velocity measurements using dualsensor probe (xprobe) over a wide range of Reynolds number in two pipe flow facilities, namely, CoLaPipe and CICLoPE. A directional calibration of the hotwire xprobe was carried out using external calibration units, Dantec Dynamics Streamline 90H02, and TSI Air Velocity Calibrator Model 1127 for the CICLoPE, and the CoLaPipe facilities, respectively. The calibration was carried out within a velocity range of 0.5–50 m/s and for a calibration angle () of range . Voltage output from both wires of the dual probe, the mean air flow velocity, and the air flow temperature were recorded for a wide range of probe angles and flow speeds. In case of an unavoidable temperature drift, instantaneous corrections were carried out during the calibration procedure as well as during measurements for temperature drifts not greater than . Utilizing the output from the xprobe and the effective cooling velocities for both wires, the sum and difference method [25], was implemented to obtain the streamwise, , and the wallnormal, velocity components, further details about the calibration procedure can be found in Ref. [16]. Two different configurations were used for the xprobe profiles. Wires in the first setup were located in plane, to measure the streamwise, , and the wallnormal/radial, , velocity components. In the second configuration, wires were located in streamwiseazimuthal, i.e. plane, to obtain the streamwise, , and spanwise, , components. However, results presented here are limited to the streamwise, and the wallnormal, i.e. , components, respectively.
To assess the direct estimation of the Reynolds shear stress (), results from measurements performed in the CoLaPipe and the CICLoPE pipe facilities are presented in both Figs. 5 and 6 for four different Reynolds numbers in CoLaPipe and five different Reynolds numbers in CICLoPE, see Table 1.
[mm]  [m/s]  
CoLaPipe  
3389  1.25  5  250  0.5239  44  30  120 
7307  1.25  5  250  1.1310  96  13  120 
11258  1.25  5  250  1.7420  148  8  120 
16200  1.25  5  250  2.5050  213  6  120 
CICLoPE  
6566  0.7  2.5  280  0.22210  10.20  68.80  160 
14671  0.7  2.5  280  0.50071  23.01  30.52  140 
22836  0.7  2.5  280  0.78086  35.86  19.57  120 
31283  0.7  2.5  280  1.06231  48.79  14.38  100 
38090  0.7  2.5  280  1.29173  59.32  11.83  80 
^{a} hot wire length, ^{b} hot wire diameter, ^{c} viscous length scale, ^{d} sampling time.
It is worth noting that all hotwire measurements were performed in a pipe section located at , where flow was ensured to be fully developed [26]. Measurements in CICLoPE pipe facility have been carried out using Dantec Streamline 90N10 CTA with custommade Platinum xwire probe, having diameters of 2.5 , and wire length of , providing an aspect ratio of [27, 16]. On the other hand, in CoLaPipe facility, all measurements have been conducted using Dantec MultichannelCTA 54N81 with commercial Dantec 90Â° xwire probes, Model 55P53. The sampling frequencies were set to 60 kHz 20 kHz with a lowpass filter at =30 kHz 10 kHz, for the CICLoPE and CoLaPIpe, respectively. The closest achievable point to the pipe wall in both facilities was approximately 50 wall units, i.e. due to geometrical constraints imposed by the xwire probe, and the pipe surface curvature [16].
For , utilizing Dantec xwire probes, the CoLaPipe data are presented in Fig. 5. Figure 5 depicts also predictions of , utilizing equation (8), for Reynolds numbers similar to experiments. The figure presents the innerscaling of the Reynolds shear stress data for four different Reynolds numbers. It is noteworthy that the data presented are not corrected for wire length () effect, where for the present working range, see Table 1. The directly measured turbulent shear stress data in Fig. 5 are on average to in agreement with predictions made using Panton’ stress relation. This observation turns to be in agreement with earlier findings made by Refs. [14, 28], utilizing hotwire anemometry. Laufer [14] observed that the total shearing stress obtained from direct fluctuation measurements was approximately 20 lower than that computed values from the mean velocity and the meanpressure gradient measurements.
Recently, GadelHak and Bandyopadhyay [11] concluded also that the directly measured turbulence shear stress is on average 10 smaller than the theoretical distribution deduced from the momentum balance and the mean flow data, in agreement with Ref. [12] who found uncertainty in estimating the Reynolds shear stress from the direct measurements, see also Ref. [13]. Similar results are obtained using the CICLoPE pipe facility with custommade xwire probes, having even better resolution for data presented in Fig. 6. With respect to those measurements in the CICLoPE facility, it should be noted – as highlighted in the source for these data [27] – that there is a clear discrepancy between the friction velocities deduced from the fit to the linear profile in the viscous sublayer, and the ones obtained through the Reynolds shear stress and pressuredrop. This discrepancy calls for further measurements probing the consistency between the different measurement methodologies as well as studies as the present one.
In the wall vicinity, i.e. below 50 wall units, no simultaneous measurements of both the streamwise, , and the wallnormal, , velocity fluctuations have been carried out due to difficulties in positioning the measuring probes with respect to the wall surface. Even away from the wall, discrepancies between the direct measure and prediction of the Reynolds shear stress were observed that might be due to uncertainties in calibrating the xwire probe, temperature drift, and locality of the wall shear stress; cf. Ref. [27] for the latter point. Moreover, movement of the xwire probe causes a considerable change in the effective angle of the wire to the mean flow, and hence a large change in velocity sensitivity [28]. One more variable that would be of considerable importance is the difficulty in identifying the zero position of the xwire probe with respect to the pipe wall.
2.4 Indirect measurements of the shear momentum transport
Equation (2) underlines a direct relationship between gradients of the viscous shear stress [], and the turbulent momentum transport [] in balance with the gradient of the mean pressure . Hence, it forms a good basis to obtain information about , in principle, from simple gradient measurements of both the streamwise mean velocity component, , and mean pressure, , along the pipe test section. Thus, accurate measures of the streamwise mean pressure, and mean velocity distributions are needed to estimate, indirectly, the Reynolds shear stress.
A single hot wire probe simultaneously with onedimensional laser Doppler velocimetry were used to measure the streamwise mean velocity at different wallnormal locations for various Reynolds numbers. It is worth renoting that the normalization of both the viscous, and the Reynolds shear stresses presented in Fig. 7 have been carried out using the wall friction velocity obtained from measurements of the mean pressure gradient, and the wall normal distance normalized with the viscous length scale .
Selected samples of both normalized stresses are presented in Figs. 7(a) (b) from hot wire anemometry and laser Doppler velocimetry, respectively. Figure 7 is a semilogarithmic plot, allowing better view of the region close to the wall relative to the rest of the total shear layer. Closer inspection of the data of the viscous and the turbulent shear stresses presented in the figure demonstrates only three distinguishable layers from the earlier mentioned fourlayer structure since not enough experimental data were obtained for . Therefore, the DNS data from Ref.[29] is used to support the fourlayer analysis. Representing data in a way shown in Fig. 7, it seems that the viscous term () plays a negligible role far away from the wall, i.e., , in contrary to its importance in viscous sublayer as indicated in section (2), as wel to describe the entire flow field; see Ref. [4]. In the wall vicinity, i.e. , the figure shows that the Reynolds shear stress is negligible. The next layer represents the balance between the viscous, and the Reynolds shear stress. One might also observe that at the location of both stresses are equal, where the turbulent energy production, , shows its maximum as will be shown in Fig. 8 in section 2.5. Next to this layer, the Reynolds shear stress is dominant, showing almost a constant behavior along this layer, and therefore its gradient indicates a zero value, see Fig. 3. Along this layer, the momentum transport is mainly accomplished by flow fluctuations. It is worth noting here that in Fig. 7, the inner limit of the log layer is considered which was often used in literature as an inner limit of the logarithmic law of the wall. Close to the wall, i.e., , and for high enough Reynolds number, i.e. , good collapse of the data was observed, indicating a Reynolds number independencey. On the other hand, the data in the core region do not collapse when the inner scaling is being used, however, the decreases because of the gradient of the streamwise mean pressure.
2.5 Turbulent kinetic energy production
Both the viscous shear stress and the turbulent momentum transport terms in equation (4), i.e., (), and (), respectively, contribute to the turbulent kinetic energy production (). Thus, the distribution of the turbulent kinetic energy production can be estimated as follows:
(12) 
At the location where both the viscous and the turbulent shear stresses are of equal importance, i.e., , equation (4) turns to read as:
(13) 
As a result, equation (12) can be rewritten as:
(14) 
For high enough , as results in:
(15) 
Alternatively, in the limit of , one can see from equation (4) that
(16) 
Within the overlap region, i.e., , and for high Reynolds number the shear momentum transport () is equivalent to Panton’ stress correlation [equation (7)], i.e., Ref. [8]. Hence, the turbulent kinetic energy production can be reconstructed as a function of the stress function correlation as follows:
(17) 
Since , see Fig. 2, it follows that an inflection point exists at , corresponding to a position at which a peak in turbulent kinetic energy production occurs, i.e. .
To further focus on the Reynolds number effect, particularly, on the turbulent kinetic energy production in pipe flow, the CoLaPipe experimental data are presented in Fig. 8 over a wide range of Reynolds number .
The data presented in the figure was estimated via the shear momentum transport (), and the gradient of the streamwise mean velocity component (). The kinetic energy production results, ), were normalized with wall variables ( ), and then presented in Fig. 8 in dimensionless form versus the normalized wallnormal locations, . For high enough Reynolds number, , Fig. 8 shows a satisfactory collapse of the turbulent kinetic energy production data, and therefore a Reynolds number independence might be concluded. Good agreement with predictions as well as with recent DNS data by [29] is observed.
It was concluded, e.g., by Marusic el al. (2010) that the dominant kinetic energy production occurs within the viscous buffer layer at , in particular, at low Reynolds numbers. This would also be justified in Fig. 8, where a peak value was obtained within the viscous sublayer at a fixed distance from the wall, i.e., . This turns to be in good agreement with the above analysis, and with Refs. [31, 13, 11, 18] and more recently with Ref. [32]. A peak value of approximately 0.25 in the turbulent kinetic energy production is observable in Fig. 8, confirming equation (15). The position of the peak in the kinetic energy production coincides with the wallnormal position at which the turbulent and the viscous shear stresses are equal.
Very small variation in the production curves, however, was observed by Panton (2001) by increasing Reynolds number, see also inset in Fig. 8. Therefore, Marusic et al. (2010) stated that it should also be noted that production curves, when plotted semilogarithmically as shown above in Fig. 8 might result in a wrong conclusion that the log region is not contribution much to the global or the bulk kinetic energy production. By looking at the inset of Fig. 8 in agreement with Marusic et al. (2010), it highlights remarkable increase in production contribution due to the inertial sublayer as the Reynolds number increases.
Hence, Marusic et al. (2010) proposed a new and more representative graphical figure, i.e. a premultiplied plot, where equal areas would indicate equal integral contributions when using semilogarithmic axes. This is shown in Fig. 9 to more highlight the contribution of the logarithmic region to the bulk energy production for sufficiently high Reynolds number. The figure is a good summary of the premultiplied turbulent energy production measured in CoLaPipe for shear Reynolds number in the range . By looking at the premultiplied energy production data presented in Fig. 9, it appears immediately that the logarithmic region contributes most to the bulk energy production, in particular, at sufficiently high Reynolds numbers. This turns out also to be in good agreement with similar conclusion made by Smits et al. [2]. The figure also shows a plausible agreement between the DNS data [29] and one of the lower Reynolds number experimental data.
3 Conclusions and Final Remarks
Recent experimental data for fully developed turbulent pipe flows have been documented in light of two measuring techniques and two independent experiments, for shear Reynolds numbers in the range , resulting in:
Discrepancies observed between the direct and indirect measures of the Reynolds shear stress attributed to uncertainties in calibrating the xwire probe, temperature drift, and unavailability of the local wall shear stress. Difficulty in identifying the zero position of xwire probe with respect to the pipe wall is to be considered.
A significant dynamical viscous effect is observed in the wall layer extending hundreds of wall units beyond the buffer layer depending on the Reynolds number. The present analysis also reinforces the universality of viscous stress gradient to the Reynolds stress gradient within both the extended buffer layer and the pipe core region.
The present set of data showed accurate evaluation for Panotn’ stress correlation in pipe flow, predicting the Reynolds shear stress, and kinetic energy production.
Examining the premultiplied kinetic energy production indicates a dominance of the inertial sublayer, contributing most to the bulk turbulent energy production, in particular, at high Reynolds number. The lower Reynolds number experimental data showed plausible agreement with the most recent DNS data Ref. [29].
The present assessment of the existing high Reynolds number pipe flow data also underlines the importance of measuring the Reynolds shear stress and friction velocity, i.e. the wall shear stress, directly, i.e. independent of each other, in order to utilize the momentum balance to ascertain the quality of the data.
Acknowledgement(s)
This work is supported by the Priority Program SPP 1881 Turbulent Superstructures of the Deutsche Forschungsgemeinschaft under grant no. EG100/241.
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