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Atmosphere
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Isotropic Turbulence: Recent Advances and Current Challenges
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Isotropic Turbulence: Recent Advances and Current Challenges

A special issue of Atmosphere (ISSN 2073-4433).

Deadline for manuscript submissions: closed (30 June 2024) | Viewed by 7089

Special Issue Editor

Prof. David McComb

E-Mail Website
Guest Editor
School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK
Interests: phenomenology of isotropic turbulence; statistical closure theory; renormalized perturbation theory; renormalization group; drag reduction

Special Issue Information

Dear Colleagues,

We have seen an enormous increase in the attention paid to isotropic turbulence in recent years. However, research in this field is divided into many different topics, each with its own group of adherents, which leads to a lack of informed consensus. In fact, it could be said that, when considered overall, the subject lacks both focus and direction. Of course, this is not really surprising, given the heterogeneous nature of the community studying it, which ranges from applied scientists and applied mathematicians to theoretical physicists and, more recently, pure mathematicians. Naturally, the topics studied exhibit a similar diversity, and, for this reason, we think that it will be helpful to divide them up into three broad strands. These are as follows:

  1. Fundamentals (thermodynamic aspects, validity of equations and the continuum limit, Taylor’s dissipation surrogate, Onsager’s conjecture and related physics, limit of infinite Reynolds numbers, Euler and Navier–Stokes equations driven by random inputs, and the relevance of intermittency).
  2. Phenomenology (conservation of energy, asymptotic behaviour with increasing Reynolds number, Kolmogorov (1941) theory, Onsager’s model and scale invariance of the energy flux, dimensionless dissipation, two-time correlations and temporal spectra, and finite Reynolds number effects).
  3. Statistical Theory (Eddy viscosity models, renormalized perturbation theories, Eulerian DIA and LET theories, Lagrangian theories, single-time Markovian models, mode elimination and sub-grid models, and renormalization group).

The aim of this Special Issue is to publish papers which can help to identify the most recent advances and what the current challenges are. Thus, we ask for submissions which have both a pedagogical and a review perspective. That is, they should be written to be intelligible to those outside your own subject area, and they should eschew experimental details and mathematical derivations alike, as these may be covered by references to your other published work.

Prof. David McComb
Guest Editor

Manuscript Submission Information

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Keywords

  • dissipation anomaly
  • Onsager’s conjecture
  • infinite Reynolds number limit
  • intermittency
  • scale-invariance of the inertial flux
  • temporal spectra
  • sweeping effects
  • finite Reynolds number effects
  • infrared divergence
  • Galilean invariance

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Published Papers (6 papers)

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Research

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16 pages, 910 KiB  
Article
Theory and Modelling of Isotropic Turbulence: From Incompressible through Increasingly Compressible Flows
by Claude Cambon
Atmosphere 2024, 15(8), 1000; https://doi.org/10.3390/atmos15081000 - 20 Aug 2024
Viewed by 852
Abstract
Homogeneous isotropic turbulence (HIT) has been a useful theoretical concept for more than fifty years of theory, modelling, and calculations. Some exact results are revisited in incompressible HIT, with special emphasis on the 4/5 Kolmogorov law. The finite Reynolds number effect (FRN), which [...] Read more.
Homogeneous isotropic turbulence (HIT) has been a useful theoretical concept for more than fifty years of theory, modelling, and calculations. Some exact results are revisited in incompressible HIT, with special emphasis on the 4/5 Kolmogorov law. The finite Reynolds number effect (FRN), which yields corrections to that law, is investigated, using both Kármán–Howarth-type equations and a statistical spectral closure of the Eddy-Damped Quasi-Normal Markovian (EDQNM)-type. This discussion offers an opportunity to give an extended review of such spectral closures, from weak turbulence, as in wave turbulence theory, to a strong one. Extensions of the 4/5 or 4/3 Kolmogorov/Monin laws to anisotropic cases, such as stably stratified and MHD turbulence, are briefly touched on. Before addressing more recent work on compressible isotropic turbulence, the simplest case of quasi-incompressible turbulence subjected to externally imposed isotropic compression or dilatation is presented. Rapid distortion theory is found to be a poor model in this isotropic case, in contrast with its relevance in strongly anisotropic flow cases. Accordingly, a fully nonlinear approach based on a rescaling of all fluctuating variables is used, in order to show its interplay with the linear operator. This opens the discussion on the cases of homogeneous incompressible turbulence, where RDT and nonlinear models are relevant, provided that anisotropy is accounted for. Finally, isotropic compressible flows of increasing complexity are considered. Recent studies using weak turbulence theory, modelling, and DNS are discussed. A final unpublished study involves interactions between the solenoidal mode, inherited from incompressible turbulence, and the acoustic and entropic modes, which are specific to the compressible problem. An approach to acoustic wave turbulence, with resonant triads, is revisited on this occasion. Full article
(This article belongs to the Special Issue Isotropic Turbulence: Recent Advances and Current Challenges)
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Review

Jump to: Research

26 pages, 603 KiB  
Review
Chaotic Measures as an Alternative to Spectral Measures for Analysing Turbulent Flow
by Richard D. J. G. Ho, Daniel Clark and Arjun Berera
Atmosphere 2024, 15(9), 1053; https://doi.org/10.3390/atmos15091053 - 30 Aug 2024
Viewed by 630
Abstract
Turbulence has associated chaotic features. In the past couple of decades, there has been growing interest in the study of these features as an alternative means of understanding turbulent systems. Our own input to this effort is in contributing to the initial studies [...] Read more.
Turbulence has associated chaotic features. In the past couple of decades, there has been growing interest in the study of these features as an alternative means of understanding turbulent systems. Our own input to this effort is in contributing to the initial studies of chaos in Eulerian flow using direct numerical simulation (DNS). In this review, we discuss the progress achieved in the turbulence community in understanding chaotic measures including our own work. A central relation between turbulence and chaos is one by Ruelle that connects the maximum Lyapunov exponent and the Reynolds number. The first DNS studies, ours amongst them, in obtaining this relation have shown the viability of chaotic simulation studies of Eulerian flow. Such chaotic measures and associated simulation methodology provides an alternative means to probe turbulent flow. Building on this, we analyze the finite-time Lyapunov exponent (FTLE) and study its fluctuations; we find that chaotic measures could be quantified accurately even at small simulation box sizes where for comparative sizes spectral measures would be inconclusive. We further highlight applications of chaotic measures in analyzing phase transition behavior in turbulent flow and two-dimensional thin-layer turbulent systems. This work shows that chaotic measures are an excellent tool that can be used alongside spectral measures in studying turbulent flow. Full article
(This article belongs to the Special Issue Isotropic Turbulence: Recent Advances and Current Challenges)
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82 pages, 17098 KiB  
Review
Statistical Dynamics and Subgrid Modelling of Turbulence: From Isotropic to Inhomogeneous
by Jorgen S. Frederiksen, Vassili Kitsios and Terence J. O’Kane
Atmosphere 2024, 15(8), 921; https://doi.org/10.3390/atmos15080921 - 31 Jul 2024
Cited by 1 | Viewed by 915
Abstract
Turbulence is the most important, ubiquitous, and difficult problem of classical physics. Feynman viewed it as essentially unsolved, without a rigorous mathematical basis to describe the statistical dynamics of this most complex of fluid motion. However, the paradigm shift came in 1959, with [...] Read more.
Turbulence is the most important, ubiquitous, and difficult problem of classical physics. Feynman viewed it as essentially unsolved, without a rigorous mathematical basis to describe the statistical dynamics of this most complex of fluid motion. However, the paradigm shift came in 1959, with the formulation of the Eulerian direct interaction approximation (DIA) closure by Kraichnan. It was based on renormalized perturbation theory, like quantum electrodynamics, and is a bare vertex theory that is manifestly realizable. Here, we review some of the subsequent exciting achievements in closure theory and subgrid modelling. We also document in some detail the progress that has been made in extending statistical dynamical turbulence theory to the real world of interactions with mean flows, waves and inhomogeneities such as topography. This includes numerically efficient inhomogeneous closures, like the realizable quasi-diagonal direct interaction approximation (QDIA), and even more efficient Markovian Inhomogeneous Closures (MICs). Recent developments include the formulation and testing of an eddy-damped Markovian anisotropic closure (EDMAC) that is realizable in interactions with transient waves but is as efficient as the eddy-damped quasi-normal Markovian (EDQNM). As well, a similarly efficient closure, the realizable eddy-damped Markovian inhomogeneous closure (EDMIC) has been developed. Moreover, we present subgrid models that cater for the complex interactions that occur in geophysical flows. Recent progress includes the determination of complete sets of subgrid terms for skilful large-eddy simulations of baroclinic inhomogeneous turbulent atmospheric and oceanic flows interacting with Rossby waves and topography. The success of these inhomogeneous closures has also led to further applications in data assimilation and ensemble prediction and generalization to quantum fields. Full article
(This article belongs to the Special Issue Isotropic Turbulence: Recent Advances and Current Challenges)
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12 pages, 317 KiB  
Review
Unsteady and Inhomogeneous Turbulent Fluctuations around Isotropic Equilibrium
by Wouter J. T. Bos
Atmosphere 2024, 15(5), 547; https://doi.org/10.3390/atmos15050547 - 29 Apr 2024
Cited by 1 | Viewed by 924
Abstract
Extracting statistics for turbulent flows directly from the Navier–Stokes equations poses a formidable challenge, particularly when dealing with unsteady or inhomogeneous flows. However, embracing Kolmogorov’s inertial range spectrum for isotropic turbulence as a dynamic equilibrium provides a conceptual starting point for perturbation theory. [...] Read more.
Extracting statistics for turbulent flows directly from the Navier–Stokes equations poses a formidable challenge, particularly when dealing with unsteady or inhomogeneous flows. However, embracing Kolmogorov’s inertial range spectrum for isotropic turbulence as a dynamic equilibrium provides a conceptual starting point for perturbation theory. We review theoretical results, combining perturbation approaches, and phenomenological turbulence closures, which allow us to gain valuable insights into the statistics of unsteady and inhomogeneous turbulence. Additionally, we extend the ideas to the case of the mixing of a passive scalar. Full article
(This article belongs to the Special Issue Isotropic Turbulence: Recent Advances and Current Challenges)
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18 pages, 882 KiB  
Review
Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence
by Shunlin Tang, Luminita Danaila and Robert Anthony Antonia
Atmosphere 2024, 15(5), 540; https://doi.org/10.3390/atmos15050540 - 28 Apr 2024
Cited by 1 | Viewed by 1236
Abstract
Since about 1997, the realisation that the finite Reynolds number (FRN) effect needs to be carefully taken into account when assessing the behaviour of small-scale statistics came to the fore. The FRN effect can be analysed either in the real domain or in [...] Read more.
Since about 1997, the realisation that the finite Reynolds number (FRN) effect needs to be carefully taken into account when assessing the behaviour of small-scale statistics came to the fore. The FRN effect can be analysed either in the real domain or in the spectral domain via the scale-by-scale energy budget equation or the transport equation for the energy spectrum. This analysis indicates that the inertial range (IR) is established only when the Taylor microscale Reynolds number Reλ is infinitely large, thus raising doubts about published power-law exponents at finite values of Reλ, for either the second-order velocity structure function (δu)2¯ or the energy spectrum. Here, we focus on the transport equation of (δu)2¯ in decaying grid turbulence, which represents a close approximation to homogeneous isotropic turbulence. The effect on the small-scales of the large-scale forcing term associated with the streamwise advection decreases as Reλ increases and finally disappears when Reλ is sufficiently large. An approach based on the dual scaling of (δu)2¯, i.e., a scaling based on the Kolmogorov scales (when the separation r is small) and another based on the integral scales (when r is large), yields (δu)2¯∼r2/3 when Reλ is infinitely large. This approach also yields (δu)n¯∼rn/3 when Reλ is infinitely large. These results seem to be supported by the trend, as Reλ increases, of available experimental data. Overall, the results for decaying grid turbulence strongly suggest that a tendency towards the predictions of K41 cannot be dismissed at least at Reynolds numbers which are currently beyond the reach of experiments and direct numerical simulations. Full article
(This article belongs to the Special Issue Isotropic Turbulence: Recent Advances and Current Challenges)
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31 pages, 6490 KiB  
Review
Some Early Studies of Isotropic Turbulence: A Review
by John Z. Shi
Atmosphere 2024, 15(4), 494; https://doi.org/10.3390/atmos15040494 - 17 Apr 2024
Viewed by 1278
Abstract
A re-examination of some early classic turbulence literature, mainly of isotropic turbulence, is given in this selective review. Some early studies, including original concepts and points, are reviewed or highlighted. Two earliest studies and six major original concepts are found: (i) Lord Kelvin’s [...] Read more.
A re-examination of some early classic turbulence literature, mainly of isotropic turbulence, is given in this selective review. Some early studies, including original concepts and points, are reviewed or highlighted. Two earliest studies and six major original concepts are found: (i) Lord Kelvin’s pioneering elementary studies of homogeneous, isotropic turbulence; (ii) Kelvin’s early introduction of Fourier Principles into turbulence studies; (iii) the Kelvin elementary concept of the direct energy cascade; (iv) the Kelvin early concept of the symmetry of turbulence; (v) the Taylor concept of the coefficient of eddy viscosity; (vi) the Taylor concept of the ‘age’ of the eddy; (vii) the Taylor–Fage–Townend concept of small eddies or microturbulence or small scale turbulence; and (viii) the Obukhov concept of a function of the inner Reynolds number (i.e., Re dependent coefficient) in both the balance equation and the energy distribution equation (the two-thirds law). Both Kelvin and Taylor should be regarded as the co-founders of the statistical theory of homogeneous, isotropic turbulence. The notion, ‘the Maxwell–Reynolds decomposition of turbulent flow velocity’, should be used. The Kolmogorov–Obukhov scaling laws are reviewed in detail. Heisenberg’s inverse seventh power spectrum is briefly reviewed. The implications or significances of these early studies, original concepts and points are briefly discussed, with special reference to their possible links with modern approaches and theories. Full article
(This article belongs to the Special Issue Isotropic Turbulence: Recent Advances and Current Challenges)
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