Dynamic Triad Interactions and Non-equilibrium Turbulence

Abstract

The classical K41 theory, based on the ideas of Kolmogorov, Richardson and Batchelor, has in recent years with accumulating evidence become subject to increased scrutiny. We elaborate on the idea that the deficiencies of the theory originate in the fundamental assumption of universal equilibrium, which in turn is the result of the basic assumption of locality of nonlinear interactions. These very fundamental assumptions are argued to have no anchoring in the governing Navier-Stokes equation. The possibility for other kinds of solutions are discussed from a historical perspective. K41 is also identified to represent an equilibrium solution that cannot predict non-equilibrium turbulence. Why the need for new perspectives on the established theory?The theoretical framework developed by Kolmogorov (including the contributing ideas of Richardson and Batchelor, often collectively referred to as the 'K41' theory [1, 2, 3]), has comprised the cornerstone of our understanding of turbulence for more than half a century. It has up until recently not been so frequently acknowledged that Kolmogorov's ideas were initially heavily questioned. In fact, it took two decades before K41 was convincingly supported by experiments; first in a turbulent round jet by Gibson [4] and secondly (and more widely acknowledged) by Grant, Stewart and Moilliet [5]. Once these empirical studies supporting K41 finally appeared, the theory has been amply supported by further experiments and simulations in some flows, while other theories have been more successful in predicting non-K41 types of flows (c.f.

Triad interactions investigated by dual vortex shedding

“…However, none of the early proposed theories for spectral transfer derives directly from the widely accepted governing equation of motion for turbulence -the NSE. Recent experimental evidence, as also pointed out early by Kraichnan (1959), hints at the weakness of the Kolmogorov picture and its inability to describe non-equilibrium turbulence, see also George (2014) and Velte and Buchhave (2021). A significant problem with K41 is that it does not take into account, for example, strong interactions between very disparate scales as caused for example by large-scale stretching of small-scale vortex filaments and vortex sheets, Kraichnan (1974).…”

“…The existence of temporal and spatial windows, which are encountered in all practical measurements, implies that the interaction between Fourier components is allowed outside the strict deltafunction phase match condition, for example within a sinc 2shaped spectral window in case of a box car window in the physical domain. A detailed study of triad interactions as they appear in real experiments is treated in Buchhave and Velte (2023) (see also Velte and Buchhave (2021)).…”

Triad interactions investigated by dual vortex shedding

“…The existence of temporal and spatial windows, which are encountered in all practical measurements, implies that the interaction between Fourier components is allowed outside the strict delta-function phase match condition, for example within a sinc 2shaped spectral window in case of a box car window in the physical domain. A detailed study of triad interactions as they appear in real experiments is treated in [2] (see also [3]).…”

“…However, none of the early proposed theories for spectral transfer derives directly from the widely accepted governing equation of motion for turbulence -the NSE. Recent experimental evidence, as also pointed out early by [10], hints at the weakness of the Kolmogorov picture and its inability to describe non-equilibrium turbulence, see also [11] and [3]. A significant problem with K41 is that it does not take into account, for example, strong interactions between very disparate scales as caused for example by large-scale stretching of small-scale vortex filaments and vortex sheets, [12].…”