Nonlinear Water Waves Induced by Vertical Disturbances Through a Navier–Stokes Solver with the Implementation of the Immersed Boundary Method

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents numerical study of near and far wave field caused by the bottom disturbances. The topic is interesting and paper is well organized. I recommend the acceptance of the manuscript after the following issues are addressed.

1. Please explain the rationale for using cases with extremely small absolute dimensions in Section 3.1—specifically, a wave height of only 0.0068 meters and a vertical grid size of merely 0.0005 meters.
2. Case 3.2 uses horizontal moving of wave paddle to generate solitary wave, has no connection with the vertical bottom disturbance.
3. Please clearly state whether the bottom disturbance function is an inherent feature of the OpenFOAM or an extended work carried out by the authors.
4. In section 2, which term accounts for the contribution of vertical bottom disturbance.
5. In Section 3.3, additional information about the model proposed by Shen and Chan (2008) should be presented. Specifically, is it a CFD-type model or another type of model? Furthermore, the discrepancies (including those in amplitude and phase) between the experimental data and the model results of Shen and Chan (2008) in Fig. 7 require further explanation. Notably, the proposed model is expected to outperform the aforementioned model, as it belongs to the CFD type and includes viscosity.
6. Line 195, stoke should be stroke?
7. The simulated results are missing in Fig.7(c) after t>15s.
8. A colorbar scale is needed in Fig.11 and Fig.12.

Author Response

Overall comments:

The manuscript presents numerical study of near and far wave field caused by the bottom disturbances. The topic is interesting and paper is well organized. I recommend the acceptance of the manuscript after the following issues are addressed.

Response to overall comments: Thanks for your generous comments. Additional modifications in the revision are marked by the yellow highlight.

 

Specified comments:

Comments 1: Please explain the rationale for using cases with extremely small absolute dimensions in Section 3.1—specifically, a wave height of only 0.0068 meters and a vertical grid size of merely 0.0005 meters.

Response 1: Thank you pointing out this issue. First, the linear wave is not sensitive to the grid size (Higuera et al., 2015), the use of a vertical grid (0.0005 m) is validated to accurately capture the free surface through the Volume-of-fluid method. Second, we have tried other grid sizes (0.00025 m and 0.001 m), and no apparent differences appear. In addition, the waves from three grids are almost identical and agree well with the theoretical solution. As a validation case, we kindly request not to include the results from other grids.

(1) Higuera, P; Losada, I.J.; Lara, J.L. Three-dimensional numerical wave generation with moving boundaries. Coast. Eng. 2015, 101: 35–47.

Comments 2: Case 3.2 uses horizontal moving of wave paddle to generate solitary wave, has no connection with the vertical bottom disturbance.

Response 2: We agree the referee’s opinion. And we decide to delete this case.

Comments 3: Please clearly state whether the bottom disturbance function is an inherent feature of the OpenFOAM or an extended work carried out by the authors.

Response 3: Thank you for the kind suggestion. We use the latest OpenFOAM version developed by Mi et al. (2024) to conduct the present simulations, and have correctly cited it in the revision.

(1) Mi, S.; Wang, M.; Avital, E.J.; Williams, J.J.; Chatjigeorgiou, I.K. An implicit Eulerian-Lagrangian model for flow-net interaction using immersed boundary method in OpenFOAM. Ocean. Eng. 2022, 264, 112843.

Comments 4: In section 2, which term accounts for the contribution of vertical bottom disturbance.

Response 4: The term F_IBM in Equation (2) is the immerse boundary force, representing the hydrodynamic forces exerted on the flow by the vertical bottom disturbance. This force is obtained by solving Equation (9) through the Bi-conjugate gradient stabilized (Bi-CGSTAB) method.

Comments 5: In Section 3.3, additional information about the model proposed by Shen and Chan (2008) should be presented. Specifically, is it a CFD-type model or another type of model? Furthermore, the discrepancies (including those in amplitude and phase) between the experimental data and the model results of Shen and Chan (2008) in Fig. 7 require further explanation. Notably, the proposed model is expected to outperform the aforementioned model, as it belongs to the CFD type and includes viscosity.

Response 5: We indeed apologize for the lack of clarity. Shen and Chan (2008) simulated this problem through an in-house code Navier-Stokes model, with the bottom disturbance being implemented through the immerse boundary method (IBM). The corresponding declaration has been added into lines 236 to 238 of the revision, as follows:

Shen and Chan (2008) reconsidered this problem through an in-house code Navier-Stokes model, with the bottom disturbance being implemented through IBM.

We acknowledge the observed discrepancies (amplitude and phase), which is due to the experimental error (Zhang and Chuang, 1996); and the associated faster wave speed of the experiments is attributed to the larger wave height. However, the close agreement between our model and that of Shen & Chan (2008) primarily serves as a validation of our CFD model, while the inherent viscous and complete physics of our model provide the essential foundation for reliable simulations of complex flow regimes beyond the scope of their model.

(1) Zhang, D.; Chwang, A.T. Numerical study of nonlinear shallow water waves produced by a submerged moving disturbance in viscous flow. Phys. Fluids 1996, 8, 147-155.

Comments 6: Line 195, stoke should be stroke?

Response 6: Thank you for pointing out this mistake. We have revised it.

Comments 7: The simulated results are missing in Fig.7(c) after t>15s.

Response 7: Thanks for your reminding. Fig.7 has been modified it.

Comments 8: A colorbar scale is needed in Fig.11 and Fig.12.

Response 8: As the referee suggested, the colorbar scale has been added.

Reviewer 2 Report

Comments and Suggestions for Authors

The article is devoted to the important and fundamental dependences of wave parameters generated by bottom movements on the parameters of these movements. The general features of the dependence of the parameters of tsunami waves on the parameters of bottom movements are well known, but the details described in the article will undoubtedly be interesting and useful to readers of the magazine. However, the style of introduction and presentation of the material is often too vague, hiding the essence of the research and the generalizations received. Specific comments to the text of the article are given below.

There are general remarks in the Introduction section (lines 31-125). Real waves are constantly mentioned in the introduction, but for clarity, it would be nice for the reader to tell what kind of reality they are – that they contain a description of turbulence, viscosity, vorticity, or that they are measured, observed, etc.

Lines 32-39 of the first paragraph of the Introduction are formal in nature. It is unclear why these particular works are mentioned. Maybe they contain the most recent reviews of the issues under consideration? It is necessary to clearly and concretely write down exactly which tasks are being solved in these works, iInstead of just mentioning these works.

48 The phrase is unclear – it is unclear what the discrepancies are and what they mean.

From what follows, it seems to become clear that this is a discrepancy between the solutions obtained and real waves, but it remains unclear what real waves are – are they measured, observed, or modeled by more complete models?

52-54 Maila and Stepanyants investigated, but what was the result of this study?

56 What is a realistic wave? Could this mean that it was observed?

64 What are real waves – measured, observed, who observed them, where and when?

72 “such” – it would be necessary to write specifically which ones.

77 What is meant by "real"?

78-86 Many laboratory experiments are mentioned, but it is not said what is contained in their results. For what purpose are these experiments mentioned here?

127 A link to the description of the OpenFOAM program is desirable.

134 Is this humor or slang – it would be nice to clarify for the uninitiated.

135 What does low speed mean? Low compared to what?

138-150 Where did these equations come from? How are constants set? Not all readers are familiar with RANS, detailed comments are necessary as well as links.

141 Why does gravity acceleration have two components – it seems that the bottom is horizontal.

220 Where formulas (12) and (13) come from, links and small comments are needed.

238 What theory are we talking about? Is this the formula (12)?

442. It would be good to clarify exactly which spatio-temporal characteristics coincide with the work of Kawai, otherwise this statement is unfounded.

 

The conclusions of the article are too general and vague.

451 The article does not show the effect of turbulence on the propagation of northwest waves. How can we prove that turbulence and vorticity are important for describing NWWs?

457 It would be good to clarify what a wide range means (how wide?). Significant compared to what? Essential signs – what are the signs?

460 Strong dependence - it is necessary to indicate how strong and in comparison with what.

467 It is unclear how forecasting applications will use this information. After all, these apps are models that automatically reproduce all of these features.

471 It would be good to indicate exactly which patterns are being demonstrated.

472 It is unclear where the shore is here – the bottom is horizontal in the experiments.

Author Response

Overall comments:

The article is devoted to the important and fundamental dependences of wave parameters generated by bottom movements on the parameters of these movements. The general features of the dependence of the parameters of tsunami waves on the parameters of bottom movements are well known, but the details described in the article will undoubtedly be interesting and useful to readers of the magazine. However, the style of introduction and presentation of the material is often too vague, hiding the essence of the research and the generalizations received. Specific comments to the text of the article are given below.

Response to overall comments: Thanks for pointing out this important issue. Improvements have been made in the revision as highlighted by the yellow marks.

 

Specified comments:

Comments 1: There are general remarks in the Introduction section (lines 31-125). Real waves are constantly mentioned in the introduction, but for clarity, it would be nice for the reader to tell what kind of reality they are – that they contain a description of turbulence, viscosity, vorticity, or that they are measured, observed, etc.

Response 1: Thank you for this insightful suggestion regarding the use of the term ‘real waves’. In the revised manuscript, we clarify that ‘real waves’ primarily means waves in a physical context that inherently encompass key fluid properties such as viscosity, turbulence, and vorticity, distinguishing them from idealized potential flow descriptions.

Comments 2: Lines 32-39 of the first paragraph of the Introduction are formal in nature. It is unclear why these particular works are mentioned. Maybe they contain the most recent reviews of the issues under consideration? It is necessary to clearly and concretely write down exactly which tasks are being solved in these works, instead of just mentioning these works.

Response 2: Thank you for the kind suggestion. We have modified the first paragraph, as follows:

Nonlinear water waves (NWWs) have garnered significant attention due to their complex generation mechanisms and associated coastal hazards. They can be generated by tide or current flowing over submarine topography (e.g., Semenov and Wu, 2020). An-other fundamental generation mechanism is sudden bottom disturbance, which has been extensively investigated through laboratory experiments and numerical simulations (e.g., Derakhti et al., 2019; Gao et al., 2021). Among these, vertical bottom disturbances represent a canonical scenario, conceptually modeling seabed earthquakes (Synolakis & Bernard, 2006; Reeve et al., 2024). Research on this mechanism has revealed that it can generate di-verse and often extreme wave forms, including solitary waves, tsunami-like waves and undular bores (e.g., Fang et al., 2020; Jing et al., 2020a; Gao et al., 2017, 2018). A thorough understanding of the generation and evolution of such NWWs is therefore of considerable engineering significance for coastal hazard mitigation.

 

Comments 3: 48 The phrase is unclear – it is unclear what the discrepancies are and what they mean.

Response 3: Thank you for pointing out this. The discrepancies include the variations of wave height, phase celerity, wave form, breaking pattern and etc. They have been explicitly declared in the revision, see lines 53 to 55, as follows:

…introduces discrepancies in wave height, phase celerity, wave form, breaking pattern and etc, which are attributable to the flow separation and energy dissipation….

 

Comments 4: From what follows, it seems to become clear that this is a discrepancy between the solutions obtained and real waves, but it remains unclear what real waves are – are they measured, observed, or modeled by more complete models?

Response 4: Thanks for the good question. Real waves include the measured in experiments and the modelled by more complete models, where both results agree well but derivate from the analytical solution. They have been explicitly declared in the revision, see lines 56 to 59, as follow:

These discrepancies between potential solutions and real waves (either generated from experiments or simulations by high-fidelity numerical solvers) have been widely noted (Zhang & Chwang, 1996; Whittaker et al., 2017; Jin et al., 2025).

 

Comments 5: 52-54 Maila and Stepanyants investigated, but what was the result of this study?

Response 5: Thank you for the kindly reminding. Their findings have been added into the revision, see lines 61 to 63, as follow:

They discovered an extra evanescent mode unique to gravity-capillary wave scattering, which is essential for satisfying the additional boundary condition at the surface introduced by capillary effects.

 

Comment 6: 56 What is a realistic wave? Could this mean that it was observed?

Response: We apologize for the unclear expression. Following Comment 4, the realistic wave has been modified into the real wave, which has the same meaning of that generated from experiments or simulations by high-fidelity numerical solvers.

 

Comments 7: 64 What are real waves – measured, observed, who observed them, where and when?

Response 7: Thank you for the kind suggestion. Similar to Comment 6, real waves are either from experiments or simulations. And we have explicitly declared them in the revision.

 

Comments 8: 72 “such” – it would be necessary to write specifically which ones.

Response 8: As the referee suggested, we have modified it as: these artificial damping-based methods.

 

Comments 9: 77 What is meant by "real"?

Response 9: It means the viscous damping is not modelled by the artificial damping but the liquid viscosity. We have revised it into “the effects of liquid viscosity”.

 

Comments 10: 78-86 Many laboratory experiments are mentioned, but it is not said what is contained in their results. For what purpose are these experiments mentioned here?

Response 10: Thank you for raising this point. The referenced laboratory experiments are cited with two clear purposes: firstly, to summarize their key findings regarding the physical phenomena our study addresses; secondly, to highlight the practical limitations of experimental methods, such as high cost and limited parametric flexibility, thereby establishing the motivation for developing our complementary numerical framework. We revise the introduction to explicitly state the purpose and key outcomes of these cited experiments. The purpose is presented in lines 98 to 100 of the revision, as follows:

…most experiments are conducted at relatively small scales and can be cost-prohibitive; and limited parametric flexibility also restricts the comprehensiveness of the experiments.

 

Comments 11: 127 A link to the description of the OpenFOAM program is desirable.

Response 11: As the referee suggested, the program description is presented in line 141 of the revision, as follows:

The OpenFOAM version proposed by Mi et al. (2024)….

(1) Mi, S.; Wang, M.; Avital, E.J.; Williams, J.J.; Chatjigeorgiou, I.K. An implicit Eulerian-Lagrangian model for flow-net interaction using immersed boundary method in OpenFOAM. Ocean. Eng. 2022, 264, 112843.

 

Comments 12: 134 Is this humor or slang – it would be nice to clarify for the uninitiated.

Response 12: We are sorry for the non-academic expression. The original statement “The fluid media are tap water and air at room temperature.” has been revised into “The fluid media are tap water and air with densities respectively being 998 kg/m³ and 1.0 kg/m³.”.

 

Comments 13: 135 What does low speed mean? Low compared to what?

Response 13: We apologize for the unclear declaration. In the present simulations, the maximal velocities of the bottom disturbance, wave particles, and wave celerity are within meters, thereby we can assume both fluids are incompressible.

 

Comments 14: 138-150 Where did these equations come from? How are constants set? Not all readers are familiar with RANS, detailed comments are necessary as well as links.

Response 14: Thank you for the suggestion. We have correctly cited the reference, as follows:

The OpenFOAM version proposed by Mi et al. (2024)….

(1) Mi, S.; Wang, M.; Avital, E.J.; Williams, J.J.; Chatjigeorgiou, I.K. An implicit Eulerian-Lagrangian model for flow-net interaction using immersed boundary method in OpenFOAM. Ocean. Eng. 2022, 264, 112843.

 

Comments 15: 141 Why does gravity acceleration have two components – it seems that the bottom is horizontal.

Response 15: Thanks for the comment. We divide the body force into components, where one (f_i) is the gravitational acceleration due to gravity, and the other (F_IBM) is the immerse boundary force, which represents the hydrodynamic forces exerted on the flow by the vertical bottom disturbance. By introducing the F_IBM, the fluid-structure interaction is proper resolved.

 

Comments 16: 220 Where formulas (12) and (13) come from, links and small comments are needed.

Response 16: The two formulas come from Lin (2007). Since Referee 1 states that Case 3.2, which uses horizontally moving wave paddle to generate solitary waves, has no connection with the vertical bottom disturbance, thereby we decide to delete this case.

(1) Lin, P. A fixed-grid model for simulation of a moving body in free surface flows. Comput. Fluids 2007, 36, 549-561.

Comments 17: 238 What theory are we talking about? Is this the formula (12)?

Response 17: Yes. Formula (12) is the theory.

 

Comments 18: It would be good to clarify exactly which spatio-temporal characteristics coincide with the work of Kawai, otherwise this statement is unfounded.

Response 18: Thank you for the suggestion. The spatio-temporal characteristics means the generated wave is at a transient state with a strain of oscillating tails rather than a standard solitary with a flat tail, which has been added into lines 415 to 416 of the revision, as follows:

…exhibiting transient spatio-temporal features of a strain of oscillating tails rather than a flat tail as the standard solitary wave…

 

Comments 19: 2. The conclusions of the article are too general and vague.

451 The article does not show the effect of turbulence on the propagation of northwest waves. How can we prove that turbulence and vorticity are important for describing NWWs?

457 It would be good to clarify what a wide range means (how wide?). Significant compared to what? Essential signs – what are the signs?

460 Strong dependence - it is necessary to indicate how strong and in comparison with what.

467 It is unclear how forecasting applications will use this information. After all, these apps are models that automatically reproduce all of these features.

471 It would be good to indicate exactly which patterns are being demonstrated.

472 It is unclear where the shore is here – the bottom is horizontal in the experiments.

Response 19: Thanks for bringing out the issue. We have modified these vague statements, and the conclusion is revised throughout, as follows:

This study numerically investigates the generation and propagation of nonlinear water waves (NWWs) by vertical bottom disturbances, employing a Reynolds-Averaged Navier-Stokes (RANS) solver. The model incorporates the k–ɛ model and the immersed boundary method (IBM) to represent the moving seabed disturbance, with the free surface captured by the Volume-of-Fluid (VOF) method. The model demonstrates robust accuracy in capturing the key features of resultant wave fields. The key findings are summarized as follows:

1. The model demonstrates high accuracy across a defined parameter space, simulating bottom disturbances with non-dimensional velocity amplitudes () ranging from 0.113 to 0.091 and non-dimensional widths (L_up/0.61) ranging from 0.328 to 1.639, validating its capability in capturing essential wave features across a broad range of disturbance velocities and widths.
2. The generated wave field exhibits strong dependence on both the disturbance duration T_d and width L_up. For instance, the decrease of T_d and the grow of L_up can independently lead to an increase in phase celerity and wave height of the leading soliton. All simulated cases evolve into dispersive wave trains whose leading crest undergoes fission into successive solitons, a hallmark of tsunami-like wave.
3. Shorter disturbance durations result in earlier fission of the leading crest into soliton trains and higher phase celerities. This inverse relationship between disturbance duration and wave celerity provides crucial insight for wave forecasting applications.
4. Larger disturbance widths generate nonlinear waves in a near-linear increase in the phase celerity of the leading wave. The amplitude of the leading soliton decreases with increasing T_d but increases with expanding L_up, revealing competing mechanisms governing wave amplitude evolution.
5. Wave energy evolution demonstrates distinct spatiotemporal patterns, with the main wave energy nonlinearly migrating from higher frequencies to lower frequencies in the offshore direction (the prorogating direction) for longer disturbances. This spectral evolution underscores the critical role of both disturbance duration and width in governing wave energy propagation characteristics.

Reviewer 3 Report

Comments and Suggestions for Authors

The topic is original and interesting, still some issues need to be resolved. The abstract is complete, comprehending a general background and methods. Main results with key, concrete values should be added to strengthen the suitability of RANS simulations.

The literature is complete and up to date, offering a stimulating discussion and enlightening the research gap the Authors intend to deal with.

LINE 130. Please justify the adoption of the k-eps turbulence model. Other models are available, e.g., k-omega, Spalart-Allmaras and their variants, etc.

LINE 134. Not clear the expression “The fluid media are tap water and air at room temperature”: you mean density = 1000 kg/m^3, but what about temperature? The process is not assumed to be temperature dependent, right?

LINE 199. Did the Authors perform a grid sensitivity analysis?

LINE 254. Is the adopted numerical scheme stable? Is stability controlled?

Conclusions support the obtained results

Check for the Reference style:

1. DOIs encouraged
2. Journal name in bold I believe.

Author Response

Overall comments:

The topic is original and interesting, still some issues need to be resolved. The abstract is complete, comprehending a general background and methods. Main results with key, concrete values should be added to strengthen the suitability of RANS simulations.

The literature is complete and up to date, offering a stimulating discussion and enlightening the research gap the Authors intend to deal with.

Response to overall comments: Thanks for your generous comments.

 

Comments 1: LINE 130. Please justify the adoption of the k-eps turbulence model. Other models are available, e.g., k-omega, Spalart-Allmaras and their variants, etc.

Response 1: Thank you for the good question. For present wavemaker problems, the standard k-ε model is chosen for its proven reliability and numerical stability in handling the turbulent free-surface flows and fluid-structure interaction of interest (Lin, 2007; Mi et al., 2024). It provides computationally efficiency and accuracy (Mi et al., 2024, the developers of the adopted OpenFOAM solver).

(1) Lin, P. A fixed-grid model for simulation of a moving body in free surface flows. Comput. Fluids 2007, 36, 549-561.

(2) Mi, S.; Wang, M.; Avital, E.J.; Williams, J.J.; Chatjigeorgiou, I.K. An implicit Eulerian-Lagrangian model for flow-net interaction using immersed boundary method in OpenFOAM. Ocean. Eng. 2022, 264, 112843.

 

Comments 2: LINE 134. Not clear the expression “The fluid media are tap water and air at room temperature”: you mean density = 1000 kg/m^3, but what about temperature? The process is not assumed to be temperature dependent, right?

Response 2: We apologize for the unclear declaration. It has been modified into “The fluid media are tap water and air with densities respectively being 998 kg/m³ and 1.0 kg/m³.”.

 

Comments 3: LINE 199. Did the Authors perform a grid sensitivity analysis?

Response 3: Thank you pointing out this issue. First, the linear and nonbreaking waves are not sensitive to the grid size (Higuera et al., 2015 and Whittaker et al., 2017), the use of a vertical grid (0.0005 m) is validated to accurately capture the free surface through the Volume-of-fluid method. Second, we have tried other grid sizes (0.00025 m and 0.001 m), and no apparent differences appear. As a validation case, we kindly request not to include the results of the sensitivity analysis.

(1) Higuera, P; Losada, I.J.; Lara, J.L. Three-dimensional numerical wave generation with moving boundaries. Coast. Eng. 2015, 101: 35–47.

(2) Whittaker, C.N.; Nokes, R.I.; Lo, H.Y.; Liu, P.F.; Davidson, M.J. Physical and numerical modelling of tsunami generation by a moving obstacle at the bottom boundary. Environ. Fluid Mech. 2017, 17, 929-958.

 

Comments 4: LINE 254. Is the adopted numerical scheme stable? Is stability controlled?

Response 4: Thank you for the reminding. We are sure that the scheme is stable, since the generated wave agree well with the theoretical results. As Referee 1 has suggested, this case is not related to nonlinear waves induced by vertical bottom disturbances, it should be deleted. So, we kindly request not to include this case and decide to delete it.

 

Comments 5: Conclusions support the obtained results.

Response 5: We thank you for the comment. The conclusion has been modified throughout.

 

Comments 6: Check for the Reference style:

DOIs encouraged

Journal name in bold I believe.

Response 6: As the referee suggested, we have revised them, as highlighted in the revision.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors discussed and took into account all the comments of the reviewer in the text. The conclusions have been completely rewritten and given a clear physical meaning. I believe that the article will undoubtedly be of interest to readers and can be published in its present form.