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Mathematics
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Turbulence Modeling
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Turbulence Modeling

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (29 February 2020) | Viewed by 12009

Special Issue Editor

Dr. Johan Anderson

E-Mail Website
Guest Editor
Department of Space, Earth and Environment, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Interests: anomalous diffusion; Tsallis entropy; nonlocal theory; Lévy noise; fractional Fokker–Plank equation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Turbulence is ubiquitous in nature, spanning many different fields of research. However, there is seldom or little interaction between different fields of research concerning turbulence modeling. This Special Issue aims at collecting current state-of-the-art modeling efforts on turbulence in, e.g., fluids and plasmas and other fields. There are areas of particular interest, since there is strong evidence from laboratory experiments, observations, and computational studies, that coherent structures can cause intermittent transport, significantly changing the dynamics. In the fluid dynamics community, there have been efforts to use machine learning and statistical methods of uncertainty quantification, and data-driven modeling to improve the model parameters in existing models and to be able to take intermittency into account. Furthermore, it is strongly suggested that these efforts may be used to find the way for the development of new models. Theoretical models accounting for or contributing to the understanding of the multiscale problem in turbulence as well as those addressing intermittency, coherent structures, and self-organization are also welcome. The Guest Editor is open to considering any paper relevant to the subject matter of the Special Issue.

Prof. Johan Anderson
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Turbulence
  • Fractional models
  • Reynolds-averaged Navier–Stokes (RANS)
  • Large Eddy simulations (LES)
  • Direct numerical simulations (DNS)
  • Statistical mechanics
  • Intermittency
  • Coherent structure
  • Multiscale analysis
  • Self-organization

Published Papers (5 papers)

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Research

9 pages, 409 KiB  
Article
Information Geometric Investigation of Solutions to the Fractional Fokker–Planck Equation
by Johan Anderson
Mathematics 2020, 8(5), 668; https://doi.org/10.3390/math8050668 - 28 Apr 2020
Viewed by 1874
Abstract
A novel method for measuring distances between statistical states as represented by probability distribution functions (PDF) has been proposed, namely the information length. The information length enables the computation of the total number of statistically different states that a system evolves through in [...] Read more.
A novel method for measuring distances between statistical states as represented by probability distribution functions (PDF) has been proposed, namely the information length. The information length enables the computation of the total number of statistically different states that a system evolves through in time. Anomalous transport can presumably be modeled fractional velocity derivatives and Langevin dynamics in a Fractional Fokker–Planck (FFP) approach. The numerical solutions or PDFs are found for varying degree of fractionality ( α ) of the stable Lévy distribution as solutions to the FFP equation. Specifically, the information length of time-dependent PDFs for a given fractional index α is computed. Full article
(This article belongs to the Special Issue Turbulence Modeling)
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13 pages, 448 KiB  
Article
Turbulence as a Network of Fourier Modes
by Özgür. D. Gürcan, Yang Li and Pierre Morel
Mathematics 2020, 8(4), 530; https://doi.org/10.3390/math8040530 - 03 Apr 2020
Cited by 5 | Viewed by 2635
Abstract
Turbulence is the duality of chaotic dynamics and hierarchical organization of a field over a large range of scales due to advective nonlinearities. Quadratic nonlinearities (e.g., advection) in real space, translates into triadic interactions in Fourier space. Those interactions can be computed using [...] Read more.
Turbulence is the duality of chaotic dynamics and hierarchical organization of a field over a large range of scales due to advective nonlinearities. Quadratic nonlinearities (e.g., advection) in real space, translates into triadic interactions in Fourier space. Those interactions can be computed using fast Fourier transforms, or other methods of computing convolution integrals. However, more generally, they can be interpreted as a network of interacting nodes, where each interaction is between a node and a pair. In this formulation, each node interacts with a list of pairs that satisfy the triadic interaction condition with that node, and the convolution becomes a sum over this list. A regular wavenumber space mesh can be written in the form of such a network. Reducing the resolution of a regular mesh and combining the nearby nodes in order to obtain the reduced network corresponding to the low resolution mesh, we can deduce the reduction rules for such a network. This perspective allows us to develop network models as approximations of various types of turbulent dynamics. Various examples, such as shell models, nested polyhedra models, or predator–prey models, are briefly discussed. A prescription for setting up a small world variants of these models are given. Full article
(This article belongs to the Special Issue Turbulence Modeling)
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12 pages, 854 KiB  
Article
Resonance Enhancement by Suitably Chosen Frequency Detuning
by Denys Dutykh and Elena Tobisch
Mathematics 2020, 8(3), 450; https://doi.org/10.3390/math8030450 - 19 Mar 2020
Cited by 1 | Viewed by 3075
Abstract
The theory of exact resonances (kinematics and dynamics) is well developed while even the very concept of detuned resonance is ambiguous and only studies of their kinematic characteristics (that is, those not depending on time) are available in the literature. In this paper, [...] Read more.
The theory of exact resonances (kinematics and dynamics) is well developed while even the very concept of detuned resonance is ambiguous and only studies of their kinematic characteristics (that is, those not depending on time) are available in the literature. In this paper, we report novel effects enforced by the resonance detuning on solutions of the dynamical system describing interactions of three spherical planetary waves. We establish that the energy variation range can significantly exceed the range of the exact resonance for suitably chosen values of the detuning. The asymmetry of system’s solutions with respect to the sign of the detuning parameter is demonstrated. Finally, a non-monotonic dependence of the energy oscillation period with respect to detuning magnitude is discovered. These results have direct implications in physics of atmosphere, e.g., for prediction of weather extremes in the Northern Hemisphere midlatitudes (Proc. Nat. Acad. Sci. USA 2016, 133(25), 6862–6867). Moreover, similar study can be conducted for a generic three-wave system taken in the Hamiltonian form which makes our results applicable for an arbitrary Hamiltonian three-wave system met in climate prediction theory, geophysical fluid dynamics, plasma physics, etc. Full article
(This article belongs to the Special Issue Turbulence Modeling)
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14 pages, 4434 KiB  
Article
Impact of Micro-Scale Stochastic Zonal Flows on the Macro-Scale Visco-Resistive Magnetohydrodynamic Modes
by Sara Moradi and Anantanarayanan Thyagaraja
Mathematics 2020, 8(3), 443; https://doi.org/10.3390/math8030443 - 18 Mar 2020
Viewed by 1837
Abstract
A model is developed to simulate micro-scale turbulence driven Zonal Flows (ZFs), and their impact on the Magnetohydrodynamic (MHD) tearing and kink modes is examined. The model is based on a stochastic representation of the micro-scale ZFs with a given Alfvén Mach number, [...] Read more.
A model is developed to simulate micro-scale turbulence driven Zonal Flows (ZFs), and their impact on the Magnetohydrodynamic (MHD) tearing and kink modes is examined. The model is based on a stochastic representation of the micro-scale ZFs with a given Alfvén Mach number, MS. Two approaches were explored: (i) passive stochastic model where the ZFs amplitudes are independent of the MHD mode amplitude, and (ii) the semi-stochastic model where the amplitudes of the ZFs have a dependence on the amplitude of the MHD mode itself. The results show that the stochastic ZFs can significantly stabilise the (2,1) and (1,1) MHD modes even at very low kinematic viscosity, where the mode is linearly unstable. Our results therefore indicate a possible mechanism for stabilisation of the MHD modes via small-scale perturbations in poloidal flow, simulating the turbulence driven ZFs. Full article
(This article belongs to the Special Issue Turbulence Modeling)
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14 pages, 5611 KiB  
Article
Information Length as a Useful Index to Understand Variability in the Global Circulation
by Eun-jin Kim, James Heseltine and Hanli Liu
Mathematics 2020, 8(2), 299; https://doi.org/10.3390/math8020299 - 24 Feb 2020
Cited by 12 | Viewed by 2084
Abstract
With improved measurement and modelling technology, variability has emerged as an essential feature in non-equilibrium processes. While traditionally, mean values and variance have been heavily used, they are not appropriate in describing extreme events where a significant deviation from mean values often occurs. [...] Read more.
With improved measurement and modelling technology, variability has emerged as an essential feature in non-equilibrium processes. While traditionally, mean values and variance have been heavily used, they are not appropriate in describing extreme events where a significant deviation from mean values often occurs. Furthermore, stationary Probability Density Functions (PDFs) miss crucial information about the dynamics associated with variability. It is thus critical to go beyond a traditional approach and deal with time-dependent PDFs. Here, we consider atmospheric data from the Whole Atmosphere Community Climate Model (WACCM) and calculate time-dependent PDFs and the information length from these PDFs, which is the total number of statistically different states that a system evolves through in time. Specifically, we consider the three cases of sampling data to investigate the distribution of information (information budget) along the altitude and longitude to gain a new perspective of understanding variabilities, correlation among different variables and regions. Time-dependent PDFs are shown to be non-Gaussian in general; the information length tends to increase with the altitude albeit in a complex form; this tendency is more robust for flows/shears than temperature. Much similarity among flows and shears in the information length is also found in comparison with the temperature. This means a strong correlation among flows/shears because of their coupling through gravity waves in this particular WACCM model. We also find the increase of the information length with the latitude and interesting hemispheric asymmetry for flows/shears/temperature, with the tendency of anti-correlation (correlation) between flows/shears and temperature at high (low) latitude. These results suggest the importance of high latitude/altitude in the information budget in the Earth’s atmosphere, the spatial gradient of the information length being a useful proxy for information flow. Full article
(This article belongs to the Special Issue Turbulence Modeling)
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