Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper presents a theoretical analysis and introduces the “Lyapunov function” which is aiming to describe the acoustic energy for finite amplitude pressure oscillations in reacting systems with a typical application in combustion chambers. A profound outcome of these phenomena is combustion instability which is quite common in low emission combustion chambers and high energy density rocket engines. These phenomena are the root cause of severe damages in combustion burners, therefore are today one of the leading challenges in developing and operating both aircraft and power-generation gas turbines.

The authors have presented in detail an analysis and the concept of the introduced function. However the paper is missing a specific application where this function would be used and produce results which then have to be evaluated.

Therefore the authors are encouraged to resubmit a modified version of their paper to the journal after this important addition.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

Thank you for review
We added Appendix B, which provides an example of the application of this function. The appendix contents are given in attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Thanks to authors for their contribution developping an energetic approach for a very difficult phenomena such as thermo-acoustic oscillations. I had another visualisation of this phenomena following Rayleigh's criteria concerning the phase angle between flame fluctuations wave and acoustic wave reflexion resultant in the same point of space. Rijke tube is another visual video helping to undertsand the effect of this phase angle which implies a positive source term in pressure fluctuations equation (Naviers-Stockes) if the phase angle is between -90° and 90°C (positive cosinus), incresaing by this way the acoustic power and vice-versa when the phase angle has a negative cosinus implying a sink term in pressure balance equation, deceasing the acoustic power. The approach here following Lyapunov  fromulation seems to be more suitable for finite element approach dealing with more mathematical concepts relatively to finite difference approach.
There some minor typographical errors:
1. 1st line of the abstract: definition instead of difinition
2. line 74: Putnam work is in refrence [13] and not in [3]
3. line 75: Carnot instead of Karnot
4. line 139: the expression reference should be (10) and not (14)?

Author Response

Thanks to review.
With all correction authors agree. All corrections were made.

Reviewer 3 Report

Comments and Suggestions for Authors

This paper proposed a physically based Lyapunov function to analyze stability in reacting flows. The study first proposed a simple model in a non-equilibrium closed system and then extended it to a moving system. The reviewer would like to recommend the paper be published if the following concerns can be addressed.

 

1. In 1. Introduction, can the authors explain what is a perspective type of fuel?

 

2. In the 2^nd paragraph of 1. Introduction, it is a little hard to understand the logic to introduce the method of vortex burning. Also, can the authors further explain why it is necessary to consider the total energy?

 

3. In 1. Introduction, can the authors further explain how the Lyapunov function is introduced? What’s the objective and motivation of this function when the authors said that “the definitions of stability criteria are usually produced from some adaptation of Lyapunov function, in the first work it was acoustic energy”?

 

4. Can the authors refine the last paragraph of 1. Introduction? For example, please list the objectives and tasks of this manuscript by using bullet points clearly if it is necessary. It sounds like the authors plan to introduce more detailed differences between this work and previous work after the authors said that “Previously this method was used in works [22 and 23], but definition of energy used in that works was only a first order polynomial …”.

 

5. In the 1^st paragraph of 4. Variational principle for study combustion instability, where is section 1? Is it 1. Introduction?

 

6. Is it possible that the authors add some toy problems (e.g., CFD, Exp. or simple artificial cases) to help readers understand how this method is applied to analyze stability in reacting flows quantitively and what’s the benefit of this novelty method?

Comments on the Quality of English Language

Minor editing of English language required.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have revised the manuscript according to the reviewer's comments. The reviewer would like to recommend it to be published in the present form.

Author Response

Thank you for appreciating our article

Author Response File: Author Response.pdf