A case of strong nonlinearity: Intermittency in highly turbulent flows

Abstract

It has long been suspected that flows of incompressible fluids at large or infinite Reynolds number (namely at small or zero viscosity) may present finite time singularities. We review briefly the theoretical situation on this point. We discuss the effect of a small viscosity on the self-similar solution of the Euler equations for inviscid fluids. Then we show that single point records of velocity fluctuations in the Modane wind tunnel display correlations between large velocities and large accelerations in full agreement with scaling laws derived from Leray's equations (1934) for self-similar singular solutions of the fluid equations. Conversely those experimental velocity-acceleration correlations are contradictory to the Kolmogorov scaling laws.To cite this article: Y. Pomeau, M. Le Berre and T. Lehner, C. R. Mecanique -(2018). RésuméUn cas de forte nonlinéarité : l'intermittence en milieu turbulentà grand nombre de Reynolds. On pense depuis longtemps que lesécoulements fluides incompressiblesà grand, sinon infini, nombre de Reynolds présentent des singularités localisées en temps et en espace. Nousétudions l'effet d'une petite viscosité sur les solutions auto-semblables deséquations des fluides. Nous montrons ensuite que des enregistrements de fluctuations de vitesse dans la soufflerie de Modane présentent des corrélations entre grandes vitesses et grandes accélérations en accord complet avec les lois d'échelle déduites des solutions auto-similaires deséquations trouvées par Leray en 1934. En revanche ces corrélations sont en contradiction avec les lois d'échelle déduites de la théorie de Kolmogorov. Pour citer cet article : Y. Pomeau, M. Le Berre and T. Lehner, C. R. Mecanique -(2018). ForewordOver the years Pierre Coullet developed an outstanding research devoted to many aspects of nonlinearity in Science. With his very sure taste he chose topics with a deep geometrical underpinning, non linearity being only one element in the structure of the scientific question. In Fluid mechanics nonlinearity and geometry concur to bring forward difficult and fascinating questions. We think first to the transition to turbulence by cascade of period doubling, predicted almost simultaneously by Pierre Coullet and Charles Tresser [1] and by Mitch Feigenbaum [2]. This scenario of transition was observed slightly afterwards in fluid experiments at Ecole normale laboratory by Jean Maurer and Albert Libchaber [3]. The understanding of the transition to turbulence with a few degrees of freedom did not end research in fluid turbulence, a difficult field where real progress has always been very slow. The next step was the realization that the transition to turbulence in large systems with many degrees of freedom belongs to the class of directed percolation [4], but this cannot be seen as the end the story. A big open question remaining in fluid turbulence was raised in the 1949 paper by Batchelor and Townsend [5] where the authors discuss observations of large velocity fluctuations they attribute in, we believe, a not fully convi...

Scaling laws in turbulence

“…This large fluctuation of the ac-celeration is more than a mathematical bug evidenced by this calculation, it is also something that is seen, although quite indirectly, in records of velocity fluctuations in a high speed wind tunnel. There, contrary to what is implied by the estimate δ u ∼ (εδ r) 1/3 , the large fluctuations of acceleration do not happen all the time, but are strongly correlated to sparse large velocity fluctuations 7 . Such a correlation contradicts Kolmogorov scaling law because in this scaling law the parameter ε is just the product of the velocity and of the acceleration.…”

“…Let us recall briefly some points in favor of their existence in turbulent flows. By analyzing time records of the velocity in the big wind tunnel of Modane, it has been found 7,8 , we believe, convincing evidence that the occurrence of such singularities explains well the observed intermittency of the signal. Our proof is based on the following arguments.…”

“…where t * is the time of the singularity, and a is a positive exponent smaller than unity. We have shown 7 that a can take the values 1/2 and 3/5 corresponding respectively to self-similar solutions which conserve the circulation and the energy in the singular domain.…”

Wave-particle duality: tribute to the memory of Yves Couder

“…Even though Kelvin model of atom is completely out of favor in atomic physics, the (non trivial) Hicks equation keeps an interest in fluid mechanics, because its localized solutions make a seed for self-similar solutions of the Euler equations blowing-up in finite time. The existence of solutions of fluid equations blowing-up self-similarly in time was imagined first by Leray [8], and such solutions are associated [9] to the phenomenon of intermittency observed in fully developed turbulence.…”

Some topological aspects of fluid dynamics

“…This provides a possible scenario for an approach to the problem of intermittency (see § 11.2 above). An approach developing similar ideas has been recently proposed by Pomeau, Le Berre & Lehner (2019).…”