2.1. Experiments

Nine flow cases covering several orders of magnitude in Reynolds number ${Re}_\tau \sim O(10^{3}-10^{6})$ and surface roughness $k_s^{+} \lesssim O(10^{4})$ are included in the present analysis. Here, $k_s$ is the equivalent sand-grain roughness and conditions are fully rough for $k_s^{+} \gtrsim$ 70 when the surface drag becomes independent of viscosity. The flow cases, previously used together in Heisel et al. (Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b), are summarized in table 1. An overview of important details is given here, and a full account of each experiment is available from the cited references in the table.

Table 1. Experimental datasets used in the analysis of prograde spanwise vortices and ISLs.

The direct numerical simulation (DNS) of a boundary layer with ${Re}_\tau = 2000$ by Sillero et al. (Reference Sillero, Jiménez and Moser2013) is included for comparison with the experimental cases. The DNS case was analysed using two-dimensional flow fields in the $x$–$z$ plane to be consistent with the PIV experiments. The variable $\Delta x$, representing the spatial resolution, refers to the streamwise grid spacing for the DNS and the vector field spacing for the PIV experiments, which all employed 50 % window overlap leading to interrogation window size 2$\Delta x$. The parameters $\eta$ and $\lambda _T$ are both dependent on wall-normal distance $z$. The normalized resolution in table 1 considers the average parameter values at $z=0.1\delta$. This reference position was chosen in terms of $\delta$ because later results are presented at matched $z/\delta$ in the outer layer. Determination of $\eta$ and $\lambda _T$ is detailed in § 2.2.

The HRNBLWT facility in table 1 refers to the High Reynolds Number Boundary Layer Wind Tunnel at the University of Melbourne. The flow cases corresponding to the HRNBLWT were measured using the high-resolution tower PIV configuration detailed in Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Marusic, Schultz and Klewicki2016). The sandpaper roughness flow cases from the same study are not included in the present analysis. In this case a higher degree of small-scale noise in the rough-wall measurements (as documented in Squire et al. Reference Squire, Morrill-Winter, Hutchins, Marusic, Schultz and Klewicki2016) precludes the possibility of accurately fitting the Oseen vortex model to extract vortex statistics, and hence these data are omitted from the present study.

The SAFL facility in table 1 is the boundary layer wind tunnel at St. Anthony Falls Laboratory, University of Minnesota. These measurements captured the bottom portion of the boundary layer thickness, including the log region and part of the wake region. For the mesh roughness cases, the tunnel floor was covered by woven wire mesh with 3 mm wire diameter and 25 mm opening size (Heisel et al. Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b).

The ASL measurements were acquired using super-large-scale PIV (SLPIV) in the lowest 20 m of the atmosphere with natural snowfall as the flow tracers. The measurements captured the roughness sublayer and bottom of the log region in near-neutral conditions at the Eolos field facility. A detailed discussion of the snowflake traceability is given in Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018). The SLPIV method cannot capture the smallest vortical motions in the ASL due to both the non-negligible snowflake inertia and the coarse spatial resolution relative to $\eta$. Yet numerous vortices were present in each SLPIV vector field with size $O$(1 m) likely augmented by coarse resolution. The vortex and ISL size statistics are included in this study, but caution must be taken in interpreting these results. Vortex and shear layer velocity statistics in the ASL, however, are shown to be in agreement with all laboratory-scale trends.

Profiles of the first- and second-order velocity statistics are shown for each flow case in figure 1. For two of the smooth-wall cases ($\times ,+$), the relatively lower wall-normal velocity statistics in figure 1(c,d) are attributed to measurement resolution as discussed in Heisel et al. (Reference Heisel, Katul, Chamecki and Guala2020a). The decreasing trends in figure 1(c,d) for the ASL case ($\bullet$) may be due to one or more of several challenges present in atmospheric field settings. For instance, uncertainty in the definition of $\delta$, statistical convergence of higher-order statistics, and the possible role of large-scale stratified motions at higher altitudes are all discussed in the original study (Heisel et al. Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018).

Figure 1. First- and second-order velocity statistics for each experimental dataset. (a) Mean streamwise velocity $U$ shown as a deficit from the free-stream condition $U_\infty$, where $\kappa ^{-1}$ is the logarithmic slope based on the von Kármán constant. (b) Streamwise variance $\langle u^{2} \rangle$. (c) Wall-normal variance $\langle w^{2} \rangle$. (d) Reynolds shear stress $-\langle u w \rangle$. Profiles are normalized by the friction velocity ${u_\tau }$ and boundary layer thickness $\delta$. Data symbols correspond to the experiments in table 1 and are shown with logarithmic spacing for clarity. The roughness sublayer is excluded from the ASL profiles.

2.2. Scaling parameters

Owing to their importance in the scaling analysis, the determination of relevant flow parameters is discussed here. Not included here is ${u_\tau }$, which is discussed in the original publications cited in table 1. For the wind tunnel studies (HRNBLWT and SAFL), the parameters were estimated using hot-wire anemometer measurements under the same flow conditions rather than the PIV measurements (Squire et al. Reference Squire, Morrill-Winter, Hutchins, Marusic, Schultz and Klewicki2016; Heisel et al. Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b). Parameter estimation for the ASL case relied on scaling assumptions which are discussed separately at the end of this section.

Many of the parameters depend on the average rate of turbulent energy dissipation $\epsilon$. The Kolmogorov scales are defined as the length $\eta \equiv (\nu ^{3}/\epsilon )^{1/4}$, velocity $u_\eta \equiv (\nu \epsilon )^{1/4}$ and time $\tau _\eta \equiv (\nu / \epsilon )^{1/2}$. Further, the definition used here for the streamwise integral length is $L = {u{{}^{\prime }}}^{3} / \epsilon$ and for the large-eddy turnover time is $T = {u{{}^{\prime }}}^{2} / \epsilon$. This integral length definition yields results similar to the autocorrelation method, based on a test using the hot-wire measurements. The turnover time $T$ is equivalent to the integral time scale definition in isotropic turbulence where $T = L / u{{}^{\prime }}$, and differs from the streamwise integral time in wall-bounded flows where $T_{int} = L/U$.

Because $\epsilon$ could not be measured directly, various estimation methods were used depending on the experiment. Dissipation was estimated as $\epsilon \approx 15 \nu \langle (\partial u / \partial x )^{2} \rangle$ assuming local isotropy with the HRNBLWT measurements (Squire et al. Reference Squire, Morrill-Winter, Hutchins, Marusic, Schultz and Klewicki2016). For the DNS and SAFL cases, the dissipation was inferred from the value of the longitudinal second-order structure function in the inertial subrange, i.e. $D_{11} = C \epsilon ^{2/3} r^{2/3}$ (Saddoughi & Veeravalli Reference Saddoughi and Veeravalli1994).

The Taylor microscale is defined here as $\lambda _T^{2} = {u{{}^{\prime }}}^{2} / \langle (\partial u / \partial x )^{2} \rangle$ and the value was estimated using this expression for all cases excluding the ASL. This definition is specific to the streamwise component, where the Taylor microscale is anisotropic in wall-bounded turbulence (Morishita, Ishihara & Kaneda Reference Morishita, Ishihara and Kaneda2019). The Taylor microscale and integral length are related as $\lambda _T / L \sim {Re}_L^{-1/2}$ with the integral Reynolds number being ${Re}_L = u{{}^{\prime }}L / \nu$ (Pope Reference Pope2000). This relationship is based on $\epsilon \sim \nu \langle (\partial u / \partial x )^{2} \rangle$, where the proportional constant is 15 for isotropic turbulence and otherwise depends on the relative contribution of each gradient term to the dissipation.

For the ASL case, the measurement resolution of both the SLPIV and a collocated sonic anemometer were too coarse for the estimation methods discussed above. The dissipation was therefore estimated as $\epsilon \approx {u_\tau }^{3} / \kappa z$, where $\kappa$ is the von Kármán constant. This estimate assumes the local shear production of turbulence in the log region is in equilibrium with the dissipation rate, which is typical for high-Reynolds-number boundary layers (Townsend Reference Townsend1961). The Kolmogorov length scale resulting from this approximation is $\eta \approx 0.7\pm 0.2$ mm. The value for $\lambda _T$ was estimated from the expression $\epsilon = 15 \nu {u{{}^{\prime }}}^{2} / \lambda _T^{2}$, which assumes local isotropy and combines the $\lambda _T$ and $\epsilon$ expressions through the shared gradient term. The resulting Taylor microscale estimate for the ASL case is $\lambda _T \approx 13$ cm. The integral length $L \sim O$(1 m) results directly from the estimated dissipation.

2.3. Spanwise vortex detection

Numerous methods exist to characterize vortices (see, e.g. Chakraborty, Balachandar & Adrian Reference Chakraborty, Balachandar and Adrian2005; Haller Reference Haller2005). The preferred method in the present analysis is to fit the local flow field to a model vortex. The primary advantage of using a model vortex is that the fitted vortex properties are not dependent on an arbitrary parameter threshold. However, a parameter threshold is still required to detect the possible vortex core regions upon which the model is fitted, and the characterization of complex vortex structures is limited to the imposed model shape. Despite these limitations, recent studies have successfully characterized large populations of vortex cores in boundary layer flows using the Oseen model (Oseen Reference Oseen1912) in cylindrical coordinates (Carlier & Stanislas Reference Carlier and Stanislas2005; Herpin et al. Reference Herpin, Stanislas and Soria2010, Reference Herpin, Stanislas, Foucaut and Coudert2013)

(2.1)\begin{equation} \boldsymbol{u} = \boldsymbol{u}{_\omega} + \frac{\varGamma}{2 {\rm \pi}r} \left[ 1 - \exp \left( - \left( \frac{ r }{ r{_\omega}} \right) ^{2} \right) \right] \boldsymbol{e}_{\theta},\end{equation}

where $\boldsymbol {u}{_\omega }$ is the advection velocity of the vortex, $\varGamma$ is the circulation, $r$ is the radial distance from the vortex centre, $r{_\omega }$ is the vortex radius and $\boldsymbol {e}_{\theta }$ is the unit vector in the azimuthal direction. The model incorporates Biot–Savart law for the velocity induced by a vortex line, i.e. $u_\theta \propto \varGamma / 2 {\rm \pi}r$, and the exponential term acts as a damping function to decrease the azimuthal velocity $u_\theta$ inside the vortex radius such that the maximum velocity occurs at $r=r{_\omega }$.

The Oseen model is consistent with the Navier–Stokes equations in its three-dimensional definition. In the formal definition, the radius is given as $r{_\omega } = \sqrt {4\nu t}$, where the radius is zero at time $t=0$ and the vortex grows in time via viscous diffusion. This is conceptually different from the Burgers vortex radius $r{_\omega } = \sqrt {4\nu / \alpha }$, where $\alpha$ is the strain rate. Rather than growing in time, the Burgers vortex size is steady due to a balance between outward diffusion and an inward radial velocity $u_r(r) = -\alpha r$ induced by axial vortex stretching (Burgers Reference Burgers1948). The Oseen model does not include a radial velocity or axial stretching. This distinction between the two models becomes important for interpreting the later results.

The vortex detection using (2.1) followed the procedure of Herpin et al. (Reference Herpin, Stanislas, Foucaut and Coudert2013), which is briefly summarized here. Possible vortex cores were identified based on flow regions with $\lambda _{ci} > 1.5 \lambda _{rms}$, where $\lambda _{ci}$ is the two-dimensional swirling strength (Adrian, Christensen & Liu Reference Adrian, Christensen and Liu2000a) and $\lambda _{rms}(z)$ is the r.m.s. swirling strength at each wall-normal distance. The threshold introduces selection bias on the vortices, but the threshold is not applied in the later model fit. Based on a test using one flow case, changing the threshold factor from 1.5$\times$ to 2$\times$ or 2.5$\times$ resulted in the detection of fewer vortices, but resulted in no statistically significant changes to the vortex properties.

A notable deviation to the original procedure was to apply a Gaussian filter to the wind tunnel (HRNBLWT and SAFL) PIV fields prior to vortex detection. The filter removed small-scale noise in the gradients which significantly improved the performance of the fitting algorithm. The size of the Gaussian filter, i.e. its standard deviation, was selected to be approximately 2$\eta$ for the reference position $z=0.1\delta$ in each case. The filter was not applied to the DNS or SLPIV vector fields. The filter would be negligibly small relative to the resolution for the ASL case, and a comparison of DNS results with and without the filter revealed no changes to the statistics of interest.

The properties of the identified core region were used as initial guesses for the six vortex model parameters: the centre position given by $x{_\omega }$ and $z{_\omega }$, the advection velocity components $u{_\omega }$ and $w{_\omega }$, the circulation $\varGamma$ and the radius $r{_\omega }$. The initial value for $\varGamma$ was based on the core area and average out-of-plane vorticity $\omega _y$ within the core region. The initial guesses for $r{_\omega }$, $x{_\omega }$ and $z{_\omega }$ were used to extract the local vector field within 2$r{_\omega }$ of the vortex centre. The extracted vector field was then fitted to (2.1) using the initial guesses and a nonlinear least-squares fitting algorithm. The algorithm was designed to update the six model parameters until the difference between the local vector field and the model vortex (i.e. the right-hand side of (2.1)) was minimized. The parameters output by the algorithm therefore describe the model vortex most closely matching the vector field within and around the vortex core. The parameters are thus inferred to represent the vortex core properties. The initial guesses informed by the detected core regions reduce the likelihood the algorithm diverged to a spurious local minimum in the model parameter space. To exclude these fits and other ill-fitted results, the properties are only considered for vortices where the coefficient of determination exceeded $R^{2}> 0.6$. Increasing the imposed minimum $R^{2}$ value reduces somewhat the convergence of the vortex statistics, but does not change the overall trends and conclusions of the study. An example vector field and closest model fit with $R^{2}= 0.9$ are shown in figure 2(a).

Figure 2. Example of a model Oseen vortex fitted to a velocity vector field, shown with the vortex advection velocity subtracted for visualization. (a) Comparison of measured velocities (black vectors) relative to the closest Oseen model (red vectors) whose size is given by the red circle. (b) Notation for the diameter $d{_\omega }$ and azimuthal velocity difference $\Delta u{_\omega }$ across the diameter of the fitted vortex.

Besides the advection velocity, the characteristic velocity of each vortex is the maximum azimuthal velocity difference $\Delta u{_\omega }$ across the vortex. This velocity, shown in figure 2(b), is twice the azimuthal velocity at the edge of the vortex $r=r{_\omega }$ due to axisymmetry and the definition of the radius. From (2.1), the velocity is

(2.2)\begin{equation} \Delta u{_\omega} = \frac{\varGamma}{{\rm \pi} r{_\omega}} ( 1 - e^{{-}1} ).\end{equation}

The properties are defined across both sides of the vortex core, i.e. as $d{_\omega }$ and $\Delta u{_\omega }$, for comparison with the ISL properties which are not in cylindrical coordinates.

The detection algorithm was conducted for all experimental datasets in table 1. Approximately 22 000 Oseen vortices were fitted for the ASL dataset, and more than 100 000 were fitted for every other case. The possible issue of selection bias due to the chosen vortex model is discussed later in § 5.3. The primary sources of uncertainty for vortex properties in the PIV experiments are spatial resolution limitations and small-scale noise. The results are focused on probability distributions of the properties rather than their mean. Artefacts due to the limitations described above occur primarily within the smallest values of each property, and conclusions are not drawn from these regions of the distributions. See Appendix A for an assessment of how measurement resolution affects the mode and tail of the probability distributions. A further benefit of a probabilistic approach is to identify the possible relevance of multiple scaling parameters. If different regions of the probability distribution are governed by separate parameters, the mean value may not reflect a simple scaling relationship. A similar probability analysis of the ISL features is unfortunately not possible for the methodology given in the following section. The analysis requires conditionally averaging which does not provide instantaneous statistics.

2.4. ISL detection

Rather than directly detecting the ISLs using a threshold on instantaneous values of $\partial u / \partial z$, we instead infer the properties of ISLs from detected interfaces of UMZs. These interfaces represent the shear regions separating the local larger-scale coherent velocity structures (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017). In the present context, the distinction between ISLs and UMZ interfaces is largely a matter of detection methodology. While the shear layers align closely with UMZ interfaces, the interfaces can also include segments where the instantaneous shear is relatively smaller (Gul et al. Reference Gul, Elsinga and Westerweel2020). These additional segments due to the detection methodology may lead to differences between ISL and UMZ interface statistics for some properties. However, the statistics presented here are limited to those where the same behaviour has been previously reported for both ISLs and UMZ interfaces, and from here on the ISL terminology is applied.

The present analysis uses the same detected UMZs from Heisel et al. (Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b). The methodology to identify UMZs and their interfaces using histograms of the streamwise velocity is detailed in the previous work. An important consideration regarding the previous study is that the detected UMZs unambiguously exhibited a dependence on wall-normal distance $z$ (Heisel et al. Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b), such that the present analysis studies the shear layers separating $z$-scaled velocity structures.

Figure 3 shows an example of ISLs and their statistics based on measurements from the ${Re}_\tau =17\ 000$ smooth-wall case in table 1. The PIV field in figure 3(a) illustrates the general, approximate organization of the streamwise velocity in the outer layer into relatively uniform regions (i.e. UMZs) separated by thin shear layers (black lines). The field in figure 3(b) is focused on the local vicinity of a single ISL to demonstrate the compiled statistics. The velocity difference between UMZs is a factor of ${u_\tau }$, and the velocity gradient occurs across a short distance centred around the wall-normal position $z_i$ of the ISL. The black dots in figure 3(b) represent the PIV vector positions near the ISL at the horizontal centre of the field. The PIV velocities were compiled at these vector positions to create a profile relative to the ISL position ($z-z_i$) and velocity ($u-u_i$ and $w-w_i$). These conditional profiles were compiled for every $x$ position of every ISL.

Figure 3. Example shear layers for the ${Re}_\tau = 17\ 000$ smooth-wall case. (a) Streamwise velocity field from figure 2 of Heisel et al. (Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b), with detected ISLs (black lines) and turbulent/non-turbulent interface (red line). (b) Velocity field in the vicinity of an ISL (black line) relative to the ISL position $z_i$ and velocity $u_i$. (c) Conditionally averaged velocity profile relative to the ISLs, indicating the average velocity difference $\Delta U_i$ and thickness $\delta {_\omega }$. The black box in the upper left of (a) indicates the size of the field in (b).

Averaging all the conditional streamwise profiles in the log region yields the profile in figure 3(c). As seen in the figure, the average velocity difference $\Delta U_i$ results from linear fits to the profile above and below the ISL (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017). One method for estimating the (vorticity) thickness $\delta {_\omega }$ is to assume the ISL is a mixing layer (Brown & Roshko Reference Brown and Roshko1974). Note the subscript ‘$\omega$’ is used for the shear layer thickness $\delta {_\omega }$ to be consistent with notation in the literature. The mixing layer method uses the maximum velocity gradient within the layer $\partial \langle u-u_i \rangle / \partial z \vert _{max}$. The gradient is highest at the centre of the layer and quickly decreases away from the centre, such that the maximum gradient is not representative of the entire layer. In practice, the measured maximum gradient also depends strongly on the spacing of the grid points $\Delta x$ in the vicinity of the layer centre, and $\delta {_\omega }$ values from this method are a function of the measurement resolution.

The thickness $\delta {_\omega }$ can more simply be estimated as the distance across which the velocity difference $\Delta U_i$ occurs (Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015). In addition to being less sensitive to measurement resolution, this estimate is more representative of the overall shear layer. This second method is used to calculate the thicknesses presented in later results. Note this method is still susceptible to resolution issues if the measurements are coarse relative to $\delta {_\omega }$, as is discussed later.

In many respects, it is important to distinguish the ISLs from the turbulent/non-turbulent interface (TNTI) separating the boundary layer flow from the free-stream condition (Elsinga & da Silva Reference Elsinga and da Silva2019), i.e. the red line in figure 3(a). Indeed, the TNTI was detected using a separate methodology based on the kinetic energy defect (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b) prior to detection of the UMZs and the internal layers (Heisel et al. Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b). However, for the specific statistics analysed in the present study, the same scaling behaviour is expected for both ISLs and the TNTI. Previous studies have shown the TNTI to have thickness $\delta {_\omega } \sim O(\lambda _T)$ and velocity $\Delta U \sim O({u_\tau })$ (Bisset, Hunt & Rogers Reference Bisset, Hunt and Rogers2002; da Silva & Taveira Reference da Silva and Taveira2010; Chauhan, Philip & Marusic Reference Chauhan, Philip and Marusic2014a; de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017), matching the ISL literature results discussed in the introduction. Later figures showing wall-normal profiles of shear layer properties include ISLs and TNTIs, such that both features contribute to values in the wake region.

3.1. Shear layer thickness

The streamwise velocity profile relative to the ISLs, shown previously in the figure 3(c) example, is compared for all flow cases in figure 4(a). The ASL measurement resolution exceeds the vertical axis limits and the profile is not visible. There is otherwise close agreement across cases with normalization by $\lambda _T$ and ${u_\tau }$. A comparison of the profiles with alternative length scale normalizations, e.g. $\nu / {u_\tau }$ and $\eta$, is presented elsewhere (e.g. de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017).

Figure 4. Estimated average thickness $\delta {_\omega }$ of ISLs normalized by $\lambda _T$. (a) Conditional average profile of streamwise velocity relative to ISLs in the log region. (b) Thickness $\delta {_\omega }$ as a function of measurement grid spacing $\Delta x$, noting the interrogation window size for the PIV cases is 2$\Delta x$. (c) Wall-normal profile of $\delta {_\omega }$. Data symbols correspond to the experimental datasets in table 1.

The average ISL thickness was calculated from the conditional profiles following the methodology discussed in § 2.4. The effect of measurement resolution on the estimated thickness is demonstrated in figure 4(b). The thickness estimate is relatively invariant to resolution for grid spacing $\Delta x \lesssim 0.1 \lambda _T$. The four SAFL PIV cases with $\Delta x \approx 0.1 \lambda _T$ have mean thickness approximately 25 % larger than the cases with smaller $\Delta x$, suggesting the effects of resolution may begin near this value. Assuming the $\lambda _T$ scaling extends to field conditions, the resolution of the ASL measurements is too coarse for an accurate estimate of the ISL thickness, and no conclusions are drawn from the $\delta {_\omega }$ value in the ASL case.

A wall-normal profile of $\delta {_\omega }$ was constructed using conditional ISL statistics in binned intervals of $z/\delta$. The resulting profile in figure 4(c) suggests a fixed relationship between $\delta {_\omega }$ and $\lambda _T$ throughout the outer layer. The proportionality $\delta {_\omega } \approx (0.3 - 0.5)\lambda _T$ and its invariance with wall-normal position are both in close agreement with previous studies (Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015; de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017). Note that normalization of $\delta {_\omega }$ by $\delta {Re}_\tau ^{-1/2}$ collapses the profiles across experiments, but the thickness varies with $z$, i.e. the profiles are not flat. The local parameter $\lambda _T(z)$ is required to achieve the observed invariance of $\delta {_\omega }(z)$ with wall-normal position. The scatter between cases in figure 4(c) is entirely consistent with the resolution findings in figure 4(b): the cases with coarser resolution relative to $\lambda _T$ have profile values closer to 0.5$\lambda _T$, and cases with higher resolution have profile values closer to 0.3$\lambda _T$. A new finding from the present results is that significant laboratory-scale roughness (i.e. $k_s^{+} \approx$ 500) does not modify the relationship between $\delta {_\omega }$ and $\lambda _T$ beyond the uncertainty due to resolution, thus confirming the scaling is not limited to smooth-wall conditions. Ebner, Mehdi & Klewicki (Reference Ebner, Mehdi and Klewicki2016) proposed that large roughness may affect the thickness of the shear layers and the overall organization of UMZs. However, the required roughness in terms of $k_s/ \delta$ for this effect to occur may be too great to maintain outer layer similarity.

3.2. Vortex diameter

As discussed in § 2.3, the prograde spanwise vortex core statistics are presented here as probability distributions rather than averages. Specifically, statistics are given as p.d.f.s, which were estimated discretely as binned histograms with amplitude scaled to achieve area unity, i.e. probability of one within the parameter space. To improve the statistical convergence of the probability tails, logarithmic spacing was used for the histogram binning intervals. The proper normalization of the histogram as a p.d.f. is then achieved by dividing the value of each bin by its respective bin width and the total number of occurrences in the histogram. Logarithmic bin intervals were only used where the scale of the abscissa is logarithmic. Combined with the large number of detected vortices in each flow case, the logarithmic binning allows for observation of six orders of magnitude in probability.

The estimated p.d.f.s of prograde vortex core diameter $p_d$ for each experimental dataset are shown in figure 5. The p.d.f.s are shown with $d{_\omega }$ normalized by both the Kolmogorov length $\eta$ (a) and the Taylor microscale $\lambda _T$ (c). Figure 5 includes vortex cores from the entire outer layer, including the logarithmic and wake regions. Prior to constructing the histograms, the fitted vortex diameters were individually normalized using the average parameters $\eta (z{_\omega })$ and $\lambda _T(z{_\omega })$ local to the vortex wall-normal position $z{_\omega }$. Later figures presenting vortex core statistics also feature contributions from throughout the outer layer and utilize the individual normalization described here. Note that limiting the statistics to vortex cores at a specific wall-normal position does not affect the figure trends or the conclusions of the study.

Figure 5. Probability distributions $p_d$ of diameter for all prograde vortex cores in the outer layer of the boundary layer: (a) p.d.f. normalized by the Kolmogorov length scale $\eta$; (b) conditional p.d.f. $p_{d/\eta }^{*} = p_{d/\eta }(d{_\omega } \mid d{_\omega } > 40\eta )$; (c) p.d.f. normalized by the Taylor microscale $\lambda _T$; (d) conditional p.d.f. $p_{d/\lambda _T}^{*} = p_{d/\lambda _T}(d{_\omega } \mid d{_\omega } > \lambda _T)$. Comparison of results in the hatched regions of (a,c) is precluded by variability in measurement resolution across cases, including the ASL case shown in the insets. The conditional p.d.f.s in (b,d) are used to evaluate the distribution tail shape parameterized by the slope $m_d$. Data symbols correspond to the experimental datasets in table 1.

Previous studies have described the most probable vortex size, i.e. the distribution mode, in terms of $\eta$. The mode for the DNS case in figure 5(a) is approximately 10$\eta$, which is between the values (8–10)$\eta$ observed by Tanahashi et al. (Reference Tanahashi, Kang, Miyamoto, Shiokawa and Miyauchi2004) and (12–13)$\eta$ by Herpin et al. (Reference Herpin, Stanislas, Foucaut and Coudert2013). While the observed range (10–20)$\eta$ in the mode across the remaining cases in figure 5(a) may be physically meaningful, it may also be attributed to variability in measurement resolution. Appendix A demonstrates that coarsening the DNS grid to $\Delta x^{+}$ values similar to the experimental cases results in the same observed range in mode position. The cross-hatched lines in figure 5 illustrate the diameter range where a strict comparison across cases is precluded by the influence of measurement resolution on the experimental cases with coarser $\Delta x$. Additionally, while there is an orders-of-magnitude difference in detected vortex size for the ASL case in the insets of figure 5(a,c), the difference is consistent with the orders-of-magnitude increase in $\Delta x$ seen in table 1. We therefore draw no conclusions regarding either the distribution mode across cases or the detected size of ASL vortex cores.

Relative to the DNS case, some of the experimental results exhibit an increased number of vortices smaller than 10$\eta$ in figure 5(a). This increased detection of small vortices is likely an artefact of residual measurement noise. While it is possible to further quantify and filter the noise (see, e.g. Foucaut, Carlier & Stanislas Reference Foucaut, Carlier and Stanislas2004), its effect is confined to the hatched region of figure 5 which is already subject to resolution bias and is excluded from further analysis.

Regarding the distribution shape, Herpin et al. (Reference Herpin, Stanislas, Foucaut and Coudert2013) showed that the size probability $p_d$ approximately follows a log-normal distribution, except with a somewhat thicker tail than the log-normal case. The figure 5 distribution tails are consistent with these findings. The tails deviate from a purely log-normal distribution and instead approximate a power law. The distribution tails shown in figure 5(b,d) are conditional p.d.f.s, e.g. $p_{d/\eta }^{*}=p_{d/\eta }(d{_\omega } \mid d{_\omega } > 40\eta )$, which describe the probability given the diameter is above the threshold shown as a dashed line in figure 5(a,c). The conditional p.d.f. is equivalent to taking $p_d$ above the threshold and rescaling to achieve area unity under the tail region. The rescaling removes the statistical contribution of the small-diameter vortices – which are affected by spatial resolution and measurement noise – to the amplitude of the normalized distribution tails. Given that the tail shape is not affected by resolution (see Appendix A), the $p_d^{*}$ plots allow for an unbiased comparison across cases for a consistent range of diameters.

The power law shape of the distribution tail is given by $p_d^{*} \propto d{_\omega } ^{-m_d}$. The collapse of the tails in figure 5(b,d) is due to a consistent slope $m_d \approx 6-7$ across cases, suggesting $m_d$ is independent of Reynolds number and the surface roughness geometries tested here. The slope is the same for the ASL case, and also does not vary with wall-normal position. The steepness of the slope $m_d \approx 6 - 7$ imposes an effective maximum value on the vortex core size, where the probability density decreases by orders of magnitude and becomes negligibly small across a narrow range of sizes. However, the relevant scaling parameter cannot be determined by the shape of the power law, as $m_d$ is independent of the normalization as seen in figure 5(b,d). In this regard, the diameter of the largest detected vortex cores may be $O(100\eta )$ as in figure 5(b) or $O(1 - 10\lambda _T)$ as in 5(d).

To discern the scaling parameter relevant to the largest vortex cores, the curvature of $p_d$ in the region preceding the power law must be considered. Given the sensitivity of the small-diameter vortex statistics to the measurement resolution (Appendix A), it is not possible to separate a physical trend in the figure 5 $p_d$ curvature from the variability of resolution across the experiments studied here. These experimental factors can be avoided by focusing the analysis on the DNS flow case. Owing to the Reynolds dependence $\eta / \lambda _T \sim {Re}_L^{-1/4}$ where ${Re}_L= u{{}^{\prime }}L / \nu$, the $d{_\omega }$ statistics for the DNS case were grouped in intervals of ${Re}_L$ based on the local mean values for $u{{}^{\prime }}$ and L at the vortex position $z{_\omega }$.

Figure 6(a,b) shows distributions of $d{_\omega }$ for eight intervals of ${Re}_L$ within the DNS case. Having already demonstrated the power law shape of the probability tail in figure 5, the abscissa is shown with linear scaling in figure 6(a,b) to highlight amplitude differences along the tail. With $\eta$ normalization, there is a clear Reynolds number trend in the distribution tail region corresponding to large diameter vortex cores. Normalization by $\lambda _T$ in figure 6(b) leads to improved agreement across the range of ${Re}_L$ represented. There is no apparent trend in Reynolds number along the tails of $p_{d/\lambda _T}$ and the remaining narrow scatter is attributed to incomplete statistical convergence.

Figure 6. Probability distributions of the vortex core diameter $d{_\omega }$ and strain rate $\alpha = 16 \nu d{_\omega }^{-2}$ inferred from the Burgers vortex. Diameter p.d.f. $p_d$ normalized by (a) the Kolmogorov length scale $\eta$ and (b) the Taylor microscale $\lambda _T$. Strain rate p.d.f. $p_\alpha$ normalized by (c) the Kolmogorov time scale $\tau _\eta$ and (d) the large-eddy turnover time $T$. Data shown are for the DNS case in intervals of the integral Reynolds number ${Re}_L$ determined by the local flow properties at the wall-normal position of the vortex. The upper right plot shows a profile of ${Re}_L$ for reference.

The general collapse of $p_{d / \lambda _T}$ in 6(b) suggests the largest vortex cores are confined by the same scaling that describes the ISL thickness. Considering also the steep slope $m_d \approx 6 - 7$ in figure 5, the results indicate that the vortex core size is bounded between $\eta$ and $\lambda _T$ (Davidson Reference Davidson2015). Note that this conclusion applies only to strongly rotating prograde spanwise vortex cores. The strict detection criteria introduce selection biases which exclude larger, weakly rotating motions such as bulges and, as discussed later in § 5.3, likely under-detects Kolmogorov-scaled vortices.

3.3. Vortex inferred rate of strain

Figure 6(b) provides promising results regarding the linkage between $\lambda _T$ and the size of the largest spanwise vortex cores. This relation is explored further here by assessing the rate of strain $\alpha$ acting on the detected vortices. The strain can be inferred using the radius $r{_\omega }$ fitted previously by the Oseen model along with the Burgers vortex radius definition $r{_\omega } = \sqrt {4\nu / \alpha }$ given in § 2.3. The vortices are described here in terms of a strain rate $\alpha$ rather than a time $t$ (as in the Oseen model) to emphasize the physical relevance of the strain field. In this sense, the present section provides insight into the straining motions corresponding to the largest vortex cores and the vortex size that results from these motions.

The relevant parameters for a strain rate comparison are the Kolmogorov time scale $\tau _\eta = \sqrt {\nu /\epsilon }$ and large-eddy turnover time scale $T = L / u{{}^{\prime }}$ introduced in § 2.2. Inserting these time scales into the expression $d{_\omega } \sim \sqrt {\nu t}$ returns the Kolmogorov length scale $\sqrt {\nu \tau _\eta } = (\nu ^{3} / \epsilon )^{1/4} = \eta$ and Taylor microscale $\sqrt {\nu T} = L / \sqrt {{Re}_L} \sim \lambda _T$, respectively. An important advantage of considering the strain rate is the amplified Reynolds dependence $\tau _\eta / T \sim {Re}_L^{-1/2}$.

As with the diameter statistics, an objective comparison of strain rates across experimental cases is precluded by the variability in experimental factors. Instead, the size analysis of DNS vortices in figure 6(a,b) is repeated for the strain rate in 6(c,d). The strain rate was estimated by inverting the Burgers vortex definition as $\alpha = 16 \nu d{_\omega }^{-2}$. Statistics for $\alpha$ are presented using the same intervals of ${Re}_L$ as for $d{_\omega }$. By using the vortex wall-normal position $z{_\omega }$ to determine the relevant parameter values, it is assumed that the net effect of non-local eddies is captured in their contribution to the local values for $T$ and ${Re}_L$. In other words, the local parameters $T(z{_\omega })$ and ${Re}_L(z{_\omega })$ are assumed to be representative of the average eddy inducing strain on the vortex.

The normalized strain rate distributions in figure 6(c,d) are consistent with the vortex diameters in 6(a,b). The Reynolds-number trend in figure 6(c) corresponds to vortices with low strain rate and large diameter. Normalization of the strain by $T$ in figure 6(d) accounts for the ${Re}_L$ dependence and yields reasonable collapse of the distributions. The normalization by $T$ also provides better agreement than the mean shear time scale $(\partial U / \partial z)^{-1}$ (not shown here) which overcompensates for the Reynolds-number trend seen in figure 6(c). Note that grouping the strain statistics in intervals of $z/\delta$ rather than ${Re}_L$ is less successful due to the non-monotonic profile of ${Re}_L(z)$ shown in the upper right plot of figure 6. The result suggests that the low-magnitude strain imposed on the detected Oseen-type vortices corresponds to a large-eddy time $T$ instead of Kolmogorov scales, and further supports $\lambda _T$ as the relevant parameter for describing the size of the largest vortex cores in the outer layer.

4.1. Shear layer streamwise velocity

The ISL average characteristic velocity $\Delta U_i$ is equivalent to the velocity difference between the UMZs bounding the shear layer. A wall-normal profile of this UMZ velocity, estimated using binned intervals of $z/\delta$, was previously given in figure 4 of Heisel et al. (Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b). The profile of $\Delta U_i$ is reproduced here as figure 7 for convenience. As noted in Heisel et al. (Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b), $\Delta U_i$ remains proportional to ${u_\tau }$ throughout the outer layer, including for the TNTI in the wake region. The velocity decreases moderately across the boundary layer. The scaling and wall-normal trend are both consistent with previous studies (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017; Gul et al. Reference Gul, Elsinga and Westerweel2020). As a reminder, probability distributions of $\Delta U_i$ are not available due to the conditional averaging technique used to estimate the parameter.

Figure 7. Wall-normal profile of the average streamwise velocity difference $\Delta U_i$ across ISLs. From figure 4 of Heisel et al. (Reference Heisel, de Silva, Hutchins, Marusic and Guala2020b). Data symbols correspond to the experimental datasets in table 1.

4.2. Vortex azimuthal velocity

The velocity probability distributions for prograde vortex cores are shown in figure 8 using the same format as the diameter p.d.f.s. The distributions again contain several decades of probability which facilitates characterization of the shape. Unlike the size statistics, the velocity distributions are shown with linear spacing of the abscissa. The distribution tails are consistently linear in this plotted format, indicative of exponential tails given by $p_u \propto \exp (-\Delta u{_\omega } / m_u)$, where $m_u$ characterizes the linear slope of the tail seen in the figure. Variability in the $p_u$ mode is in part explained by differences in spatial resolution as demonstrated in Appendix A, most notably for the ASL case. Thus, no conclusions are drawn here regarding the velocity mode, except to note the mode position may also be dependent on the vortex wall-normal position.

Figure 8. Probability distributions $p_u$ of azimuthal velocity for all prograde vortex cores: (a) p.d.f. normalized by the Kolmogorov velocity $u_\eta$; (b) conditional p.d.f. $p_{u/u_\eta }^{*} = p_{u/u_\eta }(\Delta u{_\omega } \mid \Delta u{_\omega } > 15 u_\eta )$; (c) p.d.f. normalized by the friction velocity ${u_\tau }$; (d) conditional p.d.f. $p_{u/{u_\tau }}^{*} = p_{u/{u_\tau }}(\Delta u{_\omega } \mid \Delta u{_\omega } > 3 {u_\tau })$. Comparison of results in the hatched regions of (a,c) is precluded by variability in measurement resolution across cases. The conditional p.d.f.s in (b,d) are used to evaluate the distribution tail shape parameterized by the slope $1/m_u$. Data symbols correspond to the experimental datasets in table 1.

Same as for the vortex diameter statistics, figure 8(b,d) shows conditional p.d.f.s based on the thresholds given as dashed lines. The distribution tails given by $p_u^{*}$ indicate ${u_\tau }$ to yield somewhat better agreement in the tail slope than $u_\eta$ across cases, including for the ASL case. The tail slope is given as $1/m_u$ rather than the reciprocal because the mean of an exponential distribution is defined by $m_u$. The probability $p_u$ is not fully exponential, as indicated by the presence of the mode, and $m_u$ therefore underestimates the mean in this case. However, $m_u$ is the relevant parameter for the long distribution tail and is likely important to the central moment statistics such as the average. Hereon, $m_u$ is referred to as a ‘shape parameter’ for the velocity distribution.

The value of $m_u$ was estimated for each flow case using a linear fit to the $p_u^{*}$ tails. The fits were specifically applied to the regions $\Delta u{_\omega } = (15 - 40) u_\eta$ and $(3 - 6) {u_\tau }$ where the probability statistics show better convergence. In addition to the p.d.f.s shown in figure 8 which include the vortices throughout the outer layer, $m_u$ was also estimated for velocity p.d.f.s in intervals of the wall-normal position based on the vortex centre $z{_\omega }$. The resulting $m_u$ values are shown in figure 9.

Figure 9. The shape parameter $m_u$ describing the exponential probability tail $p_u \propto \exp (-\Delta u{_\omega } / m_u)$ for the vortex azimuthal velocity: (a,b) as a function of ${Re}_\tau$ when normalized by $u_\eta$ and ${u_\tau }$, respectively; (c) wall-normal profile normalized by ${u_\tau }$. Data symbols correspond to the experimental datasets in table 1.

The shape parameter $m_u$ increases with Reynolds number ${Re}_\tau$ when normalized by $u_\eta$, and is approximately constant across three decades of ${Re}_\tau$ when normalized by ${u_\tau }$ in figure 9(b). This again supports the ${u_\tau }$ scaling for the probability tail suggested visually in figure 8. Similar to the scaling analysis for $d{_\omega }$, a weak Reynolds-number dependence is expected for the velocity scale separation. Given ${u_\tau }$ and $\delta$ are the same order as the respective integral scales $u{{}^{\prime }}$ and $L$ in the outer layer, the velocity scales are related as ${u_\tau } / u_\eta \sim {Re}_\tau ^{1/4}$. Figure 9(a) supports this relationship, where the shape parameter exhibits the same weak Reynolds-number dependence $m_u / u_\eta \sim {Re}_\tau ^{1/4}$. The velocity tail statistics are better converged with less uncertainty across cases than the diameter mode position, and the ${Re}_\tau$ trend is apparent even within the laboratory datasets in figure 9(a).

In the wall-normal profile of $m_u$ in figure 9(c), the shape parameter decreases moderately with increasing $z$. The trend matches closely with $\Delta U_i(z/\delta )$ in figure 7, where $m_u \approx 0.5 \Delta U_i$ throughout the outer layer. Given $m_u$ represents an underestimated value of the statistical mean $\Delta U{_\omega }$ for the modified exponential distributions in figure 8, the azimuthal velocity of large vortices follows the same scaling and is quantitatively similar to the shear layer velocity $\Delta U_i$, indicating a close dynamical relationship between the largest prograde vortex cores and the ISLs.

The vortex statistics presented thus far, i.e. $d{_\omega }$ and $\Delta u{_\omega }$, are not independent from one another. The relation between the vortex core size and velocity is explored in Appendix B. The appendix results show that the population of detected vortex cores within the distribution tail of $p_d$ coincide with vortices also within the tail of $p_u$. The vortex cores exhibiting ${u_\tau }$ velocity scaling are therefore the same cores for which the influence of $\lambda _T$ is observed.

4.3. Radial and wall-normal velocities

We now turn the focus to the velocity components indicating flow to and from the vortex cores and shear layers in the $x$–$z$ measurement plane. These velocities, which are relative to the advection velocity of the considered flow regions, provide insight into the temporal evolution of these features in a more Lagrangian sense. In the case of the vortex cores, the relevant component is the radial velocity $u_r$ in cylindrical coordinates relative to the vortex centre. For each detected vortex core, the local flow field around the fitted vortex centre was compiled to create a profile of $u_r(r)$ as a function of radial distance $r$ from the vortex centre. The results were filtered to only include vortex cores larger than the mode for each dataset ($d{_\omega } \gtrapprox 10 \eta \approx 0.3\lambda _T$) which are the focus of the study. A limited number of measurement points exist within the radius of the small vortices such that it is difficult to determine their radial trend. To average the profile across all the remaining vortices in a given flow case, the distance $r$ was normalized by the radius $r{_\omega }$ such that $r/r{_\omega }=1$ corresponds to the edge of each vortex core. The resulting average radial profiles of $u_r$ are provided in figure 10(a). Note that the Oseen model fitted to the vortices assumes $u_r(r) = 0$, and any deviations from zero thus indicate physical behaviour that overcomes the selection bias of the imposed model.

Figure 10. Average velocity profiles demonstrating the mean flow into the vortex cores and ISLs in the $x$–$z$ plane. (a) Vortex radial velocity $u_r(r)$ relative to the vortex centre $r=0$, normalized by its radius $r{_\omega }$ and the turnover time scale $T$. (b) Wall-normal velocity $w$ relative to the ISL velocity $w_i$ and midheight $z_i$, normalized by the Taylor microscale $\lambda _T$ and the friction velocity ${u_\tau }$. The dashed lines correspond to the average vortex radius (a) and ISL thickness (b). Data symbols correspond to the experimental datasets in table 1.

The average $u_r(r)$ profile is close to zero within the vortex core ($r/r{_\omega } < 1$) and follows a trend of increasingly negative flow outside the core. There is generally good agreement across datasets, including the ASL case, with $u_r$ normalized by $r{_\omega } / T$. Normalizing both axes by $r{_\omega }$ ensures the negative slope outside the vortex is not a function of the detected vortex size in the plot. Even though the field measurements only detect the largest vortex cores and likely augment their size to some extent due to the spatial resolution, the statistics for $\Delta u{_\omega }$ and $u_r$ in the very-high-Reynolds-number ASL case are entirely consistent with the vortex velocity trends in the laboratory-scale flows.

Assuming $1/T$ is representative of the strain rate $\alpha$ for the largest vortex cores as suggested by figure 6(d), the proportionality of $u_r$ with $1/T$ outside the vortices in figure 10(a) is consistent with the profile $u_r(r) = - \alpha r$ predicted by the Burgers vortex. The result confirms the flow field around the vortex on average moves towards the vortex core at a speed proportional to the large-eddy strain rate $1/T$. By continuity, this inward flow must be balanced by axial, out-of-plane motion as discussed further in § 5. The overall shape of the radial profile and apparent transition point at $r=r{_\omega }$ may in part be due to the fitted Oseen model or spatial resolution, and confirmation with alternative methods is required.

For the shear layers, the wall-normal velocity component $w$ indicates flow towards the layers. The shear layers have an average orientation approximately 15 degrees above horizontal (Squire Reference Squire2017), consistent with ramp-like structures, but in the present analysis the ISLs are considered horizontal for simplicity. The same conditionally averaged ISL profiles as in figure 4(a) were computed for $w$. The average profile of $w$ relative to the ISL midpoint is shown for each flow case in figure 10(b), excluding the ASL measurements whose resolution ${\rm \Delta} x$ is coarser than the range of $z - z_i$ in the plot.

The best agreement across cases is achieved using the same normalization parameters as for the streamwise velocity case: $\lambda _T$ for the relative position $z - z_i$ and ${u_\tau }$ for the relative velocity $w - w_i$. In the ISL reference frame, the average flow above the ISL is moving down into the shear layer ($w {<} w_i$) and the flow below is moving up into the shear layer ($w > w_i$). These same trends were observed in previous studies (Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015; de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017).

The normalization of the velocity $w - w_i$ by the strain parameter $\lambda _T / T$ yielded relatively less agreement across experimental datasets compared to ${u_\tau }$. There may be multiple processes resulting in the figure 10(b) profile such that a straining mechanism does not govern the scaling of the average velocity. For instance, the ISLs on average have a small net positive wall-normal velocity $W_i\sim 0.1{u_\tau }$ (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017). The average relative velocity $\langle w - w_i \rangle$ may therefore be a combined result of the ISL advection $W_i$ associated with shear stress events in addition to the local compressive flow into the ISL. For most of the flow cases, the average value of the strain parameter $\lambda _T / T$ in the log region is approximately half the average value of $W_i$, and the ratio decreases for increasing Reynolds number.

A notable deviation in figure 10(b) is the centre-most region of the profile for the DNS case where the wall-normal velocity is uniform, i.e. see the ‘$*$’ markers near $(w-w_i)^{+} \approx 0.25$. The velocity profile is asymmetric such that there is a larger velocity difference above the centre than below, and the DNS profile centre was offset to match the bulk velocity difference outside the ISL. We attribute the DNS profile shape to a possible Kolmogorov-scaled centre region of the shear layers, which is consistent with predictions (Chini et al. Reference Chini, Montemuro, White and Klewicki2017; Montemuro et al. Reference Montemuro, White, Klewicki and Chini2020). A similar uniform centre is apparent for some of the experimental cases when the profiles in figure 10(b) are further conditioned to only consider ISLs in the wake region far from the wall where the local $\eta (z)$ is larger and the resolution ${\rm \Delta} x / \eta$ is improved. The same wall-normal velocity profile shape has also been observed in previous ISL studies (Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015). A more precise analysis of this centre-most region of the shear layers is not included here, as the focus of the study is the overall thickness of the layers and their relation to $\lambda _T$.