Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution

Round 1

Reviewer 1 Report

This paper should be rejected because the only significant new results, curve fits to the mean velocity profiles found in the Princeton Superpipe experiments, are not at a high scientific level, and the paper is not written in a manner consistent with the value of the results. They are worthy of being reported in a much, much shorter communication that comes straight to the point  and does not aggrandize the accomplishment by calling the velocity profile "analytical" when it is purely empirical. 

The author develops a four-parameter curve fits for the mean turbulent velocity profile U(r) in pipe flow starting from a curve fit to the profile of the mean turbulent vorticity found from experimental data. The mean vorticity in the case of steady, fully-developed, incompressible turbulent flow of constant property fluids reduces very simply to mean vorticity= dU/dr, so it is conceptually trivial to integrate the mean vorticity profile to get the mean velocity profile.  This is neither novel nor worthy of review of lengthy discussion, yet much of this 30+ page paper does exactly that, starting from a review of fundamental text book vortex dynamics , philosophical arguments supporting starting from vorticity, and many other unnecessary matter that would be well-known to ant knowledgeable student of fluid flow. Worse, most of it is not used in any substantial way. The only work in the paper that is with reporting is the curve-fitting procedure used to represent the mean vorticity profile, the integration procedure to get the "analytical" mean velocity profile and the important quants that can be derived from the mean velocity profile: friction factor, bulk velocity and maximum velocity as functions of pipe Reynolds number. The comparisons of the curve fits for the mean vorticity and mean velocity profile with the original experimental date should also be given, although only to the extant that they demonstrate goodness of fit.

In summary, the curve fit mean velocity is useful, but curve fitting is a low-level engineering exercise   This result should be described in a 3-5 page technical note.

Author Response

First of all, thank you for rewieving of my paper.

(This paper should be rejected because the only significant new results, curve fits to the mean velocity profiles found in the Princeton Superpipe experiments, are not at a high scientific level, and the paper is not written in a manner consistent with the value of the results. They are worthy of being reported in a much, much shorter communication that comes straight to the point and does not aggrandize the accomplishment by calling the velocity profile "analytical" when it is purely empirical.)

The velocity profile is derived on basis of the known characteristics of the profile, which are pressure drop, average velocity, center line velocity, and the second derivative of the mean velocity profile in the pipe axes. The mean velocity profile function is analytic from the mathematical point of view. The definition of an analytic function is, as follows “In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable….“. From this point of wiew, in my opinion, the chosen function of  the vorticity distribution is really analytic.

The velocity profile characteristics as a function of Re, they are really empirical. It is true that this dependences have to be find from the experimental data, but the result is the analytical function of the mean velocity profile which is possible to integrate or to derivate and it is easy to use anywhere where it is necessary. There are two exponents which are determined from the velocity difference minimizing.

It seems me, that this is the advantage of this approach in comparison with the universal velocity profile in the literature [4]. The mean velocity profile is expressed in integral form in that paper, what is rather uncomfortable for practical use. There are five free parameters which are determined on the basis of the comparison with the experimental velocity profiles.

It is also mentioned in my paper that there is also a chance to find these exponets with help of the last profile characteristics. However, it is out of the scope of this paper.     

(The author develops a four-parameter curve fits for the mean turbulent velocity profile U(r) in pipe flow starting from a curve fit to the profile of the mean turbulent vorticity found from experimental data. )

I am sorry, this is not true. The fit is done to the mean velocity profile, not to the mean vorticity dristribution, to be able to find the empirical dependance of exponents on the Re. 

(The mean vorticity in the case of steady, fully-developed, incompressible turbulent flow of constant property fluids reduces very simply to mean vorticity= dU/dr, so it is conceptually trivial to integrate the mean vorticity profile to get the mean velocity profile.  This is neither novel nor worthy of review of lengthy discussion, yet much of this 30+ page paper does exactly that, starting from a review of fundamental text book vortex dynamics , philosophical arguments supporting starting from vorticity, and many other unnecessary matter that would be well-known to ant knowledgeable student of fluid flow. Worse, most of it is not used in any substantial way.)

By my opinion, it is necessary to show that this solution has the background in the vortex flow theory. Because it helps to do better flow analysis and it allows us to modify this approach to other similar solutions, for example, the mean velocity profile solution for pipes with non-circular cross-sections. By the way, I did not find anywhere this solution.

(The only work in the paper that is with reporting is the curve-fitting procedure used to represent the mean vorticity profile, the integration procedure to get the "analytical" mean velocity profile and the important quants that can be derived from the mean velocity profile: friction factor, bulk velocity and maximum velocity as functions of pipe Reynolds number. The comparisons of the curve fits for the mean vorticity and mean velocity profile with the original experimental date should also be given, although only to the extant that they demonstrate goodness of fit.

In summary, the curve fit mean velocity is useful, but curve fitting is a low-level engineering exercise   This result should be described in a 3-5 page technical note.)

Yes, I agree that this procedure of velocity profile derivation is easy and simple. However, it is a surprise for me that no one came to this solution (I did not find it anywhere). This velocity profile is better than the power law presented in every literature. The power law has principal discrepancies at the wall and in the center line.

The vorticity distribution emphasizing is the intention. When we look at the flow from this point of view then it helps us to better understand and analyse the fluid flow. For example, it helps us to define the shear boundary layer thickess. If the vorticity distribution is linear than it means that the velocity profile is parabolic. The vorticity rotor is associated with the friction force in liquid ect. It also helps us to apply this procedure for velocity profile derivation in pipes with no circular cross-section shape.

And one note on the length of the paper. I tried to give the reader all necessary information to by able to understand the idea and physic of derivation.  

Reviewer 2 Report

The paper is related to the analytical determination of the velocity profile based on the vorticity distribution on the cross-section expressed by a polynomial function. Reference is made to experimental data obtained in a high-pressure pipe (Princeton superpipe data).

The parameters of the polynomial function are determined on the basis of the boundary conditions and by the minimization  of the sum of the velocity magnitude between the analytical profile and experimental data. Empirical formulas are provided that allow the different parameters to be derived from the knowledge of average velocity, pipe radius and Reynolds number.

The paper is well written but there are some mistakes that need to be corrected, so I recommend this article to be published in Water after minor revision.

Some suggestions are given:

• Change the multiplication symbol in all formulas by centering it on the vertical.
• I suggest avoiding statements like “It is a very interesting paper” or “it is a very interesting result”. Check it throughout the manuscript.
• The symbol r is used in line 90 to represent the intensity of the vector r_k, instead in eq. (25) to represent the density (see also eq. (64) and line 435). Different symbols should be used.
• Please, modify the sentence “This is a really nice solution” in L125.
• L160: It should be “…functions. Soukup [9] tested a…”.
• In Figure 5, insert the y-axis and specify after eq. (22) that h represents viscosity.
• In line 341, “expression (44)” must be changed to “expression (43)”.
• In eq. (48) the term f_(a3) is reported as the coefficient of D₍₁₎/R²; instead the correct coefficient is f_(a4).
• In eq. (49) the term f_(b3) is reported as the coefficient of D₍₁₎/R^(K-1); instead the correct coefficient is f_(b4).
• In eq. (50) it is erroneously present v_(s) in place of v_(av); moreover it is reported as coefficient of D₍₁₎/R^(N-1)  the term f_(c3), instead the corrected coefficient is f_(c4).
• Fig. 14(b) is incorrect because it is the reproduction of Figure 14(a).
• According to Tab. 1, in Tab. 2 columns containing W_(w) and D₍₁₎ should be highlighted in grey (instead of columns containing W’_(w) and D’₍₁₎) since they represent parameters that are taken as variables in the minimization process.
• In line 437 it is said that each of the profiles in figure 15a is shifted about u⁺=3 from the previous one. Perhaps, for a better readability (see 440 line), it would be appropriate to shift the profiles by a greater amount. Similarly for Figure 15b (see lines 445-446).
• It seems that in figures 15a and 15b the variable shown on the ordinate axis is u_t instead of u⁺.
• Determining 4 parameters (W_(w) , D₍₁₎, N e K) instead of 2 (W_(w) e D₍₁₎), minimizing the sum of the velocity magnitude difference between the analytical profile and experimental data, allows to improve in a limited way the velocity profile but significantly worsens that of the normalized vorticity distribution.
• The ability of the proposed formulae to interpret experimental data other than those on which the model has been calibrated should be investigated.

Author Response

First of all, I thank you for your opinions and suggestions. I appreciate you spend a lot of time on the paper reading. All your suggestions are very valuable for me.

(Change the multiplication symbol in all formulas by centering it on the vertical.)

I changed it in all equations.

(I suggest avoiding statements like “It is a very interesting paper” or “it is a very interesting result”. Check it throughout the manuscript.)

Yes I agree. I modified it in whole paper.

(The symbol r is used in line 90 to represent the intensity of the vector r_k, instead in eq. (25) to represent the density (see also eq. (64) and line 435). Different symbols should be used.)

Yes I agree. It is fixed. (line 97)

(Please, modify the sentence “This is a really nice solution” in L125.)

Yes I agree. I modified it. (line 130)

(L160: It should be “…functions. Soukup [9] tested a…”.)

Yes I agree. It is fixed. (line 164)

(In Figure 5, insert the y-axis and specify after eq. (22) that h represents viscosity.)

It is fixed. (line 221)

(In line 341, “expression (44)” must be changed to “expression (43)”.)

Yes, it is true. It is replaced. (line 345)

(In eq. (48) the term f_(a3)is reported as the coefficient of D₍₁₎/R²; instead the correct coefficient is f_(a4).)

Yes,, I agree. It is fixed.

(In eq. (49) the term f_(b3)is reported as the coefficient of D₍₁₎/R^(K-1); instead the correct coefficient is f_(b4).)

Yes I agree. It is fixed.

(In eq. (50) it is erroneously present v_(s)in place of v_(av); moreover it is reported as coefficient of D₍₁₎/R^(N-1)  the term f_(c3), instead the corrected coefficient is f_(c4).)

Yes, I agree. It is fixed.

(14(b) is incorrect because it is the reproduction of Figure 14(a).)

Yes, I agree. It is fixed.

(According to Tab. 1, in Tab. 2 columns containing W_(w)and D₍₁₎ should be highlighted in grey (instead of columns containing W’_(w) and D’₍₁₎) since they represent parameters that are taken as variables in the minimization process.)

Yes, I agree. It is fixed.

(In line 437 it is said that each of the profiles in figure 15a is shifted about u⁺=3 from the previous one. Perhaps, for a better readability (see 440 line), it would be appropriate to shift the profiles by a greater amount. Similarly for Figure 15b (see lines 445-446).)

I fixed the shift of u⁺ to 6. (line and 455, Figures 15a and 15b)

(It seems that in figures 15a and 15b the variable shown on the ordinate axis is u_tinstead of u⁺.)

Yes, I agree. It is fixed.

(Determining 4 parameters (W_(w), D₍₁₎, N e K) instead of 2 (W_(w) e D₍₁₎), minimizing the sum of the velocity magnitude difference between the analytical profile and experimental data, allows to improve in a limited way the velocity profile but significantly worsens that of the normalized vorticity distribution.)

Yes, that is true. The question is, if there is some mistake in the basic velocity profile characteristics measurement (as a wall vorticity, or the other one). It needs to do more experiments. We are preparing a project where we want to measure the velocity profiles by 2 components PIV and by Stereo PIV.

(The ability of the proposed formulae to interpret experimental data other than those on which the model has been calibrated should be investigated.)

There is the same answer as in the previous suggestions.

Reviewer 3 Report

The present paper “Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution” deals with  the proposal of an explicit formulation to determine velocity profiles in axial symmetry inside pipes.

This is an interesting topic, and very important in pressurized systems for water distribution networks. Nevertheless, some considerations are proposed by this reviewer before publication.

Abstract. This is well presented, with the main objectives and achievements of this contribution, in this reviewer opinion, it is interesting to point out the novelty of the work, to aim readers to continue-

Introduction. Introduction presents the main objectives and importance of deriving this analytical formula, even for numerical approaches. Nevertheless, in this reviewer opinion, it would be interesting to see a structure of Introduction, Materials and Methods, Analysis of results, and Conclusions. In this case, readers cannot identify when the introduction is finished and the apportions of authors related to methodology begins.

Backgrounds. This is also a part of the introduction, in this reviewer opinion, presenting main considerations about equations; This is well described, but in this reviewer opinion, as this is a well considered problem, some more references could be included in this section.

Basic idea of velocity profile derivation. In this reviewer opinion, this is methodological and proposed by authors. This section is interesting and well described and it is important the proposal of the form of the chosen polynomial function (19). In this reviewer opinion, some more references giving validity to this proposal or indicating the novelty of this would be welcome.

Available conditions for free coefficients determining . Boundary conditions are clear and well presented. This section (as part of the methodology, in this reviewer opinion) is well described. Nevertheless, in this section, boundary condition mix with results, this could be clearer if separated in different sections.

Mean velocity profile formula derivation. In this reviewer opinion, this is a result. The proposal of the analytical function and exponents determination is well documented, with very good agreements, as indicated in Figure12. It would be interesting to see a quantification of the error in the predictions, as they seem to be very accurate with the comparison, in the different cases presented.

Analytical expression of the normalized wall vorticity Ω’(w) . This is a very accurate function, and again, in this reviewer opinion, the error (almost zero) would be interesting.

The dependence of the first vorticity derivative at the pipe axis on Re. This is a very interesting issue, and the correlation is very clear. Can be this extrapolated to low Reynolds Numbers?

The dependence of exponents K and N on the Re. This section is clear also. It would be interesting the comparison between these proposed results and others derived from references

Summary conclusion and future. In this reviewer opinion, conclusions should clearly propose the achievements and the degree of achievement of the objectives. In this reviewer opinion, no references should be included in the conclusions. Furthermore, the novelty of the research should be clearly stated in the conclusions.

Author Response

First of all, I thank you for your opinions and suggestions. I appreciate you spend a lot of time on the paper reading. All your suggestions are very valuable for me.

(Abstract. This is well presented, with the main objectives and achievements of this contribution, in this reviewer opinion, it is interesting to point out the novelty of the work, to aim readers to continue-)

I slightly extend the abstract. Line(8-22)

(Introduction. Introduction presents the main objectives and importance of deriving this analytical formula, even for numerical approaches. Nevertheless, in this reviewer opinion, it would be interesting to see a structure of Introduction, Materials and Methods, Analysis of results, and Conclusions. In this case, readers cannot identify when the introduction is finished and the apportions of authors related to methodology begins.)

I hope that the structure is not strict, because different problems need slightly different structure. I wanted the idea to fluently continue through all paper. If I distort it strictly to the given structure, then it will loss the idea continuity.  

(Backgrounds. This is also a part of the introduction, in this reviewer opinion, presenting main considerations about equations; This is well described, but in this reviewer opinion, as this is a well considered problem, some more references could be included in this section.)

Yes, it would be good, but I really do not find more references, which will be directly related to this problem. I did not say that they are not somewhere, but I did not find it. 

(Basic idea of velocity profile derivation. In this reviewer opinion, this is methodological and proposed by authors. This section is interesting and well described and it is important the proposal of the form of the chosen polynomial function (19). In this reviewer opinion, some more references giving validity to this proposal or indicating the novelty of this would be welcome.)

There is my older paper which deals with the problem of choossing of proper function for vorticity [8]. I also tested hyperbolical functions and the combination of polynomial and hyperbolical function, but it did not work well. I wrote this paper because I did not find anywhere similar method for mean velocity profile derivation based on the spatial vorticity distribution. I want to shear this idea with other researchers because there is a potential for more interesting derivations.

(Available conditions for free coefficients determining . Boundary conditions are clear and well presented. This section (as part of the methodology, in this reviewer opinion) is well described. Nevertheless, in this section, boundary condition mix with results, this could be clearer if separated in different sections.)

It is partially true. However, in the boundary conditions description there are some results which are also presented in other papers [8] but here are only slightly modified. I think that it is necessary to mention it here, because of the idea continuity. It is probably the problem with the average velocity radius. However, I describe it here to accomplish a list of all possible conditions. This condition was not used for derivation. It could be used for the derivation of exponents, but it is not described in this paper.

(Mean velocity profile formula derivation. In this reviewer opinion, this is a result. The proposal of the analytical function and exponents determination is well documented, with very good agreements, as indicated in Figure12. It would be interesting to see a quantification of the error in the predictions, as they seem to be very accurate with the comparison, in the different cases presented.)

Thank you for this recommendation. The average absolute difference percentage related to the average velocity is introduced.  (line 403-411, expression 64)

The section „Mean velocity profile prediction“ is added before the conclusion. line(527-558).

(Analytical expression of the normalized wall vorticity Ω’(w) . This is a very accurate function, and again, in this reviewer opinion, the error (almost zero) would be interesting.)

The values of Ω’(w) are rather big. It can be used similar expression as in the case of velocity profile, but in this case it has to be related to the maximal value of Ω’(w).

(The dependence of the first vorticity derivative at the pipe axis on Re. This is a very interesting issue, and the correlation is very clear. Can be this extrapolated to low Reynolds Numbers?)

We are preparing a project where we want to measure the velocity profiles by 2 components PIV and by Stereo PIV. We want to start with the Re 1000 up to 300 000.

(The dependence of exponents K and N on the Re. This section is clear also. It would be interesting the comparison between these proposed results and others derived from references)

It is possible to compare it with the power formula. There are other formulas in [2], but there are not enough information to be able to use this formula.

(Summary conclusion and future. In this reviewer opinion, conclusions should clearly propose the achievements and the degree of achievement of the objectives. In this reviewer opinion, no references should be included in the conclusions. Furthermore, the novelty of the research should be clearly stated in the conclusions.)

The Summary is modified (line 554-581). The section „Future work“ is added at the end. (line 588-602).

Reviewer 4 Report

Manuscript Number: water-1178989
Full title: Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution.
OVERVIEW
The manuscript introduces an analytical formula, relatively simple, where the free coefficients are obtained on the basis of boundary or other conditions which are depending on the Re. Using the formulation proposed by the author, the turbulent mean velocity profile is a function of the average velocity, the pipe radius R, and Re. 

1. The subject matter is actual, interesting and within the scope of the Journal Water. 
2. The title is appropriate and describes well the manuscript. 
3. The English is good but may be improved. 
4. The study does not bring breaking news about turbulent velocity profiles in pipes, but to adds some ideas and new views on the mean velocity profile.
5. It is my concern that all results are based on the experimental data PSP only. The authors should test and present the proposed methodology using data from a different experimental apparatus using water. 
6. As for the rest, I have some revisions to suggest.
In conclusion, I believe this manuscript is worthy of publication, after a major revision. 
SPECIFIC COMMENTS
Line 25: reads “The first aim is to supplement a research about the turbulent velocity profiles in pipes.”. It is absolutely redundant and by itself should not be pointed out as an objective of the research. 
Line 26: reads “Second aim is to point out the significance of the vorticity field in liquid flow.” The author should be more precise and introduce the objectives in line with the abstract. The first paragraph of the introduction should be reviewed. 
Line 38: reads “It is not so easy to find”. The English is not adequate to a scientific paper and should be reviewed. 
Line 38 reads “One very interesting paper with such experimental data is [3]. It is a very interesting paper …”. The “very interesting” is repeated to many times. The English should be reviewed. 
Line 40: where reads “These experimental data are known as Princeton superpipe data (PSP).” Later on line 248 reads “It was found, by analysis of PSP data, that there is a problem in the dependence on Re. The Reynolds number was increased by increasing of the density/pressure of the air in case of PSP measurements.” The PSP experimental apparatus and data should be better described in the paper. As it can be understood the fluid in the experiments is air, but it is not clear and explicit in the beginning of the manuscript. 
Line 87: reads “We have a situation”. The English should be reviewed not using we or I. 
Line 105: where reads “…whole cylinder and thickness…”, should read “…whole cylinder and the thickness…”.
Line 113: where reads “We assume that …”, the English should be reviewed not using we or I.
Line 116: where reads “If we want” …”, the English should be reviewed not using we or I. This situation occurs many time over the manuscript and should be reviewed. 
Line 222: where reads “Some models of the mean velocity profiles for turbulent flow”, should be more specific and refer the name of the most relevant models. 
Line 509: reads “Summary conclusion and future”, should be only Conclusions.
Line 536: The section ends with some concerns about the application to non-circular pipes. All the formulation in the paper is dependent on the radius and it is applicable only to circular pipes. The concerns about non-circular pipes should be removed or eventually go the section future work.

Author Response

First of all, I thank you for your opinions and suggestions. I appreciate you spend a lot of time on the paper reading. All your suggestions are very valuable for me.

(The manuscript introduces an analytical formula, relatively simple, where the free coefficients are obtained on the basis of boundary or other conditions which are depending on the Re. Using the formulation proposed by the author, the turbulent mean velocity profile is a function of the average velocity, the pipe radius R, and Re. 

1. The subject matter is actual, interesting and within the scope of the Journal Water. 
2. The title is appropriate and describes well the manuscript. 
3. The English is good but may be improved. 
4. The study does not bring breaking news about turbulent velocity profiles in pipes, but to adds some ideas and new views on the mean velocity profile.
5. It is my concern that all results are based on the experimental data PSP only. The authors should test and present the proposed methodology using data from a different experimental apparatus using water.)

Yes, it is true, that the mean velocity profile characteristics are derived from the PSP data only. That is the problem I mentioned in the introduction of the paper. It is not easy to gain this data somewhere. I am preparing a project where the velocity profile measurements will be done by 2D PIV and Stereo PIV (for comparison). Experiments will be done with the water and for two different pipe diameters.

(Line 25: reads “The first aim is to supplement a research about the turbulent velocity profiles in pipes.”. It is absolutely redundant and by itself should not be pointed out as an objective of the research.) 

I agree, it is changed. (line 28-30)

(Line 26: reads “Second aim is to point out the significance of the vorticity field in liquid flow.” The author should be more precise and introduce the objectives in line with the abstract. The first paragraph of the introduction should be reviewed.) 

It is done (lines 28-36)

(Line 38: reads “It is not so easy to find”. The English is not adequate to a scientific paper and should be reviewed. )

It is fixed (line 44)

(Line 38 reads “One very interesting paper with such experimental data is [3]. It is a very interesting paper …”. The “very interesting” is repeated to many times. The English should be reviewed.)

All of them are reviewed and changed.(whole paper)

(Line 40: where reads “These experimental data are known as Princeton superpipe data (PSP).” Later on line 248 reads “It was found, by analysis of PSP data, that there is a problem in the dependence on Re. The Reynolds number was increased by increasing of the density/pressure of the air in case of PSP measurements.” The PSP experimental apparatus and data should be better described in the paper. As it can be understood the fluid in the experiments is air, but it is not clear and explicit in the beginning of the manuscript.)

You are right, it was not clear. I modified the sentences. However, this experiment is completely described in [3]. I tis not a place here to describe it in detail. (Line 45-48).

(Line 87: reads “We have a situation”. The English should be reviewed not using we or I. )

It is fixed (line 94)

(Line 105: where reads “…whole cylinder and thickness…”, should read “…whole cylinder and the thickness…”.)

It is fixed (line 111)

(Line 113: where reads “We assume that …”, the English should be reviewed not using we or I.)

It is fixed (line 115)

(Line 116: where reads “If we want” …”, the English should be reviewed not using we or I. This situation occurs many time over the manuscript and should be reviewed.) 

It is fixed (line 119), whole paper)

(Line 222: where reads “Some models of the mean velocity profiles for turbulent flow”, should be more specific and refer the name of the most relevant models.)

It is fixed (line 227)

(Line 509: reads “Summary conclusion and future”, should be only Conclusions.)

It is fixed (line 558)

(Line 536: The section ends with some concerns about the application to non-circular pipes. All the formulation in the paper is dependent on the radius and it is applicable only to circular pipes. The concerns about non-circular pipes should be removed or eventually go the section future work.)

I added the section future work. (lines 586-601) This methodology can be really applied to the non-circular cross-sections, but it is true that the resulting expression will be different. The derivation procedure will be similar.

Reviewer 5 Report

The following comments are provided for the author as a reference to improve the manuscript.

1. Line 35: Please the discontinuity in the description: ”These are the discontinuity of the first derivative in the pipe axis…”. The first derivative of velocity profile in the pipe axis is continuous. I don’t know what you mean.
2. How to get Eq. (14) from Eq. (13)? I think the author needs to explain it clearly.
3. The right-hand sides of Eq. (14) and Eq. (15) are the same, but the left-hand sides are a little different. It is unfeasible.
4. Line 137: Velocity for the radius interval <0,R1> is v=γ1+γ2. It is not consistent with that in Figure 3.
5. Lines 173 and 174: “Ω=0 for 173 r=0. It means that A₍₀₎=0.” When r=0, A₍₀₎ is not necessarily equal to zero according to Eq. (18). So do even values of i. That is, A_(i) can be arbitrary for r=0.
6. Lines 177 and 178: To my knowledge, the definition of curvature radius is not the same as the second velocity profile derivative. Why did the author think they are the same?
7. Line 180: “The third derivate is also chosen not to be zero, what means that A₍₂₎≠0.” I think it is A₍₃₎≠0.
8. Lines 202 to 206: According to Line 174, A₍₀₎=0 for r=0, not on the wall(r = R).
9. Lines 222 and 223: I don’ t understand the description “Some models of the mean velocity profiles for turbulent flow have a problem with this condition [1, 2]. In their case, the Ωw has an infinite value.” Could you please explain it more clear?
10. Eq. (29) shows that the relation between the first vorticity derivative and average velocity is independent on the density. This point is different from the results as shown in Figure 7. Why?
11. In Figure 6: Why is the normalized vorticity not equal to zero when r=0? In this study, the author presented vorticity equal to zero when r=0. They are contradictory.

Similar trends happen in Figures 11 to 14.

12. Line 316: I don’t know the meaning of average velocity radius throughout the text. Please explain it.
13. Line 386: The author said “The vorticity at the wall Ω(w) is calculated from the pressure drop through the expression (24) or (26).” How did the author obtain the pressure drop measurements? No data is given. If the author used Eq. (26), then please give the value of f.
14. Lines 457 to 459: The words “underestimated” and “over estimated” are usually for the analytically calculated values based on the experimental data. Therefore, I suggest revising the descriptions for them.
15. The section from Lines 465 to 478: The expression (66) is a regression result for the experimental data so that the results in Figure 17 are in good agreement. Why did the author call it an analytical expression? Is the analytical expression the same as the analytical solution in your knowledge?
16. Again, why did the author call the regression result, Eq. (67), as analytical expression? In line 525, the author called them empirical expressions. I think it’s better.

Minor comments:

1. Line 15: “Reynold” should be “Reynolds”.
2. Line 16: “then” should be “than”.
3. Line 98: I think “ds_j” is better than “ds_i”.
4. Figure 4: Ω_w needs to be defined at its first appearance.
5. Line 204: I think “sections“ is better than “chapters”.
6. ? in Eq. (22) is undefined.
7. Line 330: The notation of radius is supposed to be ρ.
8. Some variables in Tables 1 and 2 have units. Please give them.
9. Figure 14 (b) is not vorticity distribution.
10. The notation ?? in Figure 15 (a) and (b) should be u⁺. And, y⁺ should be log y⁺ in Figure 15 (a).

Author Response

First of all, I thank you for your opinions and suggestions. I appreciate you spend a lot of time on the paper reading. All your suggestions are very valuable for me.

(Line 35: Please the discontinuity in the description: ”These are the discontinuity of the first derivative in the pipe axis…”. The first derivative of velocity profile in the pipe axis is continuous. I don’t know what you mean.)

I am trying to say that the power law formula for the mean velocity profile does not have zero the first derivative in the axis. This is not true. Real mean velocity profiles have always first derivative at the pipe axis zero. It is used as a condition. If the first derivative at the pipe axis is not zero, then it means that the profile has a sharp peak in the pipe center.

(How to get Eq. (14) from Eq. (13)? I think the author needs to explain it clearly.)

This solution is described in [7] in chapter 3.1, page 111. The exact derivation procedure will také about 3 or more pages. Therefore, it is better to refer to the literature.

(The right-hand sides of Eq. (14) and Eq. (15) are the same, but the left-hand sides are a little different. It is unfeasible.)

I am not sure what does it mean. The equations are centered in the table.

(Line 137: Velocity for the radius interval <0,R1> is v=γ1+γ2. It is not consistent with that in Figure 3.)

Yes, you are right. I fixed it. (line 142)

(Lines 173 and 174: “Ω=0 for 173 r=0. It means that A(0)=0.” When r=0, A(0) is not necessarily equal to zero according to Eq. (18). So do even values of i. That is, A(i) can be arbitrary for r=0.)

The Ω has to be zero at r=0 because of the continuous change of Ω across pipe axis. The vorticity is always zero in the axis. And on the basis of this fact the coefficient A₍₀₎ has to be zero, because r^0=1. When r is zero, then we obtain the equation Ω=A₍₀₎.r^0. When the Ω=0, then the A₍₀₎=0.

Equation (18) represents the general formula of the polynomial which has an infinite number of members. It has to be restricted. It is described in the next paragraph. Lines (171-187) After the restrictions we obtained equation (19) which is used for further processing.

(Lines 177 and 178: To my knowledge, the definition of curvature radius is not the same as the second velocity profile derivative. Why did the author think they are the same?)

Yes, it is not the same, but the radius of curvature depends on the reciprocal value of the second derivative. If the second derivative is small, then the curvature radius is big. In other words, the curvature radius is indirectly proportional to the second derivative.

I did not write it is a curvature radius, but it „determines“ the curvature radius. Line (181)

(Line 180: “The third derivate is also chosen not to be zero, what means that A(2)≠0.” I think it is A(3)≠0.)

Yes, you are right. I fixed it. (line 184)

(Lines 202 to 206: According to Line 174, A(0)=0 for r=0, not on the wall(r = R).)

Yes, you are completely right. The zero velocity at the wall is fulfilled naturally. It is apparent from the expression (43). I deleted this sentence. (line 210)

(Lines 222 and 223: I don’ t understand the description “Some models of the mean velocity profiles for turbulent flow have a proble m with this condition [1, 2]. In their case, the Ωw has an infinite value.” Could you please explain it more clear?)

It is the problem of power law formulas. They are widely used in the fundamental literature. If you make the derivative of these formulas, then they have infinite value at the wall. I slightly modified the sentence to point at the power law formulas directly. (line 227)

(Eq. (29) shows that the relation between the first vorticity derivative and average velocity is independent on the density. This point is different from the results as shown in Figure 7. Why?)

Expression (29) is derived for laminar flow and the results shown in Figure 7 are for turbulent flow.

(In Figure 6: Why is the normalized vorticity not equal to zero when r=0? In this study, the author presented vorticity equal to zero when r=0. They are contradictory.

Similar trends happen in Figures 11 to 14.)

It happens because the dashed line with circles represents the experimental data. This is a result of the measurement inaccuracy.

(Line 316: I don’t know the meaning of average velocity radius throughout the text. Please explain it.)

This is the radius, where the mean velocity magnitude equals the average velocity.

(Line 386: The author said “The vorticity at the wall Ω(w) is calculated from the pressure drop through the expression (24) or (26).” How did the author obtain the pressure drop measurements? No data is given. If the author used Eq. (26), then please give the value of f.)

This is a part of the PSP data set. When they presented the velocity profiles, they also presented the pressure drops measured for each velocity profile.

The friction factor f can be obtained, for example, from the Moody diagram, or from the empirical expressions for a given range of Re.

(Lines 457 to 459: The words “underestimated” and “over estimated” are usually for the analytically calculated values based on the experimental data. Therefore, I suggest revising the descriptions for them.)

I only want to say that the data obtained from the PSP data set does not correspond with to the data obtained from the velocity difference minimizing process for four parameters variations. It means that the wall vorticity obtained from the pressure drop measurement is smaller or bigger then the same data obtained from the minimization process.

(The section from Lines 465 to 478: The expression (66) is a regression result for the experimental data so that the results in Figure 17 are in good agreement. Why did the author call it an analytical expression? Is the analytical expression the same as the analytical solution in your knowledge?)

I am using the term „analytical“ from the mathematical point of view. However, I changed the the section title (line 483).

(Again, why did the author call the regression result, Eq. (67), as analytical expression? In line 525, the author called them empirical expressions. I think it’s better.)

Yes, you are right. I fixed it.

Round 2

Reviewer 3 Report

The present version of the document entitled “Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution” introduces the main aspects indicated in my previous review.

The novelty of the research has been pointed out along the paper, some appreciations about boundary conditions have been added and some considerations about Reynolds Number have been added, Conclusions and future work have been properly changed.

Reviewer 4 Report

The author improved the manuscript and replied to all the questions made in the first review report. In my opinion, the manuscript may be published as it is.

Reviewer 5 Report

No further comments.