A lattice Boltzmann model for the open channel flows described by the Saint-Venant equations

A new lattice Boltzmann method to simulate open channel flows with complex geometry described by a conservative form of Saint-Venant equations is developed. The Saint-Venant equations include an original treatment of the momentum equation source term. Concrete hydrostatic pressure thrust expressions are provided for rectangular, trapezoidal and irregular cross-section shapes. A D1Q3 lattice arrangement is adopted. External forces, such as bed friction and the static term, are discretized with a centred scheme. Bounce back and imposed boundary conditions are considered. To verify the proposed model, four cases are carried out: tidal flow over a regular bed in a rectangular cross-section, steady flow in a channel with horizontal and vertical contractions, steady flow over a bump in a trapezoidal channel and steady flow in a non-prismatic channel with friction. Results indicate that the proposed scheme is simple and can provide accurate predictions for open channel flows.

Minor points: Figure 9, there is no need to provide area A, since it's realted to surface elevation, results of flowrate should be given instead. Figure 12, results of flowrate should be given. There are some typos in the manuscript, for example m2/s, superscript should be used.

Do you have any ethical concerns with this paper? No
The novelty of the paper is not well described and presented. To the reader (including myself) it is hard to see what is effectively the novelty of the paper since all the methodologies, as far as I could understand, have already been presented in other research. Images are of very low quality and results not perceptible in the manuscript images.
To my opinion there are some flaws in the paper that have to be clarified regarding the novelty of the paper and as such, I cannot recommend that this paper is accepted for publication in its present form. As such my advice is to review.
Comments: -Abstracts should not have acronyms -P4 EQ5 -define alpha -P5 L20 -the subscript should be alpha -Eq 5 to Eq 24 and the whole methodology is similar to [ -P6 L11 -Zhou's scheme for the force does not conserve mass locally. You should clearly state, or at least give an insight on why you have almost no errors since the scheme you used is not conservative.
-P8 L53 -Please compute an absolute error -L2-norm or something similar is advised. Also, for each test the computational time and the machine used should be stated so that there is a comparison of efficiency between your methodology and others.
-P9 L35 -Why use v=6m/s? please explain the value of v used for this and other tests -P11 L25 -m/s1/3. -fix units -P13 Table A2 -Table is  The editors assigned to your paper ("A lattice Boltzmann model for the open channel flows described by the Saint-Venant equations") have now received comments from reviewers. We would like you to revise your paper in accordance with the referee and Associate Editor suggestions which can be found below (not including confidential reports to the Editor). Please note this decision does not guarantee eventual acceptance.
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Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. My recommendation is that major revision is needed before this manuscript can be accepted for publication. In particular, I note that each of the reviewers has requested greater clarity on the novelty of the method used. I also suggest that these are described in greater detail which may help address the novelty issue.
Each reviewer raises substantial areas for clarity and correction. I suggest these are addressed before the manuscript is returned.

Comments to Author:
Reviewers' Comments to Author: Reviewer: 1 Comments to the Author(s) Authors describe the method of simulation 1D open-channel flow by means of one of simplest LBM algorithms with the D1Q3 lattice arrangement. As a base, the system of Saint-Venant equations is used. The Saint-Venant equations are formulated in a conservative form with inclusion of two pressure terms. The idea to solve the conservative form of the Saint-Venant equations by using the LBM is by my opinion a step forward in attempt to develop robust 1D LBM solver for practical application.
Manuscript is in general correct and should be interesting for readers but it requires some minor corrections before publication: 1. The similar model has made appearance in <Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network. > Thang and Chopard et. al., J. of Comput. Phy., 229 (2010) 7373-7400. The authors should explain the difference compared with it.
2. In the second section, the governing equations had better be structured in non-dimensional form, since the original equation the LB method is trying to solve is in its dimensionless form.
3. The Chapman-Enskog expansion should be included to show exactly which governing equations can be recovered, if a brand new LB model is developed. However, considering the simplicity of the paper, the Chapman-Enskog expansion has better show to the reviewers and may not appeared in the paper. 4. Detailed description of obtaining of equilibrium distribution function in Sec. 3 may interrupt the train of thought of readers. I would recomend move everything between eq. (24) and (42) to an appendix. Fig. 4 should be greater to improve its readability.

5.
6. Some plots also require for improve their readability, e.g. Fig. 6 maight contain only one half of a channel while it is symmetric and then the vertical axis could have the sacle from, let say, 0.4 to 0.6; Fig. 7b is also nearly empty, vertical scale 1.56-1.57 would improve it.
After these corrections I recommend the paper for publication.

Reviewer: 2
Comments to the Author(s) The manuscript proposed the LBCSVE scheme for open channels flow. However, the paper is not clear about its originality and not convincing by presenting four cases of steady flow. Based on this, the reviewer do not recommend this manuscript to be published in current form. Major concerns: The originality of the LBCSVE should be discussed in the introduction. Why is this a new lattice Boltzmann method as stated in the abstract? The advantage and disadvantage of the lattice Boltzmann method should be discussed.
The numerical test only include steady state solution. Does this imply the method only apply to steady state? If not, unsteady flow should be simulated. In the numerical tests, numerical accuracy and errors should be reported, computational efficiency compared to other numerical methods (eg. Finte difference method, finite volume method)should be provided. The effects of lattice size should also be investigated.
Minor points: Figure 9, there is no need to provide area A, since it's realted to surface elevation, results of flowrate should be given instead. Figure 12, results of flowrate should be given. There are some typos in the manuscript, for example m2/s, superscript should be used.

Reviewer: 3
Comments to the Author(s) The manuscript presents a lattice Boltzmann model for the Saint-Venant equations.
This topic is of potential interest but the actual contribution in the present form is still improvable. The English of the paper is good with a coherent structure. To my opinion the structure of the paper is the correct as the classic Introduction/methods/results and discussion/conclusion is followed.
The novelty of the paper is not well described and presented. To the reader (including myself) it is hard to see what is effectively the novelty of the paper since all the methodologies, as far as I could understand, have already been presented in other research. Images are of very low quality and results not perceptible in the manuscript images.
To my opinion there are some flaws in the paper that have to be clarified regarding the novelty of the paper and as such, I cannot recommend that this paper is accepted for publication in its present form. As such my advice is to review.  Table A2 -Table is

Recommendation?
Accept as is

Comments to the Author(s)
The reviewer (former #2) is satisfied with the modification to improve its quality and recommend the manuscript to be accepted.

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Accept as is

Comments to the Author(s) Dear Authors
Thank you for having incorporated the changes and the improvements suggested in the preceding round of the revision process. I do believe that the manuscript was greatly improved in terms of readability and scientific soundness. Overall I found the manuscript to be much easier to understand, with the aims, methodology, results and significance of the work more evident to the reader. The Authors are to be congratulated for the substantial revision of their manuscript. You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience_proofs@royalsociety.org and openscience@royalsociety.org) to let us know if you are likely to be away from e-mail contact --if you are going to be away, please nominate a coauthor (if available) to manage the proofing process, and ensure they are copied into your email to the journal.
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Royal Society Open Science operates under a continuous publication model (http://bit.ly/cpFAQ). Your article will be published straight into the next open issue and this will be the final version of the paper. As such, it can be cited immediately by other researchers. As the issue version of your paper will be the only version to be published I would advise you to check your proofs thoroughly as changes cannot be made once the paper is published. Comments to the Author: Thank you for resubmitting "A lattice Boltzmann model for the open channel flows described by the Saint-Venant equations". to Royal Society Open Science. I have received 2 further reviews of your revised manuscript, which are included below and/or attached. As you can see, both reviewers are satisfied that you have addressed all points raised with the original submission and I am happy to recommend that we accept your contribution in its present form.
Reviewer comments to Author: Comments to the Author(s) The reviewer (former #2) is satisfied with the modification to improve its quality and recommend the manuscript to be accepted.

Reviewer: 3 Comments to the Author(s) Dear Authors
Thank you for having incorporated the changes and the improvements suggested in the preceding round of the revision process. I do believe that the manuscript was greatly improved in terms of readability and scientific soundness. Overall I found the manuscript to be much easier to understand, with the aims, methodology, results and significance of the work more evident to the reader. The Authors are to be congratulated for the substantial revision of their manuscript.

Response to Associate Editor:
Comments: My recommendation is that major revision is needed before this manuscript can be accepted for publication. In particular, I note that each of the reviewers has requested greater clarity on the novelty of the method used. I also suggest that these are described in greater detail which may help address the novelty issue.
Each reviewer raises substantial areas for clarity and correction. I suggest these are addressed before the manuscript is returned.
Response: Thanks so much for giving us many opportunities to revise our manuscript.
We appreciate the editor and reviewers very much for their constructive comments and suggestions. I have read the reviewer's suggestions and made corresponding revision one by one. We really carefully revise this paper marked with red color and hope to meet with your approval.  Equations are the traditional and conservative forms [1][2][3] and can be used in real rivers with arbitrary cross-section shapes. In order to calculate the hydrostatic pressure thrust I1 which was a difficulty in simulating the real rivers, the Gauss-Legendre numerical integration method was used.

Reviewer#1, Concern # 2:
In the second section, the governing equations had better be structured in non-dimensional form, since the original equation the LB method is trying to solve is in its dimensionless form.
Response: Thanks for the valuable suggestions. Originally, the LB method is do trying to solve is the dimensionless equation form. Recently, in the area of simulating the free-surface flows using LB method, many researchers have done a lot work and do not use the dimensionless form [4][5][6][7][8][9][10] . The form of governing equations in Thang et. al's work is also not dimensionless form. For the LB model for free-surface flows used the dimensional form, two stability conditions must be satisfied: (1)The relaxation time τ > 0.5 (2)The magnitude of the resultant velocity is smaller than the speed of calculated with the lattice speed:  gh v  which u is the water velocity, h is the water depth, v is the lattice speed.

Reviewer#1, Concern # 3:The Chapman-Enskog expansion should be included to
show exactly which governing equations can be recovered, if a brand new LB model is developed. However, considering the simplicity of the paper, the Chapman-Enskog expansion has better show to the reviewers and may not appeared in the paper.
Response: Thanks for the valuable suggestions. The recovery of shallow water equations which is the hyperbolic conservative form using the Chapman-Enskog expansion method have been carried out in many works [4,9,11] . Due to the similar Chapman-Enskog analysis, we do not put the Chapman-Enskog expansion in our work. The following is the Chapman-Enskog expansion for our model.
The Saint-Venant equations can be derived from the Chapman-Enskog expansion.
Assuming t  is small and equal  .
Eq.(10) in the manuscript can be expressed The Eq. (R1) is taken a Taylor expansion to the left hand side of in the time and space Substitution Eq.(R4) into Eq.(R5) leads to 1 2 Considering the 0 eq ff   and evaluating the above equation using Eq.(5), Eq. (7) and Eqs.(14)-(17) in the manuscript gives the continuity equation Reviewer#1, Concern # 5: Fig. 4 should be greater to improve its readability.
Response: Thanks for this valuable suggestion. We have replaced the original Fig. 4 with a higher resolution figure.
Reviewer#1, Concern # 6: Some plots also require for improve their readability, e.g. To my opinion there are some flaws in the paper that have to be clarified regarding the novelty of the paper and as such, I cannot recommend that this paper is accepted for publication in its present form. As such my advice is to review.  proposed by the authors [1] . We think that the computational time and efficiency was not the point of concern when modeling the one dimensional river flows. -Reviewer#3, Concern # 9:P13 Table A2 -Table is