2.1 Experimental set up and procedure

Experiments were carried out on a laboratory flume of 17.2 m long, 1 m wide, and 0.72 m deep (Figure 1). At the upstream of the flume, a tank of length 2.8 m, width 1.5 m, and depth 1.5 m were provided to regulate straight flow into the main channel. Uniform sand of 1.1 mm median grain diameter and geometric standard deviation 1.03 were utilized for bed sediment. The water was gradually released with the help of a valve from the overhead storage tank to the inlet tank and ultimately to the main flume. Since it is a recirculating flume, the water is drained into an underground tank and pumped to the upstream overhead storage tank. Flow depth and discharge in the flume were regulated with the help of the tailgate and rectangular notch situated downstream of the flume. A compound channel of 9 m length was constructed in the rectangular flume from 13 m upstream of the flume to 4 m in downstream, as shown in Figure 1. The compound channel was divided into three regions: floodplain, sloping, and the main channel region (Figure 1). The slope was made rigid with an aluminium sheet and kept at a fixed angle of 31°. Vegetation was distributed in the floodplain region staggered from 11 m to 5 m in the flume (Figure 2). The experimental cases and vegetation distribution are discussed in detail in subsequent sections.

2.1.2 Experimental cases

Two sets of experiments were conducted on the compound channel. In the first set, the vegetation zone was uniformly distributed, whereas, in the second set, it was non-uniformly distributed (Figure 2). No-vegetation cases where the flow was allowed to move without vegetation in the floodplain zone were performed for both the discharges. In the first set, the spacing between two vegetation patches was 10 cm centre to centre (c/c) from 11 m (upstream) to 5 m (downstream) in the flume. It consists of two experimental cases according to the height of vegetation in the floodplain: a single-layer and a multi-layer submerged vegetation case. In the single layer, all vegetation patch heights were 6 cm (Figure 3b). The multi-layered vegetation comprises three layers of vegetation height of 3 cm, 6 cm, and 9 cm. This varying height vegetation was distributed throughout the vegetated floodplain zone to achieve heterogeneity, as shown in Figure 3c. For convenience, the experiments in the uniform setup are denoted as U 6 cm uniform for uniform single-layer vegetation and U 3-6-9 cm submerged for uniform multi-layered vegetation in the paper. The second set of experiments was divided into three vegetation zones: sparsely vegetated, medium densely, and densely vegetated floodplain as shown in Figure 2b. Flow moves from sparsely vegetated portion upstream to denser region downstream of the flume. This was done to achieve non-uniformity in spacing in the vegetated floodplain. The spacing between two vegetation patches in sparsely, medium densely, and densely vegetated zones were 20 cm, 15 cm, and 10 cm, respectively. The cases performed for the first set were repeated for the second set, that is, single-layered and multi-layer submerged vegetation. The experiments in the non-uniform setup are denoted as NU 6 cm uniform for non-uniform single-layered vegetation and NU 3-6-9 cm submerged for non-uniform multi-layered vegetation. The average flow depth in the floodplain and main channel region of the compound channel was taken as 10 cm and 20 cm, respectively. The flow depth in the channel is kept constant for different discharges and vegetation cases by regulating the tailgate present in the downstream of the flume (Figure 1). The flow in the channel was in subcritical condition for both discharges for all the experiments. A detailed explanation of the experiments and distribution of vegetation is shown in Table 1 and Figure 2.

2.2 Data collection

3.1 Velocity

Floodplain vegetation density influences the flow behaviour in the entire cross section of the channel. This effect could be seen in the main channel of the compound channel, as seen in Figure 4. Velocity for uniform set up in the main channel does not show any particular consistent trend throughout the flow depth (Figure 4a,c). It is almost equal for $z/h>0.4$. The introduction of different vegetation density zones showed a trend where velocity profile at 5.5 m main channel section is highest whereas at 9.5 m, it shows lowest magnitude velocity. It is seen that there is more than 20% increase of depth-averaged streamwise velocity in the main channel section from 9.5 m section to 5.5 m section. The role of vegetation density can also be seen in Figure 5, where velocity profiles at various uniform and non-uniform setup sections are shown. The 5.5 m cross section was chosen to compare as vegetation density in that region (7 m to 5 m) is the same for both the setup. It is seen that in all the sections from SFPI to MC section, velocity in uniform vegetation set up is higher compared with non-uniform vegetation set up. However, the velocity at the MC section for uniform setup starts to diverge or increase from non-uniform set up at $\mathrm{z}/\mathrm{h}>0.5$. The velocity is almost the same for both vegetation setup at $\mathrm{z}/\mathrm{h}=0-0.5$. The flow in the vegetation zone in a non-uniform setup travels from a sparsely vegetated zone to a densely vegetated zone, whereas in uniform case, the whole vegetation zone is densely populated from 11 m to 5 m (Figure 2b). This decreases the velocity in the floodplain zone of uniform cases, increasing the velocity in the slope and main channel sections. Figure 5 also shows the importance of vegetation height to divert the flow towards the main channel. The velocity in the U 3-6-9 cm submerged case is greater than the U 6 cm uniform case at SFPI, SM, and SE throughout most of the flow depth. In the MC section, velocity in both U 6 cm uniform and U 3-6-9 cm submerged are almost comparable. The reason for greater velocity in slope and main channel sections for multi-layered vegetation (3-6-9 cm submerged) for most cases is the contribution of 9-cm-length vegetation. This shows that though 33% of the vegetation is 3 cm in length, the resistance in flow is mainly provided by 9-cm-length vegetation. The 9-cm vegetation provides more frontal area and flow obstruction. This ultimately decreases the flow velocity in the floodplain region and increases in the slope and main channel sections. This point can be seen in Figure 5, where the velocity of NU 3-6-9 cm submerged case is less than the NU 6 cm uniform case at the SFPI section and becomes larger at SE and MC sections.

3.2 Reynolds shear stress

The Reynolds shear stresses (RSS) help to get a better glimpse of the momentum transfer between different layers in streamwise, spanwise, and vertical directions. There is a constant momentum transfer between the floodplain and the main channel region in the compound channel because of differences in velocities and flow depth across the cross section (Dupuis et al., 2017; Proust & Nikora, 2019). Figures 6 and 7 show streamwise and transverse RSS ( $-\overline{\mathit{\text{uˊwˊ}}}$ and $-\overline{u\prime v\prime }$) for different vegetation cases at 5.5 m cross section for 35 lps. It can be seen that in all the cases, irrespective of discharge and uniformity condition, the RSS is more pronounced in the slope and the main channel sections. For higher flow depths of $\mathrm{z}/\mathrm{h}>0.4$, the streamwise RSS in uniform set up (U 6 cm uniform and U 3-6-9 cm submerged) is greater than in non-uniform cases, primarily at SE and MC sections (Figure 6e,f). However, no particular trend is observed for $\mathrm{z}/\mathrm{h}<0.4$ at both SE and MC sections. The reason might be because of flow irregularities in the lower depth of the channel. The uniform setup increases the velocity in the slope and main channel sections, as seen in Figure 4. As a result, the difference in velocity between the floodplain and main channel increases, ultimately increasing the momentum exchange compared with the non-uniform setup. The pointwise RSS profile of SE and MC (Figure 6e,f) shows that the RSS increases from $\mathrm{z}/\mathrm{h}~0.75$ to $\mathrm{z}/\mathrm{h}=0.2,$ after which it decreases till channel bed. The maximum magnitude of RSS in multi-layered vegetation (U 3-6-9 cm submerged and NU 3-6-9 cm submerged) reaches more than 0.0003 m²/s², which is greater than the homogenous vegetation height (U 6 cm uniform and NU 6 cm uniform). Negative streamwise RSS is observed for all cases, mainly in the floodplain and SFPI regions. However, the magnitude of negative RSS at the SFPI section of NU 3-6-9 cm submerged case is higher than all other cases. This point shows the presence of a strong circulation system in the region. The SFPI and SM sections are also subjected to high transverse RSS ( $-\overline{u\prime v\prime }$) for all cases, as seen in Figure 7. The white portion in slopes shows the extreme magnitude of $-\overline{\mathit{\text{uˊvˊ}}}$ and displays the importance of transverse velocity fluctuations (v $ˊ$) in overall flow in a compound channel. It is also seen that the magnitude of transverse RSS ( $-\overline{\mathit{\text{uˊvˊ}}}$) is also higher than streamwise RSS ( $-\overline{\mathit{\text{uˊwˊ}}}$) especially on the slopes.

3.3 Turbulent intensities

The turbulent intensities are given by the square root of the average of the diagonal elements of the Reynolds stress matrix. The streamwise and transverse intensities are given as ${\sigma }_u=\sqrt{\overline{u\prime u\prime }}$ and ${\sigma }_v=\sqrt{\overline{v\prime v\prime }}$, respectively. Figures 8 and 9 show the streamwise and transverse intensities, respectively, at a 5.5 m cross section for 35 lps. It is evident from the figures that both intensities are greater in uniform set up at the slopes and main channel sections as compared with non-uniform vegetation setup. The maximum streamwise and transverse intensities in the SE and MC sections are greater than 0.08 m/s for a uniform setup, whereas it is within 0.08 m/s for a non-uniform setup (Figures 8e,f and 9e,f). The magnitude of intensities is also seen to be maximum in the SFPI section as it acts as the boundary where momentum exchange occurs between the floodplain and main channel. It is also observed that the magnitude of both streamwise and transverse intensities of multi-layered vegetation (U 3-6-9 cm and NU 3-6-9 cm) at SE and MC sections is higher compared with single-layered vegetation (U 6 cm uniform and NU 6 cm uniform) throughout the flow depth. However, the percentage increase of depth-averaged intensities is different for uniform and non-uniform vegetation set up. There is a more than 15% increase of averaged streamwise intensities ( ${\sigma }_u$) at SE and MC sections in case of NU 3-6-9 cm submerged to NU 6 cm uniform whereas it is within 15% for U 3-6-9 cm submerged compared with U 6 cm uniform. Furthermore, in the case of average transverse intensity ( ${\sigma }_v$), it is more than 20% for non-uniform vegetation setup, whereas it is within 20% for uniform vegetation setup. The combined effect of varying vegetation density and height could result in greater intensity increase in the case of a non-uniform setup. On the other hand, only vegetation height acts as variable whereas vegetation spacing is constant throughout the vegetation zone in the uniform vegetation setup.

3.4 Turbulent kinetic energy (TKE)

The total TKE is the sum of the squares of turbulent intensities in streamwise, transverse, and vertical directions, and it is formulated as $k=\frac{1}{2}\left(\overline{u^{\prime }{u}^{\prime }}+\overline{v^{\prime }{v}^{\prime }}+\overline{w^{\prime }{w}^{\prime }}\right)$. The results of TKE resemble turbulent intensities to some extent, as shown in Figures 8 and 9. The magnitude of TKE is more pronounced in the slopes and main channel sections irrespective of vegetation setup (Figure 10). Like turbulent intensities, the magnitude of TKE in uniform set up is greater than that of non-uniform vegetation throughout the flow depth. The TKE magnitude in multi-layered vegetation cases for both uniform and non-uniform setup is also greater than single or constant vegetation height cases especially at slope and MC sections. However, there is more than a 30% increase of average TKE magnitude for multi-layered vegetation as compared with single-layered vegetation in non-uniform setup whereas it is within 30% for uniform vegetation setup. It is also observed in Figure 10 that TKE magnitude in the floodplain sections is comparatively less than sloping sections. It signifies that the velocity fluctuating components in different directions are more pronounced in the slopes than in the floodplain sections.

3.5 Integral scales

One of the defining features of turbulence flow is the formation of eddies. The eddy size has a wide range of scales, and the largest scale is comparable with the flow depth of the channel. This section analyses the Taylor micro-scale and Euler's scale for U 3-6-9 cm submerged and NU 3-6-9 cm submerged cases.

3.5.1 Taylor scale

The Taylor micro-scale gives the length scale of inertial subrange where inertial effects mostly govern eddy motions. It gives the average size of eddies at a particular point, and it is given as

$${\mathrm{\lambda }}_{\mathrm{T}}={\left(\frac{15\mathrm{\nu }\overline{{\mathrm{u}}^{\prime 2}}}{\mathrm{\epsilon }}\right)}^{0.5}$$ (2)

where ν is the kinematic viscosity of water and u′ is the velocity fluctuations in the streamwise direction. The value of dissipation rate ε is found as given by Krogstad and Antonia (1999):

$$\mathrm{\epsilon }=\frac{15\mathrm{\nu }}{{\overline{\mathrm{u}}}^2}\overline{{\left(\frac{\mathrm{\delta }{\mathrm{u}}^{\prime }}{\mathrm{\delta t}}\right)}^2}$$ (3)

Figure 11 shows the Taylor micro-scale of uniform and non-uniform vegetation floodplain for multi-layered vegetation cases at a 5.5 m cross section. It can be seen that in all cases, the Taylor scale governs the slope and main channel sections. The magnitude of the Taylor scale is larger in uniform multi-layered vegetation (U 3-6-9 cm submerged) compared with non-uniform multi-layered vegetation (NU 3-6-9 cm submerged) for both discharges. This trend was also previously seen in turbulence intensities and TKE analysis. It is found that there are more than 60% points above 0.004 m from slopes and main channel sections for U 3-6-9 cm submerged case at 35 lps. On the other hand, it is less than 20% points for NU 3-6-9 cm submerged case. The mean velocity of the section plays a key role in the Taylor scale, as seen in the dissipation rate equation (2). The flow in the U 3-6-9 cm submerged case is diverted more towards the slope and main channel region, ultimately affecting the average size of eddies.

3.5.2 Euler scale

The Euler scale gives the size of macro-eddies in the channel. These macro-eddies are the energy-containing eddies that disintegrate into smaller eddies whose scales are given by Taylor and Kolmogorov scales. Euler time scale should be determined to find the Euler scale, at first, which is given as

$${\mathrm{E}}_{\mathrm{T}}={\int }_0^{\infty }\mathrm{R}\left(\mathrm{t}\right)\mathrm{dt}$$ (4)

where R(t) is the autocorrelation function and dt is the lag between two consecutive autocorrelation functions. The Eulerian length scale can now be determined as

$${\mathrm{E}}_{\mathrm{L}}={\mathrm{E}}_{\mathrm{T}}\mathrm{U}$$ (5)

where U is the average velocity at a specific point of the measuring section. The results are similar to the Taylor scale, where the macro-eddies govern the slope and main channel sections. There is not much variation of Eulerian time and length scale for multi-layered vegetation cases at 20 lps discharge in the floodplain sections (Figure 12a,b). In U 3-6-9 cm submerged case, the time scale range is 0.02–0.17 s, whereas, in the case of NU 3-6-9 cm submerged, its range is 0.02–0.14 s. However, at 35 lps discharge, the two cases have considerable variation in the floodplain sections. The range for U 3-6-9 cm submerged case is 0.04–0.75 s, whereas, in the case of NU 3-6-9 cm submerged, it lies between 0.01 and 0.34 s. The Euler time scale at the SFPI section is greater than other floodplain sections for all vegetation cases as it acts as a boundary for continuous momentum exchange. The slope and the main channel sections also show greater Euler time and length scale in the U 3-6-9 cm submerged case compared with the NU 3-6-9 cm submerged case for both the discharges. The Eulerian time scale range for U 3-6-9 cm case is 0.02–0.8 s, whereas for NU 3-6-9 cm case is 0.05–0.5 s for both discharges in the slope and MC sections. This point is also evident from Figure 12c, where the Euler length scale is more pronounced in the slope and main channel sections of U 3-6-9 cm submerged case compared with NU 3-6-9 cm submerged case.

3.6 Octant analysis

Figures 13-15 show the occurrence probability of different bursting events for no-vegetation, U 6 cm uniform vegetation, and NU 6 cm uniform vegetation cases, respectively. From the figures, it can be seen that Classes I-A, III-A, I-B, and III-B are less throughout the cross section in case of no-vegetation cases. The contribution is mainly provided by the ejection (Class II-A and Class II-B) and sweep (Class IV-A and Class IV-B) events. The contribution of sweep events throughout the cross section is more than 14%, whereas outward (Class I-A and Class I-B) and inward (Class III-A and Class III-B) interaction mainly lies within 12%. This shows that sweep and ejection events are mainly present in the channel in the no-vegetation case. This scenario changes after the introduction of vegetation in the floodplain region, as shown in Figures 14 and 15. Classes I-A (internal outward interaction) and III-A (internal inward interaction) are seen to govern mainly the SFPI and SM sections. It usually lies between 14% and 17%, which is not seen in the no-vegetation case. The contribution of Classes I-A and III-A in the U 6 cm uniform case is more widespread as compared with the NU 6 cm uniform case. The majority of the points at the SFPI section in the U 6 cm uniform case contribute more than 18% in Classes I-A and III-A, whereas it is within $18\%$ for the NU 6 cm uniform case. The ejection and sweep events mainly dominate the main channel section of the cross-section. The contribution of sweep events (Class IV-A and Class IV-B) surpasses the ejection events (Figures 14 and 15). However, there is not much distinction between both types of sweep events. Another thing noticed is that Classes I-B and III-B are less than other events irrespective of any cases.

Figures 16 and 17 show the occurrence probability of different bursting events for multi-layered vegetation cases at 35 lps discharge. The results are similar to that of uniform height set up (U 6 cm uniform and NU 6 cm uniform), where the slope and main channel region mainly contribute to various bursting events. However, the contribution of Classes I-A and III-A in the SFPI section of multi-layered cases (Figures 16 and 17) is more as compared with single-layered cases (Figures 14 and 15). As seen for single-layered vegetation in Figures 14 and 15, the percentage of sweep events (Class IV-A and Class IV-B) is also greater in the main channel section of multi-layered vegetation, as shown in Figures 16 and 17. Class II-A or internal ejection class is seen on the floodplain region of both U 3-6-9 cm and NU 3-6-9 cm submerged cases. Classes I-B and III-B are less in the slope and main channel region irrespective of discharge and vegetation setup, similar to Figures 14 and 15. However, it is seen to contribute in the floodplain region of U 3-6-9 cm submerged case, as seen in Figure 16.