Singularités dans les écoulements : une nouvelle façon de penser/ Turbulence and Singularities in turbulent flows: a new way of thinking
Séjour d’étude Christophe Josserand, Paul Clavin et Yves Pomeau (Laboratoire d’Hydrodynamique, Ladhyx, (CNRS UMR 7646), Ecole Polytechnique, 91128 Palaiseau, France), du 28 juin au 3 juillet 2021.
Cette rencontre à été consacrée à une question centrale pour la mécanique des milieux continus, soit l’importance des singularités à temps fini. Nous avons examiné les singularités de Leray dans les fluides turbulents et la façon de les mettre en évidence dans les enregistrements en un point unique et comment s’assurer que la dissipation se produit là. Un autre cas de comportement auto-semblable de solutions de problèmes complexes est le cas de la transition déflagration-détonation où la rétroaction entre l’écoulement de retour couplé à la non linéarité des équations de réaction-diffusion conduit génériquement à une singularité évoluant ensuite vers une détonation.
This meeting was devoted to a question central in continuum mechanics, namely the relevance of finite time singularities there. We looked at Leray-like singularities in turbulent fluids and how to put them in evidence with records of fluctuations at a single point and how to make it sure that dissipation takes place there. Another instance of self-similar behavior of solution of a complex problem is the deflagration to detonation transition where the feedback between the back flow coupled to the non linearity of the reaction diffusion equation leads generically to a singularity evolving later into a detonation.
The problem of turbulence in fluids remains largely unsolved after many years of efforts to solve it. By solving we mean deriving from the fundamental fluid equations a way to represent accurately and even make predictions for real turbulent flows, namely a way of closing the Reynolds equation (actually derived first by Boussinesq more than 15 years before Reynolds) to get a closed set of equations permitting to find the average velocity of a turbulent flow in a given geometry (for instance to find the structure of the turbulent wake of a blunt body moving quickly in a fluid at rest at infinity), a problem considered first by Isaac Newton in the Principia and whose solution has not progressed much in the 300 plus years since. One major question in turbulence theory is to understand how dissipation enters into the game. Actually, the fundamental equations for fluid motion at large speed are without dissipative term, which is formally negligible because the viscosity responsible of dissipation in fluids is very small. The traditional way to explain dissipation in turbulent fluids is by the idea of cascade, that is by a transfer of energy from big eddies to smaller eddies that transfer by a similar mechanism their energy toward even smaller eddies and so on down to eddies of size small enough to make viscosity significant. The scale where dissipation occurs is called the Kolmogorov scale and depends on the dissipated power in the turbulent flow. However, the very existence of this scale is based on dimensional arguments and so does not validate the cascade idea obviously. This cascade theory has been repeated again and again over the years, although it is not that clearly present in Kolmogorov original paper. It was introduced by Richardson before Kolmogorov. Over the years graphical pictures of the process have been given, but it is fair to say that this is not, by far, sufficient to determine if this theory is right or wrong. Before Kolmogorov the French mathematician Jean Leray suggested another way to dissipate energy in turbulent flows by remarking that the fluid equations may have solutions collapsing to a single point at a finite time. Contrary to the ones of the cascade theory, the mathematics of this collapse are clear and pose deep and well defined problems that have been solved recently only. As discussed at LesTreilles, there is also evidence from the measurements of velocity fluctuations at a single point that the large fluctuations (of the acceleration) known to occur since 1949 and called the phenomenon of intermittency are tightly linked to the occurrence of a Leray-like singularity near the point of measurement.
Putting in evidence such point wise singularities in space and time is hard because when one records, as standard in fluid mechanics, the velocity fluctuations at a well defined location in space, this makes a priori impossible to record directly such point wise event. The chance to get a given point in space on a line is zero a priori. However, by looking at the trace of the velocity fluctuations recorded in one dimension (that combines space and time, because of the mean velocity of the flow) one can spot singularities by large fluctuations in the records due to a singularity passing nearby. The occurence of such large fluctuations has been known for a long time and given the name intermittency, without identifying their cause. We made one step further and this has been a topic of lively exchanges at Les Treilles, by combining the weak probability of passing very close to a point in space and the opposite effect of the growth of the intensity of the fluctuation as one approach the singularity. Increasing the impact of the large fluctuation in the statistics by looking at increasing powers of this fluctuations, one gets a cross over between the low probability and the growth of the amplitude a bit related to one classical explanation of the phase transitions in some models of thermodynamics.
Another result that emanated from discussions at Les Treilles is a way to put in evidence that in singularities observed as just explained there is a peak of dissipation, which makes the occurrence of singularities crucial for explaining dissipation.
Paul Clavin introduced to us  is outstanding new results on the transition from deflagration to detonation in combustion of gas mixtures. This progress takes place after many years of discussions and uncertainties on the physical mechanism behind this transition, observed since many years. The usual explanation refers to a self sustained growth of the flame surface of a deflagration wave is contradicted by the occurrence of this transition in small pipes where the flow cannot be turbulent. The correct explanation is that there is a feedback between the fluid in the pipe at the tip of the flame and in the return side flow. Mathematically this yields a set of equations relating the dynamics of the chemical reaction and the fluid flow, with a finite time singularity of the self-similar type.
We had also lively discussion by Zoom with Sergio Rica, from Santiago in Chile who could not attend our meeting because of the very strict sanitary constrains in Chile. His contribution concerned the very existence of singular solution of the Leray type for the Euler equation. He managed to map this problem into a set of hyperbolic equation with explicit singular solution leading to a singularity slightly analogous of the Riemann solution in compressible flows. This amazing result opens the way to a fairly complete understanding of the occurrence of singularities in fluid flows.
Unfortunately, it remains very hard to put in evidence Leray-like singularities in numerical simulations of turbulent flows. We  devised a fully deterministic model, the focusing nonlinear Schrodinger equation where it is known that if the non linearity is strong enough the dynamics yield finite time singularities of the Leray-type. Thanks to its (relative) simplicity this model can be studied numerically with great accuracy and we have access to its statistical properties. Amazingly, and one should say even a bit unexpectedly, we found that in this model the short range structure function displays the same transition at increasing exponents as what we observed on the hot wire signal of turbulent flows. Amazingly again, this transition is perfectly coherent with a pair correlation of the Kolmogorov-Obukhov type. This reinforce us in the belief that the observation of a smooth spectrum of the pair correlations is not a proof of the existence of a cascade in the sense of Richardson, but is fully consistent instead with dissipation in singular Leray like singularities result of a coherent (and singular) evolution from large to small scales leading to strong intermittency.
Summarizing, we had a very fruitful, interesting and friendly meeting at Les Treilles in its unique setting for doing actual progress in science.
 P. Clavin, H. Tofaili « A one-dimensional model for deflagration to detonation transition on the tip of elongated flames in tubes » Combustion and Flame 232, 111522 (2021); P.Clavin, M. Champion, « Asymptotic solutions of two fundamental problems in gaseous detonations » Combustion Science and Technology (2021) to appear.
 C. Josserand, Y. Pomeau and S. Rica « Finite-time localized singularities as a mechanism for turbulent dissipation » Phys. Rev. Fluids 5, 054607 (2020), arXiv:1910.05523v1(cond-math.stat-mech)