Introduction

River confluences are complex hydrodynamic zones where the convergence and mixing of confluent flows create substantial momentum redistribution and mass exchange among the confluent flows by producing large-scale coherent structures (Sukhodolov et al. 2010; Constantinescu et al. 2012). This flow structure plays a vital role in sediment entrainment and its transport. The confluence of the flow produces three-dimensional flow patterns and deep scour holes, causing changes in river morphology and bank erosion. Scour at confluences is caused by convective flow quickening, shear layer turbulence, and divergent pattern of near-bed flow linked to the helical movement (Leite Ribeiro et al. 2012; Yuan et al. 2018). Confluence hydrodynamic zone, a zone close to the confluence, is strongly influenced by confluence symmetry, junction angle, and the momentum flux ratio of the confluent flow (Best 1986, 1987).

Numerous studies have been carried out to investigate the sediment pattern at river confluences (Mosley 1976; Ashmore and Parker 1983; Shafai Bejestan and Hemmati 2008; Liu et al. 2012; Riley et al. 2015; Nazari-Giglou et al. 2016). Mosley (1976) set the groundwork for determining the major controlling parameters, such as junction angle, the discharge ratio etc., affecting the geomorphology at the confluence. Ashmore and Parker (1983) related the scour hole dimension with the confluence angle. Boyer et al. (2006) investigated the importance of the boundaries of the shear layer because of variation in momentum for sediment transport. Their research at the Bayonne-Berthier confluence made a different perspective of maximum sediment transport, which occurred at the edge of the shear layer rather than in the maximum turbulent zone. Best (1987) identified six zones within a confluence, including the stagnation zone in the upstream corner, the flow deflection zone, the flow separation zone immediately downstream of the confluence, the maximum velocity zone, and the flow recovery zone. The maximum velocity zone exists downstream of the confluence of the flows at the reduced cross-section close to the separation zone. Borghei et al. (2004) explained with the help of a right-angled confluence model that the scour depth increases with the discharge ratio of the tributary to the main branch and decreases with the sediment size and the tributary channel width. Leite Ribeiro et al. (2012) noticed in the right-angled confluence channel that the sediment transport in a confluence with a low‐discharge lateral channel is affected by the maximum velocity close to the bed and the turbulent kinetic energy in the shear layer.

Bryan and Kuhn (2002) observed more impact of confluence planform than the confluence angle on bed scour. Symmetry in scouring confluence was identified in a symmetrical (Y-shaped) confluence. However, asymmetrical junctions were endorsed to take more complex scours. Yuan et al. (2018) outlined the detailing of the morphology of the right-angle confluence channels for different discharge ratios. Their findings include corner scour holes, bank-attached bars, mid-channel bars, and mid-channel scour holes. Khosravinia et al. (2019) found an increment in the main channel side slope angle results in supplementary advancement in scour holes through the main and lateral channels. However, the extension of scour in the secondary channel was reduced with the rise of the side slope angle of the main channel.

A laboratory study by Best and Roy (1991) investigated the zero-degree confluence of the channels with a discordant bed and revealed that the mixing layer formed at this confluence is influenced by a separation zone that develops downstream of the entrance of the shallower channel. This interaction distorts the mixing layer, generating a complex three-dimensional flow pattern at the junction. This complex flow promotes the upward movement of water from the deeper channel into the shallower channel, enhancing rapid mixing in the shallower channel while slowing down the mixing in the deeper channel. Suspended sediments or contaminants within the deeper channel may be transferred rapidly across the flow of the shallower channel, upwelling downstream of the junction. This results in lateral movement of fluid from the deeper channel to the shallower channel. Chu and Babarutsi (1988) studied the transverse development of the turbulent mixing layers in a shallow flow downstream of the parallel confluence of the two channels, which were separated by the splitter plate. In the confinement between the free surface and the channel bed, the transverse spreading rate of the shallow mixing layer was initially twice as large as the nominal rate for the free mixing layer. Then the spreading rate reduces with distance from the splitter plate under the stabilizing influence of bed-friction and diminishes to zero in the far field region when the bed-friction number exceeds a critical value. Uijttewaal and Booij (2000) linked the mixing layer growth reduction with loss of coherence in the large turbulence structures. This loss of coherence also reduces the characteristic length-scale that establishes the lateral mixing of matter and momentum in the mixing layer. Cheng and Constantinescu (2014) investigated the development of the mixing layer downstream of the parallel channel numerically and compared with the results of Uijttewaal and Booij (2000).

There are many real-world examples of the river confluence, like confluence of Murray river and its tributary Darling at Wentworth, New South Wales, confluence of the Bhagirathi and the Alaknanda rivers at Devprayag in India, confluence of the Rhone and Arve Rivers in Geneva, Switzerland. Many researchers (Rhoads and Kenworthy 1995, 1998; Rhoads and Sukhodolov 2001, 2008; Miyawaki et al. 2010) have studied the confluence of the Kaskaskia River and the Copper Slough in east central Illinois, U.S.A. Confluence of tail race channel and spillway channel in many hydropower projects like Koldam, Kalagarh in India are real world example. Numerous studies have been carried out for open channel confluence where the bed of the merging channels is either fully rigid or fully mobile. However, as per the knowledge of the authors, as such no study is reported in the literature addressing the problem of the confluence of the rigid and mobile bed channels. In a real-world scenario, the confluence of the spillway (rigid bed) discharge and the power channel (mobile bed) discharge at the Koldam Hydropower Station and Kalagarh dam in India are the example of the confluence of the rigid and mobile bed channels.

The present study is an experimental work on the zero-degree confluence of mobile and rigid bed open channels. The finding of this study includes scour pattern, flow pattern and maximum scour depths adjacent to the confluence region for varying ratios of discharge in rigid bed channel and total discharge, length of the solid apron downstream of the rigid channel, and tailwater depth.

Experimental setup and procedure

Experimental setup

Experiments were conducted in the hydraulics laboratory of the Indian Institute of Technology Roorkee in a flume of 10 m in length, 0.60 m in width, and 0.54 m in depth. Figure 1 shows the plan and sections of the experimental set-up. In the experimental flume, rigid bed channel served as the primary channel and mobile bed channel as secondary channel. An ultrasonic flow meter was fitted in the supply pipe to measure the discharge, and valves were also provided in the supply pipe to regulate the flow. The rigid and mobile bed channels were parallel and of equal width and separated by an acrylic sheet having a thickness of 5 mm. The rigid channel ends at the confluence point and is followed up by mobile bed. A honeycomb brick grid wall was provided at the upstream end of the flume to eradicate large-size eddies in the approach flow. Flow straighteners were provided downstream of the honeycomb masonry wall to maintain the flow parallel to the flume walls. A sediment recess of 0.25 m depth and 1.0 m length was provided downstream of the confluence point. A power-driven tailgate was provided at the end of the flume to maintain tailwater level. Figure 2a shows the photographic view of the bed before the start of the run, and Fig. 2b–d demonstrates the equilibrium scour condition of the bed.

Fig. 1
figure 1

Schematic diagram of the experimental setup

Full size image
Fig. 2
figure 2

Experimental setup for the different apron length (photographic view) a L = 0, initial bed condition and equilibrium scour condition of the bed for b L = 0; c L = 0.125 m; and d L = 0.25 m

Full size image

For each experimental run, the median size of grain (d50) was taken as 1.92 mm, having a geometric standard deviation \(\left({\sigma }_{g}=\sqrt{{d}_{84}/{d}_{16}}\right)\) less than 1.4, confirming the uniformity of the grain as per Marsh et al., (2004). A total of 81 experiments run were conducted in the present study (see Table 1). Discharge in primary channel was varied from 0.01636 to 0.02345 m3 s−1 and in secondary channel 0 to 0.0029 m3 s−1. Discharge ratio is defined as the ratio of discharge in the primary channel and the total downstream discharge (i.e., Qr = Qp/Qd). Experiments were performed for three tailwater depths (dt) i.e., 0.07 m, 0.1, and 0.13 m and three solid apron length downstream of the primary channel (L) i.e. 0, 0.125 m, and 0.25 m. For each experimental run, clear water conditions prevailed. Ratio of average flow velocity and critical velocity was kept less than unity for ensuring clear water condition in the experimental test range.

Table 1 Experimental test summary
Full size table

Experimental procedure

Prior to allowing the flow in the channel, sediment bed was dressed up and covered with plywood to avoid scour in the initial operation of the experiment. When the assigned discharge was allowed in the primary and secondary channels and the tail water depth was maintained, the plywood was removed carefully. The flow velocity in the secondary channel was kept lower than critical velocity to avoid sediment movement in the secondary channel. Two scour holes were formed in the confluence zone, one at the end of the divide wall named A and second at the downstream of the primary channel named B. A point gauge of least counts of 0.1 mm was used to measure the scour profiles in the longitudinal and transverse directions of the channels. Scour was extended in the secondary channel at scour hole A. Vortex formations were found in the confluence zones. The experiment was stopped upon reaching the equilibrium stage, which was about 10 h from the beginning of the experiments. Scouring and deposition in the confluence zone is only due to the flow in the primary channel. A definition sketch of the maximum scour depth of scour hole B (ZBM), maximum scour depth of the scour hole A (ZAM) and extension of scour hole in the secondary channel (LSP) is shown in Fig. 3. Three dimensional velocities were measured using Acoustic Doppler Velocimetry (ADV) for a period of 120 s at 50 Hz frequency. The measured streamwise velocity (u), traverse velocity (v) and vertical velocity (w) were processed by removing weak signals (< 20 signal-to-noise ratio), poorly correlated signals (< 80% correlation) and despiking using the phase-space threshold method proposed by Goring and Nikora (2002).

Fig. 3
figure 3

Definition sketch of maximum scour depth of the scour hole a B; and b A

Full size image

Results and discussion

Flow pattern

Near the surface u-v velocity field is shown in Fig. 4, which depicts that flow in the primary channel is Jet-like flow, however, vortices form downstream of the secondary channel. Flow near to the interface of the primary and secondary channels return in the secondary channel and forms vortices. Such return of the flow at the interface clearly indicates transfer of the momentum from high velocity flow in the primary channel to low velocity or stagnant water in the secondary flow. The return flow led to vortex formation in the secondary channel. Figure 5 shows visualization of the flow using a floating thread. The thread follows the path of the flow and returns after traversing some distance downstream in the channel.

Fig. 4
figure 4

Flow pattern in the channels

Full size image
Fig. 5
figure 5

Visualization of the flow using a floating thread

Full size image

Scour pattern

In the experimental flume, the coordinate system was selected such that x-axis represents the streamwise distance along the length of the flume (x = 0 being the point of confluence), y-axis represents the lateral distance across the width of the flume (y = 0 being the point of confluence), and z-axis represents the depth (or height) in the vertical direction. Figure 6a–c show scour pattern for Qd = 0.02345 m3/s, Qs = 0 and tailwater depths of 0.07 m, 0.10 m, and 0.13 m, respectively. Two scour holes i.e., scour hole A and sour hole B were observed in the channels. The prime reason for developing scour hole A and extension of scour hole in the secondary channel was presence of the vortex flow near the confluence. Further downstream of the confluence, the flow loses most of the energy, and scour depth decreases. Figure 7a, b show the longitudinal scour profiles along scour hole A and scour hole B, respectively for the different tailwater depths. The extension of scour hole in the secondary channel for tailwater of 0.07 m, 0.1 m and 0.13 m were 0.3 m, 0.2 m, and 0.15 m, respectively. A small deposit of the sediment was seen downstream of the secondary channel. The extension of scour hole in the secondary channel decreases with an increase in the tailwater depth.

Fig. 6
figure 6

Scour pattern for and a dt = 0.07 m, Qd = 0.02345 m3/s, Qs = 0; b dt = 0.1 m, Qd = 0.02345 m3/s, Qs = 0; and c dt = 0.13 m, Qd = 0.02345 m3/s, Qs = 0

Full size image
Fig. 7
figure 7

Longitudinal scour profiles a along scour hole A for dt = 0.07 m, 0.1 m, and 0.13 m; and b along scour hole B for dt = 0.07 m, 0.1 m, and 0.13 m

Full size image

Figure 7a shows that the scour depth of scour hole A decreases with increment in the tailwater depth. With an increase in the tailwater depth, the location of the scour hole B shifts away from the secondary channel and moves upstream as evident from Fig. 7b. The location of scour hole B is at y = 0.06 m, 0.09 m, and 0.11 m for tailwater depths of 0.07 m, 0.10 m, and 0.13 m, respectively.

To study the effect of discharge ratio on scour pattern, the scour patterns for discharge ratios of 1.0 (Qd = 0.02165 m3/s, Qs = 0, dt = 0.1 m), 0.91 (Qd = 0.02165 m3/s, Qs = 0.0019 m3/s, dt = 0.1 m) and 0.87 (Qd = 0.02165 m3/s, Qs = 0.0029 m3/s, dt = 0.1 m) are shown in Fig. 8a–c respectively. The extension of scour hole in the secondary channel was observed 0.24 m, 0.15 m, 0.1 m for the discharge ratio 1.0, 0.91, 0.87, respectively. With a decrease in the discharge ratio, the extension of the scour hole in the secondary channel decreases. Figure 9a, b show the longitudinal scour profiles along scour hole A and scour hole B, respectively for the different discharge ratios. Scour depth of scour hole A decreases with the decrease in the discharge ratio as evident from Fig. 9a. The location of the scour hole B is at y = 0.11 m, 0.13 m, and 0.14 m for discharge ratio of 1.0, 0.91, 0.87, respectively. The scour hole B was found to be moving downstream and shifting away from the secondary channel, with the reduction in the discharge ratio as evident from Fig. 9b.

Fig. 8
figure 8

Scour pattern for a Qr = 1, Qd = 0.02165 m3/s, Qs = 0, dt = 0.1 m; b Qr = 0.91, Qd = 0.02165 m3/s, Qs = 0.0019 m3/s, dt = 0.1 m; and c Qr = 0.87, Qd = 0.02165 m3/s, Qs = 0.0029 m3/s, dt = 0.1 m

Full size image
Fig. 9
figure 9

Longitudinal scour profiles a along scour hole A for Qr = 1, 0.91, and 0.87; and b along scour hole B for Qr = 1, 0.91, and 0.87

Full size image

Souring commencing at the toe of the divide wall advances in the secondary channel, which may threaten the stability of the divide wall. Thus, reduction in scour around the divide wall is necessary to safeguard the confluence system. An attempt was made to reduce the extension of scour hole in the secondary channel by providing a solid apron of length L downstream of the primary channel. Figure 10a–c show scour patterns for dt = 0.1 m, Qd = 0.02345 m3/s, Qs = 0.0019 m3/s and L = 0, 0.125 m and 0.25 respectively. The extension of scour hole in the secondary channel was found to be 0.18 m, 0.10 m, and 0.06 m for L = 0, 0.125 m, and 0.25 m, respectively. It is concluded that an increase in the apron length reduces the extension of the scour hole in the secondary channel.

Fig. 10
figure 10

Scour pattern for a L = 0, dt = 0.1 m, Qd = 0.02345 m3/s, Qs = 0.0019 m3/s; b L = 0.125 m, dt = 0.1 m, Qd = 0.02345 m3/s, Qs = 0.0019 m3/s; and c L = 0.25 m, dt = 0.1 m, Qd = 0.02345 m3/s, Qs = 0.0019 m3/s

Full size image

Figure 11a, b show the longitudinal scour profiles along scour hole A and scour hole B, respectively, for the different apron length. A noteworthy decrease in scour depth of scour hole A was observed with an increase in apron length which is in line with the Aamir and Ahmad (2019), as evident from Fig. 11a. Figure 11b shows that the location of scour hole B is at y = 0.13 m, 0.10 m, and 0.08 m for the L = 0, 0.125 m, and 0.25 m, respectively. The location of the scour hole B shifts towards secondary channel and moves downstream with an increase in the apron length.

Fig. 11
figure 11

Longitudinal scour profiles a along scour hole A for L = 0, 0.125 m, and 0.25 m; and b along scour hole B for L = 0, 0.125 m, and 0.25 m

Full size image

Dimensional analysis

The maximum scour depth in the scour holes and extension of scour hole in the secondary channel depends on various parameters, namely, median particle size (d50), acceleration due to gravity (g), width of the channel (W), solid apron length (L), velocity at tail water level (V), discharge ratio (Qr), tail water depth (dt), density of flowing fluid (ρ), density of the bed material (ρs), dynamic viscosity of the flowing fluid (µ) and confluence angle \((\theta )\).

A functional relationship for maximum scour depth of scour hole B (ZBM) can be written as follows:

$$Z_{BM} = f\left( {d_{50} ,g,W,L,V,Q_{r} ,d_{t} ,\rho ,\rho_{s} ,\mu ,\theta } \right).$$
(1)

Detailed dimensional analysis as per Pagliara et al. 2020 (see “Appendix” for details) was performed to derive the functional relationship for ZBM. The repeating variables considered namely, V, ρ and dt, and obtained following functional relationship:

$$\frac{{Z_{BM} }}{{d_{t} }} = f\left( {\frac{L}{{d_{t} }},Q_{r} ,F_{D} = \frac{V}{{\sqrt {g\left( {\frac{{\rho_{s} }}{\rho } - 1} \right)d_{50} } }}} \right).$$
(2)

Here \({\text{F}}_{\text{D}}\) is the densimetric Froude number at the tail water level.

Following the same procedure ZAM and LSP can be expressed as:

$$\frac{{Z_{AM} }}{{d_{t} }} = f\left( {\frac{L}{{d_{t} }},Q_{r} ,F_{D} } \right),$$
(3)
$$\frac{{L_{SP} }}{{d_{t} }} = f\left( {\frac{L}{{d_{t} }},Q_{r} ,F_{D} } \right).$$
(4)

Effect of various parameters on ZBM, ZAM, and LSP

Figure 12a–c demonstrate the effects of FD, Qr, and L/dt on the ZBM. As evident from the figures ZBM increases with an increase in the FD and Qr, however, the ZBM decreases with an increase in L/dt. As shown in Fig. 13a–c, a similar trend was observed for the maximum scour depth of scour hole A (ZAM). Moreover, LSP also increases with an increase in FD and Qr and decreases with an increase in L/dt as shown in Fig. 14a–c. Due to higher velocity in the primary channel compared to the secondary channel at the confluence, high shear stress generates at the interface, such stress develops vortex flow at the confluence which in turn scours the bed. Scour downstream of the primary channel is due to jet like flow in the channel having high bed stress. High densimetric Froude number causes more scour both at confluence and downstream of the primary channel due to high velocity and low particle size.

Fig. 12
figure 12

Variation in ZBM/dt with a FD; b Qr; and c L/dt

Full size image
Fig. 13
figure 13

Variation in ZAM/dt with a FD; b Qr; and c L/dt

Full size image
Fig. 14
figure 14

Variation in LSP/dt with a FD; b Qr; and c L/dt

Full size image

Proposed equations for ZBM, ZAM, and LSP

Equations are proposed for ZBM, ZAM, and LSP using data collected in the present study and invoking least square techniques, randomly selected 80% of datasets were used to develop the equation and 20% data were used for validating the equation. The proposed equation for the maximum scour depth in scour hole B is as follows:

$$\frac{{Z_{BM} }}{{d_{t} }} = 0.54Q_{r}^{0.27} F_{D}^{1.3} \left( {1 + \frac{L}{{d_{t} }}} \right)^{ - 0.17}.$$
(5)

The coefficient of correlation (R) for Eq. (5) is 0.924. The mean absolute percentage error (MAPE) and the root mean square error (RMSE), and average absolute deviation (AAD) are 17.568, 0.205,10.98, respectively. Figure 15a shows that the scour depth computed by Eq. (5) is within ± 25% of the observed values. Equation (5) is applicable for the range of \(0.69 \le {{Z_{BM} } \mathord{\left/ {\vphantom {{Z_{BM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }} \le 2.5,0.84 \le Q_{r} \le 1.0,1.45 \le F_{D} \le 3.28,0 \le {L \mathord{\left/ {\vphantom {L {d_{t} }}} \right. \kern-0pt} {d_{t} }} \le 3.57\).

Fig. 15
figure 15

Comparison of computed ZBM, ZAM and LSP with the observed one

Full size image

For the development of an equation for the maximum scour depth in scour hole A, 80% data sets were randomly selected for equation development and 20% data sets for validation purposes. The developed equation is:

$$\frac{{Z_{AM} }}{{d_{t} }} = 0.21Q_{r}^{0.26} F_{D}^{2.17} \left( {1 + \frac{L}{{d_{t} }}} \right)^{ - 0.51}.$$
(6)

The coefficient of correlation (R) for Eq. (6) is 0.954. MAPE, RMSE and AAD are 19.243, 0.201,14.81, respectively. Figure 15b indicates that the scour depth computed by Eq. (6) is within ± 25% of the observed values. Equation (6) is applicable for the range of \(0.26 \le {{Z_{AM} } \mathord{\left/ {\vphantom {{Z_{AM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }} \le 2.77,0.84 \le Q_{r} \le 1.0,1.45 \le F_{D} \le 3.28,0 \le {L \mathord{\left/ {\vphantom {L {d_{t} }}} \right. \kern-0pt} {d_{t} }} \le 3.57\).

Randomly selected 80% data were used for developing an equation for LSP, and the remaining 20% data were used for its validation. The proposed equation for extension of scour hole in the secondary channel is:

$$\frac{{L_{SP} }}{{d_{t} }} = 0.41Q_{r}^{1.42} F_{D}^{2.13} \left( {1 + \frac{L}{{d_{t} }}} \right)^{ - 0.50}.$$
(7)

The coefficient of correlation (R) for Eq. (7) is 0.963. For Eq. (7), MAPE, RMSE and AAD are 20.191, 0.306, 14.36 respectively. Figure 15c shows that computed LSP by Eq. (7) is within ± 25% of the observed value. Equation (7) is applicable for the range of \(0.42 \le {{L_{SP} } \mathord{\left/ {\vphantom {{L_{SP} } {d_{t} }}} \right. \kern-0pt} {d_{t} }} \le 5.16,0.84 \le Q_{r} \le 1.0,1.45 \le F_{D} \le 3.28,0 \le {L \mathord{\left/ {\vphantom {L {d_{t} }}} \right. \kern-0pt} {d_{t} }} \le 3.57\).

Sensitivity analysis

Sensitivity analysis is carried out to determine the relative importance of the dimensionless independent parameters affecting the dependent parameters used in Eqs. (5, 6, and 7). For this, the average values \(\left( X \right)\) of all input variables that are Qr, FD, and \({L \mathord{\left/ {\vphantom {L {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) corresponding to \({{Z_{BM} } \mathord{\left/ {\vphantom {{Z_{BM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\),\({{Z_{AM} } \mathord{\left/ {\vphantom {{Z_{AM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) and \({{L_{SP} } \mathord{\left/ {\vphantom {{L_{SP} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) were used as suggested by Ahmed (2013). For sensitivity analysis, average value of each of these three input parameters was varied individually by \(\pm\) 10% (defined as \(\Delta X\)), and the corresponding change in the values of and were determined. In this analysis, errors in all input parameters are assumed independently. Additionally, three indices were used to define the error known as absolute sensitivity, \(AS = {{\Delta Y} \mathord{\left/ {\vphantom {{\Delta Y} {\Delta X}}} \right. \kern-0pt} {\Delta X}}\), relative error, \(RE = {{\Delta Y} \mathord{\left/ {\vphantom {{\Delta Y} Y}} \right. \kern-0pt} Y}\) and relative sensitivity, \(RS = {{X\Delta Y} \mathord{\left/ {\vphantom {{X\Delta Y} {Y\Delta X}}} \right. \kern-0pt} {Y\Delta X}}\), Where \(\Delta Y\) is the error in the output parameter, defined as the difference between output values predicted for inputs \(X\) and \(\left( {X + \Delta X} \right)\).

The sensitivity analysis for \({{Z_{BM} } \mathord{\left/ {\vphantom {{Z_{BM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\),\({{Z_{AM} } \mathord{\left/ {\vphantom {{Z_{AM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) and \({{L_{SP} } \mathord{\left/ {\vphantom {{L_{SP} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) are given in Tables 2, 3, and 4, respectively. The result implies that the most important and sensible parameter for \({{Z_{BM} } \mathord{\left/ {\vphantom {{Z_{BM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\),\({{Z_{AM} } \mathord{\left/ {\vphantom {{Z_{AM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) and \({{L_{SP} } \mathord{\left/ {\vphantom {{L_{SP} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) is FD. Qr is the second most sensible parameter for \({{Z_{BM} } \mathord{\left/ {\vphantom {{Z_{BM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) and \({{L_{SP} } \mathord{\left/ {\vphantom {{L_{SP} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) whereas \({L \mathord{\left/ {\vphantom {L {d_{t} }}} \right. \kern-0pt} {d_{t} }}\) is the least sensible parameter.

Table 2 Sensitivity analysis for \({{Z_{BM} } \mathord{\left/ {\vphantom {{Z_{BM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\)
Full size table
Table 3 Sensitivity analysis for \({{Z_{AM} } \mathord{\left/ {\vphantom {{Z_{AM} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\)
Full size table
Table 4 Sensitivity analysis for \({{L_{SP} } \mathord{\left/ {\vphantom {{L_{SP} } {d_{t} }}} \right. \kern-0pt} {d_{t} }}\)
Full size table

Conclusions

Experimental study carried out on the scour near the confluence of a rigid and a mobile bed channel reveals the formation of two scour holes, one downstream of the primary channel and other at the confluence of channels. Such scour holes diminish with the increase of tail water depth, however, increase with the ratio of discharge in primary channel and total discharge. This study also concludes that the confluences scour hole decreases with an increase in length of solid apron provided downstream of the rigid bed channel. It is found that scour hole that formed at confluence point extends towards the upstream of the secondary channel. Equations are developed for the computation of maximum scour depth of the scour holes and extension of scour hole in the secondary channel using the data collected in the present study. Computed values of the maximum scour depth of the scour holes and extension of scour hole in the secondary channel show good agreement with the observed values. Froude number of the downstream flow is found to be the most sensitive parameter both for maximum scour depth of the scour holes and extension of scour in the secondary channel. In future, the experiment shall be conducted for different widths of primary and secondary channel and for different sizes of bed material to study scale and side wall effect. The effect of roughness and rigid bed channel may also be explored.