Finite Volume Method for Transient Pipe Flow with an Air Cushion Surge Chamber Considering Unsteady Friction and Experimental Validation

Round 1

Reviewer 1 Report

1. In Fig.2, it is required to add the corresponding variables for the air chamber and bottom pipe. Check the declaration of subscript "1" and "2" in equation (7), and it is not an exact difination in both H, Q and B_P, B_M, and further, it is preferred to give the characteristic equations for bottom pipe together with equation (7).

2. Please give more details to explain how to obtain and analysis the approximate wave speed (1290m/s) for the experimental setup.

3. Fig. 4~5 gives the dynamic data from PT-4# for 4 cases, but no further analysis is provided to clarify the difference for these 4 cases. From this point, 2 cases is enough to be used for comparasive analysis. The same comment is for Fig. 6-8. Therefore, it is strongly recommeded to give the detailed analysis for each figure from Fig. 4 to Fig. 8.

4. All the variables in the given equations should be defined with the same format.

There are some writng problems in the manuscript, please read and check the English writing carefully, such as the wrong sentence “……it is extremely important to accurately simulate …… surge chamber is extremely important.”

Author Response

Part A (Reviewer #1)

The reviewer’s comment:

1. In Fig.2, it is required to add the corresponding variables for the air chamber and bottom pipe. Check the declaration of subscript "1" and "2" in equation (7), and it is not an exact difination in both H, Q and B_P, B_M, and further, it is preferred to give the characteristic equations for bottom pipe together with equation (7).

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we add the corresponding variables in Figure 2, and give the calculation formulas of B_M, B_P, C_M and C_P, as well as the characteristic equation of the bottom pipeline.

 

2. The reviewer’s comment:

Please give more details to explain how to obtain and analysis the approximate wave speed (1290m/s) for the experimental setup.

The authors’ Answer:

Totally agree with the suggestion. In the revised version, we explain in detail how to obtain and analyze the approximate wave velocity (1290 m/s) of the experimental setup. The wave velocity of the water hammer in this experiment is calculated according to the experimental data measured by the pressure sensor. According to the pressure experimental data, the time difference between any two adjacent wave peaks is recorded as 2T, and then according to the pipeline length L, water hammer wave a = 2L/T. In addition, many factors can affect the water hammer wave velocity. In order to eliminate the interference, repeated tests were carried out on different experimental conditions for many times. Finally, the average value of multiple tests was taken as the value of the water hammer wave velocity in the subsequent numerical simulation. The wave velocity of the water hammer measured by the experiment is between 1260 m/s and 1360m/s. In the numerical simulations, the wave speed is a=1290m/s.

 

3. The reviewer’s comment:

Fig. 4~5 gives the dynamic data from PT-4# for 4 cases, but no further analysis is provided to clarify the difference for these 4 cases. From this point, 2 cases is enough to be used for comparasive analysis. The same comment is for Fig. 6-8. Therefore, it is strongly recommeded to give the detailed analysis for each figure from Fig. 4 to Fig. 8.

The authors’ Answer:

Much appreciated for the excellent comment. Considering that Figures 6~8 can give the same conclusions, we delete Figure 7 (Case 2), and present the results of Case 1 (laminar flow) and Case 3 (turbulent flow). In the new version, Figures 6 (a) and 6 (b) display the results calculated by steady friction model and experimental results in Case 1 (laminar flow) and Case 3 (turbulent flow). Figure 7 gives the results calculated by numerical models (steady and unsteady friction water hammer model) and experimental results in Case 1 (laminar flow) and Case 3 (turbulent flow). In the revised version, we give the detailed analysis for each figure.

 

4. The reviewer’s comment:

All the variables in the given equations should be defined with the same format.

The authors’ Answer:

Thank you very much for the suggestion. In the revised version, we define all the variables in a given equation in the same format.

Author Response File: Author Response.pdf

Reviewer 2 Report

The work is very interesting and I read it with pleasure. However, it probably contains flaws, which I'm sure could be addressed by the authors. I regret that the lines of the article are not numbered. For this reason, in my review, I will refer to the pages. 

First of all, I think the editing of the text is a bit careless. I have listed some examples below:

- Page 2: The following sentence needs to be rephrased: "In order to realize the safe and stable operation of water systems and the realization of intelligent operations, it is extremely important to accurately simulate the transient flow of pipe system with air cushion surge chamber is extremely important.";

- I don't think there is a need to write the word "Godunov" in capitals;

- Page 3: Symbols of ni and mi coefficients are sometimes written without subscript;

- There is an inconsistency of symbols - the "V" symbol means something different in Eq. (10), (22) and (24).

- Formatting of list of references is incosistent.

I suggest that you read the entire text carefully for improved formatting and editing.

The derivations in Section 2.2 need to be expanded. There is a lack of references, explanation of some equations, explanation of some symbols (e.g., what are the "known variables at time t" CP1, CB1, CM2 and BM2 in equation (7)? The same goes for section 3.3, on boundary conditions.

Section 4 is somewhat disappointing. The measurement results are interesting, but the discussion is very limited and should be expanded. It would be appropriate to discuss why the best agreement was obtained for n = 1.0.  Why, even for n = 1.0, do the calculated pressure oscillations attenuate faster than the measured ones?  How was the complience of measurement and calculation results evaluated? Only visually, based on donwsteram pressure oscillation figures?

I assume that the symbol H0 in the tables on page 7 and 9 stands for pressure head in upstream reservoir. What was the pressure in the surge chamber? By the way, both of these tables are called "table 1". What was the pressure wave velocity in experiments with air cushion surge chamber? Did the simulations for this option assume a measured wave speed?

In my opinion, the work is interesting but should be significantly improved. Additional explanation or at least citation of relevant sources is needed for equations in Sections 2.2-2.4. The discussion in Section 4 should be expanded. I would also recommend writing in the introduction what the novelty of the paper is.

Author Response

Please see the attachment

 

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors propose to use a second-order finite-volume approach for approximating the water-hammer equations in conjunction with the consideration of unsteady friction models. A special attention is also given to the  boundary approach used for representing air cushion surge
chambers. An experimental pipe system is also considered in order to validate the previous models. The paper is well written with sufficient details. However, please find some comments or suggestions:

1) The bibliographic references concerning the finite-volume method and its use for the water-hammer equations can be improved.

2) In Eq. (3) the terms V\dxU is neglected which it is not the case in Eqs (1) and (2). Please, can the authors add comments or neglect these terms in Eqs (1) and (2)?

3) In Eq. (5), the expression of the unsteady shear stress \tau_u is given following the Zielke's model. However, the expression between \tau_u and the unsteady friction term J_u is missing.

4) The polytropic exponent is denoted by k in page 5 and by n in page 9. IT has to be homogenized.

5) The RHS of Eq. (12) seems to be incorrect. I can imagine that the authors have considered the Godunov scheme of the linear hyperbolic system associated with the water-hammer equations. Can the authors write the correct expression of the corresponding numerical flux.

6) In Eqs (15) and (16), can the authors give the expression of \overline{A}?

7) In Eq. (22), it seems for me that the subscript -1/2 has to be used instead of 1/2 for the upstream boundary of the reservoir.

8) n the computations presented in Section 4, is the valve closure assumed to be instantaneous? If it is not the case, can the authors add some details concerning the way they introduce a time delay in the valve closure modeling?

9) Page 9, three values are considered by the authors concerning the polytropic coefficient. I think that more details have to be given. For example, the choice n=1 corresponds to an isothermal process whereas n=1.4 corresponds to an adiabatic transformation. This kind of physical correspondence concerning the choice of the polytropic coefficient is required.

Finally, in my opinion the paper might deserve publication provided the previous remarks are properly addressed and reflected.

Author Response

Part C (Reviewer #3)

1. The reviewer’s comment:

The bibliographic references concerning the finite-volume method and its use for the water-hammer equations can be improved.

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we add references to the finite volume method and its use in the water hammer equation.

2. The reviewer’s comment:

In Eq. (3) the terms V\dxU is neglected which it is not the case in Eqs (1) and (2). Please, can the authors add comments or neglect these terms in Eqs (1) and (2)?

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we provide additional explanations.

3. The reviewer’s comment:

In Eq. (5), the expression of the unsteady shear stress \tau_u is given following the Zielke's model. However, the expression between \tau_u and the unsteady friction term J_u is missing.

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we add the expression for J_u.

4. The reviewer’s comment:

The polytropic exponent is denoted by k in page 5 and by n in page 9. IT has to be homogenized.

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we unify the gas polytropic index.

5. The reviewer’s comment:

The RHS of Eq. (12) seems to be incorrect. I can imagine that the authors have considered the Godunov scheme of the linear hyperbolic system associated with the water-hammer equations. Can the authors write the correct expression of the corresponding numerical flux.

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we have corrected the formula.

6. The reviewer’s comment:

In Eqs (15) and (16), can the authors give the expression of \overline{A}?

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we give an exact expression for A.

7. The reviewer’s comment:

In Eq. (22), it seems for me that the subscript -1/2 has to be used instead of 1/2 for the upstream boundary of the reservoir.

The authors’ Answer:

We have checked carefully Eq. (22) (Eq. (30) in the new version). 1/2 for the upstream boundary of the reservoir is correct.

8. The reviewer’s comment:

n the computations presented in Section 4, is the valve closure assumed to be instantaneous? If it is not the case, can the authors add some details concerning the way they introduce a time delay in the valve closure modeling?

The authors’ Answer:

Yes, the valve closure is assumed to be instantaneous. In the experiments, the valve is abruptly closed, so in the numerical simulation the valve closure is assumed to be instantaneous.

9. The reviewer’s comment:

Page 9, three values are considered by the authors concerning the polytropic coefficient. I think that more details have to be given. For example, the choice n=1 corresponds to an isothermal process whereas n=1.4 corresponds to an adiabatic transformation. This kind of physical correspondence concerning the choice of the polytropic coefficient is required.

The authors’ Answer:

Thank you very much for the suggestions. In the revised version, we give more details about this and the related discussion.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Thank you for taking info account my suggestions. In my opinion, this paper can be published in Water.