Bed shear stress distribution across a meander path

Human habitation near rivers and the associated risks of flooding necessitate reliable methods for estimating discharge capacity, a crucial aspect of flood management and planning (Sarker 2023). Rivers often adopt a meandering pattern due to various morphological factors, particularly on lowland alluvial plains, which adds complexity to predicting their behavior and flow characteristics (Gilvear et al. 2000; Sarker 2022; Sarker et al. 2023). Understanding the dynamics of meandering channels is essential for managing water resources, mitigating flood risks, and preserving ecological health.

Meandering rivers are characterized by their sinuous paths, which result from the interplay of sediment transport, flow velocity, and the geomorphological features of the landscape. These channels exhibit unique hydraulic behaviors, including variations in velocity and boundary shear stress along different sections of the meander. Accurate prediction of these variations is vital for effective river management and engineering interventions.

Recent research has focused extensively on meandering streams to understand their complex hydraulic behavior (Pradhan & Khatua 2018; Pradhan et al. 2018; Mohanta et al. 2022b; Salem et al. 2022; Ali & Maghrebi 2023; Ghosh et al. 2023; Kumar et al. 2023; Naghavi et al. 2023a; Shukla & Rhoads 2023). Advanced methodologies for estimating velocity, boundary shear stress variations, and discharge in meandering channels have been proposed, highlighting the importance of detailed analysis and modeling in these studies (Handique et al. 2022; Mohanta et al. 2022a, 2022b; Moradi et al. 2023; Naghavi et al. 2023b; Sudhir et al. 2023; Kozarek et al. 2024).

In this context, the present research aims to contribute to the existing body of knowledge by proposing a series of experiments on meandering channels with varying sinuosity and bed roughness. These experiments are designed to assess the bed shear stress distribution along the entire meandering path. The study focuses on two meandering channels with 110° and 60° crossover angles, constructed at NIT Rourkela, to observe how changes in channel geometry and bed roughness affect flow dynamics and shear stress distribution.

The study conducts an experimental investigation into how variations in bed roughness affect the distribution of bed shear stress in meandering channels. Different roughness profiles result in distinct patterns of boundary shear stress across the channel cross-sections. These effects can be observed for two different sets of meandering channels, with observations across the later sections as well as along the meandering path of a 60° meandering channel.

The sections (Figure 2) are chosen between two corresponding bend apexes. Thirteen sections, (A through M) and seven sections (A through G) are considered in the case of 110° and 60° channels, respectively. The respective crossover angles are divided equally (six divisions in 110° and three divisions in 60° channels) and extended up to the channel centerline to determine the intermediate sections at the corresponding main channels, which are perpendicular to the side banks. Due to the geometry of the 110° channel, the compound section is only considered at the bend apex section, M.

Influence of the shape and bed roughness has an enormous effect on the velocity distribution of a channel (Liu 2001; Reza et al. 2014; Sarker 2021; Sarker & Sarker 2022; Pu 2023; Rafiqii et al. 2023). Friction velocity or bed shear velocity is a significant term in the context of computing velocity profile as well as for determining the bed roughness. Friction velocity u^* is defined as the fluid velocity at an elevation of where is the elevation at zero velocity and κ is the von Kármán constant equal to 0.4.

In the turbulent logarithmic layer the measurements show that the turbulent shear stress is constant and equal to the bottom shear stress. By assuming that the mixing length is proportional to the distance to the bottom ⁠, Prandtl obtained the logarithmic velocity profile.

For the case of viscous sub-layer, the effect of bottom (or wall) roughness on the velocity distribution was first investigated for pipe flow by Nikuradse (1950) and later by Yalin (1992), Graf & Altinakar (1998) and Schlichting & Gersten (2000). The concept of equivalent grain roughness (in m) was introduced based on experimental observations, hence, flow could be classified as follows;

• • Hydraulically Smooth Flow:

• Bed roughness is much smaller than the thickness of viscous sub-layer. Therefore, the bed roughness will not affect the velocity distribution.

• • Hydraulically Rough Flow:

• Bed roughness is so large that it produces eddies close to the bottom. A viscous sub-layer does not exist, and the flow velocity is not dependent on viscosity.

• • Transitional Flow:

• The velocity distribution is affected by bed roughness and viscosity, where, is the kinematic viscosity (in m²/s) of the fluid flow.

The roughness height for the perspex sheet, gravel and artificial grass are determined as 0.0001 m, 0.012 and 0.015 m, respectively. The kinematic viscosity of water was taken as m²/s with an average value of being 0.01396, 0.01385, and 0.01340 m/s for the Smooth, Rough I and Rough II beds, respectively. The average values of are hence calculated to be in the ranges of 1.225, 145.835, and 176.316 for the perspex, gravel, and artificial grass beds, respectively. In the current study, the bed roughnesses are demarcated as hydraulically smooth and rough by use of the above classifications.

For rough boundaries, a plane or datum plane is necessary for all the measurements to be related. There are several approaches for determining this plane, i.e. the boundary datum level. Einstein & El-Samni (1949) suggested this plane to be present, at which the velocity distribution would agree with the logarithmic velocity law with κ =0.4; and observed it to be at 0.2D below the top hemispherical roughness element with diameter, ⁠. Several researchers have carried out similar experimental investigations to determine this datum level which has been illustrated by Alhamid (1991).

In the present experimental conditions, the datum level for the gravel bed, i.e. ‘Rough I’ material is undertaken as mentioned by Schlichting (1937), which suggests the use of mean roughness height as the datum level, 0.012 m in this case. For the case of ‘Rough II’, the artificial grass (with height of 0.015 m to a spacing of 0.002 m); the formulation provided by Raju & Garde (1970) is used, where the datum level is calculated to be at the top of the roughness element due to small spacing between them.

A moving bridge is fitted with a five Pitot tube arrangement with a pointer gauge. The moving bridge is traversed across the meander path to every section and respective readings are taken. The pointer gauge is utilized to find out the water surface profile across the channel width at every section. The set of Pitot tubes measure the pressure difference at every predefined location across every section. All the Pitot tubes are connected to five different manometers which are arranged on a vertical board having a spirit level. The spirit level helps to maintain the verticality of the manometers.

At the downstream, a tailgate arrangement is provided to control the depth of flow and to achieve quasi-uniform flow in the channel. Ideally, open channel flow investigations are considered under uniform flow conditions where the water surface slope or friction slope (⁠⁠) is maintained equal to bed slope (⁠⁠). Maintaining this condition is quite difficult for the case of meandering channels (Shiono et al. 1999; Terrier 2010), for which the condition of a quasi-uniform flow is maintained. For such a case, a maximum discrepancy of 2% in the value of water surface and bed slopes at different points on the meandering compound channel is maintained.

To maintain the surface slope, gauges at three different reaches are arranged, namely the second, third, and the fourth bend apex sections. M1 surface profile occurs if the depth of flow in the downstream section is greater than the upstream one. In such a case, the tailgate is opened. In the case where the upstream section has a higher depth of flow to the downstream, M2 profile is formed. In this case, the tailgate needs to be closed. The procedure is repeated such that all the three gauges give equal depths of flow. Thus, the flow can be assumed to be fully developed and uniform. Hence, a quasi-uniform flow condition is obtained where the water surface slope is parallel to the valley slope, at the meander wavelengths. These conditions are controlled for at least 4 h before conducting the experiments for each depth of flow.

There are several methods for measuring boundary shear stress, based on either direct or indirect techniques.

As observed from Figure 4(a), for a particular depth of flow, H, the discharge goes on decreasing from Series I to Series III experimentations due to the increase in the bed roughness.

An experimental investigation to determine boundary shear stress distributions was conducted on the 110° channel at the bend apex section, M, as shown in Figure 2(a). Similarly, for the 60° channel, the investigation covered compound sections from A to G, as shown in Figure 2(b).

The solid black line represents the channel bed, and the dotted blue line indicates water depth, both referenced to the primary Y-axis. Experimental data for boundary shear stress at various locations across the channel width is depicted by orange-diamond plots. The green line represents numerical boundary shear stress data obtained from the conveyance estimation system (CES), with these values referenced to the secondary Y-axis.

The CES model’s boundary shear stress plots are incorporated into the bed shear distributions for the bend apex sections (Figures 6–8, and sub-figures (a) and (g) in Figures 10–12).

For the meandering compound channels in Series I, II, and III, boundary shear distribution plots were obtained for five different relative flow depths, as shown in Figures 6–8. Across all three experimental series, an increase in the relative depth of flow resulted in higher boundary shear stress in the floodplains.

In Series I (Figure 6), there is a slight dip in bed shear stress in the meander belt region on the inner floodplain. Comparing these experimental plots to CES data reveals that CES predicts a uniform shear stress distribution across the floodplain. However, the experimental data shows higher shear stress on the outer floodplains (left-hand side) compared to the meander-belt width. This discrepancy is likely due to the channel plan depicted in Figure 2(a), where the bend apex of the subsequent meander approaches the outer floodplain of the current section, increasing shear stress in that region.

To examine the effect of differential roughness on boundary shear stress distribution, shear stress at a relative depth of is plotted for the three channel series in Figure 9. When the floodplain bed roughness changes from Series I to Series II, shear stress increases in both the main channel and floodplains. Bed shear stress is notably higher on the inner wall of the meandering main channel compared to the outer wall.

In Series III, where the roughness of the main channel is increased, the overall boundary shear stress distribution decreases. This reduction is attributed to the higher roughness of the main channel, which lowers the shear stress.

The observations highlight the complex interplay between flow depth, channel roughness, and shear stress distribution in meandering channels, providing valuable insights for better predictive models and effective river management strategies.

From the bed shear distribution plots in Figures 10–12, it is observed that shear stress is consistently higher on the inner wall of the meandering main channel compared to the outer wall along the entire meander path. In the crossover section D, shear stress peaks at the center of the main channel. The profiles show a sharp dip at the edges of the main channel bed and a sharp rise at the junction between the meandering main channel and the floodplain in almost all distribution curves.

Comparing the contour plots in Figure 13, it is evident that as the discharge of the compound channel increases, the highest bed shear stress, initially located within the meandering channel, shifts to the inner floodplain regions of sections C, D, and E. Additionally, with increasing discharge, the cross-sections inside the meandering main channel exhibit less variation in contour values, suggesting that bed shear stress distribution within the meandering channel becomes more uniform at higher discharges compared to lower discharges.

These findings highlight the complex interactions between discharge, channel morphology, and shear stress distribution in meandering channels, providing essential insights for river engineering and management.

This research is significant as it conducts experimental investigations into boundary shear stress within two meandering channels, featuring crossover angles of 110° and 60°, with varying bed roughness levels detailed in Table 2. For the 110° meandering channel, the study explores three different bed roughness variations, capturing boundary shear stress distribution at the bend apex section across varying flow depths, with five datasets for each depth.

Notably, the research thoroughly examines the entire meandering path of the 60° channel, spanning from bend apex sections A to G, providing a comprehensive analysis of shear stress variations along its meandering course. This detailed investigation offers a holistic view of how boundary shear stress changes across the channel’s path, highlighting its sensitivity to variations in flow depth and bed roughness.

Overall, the experimental findings furnish a detailed understanding of boundary shear distribution along the meandering channels, offering valuable insights into how these distributions are influenced by changes in flow depth and bed roughness. This research contributes significantly to the field of river engineering and management, providing critical data for better predictive models and effective river management strategies.

The study carries out experimental investigation of bed shear stress distribution in meandering channels with varying roughness profiles providing valuable insights into the complex interplay between channel morphology, roughness characteristics, and boundary shear stress distribution. The study meticulously examines the impact of roughness variations on bed shear stress across different flow depths in meandering channels with sinuosities of 110° and 60°.

The experimental investigation examines the impact of bed roughness and meander curvature on boundary shear stress distribution. The results show higher shear stress near the inner wall at bend apex, consistent across different bed roughness levels. Increased flow affects primarily the inner walls and crossover regions, with minimal impact on outer walls. Greater bed roughness reduces shear stress in floodplains. At lower flow depths, the highest bed shear occurs in the crossover region, shifting to the floodplains with increased flow depth. This highlights the importance of considering roughness profiles in hydraulic analysis of meandering channels.

The research contributes to the understanding of how roughness variations affect bed shear stress distribution, providing insights that can inform channel design, hydraulic modeling, erosion control strategies, and environmental impact assessments. By elucidating the relationship between roughness, sinuosity, and bed shear stress, the study offers implications for enhancing infrastructure resilience and guiding future research directions in the field of hydraulic engineering.

In conclusion, the research underscores the significance of incorporating roughness considerations in the analysis of bed shear stress distribution in meandering channels. The findings not only advance our understanding of hydraulic processes in meandering channels but also have practical implications for channel management and environmental conservation efforts.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.