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Ivchenko NA, Lebedev VV, Vergeles SS. Mixing in two-dimensional shear flow with smooth fluctuations. Phys Rev E 2024; 110:015102. [PMID: 39160955 DOI: 10.1103/physreve.110.015102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2023] [Accepted: 05/22/2024] [Indexed: 08/21/2024]
Abstract
Chaotic variations in flow speed up mixing of scalar fields via intensified stirring. This paper addresses the statistical properties of a passive scalar field mixing in a regular shear flow with random fluctuations against its background. We consider two-dimensional flow with shear component dominating over smooth fluctuations. Such flow is supposed to model passive scalar mixing, e.g., inside a large-scale coherent vortex forming in two-dimensional turbulence or in elastic turbulence in a microchannel. We examine both the decaying case and the case of the continuous forcing of the scalar variances. In both cases dynamics possesses strong intermittency, which can be characterized via the single-point moments and correlation functions calculated in our work. We present general qualitative properties of pair correlation function as well as certain quantitative results obtained in the framework of the model with fluctuations that are short correlated in time.
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2
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Kolokolov IV, Lebedev VV, Parfenyev VM. Correlations in a weakly interacting two-dimensional random flow. Phys Rev E 2024; 109:035103. [PMID: 38632784 DOI: 10.1103/physreve.109.035103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2023] [Accepted: 02/15/2024] [Indexed: 04/19/2024]
Abstract
We analytically examine fluctuations of vorticity excited by an external random force in two-dimensional fluid. We develop the perturbation theory enabling one to calculate nonlinear corrections to correlation functions of the flow fluctuations found in the linear approximation. We calculate the correction to the pair correlation function and the triple correlation function. It enables us to establish the criterion of validity of the perturbation theory for different ratios of viscosity and bottom friction. We find that the corrections to the second moment are anomalously weak in the cases of small bottom friction and small viscosity and relate the weakness to the energy and enstrophy balances. We demonstrate that at small bottom friction the triple correlation function is characterized by universal scaling behavior in some region of lengths. The developed perturbation method was verified and confirmed by direct numerical simulations.
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Affiliation(s)
- I V Kolokolov
- Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow region, Russia and National Research University Higher School of Economics, 101000, Myasnitskaya ul. 20, Moscow, Russia
| | - V V Lebedev
- Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow region, Russia and National Research University Higher School of Economics, 101000, Myasnitskaya ul. 20, Moscow, Russia
| | - V M Parfenyev
- Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow region, Russia and National Research University Higher School of Economics, 101000, Myasnitskaya ul. 20, Moscow, Russia
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3
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Svirsky A, Herbert C, Frishman A. Statistics of inhomogeneous turbulence in large-scale quasigeostrophic dynamics. Phys Rev E 2023; 108:065102. [PMID: 38243459 DOI: 10.1103/physreve.108.065102] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2023] [Accepted: 10/19/2023] [Indexed: 01/21/2024]
Abstract
A remarkable feature of two-dimensional turbulence is the transfer of energy from small to large scales. This process can result in the self-organization of the flow into large, coherent structures due to energy condensation at the largest scales. We investigate the formation of this condensate in a quasigeostropic flow in the limit of small Rossby deformation radius, namely the large-scale quasigeostrophic model. In this model potential energy is transferred up-scale while kinetic energy is transferred down-scale in a direct cascade. We focus on a jet mean flow and carry out a thorough investigation of the second-order statistics for this flow, combining a quasilinear analytical approach with direct numerical simulations. We show that the quasilinear approach applies in regions where jets are strong and is able to capture all second-order correlators in that region, including those related to the kinetic energy. This is a consequence of the blocking of the direct cascade by the mean flow in jet regions, suppressing fluctuation-fluctuation interactions. The suppression of the direct cascade is demonstrated using a local coarse-graining approach allowing us to measure space dependent interscale kinetic energy fluxes, which we show are concentrated in between jets in our simulations. We comment on the possibility of a similar direct cascade arrest in other two-dimensional flows, arguing that it is a special feature of flows in which the fluid element interactions are local in space.
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Affiliation(s)
- Anton Svirsky
- Physics Department, Technion Israel Institute of Technology, 32000 Haifa, Israel
| | - Corentin Herbert
- ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Anna Frishman
- Physics Department, Technion Israel Institute of Technology, 32000 Haifa, Israel
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4
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Svirsky A, Herbert C, Frishman A. Two-Dimensional Turbulence with Local Interactions: Statistics of the Condensate. PHYSICAL REVIEW LETTERS 2023; 131:224003. [PMID: 38101360 DOI: 10.1103/physrevlett.131.224003] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2023] [Revised: 09/03/2023] [Accepted: 10/24/2023] [Indexed: 12/17/2023]
Abstract
Two-dimensional turbulence self-organizes through a process of energy accumulation at large scales, forming a coherent flow termed a condensate. We study the condensate in a model with local dynamics, the large-scale quasigeostrophic equation, observed here for the first time. We obtain analytical results for the mean flow and the two-point, second-order correlation functions, and validate them numerically. The condensate state requires partiy+time-reversal symmetry breaking. We demonstrate distinct universal mechanisms for the even and odd correlators under this symmetry. We find that the model locality is imprinted in the small scale dynamics, which the condensate spatially confines.
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Affiliation(s)
- Anton Svirsky
- Physics Department, Technion Israel Institute of Technology, 32000 Haifa, Israel
| | - Corentin Herbert
- ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Anna Frishman
- Physics Department, Technion Israel Institute of Technology, 32000 Haifa, Israel
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5
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Rorai C, Toschi F, Pagonabarraga I. Coexistence of Active and Hydrodynamic Turbulence in Two-Dimensional Active Nematics. PHYSICAL REVIEW LETTERS 2022; 129:218001. [PMID: 36461968 DOI: 10.1103/physrevlett.129.218001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2022] [Revised: 09/29/2022] [Accepted: 10/25/2022] [Indexed: 06/17/2023]
Abstract
In active nematic liquid crystals, activity is able to drive chaotic spatiotemporal flows referred to as active turbulence. Active turbulence has been characterized through theoretical and experimental work as a low Reynolds number phenomenon. We show that, in two dimensions, the active forcing alone is able to trigger hydrodynamic turbulence leading to the coexistence of active and inertial turbulence. This type of flow develops for sufficiently active and extensile flow-aligning nematics. We observe that the combined effect of an extensile nematic and large values of the flow-aligning parameter leads to a broadening of the elastic energy spectrum that promotes a growth of kinetic energy able to trigger an inverse energy cascade.
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Affiliation(s)
- C Rorai
- CECAM, Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne (EPFL), Batochime, Avenue Forel 2, 1015 Lausanne, Switzerland
| | - F Toschi
- Department of Applied Physics, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, Netherlands
- CNR-IAC, I-00185 Rome, Italy
| | - I Pagonabarraga
- CECAM, Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne (EPFL), Batochime, Avenue Forel 2, 1015 Lausanne, Switzerland
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, C. Martí i Franquès 1, 08028 Barcelona, Spain
- University of Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, 08028 Barcelona, Spain
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6
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Parfenyev V. Profile of a two-dimensional vortex condensate beyond the universal limit. Phys Rev E 2022; 106:025102. [PMID: 36109998 DOI: 10.1103/physreve.106.025102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2022] [Accepted: 07/25/2022] [Indexed: 06/15/2023]
Abstract
It is well known that an inverse turbulent cascade in a finite (2π×2π) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length l_{f}=2π/k_{f} with k_{f} ranging from 100 to 12.5. Previously, it was found that in the asymptotic limit of small-scale forcing, the vorticity exhibits the power-law behavior Ω(r)=(3ε/α)^{1/2}r^{-1}, where r is the distance to the vortex center, α is the bottom friction coefficient, and ε is the inverse energy flux. Now we show that for a spatially homogeneous forcing with finite k_{f} the vorticity profile becomes steeper, with the difference increasing with the pumping scale but decreasing with the Reynolds number at the forcing scale. Qualitatively, this behavior is related to a decrease in the effective pumping of the coherent vortex with distance from its center. To support this statement, we perform an additional simulation with spatially localized forcing, in which the effective pumping of the coherent vortex, on the contrary, increases with r, and show that in this case the vorticity profile can be flatter than the asymptotic limit.
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Affiliation(s)
- Vladimir Parfenyev
- Landau Institute for Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova av., 142432 Chernogolovka, Russia and National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000 Moscow, Russia
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7
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Rautek P, Mlejnek M, Beyer J, Troidl J, Pfister H, Theubl T, Hadwiger M. Objective Observer-Relative Flow Visualization in Curved Spaces for Unsteady 2D Geophysical Flows. IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 2021; 27:283-293. [PMID: 33048741 DOI: 10.1109/tvcg.2020.3030454] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Computing and visualizing features in fluid flow often depends on the observer, or reference frame, relative to which the input velocity field is given. A desired property of feature detectors is therefore that they are objective, meaning independent of the input reference frame. However, the standard definition of objectivity is only given for Euclidean domains and cannot be applied in curved spaces. We build on methods from mathematical physics and Riemannian geometry to generalize objectivity to curved spaces, using the powerful notion of symmetry groups as the basis for definition. From this, we develop a general mathematical framework for the objective computation of observer fields for curved spaces, relative to which other computed measures become objective. An important property of our framework is that it works intrinsically in 2D, instead of in the 3D ambient space. This enables a direct generalization of the 2D computation via optimization of observer fields in flat space to curved domains, without having to perform optimization in 3D. We specifically develop the case of unsteady 2D geophysical flows given on spheres, such as the Earth. Our observer fields in curved spaces then enable objective feature computation as well as the visualization of the time evolution of scalar and vector fields, such that the automatically computed reference frames follow moving structures like vortices in a way that makes them appear to be steady.
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Kolokolov IV, Lebedev VV. Coherent vortex in two-dimensional turbulence: Interplay of viscosity and bottom friction. Phys Rev E 2020; 102:023108. [PMID: 32942442 DOI: 10.1103/physreve.102.023108] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2020] [Accepted: 07/21/2020] [Indexed: 11/06/2022]
Abstract
We examine coherent vortices appearing as a result of the inverse cascade of two-dimensional turbulence in a finite box in the case of pumping with arbitrary correlation time in the quasilinear regime. We demonstrate that the existence of the vortices depends on the ratio between the values of the bottom friction coefficient α and the viscous damping of the flow fluctuations at the pumping scale νk_{f}^{2} (ν is the kinematic viscosity coefficient and k_{f} is the characteristic wave vector at the pumping scale). The coherent vortices appear if νk_{f}^{2}≫α and cannot exist if νk_{f}^{2}≪α. Therefore there is a border value α∼νk_{f}^{2} separating the regions. In numerical simulations, νk_{f}^{2}/α can be arbitrary, whereas in a laboratory experiment νk_{f}^{2}/α≲1 and the coherent vortices can be observed solely near the border value of νk_{f}^{2}/α.
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Affiliation(s)
- I V Kolokolov
- Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow District, Russia.,Institute of Solid State Physics, RAS, 142432, Chernogolovka, Moscow District, Russia
| | - V V Lebedev
- Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow District, Russia
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9
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Kolokolov IV, Kostenko MM. Universal moments of accelerations in two-dimensional turbulence. Phys Rev E 2020; 101:033108. [PMID: 32289923 DOI: 10.1103/physreve.101.033108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/26/2019] [Accepted: 03/02/2020] [Indexed: 11/07/2022]
Abstract
We consider two-dimensional turbulence in the presence of a condensate. The nondiagonal correlation functions of the Lagrangian accelerations are calculated, and it is shown that they have the same universality properties as the nondiagonal correlation functions of the velocity fluctuations.
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Affiliation(s)
- Igor V Kolokolov
- L.D. Landau Institute for Theoretical Physics, Ak. Semenova 1-A, Chernogolovka 142432, Moscow region, Russia.,National Research University Higher School of Economics, Myasnitskaya 20, Moscow 101000, Russia
| | - Maria M Kostenko
- L.D. Landau Institute for Theoretical Physics, Ak. Semenova 1-A, Chernogolovka 142432, Moscow region, Russia.,Department of Physics, Saint Petersburg State University, 7/9 Universitetskaya embankment, Saint Petersburg 199034, Russia
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10
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Linkmann M, Boffetta G, Marchetti MC, Eckhardt B. Phase Transition to Large Scale Coherent Structures in Two-Dimensional Active Matter Turbulence. PHYSICAL REVIEW LETTERS 2019; 122:214503. [PMID: 31283308 DOI: 10.1103/physrevlett.122.214503] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2018] [Indexed: 06/09/2023]
Abstract
The collective motion of microswimmers in suspensions induce patterns of vortices on scales that are much larger than the characteristic size of a microswimmer, attaining a state called bacterial turbulence. Hydrodynamic turbulence acts on even larger scales and is dominated by inertial transport of energy. Using an established modification of the Navier-Stokes equation that accounts for the small-scale forcing of hydrodynamic flow by microswimmers, we study the properties of a dense suspension of microswimmers in two dimensions, where the conservation of enstrophy can drive an inverse cascade through which energy is accumulated on the largest scales. We find that the dynamical and statistical properties of the flow show a sharp transition to the formation of vortices at the largest length scale. The results show that 2D bacterial and hydrodynamic turbulence are separated by a subcritical phase transition.
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Affiliation(s)
- Moritz Linkmann
- Fachbereich Physik, Philipps-Universität of Marburg, D-35032 Marburg, Germany
| | - Guido Boffetta
- Dipartimento di Fisica and INFN, Università di Torino, via P. Giuria 1, 10125 Torino, Italy
| | - M Cristina Marchetti
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - Bruno Eckhardt
- Fachbereich Physik, Philipps-Universität of Marburg, D-35032 Marburg, Germany
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11
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Bouchet F, Rolland J, Simonnet E. Rare Event Algorithm Links Transitions in Turbulent Flows with Activated Nucleations. PHYSICAL REVIEW LETTERS 2019; 122:074502. [PMID: 30848628 DOI: 10.1103/physrevlett.122.074502] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Revised: 12/12/2018] [Indexed: 06/09/2023]
Abstract
Many turbulent flows undergo drastic and abrupt configuration changes with huge impacts. As a paradigmatic example we study the multistability of jet dynamics in a barotropic beta plane model of atmosphere dynamics. It is considered as the Ising model for Jupiter troposphere dynamics. Using the adaptive multilevel splitting, a rare event algorithm, we are able to get a very large statistics of transition paths, the extremely rare transitions from one state of the system to another. This new approach opens the way for addressing a set of questions that are out of reach through direct numerical simulations. We demonstrate for the first time the concentration of transition paths close to instantons, in a numerical simulation of genuine turbulent flows. We show that the transition is a noise-activated nucleation of vorticity bands. We address for the first time the existence of Arrhenius laws in turbulent flows. The methodology we developed shall prove useful to study many other transitions related to drastic changes for the turbulent dynamics of climate, geophysical, astrophysical, and engineering applications. This opens a new range of studies impossible so far, and bring turbulent phenomena in the realm of nonequilibrium statistical mechanics.
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Affiliation(s)
- Freddy Bouchet
- Univ Lyon, Ens de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Joran Rolland
- Univ Lyon, Ens de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France
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12
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Frishman A, Herbert C. Turbulence Statistics in a Two-Dimensional Vortex Condensate. PHYSICAL REVIEW LETTERS 2018; 120:204505. [PMID: 29864335 DOI: 10.1103/physrevlett.120.204505] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2017] [Indexed: 06/08/2023]
Abstract
Disentangling the evolution of a coherent mean-flow and turbulent fluctuations, interacting through the nonlinearity of the Navier-Stokes equations, is a central issue in fluid mechanics. It affects a wide range of flows, such as planetary atmospheres, plasmas, or wall-bounded flows, and hampers turbulence models. We consider the special case of a two-dimensional flow in a periodic box, for which the mean flow, a pair of box-size vortices called "condensate," emerges from turbulence. As was recently shown, a perturbative closure describes correctly the condensate when turbulence is excited at small scales. In this context, we obtain explicit results for the statistics of turbulence, encoded in the Reynolds stress tensor. We demonstrate that the two components of the Reynolds stress, the momentum flux and the turbulent energy, are determined by different mechanisms. It was suggested previously that the momentum flux is fixed by a balance between forcing and mean-flow advection: using unprecedently long numerical simulations, we provide the first direct evidence supporting this prediction. By contrast, combining analytical computations with numerical simulations, we show that the turbulent energy is determined only by mean-flow advection and obtain for the first time a formula describing its profile in the vortex.
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Affiliation(s)
- Anna Frishman
- Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA
- Department of Physics of Complex Systems, Weizmann Institute of Science, P.O. Box 26, Rehovot 76100, Israel
| | - Corentin Herbert
- Department of Physics of Complex Systems, Weizmann Institute of Science, P.O. Box 26, Rehovot 76100, Israel
- Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, Lyon F-69342, France
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13
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Falkovich G. Interaction between mean flow and turbulence in two dimensions. Proc Math Phys Eng Sci 2016; 472:20160287. [PMID: 27493579 DOI: 10.1098/rspa.2016.0287] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
This short note is written to call attention to an analytic approach to the interaction of developed turbulence with mean flows of simple geometry (jets and vortices). It is instructive to compare cases in two and three dimensions and see why the former are solvable and the latter are not (yet). We present the analytical solutions for two-dimensional mean flows generated by an inverse turbulent cascade on a sphere and in planar domains of different aspect ratios. These solutions are obtained in the limit of small friction when the flow is strong while turbulence can be considered weak and treated perturbatively. I then discuss when these simple solutions can be realized and when more complicated flows may appear instead. The next step of describing turbulence statistics inside a flow and directions of possible future progress are briefly discussed at the end.
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Affiliation(s)
- Gregory Falkovich
- Weizmann Institute of Science, Rehovot 76100, Israel; Institute for Information Transmission Problems, Moscow 127994, Russia
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14
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Kolokolov IV, Lebedev VV. Structure of coherent vortices generated by the inverse cascade of two-dimensional turbulence in a finite box. Phys Rev E 2016; 93:033104. [PMID: 27078444 DOI: 10.1103/physreve.93.033104] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2015] [Indexed: 06/05/2023]
Abstract
We discuss the structure and geometrical characteristics of coherent vortices appearing as a result of the inverse cascade in two-dimensional turbulence in a finite box. We demonstrate that the universal velocity profile, established by J. Laurie et al. [Phys. Rev. Lett. 113, 254503 (2014)], corresponds to the passive regime of flow fluctuations. We find the vortex core radius and the vortex size, and we argue that the amount of vortices generated in the box depends on the system parameters.
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Affiliation(s)
- I V Kolokolov
- Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow region, Russia
| | - V V Lebedev
- Landau Institute for Theoretical Physics, RAS, 142432, Chernogolovka, Moscow region, Russia
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15
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Sahoo G, Biferale L. Disentangling the triadic interactions in Navier-Stokes equations. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2015; 38:114. [PMID: 26537727 DOI: 10.1140/epje/i2015-15114-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2015] [Revised: 10/13/2015] [Accepted: 10/15/2015] [Indexed: 06/05/2023]
Abstract
We study the role of helicity in the dynamics of energy transfer in a modified version of the Navier-Stokes equations with explicit breaking of the mirror symmetry. We select different set of triads participating in the dynamics on the basis of their helicity content. In particular, we remove the negative helically polarized Fourier modes at all wave numbers except for those falling on a localized shell of wave number, |k| ~ k(m). Changing k(m) to be above or below the forcing scale, k(f), we are able to assess the energy transfer of triads belonging to different interaction classes. We observe that when the negative helical modes are present only at a wave number smaller than the forced wave numbers, an inverse energy cascade develops with an accumulation of energy on a stationary helical condensate. Vice versa, when negative helical modes are present only at a wave number larger than the forced wave numbers, a transition from backward to forward energy transfer is observed in the regime when the minority modes become energetic enough.
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Affiliation(s)
- Ganapati Sahoo
- Department of Physics and INFN, University of Rome "Tor Vergata", Via della Ricerca Scientifica 1, 00133, Rome, Italy.
| | - Luca Biferale
- Department of Physics and INFN, University of Rome "Tor Vergata", Via della Ricerca Scientifica 1, 00133, Rome, Italy
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