Fully Nonlinear Evolution of Free-Surface Waves with Constant Vorticity under Horizontal Electric Fields

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors introduced ''Fully nonlinear evolution of free-surface waves with constant vorticity under horizontal electric fields''. The paper is written in clear and understandable language. It's a very interesting problem and an interesting presentation. The research certainly contributes to the subject area. As a result, the presented work has been prepared very well and can be a starting point for further research, but I can recommend this article for publication after minor revisions. Bellow my suggestions/recommendations for the authors:

o   The motivation(s) is of this manuscript very good, but if possible, the author(s) can highlight it in a more focused manner in the abstract section.

o   Is this study applicable in all environments? that should be discussed.

o   The main point of this manuscript is plotted figures have been plotted with axes without units (for ex. Fig. 2, the vertical axis is time, what is the unit of this time sec?, min?)

o   What numerical techniques were used in the whole article investigation; this can be included in the article?

o   Some of the equations are not derived, so the authors have to quote the source of the expression.

o   The authors should cite more recent references.

 

o   In the conclusion section, the benefits that have been verified can be more explained. 

Author Response

o   The motivation(s) is of this manuscript very good, but if possible, the author(s) can highlight it in a more focused manner in the abstract section.

Thank you for your valuable comment. We have added the following sentence to the Introduction Section. We hope that the motivation is now more clear.

See attached pdf file.

o   Is this study applicable in all environments? that should be discussed.

We are thankful to reviewer for the remark. The physical model is based on the assumption of incompressible flow of an inviscid fluid moving with a linear shear flow.

The first sentence is rewritten in Section 2 by the following way:

See attached pdf file.

o   The main point of this manuscript is plotted figures have been plotted with axes without units (for ex. Fig. 2, the vertical axis is time, what is the unit of this time sec?, min?)

The quantities in all figures are measured in non-dimensional units. According to the non-dimensionalization procedure, the characteristic values of the length, time, velocity potential, and electric field potential are the following:

See attached pdf file.

o   What numerical techniques were used in the whole article investigation; this can be included in the article?

We thank reviewer for valuable comment. The first paragraph in Result Section is rewritten by the follow:

See attached pdf file.

o   Some of the equations are not derived, so the authors have to quote the source of the expression.

You are of course right. We did not derive some formulas, since their derivation is not an original result. The manuscript contains all references to known works in the field of conformal mappings.

o   The authors should cite more recent references.

 We are grateful to the reviewer for his comment. We have added the following recent references:

6. Li, D.; Li, Y.; Li, Z.; Wang, Y. Theory analyses and applications of magnetic fluids in sealing. Friction. 2023, 11, 1771–1793. doi: 10.1007/s40544-022-0676-8

 

14. Ricard, G.; Falcon, E. Transition from wave turbulence to acousticlike shock-wave regime. Phys. Rev. Fluids, 2023, 8, 014804. doi: 10.1103/PhysRevFluids.8.014804

 

24. Murashige, S.; Choi, W. Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current. J. Mar. Sci. Eng., 2021, 9(11), 1217. doi: 10.3390/jmse9111217

 

27. Lokharu, E.; Wahlèn, E.; Weber, J. On the Amplitude of Steady Water Waves with Favorable Constant Vorticity. J. Math. Fluid. Mech. 2023, 25, 58. doi: 10.1007/s00021-023-00796-6

o   In the conclusion section, the benefits that have been verified can be more explained. 

Thank you for your valuable comment. The following text has been added before Conclusion Section.

See attached pdf file.

 

 

We are grateful to the reviewer for his valuable comments. They allowed us to significantly improve our manuscript.

 Best regards, authors.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Title: A Fully Non-Linear Evolution of Free-Surface Waves with Constant Vorticity under Different Horizontal Electric Fields.

 

In this study, the authors developed mathematical models that were subsequently simulated numerically. There may be some interest in this work for mathematics audiences who are interested in hydrodynamics under applied electric fields. A couple of comments need to be addressed before this work can be considered for publication in Mathematics:

1. The introductory section should clearly state the practical applications of this study. Also, refrain from using personal pronouns like "we" throughout this document—for example, line 53, 66, 70, 78, 84, 87 etc.

2. There is a need to move lines 72-75 in the introductory section to the discussion of results.

3. This work has several missing elements, including boundary conditions, mesh structure, and mesh dependence study.

4. The predictive models developed in a 2D simulation environment are shown in Figures 2 - 14. How confident are these mathematically developed models? Experiments or literature data will suffice for validation.

Comments on the Quality of English Language

Should be improved upon.

Author Response

1. The introductory section should clearly state the practical applications of this study. Also, refrain from using personal pronouns like "we" throughout this document—for example, line53, 66, 70, 78, 84, 87 etc.

We are grateful to the reviewer for his comments. The following  sentence is added to the Introduction Section

- The practical interest in the study of electro-hydrodynamics of the surface of liquids is determined by the possibility of controlling and stabilizing hydrodynamic instabilities, [3-5].

The personal pronouns are corrected throughout the text.

2. There is a need to move lines 72-75 in the introductory section to the discussion of results.

Thank you for valuable comment. We have changed the text according to the remark.

3. This work has several missing elements, including boundary conditions, mesh structure, and mesh dependence study.

The first paragraph in Section 3 is rewritten by the follow:

see attached pdf file.

 

 

4. The predictive models developed in a 2D simulation environment are shown in Figures 2 - 14. How confident are these mathematically developed models? Experiments or literature data will suffice for validation.

Thank you for your valuable comment. Of course, our results are consistent with the analytical solution of the original equations.

The following text has been added at the end of Section 3.

see attached pdf file.

 

We are grateful to the reviewer for his valuable comments. They allowed us to significantly improve our manuscript.

 Best regards, authors.

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have carried out direct numerical simulations of a nonlinear free surface under the action of horizontal electric fields. This is a very good study and well-written. However, the reviewer has a number of comments before acceptance:

1. the authors need to mention that the Bernoulli dynamic boundary condition can be used because the flow is laminar. 

2. Are eqns 16 and 17 also solved explicitly in space as well as time?

3. How did you arrive at the 0.5e-4 time step used?

4. A sensitivity of the results to timestep and space discretisation size will be useful to readers to understand the behaviour of the flow under the electric field

5. Some form of (experimental or analytical) validation is needed for the numerical solution.

Author Response

1. The authors need to mention that the Bernoulli dynamic boundary condition can be used because the flow is laminar. 

- We are grateful to the reviewer for his comment. We have made a clarification in the text of the article.

2. Are eqns 16 and 17 also solved explicitly in space as well as time?

- You are absolutely right. Integration over time is carried out using the explicit Runge-Kutta method of fourth order accuracy. The calculation of spatial derivatives and integral operators is performed explicitly using fast Fourier transform algorithms.

3. How did you arrive at the 0.5e-4 time step used?

The time step should be comparable to the period of the fastest surface wave, i.e.,

 

see attached pdf file.

 

Where we introduced the characteristic time t₀ of the linear wave propagation. To understand the chosen value of the time step dt = 0.5∙10^-4, we plot the velocity of linear wave propagation in left figure and the characteristic time in right figure, see below, for the resolution N=2048. The figure shows only half of harmonics having physical meaning. One can see that the maximum of wave propagation is reached at highest harmonic with k = 1024. The wave period t₀ of such wave is near . One can see that the chosen time step dt = 0.5∙10^-4 is less than characteristic time of the wave propagating.

 

see attached pdf file.

Figure.1. The speed of linear wave (left figure) and characteristic value of time (right figure) are shown versus wavenumber for the parameters Ω=6, h=1, E_b=0 (blue curve) and E_b=100 (red curve).

The first paragraph in Result Section is rewritten by the follow:

see attached pdf file.

 

4. A sensitivity of the results to timestep and space discretisation size will be useful to readers to understand the behaviour of the flow under the electric field

- We are grateful to the reviewer for his comment. Our simulations show that the level of resolution does not play a sufficient role in the simulation results. The main thing is to ensure the convergence of numerical methods. In this work, we presented calculations with a relatively small spatial resolution, but for a large number of control parameters. Our goal was to give the reader an idea of the richness of fluid motion regimes in the presence of shear flow. For such a task, an extremely high grid resolution is not required.

5. Some form of (experimental or analytical) validation is needed for the numerical solution.

Thank you for your valuable comment. Of course, our results are consistent with the analytical solution of the original equations.

The following text has been added at the end of Section 3.

We are grateful to the reviewer for his valuable comments. They allowed us to significantly improve our manuscript.

 Best regards, authors.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The reviewer is very happy with the revision and must commend the speed the authors have used to produce high-quality responses. The paper is now of very good quality and may be accepted for publication. 

Author Response

We are grateful to the reviewer for the decision to accept the manuscript for publication.