Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement

Round 1

Reviewer 1 Report

This paper presented good experimental and numerical studies on the bed erosion and my suggestions are shown below:

1 The numerical method should be detailed in Abstract.

2 In section 4.3, the finite volume formulations should be given. From Fig 5, it seems to be solved by COMSOL.

3 why choose a uniform mesh of 401 points along the horizontal and 501 points in the vertical 340
direction? Is this the highest resolution?

4 In Fig 8, this model is symmetrical but the streamlines in the right below is no. Why?

5 More recent references are required.

Author Response

This paper presented good experimental and numerical studies on the bed erosion and my suggestions are shown below.

 

We would like to thank the reviewer for the constructive suggestions to improve this paper further. We have addressed all of them and modified the article accordingly. Please note that the comments are in italics, while our answers are not. The new additions in the revised manuscript are indicated in blue.

 

Comment  R1.1

The numerical method should be detailed in Abstract.

Answer:

The revised manuscript abstract includes now the following paragraph to detail the numerical method:

“The numerical techniques are based on a finite volume formulation to approximate spatial derivatives, a projection technique to calculate the pressure and velocity for each phase, and a staggered grid to avoid the spurious oscillations.”

 

Comment  R1.2

In section 4.3, the finite volume formulations should be given. From Fig 5, it seems to be solved by COMSOL.

Answer:

We do not use commercial software. The numerical method is based on an original two-phase model initially developed by Guillou, Barbry and Nguyen (2000). We refer to this paper and Patankar's book (1980) for a detailed description of the finite-volume formulation. We remark that the present group has improved this in-house code since then. The program is currently written in Fortran 90, displaying the results in Paraview software. For instance, we can refer to Uh Zapata et al. (2019) for the parallelization version of the program.   

 

Comment  R1.3

Why choose a uniform mesh of 401 points along the horizontal and 501 points in the vertical direction? Is this the highest resolution?

Answer:

Indeed, the highest resolution is given by a uniform mesh of 401 points along the horizontal and 501 points in the vertical direction. Thus, the spatial steps are given by dx=0.5 mm and dz=0.5 mm. This choice corresponds to a grid resolution close to one-grain diameter. It would not pertinent to consider finer meshes as we can not have grid resolutions more refined than the sediment diameter for Euler-Euler two-phase flow formulation. 

 

Comment  R1.4

In Fig 8, this model is symmetrical but the streamlines in the right below is no. Why?

Answer:

The velocity streamlines are not fully symmetric because Fig. 8 displays an instantaneous velocity profile of turbulent fluid. The plane jet flow shows a complex structure with some fluctuation and vortices along the jet. This behavior is in complete accordance with the experimental results. We can clearly see this oscillating behavior as the grid resolution is higher.

 

Comment  R1.5

More recent references are required.

Answer:

Due to the physics behavior's complexity, only a few approaches deal with the proposed problem. To the best of our knowledge, the more recent studies dealing with the jet erosion test case based on a two-phase flow approach are Qian et al. (2010), Kuang et al. (2013), Boyaval et al. (2018), Yuan et al. (2018), Uh Zapata et al. (2018), Benseghier et al. (2020), and Wang et al. (2021); all of them already cited in this paper.

Author Response File: Author Response.pdf

Reviewer 2 Report

This is an ambitious undertaking. Lab simulations of cratering are easy and telling, but numerical simulations are extremely challenging, for the modeling of a fluid jet cratering in sediments entail solid sediment portions, fluidized bed dynamics (regions of moving sediments regulated by many collisions), and diluted mixture (sediment particles interacting mostly with the fluid in the absence of collisions). The authors develop a multi-phase model with all dynamic regions handled in a single code to allow the tracking of the interfaces between those regions, most particularly the shape of the crater being dug by the jet (as stated in Lines 99-100 of the text). I view the study as a very commendable effort in the pursuit of an almost intractable problem. So, while the results may come a bit short in comparison with lab results (the numerical craters don't seem to have the exact same shape as their lab counterparts and don't seem to be as sharply defined - a bit like X-rays of bones), I commend the authors and recommend publication of their work as a stepping stone on the road to progress.

That said, I have minor comments (in order of appearance in the text, not in order of importance):

1. Page 4, Figure 2: Inside each figure panel, the vertical caption has a spelling error. It should be "water jet injector".  Further since "jet" and  "injector" are somewhat redundant in their inclusion of the same Latin root "jactare", one could have "Water jet nozzle".

2. Page 4, Equation (2): I think that mentioning the Reynolds number is not much more than a reflex. As the subsequent developments bear, it is not the product jet velocity times nozzle width [UJ*b] that matters but the product jet velocity times square root of nozzle width [UJ*sqrt(b) as prominent part of expression for E in Equation (5) two pages later]. This was to be expected: What defines the planar jet is not its velocity and width separately but its overall momentum flux per unit length, rho*(UJ^2)*b. Thus the proper combination of exit velocity and nozzle width is (UJ^2)*b or, alternatively, UJ*sqrt(b). Authors could remark this was to be anticipated and could mention that the Reynolds number is not pertinent in the dynamics at hand. The note can be reiterated around Line 186 where the mention "the model differs in the exponent -1/2 (instead of -1) for the L scaling." could be replaced by a mention that it is natural, if not obvious, that, if Uj goes with sqrt(b) in the numerator, then non-dimensionalization will need a sqrt(L) factor in the denominator. It's all a direct consequence of the jet being characterized by its momentum flux.

3. Page 4, Line 145: The reference to Figure 1 could come three lines earlier, before the variable symbols are quoted.

4. Page 5, Line 158: I find "The crater depth H is higher" awkward. How could 'higher' be a fitting descriptor for something that is 'deeper'? Replace by "the crater is deeper".

5. Page 5, line 159: "Furthermore," could be replaced by "As one would expect.".

6. Page 6, Line 172: The mention of  the lambda factor being positive, instead of negative, baffles me. How could the virtual origin of the jet not be up in the nozzle but actually downstream of the nozzle? Is it because the the nozzle is so long that the jet does not start opening by entrainment of surrounding fluid for some distance? Please shed a little light here.

7. Section 3 "Mathematical Model": Strangely, I do not have any comments on the harder parts of the model developments! I think that the authors did build a clever and solid model. In their place, I might not have made all the same assumptions and choices, but I rally behind theirs.

8. Section 4 "Numerical Results": Remarkable in my opinion.

 

Author Response

This is an ambitious undertaking. Lab simulations of cratering are easy and telling, but numerical simulations are extremely challenging, for the modeling of a fluid jet cratering in sediments entail solid sediment portions, fluidized bed dynamics (regions of moving sediments regulated by many collisions), and diluted mixture (sediment particles interacting mostly with the fluid in the absence of collisions). The authors develop a multi-phase model with all dynamic regions handled in a single code to allow the tracking of the interfaces between those regions, most particularly the shape of the crater being dug by the jet (as stated in Lines 99-100 of the text). I view the study as a very commendable effort in the pursuit of an almost intractable problem. So, while the results may come a bit short in comparison with lab results (the numerical craters don't seem to have the exact same shape as their lab counterparts and don't seem to be as sharply defined - a bit like X-rays of bones), I commend the authors and recommend publication of their work as a stepping stone on the road to progress. That said, I have minor comments (in order of appearance in the text, not in order of importance).

 

We completely agree with the reviewer's thoughtful observations. We want to thank you for the constructive words to improve this paper further. We have addressed all the minor comments in the following lines and modified the document accordingly. Please note that the remarks are in italics, while our answers are not. The new additions in the revised manuscript are indicated in blue.

 

Comment  R2.1

Page 4, Figure 2: Inside each figure panel, the vertical caption has a spelling error. It should be "water jet injector".  Further since "jet" and  "injector" are somewhat redundant in their inclusion of the same Latin root "jactare", one could have "Water jet nozzle".

Answer:

The revised manuscript has been modified accordingly to this suggestion.

 

Comment  R2.2

Page 4, Equation (2): I think that mentioning the Reynolds number is not much more than a reflex. As the subsequent developments bear, it is not the product jet velocity times nozzle width [UJ*b] that matters but the product jet velocity times square root of nozzle width [UJ*sqrt(b) as prominent part of expression for E in Equation (5) two pages later]. This was to be expected: What defines the planar jet is not its velocity and width separately but its overall momentum flux per unit length, rho*(UJ^2)*b. Thus the proper combination of exit velocity and nozzle width is (UJ^2)*b or, alternatively, UJ*sqrt(b). Authors could remark this was to be anticipated and could mention that the Reynolds number is not pertinent in the dynamics at hand. The note can be reiterated around Line 186 where the mention "the model differs in the exponent -1/2 (instead of -1) for the L scaling." could be replaced by a mention that it is natural, if not obvious, that, if UJ goes with sqrt(b) in the numerator, then non-dimensionalization will need a sqrt(L) factor in the denominator. It's all a direct consequence of the jet being characterized by its momentum flux.

Answer:

We agree with the reviewer that the planar jet is determined by its overall momentum flux per unit length. Indeed, the erosion parameter E defined in equation (5) is much more relevant to describe the erosion issues. However,  “although Re is not much relevant to describe the erosion phenomena, the jet Reynolds number is essential to characterize the jet regime among laminar, oscillating laminar, oscillating turbulent, and fully turbulent, as detailed by Badr et al. (2014) [10] or in Tritton [19]”. We include the previous comment after the definition of the jet Reynold number. We also added the following comment close to Line 186, as suggested by the reviewer: “Note that, if UJ goes with sqrt(b) in the numerator of equation (5), then non-dimensionalization will need a sqrt(L) factor in the denominator. It's all a direct consequence of the jet being characterized by its momentum flux. Thus, besides considering the effective jet-sediment distance, the model differs in the exponent -1/2 (instead of -1) for the L scaling.”

 

 

 

Comment  R2.3

Page 4, Line 145: The reference to Figure 1 could come three lines earlier, before the variable symbols are quoted.

Answer:

The revised manuscript has been modified accordingly to your suggestion.

 

Comment  R2.4

Page 5, Line 158: I find "The crater depth H is higher" awkward. How could 'higher' be a fitting descriptor for something that is 'deeper'? Replace by "the crater is deeper”.

Answer:

The revised manuscript has been modified accordingly to your suggestion.

 

Comment  R2.5

Page 5, line 159: "Furthermore," could be replaced by "As one would expect.”.

Answer:

The revised manuscript has been modified accordingly to your suggestion.

 

Comment  R2.6

Page 6, Line 172: The mention of  the lambda factor being positive, instead of negative, baffles me. How could the virtual origin of the jet not be up in the nozzle but actually downstream of the nozzle? Is it because the the nozzle is so long that the jet does not start opening by entrainment of surrounding fluid for some distance? Please shed a little light here.

Answer:

We agree that a long nozzle could push the virtual origin downstream. However, we mostly attribute this behavior to the 2D confined jet, as explained by Chua and Lua in [37] (https://doi.org/10.1063/1.869841). The 2D jet confinement tends to shift the position for the entrainment of surrounding fluid.

 

Comment  R2.7

Section 3 "Mathematical Model": Strangely, I do not have any comments on the harder parts of the model developments! I think that the authors did build a clever and solid model. In their place, I might not have made all the same assumptions and choices, but I rally behind theirs.

Answer:

Thank you for your comments.

 

Comment  R2.8

Section 4 "Numerical Results": Remarkable in my opinion.
Answer:

Thank you for your comments.

Author Response File: Author Response.pdf