1. Introduction
Riverbank erosion is a natural process that results in substantial economic losses. It also poses significant risks to human life. Groynes are commonly used river training structures, widely employed to maintain stable river channels and to ensure riverbank protection. These structures, which encompass levees, dams, weirs, groynes, and channel stabilization mechanisms, aim to safeguard riverbanks, regulate water flow, combat sediment build-up, and uphold the preferred dimensions of the river channel [1]. They offer effective means to manage river flow, minimize flood risks, control erosion, and reduce sedimentation issues. The groyne field provides a favorable environment for the growth of aquatic life due to the zone of reduced velocity and the sediment deposition resulting from the reduction in velocity. The groynes can take various shapes, such as I, L, and T head groynes (IHG, LHG, and THG), each with its distinct geometric characteristics that influence the flow patterns and associated scouring processes. Numerous groyne types, including T and L type groynes, have been scientifically built across the globe in accordance with various river environments; for example, spur dikes or IHGs have been provided along the Western Kosi Main Canal (WKMC), Kosi River in Bihar, India, and in the Gamka River at Calitzdorp, South Africa [2]. Six LHGs were constructed in 2012 to safeguard a solar power plant near the left bank of Solani River, Bhagwanpur (Roorkee, India) [3], and an LHG field was constructed on the left bank of the Kansas River (United States) [4]. Examining the hydrodynamics of the flow around these structures is essential to enhance our understanding of the performance of IHGs, LHGs, and THGs. From the literature, it is evident that the flow around a spur dike is divided into three zones—the main flow zone, wake zone, and mixing zone (between the two zones) [5]. Moreover, the interaction of the flow in a channel with a groyne affects the flow field and bed topography. These changes are responsible for scouring and deposition around the groyne. Such processes are referred to as dynamic feedback processes. Scouring is a critical aspect of groyne design and maintenance as it can compromise the integrity of these structures, damage the adjacent features, and alter the riverbed’s morphology. Therefore, understanding the flow patterns and scouring processes around groyne shapes is paramount. Excessive scour represents a prominent factor leading to the failure of spur dikes. Conversely, the formation of pool-riffle morphologies stemming from the scour processes introduces valuable amenities and crucial habitats for riverine species. Striking a careful equilibrium is paramount to harness the maximum positive impacts while concurrently ensuring effective scour control [6]. The groynes enhance the local biodiversity and support a wider range of species, hence increasing the ecosystem’s resilience. This boost in biodiversity is crucial for sustainability as resilient ecosystems are better able to withstand and recover from environmental stresses. Groynes contribute to the long-term stability of aquatic environments. Numerous researchers have explored the dynamics of flow and scouring around diverse groyne shapes. For instance, Zaghoul conducted experimental studies that focused on understanding how various factors, such as upstream flow conditions, sediment properties, and the shape of the spur dike, impact the maximum depth and the pattern of scour around the spur dike [7]. Kadota and Suzuki visually examined the flow patterns around T and L groynes and studied the mean and coherent flow structures [8]. Much research has been conducted on scour around abutments. Barbhuiya and Dey reviewed the various work on abutments and provided a detailed review on abutment scour, flow field, time variation of scour depth, and the formulae to estimate the scour depth [9]. Kothyari and Raju computed the temporal variation of scour around spur dikes and bridge abutments using an analogous pier model [10]. Dey and Barbhuiya investigated the temporal evolution of scouring at abutments [11]. Dey and Barbhuiya delved into velocity and studied the turbulence within a scour hole adjacent to an abutment [12]. The flow around these abutments is found to be similar to that near IHGs.
Although few researchers have investigated the flow around LHGs and THGs, the flow mechanism and comparative analysis among the groynes have not been completely established yet. LHGs and THGs have an advantage over IHGs. LHGs facilitate the localized absorption of energy by leading the face perpendicular to the flow and establishing a low flow velocity zone, which supports the sediment deposition behind the groyne. Kang and Yeo conducted experimental analyses to understand the flow characteristics of L-type groynes [13]. An experimental study to investigate the performance and efficiency of two types of groynes, IHGs and THGs, was carried out by Mall et al. [14]. The equilibrium scour depth for THG is greater as compared to an IHG. Moreover, the cost–benefit analysis also favored an IHG as the more cost-efficient option for riverbank protection. Duan et al. explored the mean flow characteristics and turbulence of flow surrounding experimental spur dikes [15], while Kumar and Ojha focused on the near-bed turbulence in the vicinity of an unsubmerged L-head groyne [16]. Kadota and Suzuki investigated the flow around IHGs, LHGs, and THGs [17]. The study unveils the alterations in the bed configurations resulting from T- and L-type groynes by comparing them with the experimental outcomes of I-type groynes. The experimental results highlight the significant variations in the maximum scour depth associated with different groyne types. Additionally, distinctive bed form changes, including the development of sand waves, were observed downstream of the groyne, indicating distinct characteristics influenced by the groyne types. The maximum scour depths in the cases of LHGs and THGs surpass those of I-type groynes. Various studies have also investigated the impact of the parallel wall angle in LHGs, which is the angle between the parallel and perpendicular walls, as conducted by Dehghani et al. [18]. To examine the flow patterns around THGs, a numerical simulation was carried out by Mortazavi Farsani [19]. The investigation considers the existence of a single dike with varying lengths perpendicular to the flow, along with different configurations including simple and T-shaped models. The findings reveal that simple dikes exhibit higher maximum scour depths and spans in comparison to T-shaped dikes. An increase in the length of the groyne relative to the flow is associated with deeper and wider scour holes. Conversely, extending the length of the dike wings tends to reduce the depth, span, and intensity of sediment scour and entrainment, except for situations with high narrowing ratios. Another study on THGs also concluded that, with an increase in the wing length of a THG, the dimensions of the scour hole decreased [20].
Most of the studies on groynes have been conducted on the scour in a sand bed, and, hence, the investigation of the flow around groynes in gravel beds remains unexplored. Kadota and Suzuki studied the physics of the flow around groynes for gravel, which is different from that of sand [8]. This can be attributed mostly to changes in bed roughness, which then influence the vortex flow field [21]. Pandey et al., examined sand gravel mixtures and discovered that sediment nonuniformity influences the variation in the scour depth [22]. It was found that the maximum scour depth for a sand gravel mixture occurred at the same location as that of the sand bed, which is the upstream tip of the groyne. However, the processes involved for scour around these groynes and the flow behavior and dynamics for the gravel bed have not been studied yet. Additionally, the flow around the groyne in the submerged case is expected to be different from the flow around the groyne in unsubmerged conditions. Kunhle et al. studied the 3D flow field and bed shear stress around a submerged IHG [23]. They observed reduced reattachment length around the IHG compared to the values reported in the unsubmerged condition. Yossef and de Vriend investigated the scour and flow pattern around both submerged and emergent (non-submerged) groynes (IHGs) in series [24]. Mehraein and Ghodsian highlighted the difference in the flow fields around fully submerged and just submerged spur dikes [25]. Moreover, the length of the downstream recirculation region is found to be dependent on both the shape and height of the spur dike. A significant difference in the nature of the turbulence among both the cases was reported. The convergence of the time-averaged velocity occurs within a short distance compared to the emerged case.
The study underscores the need to understand the flow dynamics and sediment transport around groynes in gravel beds. The primary objectives of this research are to assess and contrast the extent of scour development and create a visual representation of the scour bed morphology surrounding groynes of different shapes, namely I, L, and T head groynes, for gravel bed streams. This is completed with the motivation to provide an effective river training structure as per the utility. The flow pattern for each groyne is studied. Furthermore, a thorough cost–benefit analysis is carried out to optimize the groynes’ dimensions and to improve their cost-effectiveness. The results indicate that the maximum scour depth varies with the type of groyne. Hence, this study provides valuable insights regarding the hydrodynamics of the flow around these groynes and aids in selecting the appropriate type of groyne. The research aids the decision-making for the engineers and policymakers in selecting the appropriate groyne type based on the specific requirements.
2. Materials and Methods
2.1. Experimental Setup
An experimental setup was established at the Hydraulics Laboratory within the Civil Engineering Department at IIT Roorkee to investigate the scour and flow characteristics around I head (IHG), L head (LHG), and T head groynes (THG). This setup featured a glass-sided rectangular flume, measuring 20 m in length, 0.75 m in width, and 0.4 m deep, with a bed slope of 0.035. The flume was subdivided into three distinct sections: an upstream section, a test section, and a downstream section. The test section spanned 8 m in length and was positioned 7 m from the channel’s entrance so that fully developed flow conditions could be established [9,26]. It incorporated a uniformly graded gravel bed having D_{50} = 9.36 mm and geometric standard deviation 0.955. For each run, specific groyne was installed and attached to the right-side wall of the flume (IHG for experiment 1, LHG for experiment 2, and THG for experiment 3). The groyne was placed in the test section, at 8 m from the inlet. This was done to maintain a fully developed flow condition [27]. The length of the groyne perpendicular to the flow direction (L_{1}) was maintained at 11.25 cm to constrict 15% of the width of the channel. A constriction ratio of 15% does not induce contraction scour [18]. The length of the groyne in the flow direction (L_{2}) was kept the same as L_{1} in the case of LHG and THG. The experimental setup is shown in Figure 1, depicting the scoured bed observed at equilibrium scour condition for the three groynes.
At the start of each experiment, the sediment bed was carefully leveled using a scraper. After the installation of the groyne, the area around the groyne was leveled again. The supply of water to the flume was regulated using an inlet pipe valve. Each experiment extended over a duration of 8 h to attain the equilibrium scour bed condition. To establish the pre-determined flow conditions (depth of flow, discharge, and velocity), precise adjustments were carried out by manipulating both the inlet valve and the tailgate. The pump was initially set to a low discharge rate, and subsequent adjustments were made to the discharge from the pump and tailgate to attain the desired depth and velocity parameters. A flow depth of 0.136 m was consistently maintained, while the submergence ratio was kept as 0.83. As a precautionary measure to avert abrupt scouring induced by the flowing water, plywood was placed over the bed, with gradual removal occurring upon the successful attainment of the desired depth. The experiments were carried out under clear water conditions, adhering to the condition U/U_{c} = 0.7, ensuring U/U_{c} < 1. Here, U represents the mean approach flow velocity in the streamwise direction, and U_{c} denotes the critical velocity for sediment particles, which is determined by the formula provided by Melville [28].
$$\frac{{U}_{c}}{{u}_{*c}}}=5.75\mathit{log}\left({\displaystyle \frac{h}{{k}_{c}}}\right)+6$$
$${k}_{c}=2{d}_{50}$$
$${u}_{*c}=0.0305{{d}_{50}}^{0.5}-0.0065{{d}_{50}}^{-1},for1\mathrm{m}\mathrm{m}<{d}_{50}<100\mathrm{m}$$
The term represents critical shear velocity for sediment particles, and Equation (3) was used to compute it. In Equation (1), k_{c} is the height of roughness elements and is taken 2 times the d_{50} and was provided by Dey and Barbhuiya [11].
2.2. Data Collection and Analysis
Bed level measurements of the equilibrium scoured bed were meticulously obtained across a grid with a spacing of 0.02 m in the longitudinal and transverse directions using a pointer gauge (least count 0.01 mm). The bed level measurements serve the purpose of ascertaining scour and deposition, facilitating the identification of both maximum scour depth and deposition. The scour pattern is consequently visualized through contour plotting [16]. A diluted resin was evenly applied to stabilize the scoured bed, thus securing the equilibrium scoured bed.
Subsequently, velocity measurements were systematically collected across a grid situated on the stabilized scoured bed within the xy plane. The measurements were obtained at a specific vertical position with z/D = 0.074. Here, z represents the location within a vertical plane over the equilibrium scoured bed. Notably, a finer grid resolution was applied in the vicinity of the critical zone around the groyne, while a coarser grid was employed in other regions.
For the instantaneous velocity data collection, Vectrino +, an Acoustic Doppler Velocimeter (ADV) of Nortek, sourced from Vangkroken, Norway, was used at a 50 Hz frequency for 2 to 3 min, encompassing 3 min in the vicinity of the critical groyne area and 2 min at more distant locations. This process yielded a substantial dataset, ranging from 6000 to 9000 samples at each measurement point. The acquired velocity data underwent filtration using the Phase Space Threshold Despiking technique [29]. Detailed hydraulic parameters for the experimental runs are provided in Table 1.
To validate the accuracy of the velocity data obtained from the ADV, spectral density was plotted as a function of frequency. This step served as a potential assessment of the suitability of the chosen sampling frequency. The power spectrum was computed via the Fourier Transform of the auto-covariance function. Notably, the power spectra derived from streamwise velocity fluctuations exhibited a slope of −5/3, indicative of the presence of an inertial sub-range. Figure 2 shows the spectral density function plotted on y axis with respect to frequency. Also, rms values of streamwise velocity data were calculated for section y/B = 0.073. Using these turbulent velocity statistics, uncertainty analysis was carried out, as shown in Table 2.
2.3. Cost–Benefit Analysis
Cost–benefit analysis (CBA) is a systematic procedure employed to assess and compare the potential benefits of a given project or activity with the associated costs. The expected value of damage reduction can be termed as benefit. In this study, to assess the economic efficiency of individual I-shaped, L-shaped, and T-shaped groynes, precise cost functions were developed, considering certain variables that are significant to determine the comprehensive construction cost of groynes: transverse length (L), depth of flow (D), and groyne thickness (t).
The benefits linked to respective groynes were quantified using the bank protection lengths by each groyne type. The insights derived from this experimental study were used to compute the effective length of bank protected for I-shaped, L-shaped, and T-shaped groynes for similar flow conditions. The basic idea remained to determine the point of reattachment of flow [5]. The outcomes of this analysis provide a framework for the process of decision-making in the selection of groyne configuration and groyne dimensions. Our study aims to ascertain, within a given cost framework, which groyne offers superior benefits or, in other words, better protection. While the cost–benefit approach may face criticism for requiring the quantification of all costs and benefits in monetary terms, we contend that it constitutes a vital component of information essential for rational decision-making.
3. Results
3.1. Scour Development Pattern
Figure 3 depicts the profile of the bed level around IHG, LHG, and THG at equilibrium scour conditions. It can be noted that the blue area indicates regions of scouring, whereas the red area denotes deposition region. Here, z_{s} is the difference between the bed level at equilibrium scour condition and the bed level initially, in the plain bed. D is the depth of flow, measured at the tailgate. It can be observed from the contour plots of all three shapes that scouring begins upstream of the structure around x/L_{1} = −1.5 or −1, and this scouring extends up to x/L_{1} = 1. The scour depth, z_{s}, is normalized by D, and the directions (x and y) are normalized by the length of groyne face perpendicular to flow (L_{1}) and width of the flume (B).
The initiation of the local scour takes place at the upstream of the groyne and has been attributed primarily to the interaction between the HSV system and the downflow. The intricate details of the flow field near a groyne become more complex as scour holes, which require flow separation to create three-dimensional vortex flow, develop. Kwan and Melville found that the primary mechanism of scour (at an abutment, which can be considered anomalous to an IHG is downflow, and they also found a primary vortex that resembles the horseshoe vortex at a pier [30]. Because the abutment stops the approaching flow, a vertical pressure gradient develops on the upstream face of the abutment. The fluid rolls up by being forced downward by the pressure gradient. As a result, the primary vortex forms and grows as the scour hole develops. Additionally, the main vortex and downflow are mostly contained in the scour hole, which is located below the original bed level line. The mean values of non-dimensional scour depth for the IHG is 0.067; for the LHG the value is 0.082; and, for the THG, it turns out to be 0.075.
Here, the LHG and THG demonstrate notable greater maximum scour depths, approximately the same, while the IHG exhibits a shallower maximum scour depth. This is in sync with the findings by Kadota and Suzuki [8]. To be specific, the IHG attains a maximum scour depth of z_{s}/D = −0.21, whereas the LHG and THG attain z_{s}/D = −0.295 and z_{s}/D = −0.29, respectively. The location of the maxima also varies. In the case of IHG, it occurs near the tip due to flow separation at that critical point, and the LHG experiences maxima at the junction of L_{1} and L_{2} faces. However, THG encounters the maximum scour depth upstream of the groyne. This distinct behavior can be explained by significant downflow upstream of the groyne, resulting from the impact of the flow on the perpendicular face L_{1} [11].
Notable scouring is observed in the regions x/L_{1} = 0 to 1 and y/B = 0.15 to 0.25 for the IHG and LHG. Conversely, THG experiences major scouring at upstream location x/L_{1} = −0.5 and y/B = 0.1 to 0.15. Interestingly, even the longitudinal extent of the scour is minimal for THG. The deposition process initiates near x/L_{1} = 0.5 for the IHG and THG and near x/L_{1} = 2 for the LHG. This deposition persists until x/L_{1} = 3.5 for the IHG and LHG. However, this reduces to x/L_{1} = 3 for THG. It is essential to emphasize that, for all three groynes, the extent of deposition surpasses the extent of scour. This underlines the significance of deposition as the dominant process in the context of groyne-induced alterations to riverbed morphology.
3.2. Flow Pattern around the Groynes
The flow depth was observed to fall significantly downstream of the groyne. Figure 4 represents the distribution of the normalized streamwise velocity (u/U) around IHG, LHG, and THG. Here, u represents the streamwise velocity and is normalized by U, and both the longitudinal distance, x, and transverse distance, y, are normalized with respect to the L_{1} and B. Here, L1 is the length of the groyne face perpendicular to the flow, and B is the width of the flume. Upstream of all three groynes, there is a negative velocity zone. A significant reduction in the u/U values and the presence of negative values of u/U are evident for the sections near the groyne attached wall, i.e., for y/B = 0.04, 0.08, and 0.15, upstream of the groyne. This phenomenon is attributed to the formation of HSV system [31]. Furthermore, flow separation at the junction or tip induces a peak of u/U for y/B = 0.2 and a lateral drift of flow along the DSL path. The maximum positive streamwise velocity for the IHG, LHG, and THG are 0.85U, 0.98U, and 0.92U at (x/L_{1}, y/B) = (1.8, 0.15), (x/L_{1}, y/B) = (1.4, 0.2), and (x/L_{1}, y/B) = (2, 0.2). Fluctuations in the streamwise velocity are observed within 2 < x/L_{1} < 5 for the IHG, within 2 < x/L_{1} < 4 for the LHG, and within 2 < x/L_{1} < 4.5 for the THG. This can be attributed to the complexity of flow arising due to the interaction between the wake zone and the secondary vortices. The maximum negative streamwise velocity for the IHG, LHG, and THG are 0.3U, 0.3U, and 0.6U at (x/L_{1}, y/B) = (0.5, 0.1), (x/L_{1}, y/B) = (0.5, 0.11), and (x/L_{1}, y/B) = (−0.4, 0.12). The formation of a shear layer, marked by an abrupt change in velocity direction, is observed along the line of flow divergence. The original streamwise velocity values are re-established after x/L_{1} = 6 for the IHG, x/L_{1} = 8 for the LHG, and x/L_{1} = 7 for the THG.
According to the boundary layer hypothesis, at the separation point, the flow velocity cannot overcome the unfavorable pressure gradient, which marks the end of the separation zone. There is a discrepancy between the junction and separation points. Indeed, the velocity at the border of separation zone should be zero, and adopting the zero-streamwise-velocity isoline as the boundary is justified. Consequently, the streamwise velocity isoline is employed to mark the geometric border of the separation zone. In this method, the mainstream isoline with zero velocity serves as the boundary of the separation zone, a technique commonly referred to as the isoline method [5]. It is evident from Figure 4 that the reattachment length, i.e., the border of separation zone or the reattachment lengths for the IHG, LHG, and THG, are 1.2 L_{1}, 0.85 L_{1}, and 1.16 L_{1}, respectively. This value is much smaller compared to the reattachment length observed in other studies [11,13,27] due to the submergence condition. The observed value of reattachment length aligns with the findings reported by Kuhnle et al. for the reattachment length around an IHG, which was observed as 1.6 L_{1} [23].
In Figure 5, the distribution of the normalized transverse velocity (v/U) around the IHG, LHG, and THG is depicted. The velocity (v) is normalized by U, and both the longitudinal and transverse directions (x and y) are normalized by L1 and B, respectively. Upstream of the groyne, a strong negative transverse velocity is consistently observed in every instance. This is because the main horseshoe vortex is located upstream of the groyne.
The peak transverse velocity is observed near the groyne attached to wall, specifically at sections y/B = 0.08, 0.15, and 0.2, within the range of x/L_{1} from 0 to 1. This characteristic can be attributed to the occurrence of flow separation at the tip or junction of the groyne. The region of negative v/U near the junction or tip of the groyne is induced by recirculation vortex [13,27]. The maximum negative transverse velocity values are observed to be 0.9 U, 0.95U, and 0.45U for IHG, LHG, and THG, respectively, at (x/L_{1}, y/B) = (0, 0.15), (x/L_{1}, y/B) = (0, 0.15), and (x/L_{1}, y/B) = (−1, 0.15).
Figure 6 represents the distribution of the normalized vertical velocity (w/U) around the IHG, LHG, and THG. The velocity w is normalized by U, and the longitudinal and transverse directions (x and y) are normalized by L_{1} and B, respectively. In each of the three scenarios, a strong upward flow is observed downstream of the groyne. Additionally, in the cases of THG and LHG, a strong upward flow is observed upstream of the groyne, However, in the case of IHG, this flow is absent. It can be inferred from the plots that a negative velocity zone exists for sections y/B = 0.04, 0.11, and 0.15 upstream of the groyne. These negative values are primarily the markers of downflow phenomenon occurring at the perpendicular face of groyne. The maximum value of the negative vertical velocity are 0.15U, 0.25U, and 0.13U for IHG, LHG, and THG, respectively, at (x/L_{1}, y/B) = (0, 0.15), at (x/L_{1}, y/B) = (1.5, 0.15), and at (x/L_{1}, y/B) = (−0.5, 0.12). The elevated values of w/U signify downward flow, which occurs upstream of the structure and is subsequently transferred downstream through the HSV. This flow pattern is attributed to the intricate dynamics of the flow around the groyne structure.
3.3. Cost–Benefit Analysis (CBA)
This is a fundamental financial tool used to systematically examine the profitability of investment projects. As a part of this analysis, a detailed analysis of the incurred expenses and the anticipated benefits is conducted.
Through empirical observations on IHG, LHG, and THG, it was established that IHG effectively safeguards a length equivalent to 1.2 times its dimension (1.2 L_{1}), THG extends its protective capacity to 1.16 times the transverse length (1.16 L_{1}), and LHG protects a length 0.85 times the transverse dimension (0.85 L_{1}). These are basically the reattachment lengths for each case and are determined using 0 velocity isoline [5]. This observation serves as the foundation for calculating the benefits associated with each type of groyne, particularly regarding the length of effective bank protection.
To enhance the precision of the cost–benefit analysis, cost coefficients (C1, C2, C3, and C4) were introduced. These coefficients intricately account for the material and labor costs, along with the economic value of the structures and properties along riverbanks that can be safeguarded by these hydraulic structures. The ensuing cost–benefit ratios for the IHG, LHG, and THG are delineated below.
$$\mathrm{Cost}\u2013\mathrm{benefit}\text{}\mathrm{ratio}\text{}\mathrm{for}\text{}\mathrm{IHG},\text{}{f}_{IHG}={\displaystyle \frac{({C}_{1}+{C}_{2}d)Lt}{{1.2LC}_{4}}}$$
$$\mathrm{Cost}\u2013\mathrm{benefit}\text{}\mathrm{ratio}\text{}\mathrm{for}\text{}\mathrm{LHG},\text{}{f}_{LHG}={\displaystyle \frac{({C}_{1}+{C}_{3}d)(2Lt-{t}^{2})}{{0.85LC}_{4}}}$$
$$\mathrm{Cost}\u2013\mathrm{benefit}\text{}\mathrm{ratio}\text{}\mathrm{for}\text{}\mathrm{THG},\text{}{f}_{THG}={\displaystyle \frac{({C}_{1}+{C}_{3}d)(2Lt-{t}^{2})}{{1.16LC}_{4}}}$$
The cost–benefit function was developed, and, as can be seen from Figure 7, three scenarios were considered.
- i.
Variable L with fixed D and t:
For this scenario, we kept the depth of flow (D) at 13.6 m and the thickness (t) at 3 m constant while varying the groyne length (L) from 0 to 25 m to explore the cost–benefit function. The function for the IHG (f_{IHG}) was found to be unaffected by the variation in L (Figure 7a). As depicted in Figure 7a, it is evident that the cost–benefit function f_{THG} and f_{LHG} assumes negative values when L is less than t, presenting an unrealistic situation. The point of intersection between the plots of the two functions occurs when L equals t. In all practical applications, the cost–benefit ratios of the THG and LHG exceed that of the IHG; this indicates that the costs associated with THG and LHG are relatively higher compared to their benefits. The functions f_{THG} and f_{LHG} are found to be minimal when L equals the t. At this point of intersection, the ratio of total costs to total benefits is identical across all three cases. As the groyne length increases from L = 3 m until it equals the depth of flow (D = 13.6 m), the functions f_{THG} and f_{LHG} exhibit a steady rise. Once this value is exceeded, the functions f_{THG} and f_{LHG} remain constant.
- ii.
Variable D with fixed L and t:
In this instance, we maintained a constant groyne length (L = 11.25 m) and groyne thickness (t = 3 m) while varying the depth of flow (0 ≤ D ≤ 25 m) for analysis (Figure 7b). With a fixed groyne length and thickness, each function, f_{IHG}, f_{THG,} and f_{LHG}, demonstrates linear increases and consistently positive values as the depth of flow incrementally rises. Notably, the rate of increase in f_{LHG} is markedly steeper in comparison to that of f_{THG}, which is steeper than f_{IHG} as the D increases. This suggests IHG as more cost-effective than THG and LHG, particularly with higher depths of flow.
- iii.
Variable t with fixed L and D:
For a constant length of groynes (L = 11.25 m) and depth of flow (D = 13.6 m), an increase in the thickness of the groynes was systematically analyzed, and the functions f_{IHG}, f_{THG,} and f_{LHG}, exhibited linear increments, maintaining positive values throughout (Figure 7c). Notably, the cost–benefit functions f_{LHG} and f_{THG} consistently displayed higher values across all the considered thicknesses of groynes, indicating that the associated costs outweigh the benefits. The steep slope of the f_{LHG} and f_{THG} curves further emphasizes the pronounced increase compared to the f_{IHG} curve, underscoring the superiority of the IHG over the THG and LHG in riverbank protection with escalating groyne thickness, t.
The framework of cost–benefit analysis is mentioned in Appendix A and Appendix B. The functions developed in the section have been conducted for a Fr of 0.61. For the specific flow condition, it can be observed that, in all the scenarios, the IHG provides better cost effectiveness than the LHG and THG. This cost–benefit analysis (CBA) framework will serve as a valuable tool for field engineers as it can be adapted to their specific environmental conditions, taking into account local hydrodynamic factors and economic constraints. The implications of our findings are significant for real-world applications. This will support informed decisions about groyne installation, optimizing both cost efficiency and functional performance. This approach not only supports the design and implementation of effective groynes but also contributes to sustainable river engineering practices that balance environmental and economic considerations.
4. Conclusions
The study focused on providing valuable insights into the flow dynamics and scour patterns around IHG, LHG and THG. Our research provides crucial insights into selecting the appropriate groyne types for specific flow conditions. By analyzing the flow dynamics around I, L, and T head groynes, our study offers a comparative assessment that can guide field engineers in choosing the most effective groyne configuration.
The major conclusions of the study are as follows:
- i.
Among the three groynes, LHG and THG show the most significant scour depths, with close values of 0.295 D and 0.29 D, respectively. IHG had a maximum scour depth of 0.21 D. The maximum scour depth for IHG is achieved near the tip of the groyne and can be attributed to the flow separation that occurs at that critical point. In the case of LHG, maximum scouring is achieved near the junction of both the faces of the groyne. However, for THG, the maximum scour depth occurs upstream of the groyne. This is due to the downflow upstream of the groyne when the flow hits the perpendicular face of the groyne and the HSV system that is the outcome of the downflow. This finding emphasizes the critical role of design and shape of the groyne on the flow characteristics and scour patterns.
- ii.
The reduction in the normalized streamwise velocity and negative values is evident for the sections near groyne attached wall upstream of the groyne. This is attributed to the formation of HSV system. The peak of the normalized streamwise velocity is due to flow separation at the junction or tip.
- iii.
Shear layer formation, marked by an abrupt change in velocity direction, is observed along the line of flow divergence. The peak transverse velocity is observed near the groyne attached wall at the junction and within a small region downstream of the junction. This characteristic can be attributed to the occurrence of flow separation at the tip or junction of the groyne.
- iv.
A strong upward flow is observed downstream of the groyne for all the groynes. Additionally, a strong upward flow is observed upstream of the groyne for THG and LHG which is absent for IHG. The increase in w/U upstream signifies downward flow upstream of the groyne, which is subsequently transferred downstream through the HSV.
The reattachment length for IHG, LHG and THG is 1.2 L_{1}, 0.85 L_{1}, and 1.16 L_{1,} respectively. The difference in the submergence condition is attributed as the principal reason for this value to be smaller compared to the values observed by other studies. The cost–benefit analysis of the groynes provides insight regarding the expenses involved and the benefits expected. The results show IHG as the most cost effective groyne as compared to the THG and LHG. This analytical approach is designed to provide a quantitative framework for decision-making, facilitating a comprehensive evaluation of the economic feasibility and efficiency of the undertaking under consideration.
Understanding the variations in scour patterns and the impact of the groyne shape on riverbank stability is crucial for river engineering and management. The insights gained from this study contribute to the body of knowledge in this field and can inform future groyne design and riverbank protection strategies. This study investigates the efficacy of these river training structures in stabilizing the river channels. This study underscores the critical importance of implementing effective riverbank protection measures to ensure sustainable development. By developing a predictive model for determining the appropriate length of bank protection in the vicinity of hydraulic structures, we provide a valuable tool for engineers and decision-makers. Our model enhances the ability to design and implement targeted riverbank stabilization strategies, thereby improving the resilience of the hydraulic infrastructure. The successful application of this model can help to protect communities and ecosystems, promoting a balanced approach to development that prioritizes both human safety and environmental sustainability. Our work provides a comprehensive methodology for assessing the groyne performance under various flow conditions and offers practical tools for field engineers to apply these insights in real-world scenarios. As we strive to create resilient river systems, particularly in changing climate conditions, our research underscores the importance of selecting the most suitable groyne shape to mitigate erosion and maintain stable river channels.
Author Contributions
Conceptualization, P., M.K.M., C.S.P.O. and K.S.H.P.; Methodology, P. and S.S.; Software, P. and M.K.M.; Validation, P.; Formal analysis, P. and M.K.M.; Investigation, P., M.K.M. and S.S.; Writing—original draft, P.; Writing—review and editing, P, M.K.M., C.S.P.O. and K.S.H.P.; Visualization, C.S.P.O. and K.S.H.P.; Supervision, C.S.P.O. and K.S.H.P. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Data are available in this article.
Acknowledgments
The authors would like to thank members of Hydraulics Lab, IITR, for experimental setup provided.
Conflicts of Interest
Author Shikhar Sharma was employed by L & T Construction. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Appendix A. Cost–Benefit Analysis for I Head Groynes (IHG), L Head Groynes (LHG), and T Head Groynes (THG)
Cost estimation for IHG and THG has been discussed in Mall et al. [14]. The function has been modified according to the experimental results obtained in this study. The bank protection lengths are estimated to be 1.2 L and 1.16 L for IHG and THG, respectively. The parameters in these functions are defined as
Flow depth, D;
River width, B;
Transverse length, L;
Free board, F_{B};
Thickness of groyne, t.
Therefore, the cost–benefit function (CBF) for IHG is
$${f}_{IHG}={\displaystyle \frac{\left({C}_{1}+{C}_{2}D\right)Lt}{1.2L{C}_{4}}}$$
The CBF for THG is
$${f}_{THG}={\displaystyle \frac{({C}_{1}+{C}_{3}D)(2Lt-{t}^{2})}{1.16L{C}_{4}}}$$
The overall expense associated with constructing a single IHG can be calculated by summing the cost of concreting, cost of excavation, and cost of reinforcement.
$${Overall\_cost}_{IHG}={Final\_cost}_{concrete}+{Final\_cost}_{reinforcement}+{Final\_cost}_{excavation}$$
$${Overall\_cost}_{IHG}=\left({C}_{c}+{C}_{r}\right)\xb7Lt\left[({d}_{f}\times 0.21+1)D+{F}_{B}\right]+{C}_{e}DLt=({C}_{1}+{C}_{2}D)Lt$$
where C_{1} = F_{B} (C_{c} + C_{r}), and C_{2} = (df × 0.21 + 1)(C_{c} + C_{r}) C_{e} represent cost coefficients that vary based on the expenses associated with concrete material, reinforcement, and excavation.
The overall expense associated with constructing a single THG can be calculated by summing the cost of concreting, cost of excavation, and cost of reinforcement.
$${Overall\_cost}_{THG}={Final\_cost}_{concrete}+{Final\_cost}_{reinforcement}+{Final\_cost}_{excavation}$$
$${Overall\_cost}_{THG}=\left({C}_{c}+{C}_{r}\right)\xb7\left(2Lt-{t}^{2}\right)\xb7\left[\left({d}_{f}\times 0.29+1\right)D+{F}_{B}\right]+{C}_{e}\xb7D(2Lt-{t}^{2})\phantom{\rule{0ex}{0ex}}=({C}_{1}+{C}_{3}D)(2Lt-{t}^{2})$$
where C_{1} = F_{B} (C_{c} + C_{r}), and C_{3} = (df × 0.29 + 1)(C_{c} + C_{r}) C_{e} represent cost coefficients that vary based on the expenses associated with concrete material, reinforcement, and excavation.
Hereby, we extend the cost–benefit function for LHG.
Appendix B. Cost Estimation for LHG
The transverse length ($L$) is the length of the groyne along the direction of flow.
Depth of scour (equilibrium scour depth for LHG, Figure 3) = $0.30D$.
Appendix B.1. Cost Estimation
Appendix B.1.1. Cost of Concreting
The cost involved in concreting for LHG is estimated by calculating the volume of concrete needed for LHG, which is calculated as follows:
$${V}_{concrete}=\left(2Lt-{t}^{2}\right)\xb7\left[\right({d}_{f}\times 0.30+1)D+{F}_{B}]$$
The total cost of concreting is now calculated for LHG using ${C}_{c}$, which is cost of concrete per cubic meter:
$${Total\_cost}_{concrete}={C}_{c}\xb7\left(2Lt-{t}^{2}\right)\xb7\left[\right({d}_{f}\times 0.30+1)D+{F}_{B}]$$
Appendix B.1.2. Cost for Reinforcement
If we consider nominal reinforcement for the concrete hydraulic structure as $n\mathrm{\%}$ of the volume of concrete, the total volume of reinforcement would be
$${V}_{reinforcement}={\displaystyle \frac{n}{100}}\xb7\left(2Lt-{t}^{2}\right)\xb7\left[\right({d}_{f}\times 0.30+1)D+{F}_{B}]$$
The overall weight of reinforcement for LHG is
$${Total\_weight}_{reinforcement}={W}_{r}\xb7{\displaystyle \frac{n}{100}}\xb7\left(2Lt-{t}^{2}\right)\xb7\left[\right({d}_{f}\times 0.30+1)D+{F}_{B}]$$
where ${W}_{r}$ denotes unit weight of reinforcement bars.
The overall cost for reinforcement can be calculated as follows:
$${Overal{l}_{cost}}_{reinforcement}={R}_{r}\xb7{W}_{r}\xb7{\displaystyle \frac{n}{100}}\xb7\left(2Lt-{t}^{2}\right)\xb7\left[\left({d}_{f}\times 0.30+1\right)D+{F}_{B}\right]\phantom{\rule{0ex}{0ex}}={C}_{r}\xb7\left(2Lt-{t}^{2}\right)\xb7\left[\right({d}_{f}\times 0.30+1)D+{F}_{B}]$$
where ${R}_{r}$ represents the cost of reinforcement bars per unit weight.
C_{r} = R_{r} W_{r}$\frac{n}{100}$, represents cost coefficient.
Appendix B.1.3. Excavation Cost
Determination of the cost involved in excavation for LHG requires calculating the volume of excavation, which is
$${V}_{excavation}=\left({d}_{f}\times 0.30\right)D\xb7\left(2Lt-{t}^{2}\right)\mathrm{cubic}\mathrm{meters}.$$
The total cost of excavation amounts to
$${Total\_cost}_{excavation}={R}_{e}\xb7\left({d}_{f}\times 0.30\right)D\xb7\left(2Lt-{t}^{2}\right)={C}_{e}\xb7D(2Lt-{t}^{2})$$
where ${R}_{e}$ is the cost of excavation per cubic meter.
C_{e} = R_{e} (d_{f} × 0.30) depicts the cost coefficient.
Appendix B.2. Total Cost for Construction of LHG
The overall expense associated with constructing a single LHG is the sum of the cost of concreting, the cost of excavation, and the cost of reinforcement.
$${Overall\_cost}_{LHG}={Overall\_cost}_{concrete}+{Overall\_cost}_{reinforcement}+{Overall\_cost}_{excavation}$$
$${Overal{l}_{cost}}_{LHG}=\left({C}_{c}+{C}_{r}\right)\xb7\left(2Lt-{t}^{2}\right)\xb7\left[\left({d}_{f}\times 0.30+1\right)D+{F}_{B}\right]+{C}_{e}\xb7D\left(2Lt-{t}^{2}\right)=\left({C}_{1}+{C}_{3}D\right)\left(2Lt-{t}^{2}\right)\phantom{\rule{0ex}{0ex}}=({C}_{1}+{C}_{3}D)(2Lt-{t}^{2})$$
where C_{1} = F_{B} (C_{c} + C_{r}), and C_{3} = (df × 0.30 + 1)(C_{c} + C_{r}) C_{e} are cost coefficients, which depend on the costs of concrete material, reinforcement, and excavation.
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Figure 1. Experimental setup for (a) IHG, (b) LHG, and (c) THG.
Figure 1. Experimental setup for (a) IHG, (b) LHG, and (c) THG.
Figure 2. Power spectrum of fluctuations of streamwise velocity.
Figure 2. Power spectrum of fluctuations of streamwise velocity.
Figure 3. Scour depth around (a) IHG, (b) LHG, and (c) THG.
Figure 3. Scour depth around (a) IHG, (b) LHG, and (c) THG.
Figure 4. Dimensionless streamwise velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 4. Dimensionless streamwise velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 5. Dimensionless transverse velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 5. Dimensionless transverse velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 6. Dimensionless vertical velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 6. Dimensionless vertical velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 7. Cost–benefit functions f_{IHG} (blue), f_{LHG} (green), and f_{THG} (red) for (a) variable L with fixed D and t, (b) variable D with fixed L and t, and (c) variable t with fixed L and D.
Figure 7. Cost–benefit functions f_{IHG} (blue), f_{LHG} (green), and f_{THG} (red) for (a) variable L with fixed D and t, (b) variable D with fixed L and t, and (c) variable t with fixed L and D.
Table 1. Flow Parameters for experiments.
Table 1. Flow Parameters for experiments.
Shape | D (m) | Fr | L_{1} (m) | L_{2} (m) | C.R. |
---|---|---|---|---|---|
IHG | 0.136 | 0.61 | 0.1125 | - | 15% |
LHG | 0.136 | 0.61 | 0.1125 | 0.1125 | 15% |
THG | 0.136 | 0.61 | 0.1125 | 0.1125 | 15% |
Table 2. Turbulent velocity statistics of experimental data for section y/B = 0.77.
Table 2. Turbulent velocity statistics of experimental data for section y/B = 0.77.
x/L_{1} | $\overline{\mathit{u}}$ (m/s) | Std. Dev. (m/s) | Skewness | Kurtosis | Std. Error (m/s) |
---|---|---|---|---|---|
$-$12.50 | 0.729797 | 0.081635 | $-$0.05941 | 2.708591 | 0.000868 |
$-$8.00 | 0.751042 | 0.082159 | $-$0.15902 | 2.794846 | 0.000863 |
$-$4.00 | 0.726126 | 0.081202 | 0.007192 | 2.876749 | 0.000875 |
0.00 | 0.791727 | 0.083517 | $-$0.21658 | 3.095242 | 0.000887 |
4.00 | 0.738749 | 0.083441 | $-$0.18728 | 2.881535 | 0.000856 |
8.00 | 0.837464 | 0.081221 | $-$0.17489 | 2.771532 | 0.001299 |
11.25 | 0.862593 | 0.080529 | $-$0.25855 | 3.312264 | 0.000826 |
15.00 | 0.879171 | 0.077864 | $-$0.21254 | 2.748936 | 0.000901 |
20.00 | 0.866055 | 0.084618 | $-$0.28543 | 2.758457 | 0.000878 |
25.00 | 0.83369 | 0.082747 | $-$0.1951 | 2.837325 | 0.001033 |
30.00 | 0.734824 | 0.096966 | $-$0.24464 | 2.791556 | 0.001046 |
35.00 | 0.642727 | 0.098407 | $-$0.02617 | 2.646666 | 0.000887 |
41.00 | 0.83348 | 0.085447 | $-$0.26433 | 2.773093 | 0.000909 |
47.00 | 0.739169 | 0.092991 | $-$0.0956 | 2.712638 | 0.000987 |
53.00 | 0.733704 | 0.093498 | 0.012802 | 2.782523 | 0.000993 |
59.00 | 0.810494 | 0.097884 | $-$0.27136 | 2.715227 | 0.001044 |
65.00 | 0.711532 | 0.097301 | $-$0.19889 | 2.651946 | 0.001806 |
75.00 | 0.651122 | 0.122696 | $-$0.16853 | 2.639037 | 0.00161 |
85.00 | 0.705159 | 0.100333 | $-$0.20404 | 2.752259 | 0.000863 |
95.00 | 0.67583 | 0.100405 | $-$0.24629 | 2.750138 | 0.001311 |
110.00 | 0.67236 | 0.099111 | $-$0.16994 | 2.690638 | 0.001311 |
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