Sonic Eddy Model of the Turbulent Boundary Layer

Round 1

Reviewer 1 Report

Please find attached the comments.

Comments for author File: Comments.pdf

Author Response

The authors thank the reviewer for the thorough and detailed comments.

1.  The scales are logarithmic, as indicated in the text.  Since the transitions between regimes depend on the Reynolds number, it is difficult to provide a universal scale to the abscissa.  Just the scales are logarithmic.  So the transitions occur when M = Re^0.5 and M = Re.  The text has been modified to clarify this.
2. The details of the model for M greater than the square root of the Reynolds number are described in the original sonic eddy paper.  Such a Mach number is extremely large for turbulence at high Reynolds number.  The authors are unaware of any experiments or simulations in that domain.  The values of the ordinate at the transitions depend on the Reynolds number.
3. I am not quite sure how to change the fonts in the equations.  Perhaps the editor will be able to make a suggestion.  The subscript "i" refers to the incompressible value, as described in the text.  
4. The text has been revised to clarify this.
5. Equations (1-4) were used to reach Equation (5), with the substitution of the local speed of sound for the velocity difference, as described in the text.  For compressible flow, the sonic eddy becomes the most important, rather than the largest eddy.  Thus the local speed of sound replaces the velocity difference across the largest eddies in the equations.
6. In general, they are different constants.  This follows longstanding nomenclature, see for example Liepmann & Roshko (1957).  Since the particular values of the constants are not material to the central point of the manuscript, we would prefer not to distract the reader with unnecessary details.
7. An excellent question!  Of course, heat transfer would modify the temperature profile and thus the speed of sound.  This would change the size of the sonic eddy and hence the rate that it transports momentum.  Figures 2 and 3 in the revised manuscript illustrate the large effect of heat transfer on the skin friction and velocity fluctuations, as you point out.  
8. Two new figures have been added to address this.
9. The manuscript has been revised to clarify the subtle point that the van Driest-Morkovin density scaling should always work, even if the underlying physics is actually controlled by acoustic signaling.  According to the model, acoustic signaling controls the momentum transport and the Reynolds stress.  The latter is the density times the velocity correlation u'v'.  Thus the density field is a consequence of the acoustic signaling. 
10. Table 2 of that reference specifies the input parameter values of the test cases they studied.  It does give any information on the growth rate. 
11. The manuscript has been revised to clarify this.
12. The paragraph has been deleted.
13. Corrected.
14. Another good question.  Without knowing the local speed of sound, it is not possible to calculate the exact number.  However, the trend is in the right direction.
15. The incorrect reference has been revised.
16. The incorrect reference has been revised.
17. The manuscript has been revised to clarify that the discussion concerned the large-scale structure of the turbulence.
18. The van Driest-Morkovin view is that Mach number affects turbulence directly through the density.  If true, there would be a large effect on the incompressible boundary layer with a large density ratio.  The model predicts a very weak effect, along the lines of the Dimotakis model for density effects.  The manuscript has been revised to clarify this point.
19. The manuscript has been revised to clarify the argument.  In a nutshell, an eddy is expected to have the same transport properties no matter the proximity to a wall.  If that wasn't the case, then one is forced to address awkward questions:  At what distance from a wall does an eddy become sensitive to the density gradient?  How does the eddy know this? 

Reviewer 2 Report

In the current work, fundamentals of fluid mechanics pertaining to the sonic eddy model of the turbulent boundary layer are discussed in an elegant manner and I suggest it for publication after implementing the below things in the manuscript. The current work presents a heuristic model for compressible turbulence that assumes that Mach number effects on turbulence are solely attributable to acoustic signaling rather than to changes in density and this theory contrasts earlier theories that are based state that density changes are important. I am surprised, why plots are not included for such a nice discussion. The inclusion of results will make the paper interesting for readers.

1. Kindly include plots (xy-plots) from the published literature that demonstrate below statements mentioned in the abstract.
2. the skin friction coefficient should go as the inverse Mach number in a regime where the Mach number is larger than unity but smaller than the square root of the Reynolds number.
3. The velocity fluctuations normalized by the friction velocity should go as the inverse square root of the Mach number in the same regime. In contrast,

• The convention van Driest/Morkovin assumption is the Mach number effects are due to density variations.

2. Some of the important works of Prof. Pino Martin in the area of DNS of the turbulent boundary layer can be refereed in the introduction part of the manuscript.

https://aero.umd.edu/clark/faculty/51/Pino-Martin

4. Line 40, demonstrates with pictorial diagram reasons mentioned for the statement ‘fluid with a relatively low speed of sound compared to the speed of the turbulent motions’.
5. Line 59, ‘the sonic eddy determines the rates of entrainment and momentum transport’ Can you explain the implication of subsonic and supersonic eddy on the rates of entrainment and momentum transport in compressible flow regime is.

5. Line 85 and line 140: The effect of Mach number shows that the spreading angle of the free shear layer decreases with the increase of Mach number. What happens to the boundary layer thickness in the wall shear layer flows with an increase in Mach number?

6. Line 147: Correct typo error: [17}.

7. Include ‘In figure 2 of [17]’ in the current manuscript.

8. Line 167: Campbell (private communication), include citation of it in ref. section

9. Line 192: The authors can cite and write 1 small paragraph, about important outcomes, of recent works in the area of supersonic turbulent boundary layers in the current manuscript. Especially the effects of Mach number on the shear stress and skin friction can be discussed.

Pasha, A.A., Reddy, D.S.K., Abdulla, M.M. et al. Numerical Analysis to Evaluate the Effect of Wall Temperature on Skin Friction and Stanton Number for Turbulent Flows over a Flat Plate from Mach 2–8. Arab J Sci Eng (2021). https://doi.org/10.1007/s13369-021-06170-w

Comments for author File: Comments.pdf

Author Response

The authors thank the reviewer for many useful comments.

1. Two figures have been added.
2. The discussion has been modified to point out the particularly informative DNS results from Professor Martin's group.  Since the Introduction addresses only the theory, it was thought most appropriate to refer to her papers in the discussion section.
3. [missing?]
4. The eddy Mach number is defined to be the rotational speed divided by the local speed of sound.  It is not clear how to illustrate this in a diagram.
5. (a)  In general, the largest active eddy controls the transport rates of mass, momentum, and energy.  In a compressible flow, the finite acoustic signaling speed limits the size of the largest active eddy to one with an eddy Mach number of unity, the sonic eddy.  This text has been added to the Conclusions.  (b)  A paragraph describing unpublished data on boundary layer growth rates from Roshko and Theme has been clarified.
6. Corrected.
7. This paragraph has been deleted.
8. Corrected.
9. Thank you so much for this reference, of which we were not aware.  We have requested a scan from our library, but it has not yet arrived.  Unfortunately, the editor has set today as a deadline for our revisions.  We are reluctant to cite a reference before reading it.  The manuscript has been edited throughout to try to make it more clear, in particular the suggestion that a new experiment or simulation could resolve the central question raised here.

Reviewer 3 Report

The paper considers an original approach to the effect of compressibility on the characteristics of a turbulent boundary layer. Instead of a density correction, it is proposed to use a correction associated with limiting by acoustic signaling. The paper is based on the theory presented by one of the authors of this work about 30 years ago.
The positive side of the work is the presentation of a new perspective on the theory of a compressible turbulent boundary layer. 
But the equations obtained in the article are not strictly confirmed. In the "discussion" chapter some of the conclusions are not reasoned well enough.
For example, in paper refers to the results of numerical work [24] (Duan, L.; Beekman, I; Martin, M.P., Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number). From the point of view of the authors paper [24] confirms their theory. On the contrary from the point of view of the reviewer reference [24] demonstrates good agreement with the theory that takes into account the density correction (van Driest/Morkovin). For example Figures 6 and 7 show good data agreement over a wide range of Mach numbers. The reviewer also did not find a significant change in the topology of vortex structures (Figures 24 and 25). Perhaps the reviewer is misinterpreting the results?
From the point of view of the reviewer, in order to avoid inaccuracies in interpretation, it is necessary to add in paper some graphic information with confirmation of the dependencies obtained by the theory proposed by the authors of this article. Based on textual information alone it is difficult to draw conclusions about the usefulness of the proposed theory.
Many numerical and experimental studies of a compressible turbulent boundary layer have been carried out. Therefore the author of the paper should not have any problems with the quality confirmation of his theory.

Author Response

The authors thank the reviewer for useful comments.  Two figures have been added to compare the model predictions with experiment and computer simulations, and the text has been modified to discuss them.  The figures show that the Van Driest-Morkovin density scaling correlates well with the best DNS results.  The model less so. 

However, there are two points to make.  First, the freestream speed of sound rather than the more appropriate local speed of sound was used for the normalization.  The model may perform better using the correct speed of sound.  A second, subtle point is that the Van Driest-Morkovin density scaling should always correlate well with the results, even if the physics of the model is right.  The model asserts that acoustic signaling controls the transport of momentum, which is of course reflected in the Reynolds stresses and hence the density field.  The text of the manuscript has been modified to try to clarify these points.

The authors did not quite understand the reviewer's comment about topology.  In marked contrast to the case of the free shear layer, the large-scale structure of the boundary layer is already three dimensional, even at M=0.  So the model has nothing useful to say about topology changes in the boundary layer.  But it does address the question of correlation length.  The authors are indebted to the reviewer for the powerful suggestion to revisit Figures 24 and 25 of Duan et al. (2011).  It reminds us to note the changes in the isocorrelation length scale in the manuscript.  As the Mach number increases, the size of the sonic eddy decreases.  Happily, this is reflected in the reduced isocorrelation lengths in Duan.  The manuscript has been so modified.

Reviewer 4 Report

Summary of the manuscript:  This short communication proposes a model to characterize the effect of Mach number on momentum transport in a boundary layer by using the same physical arguments as used for free shear flows. The main idea is that the eddy must communicate with itself during a rotation in order for it to transport momentum. The main contribution of this work is that it tries to clarify the physical mechanism that drives the effect of Mach number on turbulent boundary layer. The conventional models attribute the effect of Mach number on the boundary layer to density ratio effects. However, the authors argue that the density ratio effects are known to be relatively weak in free shear layer and thus it must apply to all turbulent vortices, including those in wall flows.  

Comments:

The physical significance and the accuracy of the proposed model is not clear. The authors say that the proposed model will help explain the effect of Mach number on the skin friction and velocity fluctuations of the turbulent boundary layer. Then they introduce 'eddy Mach number' which is the eddy's rotational speed divided by the local average speed of sound, dropping the factor of pi in the interest of simplicity. However, pi is an important multiplier that will affect their analysis, as the basis of their argument is that the rotation speed of the eddy is too fast for eddy Mach number greater than 1 for the signal propagation to take place across the diameter. If we do not drop pi from the expression then this get affected by a factor >3, which is quite significant. The authors must provide more evidence why this wont effect their model and the parallels they draw with previous studies in discussions without providing any quantifiable information.  

All the subsections under discussions cite previous works and use them to loosely justify the efficacy of the model. However, in none of these sub-sections, i.e., skin friction, velocity fluctuations, growth rate, eddy celerity and topology, do the authors quantify the agreement with the experimental data. The authors rely on generic statements such as 'skin friction data go approximately inversely' (line 131), 'all that can be said is that the general trend is broadly consistent with the model' (line 137), 'number reasonable close to the value of unity...'(line 151). 'If this interpretation is correct, then it is consistent with the model' (line 162), etc. The authors need to better articulate these comparisons in quantifiable terms so that the reader can see this agreement as proposed by the authors.  

Line 150 – the authors say that 0.6 is reasonably close to 1. This needs to be better explained. What is the limit under which we can say that it is reasonably close to Mach number of unity? Can we say 0.5 is close to unity, or anything >0.3 is reasonably close?  And will this change if we take into account pi which we dropped earlier along with other simplifications? 

Line 169-174 – the authors propose the use of local speed of sound for normalization which as per their argument will bring down this number, however this decrease is neither quantified nor justified as the basis of their validation of the model.  

Statements made on lines 177 and 178 seem unrelated so it's not clear why the authors have used 'However' to connect the two statements.  

All the citations in the manuscript are > 25 years old, except for an abstract from author's own presentation and two articles on the same numerical study of hypersonic turbulent boundary layers. This numerical study is said to be in approximate accord but the agreement with the numerical data is never quantified in the manuscript.  

The use of personal communications as citations robs the reviewers and reader of the chance of verifying the stated facts and statements. However, I understand that certain journals do allow such citations and I would leave that decision to the editor. However, if the authors do intend to use such citations, they should be consistent. Its not clear why the personal communication cited on line 167 is different from the ones on line 25 and 67.  

Author Response

The authors thank the reviewer for many useful comments.

The manuscript has been extensively revised.  Two figures were added to compare model predictions with measurements and numerical simulations, along with a number of additional references.

The factor of pi was suppressed for both simplicity and because of the uncertainty in the rotation angle required for an eddy to transport momentum.  We don't that angle.  It might be closer to pi than 2 pi.  For example, in the free shear layer, we know that some fraction of the pure entrained fluid makes it all the way across the layer before mixing, but none makes it back again.  The vortex rotation angle at which an arbitrary fluid element is mixed is probably best expressed as a probability distribution, with a peak roughly around 180 degrees or less.  Of course, this example is about mass mixing rather than momentum transport, but the same basic idea may apply.  In light of our uncertainty about this angle, it would be inconsistent to over-specify its precision and therefore the precision of the definition of eddy Mach number.  In addition, there are large uncertainties in the local speed of sound.  The text of the manuscript has been revised to clarify this point.

All personal communications are now in a consistent format within the references. 

Round 2

Reviewer 1 Report

Thank you for the corrections and your answers to my questions.  The second sentence under the subsection ``growth rate'' on page 5, line 183, should be modified in a way to become clear to the reader that you are talking about the growth rate and not the boundary layer thickness itself.

Additionally, I think it is worth it if you get help from the editors to make the variables in the equations the same.  Usually, different fonts and appearances of a letter mean that they are not the same variable.

Reviewer 2 Report

The authors have addressed major issues and have restrictions in answering or addressing other queries. I accept the changes and recommend publishing.

Reviewer 4 Report

 The changes made by the authors address the comments in my review.