A 3D Smoothed Particle Hydrodynamics Study of a Non-Symmetrical Rayleigh Collapse for an Empty Cavity

Round 1

Reviewer 1 Report

In this work, the authors use SPH to perform a 3D simulation of the Rayleigh collapse of an empty bubble attached to a surface and subjected to external pressure by a big mass of water. Despite their small size, these bubble collapses have a sizeable impact on the erosion of hydraulic machinery and ships propellers. The reason for this is the strong shock waves that are produced during the collapse of the bubble and the generation of high-speed jets in the direction of the surface to which the bubble is attached.

Unfortunately, I cannot recommend the publication of this work in its present state. Overall, the text is still far from ready. It lacks a much more detailed explanation of what is done and a clearer and deeper discussion of the results. The information is not properly structured and a re-ordering of the text would be needed. Additionally, there are many typos and grammar errors that would have been easy to catch with any standard grammar checker. I just wrote down a few at the end of this review, but it is not a comprehensive list and it will require quite some editorial work in its present form.

The work is interesting and I would encourage the authors to pursue it further, but the changes to the paper and the experimental setup might be more work than a standard review process would trigger. 

In the following, I present my major and minor concerns, as well as some other suggestions.

Major remarks:

The SPH equations used in this work are what nowadays is considered "vanilla SPH" and, despite being extremely useful 20 years ago, the persistence of researchers using such an old implementation of the SPH equations is, to great extent, detrimental to the technique itself. It is well-known and documented that the equations presented in this work do not come from a Langrangian and have low accuracy when calculating gradients (see the work of Rosswog, New Astron. Rev. 53 (2009); Garcia-Senz et al., A&A 545 (2012); Cabezon et al., A&A 606 (2017); Frontiere et al., JCP 332 (2017), to mention the most relevant). The same problem can be pointed out regarding the interpolation kernel used. Pairing resistant kernels should be used nowadays in SPH simulations (see Cabezon et al. JCP 227 (2008); Dehnen & Aly MNRAS 425 (2012); Rosswog MNRAS 448 (2015)). Updating the SPH formalism and the interpolation kernel should be a priority to the authors. 

Secondary, but also important, the use of a standard artificial viscosity is one of the major complaints against SPH simulations, which dissipate too much even when there are no shocks, but shear. To that extent, switches have been proposed (see Cullen & Dehnen MNRAS 408 (2010) and Read & Hayfield MNRAS 422 (2012)). Finally, the use of standard volume elements is also nowadays something to avoid. There are better formulations using generalized volume elements that are much more proficient at treating discontinuities (see Ritchie & Thomas MNRAS 323 (2001); Saitoh & Makino ApJ 768 (2013); Hopkins MNRAS 428 (2013)). 

My major concern is that such old SPH implementations should be fading out of current research and I strongly encourage the authors to update their code. It is my opinion that works using these old implementations of SPH are becoming gradually less prone to be accepted for publications.

Admittedly, not all upgrades to SPH are going to make a difference to all problems. So, if the authors would like to check their results without a major reformatting of their code, I would recommend crossing the densities in the momentum and energy equations (that is, to use rho_i*rho_j instead of rho_i**2 and rho_j**2) and check if there is a change in the results. This formulation, despite not being completely compatible with a Lagrangian can help to avoid problems where grad(rho) becomes large, which in the case at study here means greater accuracy at the surface of the bubble and at the interaction with the wall. If there are important changes, then the authors should upgrade their code. In the case the authors decide not to upgrade their code, they should justify thoroughly (in this or future publications) why they use such an old SPH implementation.

Putting aside the upgrading of the hydrodynamics solver, in general terms, the results and overall information are very shallowly presented. The figures have very little information in their captions and could be improved considerably. Except for Figs 2 and 3, no other plot has a scale, nor is mentioned in the text the size of the domain that is plotted. Also, the analysis of the dynamics is merely qualitative and should be improved with detailed data. Besides that, only one simulation has been presented. It would be much more interesting to explore a bit for example the separation of the bubble to the domain wall (alpha) and see if their conclusions hold.

Finally, in order to validate their calculations, the authors compare their results with a previous 2D cartesian calculation. They only make a qualitative comparison, where they point out different structures being formed in both calculations, but showing plots of particle distributions side by side is not enough to validate results. There should be a quantitative comparison, for example in terms of time, energy dissipation, pressure and density profiles, fluid velocity, etc. Also, I doubt the validity of a cartesian 2D representation of this scenario. Cartesian 2D would represent a thin slab of a 3D domain, which is not the natural symmetry of this problem. Instead, a better approach would be to consider the particles rings and perform a 2D-axisymmetric simulation. This would better capture the dynamics and energy distribution of the problem and should be closer to the 3D results. 

Minor remarks:
I think section 2 is completely unnecessary. This is not a paper on the SPH technique, but an application, so going down to explain the SPH interpolant is not needed. It would be enough to mention the main characteristics and refer to the bibliography. Nevertheless, if the authors would like to introduce SPH because is not common in the field or the problem at study, more detailed work must be done. 

The problem description comes too late in the paper (Sect. 3.2). It is better for the reader to know what we want to simulate before the tools are presented. 

It is missing how the authors distribute initially the particles and how do they relax the system to get the necessary density profiles. There is no discussion whatsoever on how the initial conditions are achieved and how accurate they are, being this often one of the most complicated parts of SPH simulations.

Abstract:
The sentence "the model is compared to a similar 2D model to highlight strong and weak points of both approaches." is too ambiguous. What does "similar" mean? Same hydrodynamics solver? Same physics? Is it a 2D version of the same SPH code or third-party code? I know this will be explained in the text, but there is no reason for not being more specific in the abstract.

Intro:
L25: "when a spherical cavity is within a specific distance with the surface": which is?

Sect 2:
L59,64: As far as I know, it is not common to call this approximation by Eq.2 "Kernel representation", but "SPH interpolant". Although for most publications the distinction between the interpolant and the function to be interpolated is dropped, it is always good to be precise when defining terms at the beginning of the paper.

L64: The smoothing length is mentioned but not defined. This also links to the property of compact support that the SPH kernel typically has and it has not been mentioned. In general, the kernel definition in this paper is a bit incomplete. It should be more than just a "bell-shaped" function, but a "bell-shaped, normalized, symmetric function with compact support".

I don't think that there is a need to have two different subsections 2.1 and 2.2. Both would lie under the same umbrella of the SPH formalism.

Sect. 2.2:
L69: Particles carry physical information mostly. "Computational information" might be needed in terms of parallelization, load balancing, scalability, but within the context the authors are discussing in this section, it makes no sense and should be "physical information".

L70: "assume that an infinitesimal portion of V is occupied by a particle": The truth is that the actual volume element associated with a particle is far from infinitesimal. The standard procedure when discretizing Eq.2 is to represent dr3 as m/rho. This volume element is indeed smaller than the whole simulated domain volume, but by no means is infinitesimal in the mathematical sense. 

L73-74: This would be a good place to mention the concept of neighbors and neighboring particles, instead of just only the dry "j-th particle".

L74: h is missing an i subindex if the authors are using the gather model instead of the scatter model to define neighbors. Also, the standard SPH defines neighbor particles as those whose distance is below 2h_i. This is related to the widespread use of the Cubic Spline kernel. Modern SPH uses a more generalized definition using a constant (eta, f.e.) in front of h and defines which eta is used when the kernel is presented.

Sect. 2.3_
Eq. 5: This is an extremely old kernel, which I've barely seen in publications. Such a peaked, yet wide kernel should suffer, in my opinion, of too much sensitivity to particle disorder and strong pairing instability. The authors mention another work, where they claim to compare the effects of different kernels, but in that work, only the Lucy and the Quintic kernels were used. These are old and problematic kernels, which nowadays are completely surpassed by other interpolators such as the Wendland Kernels or the Sinc family. I don't see any reason for the kernel choice that the authors made.

"Despite being the first kernel used by Lucy in 1977, it performs well in the model presented in this work" - How do the authors know this? What does "perform well" mean? There is no quantification or qualitative description that proves that noise is not enhanced or that pairing instability is not taking place.

"the simulation does not show any form of instability", which is sometimes a red flag. Unphysical clumping of particles produces an over-pressure in contact interphases, which prevents subsonic instabilities to form that should be there otherwise in order to mix the material. 

L92: Eq. 7 is not obtained just applying Eq. 3 and 4 to 6. This formulation is obtained by including the density in the gradient operator so that a symmetric version of the momentum and energy are obtained.

L99: The Tait EOS is not the most common EOS in SPH, by any means. It might be common in some CFD simulations, but not within the technique itself. I doubt the authors can state safely, that it is more common than for example the ideal gas EOS. In any case, SPH is independent of the EOS, which is basically problem-related.

Fig.1: I see different shades of blue in the water region. I don't know if this is caused by the image or my reader. If it is the image, please try to fix it, unless it has a meaning.

L136: The radius of 0.01 mm of the green spot is meaningless if all other radii have not been defined yet.

Fig.2: Please, don't make the theoretical line so thick that it masks the differences with the experiment. Also, avoid using such big symbols. The best to see the differences is by two relatively thin lines.

L157: I don't understand the claim that at the end of the collapse there is not enough resolution. Shouldn't this be the time where the higher density is achieved? In that case, resolution is at its peak. To me, it seems more a problem regarding pairing instability when particles get too close, which artificially increases the pressure and slows down the collapse. The only other reason for losing resolution would be that a constant smoothing length is used and inadequately large for the last stage of the collapse. 

L158: The cite is missing.

Sect. 4:
Fig. 4: Why is the cavity non-spherical at least in the first snapshot? I also don't understand the color. If the pressure of the water is initially 50 MPa, why are all particles blue (i.e. at 1000 MPa)? I think that the color scale is inverted in all pressure plots.

Figs. 4, 5, 7: The authors should explain a bit more what are they representing here. Is this a cut in the Y=0 plane or a projection of all particles? Is this showing the full domain? What is the size of the box? Also, the module of the velocity is not as interesting and the radial velocity, which has a sign, and tells us also in which direction the fluid is moving.

L199: I confess it is still not completely clear for me where these structures (ring and shock) are in plots 4,5, and 7. I can guess a bit what the authors mean, but for the untrained eye, it is difficult to follow the discussion with just the information on the plots. Either an axis scale is given and the shock position and ring formation discussed within that scale, or the authors could add arrows to the plots to clarify what they mean in the text.

L204: The paper should be self-consistent. Videos are nice, but the information should be clear for the reader directly from the paper. The rotation that the authors meant, I think that is a transitory turbulent movement in the shape of a convective billow. As mentioned before, there are nice magnitudes that can help to represent this in a plot: radial velocity or vorticity. I would not say that the ring is spinning or rotating, which naturally leads the reader to think that the rotation axis is Z (which cannot be because of angular momentum conservation).

L225: "This implies that 2D simulations of Rayleigh collapse can be considered accurate up to the final phase..." This is a bold statement based on comparing qualitatively particle distributions. Much more work is needed to actually claim this.

L244-247: The information about how many particles were used, should be at the beginning of the paper, where the setup is presented. Also, just 27h for one full 3D simulation is quite fast. I don't think that a parametric study is completely impractical in 3D. In one week the authors could explore three different gammas with two different initial pressures, for example. Or six different gammas, if this is the most relevant parameter.

Typos:
Abstract: 
the model il compared -> the model is compared

Intro:
L15: in name of -> named after
L15: describe the dynamic -> describe the dynamics
L19: hydraulic machine -> hydraulic machinery / hydraulic machines

Sect. 2:
L52: "for simulate" -> "to simulate"
L69: "their own volume, mass, carrying..." -> "their own volume and mass, and carrying..."
L72: "follow" -> "follows"

Sect. 3:
L116: "with a travelling shock waves" -> "with travelling shock waves"
L118: "On contrary" -> "On the contrary"/"Conversely"
L144: "more computation intensive" -> "more computatonally intensive"
L145: "than the 2D simulations, therefore," -> "than 2D simulations. Therefore,"

Sect. 4:
L169: "problem" -> "problems"
L169: "1e-10", does this mean 10^{-10}?
L195: "This produce" -> "This produces"

Sect. 5:
L258: "high-speed het form" -> "high-speed jet from"
L262: "make cavity assumes" -> "makes the cavity assume"
L265: "new formed rig" -> "new formed ring"
L269: "the mode" -> "the model"

Author Response

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Reviewer 2 Report

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Comments for author File: Comments.pdf

Author Response

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Reviewer 3 Report

This manuscript presents a 3D particle method for a Rayleigh collapse. The application is interesting, and the results show improvements compared to 2D model. However, there are some flaws in the presentation of the method and results. Therefore, I would recommend revising the manuscript (comments below) before it could be considered for possible publication in the journal.

1- The research question is not clear. Is it just applying a 3D SPH model for this problem? Why? For improvements in the calculation of pressure field? The aim should be clearly stated in the Introduction section.

2- Section 2 (Introduction to SPH) is too long and contains unnecessary details. This section should be written more concisely.

3- All parameters should be defined under the equations. What are m, ρ, α, β, etc etc?

4- In Equation (6), ρ (density?) is out of the spatial derivatives. But in the discretized form (Equation 7), it is mixed with the particle’s volume (m/ρ), i.e. density is moved inside the spatial derivative. How is this justified and what effect will it have? A smoother pressure field?

5- What is the reason behind choosing the value of 1484 [ms] for speed of sound? This parameter is usually set to a smaller value than its real value for computational efficiency, however not smaller than a certain value to ensure that compressibility is limited to less than 1%.

6- The model is validated by comparing evolution of the dimensionless radius calculated by the model and Equation (10), but how to make sure that the model results for the pressure field in Section 4 are reliable?

7- In section 4.4, why did the 3D model provide smoother pressure? Could it be also related to the form of the discretised equations (difference with the 2D model in the discretization of the equations)? For example, see comment 4.

8- There are several writing errors and typos in the manuscript.

Author Response

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Reviewer 4 Report

In the article, the first 3D SPH model of a Rayleigh collapse for an empty cavity is proposed and is compared to a similar 2D model. The article tackles an important issue in multiphase flows/multiphase systems and therefore is suitable for ChemEngineering.

Comments:

• The article is very interesting and well prepared. I think the Authors should provide a specific purpose.
• The abstract should be corrected: the purpose of the study and conclusions should be added.
• We avoid quoting several articles at the same time eg. line 10: [1-10]; line 19: [13-15]; line 20: [16-20] without referring to the content of these works in detail. The way of citation should be fixed.
• Figs 4-8 and 10 – should have a scale of millimeters.

The experimental part of the article contains crucial information regarding the research model, method, and experimental technique. It also includes the information on the available equipment and presents the research outcomes and their detailed description. The structure of the paper is in accordance with the principles of very good scientific reports. The paper is written in good English (a few typos eg. line 6 “il” – should be “is”). The article contains 58 literature items. In the opinion of the reviewer, the article needs minor corrections. The abstract and conclusions require changes.

Author Response

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Round 2

Reviewer 1 Report

This new version of the paper has improved considerably in terms of the updating of the plots, including information to the inexistent captions, and correcting typos and grammar.

After reviewing the new version of the paper and the reply of the authors I conclude that despite the paper has improved, the work done still has serious flaws.

In their comments, the authors dismiss the stated concerns regarding the SPH implementation that they are using via quoting Occam's razor and epistemological reasons as fundamental scientific principles to excuse themselves to do the work. This is inadmissible in current scientific work. The authors complain that I give no reasons for the changes I proposed and the additional suggested work, which I disagree. Nevertheless, I already gave them a subset of the necessary bibliography which has given plenty of reasons for my concerns about their work for the last 20 years. If the authors think that these concerns do not apply to their work, they should justify this scientifically, not claiming "Pluralitas non est ponenda sine necessitate" or  "What can be asserted without evidence can also be dismissed without evidence” as they do at least five times in their reply.

The authors assume that qualitatively reasonable results justify their SPH implementation and my concern is that this is not always the case, but the authors refuse to delve further into this point, neither by a solid scientific argument nor by improving their code. Even when it implies a very small change in their equations, such as crossing the densities. Doing such a test would validate their calculations much more than their weak argumentation based just on figure 2, which corresponds to a different case than the one simulated. Because there is no analytical solution to the system that the authors want to simulate, they turn to the closest scenario (a symmetric Rayleigh collapse) with an analytical solution, which is necessary, but not sufficient to justify that their code will behave correctly when simulating their more complex asymmetric scenario. Also, they validate their code by simply visual curve comparison. There are better ways to do this, such as an L1-error calculation with respect to the theoretical prediction and compare it with other values published in the literature. This is a general trend in this paper, where affirmations such as "2D simulations of Rayleigh collapse can be considered accurate up to final phase" are based on a simple qualitative comparison of colormap plots. There is no single numeric value that the authors provide to sustain this type of affirmation. An easy example could have been simply the time evolution of the pressure which could be easily added to Fig. 3. Instead of that, the authors show side-by-side colormaps of 2D and 3D calculations which have different physical sizes, and claim that they are close enough, which the reader is simply expected to believe. We don't even know if these plots are at the same physical times, because none is provided.

As I stated in my first review, the 2D calculation is Cartesian. Therefore, a thin slab of a 3D box is simulated there, which is not the natural symmetry of the scenario. Comparing this to a 3D calculation brings little information. As I mentioned, if the authors want to actually expose the importance of dimensionality, their 3D calculation should be compared with a 2D axisymmetric calculation. Even more, as stated by the authors themselves in their reply, there are already 2D axisymmetric calculations in the literature, which would be interesting to compare with (or at least justify why not compare with them). Hence the comparison with a 2D Cartesian simulation in this work was a choice. There is no reasoning given behind this choice, which leads me to think that the authors simply ignored better-suited models to compare with, in favor of citing their own work.

The SPH introduction (old section 2.1 now merged with 2.2) is still extremely weak and outdated. The authors refused to change it. Apart from a few cosmetic changes here and there it is basically the same, hence it remains shallow and unnecessary.

The authors do not answer adequately the question about the initial conditions. They still don't state how are the particles initially distributed. Are they in a mesh? Are they in a glass-like configuration? Despite what the authors think, non-equilibrium systems can also be relaxed if needed, for example via an angular relaxation. But in their problem, this is even easier because the non-equilibrium configuration is imposed after relaxation. The authors have a domain with constant density and pressure. This can be relaxed without problems (or set via a mesh of particles) and afterward carve the cavity needed for the scenario. Because the authors give no information we cannot know how close to the expected value is their initial density profile in the blue region of Fig. 1.

According to their answer to my question, the authors do not know what is the SPH tensile instability, which means that they didn't check if this was taking place or not, as the cause for the prevention for other instabilities to form. This links to my question of why the authors use such an outdated SPH kernel. Their answer points again to Fig 2 as the only justification, which again is calculated in a scenario where the collapse is symmetric and there are no boundaries, and to non-scientific epistemological reasons.

After considering the reasons exposed above and the answers of the authors, I cannot recommend the publication of this work.

Author Response

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Reviewer 2 Report

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Author Response

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Reviewer 3 Report

-

Author Response

We could not see any comment by Reviewer 3. We assume that he/she was satisfied by our previous response.

Round 3

Reviewer 2 Report

No further comments.