2.1. Recent large-scale experiments

For these recent datasets, we experimentally investigated moderately strong APG TBLs at high Reynolds numbers using two measurement techniques, under matched conditions. This is made possible by recent modifications of the large Melbourne wind tunnel test section (Deshpande et al. Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023), shown schematically in figure 1(a). Low-porosity screens affixed to the outlet are used to raise the test section static pressure, while air bleed slots along the ceiling are opened (solid arrows in figure 1 a) or restricted/closed (dashed arrows in figure 1 a) to create a user controlled streamwise pressure gradient profile. For each case, the inlet unit Reynolds number ( $U_{\infty }(x=0)/\nu =8.7E5$ m $^{-1}$ ) was held constant. In figure 1(a) the streamwise profiles of the pressure coefficient, $C_{P}(x) = 1-U^2_{\infty }(x)/U_{\infty }^{2}(x = 0)$ , measured for two pressure gradient cases, one nominally ZPG and one with a mild APG, are overlaid for reference.

Figure 1.

(a) Schematic of modified Melbourne large wind tunnel facility, adapted from Deshpande et al. (Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023). (b) Schematic of PIV set-up adapted from Marusic et al. (Reference Marusic, Abu Rowin, Lozier, Lindić, Zarei and Deshpande2024). Snapshots of instantaneous streamwise velocity for (c) ZPG and (d) APG cases across the full TBL, made possible by stitching individual flow fields from the four PIV cameras (C1–C4).

A single hot-wire sensor was used in the first experiments (referred to as MELB2) to measure time-resolved velocity statistics across the TBL at the selected measurement location of $x=17.5$ m. The sensor was made in-house with a diameter of $d=2.5$ µm and a nominal length of $l=0.5$ mm ( $l^{+} \approx 11$ ). The hot-wire sampling frequency was $f_{s}=50$ kHz ( $t^{+} \approx 0.3$ ) and the total sampling time ( $T_{s}$ ) was set such that $T_{s}U_{\infty }(x)/\delta _{99}\gt 20\,000$ for each case, to reach a reasonable level of statistical convergence. In each experiment, two independent profiles were acquired with unique wall-normal resolutions (i.e. unique spacings between wall-normal measurement locations). The first profile followed a traditional logarithmic spacing, with $44$ total wall-normal measurement locations beginning near the wall and terminating in the free stream (square symbols in table 2). In contrast, the second profile had $30$ linearly spaced wall-normal measurement locations restricted to the far outer region ( $0.9\lesssim z/\delta _{99} \lesssim 1.4$ ), leading to more data points near the TBL edge compared with the traditional profile (circle symbols in table 2). The combination of these profiles then allows for the evaluation of conventional measurement practices, but also provides highly resolved measurements near the TBL edge for comparison of the various TBL thickness definitions. Calibration of the hot-wire probe was performed before and after each experiment to account for ambient drift over the long measurement duration; however, hot-wire drift was confirmed to be negligible in these experiments. Due to the optical access at this streamwise measurement location, the friction velocity was obtained directly from oil-film interferometry for each case. Further details of the hot-wire measurements can be found in Marusic et al. (Reference Marusic, Abu Rowin, Lozier, Lindić, Zarei and Deshpande2024).

Complementing the hot-wire measurements, high-resolution PIV measurements (referred to as MELB1) were also conducted to capture detailed velocity fields across the entire boundary layer for both the ZPG and APG TBLs centred about a matched streamwise location of $x \approx 17.5$ m, as indicated in figure 1(a). The set-up utilised four vertically staggered Imager CX-25 cameras (figure 1 b) with $5312\times 4608$ pixel complementary metal–oxide semiconductor (CMOS) sensors and Tamron SP AF 180 mm macro lenses set at $f/11$ , achieving a digital resolution of 22 µm/pixel. The positioning of the cameras ensured complete TBL coverage, with $\delta _{99}$ for both the ZPG and APG TBLs lying within the field of view (FOV) of the middle cameras (figure 1 c). A dual-pulse Nd:YAG laser (InnoLas SpitLight Compact PIV 400) with a 2 mm thick laser sheet (corresponding to a viscous-scaled thickness of $\sim 40$ – $50$ ) illuminated the flow that was seeded with 1–2 m particles, while synchronisation was handled by a programmable timing unit (PTU X, LaVision GmbH) via DaVis 10.1 software. The final stitched FOV (figure 1 b) measured $104 \times 441$ mm² in the streamwise and wall-normal directions ( $x \times z$ ).

A two-dimensional dot target was used for camera calibration, and a minimum intensity subtraction technique enhanced image quality. Multi-pass cross-correlation was applied with a final interrogation window size of $24 \times 24$ pixels ( $0.53 \times 0.53$ mm²) and 50 % overlap, yielding viscous-scaled spatial resolutions of $18 \times 18$ for the ZPG and $11 \times 11$ for the APG case ( $x^{+} \times z^{+}$ ). A sample of the instantaneous streamwise velocity field for both the ZPG and APG case are given in figure 1(c,d), respectively. Further details of the PIV set-up can be found in Marusic et al. (Reference Marusic, Abu Rowin, Lozier, Lindić, Zarei and Deshpande2024) and Lindić et al. (Reference Lindić, Abu Rowin, Deshpande and Marusic2025), which also give details of the specific TNTI detection methodology adopted for the present study, and the challenges associated with adopting other methodologies for experimental datasets.

2.2. Published datasets

A set of published datasets were also considered, supplementing the current analysis with different experimental/numerical techniques and unique combinations of non-canonical effects. These datasets are described briefly, with the relevant citations provided for further details.

The datasets referred to as MELB3 and MELB4 in table 2 are from a series of previously published experimental studies documenting the effects of surface roughness on high friction Reynolds number ( ${\textit{Re}}_{\tau }\equiv U_\tau \delta _{99}/\nu$ ) TBLs, which were conducted in the same facility as MELB1 and MELB2. In both of these studies, surface roughness was introduced by covering the entire bottom wall of the test section with a single sheet of sandpaper. The surface roughness is quantified by an equivalent sand grain roughness Reynolds number, $k_{S}^{+}=k_{S}U_{\tau }/\nu$ . Specifically, the dataset MELB3 is a set of PIV measurements of ZPG smooth- and rough-wall TBLs documented in Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Marusic, Schultz and Klewicki2016a ). While, the dataset MELB4 is associated with hot-wire measurements of ZPG rough-wall TBLs documented in Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016b ), both of which can be directly consulted for further details on the experimental set-up.

Datasets with the prefix USNA in table 2 correspond to laser Doppler velocimetry (LDV) measurements conducted in the US Naval Academy water channel. Here, a range of pressure gradient conditions, starting from a favourable-pressure gradient (FPG) and ending with an APG, were introduced by adjusting four flat plates along the upper wall of the channel. Additionally, varying levels of surface roughness were introduced through interchangeable plates, which made up the bottom wall of the channel. Datasets USNA1, USNA2 and USNA3 respectively correspond to LDV measurements of smooth-wall pressure gradient TBLs (Volino Reference Volino2020), ZPG rough-wall TBLs (Volino & Schultz Reference Volino and Schultz2022) and rough-wall pressure gradient TBLs (Volino & Schultz Reference Volino and Schultz2023).

Additionally, two numerical simulations of canonical TBLs were also considered, supplementing the current analysis with a wider range of measurement resolutions and flow conditions. Datasets referred to as UPM and KTH in table 2 correspond to a direct numerical simulation (DNS) and a well-resolved large-eddy simulation (LES) of a smooth-wall ZPG TBL, respectively. Details of the DNS and LES are documented in Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) and Eitel-Amor, Örlü & Schlatter (Reference Eitel-Amor, Örlü and Schlatter2014), respectively.

3.1. Physical insights

An example of a representative skewness profile for a canonical TBL, demonstrating the significance of this sign change, can be seen in figure 2(a). The green symbol (and green dotted lines in figure 2 d–f) indicates the wall-normal location where the skewness changes from negative to positive, labelled as ‘B’. The red and blue symbols (labelled ‘C’ and ‘A’) indicate the wall-normal locations of the outer positive and negative peaks in the skewness profile, respectively, which bound the zero-crossing. Figure 2(b,c) demonstrates the flow phenomenology associated with the skewness of streamwise velocity fluctuations. Here the background colour represents the instantaneous streamwise velocity magnitude and the arrows represent instantaneous fluctuations in the streamwise and wall-normal velocity. Following Lindić et al. (Reference Lindić, Abu Rowin, Deshpande and Marusic2025), the local turbulent kinetic energy (LKE, $\tilde {k}$ ) in a frame of reference moving with the free-stream velocity, over a $3 \times 3$ grid is defined as

(3.2) \begin{equation} \tilde {k}=100 \times \frac {1}{9 U_{\infty }^2} \sum _{m,n=-1}^1\Big [\big (\tilde {U}_{m, n}-U_{\infty }\big )^2+\big (\widetilde {W}_{m,n}-W_{\infty }\big )^2\Big ], \end{equation}

and the instantaneous interface coordinates $[x_{i},z_{i}]$ can be defined by a contour of the LKE where $\tilde {k}(x_{i},z_{i})=0.2$ , represented by solid black lines in figure 2(b,c). Fluctuations above this interface are typically weak, and the instantaneous velocity is typically either equal to, or slightly greater than, the free-stream velocity ( $\tilde {U}\gtrsim U_\infty$ ). The latter is attributed to a local acceleration of the streamwise velocity above the evolving interface/bulges. This acceleration was observed experimentally, over a small wall-normal extent above the LKE-defined interface, in approximately 40 % of snapshots for both ZPG and APG TBLs, like those shown in figure 2(b,c). In the other snapshots, the instantaneous velocity in the region above the interface was indistinguishable from the free-stream velocity. Below the interface, there are strong turbulent fluctuations, and the instantaneous velocity is consistently lower than the free-stream velocity. While the relative position of these velocity fluctuations/features, with respect to the interface, may vary with interface detection method (e.g. LKE in this case; Lindić et al. Reference Lindić, Abu Rowin, Deshpande and Marusic2025) they are still expected to be highly correlated with the interface topology. Additionally, the commonality of these flow features between the ZPG and APG TBLs is not surprising given the qualitatively similar coherent flow structures/energy dynamics in their respective far outer regions (Lee Reference Lee2017; Deshpande & Vinuesa Reference Deshpande and Vinuesa2024). These flow features were also observed in a ZPG rough-wall TBL using the PIV dataset MELB3 from Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Marusic, Schultz and Klewicki2016a ), though not shown here for conciseness.

Figure 2.

(a) Profile of streamwise velocity skewness in the outer region of a TBL. (b,c) Instantaneous LKE interface (black lines) imposed on the instantaneous streamwise velocity field (colours) with instantaneous fluctuation vectors (arrows) overlaid. (d,e) Probability distribution of the interface location in the outer region of a TBL with contours of zero skewness overlaid. Data are from the MELB1 (a,b,d) ZPG and (c,e) APG cases. ( f) Schematic of instantaneous flow phenomenology associated with the characteristic wall-normal variation of streamwise velocity as shown in panels (a,b,c).

Figure 2( f) schematically relates the wall-normal variation in TBL flow features with the wall-normal profile of the skewness of streamwise velocity fluctuations. For instance, positive skewness (red) arises in a region where the flow is primarily free stream, with intermittent accelerations of the flow as the turbulent bulges pass below, through the far outer region. Alternatively, negative skewness (blue) appears in a region where the flow experiences free-stream flow with intermittent, turbulent, low instantaneous velocity events, i.e. events associated with the turbulent eddies within the TBL. Finally, there is a region in which the flow sharply transitions from the negative to positive skewness state with increasing wall-normal distance, creating a zero-crossing that is identified and used in (3.1) as the metric by which to estimate the boundary layer thickness, $\delta _{S}$ . In figure 2(d,e), 2000 instantaneous LKE interfaces are ensembled averaged to create a two-dimensional probability distribution for the interface, $P(x_{i},z_{i})$ , with an overlaid contour of locally zero skewness (green dotted line). While interfaces can (infrequently) exceed the contour of zero skewness, the zero-skewness contour acts as a nominal indicator of the expected uppermost extent of the interface, on average. In this way the definition of skewness proposed here is phenomenologically similar to (1.6), but does not rely on thresholds and can be readily applied to single-point measurements of the streamwise component of velocity (even retroactively). Additionally, following the theoretical framework of Phillips (Reference Phillips1955), the skewness of velocity fluctuations parallel to the mean TNTI (i.e. in a rotated frame of reference accounting for TBL growth) were also found to be positive for $z \geq {\delta _{S}}$ , akin to the skewness of streamwise velocity fluctuations considered here. This observation reaffirms that the characteristic velocity fluctuations in the far outer region of the TBL (described above) have a strong relationship with the local TNTI dynamics (rather than mean TNTI location alone). These results also suggest Phillips’ model for velocity fluctuations outside the turbulent boundary should perhaps be tested relative to the instantaneous TNTI, which is however beyond the scope of the present work.

3.2. Attached eddy modelling

The generic nature of these flow features can further be demonstrated by considering the simulations of Deshpande et al. (Reference Deshpande, de Silva, Lee, Monty and Marusic2021) that incorporated empirically obtained geometric scaling laws into the classical attached eddy model (AEM) (Perry & Chong Reference Perry and Chong1982; Marusic & Monty Reference Marusic and Monty2019) to simulate the coherent large-scale flow structures typical of ZPG TBLs. Specifically, this model consisted of a data-driven distribution of three-dimensional ‘packets’ of $\varLambda$ eddies, representing geometrically self-similar eddies and $\delta$ -scaled large-scale motions (LSMs), but did not include information related to scale-specific nonlinear interactions (Duvvuri & McKeon Reference Duvvuri and McKeon2015; Lozier et al. Reference Lozier, Marusic and Deshpande2024a ). A snapshot of the instantaneous velocity fields from one such simulation can be seen in figure 3(c) along with the skewness profiles from a set of simulations with varying ${\textit{Re}}_{\tau }$ in figure 3(a). Key characteristics of both the instantaneous velocity field from figure 2(b) (e.g. a slight acceleration above the largest eddies with lower-velocity motions below) and the skewness profile from figure 2(a) (e.g. the distinct positive and negative peaks) can also be seen in these AEM results, reaffirming the phenomenological description behind the $\delta _{S}$ definition, and its broad applicability. For comparison, scale-dependent filtering ( $u_{\textit{filtered}} = u \lbrack T^{+} \gt 350 \rbrack$ ; Lozier et al. Reference Lozier, Marusic and Deshpande2024a ) was also applied to relevant experimental data in order to isolate the contributions of LSMs to the measured instantaneous velocity fields and skewness profiles shown in figure 2. Filtered skewness profiles from Lozier et al. (Reference Lozier, Marusic and Deshpande2024a ) and a filtered PIV snapshot reproduced from figure 2(b) are shown in figures 3(b) and 3(d), respectively. Critically, the key characteristics of the instantaneous velocity field and skewness profiles are retained after filtering, and agree well with the AEM results. These results collectively reinforce the dependence of the skewness profile on the influence of LSMs (which grows with ${\textit{Re}}_{\tau }$ ), and support the physical insights that led to the establishment of this new $\delta _{S}$ definition. However, highly resolved DNS are also recommended to investigate this relationship between TNTI dynamics and skewness in greater detail (Borrell & Jiménez Reference Borrell and Jiménez2016; Zecchetto & da Silva Reference Zecchetto and da Silva2021).

Figure 3.

(a) Skewness profiles and (c) a snapshot of the instantaneous velocity field from Deshpande et al. (Reference Deshpande, de Silva, Lee, Monty and Marusic2021). (b) Filtered skewness profiles from Lozier et al. (Reference Lozier, Marusic and Deshpande2024a ). (d) Filtered PIV snapshot of the instantaneous velocity field from figure 2(b).

3.3. Comparison with other definitions

To compare past $\delta$ definitions with this new $\delta _{S}$ definition, figure 4(a) shows a diagnostic style plot (Alfredsson et al. Reference Alfredsson, Segalini and Örlü2011; Vinuesa et al. Reference Vinuesa, Bobke, Örlü and Schlatter2016) with an ensemble of published smooth-wall ZPG (i.e. canonical) TBL datasets, from both numerical and experimental studies, encompassing a broad range of wall-normal resolutions. The vertical dashed red line shows the threshold where the mean velocity is equal to 99 % of the free-stream velocity (i.e. $\delta _{99}$ ), while the horizontal blue dashed line shows the turbulence intensity threshold used to find the boundary layer thickness $\delta _{D}$ (Vinuesa et al. Reference Vinuesa, Bobke, Örlü and Schlatter2016). All the ZPG datasets appear to pass through the intersection of these two thresholds, confirming that, $\delta _{99}=\delta _{D}$ for ZPG TBLs, consistent with Vinuesa et al. (Reference Vinuesa, Bobke, Örlü and Schlatter2016). In fact, this relationship was also found to hold for the non-canonical datasets considered in this study (see table 2) and as such, we will use (1.3) to find $\delta _{99}$ , for consistency, dropping the $\delta _{D}$ terminology from here on out. Figure 4(a) also demonstrates some key limitations of these methods, associated in particular with the asymptotic nature of first- and second-order statistics in ZPG TBLs. For instance, as described in $\S$ 1, the $\delta _{99}$ definition is predicated on the assumption that the mean velocity profile monotonically approaches the free-stream velocity with increasing wall-normal distance ( $z$ ) through the outer region. While this is true for canonical TBLs (see figure 4 a), this behaviour has been shown to differ in the case of non-canonical TBLs, such as APG TBLs (Vinuesa et al. Reference Vinuesa, Bobke, Örlü and Schlatter2016; Griffin et al. Reference Griffin, Fu and Moin2021). Similarly, the diagnostic plot method also relies on the streamwise turbulence intensity decaying towards a relatively low free-stream turbulence intensity level as the wall-normal distance increases through the outer region. If the free-stream turbulence intensity is near or above the originally prescribed threshold (2 % from Vinuesa et al. Reference Vinuesa, Bobke, Örlü and Schlatter2016), determining the boundary layer thickness may require modification of the threshold accordingly. Additionally, the first- and second-order statistics do not capture all the important physics happening in the outer region, which is evident on comparing figures 2 and 4(a).

Figure 4.

Comparison of experimental and numerical ZPG TBL statistics with varying wall-normal resolutions. (a) Diagnostic style plot used to find $\delta _{D}$ following the methodology of Vinuesa et al. (Reference Vinuesa, Bobke, Örlü and Schlatter2016) (analogous to $\delta _{99}$ ). (b) Relationship between turbulence intensity and skewness of streamwise velocity fluctuations in the outer region of the TBL. Wall-normal profiles of skewness normalised by $\delta _{S}$ in (c) logarithmic scaling and (d) linear scaling. The magenta curve represents a generalised form of the normalised skewness profile (3.3) fit to DNS data from Sillero et al. (Reference Sillero, Jiménez and Moser2013).

To further demonstrate this, the vertical axis in figure 4(b) shows the streamwise turbulence intensity, like figure 4(a), however, the normalised mean streamwise velocity on the horizontal axis has been replaced with the streamwise velocity skewness. As the streamwise turbulence intensity decreases (corresponding to increasing wall-normal distance indicated by the vertical arrow), significant and non-monotonic variations in skewness are observed. This remains true even as the streamwise turbulence intensity approaches the free-stream turbulence intensity level, $ \sqrt {\overline {u^{2}_{\infty }}} \textstyle /U_{\infty }=0.003$ for the experimental datasets shown here (marked by the horizontal black dashed line in figure 4(b) inset). This indicates that there are still significant skewness-contributing events occurring near the TBL edge, as described in figure 2, which give rise to the unique profile of skewness seen in the far outer region of the TBL, and which can not be captured by first- or second-order statistics alone (figure 4 a). To this end, the wall-normal profiles of skewness for each case are shown in figure 4(c,d). Here, the wall-normal distances have been normalised using $\delta _{S}$ following (3.1). From these figures we can see that, for smooth-wall ZPG TBLs, the skewness profiles all agree reasonably well both above and below the point of $z={\delta _{S}}$ . Visually identifying the zero-crossing in the skewness profiles here is intuitive. However, for the experimental datasets, noise and/or large wall-normal spacing between points can make determination of the outer zero-crossing of the skewness profile relatively more challenging as compared with the numerical datasets. To assist with this effort, we propose fitting a model equation to the outer region of the skewness profile, to extract the location of the zero-crossing more consistently. The magenta lines in figure 4(c,d) represent a Fourier model of the skewness profile in the far outer region of the TBL, given by

(3.3) \begin{equation} \overline {\overline {u^{3}}} = a_{0} + \sum _{n=1}^{3} a_{n}\cos \left(n\omega \frac {z}{{\delta _{S}}}\right) + b_{n}\sin \left(n\omega \frac {z}{{\delta _{S}}}\right)\!, \end{equation}

which has been fitted to the DNS skewness profile within our region of interest, $0.8\leq z/{\delta _{S}} \leq 1.1$ . The resulting coefficients were: $a_{0}=-1.06, \,a_{1}=0.67, \,a_{2}=-0.01, \,a_{3}=0.08, \,b_{1}=1.82, \,b_{2}=0.23, \,b_{3}=-0.05$ and $\omega =12.73$ . The ZPG TBLs considered here all agree well with this Fourier model, and going forward we consider this Fourier model as a generalised representation of the skewness profile in the far outer region of a smooth-wall ZPG TBL. In this way we can fit all datasets considered here to this model in order to extract $\delta _{S}$ , as done in figure 4(c,d), and all subsequent analyses. Additionally, this method involving the Fourier model will be directly compared with simple linear interpolation in $\S$ 4.

Interestingly, it can be noted in figure 4(d) that the positive peak in the skewness profile, beyond the boundary layer thickness ( $z/{\delta _{S}}\gt 1$ ), is lower in magnitude for the experimental datasets as compared with the numerical datasets (and subsequently the Fourier model). This difference is likely a result of Gaussian free-stream turbulence that, in a region with some skewness-contributing events (positive or negative), will tend to bring the measured skewness towards zero. However, even with differences in the peak amplitudes, the experimental data still shows a shape that is consistent with the numerical data and the Fourier model. This also confirms that Gaussian free-stream turbulence and/or measurement noise should not change the location of the zero-crossing (and consequently $\delta _{S}$ ) since it does not contribute strong positive or negative skewness to the velocity signal. Along similar lines, the agreement of the various experimental profiles also indicates that they have reasonable statistical convergence, which is important when considering higher-order statistics such as skewness. However, it is noteworthy that while the hot-wire measurements achieve convergence temporally (i.e. through long sampling times), the PIV measurements achieve a similar level of convergence through spatial and ensemble averaging. Additionally, figure 4(d) demonstrates the effect of measurement wall-normal resolution ( $\Delta z$ ) on the accuracy of resolving the skewness profile and its zero-crossing. The experimental and numerical datasets with small wall-normal resolutions (i.e. $\Delta z^{+}\leq 125$ and $\Delta z/{\delta _{S}} \leq 0.017$ ) appear to follow the Fourier model well. Note that the hot-wire dataset with the poorest spatial resolution (square symbols) is typical of experiments with logarithmically spaced measurement points, where the distance between measurement points is large in the far outer region. However, there are still multiple points within the region of interest (i.e. $0.8 \leq z/{\delta _{S}} \leq 1.1$ ) and fitting these points to the model results in a good estimate of $\delta _{S}$ , consistent with that obtained by fitting the model to better resolved hot-wire statistics. Next, we apply our new definition of the boundary layer thickness (3.1) and compare it with past definitions used in the literature. Later, in $\S$ 4 we also demonstrate the robustness of our $\delta _{S}$ determination method (as described above) on a range of previously published, single-point experimental datasets in both canonical and non-canonical TBLs.

Figure 5.

Two-dimensional fields of relevant statistics from PIV measurements of (top) ZPG and (bottom) APG TBLs with solid contours of the TBL thickness overlaid based on (a) $\delta _{99}$ , $\varDelta _{1.25}$ , (b) $\delta _{\textit{u}w}$ , (c) $\delta _{\textit{TNTI}}$ , and (d) $\delta _{S}$ definitions. The black dotted lines overlaid in (c) represent p.d.f.s of the TNTI height (1.5).

Plots in figure 5 compare our new definition of the TBL thickness with other commonly used definitions of the boundary layer thickness (as summarised in table 1) for both ZPG and APG TBLs. It is clear that $\delta _{S}$ (shown in figure 5 d) is larger (farther from the wall) than $\delta _{99}$ (shown in figure 5 a) for both the ZPG and APG cases. However, for the ZPG case, $\delta _{S}$ agrees reasonably well with $\varDelta _{1.25}$ (i.e. $1.25\delta _{99}$ , figure 5 a), consistent with the relationship established in Baxerres et al. (Reference Baxerres, Vinuesa and Nagib2024) using a composite profile of the mean velocity. In the case of APG TBLs, however, there is currently a lack of a universal composite profile for the outer region of the mean velocity profile that could be used to find an equivalent $\varDelta$ parameter. In both the ZPG and APG cases, $\delta _{S}$ is also larger than $\delta _{\textit{u}w}$ (shown in figure 5 b) by $\approx 11\,\%$ , but it is comparable, suggesting that a slightly different threshold of $\overline {uw}$ may result in the same boundary layer thickness. Similarly, the boundary layer thickness found using the p.d.f. of the TNTI height ( $\delta _{\textit{TNTI}}$ ) agrees well with $\delta _{S}$ ( $\approx 4\,\%$ difference) for both the ZPG and APG TBLs shown in figure 5(c). These results show that this new definition of boundary layer thickness, which is motivated by characteristic TBL physics (depicted in figure 2 f), yields results comparable to previously used definitions, but without the use of thresholds.

Figure 6.

Wall-normal profiles of (a) turbulence intensity and (c) skewness of streamwise velocity from high-resolution hot-wire measurements of ZPG (in black) and APG (in red) TBLs. The p.d.f. of streamwise velocity fluctuations, as a function of wall-normal distance, in the outer region of the ZPG TBL normalised by (b) the free-stream velocity and (d) local turbulence intensity. (e) Effect of sampling time on the magnitude of skewness measured at wall-normal locations corresponding to the negative peak, zero-crossing and positive peak in the ZPG TBL skewness profile (labelled ‘A’, ‘B’ and ‘C’, respectively). ( f) Time series excerpt of the instantaneous velocity measured near the zero-crossing.

To complement the results presented in figure 5, hot-wire measurements of the outer region, with a dense linear wall-normal spacing, were conducted for both the ZPG (black) and APG TBLs (red), at conditions matched to the PIV experiments, and the results are shown in figure 6. These experiments are used to further examine the skewness profile in the outer region and demonstrate how the new $\delta _{S}$ definition can be applied to conventional single-point measurements (and retroactively to previously acquired datasets). Figure 6(a) shows the normalised turbulence intensity for both cases as a function of wall-normal distance, with $\delta _{99}$ being estimated using (1.3) (conventional threshold shown as a dashed blue line). Similarly, figure 6(c) shows the wall-normal profile of skewness for both cases, with $\delta _{S}$ estimated by fitting the skewness profiles with the Fourier model ((3.3), zero skewness is highlighted with a dash dotted green line). For the ZPG case, $\delta _{S}$ and $\varDelta _{1.25}$ are compared directly, demonstrating good agreement once again. Additionally, the relative error in the skewness measurement at critical wall-normal locations in the profile can be estimated by examining how the magnitude of skewness changes with increasing sampling time, as shown in figure 6(e). Select measurement points corresponding to the negative peak, zero-crossing and positive peak in the skewness profile have been labelled as A, B and C, respectively, following figure 2. For the point aligned with the negative peak (A), the skewness magnitude appears to be monotonically converging for sampling times greater than $\approx 8000$ large-eddy turnover times ( $\delta _{99}/U_{\infty }$ ). However, for the points near the zero-crossing and aligned with the positive peak, the skewness magnitude is highly sensitive to infrequent but significant skewness-contributing events resulting in a relatively high uncertainty even after a cumulative sampling time of $12\,000$ large-eddy turnovers. To highlight this, an excerpt of the instantaneous velocity time series, which aligns with a sharp jump in skewness magnitude in figure 6(e), is shown in figure 6( f). Even with an accumulated sampling time of nearly $7300$ large-eddy turnovers, adding this additional sample of around $100$ turnover times, which contains several large fluctuations (i.e. skewness-contributing events), results in the significant jump in skewness magnitude observed in figure 6(e). This time series excerpt also reaffirms that, near the zero-crossing, the instantaneous velocity can intermittently exceed the free-stream velocity magnitude and/or drop significantly below the local mean consistent with the observations/model presented in § 3.1 and figure 2. Despite a persistent uncertainty in skewness around the zero-crossing, the estimation of $\delta _{S}$ is accurate when fitting these profiles with the Fourier model, as seen in figure 6(b), as the fitting method considers many points between the negative and positive peaks, which mitigates this uncertainty. It should also be reiterated that while the hot-wire measurements shown here achieve reasonable statistical convergence temporally (i.e. through sufficiently long sampling times), other measurements can alternatively achieve statistical convergence through spatial and/or ensemble averaging. The p.d.f.s of streamwise velocity fluctuations across the outer region are also shown in figure 6(b,d). In figure 6(b) the velocity fluctuations are normalised by the free-stream velocity, with the extent of distribution at the positive and negative peaks (A and C) highlighted to bring out the ‘tailedness’ responsible for the skewness magnitude. In figure 6(d) the fluctuations have been normalised by the turbulence intensity to explicitly bring out the negative and positive peaks, and demonstrate the recovery of $u$ fluctuations to the expected Gaussian distribution in the free stream. It should also be highlighted that while the distribution clearly transitions from negative to positive, the distribution is not Gaussian at locations near the zero-crossing. The same has also been confirmed based on the locally large kurtosis magnitudes measured at and near the zero-crossing, which are not been shown here for brevity.

4.1. Limitations and recommendations

Despite the demonstrated robustness in the methodologies adopted to implement the new $\delta _{S}$ definition, there are several limitations that have been mentioned previously and they will be discussed in further detail here. Accordingly, minor modifications are also recommended to be implemented in future experimental/numerical studies to enable and enhance the accuracy of the estimation of $\delta _{S}$ following the methods introduced here.

The first limitation is related to high levels of measurement noise and/or free-stream turbulence intensity. Figure 10 shows the effect of Gaussian white noise added to experimental velocity time series for a high-Reynolds-number ZPG TBL (MELB2) at varying levels of signal-to-noise ratio (SNR). In figure 10(a) increasing SNR reduces the magnitude of the negative peak in the skewness and also raises the apparent free-stream turbulence intensity. In figure 10(b) it can also be seen that the artificial noise significantly reduces the magnitude of the positive peak in the skewness. However, due to the Gaussian behaviour of the artificial noise (i.e. zero skewness is contributed), there is minimal effect observed on the location of the zero-crossing, and the same $\delta _{S}$ can be extracted from the profiles with added noise, as compared with the baseline. The effect of noise can be further confirmed by considering the skewness profiles from the LDV measurements in figure 7(b,e), which have relatively high measurement noise (as compared with the baseline hot-wire measurements), and are comparable to the profiles in figure 10(b) with artificially added noise. Additionally, when free-stream turbulence is sampled in the intermittent and free-stream regions, an effect akin to adding Gaussian noise is expected on the skewness profile. This is highlighted in figure 10(b) where the compounding effects of the free-stream turbulence (although small for the baseline hot-wire case) and artificial Gaussian noise lower the magnitude of the positive peak significantly. As such, it is recommended that both measurement noise and free-stream turbulence levels be minimised so the crucial part of the skewness profile in the outer region can be resolved. It should be noted that at the other extreme, a sign change in the outer skewness profile is not observed in TBLs that have very high (induced) free-stream turbulence intensities (ranging from 8 % to 13 %) as shown in the Appendix (Hearst et al. Reference Hearst, de Silva, Dogan and Ganapathisubramani2021). At these levels of free-stream turbulence intensity, it appears that the interface physics differs from the description given in $\S$ 3 (and figure 2), thereby making the application of our $\delta _{S}$ definition unsuitable for TBL flows with very high free-stream turbulence intensity.

A second limitation is related to errors in the measurement of skewness, i.e. the convergence of higher-order statistics. For example, a relatively long sampling time is needed to achieve reasonable convergence of higher-order statistics, such as skewness, in conventional experimental measurements. This is especially true for measurement points around the skewness zero-crossing, as shown in figure 6(c,e,f), where infrequent, but significant, skewness-contributing events can alter the skewness magnitude when it is small. However, as long as this error is reasonable (i.e. much less than the magnitude of the positive/negative peaks in the skewness), minimal adverse effects are expected on the estimation of $\delta _{S}$ , as confirmed by figure 6(c). Conveniently, the statistical convergence of the skewness can be tested empirically, and improved through temporal, spatial and/or ensemble averaging of data. To this end, we recommend future measurements be designed with the convergence of higher-order statistics in mind to ensure a flow representative profile of the skewness.

A third limitation is related to the wall-normal spatial resolution of measurements, especially for conventional experimental measurement techniques. Moderate resolution, typical of conventional log-spaced experimental measurements, was found to be adequate for estimating the zero-crossing, as confirmed by figures 4 and 7. For reference, there were typically three measurement locations that fell between the negative and positive peaks in the skewness profile in these cases. However, with a higher spatial resolution (i.e. more measurement locations in the outer region; see figure 6), the skewness profile will be better resolved, which is expected to lead to a more accurate estimation of the location of the zero-crossing. To quantify this, the skewness measurements of the ZPG TBL with high spatial resolution shown in figure 6 were considered as a baseline. Varying numbers of these measurement locations, which fall between the positive and negative peaks in the skewness profile, ranging from three up to the full available set, were then repeatedly selected at random and used to estimate $\delta _{S}$ . For the minimum resolution, i.e. using only three randomly distributed measurement locations between the positive and negative peaks, the simulated root-mean-square deviation of the measured $\delta _{S}$ , from the baseline value of $\delta _{S}$ , was found to be around 1 %. However, increasing the number of random locations considered, up to just five, nearly halved this simulated root-mean-square deviation. As such, we recommend making as many measurements as practical in the outer region (with particular focus on the region above $\delta _{99}$ ), ideally with 3–5 measurements at wall-normal locations falling between the negative and positive peaks in the skewness profile, to ensure high confidence when estimating the zero-crossing in the skewness profile using the proposed fitting method.

A final consideration, for future/previous numerical studies and low-order models in particular, is the requirement to resolve large-scale turbulent velocity fluctuations to some extent in order to compute the skewness. For example, it is not possible to extract skewness information from Reynolds-averaged Navier--Stokes (RANS) simulations without the incorporation of an additional model for the skewness specifically, or a supplementary eddy-resolved simulation. However, in § 3.2 the results from an AEM (Deshpande et al. Reference Deshpande, de Silva, Lee, Monty and Marusic2021) are presented to demonstrate that even a relatively simple model, which statistically models the large, inertia-dominated eddies, without consideration of the nonlinear interactions, can still resolve the outer skewness profile for the purposes of determining $\delta _{S}$ . There are, however, considerable differences observed at wall-normal locations near the wall between the skewness profiles from the AEM (figure 3 a) and experimental measurements (figure 4 c), which can be attributed to the disconsideration of small (viscosity-dominated) eddies and their nonlinear interactions with the attached eddies in the AEM. It is also noted that consideration of additional TBL dynamics, not included in the AEM, such as intermittent small-scale turbulence, free-stream turbulence and TNTI dynamics, will further contribute to the skewness profile in the outer region. However, these simulations confirm the significance of LSMs in generating the unique features of the skewness profile in the TBL outer region, and provide guidance for the modelling required to reasonably resolve the outer skewness profile. Further investigation of the relationship between $\delta _{S}$ , and other definitions for the TBL thickness (such as $\delta _{99}$ ), could also be useful for cases (such as RANS or laminar boundary layers) where some mean turbulence statistics are known, but the skewness is not.

Fortunately, it should be noted that many previously acquired datasets are already sufficient, with respect to the above limitations, to be reprocessed using this new definition/method, for comparison sake. However, the recommendations given here can also be used to design future experimental/numerical studies with the intention of applying this method to determine an appropriate and representative TBL thickness. Additionally, two different methods for estimating $\delta _{S}$ have been compared here, both of which were found to give reasonable results under the right conditions. As such, it is left to each analyst to choose an appropriate method of locating the skewness zero-crossing, based on the unique characteristics of their datasets/statistics. Finally, it is noted that the consideration of other turbulent flows of interest, such as jets or wakes, is beyond the current scope of this work. It is presently unclear if other turbulent shear flows will have similar and/or distinct skewness profiles with features that can be used to define a characteristic length scale in a way similar to the TBL. To this end, the results presented here can be used as motivation to revisit the definitions of characteristic length scales in other turbulent shear flows.