August  2005, 5(3): 659-686. doi: 10.3934/dcdsb.2005.5.659

Multiresolution analysis for 2D turbulence. Part 1: Wavelets vs cosine packets, a comparative study

1. 

Université Bordeaux 1, IMB, CNRS UMR 5466, INRIA projet MC2, 351, Cours de la Libération, 33405 Talence Cedex, France

Received  April 2004 Revised  August 2004 Published  May 2005

The widely accepted theory of two-dimensional turbulence predicts a direct downscale enstrophy cascade with an energy spectrum behaving like $k^{-3}$ and an inverse upscale energy cascade with a $k^{-5/3}$ decay. Nevertheless, this theory is in fact an idealization valid only in an infinite domain in the limit of infinite Reynolds numbers, and is almost impossible to reproduce numerically. A more complete theoretical framework for the two-dimensional turbulence has been recently proposed by Tung et al . This theory seems to be more consistent with experimental observations, and numerical simulations than the classical one developed by Kraichnan, Leith and Batchelor.
Multiresolution methods like the wavelet packets or the cosine packets, well known in signal decomposition, can be used for the 2D turbulence analysis. Wavelet or cosine decompositions are more and more used in physical applications and in particular in fluid mechanics. Following the works of M. Farge et al , we present a numerical and qualitative study of a two-dimensional turbulence fluid using these methods. The decompositions allow to separate the fluid in two parts which are analyzed and the corresponding energy spectra are computed. In the first part of this paper, the methods are presented and the numerical results are briefly compared to the theoretical spectra predicted by the both theories. A more detailed study, using only wavelet packets decompositions and based on numerical and experimental data, will be carried out and the results will be reported in the second part of the paper. A tentative of physical interpretation of the different components of the flow will be also proposed.
Citation: Patrick Fischer. Multiresolution analysis for 2D turbulence. Part 1: Wavelets vs cosine packets, a comparative study. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 659-686. doi: 10.3934/dcdsb.2005.5.659
[1]

Patrick Fischer, Charles-Henri Bruneau, Hamid Kellay. Multiresolution analysis for 2D turbulence. part 2: A physical interpretation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 717-734. doi: 10.3934/dcdsb.2007.7.717

[2]

Ka Kit Tung, Wendell Welch Orlando. On the differences between 2D and QG turbulence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 145-162. doi: 10.3934/dcdsb.2003.3.145

[3]

S. Danilov. Non-universal features of forced 2D turbulence in the energy and enstrophy ranges. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 67-78. doi: 10.3934/dcdsb.2005.5.67

[4]

Marcel Lesieur. Two-point closure based large-eddy simulations in turbulence. Part 2: Inhomogeneous cases. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 227-241. doi: 10.3934/dcds.2010.28.227

[5]

Nusret Balci, Ciprian Foias, M. S Jolly, Ricardo Rosa. On universal relations in 2-D turbulence. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1327-1351. doi: 10.3934/dcds.2010.27.1327

[6]

Eleftherios Gkioulekas, Ka Kit Tung. On the double cascades of energy and enstrophy in two dimensional turbulence. Part 2. Approach to the KLB limit and interpretation of experimental evidence. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 103-124. doi: 10.3934/dcdsb.2005.5.103

[7]

Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685

[8]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395

[9]

Géry de Saxcé, Claude Vallée. Structure of the space of 2D elasticity tensors. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1525-1537. doi: 10.3934/dcdss.2013.6.1525

[10]

Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297

[11]

Bernd Kawohl, Guido Sweers. On a formula for sets of constant width in 2d. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2117-2131. doi: 10.3934/cpaa.2019095

[12]

Brian Ryals, Robert J. Sacker. Global stability in the 2D Ricker equation revisited. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 585-604. doi: 10.3934/dcdsb.2017028

[13]

Leonardo Kosloff, Tomas Schonbek. Existence and decay of solutions of the 2D QG equation in the presence of an obstacle. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1025-1043. doi: 10.3934/dcdss.2014.7.1025

[14]

Yuri N. Fedorov, Luis C. García-Naranjo, Joris Vankerschaver. The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4017-4040. doi: 10.3934/dcds.2013.33.4017

[15]

Makram Hamouda, Chang-Yeol Jung, Roger Temam. Boundary layers for the 2D linearized primitive equations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 335-359. doi: 10.3934/cpaa.2009.8.335

[16]

A. Rousseau, Roger Temam, J. Tribbia. Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1257-1276. doi: 10.3934/dcds.2005.13.1257

[17]

Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413

[18]

Tian Ma, Shouhong Wang. Structure of 2D incompressible flows with the Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 29-41. doi: 10.3934/dcdsb.2001.1.29

[19]

Jutta Bikowski, Jennifer L. Mueller. 2D EIT reconstructions using Calderon's method. Inverse Problems & Imaging, 2008, 2 (1) : 43-61. doi: 10.3934/ipi.2008.2.43

[20]

Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]