4.1. Validation

Figure 2 presents profiles of $(a)$ the normalised streamwise velocity $\langle \bar {u}\rangle _x^+$ , $(b)$ the root-mean-square streamwise velocity fluctuations $u_{\mathit{rms}}^+$ and $(c)$ the root-mean-square vertical velocity fluctuation $w_{\mathit{rms}}^+$ at the channel centreline ( $y/B=0.5$ ) in the channel as predicted by the LES, together with DNS (Yao et al. Reference Yao, Chen and Hussain2022) and experimental (Duan et al. Reference Duan, Chen, Li and Zhong2020) data. The mean velocity profile predicted by the LES agrees very well with both the DNS and experimental profiles, all showing features typical of canonical wall-bounded turbulence, with a viscous sublayer at $z^+\lt 5$ , a buffer layer within $5\lt z^+\lt 100$ and a logarithmic layer at $z^+\gt 100$ (Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010). In addition, the LES-predicted streamwise and vertical velocity fluctuations (figure 2 $b{,}c$ ) peak at $z^+\approx 15$ and $z^+\approx 100$ , respectively, which is comparable to other wall-bounded turbulent flows (e.g. Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010; Lee & Moser Reference Lee and Moser2015). For the velocity fluctuations, the LES results agree well with both the DNS and experimental data in terms of the overall distribution, although the magnitudes of the $ u_{\mathit{rms}}^+$ and $w_{\mathit{rms}}^+$ peaks show minor differences between LES and EXP, that are within $10\,\%$ . Overall, the close agreement among LES, DNS and experiments supports the validity of the present LES approach.

4.2. Mean flow structure

Figure 3.

Profiles of the normalised streamwise time-averaged velocity $(a)\,\langle \bar {u}\rangle ^+$ , and root-mean-square of velocity fluctuations: $(b)\,u_{\mathit{rms}}^+$ , $(c)\,v_{\mathit{rms}}^+$ and $(d)\,w_{\mathit{rms}}^+$ at the centreline for the four cases. The grey dashed and dashed-dot lines in panel $(a)$ denote the linear law $u^+=z^+$ and the logarithmic law $u^+=(1/\kappa )\ln (z^+)+A$ , respectively, where $\kappa =0.39$ is the von Kármán constant and $A=4.6$ , following Yao et al. (Reference Yao, Chen and Hussain2022).

Figure 3 shows profiles of $(a)$ the normalised streamwise velocity $\langle \bar {u}\rangle _x^+$ , and the root-mean-square velocity fluctuations: $(b)\,u_{\mathit{rms}}^+$ , $(c)\,v_{\mathit{rms}}^+$ and $(d)\,w_{\mathit{rms}}^+$ along the respective channel centreline for the four simulation cases (table 1). The time-averaged velocity profiles collapse onto each other in the inner regions ( $z^+\lt 400$ ), while the ‘velocity dip’ is significant in the AR23 case near the water surface and completely disappears in the AR86 and AR129 cases (figure 3 $a$ ). These findings align well with the early experimental observations in OCFs reported by Nezu et al. (Reference Nezu, Nakagawa and Tominaga1985), who pointed out that the velocity dip is absent in channels with . Regarding the turbulent fluctuations, the vertical distances ( $z^+$ ) for the peaks of the three components are almost the same across the four cases, with their values decreasing as increases from 4.6 to 12.9. This is attributed to the decline in turbulent kinetic energy (TKE) production at higher , as shown in figure 4. However, case AR23 exhibits a different trend, having a smaller maximum compared with AR46 in all three fluctuation components due to lower TKE production (figure 4 $c$ ) with respect to that in AR46. Figure 4( $a$ ) demonstrates that the differences in the vertical velocity gradient remain invisible across all cases, while the peak of Reynolds shear stress in case AR23 is approximately 10 % lower than in AR46 (figure 4 $b$ ). Consequently, in the AR23 case, the attenuation of the Reynolds shear stress directly dictates the net reduction in TKE production (figure 4 $c$ ). It is also noted that $u_{\mathit{rms}}^+$ and $v_{\mathit{rms}}^+$ increase at larger $z^+$ in the immediate vicinity of the free surface. This is attributed to the slightly higher TKE convection and turbulent transport in the spanwise direction very close to the water surface (not shown for brevity).

Figure 4.

Profiles of ( $a$ ) the vertical gradient of the time- and streamwise-averaged velocity, $\partial \langle \overline {u}\rangle _x/\partial z$ , normalised by $u_*/h$ , ( $b$ ) the Reynolds shear stress, $-\langle \overline {u'w'} \rangle$ , normalised by $u_*^2$ and ( $c$ ) the TKE production, $\langle \textit{TKE}_p \rangle =-\langle \overline {u'w'}\rangle \times ({\partial \langle \overline {u}\rangle _x}/{\partial z})$ , normalised by $u_*^3/h$ , along the channel centreline for the four cases. The legends are the same as in figure 3.

Figure 5.

Contours of the time- and streamwise-averaged velocity normalised by the bulk velocity $\langle \overline {u}\rangle _x/U_b$ for $(a)$ AR23, $(b)$ AR46, $(c)$ AR86, $(d)$ AR129. The solid lines represent the velocity maximum lines.

Contours of the time- and streamwise-averaged velocity, $\langle \overline {u}\rangle _{x}/U_b$ , are shown in figure 5. For cases AR46, AR86 and AR129, the deviations in $\langle \overline {u}\rangle _{x}/U_b$ between any two symmetric positions relative to the channel centreline are less than $1\,\%$ , confirming the inherent symmetry of the flow about the channel centreline and that the level of data convergence is sufficient. Therefore, only half of the cross-section is presented in figure 5 and subsequent figures except AR23. For case AR23, the maximum velocity occurs approximately $0.3h$ below the water surface at the centreline. This dip distance is slightly smaller than those reported for OCFs with : for example, Nezu et al. (Reference Nezu, Nakagawa and Tominaga1985) and Yu et al. (Reference Yu, Modesti and Pirozzoli2023) provided values of approximately $0.368h$ and $0.35h$ , respectively. Given a negative correlation between dip distances and demonstrated by Yu et al. (Reference Yu, Modesti and Pirozzoli2023) and the slightly larger aspect ratio in case AR23, the location of the velocity dip observed here is considered consistent with previous studies. The isolines in the vicinity of the bottom wall bisector bulge downwards, a result of the bottom corner vortex, which is consistent with the conventional knowledge (e.g. Prandtl Reference Prandtl1952) and recent DNS at of Yu et al. (Reference Yu, Modesti and Pirozzoli2023). For higher cases, the velocity dip at the channel centreline disappears, yet the bulging towards the corners still exists. Following Nezu et al. (Reference Nezu, Nakagawa and Tominaga1985), a velocity maximum line, $z_{\textit{max}}/h$ , at which $\langle \overline {u}\rangle _{x}/U_b$ attains its maximum at each lateral position, is examined and shown as solid lines in figure 5. The profiles of these maximum velocity lines for each case are nearly the same against the distance from the sidewall ( $y/h$ ). For instance, the profile of AR129 is nearly identical to that of AR86, and both coincide with AR46 up to $y/h=2.3$ . The shapes and lateral extents of these maximum velocity lines agree with those reported by Nezu et al. (Reference Nezu, Nakagawa and Tominaga1985) at comparable aspect ratios (e.g. a lateral extent of approximately $2.5h$ at compared with $2.46h$ in case AR86).

Figure 6.

Contours of the time- and streamwise-averaged secondary flow strength normalised by the bulk velocity $U_s/U_b$ for $(a)$ AR23, $(b)$ AR46, $(c)$ AR86, $(d)$ AR129. The time-averaged secondary flow vectors are superimposed for all cases.

Figure 6 presents contours of the normalised time- and streamwise-averaged secondary flow strength, $U_s/U_b$ for the four cases. Here, the secondary flow strength is calculated as $U_s=\sqrt {\langle \overline {v}\rangle _{x}^2+\langle \overline {w}\rangle _{x}^2}$ (Liu et al. Reference Liu, Stoesser and Fang2022a ). Vectors of the time-averaged secondary flow are also added, with a reference vector of magnitude $0.05U_b$ . The two prominent corner vortices are observed in all four cases, i.e. one near the water surface (hereafter referred to as the surface vortex, SV) and the other near the sidewall–bed corner (hereafter referred to as the bottom vortex, BV). The BV is similar in size and strength regardless of the aspect ratio and is confined within $y/h\lt 0.8$ and $z/h\lt 0.6$ , with a maximum $U_s$ of approximately $2.5\,\%U_b$ , which is consistent with previous findings (Nezu et al. Reference Nezu, Nakagawa and Tominaga1985; Sakai Reference Sakai2019; Yu et al. Reference Yu, Modesti and Pirozzoli2023). By contrast, the size of the SV in cases AR46, AR86, AR129 is substantially larger than that in case AR23, attaining a spanwise width of roughly 2.3 $h$ . This agrees with the OCF measurements at reported by Nezu et al. (Reference Nezu, Nakagawa and Tominaga1985) and Onitsuka & Nezu (Reference Onitsuka and Nezu2001). Meanwhile, the secondary flow is strongest in case AR23 (figure 6 $a$ ) among all cases, where the maximum SC strength reaches $U_s/U_b\approx 0.065$ at the water surface, similar to values previously reported for narrow OCFs (Liu et al. Reference Liu, Stoesser and Fang2022a ; Yu et al. Reference Yu, Modesti and Pirozzoli2023), and consistent with laboratory experiments of Tominaga et al. (Reference Tominaga, Nezu, Ezaki and Nakagawa1989), Nezu & Nakagawa (Reference Nezu and Nakagawa1993) and Zampiron et al. (Reference Zampiron, Cameron and Nikora2020). In addition, a weak vortex spanning the entire water depth is observed at $y/h \approx 2.7$ in the wide channels, cases AR86 and AR129 (figure 6 $c$ , $d$ ). A similar secondary flow was also identified in OCF at $y/h \approx 2.5$ with by Nezu et al. (Reference Nezu, Nakagawa and Tominaga1985). In the two wide cases, SCs are absent near the channel centreline, in agreement with the absence of the velocity dip in figure 5(c,d). As demonstrated by simulations with a free surface in Appendix B, the topology and position of the SV remain robust, although the free-slip boundary condition modestly underestimates its maximum strength.

Figure 7.

Contours of the normalised spanwise shear stress for $(a)$ AR23, $(b)$ AR46, $(c)$ AR86, $(d)$ AR129 in the cross-section. Vectors denote the time-averaged secondary flow.

In order to further elucidate the role of SCs in transporting turbulent momentum, the contours of the normalised Reynolds shear stresses $-\overline {u'v'}$ and $-\overline {u'w'}$ are examined, and these are presented in figures 7 and 8. The highest (absolute) values of $-\overline {u'v'}$ are observed near the sidewalls in all cases. The BV exerts a marked influence on $-\overline {u'v'}$ , characterised by a minimum of $-\overline {u'v'}$ in the centre of the BV. Additionally, a region of elevated levels of $-\overline {u'v'}$ appears on the side of the vortex away from the sidewall, located at $y/h \approx 0.8$ in the case and $y/h \approx 1$ in the other cases. These regions behave as turbulent shear zones that induce localised velocity reduction (figure 5). Similar moderate $-\overline {u'v'}$ regions and corresponding velocity reductions were also reported in the DNS by Yu et al. (Reference Yu, Modesti and Pirozzoli2023). In contrast, $-\overline {u'v'}$ remains consistently low throughout the region influenced by the SV, with normalised values of $-\overline {u'v'}/u_*^{2} \approx 0.2$ . In cases AR86 and AR129, the weak depth-spanning SC near $y/h = 2.7$ gives rise to a zone of slightly negative $-\overline {u'v'}$ spanning the entire water column.

The distribution of $-\overline {u'w'}$ is primarily characterised by a vertical gradient, modulated by local perturbations due to the SCs near the sidewalls. In regions dominated by the SV, a zone of negative $-\overline {u'w'}$ emerges, spanning the entire channel width in AR23 and extending to approximately $y/h = 2.1$ away from the walls in the other cases. The presence of a near-surface region of negative $-\overline {u'w'}$ reflects the presence of a velocity maximum below the water surface (figure 5), the spanwise line of which serves as a separation line between positive and negative Reynolds stresses $-\overline {u'w'}$ . Near the wall–water-surface corner, a very small area of positive $-\overline {u'w'}$ is observed, consistent with observations in partially filled pipes (Brosda & Manhart Reference Brosda and Manhart2022) and narrow open channels (Yu et al. Reference Yu, Modesti and Pirozzoli2023). Furthermore, the BV generates a localised hump in $-\overline {u'w'}$ near the bed, located at $y/h \approx 0.7$ in case AR23 and $y/h \approx 1.0$ in the other cases. The co-occurrence of elevated $-\overline {u'v'}$ and $-\overline {u'w'}$ at these locations indicates strong turbulence production driven by LSMs, as is discussed further in § 4.4.

Figure 8.

Contours of the normalised vertical shear stress for $(a)$ AR23, $(b)$ AR46, $(c)$ AR86, $(d)$ AR129 in the cross-section. The solid lines represent the $\overline {u'w'}=0$ .

4.3. Instantaneous flow

Contours of the normalised streamwise velocity fluctuations in a horizontal ( $x{-}y$ ) plane at $z/h=0.5$ for various aspect ratios are presented in figure 9, illustrating qualitative differences in the coherence of velocity fields. The full channel width is shown here, since the instantaneous field is not symmetric about the centreline. As the streaks meander and evolve in time, these snapshots serve only to illustrate the structures qualitatively. In the cases with and $8.6$ (figure 9 $d{,}c$ ), elongated, meandering streaks of alternating high- and low-momentum fluid, which typically correspond to VLSMs, are clearly visible. These structures resemble those reported in previous studies of turbulent boundary layers and OCFs (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007; Cameron et al. Reference Cameron, Nikora and Stewart2017; Zampiron et al. Reference Zampiron, Cameron and Nikora2020). In contrast, the contours in the case show little evidence of VLSMs. In the case, visualised structures appear to be longer than in the narrowest-channel flow, with some strong low-momentum streaks present in the vicinity of the channel centre (e.g. in the range $x/h=20{-}35$ ), with associated high-momentum streaks being somewhat weaker. The length scale of the streaks in the channels with cannot be reliably determined from the instantaneous flow fields alone, and therefore, spectral analysis is employed in the following section to quantitatively characterise the structures’ dominant wavelengths.

Figure 9.

Contours of the instantaneous normalised streamwise velocity fluctuation at $z/h=0.5$ over the full channel width in channels with $(a)$ AR23, $(b)$ AR46, $(c)$ AR86, $(d)$ AR129.

4.4. Spectral characteristics

Figure 10.

Contours of pre-multiplied spectra of streamwise velocity fluctuations for the four cases at increasing distance from the sidewall: ( $a,c,f,j$ ) $y/h=0.5$ , ( $b,d,g,k$ ) $y/h=1.0$ , ( $e,h,l$ ) $y/h=2.0$ and ( $i,m$ ) $y/B=0.5$ (channel centreline). Peaks in the spectra are highlighted as: solid line denoting LSMs, long-dash line denoting VLSMs and dash-dotted line denoting SCI. Where $k_x=2\pi /\lambda _x$ is the streamwise wavenumber, $\varPhi _{uu}$ is the velocity spectrum. Values of $k_x\varPhi _{uu}/u_*^2\lt 0.1$ are masked.

Pre-multiplied spectra of the streamwise velocity component, $k_x\varPhi _{uu}/u_*^2$ , where $k_x=2\pi /\lambda _x$ is the streamwise wavenumber and $\lambda _x$ is the corresponding wavelength (e.g. Kim & Adrian Reference Kim and Adrian1999; Hutchins & Marusic Reference Hutchins and Marusic2007; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009), are computed using instantaneous velocities in longitudinal planes ( $x{-}z$ planes) at spanwise distances $y/h$ = 0.5, 1.0, 2.0 and $y/B=0.5$ away from the sidewall. Note that the spectra for the $y/B=0.5$ plane are calculated only for the cases AR86 and AR129. Approximately 1000 instantaneous snapshots are collected at uniform time interval in each of the longitudinal planes. The spectra are initially obtained for each time instant and subsequently ensemble averaged to produce the smoothed spectral contours presented in figure 10. For $y/h=0.5$ and $ z/h \lt 0.6$ (figure 10 $a{,} c{,} h{,} j$ ), which corresponds to the region occupied by the BV (figure 7 $a{-}d$ ), all cases exhibit a spectral peak at approximately $\lambda _x\approx 6h$ . Therefore, this wavelength is likely to be associated with SCI, as defined in Zampiron et al. (Reference Zampiron, Cameron and Nikora2020). Noticeably, the wavelength of the SCI appears to be larger than that of the LSMs (solid lines), i.e. $\lambda _x\approx 3h$ for all cases in figure 10 ( $b{,} d{,} g{,} k$ ). It is hypothesised that the spectral peak at $\lambda _x\approx 6h$ represents an instability associated with sidewall-generated SCs, and more evidence of this SCI is provided in § 4.5. At $y/h=0.5$ , another region of elevated $k_x\varPhi _{uu}/u_*^2$ emerges at $z/h\gt 0.8$ . Its characteristic wavelengths are identified with the aid of figure 22 in Appendix D, in which $k_x\varPhi _{uu}$ is normalised by the $z$ -dependent streamwise Reynolds normal stress. Figure 22 shows that the dominant wavelengths range from approximately 3 $h$ –4 $h$ and decrease with distance from the free surface, down to $z/h\approx 0.7$ . Within $0.6\lesssim z/h\lesssim 0.7$ , a deficit in the pre-multiplied spectral energy is observed in the $y/h=0.5$ plane. This deficit is likely associated with the reduced TKE production there (Appendix E; figure 23). Conversely, the TKE production rate increases toward the free surface (figure 23), further supporting the elevated $k_x\varPhi _{uu}/u_*^2$ observed at $z/h\gt 0.8$ (figure 10 $a{,} c{,} h{,} j$ ).

For , another spectral peak emerges at approximately $\lambda _x\approx 20h$ at $y/h=0.5$ , suggesting that VLSMs extend into this near-sidewall region. These signatures may arise either from the local generation of VLSMs independently of the SCIs or from the sidewall-ward transport of VLSMs generated near the channel centre. Future investigations are required to assess the relative importance of these two mechanisms. The wavelengths of the SCI and VLSMs agree with what was observed in OCFs recently. For example, wavelengths of VLSMs reported in Zampiron et al. (Reference Zampiron, Cameron and Nikora2020) and Duan et al. (Reference Duan, Chen, Li and Zhong2020) are $\lambda _x=19.7{-}25.3h$ and ${\approx} 20h$ , respectively, with more similar estimates given in a recent summary on VLSM wavelengths in smooth-bed OCFs in Cerino et al. (Reference Cerino, Zampiron, Proust, Berni and Nikora2024). The wavelengths of SCIs found by Zampiron et al. (Reference Zampiron, Cameron and Nikora2020) and Luo et al. (Reference Luo, Stoesser, Cameron, Nikora, Zampiron and Patella2023) are $\lambda _x=2.7{-}5h$ and $4{-}10h$ , respectively, which are consistent with the current estimate.

Away from the sidewalls (figure 10 $e{,} h{,} i{,} l{,} m$ ), the maximum values in the velocity spectra near the water surface gradually diminish, accompanied by an expansion of the high-energy regions near the bed. The high-energy VLSM regions reach almost to the water surface, most noticeably at the channel centreline for the case. The spectral signatures of SCIs become insignificant in planes with $y/h \ge 1.0$ across all cases. Instead, a shorter wavelength peak (the solid lines) emerges in these planes. This spectral peak represents the LSMs originating from the bottom walls, with a typical wavelength ( $\lambda _x\approx 3h$ ) similar to those reported for OCFs (e.g. Cameron et al. Reference Cameron, Nikora and Stewart2017) and other wall-bounded flows (Kim & Adrian Reference Kim and Adrian1999; Hutchins & Marusic Reference Hutchins and Marusic2007; Smits et al. Reference Smits, McKeon and Marusic2011). It is also noteworthy that the wavelength of VLSMs in the AR46 case is approximately $10h$ at $y/h=2$ , which is nearly half of that for the large cases. A similar trend is also reported by Peruzzi et al. (Reference Peruzzi, Poggi, Ridolfi and Manes2020), who found that the VLSM wavelength in OCFs at (also ${\sim} 10h$ ) is approximately half of that at .

Figure 11.

Pre-multiplied spectra of streamwise velocity at fixed vertical locations at ( $a$ ) $z/h=0.2$ , ( $b$ ) $z/h=0.5$ , ( $c$ ) $z/h=0.75$ , ( $d$ ) $z/h=0.9$ for four cases, spectra for successive are offset vertically by $0.5$ ( $0.3$ in panel $c$ ). The dashed and solid lines denote the spectra based on velocities at $y/h = 0.5$ and the channel centreline, respectively, for all four cases. The legends are the same as in figure 3.

In figure 11, pre-multiplied spectra at various elevations including ( $a$ ) $z/h=0.2$ , ( $b$ ) $z/h=0.5$ , ( $c$ ) $z/h=0.75$ , ( $d$ ) $z/h=0.9$ are plotted for the four cases. The solid and dashed lines represent spectra at the channel centreline and near the sidewall ( $y/h=0.5$ ), respectively. At low elevations ( $z/h = 0.2$ , figure 11 $a$ ), the spectral energy in the channel centre is higher than near the sidewall across all ARs. This arises from stronger turbulence production due to higher near-bed velocity gradients (figure 5) and higher Reynolds shear stress (figure 8). However, at the highest elevation ( $z/h=0.9$ , figure 11 $d$ ), this trend reverses: the turbulent energy near the sidewall noticeably exceeds that in the centre, suggesting intensified streamwise fluctuations near the water surface driven by SCs. For the sidewall spectra, the dominant wavelength is $\lambda _x\approx 6h$ across most and elevations, except at $z/h=0.2$ in AR23 and at $z/h=0.75$ , where the peak is less distinct. Whereas for the centreline spectra, VLSMs with $\lambda _x \approx 20h$ are identified at all elevations for the two higher cases, although the peak becomes broader and weaker towards the free surface. At , large-wavelength peaks are noticeable within $z/h \le 0.5$ (figure 11 $a$ , $b$ ), but the wavelength is approximately $10h$ . There are also LSMs signatures in the pre-multiplied spectra in the channel centre, especially visible in the near-bed region of flow cases with larger aspect ratios.

Figure 12.

Spanwise variation of pre-multiplied spectra of streamwise velocity fluctuation at three wall-normal elevations: $(a,d,g,j)$ $z/h=0.2$ , $(b,e,h,k)$ $z/h=0.5$ and $(c,f,i,l)$ $z/h=0.8$ , with rows corresponding to AR23, AR46, AR86 and AR129, respectively. The solid, dash-dotted and dashed lines denote the LSMs, SCIs and VLSMs respectively. The solid and dashed lines originate at $y/h=3.0$ and extend towards the sidewall, whereas the dash-dotted lines originate at the sidewall and extend towards the channel centre. The spanwise extent of each line is determined by its intersection with the isoline corresponding to a $z/h$ -dependent threshold of $k_x\varPhi _{uu}/u_*^2$ . The threshold values are set to $k_x\varPhi _{uu}/u_*^2$ = $0.7$ , $0.4$ and $0.2$ at $z/h=0.2$ , $z/h=0.5$ , $z/h=0.8$ , respectively. Values of $k_x\varPhi _{uu}/u_*^2\lt 0.1$ are masked.

Figures 10 and 11 suggest a progressive transition in the dominant motions from SCIs near the sidewalls to VLSMs toward the channel centre in cases with . To further elucidate the spanwise organisation of energetic motions, figure 12 presents the pre-multiplied spectra of streamwise velocity as functions of the streamwise wavelength $\lambda _x/h$ and the spanwise coordinate $y/h$ at three representative vertical locations ( $z/h = 0.2$ , $0.5$ and $0.8$ ). This analysis reveals the lateral distributions of the energy of velocity fluctuations and their characteristic wavelengths across the channel width. Unlike figures 10 and 11, where spectra are computed in the streamwise wavenumber domain, the data in figure 12 are obtained in the frequency domain. Using the 40 FT velocity time series collected at $x/h=10.0$ , the frequency spectra are obtained at uniform spatial intervals defined by the background (coarse) grid resolution (i.e. at every two grid points in the cross-section). To convert frequency to streamwise wavelengths, Taylor’s frozen-turbulence hypothesis is employed using $\lambda _x=\langle \overline {u}\rangle _x/f$ , where $f$ is the frequency. The applicability of this conversion is assessed in Appendix F (figure 24), where direct comparisons with spatial spectra show close agreement over the energy-containing wavelength range, thereby validating the use of Taylor’s hypothesis.

At all vertical locations and across all cases, the pre-multiplied spectra exhibit a clear two-zone lateral structure, consisting of a near-sidewall region and a region farther from the sidewall. A transient zone at $y/h \approx 0.2-0.4$ of reduced spectral energy separates these two regions. Near the bed (figure 12 $a$ , $d$ , $g$ , $j$ ), the near-sidewall spectral zone remains confined within $y/h \le 0.2$ and shows no distinct energetic wavelengths. In contrast, SCI-associated wavelengths ( $\lambda _x/h \approx 6$ , dash-dotted circles) appear as a local spectral bulge within $0.3 \le y/h \le 0.6$ for all cases. Farther away from the sidewall, the dominant spectral energy shifts to the LSM-associated scale (solid lines), $\lambda _x/h \approx 3$ . At half-water depth ( $z/h$ = 0.5 in figure 12 $b$ , $e$ , $h$ , $k$ ), the lateral arrangement of the LSM-associated wavelength bands resembles that observed near the bed, albeit with overall reduced spectral energy. For AR86 and AR129, an additional spectral feature emerges at $y/h \gtrsim 2.1$ , i.e. spectral hills at wavelengths at $\lambda _x/h \approx 20$ , corresponding to VLSMs. However, the SCI-associated bulges within $0.3 \le y/h \le 0.6$ disappear. Concurrently, a new $ \lambda _x/h \approx 6$ spectral ridge develops adjacent to the sidewall and extends to $y/h \lesssim 0.4$ (figure 12 $b$ , $e$ , $h$ , $k$ ). The lateral extension of this ridge and the spectral energy remain at the highest elevation $z/h$ = 0.8 (figure 12 $c$ , $f$ , $i$ , $l$ ), indicating a consistent SCI zone at $z/h\gt 0.5$ near the sidewall. In addition, the LSM band weakens substantially at $z/h=0.8$ and no longer forms a distinct spectral peak (a weak LSM signature nonetheless persists, e.g. in AR86), so no LSM line is drawn at this elevation; the VLSM footprints, by contrast, remain noticeable.

Figure 13.

Spanwise distributions of the normalised pre-multiplied streamwise velocity spectra for $(a)$ AR23, $(b)$ AR46, $(c)$ AR86 and $(d)$ AR129. For each case, the upper and lower subpanels show depth averaging over the sampled elevations within the upper ( $0.6\lt z/h\lt 1$ ) and lower ( $0.1\lt z/h\lt 0.6$ ) depth ranges, respectively. The orange solid and magenta dash-dotted lines denote the spectral energy evaluated at $\lambda _x=3h$ (LSM scale) and $\lambda _x=6h$ (SCI scale), respectively. The cyan dashed lines denote the long-wavelength contribution evaluated at $\lambda _x/h=10$ for AR23 and AR46 and at $\lambda _x/h=20$ for AR86 and AR129.

To quantitatively determine the locations of transitions among SCIs, LSMs and VLSMs, figure 13 presents the spanwise distributions of the pre-multiplied spectra averaged separately over the lower layer ( $0.1\lt z/h\lt 0.6$ ) and the upper layer ( $0.6\lt z/h\lt 1$ ). The spectra are evaluated at three selected wavelengths, i.e. at $\lambda _x/h = 3.0$ ( $\langle k_x\varPhi _{uu}\rangle _z|_{\lambda _x=3h}$ ), $\lambda _x/h = 6.0$ ( $\langle k_x\varPhi _{uu}\rangle _z|_{\lambda _x=6h}$ ) and $\lambda _x/h =$ 10.0 ( $\langle k_x\varPhi _{uu}\rangle _z|_{\lambda _x=10h}$ in AR23 and AR46) or 20.0 ( $\langle k_x\varPhi _{uu}\rangle _z|_{\lambda _x=20h}$ in AR86 and AR129). These wavelength-dependent pre-multiplied spectra represent the contributions of LSMs, SCIs and VLSMs to the streamwise velocity fluctuations at different spanwise locations and vertical regions. Across all cases, the spectral energy averaged over $z/h\lt 0.6$ is generally higher than that averaged over $z/h\gt 0.6$ , particularly in the channel centre region, as TKE is primarily produced near beds (figures 4 c and 23). The contribution of VLSMs is markedly lower than that of LSMs and SCIs at both $z/h\lt 0.6$ and $z/h\gt 0.6$ across all spanwise locations in AR23 (figure 13 $a$ ), within $y/h\lt 0.6$ in AR46 (figure 13 $b$ ) and within $y/h\lesssim 2.4$ in AR86 and AR129 (figure 13 $c$ , $d$ ). Beyond $y/h=0.6$ , the contribution at $\lambda _x/h=10$ becomes roughly comparable to those at the LSM- and SCI-associated wavelengths in AR46. Whereas, in AR86 and AR129, the spectral energy at $\lambda _x/h=20$ exceeds that of the other two since $y/h \approx 2.4$ . While the spectral magnitudes of LSMs and SCIs are comparable at $z/h\gt 0.6$ across all cases (the upper layer in figure 13(a–d)). Their relative energy contributions exhibit a spanwise dependence at $z/h\lt 0.6$ (the lower layer). Near the wall–bed corner ( $y/h\lt 0.5$ and $z/h\lt 0.6$ in AR23 and $y/h\lt 0.6$ and $z/h\lt 0.6$ in the other three cases), the SCI-associated spectral energy is slightly higher than the LSM- and VLSM-associated contributions across all cases. Beyond $y/h\approx 0.6$ , the LSM-associated contribution becomes slightly higher than that of SCI until the VLSM-associated contribution becomes increasingly important. Taken together, figures 12 and 13 establish a consistent picture in which the organisation of energetic turbulent motions transitions from a SCI-dominated near-sidewall dynamics to LSM- and ultimately VLSM/LSM-dominated behaviour, particularly at $z/h\lt 0.6$ .

4.5. Phase-averaged statistics

In order to visualise flow features that correspond to the respective turbulence structures, SCIs and VLSMs, a phase-averaging procedure is applied to the velocities in the cross-section (Luo et al. Reference Luo, Stoesser, Cameron, Nikora, Zampiron and Patella2023; Tang et al. Reference Tang, Stoesser, Huang, Liu and Fang2025). The procedure involves (i) subtracting the local time-averaged streamwise velocity from the instantaneous velocity to obtain the streamwise velocity fluctuation and applying a Gaussian filter to the streamwise velocity fluctuation at selected locations, (ii) identifying local maxima and minima of the filtered fluctuation signal and their corresponding frequencies, (iii) dividing a maximum-minimum-maximum cycle into six phases (snapshots) by six equal time intervals and (iv) calculating the respective phase-averaged velocity in the entire cross-section (Appendix G). The phase-averaged velocity fluctuations are denoted as $\dot {u}'$ , $\dot {v}'$ and $\dot {w}'$ . Figure 14 shows an example of the raw and Gaussian filtered streamwise velocity fluctuation $(a)$ at the centre of BV ( $y/h=0.5$ and $z/h=0.2$ ) and $(b)$ at the channel centreline ( $y/B=0.5$ and $z/h=0.2$ ) in case AR86 and their respective velocity spectra ( $c$ ) and ( $d$ ). The grey dashed lines in figures 14 $(a)$ and 14 $(b)$ are two sinusoidal curves with wavelengths of $\lambda _x/h=6$ and $\lambda _x/h=20$ , respectively. The Gaussian filter is applied to remove the LSM-scale and smaller-scale fluctuations, so that the SCI (and the VLSM at the centreline) can be reliably identified and used as the phase reference. The width (i.e. standard deviation) of the Gaussian filter is chosen to be $0.6h$ . The velocity spectra of the original and Gaussian-filtered signals (figure 14 $c{,}d$ ) show that the spectral density of LSMs ( $fh/U_b \approx 0.33$ ) reduces by approximately one order of magnitude, while the spectral densities of SCI and VLSMs are largely retained ( $fh/U_b \le 0.16$ ). For the SCI (figure 14 $a$ ), a total of 480 local maxima and minima (blue open symbols) are identified during the ${\sim} 40$ FT duration data (only the first 10 FT of data are shown in figure 14), yielding a periodicity of $\langle \overline {u}\rangle _{x}\Delta t=6h$ (in other words, $\lambda _x/h=6$ ), while for the VLSMs, 84 cycles are identified (figure 14 $b$ ), corresponding to a wavelength of approximately $20h$ . The number of cycles identified for the other three cases (not shown for brevity) also agrees with the corresponding wavelengths in figure 10. At the centre of BV, all cases exhibit a range of 155–240 cycles. At the channel centreline, case AR23 detects 87 cycles, while cases AR46 and AR129 show 88 and 85 cycles, respectively.

Figure 14.

The LES-computed and Gaussian-filtered velocity signals for case AR86 $(a)$ at the centre of the BV and $(b)$ in the channel centreline, at the same level as in $(a)$ . Power spectral density (PSD) of the streamwise velocity fluctuation $(c)$ at the centre of the BV and $(d)$ in the channel centreline for the same case. The grey dashed lines are sinusoidal reference curves with wavelengths of $\lambda _x/h=6$ in ( $a$ ) and $\lambda _x/h=20$ in ( $b$ ).

Figure 15 illustrates the phase-averaged in-plane velocity vectors ( $\dot {v}, \dot {w}$ ), defined as $\dot {v}=\dot {v}'+\overline {v}$ and $\dot {w}=\dot {w}'+\overline {w}$ , together with the phase-averaged streamwise velocity fluctuation ( $\dot {u}'$ ) of case AR46. The first column in figure 15 shows results obtained by applying phase averaging at the centre of BV, whereas the second column corresponds to phase averaging at the channel centreline. The instantaneous centres of the BV and SV are marked by red and blue dots, respectively, in the first column. The inset in figure 15 $(a)$ shows the BV-centre positions over all phases and clearly reveals a clockwise, periodic back-and-forth cycling motion. When $\dot {u}$ at $y/h = 0.5$ , $z/h = 0.2$ reaches its maximum (figure 15 $a$ ), the BV centre is farthest from the sidewall at $y/h = 0.58$ , and the vortex reaches its maximum size, extending to approximately $y/h \approx 1.0$ and $z/h \approx 0.6$ . Owing to this enlarged BV, high-momentum fluid from the outer layer is entrained downward towards the bed along the bed–sidewall corner bisector, producing a region of positive $\dot {u}'$ in the left part of the BV. Meanwhile, low-momentum fluid near the corner is ejected upward around $y/h \approx 1.0$ . Thereafter, the BV contracts and its centre shifts towards the corner region, reaching a minimum position at $y/h = 0.34$ with its size shrinking to nearly half (figure 15 $d$ ). At this phase, $\dot {u}$ reaches its minimum and the surrounding streamwise velocity is uniformly reduced ( $\dot {u}'\lt 0$ ). Notably, the velocity on the right side of the BV increases because of the influx of high-momentum fluid from the SV (figure 15 $c{,}d$ ). As the cycle progresses, a re-expansion of the BV and a retreat of its centre away from the sidewall during the ascending half of the $\dot {u}$ cycle occurs. The vertical displacement of the BV centre is relatively small, approximately one third of its maximum spanwise displacement.

Figure 15.

Contours of the phase-averaged streamwise velocity fluctuation deduced from Gaussian-filtered velocity signals and vectors of the secondary flow ( $\dot {v}/U_b, \dot {w}/U_b$ ) in case AR46. Panels $(a{-}f)$ and $(g{-}l)$ are the six successive phases (equal time intervals over one cycle), with $(a)$ at the maximum and $(d)$ at the minimum, of the phase averaging at the centre of BV and at the channel centreline, respectively. The movement of the corner SC cell centre within one period of SC meandering is shown in an insert in $(a)$ .

The cycling motion of the BV induces a corresponding division–fusion cycle in the SVs. In figure 15 ( $a{,}b{,}e{,}f$ ) (where they exhibit relatively large $\dot {v}$ and $\dot {w}$ from the bed to the channel centre), two distinct SVs emerge: an inner SV at $y/h \approx 0.5$ and an outer SV at $y/h \approx 1.3$ . This likely arises because the enlarged BV occupies a substantial near-sidewall space, limiting the development of a large single SV and resulting in two smaller, spatially constrained vortices. In contrast, when BV contracts (figure 15 $c{,}d$ ) and releases space for SV development, the inner and outer SVs merge into one single, larger SV, whose size and shape resemble the time-averaged SV in figure 6. The clockwise, periodic back-and-forth cycle of the BV and its associated division–fusion cycle in the SVs described above appear again in higher cases (e.g. in AR86; figure 16). This finding suggests the ubiquitous occurrence of these two cycles in wide rectangular open channels. Furthermore, near the sidewall at $y/h \approx 0.2$ and $z/h \gtrsim 0.5$ , a local region of streamwise velocity fluctuations with an opposite sign to that of the instantaneous BV core is observed (figure 15 $a{,}d$ and 16 $a{,}d$ ).

Figure 16.

Contours of the phase-averaged streamwise velocity fluctuation deduced from Gaussian-filtered velocity signals and vectors of the secondary flow ( $\dot {v}/U_b, \dot {w}/U_b$ ) in case AR86. Panels $(a{-}f)$ and $(g{-}l)$ are the six successive phases (equal time intervals over one cycle), with $(a)$ at the maximum and $(d)$ at the minimum, of the phase averaging at the centre of BV and at the channel centreline, respectively. The movement of corner SC cell centre within one period of SC meandering is shown in an insert in $(a)$ .

Figure 17.

Contours of the phase-averaged streamwise velocity fluctuation deduced from Gaussian-filtered velocity signals and vectors of the secondary flow ( $\dot {v}/U_b, \dot {w}/U_b$ ) in case AR23. Panels $(a{-}f)$ and $(g{-}l)$ are the six successive phases (equal time intervals over one cycle), with $(a)$ at the maximum and $(d)$ at the minimum, of the phase averaging at the centre of BV and at the channel centreline, respectively. The movement of corner SC cell centre within one period of SC meandering is shown in an insert in $(a)$ .

When examining the phase-averaged velocities at the centreline (figure 15 $g{-}l$ ), instantaneous near-bed vortices are observed only in phases of maximum (figure 15 $g$ ) and minimum (figure 15 $j$ ) velocities. When $\dot {u}$ at the centreline is near its peak, intensified downward motion transports high-momentum fluid towards the bed, giving rise to a pair of counter-rotating vortices near the sampling point ( $y/B = 1/2$ , $z/h = 0.2$ ). This vortex pair produces a region of positive $\dot {u}'$ flanked by two zones of negative $\dot {u}'$ . In contrast, at the minimum phase of $\dot {u}$ , a new pair of counter-rotating vortices forms with a reversed sense of rotation. These vortices jointly induce an upward flow near the sampling point, resulting in $\dot {u}'$ distributions of opposite sign compared with those at the maximum velocity phase. The remaining four phases represent transitional states between the velocity maximum and minimum phases, characterised by the absence of near-bed vortices and reduced zones of elevated $\dot {u}'$ . In AR46, the vortex pairs in figure 15( $g{,}j$ ) occupy only approximately half of the water depth due to the persistent presence of SV. At higher AR, these pairs span almost the entire water depth (figure 16 $g{,}j$ ) and are also present in some of the transient stages (figure 16 $h{,}l$ ).

Figure 15 suggests that the SCI of BV is associated with periodic oscillations of its position and size, while the VLSMs at the channel centreline are linked to the formation and decay of two pairs of counter-rotating vortices with opposite senses of rotation. The SVs, especially the outer SV, are influenced by both mechanisms, thereby forming an indirect link between the SCI and VLSMs. This linkage is absent at higher (figure 16) and strengthens at . For example, the minimum $\dot {u}$ phases at the BV centre (figure 17 $d{,}e$ ) share similar secondary vortices with the maximum $\dot {u}$ phases at the centreline (figure 17 $g{,}h$ ): (i) small and near-sidewall BV, (ii) strong downward motion near the centreline, (iii) an additional near-bed vortex adjacent to the BV and (iv) zones of negative and positive $\dot {u}'$ around $y/h$ = 0.5 and $y/h$ = 1.0, respectively.

To further investigate the spatial coherence of these flow structures, two-point correlations of the streamwise velocity fluctuations are computed as

(4.1) \begin{equation} R_{uu}(y_r,z_r,\Delta y,\Delta z) = \frac {\overline {u'(y_r,z_r)u'(y_r+\Delta y,z_r+\Delta z)}}{u_{\mathit{rms}}(y_r,z_r) u_{\mathit{rms}}(y_r+\Delta y,z_r+\Delta z)}, \end{equation}

where $(y_r,z_r)$ denotes the location of reference points. Figure 18 presents the contours of $R_{uu}$ with reference points chosen at the centre of the BV (reference points at $y/h=0.5$ , $z/h=0.2$ ; panels $a,c$ ) and at the channel centreline (reference points at $y/B=0.5$ , $z/h=0.2$ ; panels $b,d$ ). Focusing on the sidewall region (figure 18 $a{,}c$ ), the correlation map reveals a strong positive core coincident with the BV, flanked by regions of negative correlation. Crucially, a distinct region of negative correlation is observed adjacent to the sidewall ( $y/h\lt 0.3,z/h\gt 0.5$ ), with their locations coincident with the local region opposite-signed $\dot {u}'$ observed in figure 15 $d$ and 16 $d$ , directly explains the spectral concentration at $\lambda _x\approx 6h$ identified in the near-wall region ( $y/h\lt 0.3$ , $z/h\gt 0.5$ ) in figure 12.

In the channel centre, the correlation contours (figure 18 $b{,}d$ ) reveal negative correlations between velocity fluctuations at spanwise locations approximately 1 $h$ from the reference point. The slight departure of these contours from symmetry about the channel centreline is attributed to finite statistical convergence, as $R_{uu}$ was estimated from approximately $10^4$ time-resolved cross-sectional velocity fields collected over 40 FTs. Nevertheless, the principal correlation patterns remain clearly identifiable. The vertical extent of these structures exhibits a remarkable dependence on . For the narrower-channel flows (figure 18 $b$ ), the correlation structures are vertically confined, occupying only half of the water depth ( $z/h\lt 0.6$ ). This suggests that in lower channels, the development of full-depth VLSMs is inhibited, likely due to the spatial constraints imposed by the sidewall SCs. In contrast, for the wide-channel case (figure 18 $d$ ), the structures are elongated and span the entire water depth, indicating the presence of fully developed VLSMs consistent with observations in wide OCFs (Cameron et al. Reference Cameron, Nikora and Stewart2017).

Figure 18.

Contours of $R_{uu}$ for ( $a,b$ ) AR23 and ( $c,d$ ) AR129. White dots denote the two-point correlation reference points at ( $a,c$ ) $z/h=0.2$ , $y/h=0.5$ and ( $b,d$ ) $z/h=0.2$ , $y/B=0.5$ . Solid and dashed lines represent $R_{uu}=0.1$ and $R_{uu}=-0.1$ , respectively.

Figure 19.

Conceptual schematic of the organisation of flow structure across the channel width. Panels $(a,b)$ are wider-channel flows and $(c,d)$ are narrower-channel flows. The top row $(a,c)$ and bottom row $(b,d)$ illustrate the expansion and contraction phases of the BV cycles, respectively. The light-blue dashed loops in the transitional region $0.6\lt y/h\lt 2.4$ denote vertically constrained, incompletely developed VLSMs.

4.6. Conceptualisation of the interrelations between SCs, LSMs and VLSMs

Based on the preceding spectral, phase-averaging analyses, a schematic representation of the interactions between SCIs and VLSMs at different aspect ratios is proposed in figure 19. Within the near-sidewall region ( $y/h \lt 0.6$ ), the dominant flow structure is the BV located near the sidewall–bed corner. Rather than remaining ‘frozen’ at its time-averaged position, the BV undergoes a clockwise, periodic back-and-forth oscillation about $y/h \approx 0.5$ , which represents the action of the SCIs. When the BV is farthest from the sidewall, it reaches its maximum size and strength, enabling it to transport high-momentum fluid from mid-depth ( $z/h \approx 0.5$ ) towards the sidewall–bed corner, leading to a local flow acceleration (figures 19 $a$ , 16 $a$ ). In contrast, when the BV moves closest to the sidewall, it becomes smallest and weakest, accompanied by a local velocity deficit near the sidewall–bed corner (figure 19 $b$ ). This periodic acceleration and deceleration, i.e. the streamwise velocity modulation, explains the spectral peak at $\lambda _x/h \approx 6.0$ observed in the pre-multiplied spectra. The lateral oscillation and size variation of the BV simultaneously induce periodic velocity fluctuations at $\lambda _x/h \approx 6.0$ in the near-sidewall region around $z/h \approx 0.5$ (figures 12 $b$ and 16 $a{-}f$ ). These fluctuations are first transported towards the sidewall–water-surface corner by the inner SV and then advected along the free surface towards the channel centre, during which their energy gradually dissipates and their characteristic scale decreases (figures 14 and 12). Meanwhile, the outer SV migrates synchronously with the expansion and contraction of the BV: it merges with the inner SV when the BV contracts (figure 19 $b$ ) and separates from it when the BV expands (figure 19 $a$ ).

When the aspect ratio is large, the channel centre region ( $y/h \gt 2.4$ ) develops full-depth, counter-rotating instantaneous vortices (figure 19). The rotation direction of these vortices periodically reverses about the centreline. This behaviour is consistent with previous observations of VLSMs, indicating spanwise meandering of full-depth streamwise vortices (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007; Zampiron et al. 2020, 2024b ).

The region $0.6 \lt y/h \lt 2.4$ can be regarded as a transitional zone in which SCIs, LSMs and VLSMs coexist. Beyond the near-sidewall SCI-dominated region, the LSM-associated spectral energy generally exceeds that associated with SCIs and the distinct peak at $\lambda _x/h\approx 3$ demonstrates that LSMs provide the principal energetic contribution over much of this region (figures 12 and 13). For sufficiently large , VLSM-associated energy is also detectable within this region, although it remains lower than, or comparable to, the LSM contribution and does not form the full-depth vortices observed further from the sidewall. These vertically constrained, incompletely developed VLSM-scale vortices are represented by the light-blue dashed loops in figure 19 $(a{,}b)$ . Thus, $y/h\approx 2.4$ marks the approximate inner boundary of the fully developed VLSM/LSM-dominated region, rather than a threshold below which no VLSMs exist.

From this perspective, the cross-section of a high- OCF can be divided into three lateral regions characterised by distinct large-scale structures: (i) an SCI-dominated region adjacent to the sidewall ( $y/h \lt 0.6$ ), (ii) a transient region ( $0.6\lt y/h\lt 2.4$ ) and (iii) a VLSM/LSM-dominated core region ( $y/h \gt 2.4$ ). In narrow channels (e.g. case AR23), the SCI-dominated region is confined to $y/h \lt 0.5$ , while the transient region transitions into an LSM-dominated region at larger $y/h$ , where VLSMs are absent. For the intermediate aspect ratios, fully developed VLSMs are absent, although shorter-wavelength LSMs may still occur. By comparing the simulated cases ( and ) with flume experimental data over smooth-bed OCFs at the specific aspect ratios ( and adopted in Peruzzi et al. (Reference Peruzzi, Poggi, Ridolfi and Manes2020)), it can be concluded that the lower threshold for the emergence of VLSMs lies between and . Fully developed VLSMs emerge within , becoming fully independent of at its larger values (Cerino et al. Reference Cerino, Zampiron, Proust, Berni and Nikora2024). This dependence of VLSMs on appears to be a feature of smooth-bed flows. Over rough beds, however, VLSM length also depends on the channel width, roughness height and viscous scales (Cameron et al. Reference Cameron, Nikora and Stewart2017; Cerino et al. Reference Cerino, Zampiron, Proust, Berni and Nikora2024), and these additional scaling effects warrant further investigation.

Using the extensive experimental data, Nezu & Nakagawa (Reference Nezu and Nakagawa1993) classified OCFs as narrow or wide, based on the presence or absence of a velocity dip in velocity distribution. They identified a threshold of that separates these two flow conditions. The present study may be regarded as an extension of the Nezu & Nakagawa (Reference Nezu and Nakagawa1993) framework by providing physical insights into the transition from narrow to wide OCFs and identifying three spanwise regions with distinctly different flow dynamics. Notably, the threshold for the full development of VLSMs () agrees with the Nezu & Nakagawa (Reference Nezu and Nakagawa1993) threshold, suggesting that the absence of a velocity dip may serve as an indicator of the presence of fully developed VLSMs. Enhanced physical content of the criterion suggests that this indicator might be more broadly applicable to OCF scenarios, such as the confluence of two OCF at mild or zero angles (Proust et al. Reference Proust, Cérino, Berni and Nikora2025), although this remains to be verified.

Importantly, at , shorter-wavelength VLSMs may still occur. As noted by Duan et al. (Reference Duan, Zhong, Wang, Zhang and Li2021), VLSMs can contribute up to $20\,\%$ of the bed shear stress, and sediment entrainment near the bed is closely linked to their dynamics (Cameron et al. Reference Cameron, Nikora and Witz2020). Whether such shorter-wavelength VLSM influence sediment transport and bed morphology therefore warrants further investigation. Here, it may be speculated that shorter-wavelength VLSMs may lead to the formation of streamwise ridges, given that the typical spanwise spacing of such ridges is approximately $2h$ (Zampiron et al. Reference Zampiron, Cameron and Nikora2020; Luo et al. Reference Luo, Stoesser, Cameron, Nikora, Zampiron and Patella2023; Wang et al. Reference Wang, Kong, Zhong, Yang, Zhang and Wu2025), which coincides with the LSM-dominated region near the sidewalls where shorter VLSMs can still be generated.