Numerical Simulation of Breaking Wave Loading on Standing Circular Cylinders with Different Transverse Inclined Angles

Round 1

Reviewer 1 Report

Density symbol should be included in the dynamic pressure term of Eq. 17.

Caption of Fig. 10 should be corrected as "Figure 10. Comparison of the peak values of the free surface elevations around..."

Correction is needed for the minor errors aforementioned.

This paper presents a numerical simulation result for the breaking wave past a standing cylinder with various transverse inclined angles. With the open-source CFD toolbox waves2Foam based on OpenFOAM being utilized, a successful validation through comparing the surface elevation and horizontal breaking wave force with the existing experimental data was made for the numerical model employing the unsteady Reynolds-Averaged Navier-Stokes governing equations with the k-ω SST turbulence model and the Volume of Fluid method in the treatment of air-water interface. A parametric study was conducted primarily with the transverse inclined angles (θ in Y-Z plane) of the standing cylinder: the characteristics on the free surface elevation and wave force were observed definitely with varying θ.

The numerical model employed and the results evaluated in this paper may give a useful information to the relevant R&D communities.

Author Response

The authors thank the reviewers for the feedback for our manuscript. Below are our answers to the comments and questions of Reviewer 1. The answers are in blue and the amendments in the revised manuscript are in red.

This paper presents a numerical simulation result for the breaking wave past a standing cylinder with various transverse inclined angles. With the open-source CFD toolbox waves2Foam based on OpenFOAM being utilized, a successful validation through comparing the surface elevation and horizontal breaking wave force with the existing experimental data was made for the numerical model employing the unsteady Reynolds-Averaged Navier-Stokes governing equations with the k-ω SST turbulence model and the Volume of Fluid method in the treatment of air-water interface. A parametric study was conducted primarily with the transverse inclined angles (θ in Y-Z plane) of the standing cylinder: the characteristics on the free surface elevation and wave force were observed definitely with varying θ.
The numerical model employed and the results evaluated in this paper may give a useful information to the relevant R&D communities.
1. Density symbol should be included in the dynamic pressure term of Eq. 17.
Reply:
Forsubsonic incompressible flow, the density is not included in the patch pressure in OpenFOAM (see the link: https://www.openfoam.com/documentation/guides/latest/doc/ guide-bcs-inlet-outlet-total-pressure.html).

Patch pressure described by subtracting the dynamic pressure from the total pressure, where the flux has dimensions of m3/s.

2. Caption of Fig. 10 should be corrected as "Figure 10. Comparison of the peak values of the free surface elevations around..."

Reply:
Thanksfor youradvice. The error has been corrected in the revised manuscript, shown as follows:

Figure 10. Comparison of the peak values of the free surface elevations around the cylinder with different transverse inclined angles ( ).

Author Response File: Author Response.docx

Reviewer 2 Report

In this paper, the authors investigate wave breaking around inclined cylinders using URANS and VOF numerical computations using the open-source OpenFOAM solver. Overall, the article is well written and presents some very interesting results, but which are not fully analysed. Therefore, the following comments should be addressed:

1) Overall the paper includes a good survey of previous work on the topic, but it is surprising to see that no work done with the smoothed particle hydrodynamics approach was cited, method widely used in the field of free-surface numerical simulations.

2) The reviewer freely admits being no expert in off-shore instalations (wind turbines, platforms, bridge pillars, etc.). It is difficult to visualise how such structures would be built using inclined pillars, as opposed to today's vertical structures. The authors could write a brief observation on the impact of their work.

3) In addition to the details related to Menter's SST turbulence model, the authors should provide more detail on the VOF method (for example, it is not mentioned that a single set of momentum equations is used for the two phases, as opposed to other multiphase algorithms).

4) Why set the outlet velocity for water as zero and what wave damping mechanism is used in the absorbtion zone?

5) In general, the authors comment on what can be observed in the graphs, but offer no in-depth insight into their results throughout the discussion. The authors should make better use of the simulation data in order to provide more in-depth explanations for some of the behaviours observed. For example:

5a) The authors should comment more on the physical mechanisms which make the free surface elevation increase at WG3 when the cylinder angle is increased above 30 deg.

5b) Comment more on why the high-order force is much larger for the vertical cylinder but appears to be relatively independent of the inclination angle for the other angles chosen. Why is the low-order force independent of inclination angle?

6) It would be very interesting to see how much of the impact force is due to pressure forces alone and if significant turbulence generation occurs in this scenario.

Author Response

The authors thank the reviewers for the feedback for our manuscript. Below are our answers to the comments and questions of Reviewer 2. The answers are in blue and the amendments in the revised manuscript are in red.
Reviewer 2
In this paper, the authors investigate wave breaking around inclined cylinders using URANS and VOF numerical computations using the open-source OpenFOAM solver. Overall, the article is well written and presents some very interesting results, but which are not fully analysed. Therefore, the following comments should be addressed:

1. Overall the paper includes a good survey of previous work on the topic, but it is surprising to see that no work done with the smoothed particle hydrodynamics approach was cited, method widely used in the field of free-surface numerical simulations.

Reply:
Thanks for your advice. The literature with the SPH method has been included in the introduction, see the follows:

Shao[1]employed the incompressible smoothed particle hydrodynamic (ISPH) method coupled with turbulence model to simulate the spilling and plunging waves by solving RANS equations. It was found that the SPH method can accurately capture the free surfaces without the numerical diffusion compared with the Euler grid method.Chowet al.[2]simulated non-breaking and breaking waves past a vertical cylinder with the improved ISPH method. They found that the maximum loading on the structure was related to the free surface elevation and steepness of the wave across the cylinder. Local pressures on the cylinder near the free surface under breaking waves are higher than those of non-breakingwaves.

2. The reviewer freely admits being no expert in off-shore instalations (wind turbines, platforms, bridge pillars, etc.). It is difficult to visualise how such structures would be built using inclined pillars, as opposed to today's vertical structures. The authors could write a brief observation on the impact of their work.

Reply:
As can be seen from the pictures below, inclined cylinders are commonly used as support structures in wind turbines, platforms and large floating bridges.In consideration of copyright, the author did not insert these pictures into the manuscript. Therefore, accurately predicting the wave impact forces on standing cylindrical structures with different inclined angles is very important to ensure the stability and safety of the structures. The impact of the present work has beenincluded in the introduction, see the follows:
Many coastal and offshore structures consist of cylinders with different inclined angles, such as supporting structures of wind turbines, platforms, and large floating bridges. The breaking waves normally hit the structures within a very short duration and cause a large wave impact force, which can lead to a significant damage to the structures. Therefore, accurately predicting the wave impact forces on standing cylindrical structures with different inclined angles is of great concern for the design of coastal and offshore structures.
However, for assessmenton the stability and safety of the structures during the engineering design process, it is important tounderstand the interaction between the breaking waves and the cylinders with different inclineddirections. Therefore, the primary purpose of the present study is to investigate the characteristics ofbreaking wave forces on a standing cylinder with different transverse inclined angles in orientationnormal to the wave propagation.

https://www.wbur.org/onpoint/2017/02/13/wind-power-energy-america https://www.offshoreenergytoday.com/haven-rig-starts-johan-sverdrup-job/

https://www.iro-catalogue.nl/company_14http://moura.nl/Floating-project.html

https://www.newcivilengineer.com/archive/floating-solutions-norways-extraordinary-bridges-plan-15-04-2016/
https://www.mabeybridge.com/projects/berbice-river-floating-bridge

3. In addition to the details related to Menter's SST turbulence model, the authors should provide more detail on the VOF method (for example, it is not mentioned that a single set of momentum equations is used for the two phases, as opposed to other multiphase algorithms).

Reply:
In the VOF method, thegoverning equations (including momentum equation)for the two phases can be expressed as:
(1)
(2)
The density and kinematic viscosity at the interface can be obtained based on the formulas as follows:
(13)
where and are the densities of air and water; and denote the kinematic viscosity coefficients of air and water.
The volume fraction is solved by the following advection equation:
(14)
The more details can be seen in Section 2.1 and Section 2.2 of the manuscript.

4. Why set the outlet velocity for water as zero and what wave damping mechanism is used in the absorbtion zone?

Reply:
Because there is no fluid flowing in from the inlet boundary. According to the law of conservation of mass, no fluid should flow out at the outlet boundary. In this paper, the relaxation method from Jacobsen et al. [3] is used to generate and absorb waves in thenumerical wave tank.Relaxation zones are implemented to avoid wave reflection from the outlet boundary.The mechanism can be seen in Section 2.3of the revisedmanuscript, shown as follows:

2.3 Numerical wave tank
A relaxation method presented by Mayer et al. [4] is used to generate and absorb waves in the numerical wave tank. The relaxation algorithm [3] used to update the wave height and the velocity in the generation zone and absorption zone is expressed as:
(15)
where is either or , and the weighting factor is defined by:
, (16)
5. In general, the authors comment on what can be observed in the graphs, but offer no in-depth insight into their results throughout the discussion. The authors should make better use of the simulation data in order to provide more in-depth explanations for some of the behaviours observed. For example:

5a. The authors should comment more on the physical mechanisms which make the free surface elevation increase at WG3 when the cylinder angle is increased above 30 deg.

Reply:
Thanks for your advice.The supplementary comments have been added in Section 4.2 of the revised manuscript, shown as follows:

The free surface elevation at WG1 decreases first and then increasesslightly.The minimum surface elevation is obtained at .The free surface elevation at WG3decreasesgraduallywith varyingfrom to andincreaseswhen increases from to .This can be explained as given below. When the breaking wave passes a standing cylinder,it will produceachute-like jet at the back of the cylinder. The height of the chute-like jet is related to the transverse inclined angle of the cylinder and decreases with increasing (see Figures15(c) and 16(c)).It is the reason that the free surface elevationat WG3 reduces first.With the further increase of , the cylinder's resistance to waves in the vertical direction is weaker, and waves can propagate more easily through the cylinder, which increases the surface elevation.

5b. Comment more on why the high-order force is much larger for the vertical cylinder but appears to be relatively independent of the inclination angle for the other angles chosen. Why is the low-order force independent of inclination angle?

Reply:
It should be noted that Figure 14 shows thenormalized wave forces on the cylinder. As shown in Table 3, both the high-order wavesforces and low-order wave forces increase withthe increasing except .The main component of the high-order wave force is from the wave impact force (see the main peaks in Figure 13) on the cylinder. Therefore, under the same incident waves, the high-order wave force is mainly dependent on the maximum pressure on the cylinder. Comparing Figure 15(b) and Figure 16(b), the maximum pressure on the vertical cylinder is higher than that on the inclined cylinder of . Therefore, the larger high-order wave force on the cylinder is observedat . While the low-order wave force mainly depends on the area of the cylinder in contact with the wave. The larger submerged area gives the large low-order wave force when increases. For the case of , the free surface elevation is the highest due to the large wave reflection from the cylinder as shownin Figure 10, which causes a large wet surface area. Therefore, the large low-order wave force is generated at . The supplementary comments are added in Section 4.2 of the revised manuscript, shown as follows：

As shown in Figure 13, the main peak of the high-order wave force corresponds the peak of the impact force. Therefore, it is considered that the main component of the high-order wave force is from the wave impactforce on the cylinder. Under the sameincident waves, the high-order wave force is mainly dependent on the maximum pressure on the cylinder. Comparing Figure 15(b) and Figure 16(b), the maximum pressure on the vertical cylinder is higher than that on the inclined cylinder with . Therefore, the larger high-order wave force on the vertical cylinder is observedat . While the low-order wave force mainly depends on the area of the cylinder in contact with the wave. The larger submerged area gives the large low-order wave force when increases. For the case of vertical cylinder( ), the free surface elevation is the highest due to the large wave reflection from the cylinder as shownin Figure 10, which leads toalarger wet surface area as compared to the other transverse inclined angles. Therefore, the large low-order wave force is generated at .

6. It would be very interesting to see how much of the impact force is due to pressure forces alone and if significant turbulence generation occurs in this scenario.

Reply:
The peak impact wave force mainly comes from the pressure forces on the cylinder. This is due to that the breaking impact occurs in a very short duration, and turbulence cannot have sufficient time to develop and influence the impact force significantly. The reason for including the turbulence model in the present study is two-fold. Firstly, the turbulence model can help to predict breaking point more accurately by introducing relatively physical diffusion and dissipation, which has a significant impact on the force acting on the cylinder. Moreover, the fluid becomes turbulent and irregular after wave breaking behind the cylinder. The application of the turbulence model can predict the secondary loads and the characteristics of flow field in surf zonemore accurately.

Author Response File: Author Response.docx

Reviewer 3 Report

 The paper is well written and the first part of the paper present the problem very well with a detailed description on the practical application.

A few mistakes can be found in the text:

line 239: ... around the measure locations ... line 294: ... and then increase with the increases of the .... line 294: ... inclined angle, and the minimum ...

As a supplementary comment which can be discussed also in the paper, up to now have been studied another shapes of the surface or only circular ones ? 

Author Response

The authors thank the reviewers for the feedback for our manuscript. Below are our answers to the comments and questions of Reviewer 3. The answers are in blue and the amendments in the revised manuscript are in red.
Reviewer 3
The paper is well written and the first part of the paper present the problem very well with a detailed description on the practical application.A few mistakes can be found in the text:

1. line 239: ... around the measure locations ... line 294: ... and then increase with the increases of the .... line 294: ... inclined angle, and the minimum ...

Reply:
Thanks for your advice. These mistakes have been corrected in the revised manuscript, shown as follows

The interactions between the wave tank sidewall and the wave gauges, as well as the interactions between different wave gauges, can affect the free surface around the measurement locations.

The free surface elevation at WG1 decreases first and then increasesslightly.The minimum surface elevation is obtained at .The free surface elevation at WG3decreasesgradually with varyingfrom to andincreaseswhen increases from to .This can be explained as given below. When the breaking wave passes a standing cylinder, it will produce a chute-like jet at the back of the cylinder. The height of the chute-like jet is related to the transverse inclined angle of the cylinder and decreases with increasing (see Figures 15(c) and 16(c)). It is the reason that the free surface elevation at WG3 reduces first.With the further increase of , the cylinder's resistance to waves in the vertical direction is weaker, and waves can propagate more easily through the cylinder, which increases the surface elevation.

2. As a supplementary comment which can be discussed also in the paper, up to now have been studied another shapes of the surface or only circular ones ?

Reply:
We agree with the reviewer. At present, we only focus on the interaction between the breaking wave and the standing circular cylinders with different transverse inclined angles. It would be very interesting to study the breaking wave past structures with different shapes of the surface. Therefore, future work is added at the end of the paper, shown as follows:

In future work, the breaking wave past two cylinders in tandem will be conducted to study the effect of the distance between two cylinders and transverse inclined angles on the forces. Meanwhile,the breaking wave past the structures with different shapes will be also investigated.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The authors have successfully addressed the comments.