The turbulent cascade in five dimensions

Abstract

To the naked eye, turbulent flows exhibit whirls of many different sizes. To each size, or scale, corresponds a fraction of the total energy resulting from a cascade in five dimensions: scale, time and three-dimensional space. Understanding this process is critical to strategies for modeling geophysical and industrial flows. By tracking the flow regions containing energy in different scales, we have detected the statistical predominance of a cross-scale link whereby fluid lumps of energy at scale ∆ appear within lumps of scale 2∆ and die within those of scale ∆/2. Our approach uncovers the energy cascade in a simple water-like fluid, offering insights for turbulence models while paving the way for similar analyses in conducting fluids, quantum fluids and plasmas.Perhaps no other area of physics research has borne the influence of a rhyming verse more than turbulence, where Richardsons "Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity" (1) is embedded in the seminal theory of Kolmogorov, Onsager, von Weizscker and Heisenberg (2,3,4,5). The last three physicists transcribed the phenomenology in terms of wave numbers, which were to become the predominant tool in theoretical studies of the energy cascade (6,7,8,9,10). Consequently, scale and wavenumber became almost interchangeable concepts. A crucial point in the development of theories was the scale locality of the cascade, understood in terms of how close wavenumbers are when energy is exchanged between them (11). Since the advent of computer simulations, the locality of these wavenumber interactions has been controversial, with studies claiming evidence in favor of (12) or against (13) it. Rigorous explanations proposed for these discrepancies (14, 15) advocate for the classic scale-local view of the cascade. The debate, however, has turned predominantly around the equivalence between wavenumber and scale, ruling out any possibility of attributing the ongoing cascade to specific whirls visible where the flow actually evolves: the real space. Furthermore, computer simulations of industrial and atmospheric flows are carried out on numerical grids representing physical space and rely heavily on the modeling Studies of the interscale energy transfer based on real-space quantities share one of two limitations. They either focus on a subset of the source or sink terms responsible for the changes in energy at a point (17, 18), or they make no use of time thus precluding any dynamical information or knowledge of causality (19,20,21,22,23). Often both limitations are combined. A noteworthy exception found a delay in the peak of the correlation between energy at two different scales when following the larger-scale flow (24), suggesting that eddy structures transfer their energy to smaller scales. In the wake of that study, we aimed to follow individual eddy structures. This has become possible with modern data-storage facilities where flow simulations are preserved in a movie-like manner. Such data sets...

The coherent structure of the kinetic energy transfer in shear turbulence

“…Finally, the characterisation of the cascade presented in this study is static and do not contain dynamic information on how the kinetic energy is transferred in time between hairpins at different scales. Therefore, our work is just the starting point for future investigations and opens new venues for the analysis of the time-resolved structure of the energy cascade (Cardesa et al 2017…”

“…Turbulence exhibits a wide range of flow scales, whose interactions are far from understood (Cardesa et al 2017). These interactions are responsible for the cascading of kinetic energy from large eddies to the smallest eddies, where the energy is finally dissipated (Richardson 1922;Obukhov 1941;Kolmogorov 1941Kolmogorov , 1962Aoyama et al 2005;Falkovich 2009).…”

Lagrangian statistics of pressure fluctuation events in homogeneous isotropic turbulence

“…Two values were considered: Re λ = 150, obtained from the database maintained by J. Jiménez and coworkers at Univ. Politécnica Madrid 16 , and Re λ = 418, obtained from the Johns Hopkins Turbulence Databases [17][18][19] . Notice that, under this scaling, the Kolmogorov scales 20 are given by…”

Machine-aided turbulence theory

“…The structural view of turbulent shear flows has recently been reviewed by Brown & Roshko (2012) away from walls, and by Jiménez (2018) near them. Curiously, the structures of the nominally simpler case of homogeneous turbulence have been investigated less, and, while vortices have been considered often (Vincent & Meneguzzi, 1991;Jiménez et al, 1993), there is clear evidence of larger kinetic-energy structures (Cardesa et al, 2017) which are not necessarily related to the vorticity.…”