Approximate Analytical Models of Shock-Wave Structure at Steady Mach Reflection
Round 1
Reviewer 1 Report
In the manuscript "Approximate analytical models of shock-wave structure
at steady Mach reflection" by Mikhail V. Chernyshov, Anna S. Kapralova
and Karina E. Savelova, a new complex analytical model for supersonic
flow with Mach reflection based on flow averaging downstream
curved Mach shock is worked out. Results are interesting and useful.
They are of interest for supersonic gas jet flow studies, for design
of convergent supersonic inlets, supersonic nozzles, prospective ramjets
and other aerospace facilities, as stated by authors. Article is well
written and results deserve to be published.
My remark are figure captions. They are very poor. Authors should
better explain the figures. Also, for example Fig. 4. they should
explain notation. E.g.: strength of the incident shock -->
strength of the incident shock (J1), and say what represent the numbers.
In the text, for example line 268: region BTH --> region BTH (see Fig...
some typos:
line 133: ant --> and
line 319: on form --> in form
Author Response
We are grateful to the Reviewer for high estimation of our study and also for useful remarks. We have read again the text of the paper and did our best to correct grammatical errors and typos, including that the Reviewer pointed. We also extended the figure captions, trying to explain every designation, and added several references to figures in the text of this paper (including places that the Reviewer marked).
Reviewer 2 Report
In the paper, a semi-analytic method to evaluate Mach shock height in planar steady supersonic flows is presented.
Results have been reported in a single case of overexpanded jet flow for flow Mach number M=5 and various angles of incident shock inclination.
The method could be useful for benchmarking fluid-dynamic codes, but in order to evaluate its validity, a larger number of simulations and comparisons should be made. Authors claim in abstract that methods of quick approximation have been developed since 1980 but a more precise comparison should be made with results obtained by using these methods.
The numerical procedure is sufficiently explained but a deeper discussion must be made on the advantages and the drawbacks of this semi-analytic and numerical procedure. Numerical instabilities, computation time and other numerical aspects must be given explaining the overall iterative procedure.
Finally, a wider discussion must be made in the introduction section on the real effect that can arise in shock waves (plasma formation for example), referring to recent advances on this topic.
Author Response
The main goal of this paper was to present new analytical method. The results of analysis of various flows with Mach reflection should be a subject of the following studies based on this publication and citing it.
Nevertheless, we added a comparison between our method, methods proposed earlier and experimental data concerned with flow in narrowing channel between two wedges (see Figure 5).
The comparison between our analytical model and preceding ones are done in text. For example,
“Comparison with the experimental results [12, 13] reveals large (up to 50-100%) errors in estimations provided by [41]” (page 7);
“Approximation [42] of flow in region II by Grib-Ryabinin method [50] based on Lagrangian tangent transformation complicated the mathematical model sufficiently, but difference between results of [41] and [42] is only 2-4% as it is demonstrated in [51], so the errors of Mach stem size estimation are basically preserved” (page 7);
“Results achieved by numerical method of characteristics for all supersonic flowfield [47] are presented at the last string of Table 1; they differ approximately on 0,5%. Authors also realized the second-order method of characteristics for the supersonic part of the flow. Results of comparison between authors’ numerical and analytical data are presented in Figure 4, and difference between them also does not exceed 1%. Consequently, proposing approximate analytical method has very large accuracy. For example, at the similar problem of Mach reflection in planar narrowing channel imitating the air inlet [41, 42] results of actual approximate analytical solutions differs from numerical and experimental data on 30-40% and even on 90%” (pages 13-14)
“Results shown in Figure 5 lie within measurement errors of the experimental data of [12] whilst the results of the analytical model [28] differ from them on 20-25%, and discrepancies of approximate models [41] and [42] are even more” (page 15)
Please see also comparison provided in Figure 5.
Speaking about numerical instabilities, computational time and other numerical aspects, it seemed to us excessive because we saw no numerical instabilities at reasonably given flow parameters. Also, it seems evident that it is 102-103 times more quickly to solve boundary-value problem for ordinary differential equation can be solved than the analogous problem for system of partial differential equation (the method of characteristics) or to calculate the whole flowfield by any shock-capturing method. We added the following phrases into the end of the subsection 2.7 (page 13):
“Our calculations demonstrated that there are no numerical instabilities at reasonably given flow parameters, at least, in overexpanded supersonic jet flow with Mach number M>2 when incident shock strength satisfies the inequality (8). Selection of Mach stem size with ordinary PC and chosen widespread software (MATLAB 2017) takes about one minute, while it takes several our to select Mach stem size applying the second-order method of characteristics at every step. Direct CFD calculation with well-known hydrocodes (for example, ANSYS Fluent applied in [32] to similar problem) takes about two hours at the same PC, but it does not allow estimate Mach stem size and shape of other gasodynamic discontinuities due to insufficient shock resolution of widely used shock-capturing methods.”
The proposed model does not include the real gas effects, but can be developed for it in the further studies. Authors work on taking into account, for example, impulse heat efflux downstream the Mach disc and decrease of gas “effective” adiabatic index. So we include the following text into the final part of the Introduction section (page 2):
“The proposed model does not take into account the real gas effects, impulse energy efflux due to combustion and detonation initialized by high temperature growth behind the Mach stem [18, 19], or, for example, plasma formation. Impulse energy influx can be included into this mathematical model applying ZDN (Zel’dovich– Döring – von Neumann) detonation model and corresponding relations for Mach shock wave with energy efflux [33, 34]. As preliminary results show, heat efflux can lead to formation of Mach reflection at situations where only regular reflection is theoretically possible in flows without chemical reactions. Heat efflux also leads to sufficient increase of Mach stem size. Real gas effects that result in decrease of the “effective” gas adiabatic index at Mach stem shifts the regular / Mach transition border at the same direction and also increases Mach stem height, but not so sufficiently. Ionization and plasma formation in similar flows with Mach reflection can be subjects of the further studies”
We plan to publish a paper on influence of heat efflux and real gas effects, maybe, in journal “Fluids” or any other journal this year (now we are close to finish of that manuscript).
Reviewer 3 Report
I liked the work, I recommend it for publication.
The main remark: the abstract, introduction and conclusion says about the comparison of the model results with experimental data. There is no such thing in the paper. There is a comparison with the method of characteristics (tab1 and fig.4), which itself is in good agreement with experiment. But then either it is necessary to write this way and remove the words about the experimental comparison, or to describe in more detail, or to present a comparison with the experiment.
Also, in the titles to Fig. 4 and Tab.1, it is necessary to indicate where the results of numerical modeling were taken from ([44]). Some readers are primarily interested in comparison, only then they look at the text.
I recommend making Fig. 1b clearer, it contains important symbols, but is blurry and pixelated.
It is necessary to reveal the abbreviation "HE" in line 23 (or remove and write in words).
Maybe it makes sense to redraw Fig. 1 for section 2.4 with new designations? Otherwise, some of them may cause confusion for the reader. Such as J2B,J5,tetta4,8... - eqs. (20), (21)... For some It is not so easy to understand what is meant.
Author Response
At the very first, we must thank the Reviewer for his attentive reading of our paper and good opinion.
The main goal of this paper was to present new analytical method. The results of analysis of various flows with Mach reflection should be a subject of the following studies based on this publication and citing it.
Nevertheless, we added a comparison between our method and experimental data concerned with flow in narrowing channel between two wedges (see Figure 5).
We also added two phrases to the beginning of “Results” section which mentioned the comparison of Table 1 with experimental data of [44] (now – [47]), and comparison of Figure 4 with authors’ own numerical experiment.
Figure 1,b is vectored graphic made in “Adobe Illustrator” program. We are sure that it should look better at pages of the journal.
Abbreviation “HE” (high explosives) is written in words.
We also greatly extended the figure captions, trying to explain every notation and symbol, and added several references to figures at the text of the paper. So we hope that there is no need to redraw Figure 1 with new designations at the end of the paper. Above all, we are afraid that that it can lead to misunderstanding rather than to explanation.
Round 2
Reviewer 2 Report
Authors addressed all the comments and improved the work, adding new comparisons and clarifying sentences.
Finally, they could add some references regarding the discussion of future developments in the case of real gas and chemical non-quilibrium plasma in supersonic or hypersonic regimes. As an example, works by
- Prakash, N. Parsons, et al., J. Comput. Phys. 230 (23) (2011) 8474–8507 or
- Pepe et al., Computer Physics Communicartion, volume 196, 2015, 179-193.
Author Response
Dear colleague,
we became familiar with both papers your recommended, and decided to include corresponding references on them both to the Introduction. Now there are References [33] and [34]. We hope that those studies can be useful for us in future for comparison of analytical and numerical results.
