2.1 Facilities

The present data were collected as part of a collaborative effort between the authors at the Center for International Collaboration in Long Pipe Experiments (CICLoPE) and the Flow Physics Facility (FPF) – respectively the largest-scale turbulent pipe flow and zero pressure gradient boundary layer facilities in existence. The former is a closed-loop system that generates a fully developed turbulent pipe flow in a 90 cm diameter test section over a development length of 110.9 m (i.e. a length-to-diameter ratio of 123.2). The loop includes a heat exchanger which keeps the flow temperature constant to within $\pm 0.2\,^{\circ }\text{C}$ , even for measurement durations in excess of 9 h. A detailed design of CICLoPE can be found in Talamelli et al. (Reference Talamelli, Persiani, Fransson, Alfredsson, Johansson, Nagib, Rüedi, Sreenivasan and Monkewitz2009), and initial velocity measurements are reported in Örlü et al. (Reference Örlü, Fiorini, Segalini, Bellani, Talamelli and Alfredsson2017). The FPF, first characterized in Vincenti et al. (Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013), is an open circuit zero pressure gradient wind tunnel in which the boundary layer grows continuously over a streamwise development length of 72 m, ultimately achieving boundary layer heights of up to 75 cm. The spatial development of the boundary layer over this long fetch permits the outer flow scale to be set to any value up to the maximum by establishing a fixed measurement station at the corresponding streamwise location. The friction velocity at any streamwise location is constant to within within 0.5 % for the central 5 m of the total 6 m test section span, while the sloped ceiling maintains the free-stream velocity as constant to within $\pm 1\,\%$ over the range used herein (Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013).

Both facilities are ideal for high-fidelity measurements of high Reynolds number flows, as their physical size allows for the generation of a wide range of energy-containing scales without the smallest of those being unresolvable via conventional measurement techniques. The two facilities are also particularly well suited for direct flow comparisons with one another, as the operational flow speeds and physical dimensions make it possible to simultaneously match both inner and outer flow scales at considerable Reynolds numbers.

It is worth noting that the open-circuit design of the FPF presents additional experimental challenges relative to smaller, indoor (or closed-loop) facilities. As the inflow is drawn from the atmosphere, compensation is needed for the calibration drift associated with changes in atmospheric temperature over the course of each measurement. The FPF data also show slight departures from canonical behaviour in the wake of the generated boundary layer (e.g. see Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013). Although we do not believe that these factors impact the conclusions of this study, additional boundary layer measurements collected in the High Reynolds Number Boundary Layer Wind Tunnel (HRNBLWT) at the University of Melbourne (e.g. see Kulandaivelu Reference Kulandaivelu2012) are included in appendix A for comparison. The HRNBLWT is an indoor open-circuit ZPG boundary layer wind tunnel with a streamwise development length of 27 m, which allows for generation of a boundary layer up to 35 cm thick. As such, to achieve matched spatial resolution with the FPF and CICLoPE measurements, the HRNBLWT measurements are collected at Reynolds numbers approximately $2/3$ as large as those obtained at the FPF and the CICLoPE.

2.2 Measurement probe

All of the data presented herein were acquired via a multi-element hot-wire anemometry probe consisting of 8 independent sensing elements. The design of this probe and its capacity to capture key aspects of the velocity and vorticity time series in turbulent boundary layers are discussed in detail in Zimmerman, Morrill-Winter & Klewicki (Reference Zimmerman, Morrill-Winter and Klewicki2017). The arrangement of the sensing elements, shown in figure 2, is similar to the arrangement deployed by Antonia, Zhou & Zhu (Reference Antonia, Zhou and Zhu1998) in a grid-generated turbulent flow. Several modifications were made to this design to reduce the overall measurement volume and better suit operation in wall-bounded flows. These include a reduction of the relative spacing between sub-arrays (a) and (b) to prioritize resolution of the $x_{2}$ gradients, and the use of gold-plated tungsten wire in place of platinum-core Wollaston wire.

Figure 2.

(a) Probe schematic with relative dimensions. (b) Front-on picture of actual probe. Labels (a–d) in (a) refer to $\times$ -array ‘sub-arrays’ as referenced throughout text. Probe centroid is indicated by $\oplus$ . Reference length $l_{w_{p}}$ is the sensor length $l_{w}$ projected into the $x_{2}$ – $x_{3}$ plane, which for this study is fixed at 0.8 mm.

For illustrative purposes, it is useful to describe the present probe as being composed of four individual $\times$ -wire sub-arrays. The probe schematic shown in figure 2 is consistent with this description and demonstrates one way in which both the velocity and vorticity vectors may be obtained about the centroid of the measurement volume. In contrast to some other multi-element hot-wire probes deployed in wall-bounded flows (e.g. see the review of Wallace & Vukoslavčević (Reference Wallace and Vukoslavčević2010)), the individual sub-array centroids of the present probe are symmetric about the overall measurement volume centroid. The advantage of this symmetry is that all gradient estimates (and thus vorticity component estimates) can be obtained via central finite differences about a single common point. Another advantage of the present design is the focus on resolving the vorticity vector specifically rather than the entire velocity gradient tensor. Forgoing measurement of two normal gradients ( $\unicode[STIX]{x2202}u_{2}/\unicode[STIX]{x2202}x_{2}$ and $\unicode[STIX]{x2202}u_{3}/\unicode[STIX]{x2202}x_{3}$ ) eliminates the practical requirement for each sub-array to estimate all three components of velocity simultaneously, the merits of which are evidenced by the velocity component variances reported in Zimmerman et al. (Reference Zimmerman, Morrill-Winter and Klewicki2017).

2.3 Calibration

Data collected from a two-step in situ calibration procedure are combined to characterize the response of each sensor to a range of flow angles and speeds expected to be encountered in the profile scans. In the first procedure, the sensors are traversed to a position where they will encounter quasi-uniform flow ( $x_{2}>\unicode[STIX]{x1D6FF}_{99}$ in the boundary layer and $x_{2}=0.93R$ in the pipe). Note that the mean streamwise velocity at $x_{2}=0.93R$ (the location corresponding to the maximum extent of the traversing apparatus) differs from the centreline velocity by less than 0.3 % according to the DNS dataset of Chin et al. (Reference Chin, Monty and Ooi2014). The sensors are then exposed to between 9 and 11 flow speeds ranging from approximately $1~\text{m}~\text{s}^{-1}$ at the low end to $1.25U_{o}$ at the high end, where $U_{o}$ is the velocity in either the free stream or at the pipe centreline. The flow speeds in both cases are measured by Pitot-static tubes. A third-order polynomial is then fitted to the median hot-wire voltage versus median flow speed data points. The median is used rather than the mean to remove the influence of non-uniformity in the calibrating flow, (e.g. non-zero turbulence intensity in the pipe centreline). This procedure is performed before and after every profile scan, providing two reference points for temperature-based interpolation of a single response curve for each profile measurement $x_{2}$ position. A key advantage of this procedure is that the flow incidence is known to be $0^{\circ }$ relative to the position of the probe during the actual profile scan.

The second step of the calibration procedure utilizes an in-house built articulating jet first described in Morrill-Winter & Klewicki (Reference Morrill-Winter and Klewicki2013) to generate uniform flow at both yaw and pitch angles across the same range of flow speeds as the quasi-uniform tunnel calibrations described above. Data are collected at thirteen pitch and thirteen yaw angles at each speed. Tangential cooling coefficients $k$ (Jorgensen Reference Jorgensen1971), and effective cooling angles $\unicode[STIX]{x1D6FC}$ (Bradshaw Reference Bradshaw1971), are determined for each sensor at each speed and used in the following expression to describe the sensor response:

(2.1) $$\begin{eqnarray}\displaystyle u_{e}^{2}=\underbrace{(u_{1}\sin \unicode[STIX]{x1D6FC}-u_{i}\cos \unicode[STIX]{x1D6FC})}_{u_{N}}^{2}+k^{2}\underbrace{(u_{1}\cos \unicode[STIX]{x1D6FC}+u_{i}\sin \unicode[STIX]{x1D6FC})}_{u_{T}}^{2}, & & \displaystyle\end{eqnarray}$$

where $u_{e}$ is the ‘effective’ cooling velocity and $u_{i}$ is either $u_{2}$ or $u_{3}$ (depending on the orientation of the sensor), and the subscripts $N$ and $T$ refer to the directions normal and tangential to the sensor, respectively. The original expression suggested by Jorgensen (Reference Jorgensen1971) includes another term that describes the effect of ‘bi-normal’ cooling velocity, but this term is neglected here (Zimmerman et al. Reference Zimmerman, Morrill-Winter and Klewicki2017). The use of (2.1) with speed-dependent cooling coefficients collapses the jet calibration data onto a single curve with errors typically less than 1 % for speeds above $2~\text{m}~\text{s}^{-1}$ . While the jet calibration is theoretically sufficient to describe the entire probe response, there are two practical difficulties associated with the jet calibration that necessitate the tunnel calibration (i.e. the first step). First, it was shown in Zimmerman et al. (Reference Zimmerman, Morrill-Winter and Klewicki2017) that even very minor misalignment of the jet relative to the mean flow could result in substantial errors in reported Reynolds shear stress components. Second, the fan which generates the calibrating jet flow can heat the air by several degrees Celsius, resulting in a shifted speed response curve. Both of these issues are circumvented by forcing the jet calibration surface to fall upon (at $0^{\circ }$ nominal flow incidence) the tunnel calibration curve.

2.4 Data reduction

Raw sensor output is reduced to velocity and shear gradients about the centroid of the measurement volume by solving two closed four-equation four-unknown systems. These systems, obtained from (2.1), and given in (2.2) and (2.3), are first-order Taylor series expansions of two-dimensional velocity in the direction normal to the velocity plane for each set of four coplanar sensing elements.

(2.2) $$\begin{eqnarray}\displaystyle u_{e_{j}}^{2} & = & \displaystyle \left[\left(u_{1}+h_{j}\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{2}}\right)\sin (\unicode[STIX]{x1D6FC}_{j})-\left(u_{3}+h_{j}\frac{\unicode[STIX]{x2202}u_{3}}{\unicode[STIX]{x2202}x_{2}}\right)\cos (\unicode[STIX]{x1D6FC}_{j})\right]^{2}+\cdots \nonumber\\ \displaystyle & & \displaystyle k_{j}^{2}\left[\left(u_{1}+h_{j}\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{2}}\right)\cos (\unicode[STIX]{x1D6FC}_{j})+\left(u_{3}+h_{j}\frac{\unicode[STIX]{x2202}u_{3}}{\unicode[STIX]{x2202}x_{2}}\right)\sin (\unicode[STIX]{x1D6FC}_{j})\right]^{2},\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle u_{e_{j}}^{2}=\left[\left(u_{1}+h_{j}\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{3}}\right)\sin (\unicode[STIX]{x1D6FC}_{j})-\left(u_{2}+h_{j}\frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}x_{3}}\right)\cos (\unicode[STIX]{x1D6FC}_{j})\right]^{2}+\cdots & & \displaystyle \nonumber\\ \displaystyle k_{j}^{2}\left[\left(u_{1}+h_{j}\frac{\unicode[STIX]{x2202}u_{1}}{\unicode[STIX]{x2202}x_{3}}\right)\cos (\unicode[STIX]{x1D6FC}_{j})+\left(u_{2}+h_{j}\frac{\unicode[STIX]{x2202}u_{2}}{\unicode[STIX]{x2202}x_{3}}\right)\sin (\unicode[STIX]{x1D6FC}_{j})\right]^{2}. & & \displaystyle\end{eqnarray}$$

Equation (2.2) is thus appropriate for the four wires oriented in sub-arrays $a$ and $b$ , and (2.3) is appropriate for those in $c$ and $d$ . Initial guesses are produced from $\times$ -array outputs of sub-arrays $a$ , $b$ , $c$ and $d$ from figure 2. These initial $\times$ -array outputs are obtained via lookup tables that are populated based on (2.1) and the third-order polynomial fit relating voltage to $u_{e}$ mentioned above. Note that the index $j$ in (2.2) and (2.3) indicates an individual sensing element (1–8), and $h_{j}$ indicates the separation between sensor ‘ $j$ ’ and the centroid in the direction of the Taylor series expansion. For example, if elements 1 and 2 are in sub-array $a$ , then $h_{1}=(3/4)l_{w_{p}}$ and $h_{2}=(1/4)l_{w_{p}}$ (see figure 2).

At each measurement time instant, we obtain $u_{e_{j}}$ for each wire based on its voltage output and $k_{j}$ , and $\unicode[STIX]{x1D6FC}_{j}$ based on the initial $\times$ -array solution. These values are substituted into (2.2) and (2.3) along with the initial $\times$ -array solutions for the velocities and their gradients, and the systems are solved via an iterative nonlinear least-squares algorithm. The output of this solution method is the three velocity components and the four cross-stream shear gradients $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{2}$ , $\unicode[STIX]{x2202}u_{3}/\unicode[STIX]{x2202}x_{2}$ , $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{3}$ , and $\unicode[STIX]{x2202}u_{2}/\unicode[STIX]{x2202}x_{3}$ . The advantage of the systems given by (2.2) and (2.3) compared to a typical $\times$ -array method is that the assumption of uniform flow across $\times$ -wire pairs is relaxed to one that allows a linear velocity gradient across the measurement volume. This reduces the aliasing associated with non-uniform flow across the $\times$ -wire domain (Zimmerman et al. Reference Zimmerman, Morrill-Winter and Klewicki2017). The remaining shear gradients not given by (2.2) or (2.3) (i.e. those taken in the streamwise direction) are estimated via Taylor’s frozen turbulence hypothesis, using the local mean velocity as the convection velocity. These six shear gradients are then used to compute all three instantaneous vorticity components.

All statistics presented herein are computed from velocity time series obtained via (2.2) and (2.3) with the exception of the statistical moment profiles of $u_{2}$ and $u_{1}u_{2}$ , which are obtained via the same lookup-table approach as is used to produce initial guesses for (2.2) and (2.3). Although solving (2.3) produces a higher-fidelity estimate of $u_{2}$ about the probe centroid than any linear combination of outputs from sub-arrays $c$ and $d$ , there is no particular benefit associated with computing $u_{2}$ statistics from a centralized measurement. The probe-centred signal is, however, preferable for the calculation of the $\unicode[STIX]{x2202}u_{2}/\unicode[STIX]{x2202}x_{1}$ component of $\unicode[STIX]{x1D714}_{3}$ (for example), since the collocation of the $\unicode[STIX]{x2202}u_{2}/\unicode[STIX]{x2202}x_{1}$ and $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{2}$ gradient estimates significantly improves the fidelity of the overall $\unicode[STIX]{x1D714}_{3}$ estimate (Zimmerman et al. Reference Zimmerman, Morrill-Winter and Klewicki2017).

2.5 Measurement parameters

The relevant measurement parameters of this study are summarized in table 1. The friction velocity $U_{\unicode[STIX]{x1D70F}}$ in the boundary layer case is determined by the composite fit of Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2009) with the von Kármán and intercept constants chosen as $\unicode[STIX]{x1D705}=0.39$ and $B=4.3$ , respectively (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). An analogue of the standard $\unicode[STIX]{x1D6FF}_{99}$ is used due to the slight non-zero curvature of the mean velocity profile in the FPF boundary layer free stream (cf. figure 3 in the next section). Since the inner-normalized streamwise velocity variance profile as a function of $x_{2}/\unicode[STIX]{x1D6FF}$ follows a single $Re$ -independent curve in the vicinity of $\unicode[STIX]{x1D6FF}_{99}$ , at least over the $Re$ -range considered herein (e.g. see Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015), $\unicode[STIX]{x1D6FF}_{99}$ is identified in the present ZPG cases as the position where the inner-normalized streamwise variance equals 0.257 – its value at the position corresponding to $U=0.99U_{o}$ based on the DNS results of Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013). Throughout the rest of this text, $\unicode[STIX]{x1D6FF}_{99}$ will refer to this analogue definition. The friction velocity in the pipe is obtained from direct measurements of the pressure drop using 18 ports located along the entire working section. Wall position in both measurements is first determined with a microscope to within $\pm 0.1$  mm and subsequently tracked via an optical encoder on the traversing apparatus. Integration of the measured mean velocity profiles yields the average, or bulk velocity, from which (along with the measured mean pressure gradient) the friction factor $\unicode[STIX]{x1D706}$ and Reynolds number $Re_{D}$ based on pipe diameter and bulk velocity are obtained. The present measured values of $\unicode[STIX]{x1D706}$ (for each measured $Re_{D}$ ) are all within 0.75 % of those based on the curve suggested by McKeon et al. (Reference McKeon, Swanson, Zagarola, Donnelly and Smits2004) at the corresponding values of $Re_{D}$ .

Also summarized in table 1 are the numerical data sets used for comparison. These include the boundary layer DNS of Sillero et al. (Reference Sillero, Jiménez and Moser2013), the pipe DNS of Chin et al. (Reference Chin, Monty and Ooi2014) and a computer simulation of our probe when exposed to the six DNS flow volumes made available by Sillero et al. (Reference Sillero, Jiménez and Moser2013). This simulation, or ‘synthetic experiment’, seeks to predict the effects of physical scale, probe geometry and data reduction method on each measured statistic. More detail on the synthetic experiment is available in Zimmerman et al. (Reference Zimmerman, Morrill-Winter and Klewicki2017). Statistics from the fields of Sillero et al. (Reference Sillero, Jiménez and Moser2013) that are not published online, such as velocity fluctuation kurtosis and vorticity skewness/kurtosis, are computed from the six available fields, and so may not be fully converged. Pipe DNS statistics are limited to only those which were published in Chin et al. (Reference Chin, Monty and Ooi2014). Pipe synthetic experimental ‘results’ are not computed directly, but rather we normalize the pipe DNS statistics with the ratio of boundary layer simulation statistics to boundary layer DNS statistics.

Table 1.

Summary of present experiments and DNS comparisons. Boundary layer and pipe DNS respectively from the datasets of Sillero et al. (Reference Sillero, Jiménez and Moser2013) and Chin et al. (Reference Chin, Monty and Ooi2014).

$^{a}$ Measurement sample times correspond to all samples for each case, with the exception of four selected $x_{2}$ locations in the pipe cases for which longer samples were collected – these longer sample times (given in parentheses) correspond to the centremost location, and (near) the start, middle and end of the log layer (i.e. $x_{2}^{+}=0.93\unicode[STIX]{x1D6FF}^{+}$ , ${\approx}2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ , ${\approx}0.15\unicode[STIX]{x1D6FF}^{+}$ and ${\approx}(2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}\times 0.15\unicode[STIX]{x1D6FF}^{+})^{1/2}$ ).

$^{b}$ Pipe synthetic experiment based on boundary layer results, see text for details. Outer scale $\unicode[STIX]{x1D6FF}$ refers to pipe radius or the analogue to $\unicode[STIX]{x1D6FF}_{99}$ (defined in § 2.5), where applicable.

3.1 Streamwise

Figures 3(a) and 3(b) show the mean streamwise velocity $U_{1}$ in log-law and defect form, respectively. As noted in § 2.5, $U_{\unicode[STIX]{x1D70F}}$ is determined for the boundary layer by fitting the measured points to the composite profile of Chauhan et al. (Reference Chauhan, Monkewitz and Nagib2009) with a von Kármán constant $\unicode[STIX]{x1D705}=0.39$ and an intercept $B=4.3$ . Thus, agreement between the boundary layer cases and the boundary layer DNS of Sillero et al. (Reference Sillero, Jiménez and Moser2013) in the log layer is essentially prescribed. It is for this reason that the exact slopes of the profiles are not compared herein. Plotting the mean velocity in defect form, as in figure 3(b), reveals slight departures from canonical behaviour in the FPF boundary layer cases. For the purposes of this plot, the boundary layer free-stream velocity $U_{o}$ is chosen to force the log-law portions of each curve to lie on the expected ZPG boundary layer curve. This reveals that the two lower- $Re_{\unicode[STIX]{x1D70F}}$ boundary layer measurements show good agreement in the wake with the DNS of Sillero et al. (Reference Sillero, Jiménez and Moser2013) out to $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.8$ , at which point $U_{1}$ reaches a maximum and begins to decrease. In addition to exhibiting the same local maxima in $U_{1}$ , the wake of the highest- $Re_{\unicode[STIX]{x1D70F}}$ boundary layer case is also slightly weaker than that of a canonical ZPG boundary layer wake. In spite of these departures from the expected $U_{1}$ profile shape, the turbulence statistics shown in the figures that follow (i.e. figures 4–11) show close agreement with the boundary layer DNS as well as with the (lower- $Re_{\unicode[STIX]{x1D70F}}$ ) measurements collected at the HRNBLWT (shown in figures 15–19 in appendix A), even for $x_{2}/\unicode[STIX]{x1D6FF}_{99}>1$ . The present pipe measurements are virtually indistinguishable from the DNS curve of Chin et al. (Reference Chin, Monty and Ooi2014) through the log layer and wake. Likewise, the HRNBLWT $U_{1}$ profiles shown in figure 15 in appendix A show very close agreement with the DNS curve of Sillero et al. (Reference Sillero, Jiménez and Moser2013).

Figure 3.

(a) Mean streamwise velocity in log-law form. Experimental profiles plotted as solid lines capped by symbols for clarity. Symbols at start/end of each line correspond to table 1. (b) Mean streamwise velocity in defect form, with $U_{o}$ chosen for the boundary layer cases to force agreement (for illustrative purposes) with log law (see text for details).

Figures 4(a), 4(b), and 4(c), respectively show the variance, skewness coefficient and kurtosis coefficient profiles of the fluctuating streamwise velocity $u_{1}$ for all present data and DNS. Apart from the expected difference in $\overline{u_{1}^{2}}^{+}$ near $x_{2}/\unicode[STIX]{x1D6FF}\approx 1$ , the $u_{1}$ variance profiles do not exhibit any systematic differences that are distinguishable with the present dataset. The similarity between the $\overline{u_{1}^{2}}^{+}$ profiles of the two flows is in agreement with Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009), who presented streamwise velocity statistics up to the fourth order for pipe, channel and boundary layer flows. The HRNBLWT $u_{1}$ variance profiles, shown in figure 16(a) in appendix A, are also very similar in shape to the pipe profiles, although they do indicate that the boundary layer features slightly higher $u_{1}$ variance than the pipe over the range $0.3\gtrsim x_{2}/\unicode[STIX]{x1D6FF}\gtrsim 0.8$ . The difference being slight, however, means that this conclusion is particularly sensitive to the choice of outer length scale for the boundary layer. Still, it will be shown below that the enstrophy and the other components of the Reynolds stress tensor are all higher in the boundary layer than the pipe over approximately the same range.

Figure 4.

(a–c) Fluctuating streamwise velocity variance, skewness and kurtosis, respectively. Logarithmic line of Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) (i.e. $\overline{u_{1}^{2}}/U_{\unicode[STIX]{x1D70F}}^{2}=1.95-1.26\log (x_{2}/\unicode[STIX]{x1D6FF}_{99})+\log (1.15)$ ) included in (a) for reference (note the additional $\log (1.15)$ constant accounts for differing definitions of $\unicode[STIX]{x1D6FF}$ ). The dashed lines in (b) and (c) represent the Gaussian values of the plotted statistics.

The streamwise velocity skewness coefficient profiles of the pipe and boundary layer are very similar from the wall until at least $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.5$ . The profiles of both flows grow more negative at a rate that is approximately logarithmic across the sub-domain where the mean velocity is logarithmic. Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009) also observed this similarity out to $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.5$ , but remarked that ‘it could be argued that the boundary layer skewness exhibits a slightly different trajectory for $x_{2}^{+}\gtrsim 200$ ’. Indeed, the boundary layer $u_{1}$ exhibits (on average) slightly less negative skewness than the pipe in the vicinity of $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.1$ . The same is clearly true of the boundary layer data acquired in the HRNBLWT, as shown in figure 17 in appendix A. This difference is part of a general trend: the boundary layer exhibits higher variance and probability density functions that are less dominated by extreme events for most measured quantities near the outer edge of the log layer/wallward edge of the wake. It is surmised in § 5 that this results (at least in part) from the difference between intermittency associated with the turbulent/non-turbulent interface (TNTI) in the boundary layer and that associated with the turbulent/quiescent-core interface in the pipe.

As with the skewness coefficient profiles, the pipe and boundary layer kurtosis profiles remain very similar moving outward from the wall until the emergence of a super-Gaussian peak in the wake of the boundary layer that far exceeds the more modest peak in the pipe (see figure 4 c). The HRNBLWT measurements, shown in figure 18 in appendix A, suggest that the boundary layer kurtosis is slightly lower than that of the pipe in the range $0.3\gtrsim x_{2}/\unicode[STIX]{x1D6FF}\gtrsim 0.5$ . Again, this is consistent with the differences between the intermittency in the pipe and boundary layer. Both profile sets remain sub-Gaussian from the nearest wall measured points until approximately $0.5\unicode[STIX]{x1D6FF}$ . The pipe centreline kurtosis is higher in the lowest- $Re_{\unicode[STIX]{x1D70F}}$ case than the two higher- $Re_{\unicode[STIX]{x1D70F}}$ cases. This higher value is observed individually by all sub-arrays, and at a position where the time-record length exceeded 25 000 radius turnover times. Thus, it is unlikely that this observation results from spurious probe behaviour or insufficient statistical convergence. As no two pipe measurements are collected at the same $Re_{\unicode[STIX]{x1D70F}}$ or sensor resolution, the cause of the difference in centreline kurtosis is left unclear.

3.2 Wall normal

Figure 5.

(a–c) Wall-normal velocity variance, skewness and kurtosis, respectively. Shaded region in (a) corresponds to the range of inner log-layer boundaries (i.e. $x_{2}^{+}\approx 2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ ) for the present experiments. Dashed black lines indicate Gaussian values of the plotted statistics.

Figure 5 shows the variance, skewness and kurtosis profiles of the wall-normal velocity component. Based on the DNS-based synthetic probe predictions for the experimental data in figure 5(a), the wall-normal velocity variance is expected to suffer noticeable attenuation much farther from the wall than the streamwise velocity variance. Thus, while a slight positive slope is observed in the pipe $u_{2}$ variance across the domain where the mean velocity is logarithmic, it is unlikely that this trend would be observed in the absence of spatial filtering. Still, if either the absolute or proportional attenuation of the boundary layer and pipe $u_{2}$ signals are equal, the present data indicate that the $u_{2}$ variance differs between the two flows starting at least at the inner edge of the log layer. The same conclusion is reached via inspection of the HRNBLWT data, as shown in figure 16 in appendix A. According to both the experimental and DNS data, the difference in profiles is the most pronounced in the outer region $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.2$ . At this location, the boundary layer case exhibits an outer peak that, according to Morrill-Winter et al. (Reference Morrill-Winter, Klewicki, Baidya and Marusic2015), continues to grow with increasing $Re_{\unicode[STIX]{x1D70F}}$ . The present pipe data does not exhibit an outer peak or any obvious trend with $Re_{\unicode[STIX]{x1D70F}}$ , and is of considerably lower magnitude than the boundary layer cases, in agreement with the findings of Jiménez & Hoyas (Reference Jiménez and Hoyas2008), which were observed at lower $Re_{\unicode[STIX]{x1D70F}}$ .

Figure 6.

(a,b) Spanwise/azimuthal velocity variance and kurtosis profiles. Note the inclusion of the channel DNS variance profiles of Hoyas & Jiménez (Reference Hoyas and Jiménez2006) and Lee & Moser (Reference Lee and Moser2015), indicated by ‘

’ and ‘

’, respectively. Log lines in (a) correspond to $\overline{u_{3}^{2}}/U_{\unicode[STIX]{x1D70F}}^{2}=1.52-0.34\log (x_{2}/\unicode[STIX]{x1D6FF}_{99})$ for the boundary layer measurements and $\overline{u_{3}^{2}}/U_{\unicode[STIX]{x1D70F}}^{2}=1.11-0.42\log (x_{2}/\unicode[STIX]{x1D6FF})$ for the pipe measurements. Shaded region in (a) corresponds to the range of inner log-layer boundaries (i.e. $x_{2}^{+}\approx 2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ ) for the present experiments. Dashed black line in (b) corresponds to Gaussian value the kurtosis.

As with the streamwise velocity, the pipe and boundary layer skewness and kurtosis profiles for $u_{2}$ (shown in figure 5 b,c) are very similar from the near wall to the onset of the wake region. The synthetic probe results predict that combined probe filtering/aliasing effects near the wall result in positive near-wall skewness, rather than the negative values reported by the fully resolved DNS. This effect is indeed observed in both the pipe and boundary layer experimental results. Once this effect becomes negligible, the DNS, pipe and boundary layer skewness profiles show very close agreement over the domain where the mean velocity is logarithmic. In all cases, the wall-normal fluctuations have an approximately constant skewness coefficient on this domain, varying only between 0.1 and 0.15. The pipe and boundary layer cases both exhibit a positive peak in the $u_{2}$ skewness in the wake region, although the magnitude of the boundary layer peak far exceeds that of the pipe. The existence of this skewness peak in the boundary layer is not altogether surprising given the non-zero skewness in the log layer and the presumed tendency of free-stream intermittency to increase the probability density of $u_{2}\approx 0$ . Although there is no source of truly non-turbulent ‘free stream’ in fully developed pipe flow, the intermittency associated with the varying boundary of the quiescent core (Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014) would also presumably increase the probability density of $u_{2}\approx 0$ (as well as of $u_{3}\approx 0$ ), and thus produce the observed outer peak in the pipe $u_{2}$ skewness profiles. Indeed, all of the velocity fluctuation skewness and kurtosis profiles presented herein exhibit a tendency to increase in magnitude as one moves outward from $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.3$ , further supporting the existence of a quiescent core in the pipe. As with the $u_{1}$ component, the measurements of $S_{u_{2}}$ collected in the HRNBLWT (shown in figure 17 in appendix A) indicate that the boundary layer skewness is slightly more Gaussian (closer to zero) near the outer edge of the log layer/inner edge of the wake. Again, this is consistent with the differences shown in § 5 between the intermittency associated with the boundary layer TNTI and with the quiescent core in the pipe.

In contrast to the streamwise velocity fluctuations, the kurtosis of the $u_{2}$ fluctuations is super-Gaussian across the entire flow domain of both the pipe and the boundary layer. The kurtosis profiles of both flows exhibit an increase in magnitude as one moves from $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.3$ towards the centreline, although the magnitude increase in the boundary layer far exceeds that of the pipe. While the difference between the $u_{2}$ skewness profiles of the two flows is detectable as close to the wall as $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.35$ , the kurtosis profiles do not appear to rapidly diverge until $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.55$ , which is close to the point at which the $u_{2}$ variance profiles intersect. Using the ‘intermittency factor’ $\unicode[STIX]{x1D6FE}$ (defined as the time fraction of non-turbulent flow) to account for the effects of external intermittency, Schubauer (Reference Schubauer1954) found close agreement in the kinetic energy between the pipe and the turbulent portion of a boundary layer in the region above $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.6$ . The author argued based on this agreement that the distribution of turbulent energy in a pipe and turbulent portion of a boundary layer are most likely the same in this region. The fact that the point at which the boundary layer and pipe $u_{2}$ variances are equal is approximately coincident with the point at which the kurtosis profiles begin to diverge is consistent with this hypothesis (at least for $x_{2}/\unicode[STIX]{x1D6FF}\gtrsim 0.6$ ).

Figure 7.

(a–e) Reynolds shear stress mean, r.m.s., skewness coefficient, kurtosis coefficient and correlation coefficient, respectively.

3.3 Spanwise/azimuthal

As the odd moments of the spanwise/azimuthal velocity are identically zero in both pipes and boundary layers in theory (and to within experimental error in actuality), figure 6 shows only the even-moment statistics of $u_{3}$ . Similar to $u_{2}$ , the $u_{3}$ boundary layer variance profiles show a sharp outer slope change, while the slope change in the pipe profiles is less pronounced. These slope changes occur at approximately $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.3$ for both the DNS and experimental results, which is approximately coincident with the outer ‘bump’ feature in the $u_{2}$ variance profile, and is in agreement with the findings of Jiménez & Hoyas (Reference Jiménez and Hoyas2008) at lower $Re_{\unicode[STIX]{x1D70F}}$ . Further, this location is near the lower $x_{2}/\unicode[STIX]{x1D6FF}$ limit for turbulent/non-turbulent intermittency suggested by Chauhan, Philip & Marusic (Reference Chauhan, Philip and Marusic2014a and Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ). Thus, the ‘knee’ in the profile and subsequent rapid decay of $u_{3}$ variance is caused (at least in part) by an increasing time fraction of signal containing quasi-irrotational flow with near-zero spanwise velocity as one moves above $x_{2}/\unicode[STIX]{x1D6FF}\simeq 0.3$ .

That the $u_{3}$ variance is higher in magnitude (and features a different slope) in the boundary layer than in the pipe and channel below $0.3\unicode[STIX]{x1D6FF}$ is apparently a separate issue. Log lines of best fit computed from the present boundary layer and pipe $u_{3}$ variance data over the range $2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}<x_{2}^{+}<0.15\unicode[STIX]{x1D6FF}^{+}$ have slopes of $-0.34$ and $-0.42$ , respectively. The HRNBLWT $u_{3}$ variance measurements, shown in figure 16 in appendix A, feature a log-layer fit of $\overline{u_{3}^{2}}/U_{\unicode[STIX]{x1D70F}}^{2}=1.66-0.26\,\text{log}(x_{2}/\unicode[STIX]{x1D6FF})$ . If the difference between the pipe and boundary layer cases were predominantly related to flow geometry, one would expect the channel flow profile to closely resemble the boundary layer profile. Instead, the channel flow profile of Hoyas & Jiménez (Reference Hoyas and Jiménez2006) in figure 6 is virtually indistinguishable from the pipe flow profile of Chin et al. (Reference Chin, Monty and Ooi2014) at the same $Re_{\unicode[STIX]{x1D70F}}$ . The $Re_{\unicode[STIX]{x1D70F}}\approx 5200$ channel DNS of Lee & Moser (Reference Lee and Moser2015) is also shown in figure 6(a) for comparison. Although the higher $Re_{\unicode[STIX]{x1D70F}}$ DNS features slightly higher $u_{3}$ variance in the outer region than the $Re_{\unicode[STIX]{x1D70F}}\approx 2000$ pipe and channel DNS, it is still much closer to these cases than it is to the boundary layer DNS of Sillero et al. (Reference Sillero, Jiménez and Moser2013). Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010) suggested that the discrepancies between channel and boundary layer cases in both the $u_{2}$ and $u_{3}$ variance profiles are caused by higher pressure fluctuation root mean square (r.m.s.) in the boundary layer, resulting in an increased redistribution of the $u_{1}$ energy to the $u_{2}$ and $u_{3}$ components. This explanation is also appropriate for pipe flow as noted by Chin et al. (Reference Chin, Monty and Ooi2014), because the r.m.s. profiles of the pressure fluctuations in the pipe and channel also closely resemble one another, at least at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ . Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010) showed for a boundary layer and channel both at $Re_{\unicode[STIX]{x1D70F}}\approx 550$ that the difference in pressure r.m.s. in this region is due almost exclusively to the negative fluctuations, which are generally associated with the ‘cores’ of vortices. In support of this notion, it will be shown in §§ 4 and 5 that the boundary layer features higher mean enstrophy (at the resolved scales) and less quasi-irrotational flow in the same region.

Like the $u_{2}$ fluctuations, the $u_{3}$ fluctuations exhibit slightly super-Gaussian kurtosis throughout the entire flow domain of both pipes and boundary layers (see figure 6 b). The kurtosis for all three pipe profiles in the domain where the mean velocity is logarithmic remains constant at approximately 3.35 to within the scatter of the data. In contrast, all three boundary layer kurtosis profiles trend toward the Gaussian value of 3 with increasing distance from the wall within the log layer, moving from $\simeq 3.35$ to $\simeq 3.25$ . The local minimum of the boundary layer kurtosis profiles corresponds approximately to the location of the ‘knee’ in the variance profiles (see figure 6 a). The difference between the pipe and boundary layer $u_{3}$ kurtosis is also clear when using the HRNBLWT measurements as in figure 18(c) in appendix A. This feature, again, is consistent with the difference in the time fractions of ‘fully’ turbulent flow (as opposed to quiescent or non-turbulent) near the outer edge of the log layer that will be discussed further in § 5.

As is the case with $u_{2}$ fluctuations, the point at which the boundary layer and pipe $u_{3}$ variance profiles intersect is approximately coincident with the point at which the kurtosis profiles rapidly diverge, with both features occurring near $x_{2}/\unicode[STIX]{x1D6FF}\approx 0.6$ . Again, this is consistent with the hypothesis of Schubauer (Reference Schubauer1954) that the distribution of turbulent energy between the pipe and the turbulent patches of the boundary layer is the same over this region.

3.4 Reynolds shear stress

Owing in part to its strong dependence on probe alignment (Zimmerman et al. Reference Zimmerman, Morrill-Winter and Klewicki2017), the mean Reynolds shear stress is one of the more difficult statistics to accurately measure. The magnitude of all present kurtosis profiles of the instantaneous $u_{1}u_{2}$ signal reveal its highly intermittent nature, and indicate that the mean value is composed of a delicate balance of instantaneous motions that often far exceed the magnitude of the mean in both directions. That said, the present mean Reynolds stress profiles in the boundary layer (see figure 7 a) do exhibit the expected outer region deviation from the pipe cases in the same region as the observed boundary layer ‘knees’ in both the $\overline{u_{2}^{2}}$ and $\overline{u_{3}^{2}}$ profiles. This difference is shown more clearly in the HRNBLWT measurements (cf. figure 16 d in appendix A).

The slightly higher mean Reynolds stress values in the outer region of the boundary layer in figure 7(a) are matched by slightly higher signal r.m.s. boundary layer profiles compared to the pipe profiles as shown in figure 7(b). In agreement with the findings of Morrill-Winter (Reference Morrill-Winter2016) and the computation from the available DNS fields, the Reynolds shear stress signal is negatively skewed across the entire flow domain with a coefficient of $S_{u_{1}u_{2}}\approx -1.6$ in the region where the mean velocity is logarithmic (see figure 7 c). The skewness and kurtosis profiles for the pipe experiments closely resemble those of the boundary layer out to $x_{2}/\unicode[STIX]{x1D6FF}\simeq 0.5$ . As with the constituent components $u_{1}$ and $u_{2}$ , the HRNBLWT measurements indicate that the $u_{1}u_{2}$ skewness and kurtosis magnitudes are slightly lower in the boundary layer than in the pipe near the outer edge of the log layer. This is also shown by the FPF measurements (at least in a mean sense), albeit less convincingly. This slight discrepancy is likely related to the departures from the canonical ZPG wake discussed in § 3.1.

The $(u_{1}u_{2})^{\prime }$ skewness reaches a negative peak in the wake regions of both the pipe and boundary layer. As with the skewness profiles of the constituent velocity components, the peak magnitude in the boundary layer far exceeds that of the pipe. As noted above, the instantaneous fluctuating $u_{1}u_{2}$ signal is characterized by extreme events, resulting in a kurtosis greater than 10 across the entire flow domain (see figure 7 d). The pipe and boundary layer kurtosis profiles appear to match one another everywhere except in the wake region. Here, a substantial increase in the pipe profiles is outpaced by an even more substantial peak in the boundary layer profiles.

The correlation coefficient $\unicode[STIX]{x1D70C}_{u_{1}u_{2}}$ is shown in figure 7(e). While the present experimental results are of lower magnitude than the synthetic experiment and DNS, a slight decrease in magnitude is expected with increasing $Re_{\unicode[STIX]{x1D70F}}$ . This decrease in magnitude is expected since (at least) the $u_{1}$ fluctuations are known to increase in strength with increasing $Re_{\unicode[STIX]{x1D70F}}$ (albeit slowly) in the region where the mean Reynolds shear stress remains close to $-U_{\unicode[STIX]{x1D70F}}^{2}$ . The correlation coefficient in the pipe remains fairly constant over the majority of the flow domain (in logarithmic space) before turning sharply toward zero, passing through $-0.3$ at $x_{2}/\unicode[STIX]{x1D6FF}\simeq 0.7$ . In contrast, the boundary layer correlation coefficient profiles slope gently away from zero for the majority of the flow domain before turning sharply toward zero, passing through $-0.3$ at $x_{2}/\unicode[STIX]{x1D6FF}_{99}\simeq 0.8{-}1$ .

Figure 8.

(a) Inner-normalized streamwise vorticity r.m.s. and (b) kurtosis coefficient profiles. Inset to (a) depicts pipe profiles as lines without symbols and boundary layer profiles as symbols with lines for clarity.

Figure 9.

(a) Inner-normalized wall-normal vorticity r.m.s. and (b) kurtosis coefficient profiles. Inset to (a) depicts pipe profiles as lines without symbols and boundary layer profiles as symbols with lines for clarity.

Figure 10.

(a) Inner-normalized spanwise/azimuthal vorticity r.m.s., (b) skewness coefficient and (c) kurtosis coefficient profiles. Inset to (a) depicts pipe profiles as lines without symbols and boundary layer profiles as symbols with lines for clarity.

Figure 11.

(a) Inner-normalized turbulence enstrophy profiles plotted as solid lines capped by symbols for clarity. Symbols at start/end of each line correspond to table 1. (b) Ratio of turbulence enstrophy between boundary layers and pipes of approximately matched $Re_{\unicode[STIX]{x1D70F}}$ and spatial resolution. Symbol shape corresponds to spatial resolution in accordance with the trend established in table 1.