3.1 Bulk statistics

In this section we confirm the two-dimensionality of the studied flows in the measurement area, examine the potential existence of a logarithmic scaling region in the mean velocity profile, and look for potential effects of flow relative submergence on bulk statistics up to order four. These statistics provide background information on the studied flows as bulk characteristics are likely to be associated with features of LSMs and VLSMs identified in the following sections.

Our experiments were conducted with large flow aspect ratios ( $B/H\gtrapprox 10$ ), which according to traditional wisdom (e.g. Nezu & Nakagawa Reference Nezu and Nakagawa1993) should be sufficient to establish two-dimensional flow conditions at the flume centre, i.e. $\unicode[STIX]{x2202}^{-}/\unicode[STIX]{x2202}y=\bar{v}=\bar{w}=0$ , where the overbar represents time averaging. With the long record length and small sampling error in this study, however, we are able to detect small-amplitude transverse fluctuations with period $2H$ in all time-averaged velocity statistics consistent with the presence of secondary currents (e.g. Nezu & Nakagawa Reference Nezu and Nakagawa1993). In order to reduce the dataset and suppress weak transverse fluctuations, we incorporate spatial averaging over a transverse domain of width $n\times 2\times H$ ( $n=3,2,1,1,1$ for $\text{H}030,\ldots ,\text{H}120$ , respectively). With this averaging, denoted here with angle brackets, we can reasonably expect that far from the sidewall the relation $\unicode[STIX]{x2202}\langle \bar{\phantom{v}}\rangle /\unicode[STIX]{x2202}y\approx \langle \bar{v}\rangle \approx \langle \bar{w}\rangle \approx 0$ will hold. In this case the double-averaged momentum conservation equation (e.g. Nikora et al. Reference Nikora, McEwan, McLean, Coleman, Pokrajac and Walters2007) in the streamwise direction for the flow region above the roughness tops reduces to

(3.1) $$\begin{eqnarray}-\unicode[STIX]{x1D70C}gS_{0}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\left\{-\unicode[STIX]{x1D70C}\langle \tilde{\bar{u}}\tilde{\bar{w}}\rangle -\unicode[STIX]{x1D70C}\langle \overline{u^{\prime }w^{\prime }}\rangle +\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\langle \bar{u}\rangle }{\unicode[STIX]{x2202}z}\right\},\end{eqnarray}$$

Figure 3. (a–e) Bulk statistics and (f) distribution of ‘instantaneous’ velocity fluctuations at $z/H=0.5$ . Symbols shown every tenth measurement point for clarity.

where $\unicode[STIX]{x1D70C}$ is fluid density, $\unicode[STIX]{x1D707}$ is fluid dynamic viscosity and we decompose the $i\text{th}$ instantaneous velocity component as $u_{i}=\langle \bar{u}_{i}\rangle +\tilde{\bar{u}}_{i}+u_{i}^{\prime }$ , where tilde and prime symbols represent space and time fluctuations, respectively. Away from the bed, the form-induced stress $-\unicode[STIX]{x1D70C}\langle \tilde{\bar{u}}\tilde{\bar{w}}\rangle$ can be used as a measure of the contribution of secondary currents to the total momentum flux. Figure 3(c) indicates that for our experiments $\langle \tilde{\bar{u}}\tilde{\bar{w}}\rangle /u_{\ast }^{2}$ is quite small ( ${\approx}-\!0.006$ at $z/H=0.5$ ) and $\langle \overline{u^{\prime }w^{\prime }}\rangle /u_{\ast }^{2}$ approaches the linear trend expected for two-dimensional OCF. We therefore conclude that secondary currents make only a minimal (essentially negligible) contribution to the total momentum flux and thus the studied flow region can safely be treated as two-dimensional.

The overlap region of wall-bounded flows where the mean velocity is expected to scale with the log of wall-normal distance remains the subject of intensive theoretical and experimental research (e.g. Adrian & Marusic Reference Adrian and Marusic2012). Debate has largely centred around the limits of the log region and the appropriate value of the von Kármán coefficient ( $\unicode[STIX]{x1D705}$ ) and its potential variation between different flow types. The data suggest that the von Kármán coefficient is probably not universal and for smooth walls has a value of around 0.41 for pipe flow, 0.38 for boundary layers, 0.37 for closed channels (Nagib & Chauhan Reference Nagib and Chauhan2008), and 0.41 for open channels (Nezu & Nakagawa Reference Nezu and Nakagawa1993). It should be noted, however, that experimental uncertainty for the value of $\unicode[STIX]{x1D705}$ is still high (e.g. $\pm$ 0.02, Bailey et al. Reference Bailey, Vallikivi, Hultmark and Smits2014) and its estimates will continue to be refined as new and more accurate data become available. For rough-bed flows, analysis of the log scaling region is further complicated by an additional free parameter in the equation for mean velocity, the so-called zero-plane displacement ( $d$ ), which sets the ‘virtual’ bed level for the log law. Nikora et al. (Reference Nikora, Koll, McLean, Dittrich and Aberle2002) proposed that conceptually the zero-plane displacement could be considered as the level that large-scale attached eddies ‘feel’ as the bed and, therefore, their dimensions should scale linearly with the distance from this location. The differential form of the log law incorporating the zero-plane displacement can be written as $(\unicode[STIX]{x2202}\bar{u}/\unicode[STIX]{x2202}z)^{-1}=\unicode[STIX]{x1D705}(z+d)/u_{\ast }$ (Nikora et al. Reference Nikora, Koll, McLean, Dittrich and Aberle2002), where $z=0$ corresponds to the level of the roughness tops. This ‘diagnostic’ form of the log law is preferable for analysing experimental data, as small deviations from the logarithmic behaviour are easy to detect. The linear relation between the inverse velocity derivative and the vertical coordinate in $(\unicode[STIX]{x2202}\bar{u}/\unicode[STIX]{x2202}z)^{-1}=\unicode[STIX]{x1D705}(z+d)/u_{\ast }$ also provides a simple and unambiguous technique for estimation of $d$ and $\unicode[STIX]{x1D705}$ (Nikora et al. Reference Nikora, Koll, McLean, Dittrich and Aberle2002). Thus, the relation $(\unicode[STIX]{x2202}\bar{u}/\unicode[STIX]{x2202}z)^{-1}=\unicode[STIX]{x1D705}(z+d)/u_{\ast }$ has been used in our study first to identify the existence of the logarithmic region and its bounds, and then to obtain values of $d$ and $\unicode[STIX]{x1D705}$ . Before we summarise the outcomes of our analysis, we should mention that for flows with low relative submergence (small $H/D$ ) the log layer may be disrupted by the roughness elements and appear with modified $\unicode[STIX]{x1D705}$ (e.g. Dittrich & Koll Reference Dittrich and Koll1997); for sufficiently small $H/D$ the log layer is not expected to exist at all (e.g. Jiménez Reference Jiménez2004). Owing to a lack of quality experimental data for low-submergence OCF, the flow conditions required for a logarithmic layer to develop have not yet been established and thus our data may help with clarifying this matter. Figure 3(a) shows that the approximation of the measured data with the logarithmic equation is highly accurate, at least within $z/D>0.5$ and $z/H<0.5$ . Zero-plane displacements, estimated using a regression fit to the inverse velocity gradient between $z/D>0.5$ and $z/H<0.4$ , indicate a ‘virtual’ bed level to be just below the roughness tops, i.e. $d=1.8$ , 1.8, 1.6, 1.1, 2.0 mm for $\text{H}030,\ldots ,\text{H}120$ , respectively. No clear trend in the obtained values of the zero-plane displacement is observed, suggesting that variation in estimates of $d$ with relative submergence is probably due to experimental uncertainties. It is somewhat surprising, given the low submergence of the studied flows, to find that the present data support a logarithmic range with $\unicode[STIX]{x1D705}=0.38$ extending to $z/H\approx 0.5$ (figure 3 a), well beyond the $z/H\approx 0.15$ upper bound expected in other flow types. The extended range of the log scaling observed here suggests that the low-submergence log law may develop due to different physical mechanisms and require different theoretical justification compared to its high-submergence and smooth-bed counterparts. These differences may be further explored as new accurate data at other flow conditions and different roughness types become available.

Second-order statistics $\langle \overline{u^{\prime }u^{\prime }}\rangle$ , $\langle \overline{v^{\prime }v^{\prime }}\rangle$ and $\langle \overline{w^{\prime }w^{\prime }}\rangle$ (figure 3 b) indicate a trend of increasing variance in the near-bed region with growing submergence. Such an effect has been noted before (e.g. Bayazit Reference Bayazit1976; Dittrich & Koll Reference Dittrich and Koll1997) although the physical mechanisms behind this trend remain unclear. Velocity skewness and kurtosis (figure 3 d,e) collapse onto common curves for all experiments, suggesting that these distributions might be independent of relative submergence. Skewness and kurtosis data for open channels are relatively scarce, but we find qualitative agreement between our measurements and those of Hurther & Lemmin (Reference Hurther and Lemmin2000), Nikora & Goring (Reference Nikora and Goring2000b ) and Bigillon, Niño & Garcia (Reference Bigillon, Niño and Garcia2006). The ratio between streamwise and bed-normal velocity skewness approaches a constant value of $-1.05\pm 0.05$ near the mid-depth of the flow (figure 3 d) consistent with the assertion of Raupach (Reference Raupach1981) of a proportional relationship between third moments.

3.2 Large-scale streaks

The pseudo-instantaneous velocity field (figure 3 f) reconstructed from time-series records using Taylor’s frozen turbulence hypothesis is composed of regions of high- and low-momentum fluid alternating in the transverse direction and elongated in the streamwise direction, visually very similar to the superstructures identified by Hutchins & Marusic (Reference Hutchins and Marusic2007) in their boundary layer studies. It is not easy to directly identify length scales of the large streaks from the ‘instantaneous’ flow fields, as they tend to meander, fade and merge with other features without providing discernible start and end points. In the next section we will use correlation functions and spectra to extract characteristic scales and examine the potential influence of relative submergence.

3.3 Correlation functions and spectra

Selected components of the two-point correlation tensor,

(3.2) $$\begin{eqnarray}Cu_{i}u_{j}(\unicode[STIX]{x0394}y,\unicode[STIX]{x0394}z,z_{0})=\frac{\displaystyle \sum ^{y}\displaystyle \sum ^{t}u_{i}^{\prime }(y,z_{0},t)u_{j}^{\prime }(y+\unicode[STIX]{x0394}y,z_{0}+\unicode[STIX]{x0394}z,t)}{\left(\displaystyle \sum ^{y}\displaystyle \sum ^{t}[u_{i}^{\prime }(y,z_{0},t)]^{2}\right)^{0.5}\left(\displaystyle \sum ^{y}\displaystyle \sum ^{t}[u_{j}^{\prime }(y+\unicode[STIX]{x0394}y,z_{0}+\unicode[STIX]{x0394}z,t)]^{2}\right)^{0.5}},\end{eqnarray}$$

calculated assuming homogeneity in the transverse direction are plotted in figure 4 for $z_{0}=0.5H$ ; where $t$ is a time index, $z_{0}$ is a reference elevation, and $\unicode[STIX]{x1D6E5}_{y}$ and $\unicode[STIX]{x1D6E5}_{z}$ are displacement increments. The $-C_{uw}$ and $-C_{uv}$ components when plotted as vectors (as in Marusic & Hutchins Reference Marusic and Hutchins2008) indicate that fluctuations in the streamwise velocity component are correlated with a pattern of repeating counter-rotating depth-scale cells, which may be associated with the large-scale streaks identified in figure 3(f). Note that we adopt vectors ( $-C_{uw}$ , $-C_{uv}$ ) for consistency with Marusic & Hutchins (Reference Marusic and Hutchins2008), who pointed out that such a vector map is approximately equivalent to the linear stochastic estimate where the condition vector is a negative sign $u$ event at the reference point. As it is important to show that the counter-rotating cell pattern repeats (there are four cells visible in figure 4), we show vectors ( $-C_{uw}$ , $-C_{uv}$ ) with unit magnitudes to prevent attenuating visual effects due to a wide range of correlation values. Figure 4 shows that, with increasing submergence, the transverse separation of the cells becomes progressively smaller and the central lobe of the $C_{uu}$ correlation shrinks. The counter-rotating depth-scale cells are reminiscent of secondary current flow patterns that appear near sidewalls in open channels (e.g. Nezu & Nakagawa Reference Nezu and Nakagawa1993). The origin, dynamics and energetics of secondary currents in OCF are still the subject of ongoing research and it is possible that they form due to a preferential alignment of meandering VLSMs caused by the channel sidewalls (see also Adrian & Marusic (Reference Adrian and Marusic2012) for discussion of potential links between VLSMs and secondary currents).

Figure 4. Two-point correlation functions $C_{uu}$ , $C_{uw}$ and $C_{uv}$ for $z_{0}/H=0.5$ . Note that $C_{uw}$ and $C_{uv}$ are shown as vectors ( $-C_{uw}$ , $-C_{uv}$ ) that have unit magnitude for a sharper identification of rotational patterns.

Figure 5. (a–e) Pre-multiplied spectra; amplitude of the (f) LSM and (g) VLSM spectral peaks; (h) wavelengths of the LSM and VLSM peaks; and (i) maximum wavelength of the VLSM versus flow aspect ratio. Symbols are defined in figure 3. Dashed line in panel (h) corresponds to $\unicode[STIX]{x1D706}_{x_{VLSM}}/H=23(z/H)^{3/7}$ proposed by Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009).

To identify the prevailing scales in the flow field, we used pre-multiplied wavenumber spectra as in other LSM and VLSM studies (e.g. Kim & Adrian Reference Kim and Adrian1999; Hutchins & Marusic Reference Hutchins and Marusic2007; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009). The pre-multiplied spectra for the streamwise velocity component ( $k_{x}F_{uu}(k_{x})/u_{\ast }^{2}$ , where $k_{x}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{x}$ and $\unicode[STIX]{x1D706}_{x}$ is streamwise wavelength) were calculated in the frequency domain and averaged over $225\times$ 32 s windows with, assuming homogeneity, additional averaging over the transverse extent of the measurement domain. Taylor’s frozen turbulence hypothesis was used to transform the frequency spectra to the wavenumber domain using a convection velocity equal to the local mean velocity $\langle \bar{u}\rangle$ . Recognising the potential limitations of applying Taylor’s hypothesis at large scales and uncertainty as to the appropriate value of the convection velocity, which may diverge from the local mean velocity in the region $z/H<0.1$ (Nikora & Goring Reference Nikora and Goring2000a ; Cameron & Nikora Reference Cameron and Nikora2008; Stewart Reference Stewart2014), the spectral estimates presented here may be distorted from the true wavenumber spectra. Nevertheless, our methodology is consistent with the approach used in other studies (e.g. Kim & Adrian Reference Kim and Adrian1999; Hutchins & Marusic Reference Hutchins and Marusic2007). The spectra (figure 5) reveal two characteristic length scales in the outer flow and resemble spectra measured in pipes and closed channels with smooth-wall conditions (e.g. Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009). Following the terminology of Kim & Adrian (Reference Kim and Adrian1999), we will refer to flow structures associated with the shorter of these length scales as LSMs and structures associated with the larger length scale as VLSMs. The length scales determined from spectral peaks may not directly relate to the physical length of turbulent structures due to their transverse meandering (Hutchins & Marusic Reference Hutchins and Marusic2007) but, nevertheless, the spectral length scale provides a convenient and consistent reference to compare with other studies. Near the bed ( $z/H<0.1$ ) the spectral maps are dominated by a single length scale and experiments with larger $H$ suggest a bifurcation point where the distinct LSM and VLSM scales emerge. Peak amplitudes of the pre-multiplied spectra at the LSM and VLSM wavelengths ( $\unicode[STIX]{x1D706}_{x_{LSM}}$ and $\unicode[STIX]{x1D706}_{x_{VLSM}}$ respectively) are approximately the same (figure 5 f,g). Amplitude dependence on flow submergence is observed near the bed, consistent with the trend noted earlier for the bulk variance $\langle \overline{u^{\prime }u^{\prime }}\rangle$ . Combining the normalised LSM and VLSM length scales for all experiments (figure 5 h), it is surprising to find that, although estimates of the LSM length scale collapse onto a single curve, the estimates of VLSM length scale are stratified according to flow submergence (or equivalently to flow aspect ratio). In addition to the measured data, figure 5(h) also shows an empirical relationship $\unicode[STIX]{x1D706}_{x_{VLSM}}/H=23(z/H)^{3/7}$ proposed in Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009) based on the available data for pipe and closed-channel flows. As one can see in figure 5(h), the Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009) relation represents a lower envelope curve that is close to the data corresponding to the highest submergence H120 (table 1).

Plotting the maximum of each $\unicode[STIX]{x1D706}_{x_{VLSM}}/H$ distribution (figure 5 i), the VLSMs appear to scale according to $\text{max}[\unicode[STIX]{x1D706}_{x_{VLSM}}/H]=5.7[B/H]^{0.60}=75[H/D]^{-0.60}=3500[H^{+}]^{-0.60}$ , where, given that the ratios $B/D$ and $Du_{\ast }/\unicode[STIX]{x1D708}$ are constant for all experiments, all three depth normalisations ( $B/H$ , $H/D$ , $H^{+}$ ) have equivalent empirical fit. Although further data are required to identify the most appropriate scaling for the VLSMs, we can speculate that $H/D$ scaling does not seem likely given that the smaller LSMs scale independently of the relative submergence (figure 5 h). Reynolds-number ( $H^{+}$ ) effects have been assessed in other flow types and found to have minimal influence on VLSM length scale (e.g. Balakumar & Adrian Reference Balakumar and Adrian2007) though the amplitudes of the pre-multiplied spectra have been found to exhibit Reynolds-number dependence (Hutchins & Marusic Reference Hutchins and Marusic2007). Flow aspect ratio ( $B/H$ ) is perhaps the most plausible scaling given that $B$ and $\unicode[STIX]{x1D706}_{x_{VLSM}}$ are of the same order of magnitude and keeping in mind the potential constraining effect of the flow width.

Figure 6. (a–e) Pre-multiplied two-dimensional spectra (contours) and (f) pre-multiplied one-dimensional spectra. Symbols as in figure 3. The 95 % confidence interval for the one-dimensional spectra is approximately $0.94\,\times$ to $1.07\,\times$ $k_{x}F_{uu}(k_{x})/u_{\ast }^{2}$ .

The two-dimensional pre-multiplied spectrum $k_{x}k_{y}F_{uu}(k_{x},k_{y})/\langle \overline{u^{\prime }u^{\prime }}\rangle$ indicates the distribution of energy across streamwise ( $\unicode[STIX]{x1D706}_{x}$ ) and transverse ( $\unicode[STIX]{x1D706}_{y}$ ) wavelengths. We calculate the two-dimensional spectrum in the hybrid transverse wavenumber–frequency domain and subsequently transform the frequency to equivalent streamwise wavenumber using Taylor’s hypothesis. The contribution of VLSMs is readily apparent in the two-dimensional spectra (figure 6), as they concentrate energy into a narrow range of transverse scales around $\unicode[STIX]{x1D706}_{y}=2H$ for the shallower depth experiments $(\text{H}030,\ldots ,\text{H}070)$ . For the deeper flow experiments (H095, H120), the transverse wavelength of the VLSM appears to be shorter, but this may be due to the limited resolution in $k_{y}$ for these experiments (as the PIV measurement window becomes smaller in terms of $H$ ). The transverse scale of the LSM is elevation-dependent and increases along with the streamwise length scale with increasing $z$ . The VLSM transverse scale in contrast appears to be constant for all  $z$ .

Data presented here indicate that, in rough-bed OCFs, VLSMs scale differently to LSMs in terms of both their transverse and streamwise length scales. These scaling differences suggest that VLSMs may form independently of LSMs rather than as a streamwise alignment of the smaller structures. Further experimental data are required to resolve scaling and formation mechanisms of LSMs and VLSMs in rough-bed OCFs.