Numerical Simulation Study on Hydraulic Characteristics of Asymmetric and Symmetric Triangular Labyrinth Weirs: A Comparative Analysis

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this study, the authors investigated asymmetric cycle conditions of a symmetric triangular labyrinth weir using computational fluid dynamics (CFD). The weir geometry was modeled based on a symmetric triangular labyrinth weir reported in the literature. The methodologies and approaches employed in the study were appropriate and correctly implemented. The paper is well organized, and the results appear to be robust. It is commendable that the authors included numerical model validation and mesh sensitivity analyses. Overall, the study is of good quality and worthy of publication, though minor revisions are recommended based on the comments provided below.

1. Which mathematical model was used for free surface flow? For example, if a TrueVOF scheme was used, provide its theory. Was a single-phase or two-phase (air-water) flow model used? Pleas give more detail undet the title of Control Equations
2. Why was RNG k-epsilon model prefered? was tried the other turbulence model for test of the models? or any are there literature recommendation to select the turbulence model?
3. The boundary conditions can be given as a table together with the range of their variable and parameters.
4. What is the reference for GCI approaches. Please give a reference for this.
5. Please refer to solution convergence for Fig. 5.
6. Explain the relationship between nappe adhesion and sub-nappe pressures at low water heads seen in Figure 6, giving a contours of the pressure distributions if possible.
7. What should be the minimum crest water head so that the effect of surface tension can be neglected? Please specify with reference to the literature.
8. Figures 9 and 10 show that the symmetrical condition provides significantly better weir efficiency and performance than the asymmetrical condition. So why use Asymmetric Labyrinth Triangle Weirs? What is the authors' defense of this issue in their work?

Author Response

Comments 1: Which mathematical model was used for free surface flow? For example, if a TrueVOF scheme was used, provide its theory. Was a single-phase or two-phase (air-water) flow model used? Please give more detail under the title of Control Equations.

Response 1: I apologize for any confusion. We have expanded the "Control Equations" section in the revised manuscript. A single-phase fluid model is employed in this study, assuming the fluid to be incompressible, with an emphasis on the free surface. In this model, the fluid volume fraction is used to describe the distribution of the liquid and gas phases. When the volume fraction F = 1, the computational cell is completely filled with liquid; when F = 0, the cell is filled with air; and when 0 < F < 1, the cell contains the free surface region. The Volume of Fluid (VOF) method is used to capture the free surface, following the approach proposed by Hirt and Nichols [33]. The method tracks the free surface by solving a transport equation for the volume fraction of the fluid in each computational cell. The basic assumption of the VOF method is that in regions where the volume fraction F = 0 (i.e., air), the pressure and density are uniform, and the inertia of the gas phase is negligible compared to the liquid phase. Therefore, this study employs a single-phase flow model, and the governing equations do not involve a two-phase flow treatment. The manuscript has been revised and supplemented accordingly, from lines 180 to 190.

 

Comments 2: Why was RNG k-epsilon model preferred? Was the other turbulence model tried for test of the models? Or are there literature recommendations to select the turbulence model?

Response 2: Thank you for your inquiry. This study employed the RNG k-ε turbulence model, which is based on the improved Renormalization Group (RNG) method to solve turbulence problems. It is capable of effectively simulating low-intensity turbulence, strong shear flows, and high-velocity swirling flows, making it particularly suitable for flow simulations on highly refined computational meshes. The RNG k-ε model demonstrates high accuracy and computational efficiency when handling these complex flows. The RNG k-ε model has been widely applied in fluid dynamics and hydraulic engineering, and is recommended in various fields for providing more accurate and efficient solutions. Literature indicates that this model is particularly well-suited for describing flows with strong shear and complex turbulence structures, and is able to capture small-scale eddies and large turbulent vortices in the flow [37-41]. The manuscript has been revised and supplemented accordingly, as seen in lines 209 to 214.

[37] Mei, J.; Zhou, Y.; Xu, K.; et al. Energy Dissipation on Inclined Stepped Spillways. Water 2025, 17 (2), 251. https://doi.org/10.3390/w17020251.

[38] Mirkhorli, P.; Ghaderi, A.; MohammadNezhad, H.; et al. Hydraulic Behavior and Energy Dissipation in Piano Key Weirs vs. Rectangular Labyrinth Weirs: A Comparative Study. Flow Meas. Instrum. 2025, 102, 102830. https://doi.org/10.1016/j.flowmeasinst.2025.102830.

[39] Roy Biswas, T.; Singh, P.; Sen, D. Submerged Flow over Barrage Weirs: A Computational Fluid Dynamics Model Study. J. Irrig. Drain. Eng. 2021, 147 (12), 04021058. https://doi.org/10.1061/(ASCE)IR.1943-4774.0001634.

[40] Safarzadeh, A.; Noroozi, B. 3D Hydrodynamics of Trapezoidal Piano Key Spillways. Int. J. Civ. Eng. 2017, 15 (1), 89–101. https://doi.org/10.1007/s40999-016-0100-8.

[41] Arjenaki, M. O.; Sanayei, H. R. Z. Numerical Investigation of Energy Dissipation Rate in Stepped Spillways with Lateral Slopes Using Experimental Model Development Approach. Model. Earth Syst. Environ. 2020, 6 (2), 605–616. https://doi.org/10.1007/s40808-020-00714-z.

 

Comments 3: The boundary conditions can be given as a table together with the range of their variable and parameters.

Response 3: We agree with this suggestion. We have revised the manuscript to include a table summarizing the boundary conditions and specified the relevant ranges for each parameter. We have clearly listed the boundary conditions for each mesh block, including pressure inlet, free outflow boundary, wall boundaries, air inlet, and symmetry conditions. These are now organized in an intuitive and easy-to-understand table format. We have specified the range for the free-surface elevation at the channel inlet, which is prescribed between 0.013 m and 0.05 m. The manuscript has been revised and supplemented accordingly, as seen in lines 224 to 234.

 

Comments 4: What is the reference for GCI approaches? Please give a reference for this.

Response 4: Thanks for your comments. We have added the relevant references for the Grid Convergence Index (GCI) method and provided supporting citations in the revised manuscript. To ensure the reliability of the numerical simulations, we performed a Grid Convergence Index (GCI) analysis and experimental validation. The GCI method quantifies the discretization error and assesses how closely the numerical solution approaches the asymptotic value. We have included references to several studies that have applied the GCI method to assess the grid independence of numerical results [42-45]. In the revised section, we clearly cite the relevant literature to support the use of the GCI method for grid independence analysis in this study. The manuscript has been revised and supplemented accordingly, as seen in lines 239 to 240.

[42] Shen, G.; Cao, D.; Li, S.; et al. Numerical and Sensitivity Analysis of Hydraulic Characteristics of Triangular Labyrinth Side Weir. Flow Meas. Instrum. 2024, 100, 102686. https://doi.org/10.1016/j.flowmeasinst.2024.102686.

[43] Celik, I. B.; Ghia, U.; Roache, P. J.; et al. Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications. J. Fluids Eng.-Trans. ASME 2008, 130 (7). https://doi.org/10.1115/1.2960953.

[44] Soydan Oksal, N. G.; Akoz, M. S.; Simsek, O. Numerical Modelling of Trapezoidal Weir Flow with RANS, LES, and DES Models. Sādhanā 2020, 45 (1), 91. https://doi.org/10.1007/s12046-020-01332-2.

[45] Oksal, N. G. S.; Akoz, M. S.; Simsek, O. Experimental Analysis of Flow Characteristics over Hydrofoil Weirs. Flow Meas. Instrum. 2021, 79, 101867. https://doi.org/10.1016/j.flowmeasinst.2020.101867.

 

Comments 5: Please refer to solution convergence for Fig. 5.

Response 5: Thank you for your suggestions. We have provided a more detailed explanation of the solution convergence process and specifically addressed Figure 5. we enhanced the description of how the convergence of the numerical solution was evaluated. The discharge variation at a specified cross-section was rigorously monitored throughout the simulation to track the convergence of the steady-state solution. We have further clarified the role of the discharge time series shown in Figure 5 as a key indicator for assessing computational convergence. To ensure that all initial transient effects were completely eliminated, the simulation was run for a sufficiently long duration. After an initial development phase of approximately 2 to 3 seconds, the discharge curves for both conditions exhibited an asymptotic convergence trend, gradually approaching a constant value. Around 6 seconds, the temporal variation in discharge became negligible, with fluctuations confined within ±0.5% of the mean value. The high stability of the discharge variation at the monitoring section indicates that the flow field had achieved full convergence, yielding a reliable steady-state solution. The manuscript has been revised and supplemented accordingly, as seen in lines 267 to 279.

 

Comments 6: Explain the relationship between nappe adhesion and sub-nappe pressures at low water heads seen in Figure 6, giving contours of the pressure distributions if possible.

Response 6: In the revised manuscript, we have provided a detailed explanation of the relationship between nappe adhesion and sub-nappe pressures under low water head conditions. Specifically, when the flow passes over the weir crest, the nappe is relatively thin, and the water depth is shallow. Although the flow velocity at the weir crest is not high, the flow experiences some acceleration, which leads to a reduction in static pressure, thereby generating negative pressure. We have explained the mechanism behind the formation of the negative pressure zone and discussed how the limitations on flow velocity under low head conditions, as well as nappe adhesion, influence the extent and distribution of the negative pressure zone. Additionally, we have improved the original pressure distribution contour plots, notably by increasing the font size to enhance the readability and clarity of the figures. These adjustments ensure that the details of the contour plots are more legible, particularly in terms of presenting the pressure distribution more clearly. The relevant arguments are presented in the manuscript in lines 306 to 314 and lines 323 to 324.

 

Comments 7: What should be the minimum crest water head so that the effect of surface tension can be neglected? Please specify with reference to the literature.

Response 7: Thank you for your suggestion. We have elaborated on the effect of surface tension in the revised manuscript and provided relevant references from the literature. We also clarified the minimum crest water head at which surface tension effects can be neglected. According to Tullis et al. [35], the influence of surface tension is more significant at very low headwater conditions, particularly when the headwater is below a critical threshold, at which point these effects become more pronounced. We have specified that the minimum relative headwater in this study is H₀/P = 0.13, which exceeds the critical threshold of 0.12 given by Tullis et al. [35] for their largest model scale. Therefore, the effects of surface tension are considered negligible, and any resulting scale effects are expected to remain within ±5%, consistent with observations for piano key weirs with flat crests [35,36].The manuscript has been revised and supplemented accordingly, as seen in lines 199 to 205.

[35] Tullis, B. P.; Crookston, B. M.; Young, N. Scale Effects in Free-Flow Nonlinear Weir Head-Discharge Relationships. J. Hydraul. Eng. 2020, 146 (2), 04019056. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001661.

[36] Tullis, B. P.; Young, N.; Crookston, B. M. Physical Modeling Size-Scale Effects for Labyrinth Weirs with Half-Round Crests. In Labyrinth and Piano Key Weirs III; CRC Press: Boca Raton, FL, 2017; pp. 185–192. https://doi.org/10.1201/9781315169064-26.

 

Comments 8: Figures 9 and 10 show that the symmetrical condition provides significantly better weir efficiency and performance than the asymmetrical condition. So why use Asymmetric Labyrinth Triangle Weirs? What is the authors' defense of this issue in their work?

Response 8: Thank you for your insightful comment. We acknowledge that the symmetrical labyrinth weirs (STLWs) show higher efficiency and performance in terms of discharge capacity under idealized conditions as presented in Figures 9 and 10. However, the motivation for studying asymmetric labyrinth weirs (ATLWs) stems from the practical need to address real-world hydraulic challenges that arise from complex topographies and varying flow conditions, which often cannot be effectively addressed by symmetric designs alone. While symmetric labyrinth weirs exhibit excellent performance under ideal conditions (uniform flow, regular topography), their application in real-world scenarios can be limited. In particular, in flood control systems and water resource management, the natural environment often presents non-uniform flow conditions and complex terrain that may not be well-suited for symmetric designs. ATLWs offer a more adaptable solution as their asymmetric geometry allows for better integration into such environments. By allowing for flexibility in design, ATLWs can potentially enhance the overall flow management, especially in areas with varied inflow patterns, irregular weir geometries, and varying flow rates. The relevant arguments are presented in the manuscript in lines 65 to 70 and lines 124 to 129.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1. Line 28-29: could you please mention the percentage compared to STLWs?
2. Line 50-51: “STLWs)—including triangular, trapezoidal, rectangular, and semi-circular configurations—” should be in parenthesis.
3. Line 52: you have not mentioned the end point of n. is it infinite ? Moreover, please mention and focus this pint about which researchers used which value of n? and what is the impact of this parameter on other parameters that you are going to study. Please be specific.
4. In your whole manuscript, “—” this sign was used so many times. Can you please define it ? What is the definition of this sign? Why did you use so many times in your manuscript? (remove this sign from your manuscript)
5. Line 62-66: here you mention again about the geometric parameters. But in the previous paragraph Line 52, you already mentioned about “n”. Please revise it and ensure your fluency.
6. Line 62-102: you summarized some studies, but all are very inconsistent, without concluding remarks. What was the study gap in all this research. In the introduction, if you mentioned some literature, it doesn’t mean that this study was done by ABC. The amin purpose is to highlight. Please conclude this paragraph by mentioning all the points that have not considered by the studies mentioned.
7. Line 62-102: All are considering STLWs? If yes, then none of the study till now has not been done by any researcher ?
8. Just like streamlines figure, you add red color arrows to highlight the changes. Like that, please add in figure 6 to highlight the important changes in the pressure.
9. I am not sure which approach you utilized? Either rigid lid assumption or VOF? Please mention clearly including abstract.
10. Even in the introduction you have not highlighted this important factor,
11. To strengthen your introduction, I would suggest, “Numerical modeling of flow dynamics around L‑shaped and T‑shaped dikes with varying geometric configurations and wing arrangements”

Author Response

Comments 1: Line 28–29: could you please mention the percentage compared to STLWs?

Response 1: Thank you for your suggestion. We have added a percentage comparison in the revised manuscript to address the performance differences between ATLWs and STLWs under high headwater conditions. Based on the numerical simulation results, we have provided a detailed comparison of the discharge coefficient (C_d) between STLWs and ATLWs at high headwater conditions. Specifically, we found that the Cd of STLWs is approximately 5.4% to 14.3% higher than that of ATLWs under these conditions. The manuscript has been revised and supplemented accordingly, as seen in lines 28 to 31.

 

Comments 2: Line 50–51: "STLWs)---including triangular, trapezoidal, rectangular, and semi-circular configurations---" should be in parenthesis.

Response 2: We apologize for this formatting oversight. We have made the corresponding revisions in the manuscript. The part "STLWs— including triangular, trapezoidal, rectangular, and semi-circular configurations—" has been moved into parentheses to enhance clarity and accuracy. Following this suggestion, we have replaced the dashes with parentheses to ensure proper punctuation usage and to make the sentence structure more concise and clear. The manuscript has been revised and supplemented accordingly, as seen in lines 55 to 57.

 

Comments 3: Line 52: you have not mentioned the end point of n. is it infinite? Moreover, please mention and focus this point about which researchers used which value of n? and what is the impact of this parameter on the other parameters that you are going to study. Please be specific.

Response 3: I apologize for any confusion caused. In this study, n represents the number of cycles in the labyrinth weir. We did not set n to infinity but instead tested a reasonable range of cycle numbers within the scope of numerical simulations. The cycle numbers considered ranged from n=1 to n=5. Based on the practical experience from literature [15-19], the value of n generally does not exceed 5, as the discharge coefficient (C_d) tends to stabilize and no longer increases significantly beyond this point. Therefore, our focus is on asymmetric triangular labyrinth weirs with non-integer cycle numbers (n = 1.5, 2.5, 3.5) and symmetric triangular labyrinth weirs with integer cycle numbers (n = 2, 3), as mentioned in Section 2.1 (Computational Domain and Mesh). This approach allows for a direct comparison between asymmetric and symmetric configurations, while also enabling the investigation of the effects of cycle number within a practically relevant range. Many studies have focused on testing different n values. For example, Tacail et al. [15] studied labyrinth weirs with n=2 and n=3; Namazi and Mozaffari [16] investigated labyrinth weirs with n=1 and n=2; Emiroglu et al. [17] also studied labyrinth weirs with n=1 and n=2; Khode et al. [18] analyzed labyrinth weirs with n=2; and Christensen [19] studied labyrinth weirs with n≥5. Regarding the impact of cycle number n on the discharge coefficient (C_d), the studies by Tacail et al. [15] and Emiroglu et al. [17] both indicate that the C_d decreases as n increases.

 

[15] Tacail, F. G.; Even, B.; Babb, A. Case Study of a Labyrinth weir Spillway. Can. J. Civ. Eng. 1990, 17 (1), 1–7. https://doi.org/10.1139/l90-001.

[16] Namazi, F. S. A.; Mozaffari, J. Investigation of Labyrinth Weirs Discharge Coefficient with the Same Length. Flow Meas. Instrum. 2023, 94 (1), 102468. https://doi.org/10.1016/j.flowmeasinst.2023.102468.

[17] Emiroglu, M. E.; Aydin, M. C.; Kaya, N. Discharge Characteristics of a Trapezoidal Labyrinth Side Weir with One and Two Cycles in Subcritical Flow. J. Irrig. Drain Eng. 2014, 140 (5), 04014007. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000709.

[18] Khode, B. V.; Tembhurkar, A. R.; Porey, P. D.; Ingle, R. N. Experimental Studies on Flow over Labyrinth Weir. J. Irrig. Drain Eng. 2012, 138 (6), 548–552. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000336.

[19] Christensen, N. A. Flow Characteristics of Arced Labyrinth Weirs. M.S. Thesis, Utah State University, Logan, UT, 2013. https://doi.org/10.26076/a055-29f4.

 

Comments 4: In your whole manuscript, "---" this sign was used so many times. Can you please define it? What is the definition of this sign? Why did you use so many times in your manuscript? (remove this sign from your manuscript)

Response 4: We apologize for the inconsistent and excessive use of the em-dash throughout the original manuscript. All instances of the standalone em-dash have been removed. The affected sentences have been carefully revised, typically by using commas, parentheses, or by restructuring the sentence for better clarity and formal academic tone. The relevant arguments are presented in the manuscript in lines 55 to 57 and lines75 to 78.

 

Comments 5: Line 62–66: here you mention again about the geometric parameters. But in the previous paragraph Line 52, you already mentioned about "n". Please revise it and ensure your fluency.

Response 5: Thanks for your comments. We have followed the reviewer's suggestion by removing the repetitive content in lines 62-66 that was already mentioned in line 52, and rephrasing the reference to "n" to ensure the conciseness and logical consistency of the manuscript. The manuscript has been revised and supplemented accordingly, as seen in lines 58 to 59.

 

Comments 6:  Line 62–102: you summarized some studies, but all are very inconsistent, without concluding remarks. What was the study gap in all this research? In the introduction, if you mentioned some literature, it doesn't mean that this study was done by ABC. The main purpose is to highlight. Please conclude this paragraph by mentioning all the points that have not considered by the studies mentioned.

Response 6: We have added a concluding paragraph at the end of the Introduction (Lines 124–129) that explicitly states the research gap: previous studies have focused largely on symmetric weirs under idealized conditions, with limited attention to asymmetric frontal weirs and their performance in complex, non-uniform flow environments. Our study aims to fill this gap.

 

Comments 7: Line 62–102: All are considering STLWs? If yes, then none of the study till now has not been done by any researcher?

Response 7: Thank you for your inquiry. We have revised the manuscript to clearly differentiate between research on symmetric (STLWs) and asymmetric labyrinth weirs (ATLWs). Specifically, we have clarified that most previous studies have focused on symmetric configurations and highlighted the research gap regarding asymmetric frontal labyrinth weirs. We emphasize that while studies on ATLWs remain limited, existing research, such as those by Karimi et al. [29] and Borghei and Parvaneh [30], has investigated asymmetric labyrinth side weirs. However, these studies are confined to side weirs with lateral inflow and do not address the hydraulic characteristics of asymmetric frontal labyrinth weirs, which remains a significant research gap in the literature. The manuscript has been revised and supplemented accordingly, as seen in lines 117 to 124.

 

[29] Karimi, M.; Ghazizadeh, M. J.; Saneie, M.; Attari, J. Flow Characteristics over Asymmetric Triangular Labyrinth Side Weirs. Flow Meas. Instrum. 2019, 68, 101574. https://doi.org/10.1016/j.flowmeasinst.2019.101574.

[30] Borghei, S. M.; Parvaneh, A. Discharge Characteristics of a Modified Oblique Side Weir in Subcritical Flow. Flow Meas. Instrum. 2011, 22 (5), 370–376. https://doi.org/10.1016/j.flowmeasinst.2011.04.009.

 

Comments 8: Just like streamlines figure, you add red color arrows to highlight the changes. Like that, please add in figure 6 to highlight the important changes in the pressure.

Response 8: Thank you for your suggestions. We have improved the original pressure distribution contour plots, particularly by increasing the font size to enhance the readability and clarity of the figures. These adjustments ensure that the details of the contour plots are more legible, particularly in terms of presenting the pressure distribution more clearly.

 

Comments 9: I am not sure which approach you utilized? Either rigid lid assumption or VOF? Please mention clearly including abstract.

Response 9: We apologize for the lack of clarity. The numerical model employed the Volume of Fluid (VOF) method for free-surface tracking. The rigid lid assumption was not used. To ensure this is clear, we have explicitly mentioned "VOF" in the abstract (Lines 20 to 22) and provided a detailed description of the method and the single-phase modelling approach in Section 2.2.1 (Lines 185 to 190).

 

Comments 10: Even in the introduction you have not highlighted this important factor,

Response 10: Thanks for your comments. We have revised the introduction to better highlight the key factors and methods used in our study, particularly the application of the RNG k-ε turbulence model and the Volume of Fluid (VOF) method for simulating flow dynamics. These approaches are essential for accurately capturing the complex behaviors of flow around the labyrinth weirs. The revised introduction now includes a more detailed explanation of these methods. The manuscript has been revised and supplemented accordingly, as seen in lines 137 to 144.

 

Comments 11: To strengthen your introduction, I would suggest, "Numerical modeling of flow dynamics around L‑shaped and T‑shaped dikes with varying geometric configurations and wing arrangements"

Response 11: We appreciate the reviewer’s valuable suggestion. We have carefully read and thoroughly studied the article you mentioned. While it offers valuable insights into fluid dynamics and numerical simulation of hydraulic structures, the focus of the study differs from the main emphasis of our research. Our study primarily investigates the hydraulic performance of asymmetric labyrinth weirs, with a specific focus on the impact of geometric parameters on flow characteristics and discharge capacity under complex flow conditions. Although the referenced article is highly informative, its focus does not fully align with our research objectives. As such, we did not directly reference it in our current study. However, we will continue to monitor developments in this area and consider incorporating relevant insights into future research to further enrich our theoretical framework.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Further, i don't have any comment.