Interactions of Solitary Wave with a Submerged Step: Experiments and Simulations

Round 1

Reviewer 1 Report

This is an interesting paper on a classic problem of hydrodynamics, the transmission of solitary waves over a shelf. The paper presents some new laboratory data and two different numerical solutions and attempts to draw conclusions as to the level of approximation needed to best model the transmitted wave.

  I have made numerous stylists comments on the manuscript, here I will list some major concerns.    1) I couldn’t find the classic work of Goring and Raichlen 1992 referenced ( https://doi.org/10.1061/(ASCE)0733-950X(1992)118:1(43)). They have laboratory data, as well as analytical predictions as the evolution of solitary waves over shelves. I quote from their abstract "It is found that reflection can be described well by a linear theory for relatively small incident waves and/or relative depths, but nonlinear effects become important as wave heights and relative depths increase”.    2) Couldn’t find references to Kanoglu and Synolakis (1998) J. Fluid Mech. Vol 374. I quote from their study "We develop a general solution method for deter- mining the amplification factor of different ocean topographies consisting of linearly varying and constant-depth segments to study how spectral distributions evolve over bathymetry, and apply our results to study the evolution of solitary waves.”   3) At a depth of 10cm, surface tension effects may be important in the lab experiments - particularly in the high frequency waves in the tail of the transmitted solitary wave. These are not modelled in the two codes used. Please discuss briefly.   4) The authors appear to name every wave crest in the tail of transmitted or reflected waves a soliton.  It may be, but we don’t know. The authors need to extend the computational domain so that the  waves can propagate longer numerically, and it can be determined if and how many solitons emerge.   5) The first sentence of the manuscript attributed to ref(1) is fairly generic and does not need a reference. In ref (10), use the journal paper of the same title "Validation and Verification of Tsunami Numerical Models, Synolakis et al, Pageoph, 2008  https://doi.org/10.1007/978-3-0346-0057-6_11   6) The discussion of the differences between the FD and FE model needs to be enhanced - I have been studying hydrodynamics for 30 years and couldn’t really understand which approximations of the equations of motion are used. In water waves, there are two approximations in use, the shallow-water wave equations and the Boussinesq-type equations, short of the Navier-Stokes equations. The authors refer extensively to other references, but the codes they refer to are not known and appear not benchmarked. If they referred to MOST or COULWAVE or VOLNAC or TUNAMI-DANCE it wouldn’t be necessary to discuss in length. But for codes which are less known a few more sentences are needed. See the article by Behrens and Dias (2015) in https://royalsocietypublishing.org/toc/rsta/2015/373/2053.

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Reviewer 2 Report

Review of ”Interactions of Solitary Wave with a Submerged Step:
Experiments and Simulations”
by Wei-Ting Chao , Shin-Jye Liang , Chih-Chieh Young and
Chao-Lung Ting
Submitted for publication in Water

1 General Comments
This work aims at the study of the interaction of solitary waves with a submerged step. The authors conduct
a series of experiments and use the results to validate two existing numerical models. The parameters under
investigation are the relative step width, height and wave amplitude. Results on the wave reflection and
transmission coefficients are also obtained and discussed.
The main contribution of the present work is the quantification of the energy dissipation in terms of the
aforementioned parameters, resulting from a multivariate regression on the obtained (numerical) results. It
is shown that the height of the step is the most important factor and a formula for the dissipation is provided
(Eq. (8)) which may be useful for design purposes.
Overall, we may conclude that the work is potentially useful for the coastal engineering community.
However, the presentation of the results is quite poor and needs improvement. Although I consider this
manuscript with potential for publication, I would like to point out some, hopefully constructive, aspects
and remarks that should be addressed/clarified.

2 Remarks
1. The general structure of the introduction is understandable and several relevant works are documented.
However, the cited literature could be enriched with more recent relevant articles. For example, in the
second paragraph (line 41-48), you mention interesting phenomena of solitary wave transformation due to
bottom topography, citing classical works on run-up ([12]) and regular waves ([14]). Several interesting
purely solitary wave-bottom interactions are simulated in (Papoutsellis et al., 2018). Another article, quite
relevant to the numerical part of you work is (Vaziri et al., 2011). Other recent works investigating step-like
bathymetries include (Chen et al., 2017), (Athanassoulis et al., 2019) (potential flow), (Ji et al., 2017) (multiphase
flow), (Wu and Hsiao, 2017) (experiment and RANS), (Li et al., 2012), (Han and Dong, 2020) (SPH).

2. The experimental configuration is well-described. Figure 1 (a) shows the setup of the experiment while
Figure 1 (b) shows the geometry used in the simulations (according to the caption). However, it is not
referenced in the text. Also, in this figure, two dashed lines are shown of slope 1/2. These are not explained
in the text. Moreover, I think figure 2 (a) could be clarified. You mention that black pixels represent air.
In the figure, the black colour is visible also below the red curve (free surface).

3. Concerning the video measurement technique, the relation (similarities/differences) of the present set-up with previous studies should be clarified.

4. In Section 3, where the numerical models are described, are there any developments in relation with the cited references. If this is the case, they should be mentioned in a clear way. For example, in lines152-154 you mention a ”special treatment” for the top-layer pressure boundary condition in the σ-model. Could you be more specific or at least point to a relevant work? Moreover, you mention the use of inflow boundary conditions for the σ-model. This is somewhat unusual for initiating a solitary wave simulation
and deserves some discussion.

5. The second paragraph of Section 4 describes the treatment of the second reflection. Although, the description of the phenomenon is understandable, it is not very clear how it relates with the calculation of the reflection coefficient. How HR is found? How the amplitudes of the first and second reflected waves are compared? Also, in line 211, you mention positions 3 and 4. Please refer to figure 1. Moreover, HT in Eq. (2) is not properly defined. How is it calculated in terms of the amplitudes of the fission solitary waves (of different height) that are transmitted after the interaction? It would be interesting (and straightforward) to calculate the transmitted and reflected wave energy, as e.g. in (Papoutsellis et al., 2018, Section 6.4), and
compare it with the initial one.

6. In order to validate the numerical models, the authors consider the test of the steady solitary wave propagation over flat bottom. They give as initial conditions a solution that seems to be the KdV solitary wave (free-surface profile) η (Eq. (4) should be checked) but they also write down two velocities U(x, t) and W(x, t). Apparently, these are the (shallow water) horizontal and vertical velocities on the free surface but this is not stated right after their appearance. This is clarified in the third paragraph and in the caption of
figure 3. Also, I think a few words on how these initial velocities are implemented in the numerical models
would be useful.

6. The second paragraph of Section 5 describes the numerical parameters used in the simulation but needs some rephrasing in order to be more readable. Also for the FD model, you mention 5 uniform grids. Is it 5 vertical layers? It seems somewhat arbitrary as a choice. What happens when the number of layers is 2 or 7, for example?

7. In Figure 3 (c), some small amplitude fluctuations are visible in the dynamic pressure in the form of a dispersive trail of the main solitary wave form computed by FD. Can you comment on that? Can this mean that this model does not have sufficient dispersion? Is it due to the choice of initial conditions? Is maybe the number of vertical layers insufficient? In my opinion, the evolution of the free surface should also be included in this figure. Also, you mention that H = 0.009 m and h = 0.1 m, however in the figure title you state H/h = 0.138.

8. In figure 4, the simulated free surface is compared with experimental data in the region (7,9.25). However, in the text you mention the ROE is (5,10.2).
8. In figure 5 (b), the obvious differences between the two simulations are not discussed.

9. In figure 6 (c), simulation results are provided only for d/h = 0.5 and 0.7 with significant differences from the experimental measurements due to bottom friction and viscous effects not taken into account into the models. Do these discrepancies persist when d/h is smaller. In my opinion, the results of one more simulation, with d/h = 0.2 for example, should be included. Also, how the is the dissipation coefficient calculated in the models? If viscosity and bottom friction is absent, what is the underlying dissipation mechanism?

10. In lines 415-417, your comment about the comparison of your results with those of Lin [16] is not very understandable.

11. You conclude that the FD outperforms the FE model in terms of computational accuracy and efficiency. You also state that the FE model is computationally more expensive. This is somewhat confusing
since the FE is a single layer model. Could you provide the ratio of CPU times for an indicative case? Also, the statement “To improve the accuracy and extend the applicability scope of the FE, the multi-layer coordinate model is suggested for future study.” needs some clarification.

3 Further comments
The quality of the writing can be improved in many places in the manuscript. For instance,
• line 15: “solitary conditions” should be ”solitary wave conditions”.
• line 43: “interesting phenomena occurred” should be “interesting phenomena that occur”.
• line 51 “passing through” should be ”passing over”.
• line 78 “which” should be “where”.
• line 146 “solved” should be “solves”.
• line 292 “viscous” should be “viscosity”.
• line 430 “which not” should be “which was not”.
The above comments and remarks are intended to contribute to the improvement of the manuscript and the
clarification of its messages. However, they do not exhaust all corrections to be made.

References
Athanassoulis, G., Mavroeidis, C., Koutsogiannakis, P., and Papoutsellis, C. (2019). A numerical study
of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem
trenches. J. Ocean Eng. Mar. Energy, 5:311––331.
Chen, L.-F., Ning, D.-Z., Teng, B., and Zhao, M. (2017). Numerical and experimental investigation of
nonlinear wave-current propagation over a submerged breakwater. Journal of Engineering Mechanics,
143(9):04017061.
Han, X. and Dong, S. (2020). Interaction of solitary wave with submerged breakwater by smoothed particle
hydrodynamics. Ocean Engineering, 216:108108.
Ji, Q., Dong, S., Luo, X., and Guedes Soares, C. (2017). Wave transformation over submerged breakwaters
by the constrained interpolation profile method. Ocean Engineering, 136:294–303.
Li, J., Liu, H., Gong, K., Tan, S. K., and Shao, S. (2012). Sph modeling of solitary wave fissions over uneven
bottoms. Coastal Engineering, 60:261–275.
Papoutsellis, C., Charalampopoulos, A., and Athanassoulis, G. (2018). Implementation of a fully nonlinear
hamiltonian coupled-mode theory, and application to solitary wave problems over bathymetry. European
Journal of Mechanics - B/Fluids, 72:199–224.
Vaziri, N., Chern, M.-J., and Borthwick, A. G. (2011). A pseudospectral σ-transformation model of solitary
waves in a tank with uneven bed. Computers and Fluids, 49(1):197–202.
Wu, Y.-T. and Hsiao, S.-C. (2017). Propagation of solitary waves over double submerged barriers. Water,
9(12).
3

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Reviewer 3 Report

The paper under review deals with an important problem of the solitary wave (soliton) transformation going through the underwater barrier. It contains both the laboratory experiment results with low-amplitude waves and the calculations, using two non-hydrostatic models, that demonstrate excellent agreement with each other. In fact, half of the reviewed paper contains the description of test experiments and the calculations of wave motion in a channel of constant depth, used to verify the models. Then the data on the wave transformation over the submerged obstacle are given. The results demonstrate both the formation of secondary solitons in the transmitted wave and the wave reflection from both slopes of the underwater obstacle. I understand the difficulties of comparing the results obtained with analytics since the slopes of the bottom obstacle are not vertical, therefore, I can only wish the authors to carry out such studies in the future. My opinion about the given paper is the most favorable, although there are several minor remarks.

1. The authors give analytical formulas for the velocity field in a single traveling wave and speak of a similar formula for dynamic pressure without giving it. It would be nice to present it in the paper. I know the analytical formula for the bottom pressure under a soliton (Pelinovsky, EN, Kuznetsov, KI, Touboul, J., and Kurkin, AA Bottom pressure caused by passage of a solitary wave within the strongly nonlinear Green – Naghdi model. Doklady Physics, 2015, vol. 60, No. 4, 171–174), but the hydrostatic component is not subtracted there. It would be interesting to know what formula was used by the authors and within what theory framework it was derived.
2. Although the problem of the soliton transformation on a vertical step has a long history, which is well presented in the reviewed paper), I would like to draw the authors’ attention to two papers (Nakoulima, O., Zahibo, N, Pelinovsky, E., Talipova, T., and Kurkin, A. Solitary wave dynamics in shallow water above periodic bottom. Chaos, 2005, vol. 15, No. 3, 037107; and Pelinovsky, E., Choi, BH, Talipova, T., Woo, SB, and Kim, DC Solitary wave transformation on the underwater step: theory and numerical experiments. Applied Math Computations, 2010, vol. 217, No. 4, 1704-1718), which compare the analytical theory with numerical results.
3. Finally, I would like to note that the authors use the paragraph indentation after giving each formula which breaks the sentence structure (it is embedded in the Microsoft Word options which are often used thoughtlessly).

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Round 2

Reviewer 1 Report

The ms is quite improved, but still there are  still inadvertent misspellings and grammar errors - I sent an annotated manuscript to Dr. Markovic. 

A basic remaining issue is the use of the term "soliton" for the tail of waves. One needs to prove that the waves are actually solutions, for example, by extending the numerical flow field and watching the waves evolve far from the step. Without it, the authors should call the waves they observe "soliton like".

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Reviewer 2 Report

First, I would like to thank the authors for their response. All issues raised are addressed in their report and the revised manuscript. I believe the revised manuscript merits publication. I have only some minor comments:

1. The use of the so-called "virtual slope" is not standard. The authors mention the reference [52] but I do not think it is enough. Is the bottom a trapezoid of angle ½ in the simulation? If this is not the case, then showing that slope in the figure is a bit misleading. Also, how does this virtual slope apply to the shallow water (one layer) model?

2. Some author names initials and full names in the reference list do not appear as the official citation. Please check.

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