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Cascades and Transitions in Turbulent Flows

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Cascades and Transitions in Turbulent Flows Cascades and transitions in turbulent flows1 A. Alexakisa, L. Biferaleb a Laboratoire de physique statistique, D´epartementde physique de l’ Ecole´ normale sup´erieure, PSL Research University, Universit´eParis Diderot, Sorbonne Paris Cit´e,Sorbonne Universit´es,UPMC Univ. Paris 06, CNRS, 75005 Paris, France b Department of Physics and INFN University of Rome ’Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Rome, Italy Abstract Turbulent flows are characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal three and two dimensional homo- geneous and isotropic cases characterized by the presence of a strictly direct and inverse energy cascade, respectively. New phenomena appear that alter the global and local trans- fer properties. In this review, we provide a critical summary of historical and recent works from a unified point of view and we present a classification of all known transfer mecha- nisms. Beside the classical cases of direct and inverse energy cascades, the different sce- narios include: split cascades for which an invariant flows both to small and large scales si- multaneously, multiple/dual cascades of different quantities, bi-directional cascades where direct and inverse transfers of the same invariant coexist in the same scale-range and finally equilibrium states where no cascades are present, including the case when a large scale condensate is formed. We classify all possible transitions from one scenario to another as the control parameters are changed and we analyse when and why different configurations are observed. Our discussion is based on a set of paradigmatic applications: helical tur- bulence, rotating and/or stratified flows, magnetohydrodynamics (MHD) turbulence, and passive/active scalars where the transfer properties are altered as one changes the embed- ding dimensions, the thickness of the domain or other relevant control parameters, as, e.g., the Reynolds, Rossby, Froude, Peclet,´ or Alfven´ numbers. We briefly discuss the presence of anomalous scaling laws in 3D hydrodynamics and in other configurations, in connec- tion with the intermittent nature of the energy dissipation in configuration space. A quick overview is also provided concerning the importance of cascades in other applications such as bounded flows, quantum fluids, relativistic and compressible turbulence, and active mat- ter, together with a discussion of the implications for turbulent modelling. Finally, we present a series of open problems and challenges that future work needs to address. Keywords: Homogeneous and isotropic turbulence, two dimensional turbulence, wave arXiv:1808.06186v2 [physics.flu-dyn] 9 Oct 2018 turbulence, rotating flows, thick layers, MHD, turbulent diffusion, passive and active scalars, stratified flows, helicity, convection, Lagrangian turbulence, Richardson cascade, inverse energy cascade, direct energy cascade, intermittency, anomalous scaling laws, energy condensate, absolute equilibrium. 1Post print version of the article published on Physics Reports 2018. https://doi.org/10.1016/j.physrep.2018.08.001 Preprint submitted to Physics Reports October 10, 2018 Contents 1 Introduction 4 2 Theoretical Setup 6 2.1 Dynamical equations and control parameters . .7 2.2 Inviscid Invariants and balance equations . .7 2.3 Fourier space representation . .8 2.4 Configuration space representation . 11 2.5 Scale-filtered representations . 13 3 Definitions of Turbulent Cascades 14 3.1 3D Direct Energy Cascade and the Kolmogorov 1941 theory . 14 3.1.1 Fourier space . 15 3.1.2 Configuration space . 16 3.2 2D Inverse Energy Cascade and the Batchelor-Kraichnan theory . 17 3.2.1 Fourier space . 18 3.2.2 Configuration space . 22 3.3 Split Energy Cascade . 22 3.4 Statistical Equilibrium . 24 3.5 Multiple invariants . 27 3.6 Definitions . 28 3.6.1 Forward, Inverse & Split cascades . 29 3.6.2 Partial Fluxes and bidirectional cascades . 31 3.6.3 Dual and Multiple cascades . 31 3.6.4 Finite domains and Asymptotically Equilibrium states . 32 3.7 Classification of Cascade transitions . 33 4 Applications 35 4.1 Helicity . 36 4.1.1 Homogeneous and Isotropic Helical Turbulence . 37 4.1.2 Inverse energy cascades in homochiral turbulence . 40 4.1.3 Energy cascade transitions in Helical variants of the Navier-Stokes equations . 44 4.1.4 Summary . 46 4.2 Turbulence in layers of finite thickness . 47 4.2.1 Formulation . 47 4.2.2 Extremely thin layers, H Hc .................... 52 4.2.3 Critical layer thickness, H ∼ Hc ................... 52 4.2.4 Finite horizontal size and finite Reynolds . 55 4.2.5 Summary . 57 4.3 Rotation . 57 4.3.1 Formulation . 57 4.3.2 Decomposition in the 2D3C and 3D manifolds . 60 4.3.3 Rotating turbulence at finite thickness, H, and at changing Re . 62 4.3.4 Dependency on the forcing . 65 4.3.5 Finite horizontal size . 66 4.3.6 Summary . 68 4.4 Stratification . 68 4.4.1 Stable stratification . 69 4.4.2 Stratified flows at finite thickness, H ................. 72 4.4.3 2D vertically stratified flows . 73 4.4.4 Unstable stratification . 73 2 4.4.5 Summary . 74 4.5 Rotating and stratified flows . 74 4.5.1 Summary . 77 4.6 MHD (3D and 2D) . 78 4.6.1 Formulation and invariants for 3D MHD . 78 4.6.2 Isotropic 3D MHD turbulence . 80 4.6.3 Anisotropic MHD . 81 4.6.4 2D MHD . 84 4.6.5 Quasi-static MHD flows . 86 4.6.6 Magnetic helicity . 87 4.6.7 Summary . 89 4.7 Passive and active scalars . 89 4.7.1 Background definitions . 90 4.7.2 Passive scalar cascades . 91 4.7.3 Active scalar cascades . 93 4.7.4 Active vs Passive scalar cascades in Lagrangian coordinates . 95 4.7.5 Compressible flows and transitions to inverse cascades . 97 4.7.6 Summary . 98 5 Further topics about cascades 99 5.1 Wave turbulence . 99 5.2 Intermittency and multi-scaling . 101 5.2.1 Turbulence under unconventional Galerkin truncation . 105 5.2.2 Scaling in anisotropic flows . 107 5.3 Modeling . 108 5.3.1 Large Eddy Simulations . 108 5.3.2 Shell Models . 109 5.3.3 Eddy Dumped Quasi Normal Markovian Approximation . 111 5.4 Cascades in bounded flows and with multi-scale injections . 111 5.5 Guided tour across different turbulent systems . 113 5.5.1 Gravito-Capillary surface waves . 113 5.5.2 Quantum turbulence . 114 5.5.3 Elastic wave turbulence . 115 5.5.4 Compressible and Relativistic turbulence . 115 5.5.5 Active matter . 116 6 Conclusions and open problems 116 7 Acknowledgements 119 3 1. Introduction “Big whorls have little whorls that feed on their velocity, and little whorls have smaller whorls and so on to viscosity - in the molecular sense”. This is the celebrated poem com- posed by Lewis Fry Richardson in 1922 [1], where it was proposed that turbulent cascades are the fundamental driving mechanism of the atmosphere, moving energy from the large injections scales down to the dissipative small scales. It is a visionary way to summarize many fundamental aspects of the turbulent energy transfer which is empirically observed in the three dimensional Navier-Stokes equations (NSE). The Richardson cascade descrip- tion, first quantified by A.N. Kolmogorov [2], constitutes the most fundamental concept of turbulence theory. In more words, in a turbulent flow, the energy externally injected is redistributed among length scales due to non-linear eddy (whorls) interactions. If this energy is removed from the flow (or accumulated) at a scale `out significantly different than the injection scale, `in, a cascade can build up with a continuous transfer of energy from `in to `out. Depending on the system, the energy transfer can be towards the small scales and/or towards the large scales, leading to what it is referred to as a forward or inverse cascade, respectively. In three dimensional (3D) homogeneous and isotropic turbulence (HIT) the energy is transferred to the small scales while in two dimensions (2D) it cas- cades to the large scales. Despite the simplicity of the cascade description, after almost 100 years, we are still fighting to define the exact terms of the game and fail to have a com- plete statistical description even for the simplest case of HIT. When and why Richardson’s cascade is correct? What happens when it fails? These and many other questions have chal- lenged mathematicians, physicists and engineers for more than a century without reaching, to this date, clear answers, rightfully titling turbulence as the last open problem of classical physics. There are various text books written over the years reviewing both findings and open questions of different aspects of turbulence [3–5]. A recent overview can be found in [6], where the historical developments in engineering, mathematical, and physical sciences have been analysed from the ”shoulders of twelve historical fathers of turbulence research”. The situation, however, can be very complex. First, any inviscid invariant of the system is also subject to a non-linear transfer. The interaction of the transfer of different quantities plays an important role in determining the direction of their cascade. For example, it is the conservation of enstrophy that forces an inverse cascade of energy in 2D. In 3D, the second invariant is helicity and it is not sign-definite and all experimental investigations, numeri- cal simulations and phenomenological theories indicate that both energy and helicity have a simultaneous mean transfer to the small scales. However, this is an empirical observation and it is not proven from basic principles yet. The dual cascade of helicity and energy is important for 3D hydrodynamical turbulence [7–10]. In the latter case, recent studies have shown that even in the idealized case of 3D HIT, there exists a bi-directional trans- fer of energy with some helical-Fourier channels that cascade energy forward and others that transfer energy backward [11–21], with potential applications to rotating turbulence also.
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