Numerical simulation of energy cascade in turbulent fluids
DOI:
https://doi.org/10.17268/sel.mat.2022.02.07Keywords:
Navier Stokes Equations, energy transfer, shell modelsAbstract
Transfer of energy from large to small scales in turbulent flows is described as a flux of energy from small wave numbers to large wave numbers in the spectral representation of the Navier-Stokes equation. The problem of resolving the relevant scales in the flow corresponds in the spectral representation to determining the spectral truncation at large wave numbers. The effective number of degrees of freedom in the flow depends on the Reynolds number and a numerical simulation of the Navier Stokes equation for high Reynolds numbers is impractical without some sort of reduction of the number of degrees of freedom. The shell models are constructed to obey the same conservation laws and symmetries as the Navier Stokes equation as an alternative. In this work we present the Sabra shell model for the study of the energy cascade of turbulence and we will show numerically that the energy dissipation is approximately -1/3 which is in agreement with the theory K41.
References
Ditlevsen PD. Turbulence and Shell Models, Cambridge University Press 2011.
L’vov VS, Podivilov E, Pomyalov A, Procaccia I, Vandembroueq D. Improved shell model of turbulence. Phys. Rev. E. 1998; 58(2):1811–22.
Lorenz EN. Low order models representing realizations of turbulence. J. Fluid Mech. 1972; 55:545–63.
Gledzer EB. System of hydrodynamic type admitting two quadratic integrals of motion
Dokl. Akad. Nauk SSSR. 1973; 209(5):1046–1048.
Yamada M, Ohkitani K. Lyapunov Spectrum of a Chaotic Model of Three-Dimensional Turbulence. J. Phys. Soc. Jpn. 1987: 56:4210-4213.
Jensen MH, Paladin G, Vulpiani A. Intermittency in a cascade model for three-dimensional turbulence Phys. Rev. A. 1991; 43:798.
Pissarenko D, Biferale L, Courvoisier D, Frisch U, Vergassola M. Further results on multifractality in shell models. Phys. Fluids A. Fluid Dynamics; 1993; 5:2533.
Benzi R, Biferale L, Parisi G. On intermittency in a cascade model for turbulence. Physica D. 1993; 65:163-171.
Obukhov AM. On some general characteristics of the equations of the dynamics of
the atmosphere. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana. 1971; 7:695–704.
Ditlevsen PD. Symmetries, invariants, and cascades in a shell model of turbulence. Phys. Rev. E. 2000; 62:484-9.
Kolmogoroff AN. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR. 1941; 30:301–305.
Batchelor GK. An Introduction to Fluid Dynamics. Cambridge University Press. 1967.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Selecciones Matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
