Competing Lagrangians for incompressible and compressible viscous flow

A recently proposed variational principle with a discontinuous Lagrangian for viscous flow is reinterpreted against the background of stochastic variational descriptions of dissipative systems, underpinning its physical basis from a different viewpoint. It is shown that additional non-classical contributions to the friction force occurring in the momentum balance vanish by time averaging. Accordingly, the discontinuous Lagrangian can alternatively be understood from the standpoint of an analogous deterministic model for irreversible processes of stochastic character. A comparison is made with established stochastic variational descriptions and an alternative deterministic approach based on a first integral of Navier–Stokes equations is undertaken. The applicability of the discontinuous Lagrangian approach for different Reynolds number regimes is discussed considering the Kolmogorov time scale. A generalization for compressible flow is elaborated and its use demonstrated for damped sound waves.

They then show that the additional non-classical contributions vanish when time averaged, in the assumption that the phases of the thermal fluctuation vary much faster in time compared to the characteristic timescales of the flow. This is the only novel result presented, as far as I can see. Expanding on the validity of this assumption from the physical perspective of spatial scales would strengthen this work, as splitting eq. 2.7 just into fast and slow terms leads to the recovery of the classical NS in a seemingly trivial way. Basically, why do I need eq 2.7 in the first place if the flow (as in all scales up to the Kolmogorov scale) is always much slower than the phases of the thermal fluctuations? It would be interesting to list the spatial scale (compared to the Kolmogorov scale) at which the timescale of the non-equilibrium restoration force is order unity to the other terms.
The comparison with stochastic variational description is welcomed, but it's quite light.
While I enjoyed reading this articled, I find it to be more of a comment on their previous work than a standalone research article. The authors should greatly expand on the "deterministic versus stochastic approaches" aspect of the work. At this point, I do not recommend publication of this work.

Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? No

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Major revision is needed (please make suggestions in comments)

22-Jan-2018
Dear Dr Scholle: Manuscript ID RSOS-172136 entitled "Lagrangians for viscous flow: deterministic versus stochastic approaches" which you submitted to Royal Society Open Science, has been reviewed. The comments from reviewer(s) are included at the bottom of this letter.
In view of the criticisms of the reviewer(s), I must decline the manuscript for publication in Royal Society Open Science at this time. However, a new manuscript may be submitted which takes into consideration these comments.
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I look forward to a resubmission. I have now heard from the reviewers. Both indicate that whilst there is merit in the ideas presented in this paper, the material at hand would require a complete rework to warrant publication. Both referees give indication that the authors may want to consider if submitting a further paper based on their ideas.
Reviewer comments to Author: Reviewer: 1 Comments to the Author(s) The authors obtain an extended Navier-Stokes (NS) type equation, which reduces to the classical NS eq. in the appropriate limit, using a variational formalism and a proposed Lagrangian density. This is an approach that I found very interesting. However this was introduced by two of the authors in an article from 2017, in the same journal (Ref. 12).
They then show that the additional non-classical contributions vanish when time averaged, in the assumption that the phases of the thermal fluctuation vary much faster in time compared to the characteristic timescales of the flow. This is the only novel result presented, as far as I can see. Expanding on the validity of this assumption from the physical perspective of spatial scales would strengthen this work, as splitting eq. 2.7 just into fast and slow terms leads to the recovery of the classical NS in a seemingly trivial way. Basically, why do I need eq 2.7 in the first place if the flow (as in all scales up to the Kolmogorov scale) is always much slower than the phases of the thermal fluctuations? It would be interesting to list the spatial scale (compared to the Kolmogorov scale) at which the timescale of the non-equilibrium restoration force is order unity to the other terms.
The comparison with stochastic variational description is welcomed, but it's quite light.
While I enjoyed reading this articled, I find it to be more of a comment on their previous work than a standalone research article. The authors should greatly expand on the "deterministic versus stochastic approaches" aspect of the work. At this point, I do not recommend publication of this work.

Reviewer: 2
Comments to the Author(s) Attached.

Is the language acceptable? Yes
Is it clear how to access all supporting data? Not Applicable

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation? Accept as is
Comments to the Author(s) I have read the new version of the article and I have no detailed comments to make.
The article is now in a state that can stand on its own (this has nothing to do with the number of pages, just with the work presented in those pages) and I consider it's worth publication in its current form.
The impact of the non-equilibrium terms for different flow regimes may yield novel results or give trivial answers. The community at large will ultimately decide on the practical relevance of the approach. However, from a mathematical perspective, I fully agree with the authors on their point: "... the question is not if this equation is needed or not, but the question is if an alternative Lagrangian exists that provides an equation of motion without additional non-equilibrium terms." Decision letter (RSOS-181595.R0)

07-Dec-2018
Dear Dr Scholle, I am pleased to inform you that your manuscript entitled "Competing Lagrangians for incompressible and compressible viscous flow" is now accepted for publication in Royal Society Open Science.
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Articles are normally press released. For this to be effective we set an embargo on news coverage corresponding to the publication date of the article. We request that news media and the authors do not publish stories ahead of this embargo (when final version of the article is available). The article is now in a state that can stand on its own (this has nothing to do with the number of pages, just with the work presented in those pages) and I consider it's worth publication in its current form.
The impact of the non-equilibrium terms for different flow regimes may yield novel results or give trivial answers. The community at large will ultimately decide on the practical relevance of the approach. However, from a mathematical perspective, I fully agree with the authors on their point: "... the question is not if this equation is needed or not, but the question is if an alternative Lagrangian exists that provides an equation of motion without additional non-equilibrium terms." Follow Royal Society Publishing on Twitter: @RSocPublishing Follow Royal Society Publishing on Facebook: https://www.facebook.com/RoyalSocietyPublishing.FanPage/ Read Royal Society Publishing's blog: https://blogs.royalsociety.org/publishing/ The goal of the set of equation that this group investigates is to modify the usual incompressible viscous Navier-Stokes equations so that there is a second reservoir for the dissipated energy, is not strictly dissipative and might have a Hamiltonian formulation. They use Clebsch variables to represent where the energy removed by viscous effects goes and in this paper, they have added stochastic effects.
What is missing from the beginning of this paper is a physical motivation for introducing these equations, which seem to be a variation upon, or parameterisation of, the Navier-Stokes-Duhem equations, without any reference to what those are. From their earlier paper, they are in a contribution by Olsson that I don't have ready access to. Also from their earlier paper, the introduction of Clebsch variables might be relevant to a fluid with dislocations.
As far as this being an extension of true incompressible viscous Navier-Stokes equation, I would like to believe that an extension with a Stokes viscosity and a Hamiltonian formulation exists where the loss of kinetic energy through the viscous terms can be replaced by suitable additional physics.
Other than going back to the compressible Navier-Stokes equation, I don't believe there is and don't believe that adding a Clebsch pair, inspired by the Madelung transformation for quantum media, which can generate a phase velocity and even a pseudovorticity in the manner of Berry, plus some stochasticity, is the solution.
One fundamental problem with this approach is the question of helicity. What role helicity plays in turbulent flows has never been adequately addressed, except it is there in observed vortical structures and cannot be represented by a single Clebsch pair. Due to the extra potential needed to ensure incompressibility, the integrated helicity for a single Clebsch pair is identically zero. Which means for a physical fluid, multiple Clebsch pairs will be needed. If they pose their solutions as solutions for a Duhem liquid, which I have never hear of unless it is something like Silly Putty, I will listen. However, as currently posed, I am against publication.

Authors' note to the Editor
We are grateful to the editor for providing us with the opportunity to revise and resubmit our manuscript and thank the two referees for their very constructive and insightful comments, enabling us as carry out a complete revision of the work, both rigorous and thorough, leading to a far superior article via which the essence of our work is more clearly articulated.
We are confident, based on the thorough revision carried out, that the new version of the manuscript addresses the concerns raised by the referees in full, showing the connection to existing approaches/theories.

Authors' reply to the Referees
We would like to begin by thanking the referee for their frank and honest opinion as to the content of our manuscript and for taking the time to point us in the right direction as to its essential improvement.

Reviewer: 1
The authors obtain an extended Navier-Stokes (NS) type equation, which reduces to the classical NS eq. in the appropriate limit, using a variational formalism and a proposed Lagrangian density. This is an approach that I found very interesting. However this was introduced by two of the authors in an article from 2017, in the same journal (Ref. 12).
Reviewer's comment: They then show that the additional non-classical contributions vanish when time averaged, in the assumption that the phases of the thermal fluctuation vary much faster in time compared to the characteristic timescales of the flow. This is the only novel result presented, as far as I can see. Expanding on the validity of this assumption from the physical perspective of spatial scales would strengthen this work, as splitting eq. 2.7 just into fast and slow terms leads to the recovery of the classical NS in a seemingly trivial way.
Authors' reply: We thank the reviewer for these fruitful hints. We agree that spatial averaging would provide an interesting alternative to time averaging and would strengthen this work, while expecting the same result as in turbulence theory due to ergodic hypothesis. We started investigating in this variant but need more time to work it out. The main reason for this is that the discontinuities appearing in the theory become manifest as wave-like structures on a microscopic scale. By spatial averaging they become micro-solitons the detailed analysis of which is required before serious conclusions about their physical meaning can be drawn. The necessary studies of the solutions of the corresponding evolution equation for the thermal excitation will require some time, but we are confident to provide results in forthcoming papers. At least 1 Appendix B we have discussed the idea of spatial averaging in Sec. 2(c) and in the outlook section 5.
Reviewer's comment: Basically, why do I need eq 2.7 in the first place if the flow (as in all scales up to the Kolmogorov scale) is always much slower than the phases of the thermal fluctuations? It would be interesting to list the spatial scale (compared to the Kolmogorov scale) at which the timescale of the nonequilibrium restoration force is order unity to the other terms. The comparison with stochastic variational description is welcomed, but it's quite light.
Authors' reply: The equation of motion 2.9 (2.7 in the original manuscript) results from the Lagrangian 2.1 by variation, so the question is not if this equation is needed or not, but the question is if an alternative Lagrangian exists that provides an equation of motion without additional non-equilibrium terms. Based on prior works, from Millikan to our RSOS paper from 2017, no alternative Lagrangian delivering full Navier-Stokes equations (without introducing additional fields without physical meaning in the sense of a weak formulation) is available, apart from the one provided in the framework of the first integral approach in Sec. 2(d).
However, the question arises if the non-equilibrium terms occurring in the equation of motion are relevant in turbulent considering the time Kolmogorov scale τ η . At least, by 2π/τ η > ω 0 a rough criterium is given for a Reynolds number regime where the thermal fluctuations and the no-classical terms occurring in the equations of motion should be without any relevance. This opens the perspective to use the discontinuous Lagrangian approach directly without time averaging, making the implementation of it easier. Without any doubt it would have been more consequent and satisfying to list a separate spatial scale for comparison with the Kolmogorov scale. Unfortunately an application for funding a research project related to the application of our unconventional methods to turbulent flow has been refused by the DFG, so our research capacities are quite restricted and we are sorry that we can address your suggestion only in this minimalistic manner. At least our rough considerations has been added in a separate section 2(e), where also the use of the alternative approaches is discussed at different Re-regimes. We are thankful to the reviewer pointing on this relevant question.
Reviewer's comment: While I enjoyed reading this articled, I find it to be more of a comment on their previous work than a standalone research article. The authors should greatly expand on the "deterministic versus stochastic approaches" aspect of the work. At this point, I do not recommend publication of this work.