2.1. Open-channel flume and flow condition

Experiments were conducted in the $B=1.18\,\text{m}$ wide and $18.0\,\text{m}$ long glass-sided Aberdeen Open-Channel Facility. The entire channel bed and part of the entrance tank (figure 1 a) were covered with a single layer of glass spheres of nominal diameter $16\,\text{mm}$ , sieve selected to a $15.9$ – $16.1\,\text{mm}$ size range. The spheres were installed in a $16.2\,\text{mm}$ -pitch hexagonal packing arrangement using perforated stainless-steel plates, with the rows oriented in the transverse direction. A single flow condition (table 1) was employed to maximise the number of streamwise positions covered and was chosen as representative based on previous experiments reported by Cameron et al. (Reference Cameron, Nikora and Stewart2017), closely corresponding to the ‘H050’ case described in their study. Since that study, the flume entrance tank has been upgraded, including modifications to bed curvature, honeycomb screens and streamwise-oriented vanes (figure 1 a) to damp spurious water-surface fluctuations propagating along the channel. In addition, the vertical-slat exit weir used by Cameron et al. (Reference Cameron, Nikora and Stewart2017) has been replaced with a servo-controlled tilting floor designed to reduce the reflection of water-surface fluctuations back into the channel.

The chosen flow depth, $H_t = 52.0\,\text{mm}$ (table 1, figure 1 b), enabled measurements up to $x/H_t=300$ , while maintaining a friction Reynolds number, $H^+ = \textit{Hu}_*/\nu = 2140$ , where $\textit{u}_*$ is shear velocity and $\nu$ is fluid kinematic viscosity. This value is comparable to previous VLSM studies in other types of wall-bounded flows (e.g. Kim & Adrian Reference Kim and Adrian1999; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009). The roughness Reynolds number, $D^+ = \textit{Du}_*/\nu = 619$ , and the Froude number, $\textit{Fr} = 0.58$ (table 1), indicate a hydraulically fully rough, subcritical flow. The water temperature was stabilised at $17\,^{\circ }\mathrm{C}$ using a closed-loop water chiller. We set $z=0$ at the plane of the roughness crests (figure 1 b) and define the mean-flow depth as

(2.1) \begin{equation} H = \int _{-D/2}^{z_{ws}} \varphi (z)\,\mathrm{d}z \approx H_t+0.20D, \end{equation}

where $H_t$ is the flow depth above the crests, $z_{ws}$ is the mean water-surface elevation and $\varphi (z)=V_{\!f}/V_0$ is the roughness-geometry function (fluid volume fraction) evaluated within a thin bed-parallel layer at elevation $z$ , with $V_{\!f}(z)$ and $V_0(z)$ denoting the fluid and total volumes, respectively (e.g. Nikora et al. Reference Nikora, McEwan, McLean, Coleman, Pokrajac and Walters2007). The lower limit of integration, $-D/2$ , is chosen as the level at which $\varphi (z)$ attains its local minimum, i.e. the plane through the sphere centres. Note that we adopt $H_t$ (rather than $H$ ) to normalise coordinates (e.g. $x/H_t$ and $z/H_t$ ), because $H_t$ is directly measurable from the sphere tops to the mean water surface; this choice is also consistent with earlier work (e.g. Cameron et al. Reference Cameron, Nikora and Stewart2017).

Figure 2.

(a) Four-camera stereoscopic PIV configuration, (b) custom-built semi-immersible lens and (c) Scheimpflug camera mount.

2.2. Stereoscopic PIV

An in-house-developed four-camera stereoscopic PIV system was used to measure all three velocity components in a cross-flow plane spanning $450\,\text{mm}$ in the transverse direction (figure 2 a) and covering the entire flow depth. A pulsed light sheet of width $\delta _{LS}=2.6\,\text{mm}$ (defined here as the $4\sigma$ width of the Gaussian intensity profile) was generated by a Litron ‘Plasma 75-100 PIV’ diode-pumped Nd:YAG laser (pulse energy $75\,\text{mJ}$ at $532\,\text{nm}$ ), shaped using a series of spherical and cylindrical lenses. Images of $30$ – $40$ μm-diameter, neutrally buoyant, silver-coated glass tracer particles were recorded using JAI ‘Spark SP-12000M-CXP4’ cameras at a resolution of $4096 \times 900$ pixels and a frame rate of $100$ image pairs per second. The interframe time was $\Delta t = 1.2\,\text{ms}$ , giving a representative relative out-of-plane displacement of $U\Delta t/\delta _{LS} \approx 0.2$ when scaled with the bulk velocity and light-sheet width. This lies within a well-behaved range where the error associated with out-of-plane displacement follows a predictable trend and remains below the large-displacement regime in which the error approaches exponential scaling (Cameron Reference Cameron2022). For this study, we designed a new five-element, $f/11$ , semi-immersible imaging lens (figure 2 b), based on an earlier design reported in Nikora et al. (Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019). A prism with an $8\,\text{mm}$ side length, mounted at the front of the lens, penetrates the free surface, redirecting the field of view of the downward-facing cameras towards the upstream measurement region (figure 2). Once aligned and focused, the camera system could be positioned anywhere within the channel using a motorised multi-axis traverse.

The PIV measurements were conducted for 2 h at each of the streamwise positions $x/H_t = 5$ , 25, 50, 100, 200 and 300, where $x = 0$ marks the start of the flat section of the flume (figure 1 a). The glass-sphere roughness started 23 flow depths upstream of this location, so boundary-layer development had already begun before $x/H_t = 0$ .

Image processing was performed using an iterative deformation method (IDM) cross-correlation algorithm (Cameron Reference Cameron2011), with $64 \times 64$ pixel interrogation regions weighted by a Blackman window function, evaluated on a $12 \times 12\,\text{pixel}$ grid. The stereoscopic reconstruction procedure involved first dewarping the camera images onto a common grid, then computing per-camera velocity vectors using IDM and finally reconstructing the three-component velocity fields. The resulting impulse response, representing the effective sampling volume, had a full width at half-maximum of approximately $1.1 \times 2.4 \times 2.6\,\mathrm{mm}$ in the streamwise, bed-normal and transverse directions, respectively. Redundancy in the reconstruction equations was exploited to reduce noise contributions to computed velocity statistics. Both the dewarping and reconstruction phases relied on a 13-parameter pinhole calibration model for each camera (e.g. Tsai Reference Tsai1987), with parameters estimated from images of a dot-grid target. The orientation of the coordinate axes is shown in figures 1 and 2.

3.1. Bulk turbulence statistics

Figure 3 shows the development of double-averaged streamwise velocity $\langle \bar {u} \rangle$ , turbulent stresses $\langle \overline {u_i'u_{\!j}'} \rangle$ , dispersive stresses $\langle \tilde {\bar {u}}_i \tilde {\bar {u}}_{\!j}\rangle$ , skewness and kurtosis. These statistics were obtained at each elevation by spatial averaging across a $450\,\text{mm}\,(\approx 9H_t)$ transverse domain and temporally averaging over the 2 h measurement period. At each measurement plane, with the fixed streamwise coordinate omitted for compactness, the velocity was decomposed as

(3.1) \begin{equation} u_i(y,z,t)=\langle \bar {u}_i \rangle (z)+\tilde {\bar {u}}_i(y,z)+u_i'(y,z,t), \end{equation}

where $\langle \bar {u}_i \rangle (z)$ is the spanwise average of the time-averaged velocity at elevation $z$ , $\tilde {\bar {u}}_i(y,z)=\bar {u}_i(y,z)-\langle \bar {u}_i \rangle (z)$ is the spatial fluctuation of the time-averaged field and $u_i'(y,z,t)=u_i(y,z,t)-\bar {u}_i(y,z)$ is the temporal fluctuation. Spatially averaged skewness and kurtosis were computed as $S_{u_i} = {\langle \overline {u_i'u_i'u_i'} \rangle } / {\langle \overline {u_i'u_i'} \rangle ^{3/2}}$ and $K_{u_i} = {\langle \overline {u_i'u_i'u_i'u_i'} \rangle } / {\langle \overline {u_i'u_i'} \rangle ^{2}}-3$ , respectively.

Figure 3.

Development of mean flow and turbulence along the channel: (a) double-averaged streamwise velocity, (b) spatially averaged Reynolds shear stress, (c) dispersive shear stress, (d–f) turbulent normal stresses, (g–i) skewness and (j–l) kurtosis of velocity components.

Figure 3. Long description

Panel A: A line graph shows the double-averaged streamwise velocity. The vertical axis is labeled z/Ht and the horizontal axis is labeled <u>/u*. Different colored lines represent various x/Ht values. Panel B: A line graph displays the spatially averaged Reynolds shear stress. The vertical axis is labeled z/Ht and the horizontal axis is labeled -<u’w’>/u*2. Different colored lines represent various x/Ht values. Panel C: A line graph illustrates the dispersive shear stress. The vertical axis is labeled z/Ht and the horizontal axis is labeled -<u~w~>/u*2. Different colored lines represent various x/Ht values. Panel D: A line graph shows the turbulent normal stress in the streamwise direction. The vertical axis is labeled z/Ht and the horizontal axis is labeled <u’u’>/u*2. Different colored lines represent various x/Ht values. Panel E: A line graph depicts the turbulent normal stress in the transverse direction. The vertical axis is labeled z/Ht and the horizontal axis is labeled <v’v’>/u*2. Different colored lines represent various x/Ht values. Panel F: A line graph shows the turbulent normal stress in the bed-normal direction. The vertical axis is labeled z/Ht and the horizontal axis is labeled <w’w’>/u*2. Different colored lines represent various x/Ht values. Panel G: A line graph illustrates the skewness of the streamwise velocity component. The vertical axis is labeled z/Ht and the horizontal axis is labeled Su. Different colored lines represent various x/Ht values. Panel H: A line graph depicts the skewness of the transverse velocity component. The vertical axis is labeled z/Ht and the horizontal axis is labeled Sv. Different colored lines represent various x/Ht values. Panel I: A line graph shows the skewness of the bed-normal velocity component. The vertical axis is labeled z/Ht and the horizontal axis is labeled Sw. Different colored lines represent various x/Ht values. Panel J: A line graph illustrates the kurtosis of the streamwise velocity component. The vertical axis is labeled z/Ht and the horizontal axis is labeled Ku. Different colored lines represent various x/Ht values. Panel K: A line graph depicts the kurtosis of the transverse velocity component. The vertical axis is labeled z/Ht and the horizontal axis is labeled Kv. Different colored lines represent various x/Ht values. Panel L: A line graph shows the kurtosis of the bed-normal velocity component. The vertical axis is labeled z/Ht and the horizontal axis is labeled Kw. Different colored lines represent various x/Ht values.

At the first measurement cross-section ( $x/H_t=5$ ), the data show that the developing boundary layer extends up to $z/H_t \approx 0.7$ . Above this height, both the mean velocity gradient and the turbulent stresses approach zero (figures 3 a, 3 b, 3 d–f). Further downstream, the boundary-layer thickness increases, reaching the free surface by $x/H_t=25$ . This development is also reflected in the streamwise-velocity skewness (figure 3 g), whose change of sign from negative to positive provides a practical estimate of the boundary-layer edge, consistent with the definition proposed for turbulent boundary layers by Lozier et al. (Reference Lozier, Deshpande, Zarei, Lindić, Abu Rowin and Marusic2025).

By $x/H_t=100$ , the mean streamwise velocity, the turbulent normal stresses $\langle \overline {v'v'} \rangle$ and $\langle \overline {w'w'} \rangle$ and the higher-order statistics have generally converged. The Reynolds shear stress $-\langle \overline {u'w'} \rangle$ , the dispersive shear stress $-\langle \tilde {\bar {u}} \tilde {\bar {w}} \rangle$ and the streamwise-velocity variance $\langle \overline {u'u'} \rangle$ continue to evolve up to the final measurement section at $x/H_t=300$ . The dispersive stress $-\langle \tilde {\bar {u}} \tilde {\bar {w}} \rangle$ above the near-bed region indicates a contribution to the momentum flux from SCs, which intensifies between $x/H_t=5$ and $x/H_t=25$ , and then decreases monotonically towards the end of the flume. A similar decreasing trend in dispersive stresses associated with near-sidewall SCs was also reported by Zampiron et al. (Reference Zampiron, Cameron, Stewart, Marusic and Nikora2023).

Figure 4.

Spatial fluctuations of the mean vertical velocity ( $\tilde {\bar {w}}$ ) normalised by the bulk velocity $U$ , illustrating the development of SCs along the channel. The lateral relative coordinate $y'=0$ marks the approximate centreline of the flume.

3.2. Secondary currents

Figure 4 shows contours of the mean vertical velocity deviation $\tilde {\bar {w}}$ , revealing alternating regions of upflow and downflow consistent with counter-rotating SC cells. Near the flume entrance ( $x/H_t=5$ ), the cells are closely spaced with an average transverse wavelength of $\lambda _{\!y} \approx 1H_t$ , are irregular in strength and are strongest within the boundary layer ( $z/H_t \lt 0.7$ ). Further downstream, the cells grow and merge with neighbouring cells, stabilising at a transverse wavelength of $\lambda _{\!y} \approx 2.2H_t$ while extending across the full flow depth. The near-bed ( $z/H_t\lt 0.1$ ) pattern at $x/H_t=200$ (figure 4) is a local feature associated with upflow around individual spheres and is confined to a thin layer adjacent to the roughness crests. It is not evident in other cross-sections because the PIV plane intersects the roughness array at different offsets relative to the sphere tops. Because this feature is restricted to $z/H_t\lt 0.1$ , it does not influence the interpretation of the SC field in the outer flow.

The SC-strength metric

(3.2) \begin{equation} \varPsi _{\tilde {\bar {u}}_i} = \left [ \frac {1}{A}\iint _{A} \tilde {\bar {u}}_i^{\,2}(y,z)\, \text{d}y\,\text{d}z \right ]^{1/2}, \end{equation}

where $A$ is the PIV measurement domain, was adopted because it provides a direct and robust whole-domain measure of the magnitude of the SC-related mean-flow deviation. Compared with more local or derivative-based measures, for example those based on vorticity or circulation, it avoids spatial differentiation and the need to identify individual cell centres or boundaries, making it well suited to experimental PIV data. It is plotted for all velocity components in figure 5. This confirms that, in a cross-sectional average, the SC strength increases between $x/H_t=5$ and $x/H_t=25$ , and then decreases towards $x/H_t=300$ .

By the end of the flume, the SCs are weak, contributing less than 1.3 % of the total vertical flux of streamwise momentum ( $\approx -\langle \overline {u'w'} \rangle - \langle \tilde {\bar {u}} \tilde {\bar {w}} \rangle$ ), with maximum relative mean vertical velocities of $\tilde {\bar {w}}/U \approx 0.006$ ( $U$ is bulk velocity), consistent with typical values reported for wide channels ( $B/H \gg 5$ ; Nezu & Nakagawa Reference Nezu and Nakagawa1993). The SCs likely arise naturally, either from sidewall effects or from local production by the aligned spherical roughness owing to turbulence anisotropy and heterogeneity. Their confinement to the developing boundary layer, together with their growth over the first $25H_t$ , argues against entrance-tank artefacts associated with bed curvature or the streamwise flow-straightening vanes. Nevertheless, a contribution from these features cannot be entirely excluded.

Figure 5.

Relative strength of SCs along the flume, quantified by the SC-strength metric $\varPsi _{\tilde {\bar {u}}_i}$ defined in (3.2): (a) streamwise component, (b) transverse component, (c) bed-normal component. Symbols are defined in figure 3.

Figure 6.

Pre-multiplied streamwise-velocity spectra: (a), 1-D spectra; (b), 2-D spectra at $z/H_t=0.5$ , illustrating the development of VLSMs along the channel. Solid and dashed lines mark the local maxima associated with LSMs and VLSMs, respectively, while the dash-dot line indicates $\lambda _{\!y} = 2H_t$ . The ‘+’ symbols denote the locations of maximum pre-multiplied 2-D spectral magnitude.

3.3. Velocity spectra

To further examine the development of turbulence along the channel, we present one-dimensional (1-D) pre-multiplied spectra $k_xF_{uu}^{1\hbox{-}\textrm{D}}$ and mid-depth two-dimensional (2-D) pre-multiplied spectra $k_xk_{\!y}F_{uu}^{2\hbox{-}\textrm{D}}$ of the streamwise-velocity component in figure 6. Here, $k_x$ and $k_y$ are the streamwise and transverse wavenumbers, respectively, and $F_{uu}^{1\text{-D}}$ and $F_{uu}^{2\text{-D}}$ are the corresponding one- and two-dimensional spectral densities of the streamwise velocity fluctuations. The 1-D spectra were computed at each elevation by ensemble averaging 350 Hann-weighted time windows of approximately 40 s duration with 50 % overlap, and were subsequently averaged across 412 spanwise points. The 2-D spectra were obtained by averaging over 700 Hann-weighted space–time windows spanning 450 mm in the spanwise direction and approximately 20 s in time, with 50 % temporal overlap. In both cases, the double-averaged streamwise velocity $\langle \bar {u} \rangle (z)$ was used to convert frequency to streamwise wavenumber, as

(3.3) \begin{equation} k_x = \frac {2\pi f}{\langle \bar {u} \rangle (z)} = \frac {2\pi }{\lambda _x}. \end{equation}

The resulting $k_x$ should be interpreted as an approximation, as possible scale-dependence and spanwise variability of the convection velocity are not accounted for.

Figure 7.

(a–b) Amplitude of the LSM and VLSM spectral peaks. (c–d) Corresponding wavelengths of the LSM and VLSM peaks.

Figure 7. Long description

The image contains four line graphs labeled (a), (b), (c), and (d). Each graph shows data for different streamwise positions normalized by the flow depth (x/Ht) with values 5, 25, 50, 100, 200, and 300. Panel A: This line graph shows the amplitude of the large-scale motion (LSM) and very-large-scale motion (VLSM) spectral peaks normalized by the friction velocity squared (kxF1Duu(kx,LSM)/u*^2) on the vertical axis and the normalized height (z/Ht) on the horizontal axis. Different markers and colors represent different streamwise positions. The graph shows a general decrease in amplitude with increasing height. Panel B: This line graph shows the normalized amplitude of the LSM and VLSM spectral peaks (kxF1Duu(kx,LSM)/u*^2) on the vertical axis and the normalized height (z/Ht) on the horizontal axis. Different markers and colors represent different streamwise positions. The graph shows a similar trend of decreasing amplitude with increasing height. Panel C: This line graph shows the corresponding wavelengths of the LSM and VLSM peaks normalized by the flow depth (λx,LSM/Ht) on the vertical axis and the normalized height (z/Ht) on the horizontal axis. Different markers and colors represent different streamwise positions. The graph shows an increase in wavelength with increasing height. Panel D: This line graph shows the corresponding wavelengths of the LSM and VLSM peaks normalized by the flow depth (λx,LSM/Ht) on the vertical axis and the normalized height (z/Ht) on the horizontal axis. Different markers and colors represent different streamwise positions. The graph shows a similar trend of increasing wavelength with increasing height.

The pre-multiplied 1-D spectra (figure 6, left column) show that VLSMs are not detectable near the flume entrance ( $x/H_t=5$ ), but develop progressively along the channel, first appearing as a long-wavelength shoulder at $x/H_t=25$ and $50$ . By $x/H_t=100$ , they have fully emerged as a second local maximum (marked by dashed lines in figure 6), distinct from the shorter-wavelength peak associated with LSMs (solid lines). By extracting the corresponding wavelengths and magnitudes of the pre-multiplied spectral ridges (figure 7), it is evident that the streamwise extent of the VLSMs ( $\lambda _{x,\textit{VLSM}}$ ) continues to grow up to the last measurement cross-section at $x/H_t=300$ , whereas their contribution to the spectral magnitude, $k_xF_{uu}^{1\hbox{-}\textrm{D}}(k_{x,\textit{VLSM}})$ , increases only slowly between $x/H_t=100$ and $300$ . In contrast, both the magnitude $k_xF_{uu}^{1\hbox{-}\textrm{D}}(k_{x,\textit{LSM}})$ and the streamwise wavelength $\lambda _{x,\textit{LSM}}$ of the LSMs stabilise by $x/H_t=100$ . These findings reveal that VLSM scaling depends not only on flow aspect ratio but also on streamwise development, a factor often overlooked in previous studies.

The pre-multiplied 2-D spectra (figure 6, right column) exhibit a trend of VLSMs lengthening along the channel similar to that observed in the 1-D spectra, with their transverse wavelength stabilising at $\lambda _{y, VLSM}/H_t \approx 2$ by $x/H_t = 100$ . However, the VLSM streamwise wavelengths identified in the 2-D spectra are slightly longer than those detected in the 1-D spectra. This likely occurs because the 2-D spectra isolate VLSMs more effectively, whereas the 1-D spectra integrate energy contributions across all transverse wavelengths. This observation may also explain the elevation dependence of the apparent streamwise wavelength of VLSMs inferred from the 1-D spectra (figure 7 d), which is further influenced by uncertainties in the assumed convection velocity.

Together, figures 4, 5, 6 and 7 help explain the earlier observations (figure 3) of dispersive and turbulent stresses continuing to evolve up to $x/H_t=300$ . This can be interpreted as the combined influence of decaying SCs and the progressive strengthening of VLSMs along the channel. This naturally raises the question of a possible connection between SC decay and VLSM growth. To explore this possibility, we examine the transverse wavenumber spectra of the streamwise-velocity component at the three measurement sections where VLSMs are clearly identifiable: $x/H_t = 100$ , $200$ and $300$ .

The 1-D pre-multiplied spanwise spectrum of the streamwise velocity is defined as

(3.4) \begin{equation} k_{\!y}F_{uu}^{1\hbox{-}\textrm{D}}(k_{\!y}) = k_{\!y}\,\overline { \mathcal{F}_{\!y}\{u\}(k_{\!y},t)\, \mathcal{F}_{\!y}\{u\}^{*}(k_{\!y},t) }, \end{equation}

where $\mathcal{F}_{\!y}\{\boldsymbol{\cdot }\}$ denotes the Fourier transform in the spanwise coordinate $y$ , $\overline {(\boldsymbol{\cdot })}$ denotes ensemble averaging and $(\boldsymbol{\cdot })^*$ denotes complex conjugation. Using the decomposition in (3.1) and the linearity of the Fourier transform, we obtain

(3.5) \begin{equation} \mathcal{F}_{\!y}\{u\} = \mathcal{F}_{\!y}\{\langle \bar {u}\rangle \} + \mathcal{F}_{\!y}\{\tilde {\bar {u}}\} + \mathcal{F}_{\!y}\{u'\}. \end{equation}

Substituting (3.5) into (3.4) yields contributions from the double-averaged, dispersive and turbulent components, together with mixed terms. Following the usual double-averaging rules for well-behaved variables (Nikora et al. Reference Nikora, McEwan, McLean, Coleman, Pokrajac and Walters2007) and using the linearity of the Fourier transform, these mixed terms approach zero under sufficient averaging and are therefore neglected here. Furthermore, $\mathcal{F}_{\!y}\{\langle \bar {u}\rangle \}$ is neglected, as it contributes primarily at $k_{\!y}=0$ . The measured spectrum is therefore partitioned as

(3.6) \begin{equation} k_{\!y}F_{uu}^{1\hbox{-}\textrm{D}}(k_{\!y}) = k_{\!y} F_{\tilde {\bar {u}}\tilde {\bar {u}}}^{1\hbox{-}\textrm{D}}(k_{\!y}) + k_{\!y}F_{u'u'}^{1\hbox{-}\textrm{D}}(k_{\!y}). \end{equation}

Here, $k_{\!y} F_{\tilde {\bar {u}}\tilde {\bar {u}}}^{1\hbox{-}\textrm{D}}(k_{\!y})$ represents the dispersive contribution associated with the spanwise structure of the time-averaged velocity field, whereas $k_{\!y}F_{u'u'}^{1\hbox{-}\textrm{D}}$ represents the turbulent contribution associated with temporal fluctuations about the local mean.

Both spectra were computed using a 1-D Hann-weighted spatial window of width 450 mm, spanning the full transverse measurement domain. For the dispersive spectra, the window was applied once to the spatial fluctuation velocity field $\tilde {\bar {u}}(y)$ , whereas for the turbulent spectra it was applied at each time step prior to ensemble averaging over the 2 h record ( $\approx 720\,000$ samples).

Figure 8.

Transverse wavenumber spectra of streamwise velocity: (a) dispersive SC contribution, (b) turbulent contribution and (c) combined SCs and turbulence, measured at $z/H_t = 0.5$ . The vertical dash-dot line corresponds to $\lambda _{\!y}=2H_t$ . Line colours as in figure 7.

The transverse spectra of the mean field provide a spectral signature of the SC organisation. The mid-depth spectra (figure 8) show that SC energy is concentrated in a narrow band around $\lambda _{\!y}/H_t \approx 2.2$ and that this energy decays downstream, consistent with the previously noted weakening of SCs (figures 4 and 5). In contrast, the turbulent spectra show contributions across a broad range of scales, but their maximum close to $\lambda _{\!y}/H_t \approx 2$ suggests that VLSMs provide the dominant contribution (consistent with the 2-D spectra in figure 6). In addition, the magnitude of the turbulent spectra increases with streamwise distance, as previously observed for the growth of VLSMs. The combined spectra $k_{\!y}F_{uu}^{1\hbox{-}\textrm{D}}$ progressively collapse towards a common curve downstream, with some evolution remaining between $x/H_t=100$ and $200$ , while showing excellent collapse by $x/H_t=200$ and $300$ . This behaviour suggests a downstream redistribution of spectral energy, whereby the dispersive SC contribution weakens as the turbulent VLSM contribution strengthens. A possible explanation is that both SCs and VLSMs originate from a common system of depth-scale counter-rotating vortices. This possibility is examined in the next section by analysing statistics of the lateral phase.

3.4. Lateral meandering and phase statistics

To characterise the instantaneous lateral position of the dominant spanwise structure and to quantify its temporal meandering, we computed the instantaneous phase of the Fourier component at the dominant spanwise wavelength ( $\lambda _{\!y} \approx 2.2H_t$ ). The complex Fourier coefficients were defined as

(3.7) \begin{equation} \hat {u}(t;\lambda _{\!y})=\int u(y,t)\,e^{-i k_{\!y} y}\,\mathrm{d}y,\quad \hat {\tilde {u}}(\lambda _{\!y})=\int \tilde {\bar {u}}(y)\,e^{-i k_{\!y} y}\,\mathrm{d}y,\quad k_{\!y}=\frac {2\pi }{\lambda _{\!y}}, \end{equation}

and the wrapped relative phase was then obtained as the argument of the conjugate product

(3.8) \begin{equation} \phi (t;\lambda _{\!y})=\arg \!\big (\hat {u}(t;\lambda _{\!y})\,\hat {\tilde {u}}(\lambda _{\!y})^{*}\big ),\quad \phi \in (-\pi ,\pi ]. \end{equation}

Here, hat notation is used for compactness: $\hat {u}(t;\lambda _{\!y})$ denotes the Fourier coefficient $\mathcal{F}_{\!y}\{u\}(k_{\!y},t)$ evaluated at $k_{\!y}=2\pi /\lambda _{\!y}$ , and similarly for $\hat {\tilde {u}}(\lambda _{\!y})$ . This conjugate product anchors the phase to the SC footprint by subtracting its (time-independent) phase, i.e. it yields the wrapped phase difference $\arg (\hat {u})-\arg (\hat {\tilde {u}})$ , independent of the spatial origin. Thus, $\phi =0$ corresponds to an instantaneous structure aligned with the SC footprint. In practice, the integrals in (3.7) were evaluated over the measured spanwise domain on the discrete PIV grid, using the same Hann windowing procedure as employed for the spanwise wavenumber spectra. The phase $\phi$ is computed from the instantaneous velocity $u(y,t)$ without decomposing it into dispersive and turbulent parts, hence the relative phase contains contributions from both. A key advantage of the phase-based diagnostic is that it is directly connected to the spectral framework through which VLSMs are commonly identified, avoids the threshold dependence and interpretive ambiguity often associated with flow visualisations, and provides a compact quantitative description of lateral meandering.

The probability density function $p(\phi )$ was estimated at the mid-depth elevation using a circular histogram on $(-\pi ,\pi ]$ , discretised into 22 equally spaced bins ( $\Delta \phi = 2\pi /22 \approx 0.29\,\mathrm{rad}$ ) and normalised to unit area. The resulting distributions (figure 9 a) are unimodal, approximately symmetric and centred close to zero phase, which is consistent with the interpretation that SCs and VLSMs derive from the same instantaneous spanwise structure. A uniform distribution on $(-\pi ,\pi ]$ would correspond to no preferred phase relationship between the instantaneous and dispersive components, whereas distinct structures with different preferred alignments would be expected to produce either a multimodal distribution or a persistent offset from zero phase. The probability distribution widens with increasing downstream distance, reflecting enhanced lateral meandering and a progressive broadening towards uniformity (horizontal dash-dot line in figure 9(a)). The qualitative form of $p(\phi )$ and its downstream broadening were not strongly dependent on the sampling elevation within the outer flow, nor on modest variations of $\lambda _{\!y}$ about the spectral peak (results omitted for brevity).

Figure 9.

(a) Probability distribution of lateral phase $\phi$ for wavelength $\lambda _{\!y} = 2.2H_t$ at $z/H_t = 0.5$ . The horizontal dash-dot line denotes the uniform distribution corresponding to no preferred phase relationship. (b) Lateral meandering scale $\sigma _{\!y}$ versus streamwise wavelength of VLSMs. Symbols as in figure 7.

To quantify the downstream broadening of $p(\phi )$ , we use circular statistics (e.g. Fisher Reference Fisher1993) to compute a lateral meandering scale $\sigma _{\!y}$ defined by

(3.9) \begin{equation} \sigma _{\!y} = \frac {\lambda _{\!y}}{2\pi }\,s_a, \quad s_a = \sqrt {\,2\bigl (1-\bar R\bigr )} , \end{equation}

with unit phasors

(3.10) \begin{equation} z_i \equiv \frac {\hat {u}(t_i;\lambda _{\!y})\,\hat {\tilde {u}}(\lambda _{\!y})^{*}} {\,\bigl |\hat {u}(t_i;\lambda _{\!y})\,\hat {\tilde {u}}(\lambda _{\!y})^{*}\bigr |\,} = e^{\,i\phi (t_i;\lambda _{\!y})}, \end{equation}

and mean resultant length

(3.11) \begin{equation} \bar R = \left |\frac {1}{N}\sum _{i=1}^{N} z_i\right |\!, \end{equation}

where $s_a$ is the angular deviation (radians) computed from $\bar R\in [0,1]$ . Because $\phi$ is circular on $(-\pi ,\pi ]$ , we use the circular dispersion based on $\bar R$ rather than a linear standard deviation, since the circular measure is invariant to phase wrapping. The mean resultant length $\bar R$ is the magnitude of the average unit phasor: $\bar R=1$ when all relative phases $\phi (t_i;\lambda _{\!y})$ are equal (instantaneous structures consistently aligned with the SC footprint), and $\bar R\to 0$ for a uniform phase distribution on $(-\pi ,\pi ]$ . We use $N$ ( $\approx 720\,000$ ) to denote the number of samples; $\hat {u}(t_i;\lambda _{\!y})$ and $\hat {\tilde {u}}(\lambda _{\!y})$ are the Fourier coefficients of the instantaneous and dispersive velocity fields defined previously; $t_i$ is the time of the ith sample ( $i=1,\ldots ,N$ ), and $\lambda _{\!y}=2.2H_t$ is the spanwise wavelength of interest. Thus, $\sigma _{\!y}$ provides a physically interpretable estimate of the typical lateral displacement of VLSMs. This estimate is independent of the constant SC reference phase, which only shifts $p(\phi )$ without changing its shape.

Figure 9(b) confirms that the lateral meandering scale $\sigma _{\!y}$ increases downstream and is approximately linearly related to the VLSM streamwise wavelength $\lambda _{x,\textit{VLSM}}$ over the sampled range. Thus, lateral meandering (quantified by $\sigma _{\!y}$ ) emerges as a key parameter that appears to influence the apparent streamwise wavelength of VLSMs and modulates the balance of spectral energy between SCs and VLSMs at the dominant spanwise wavelength. When $\sigma _{\!y}$ is small, the relative phases cluster near zero and the dispersive SC component dominates. As $\sigma _{\!y}$ grows downstream, the phase distribution broadens, the SC contribution weakens and VLSMs lengthen while taking a larger share of the spectral energy at the dominant spanwise wavelength.