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Table of contents
Remerciements
Introduction
I Context and open questions
I.1 Physical concepts
I.1.1 Turbulent fluid flows
I.1.2 Filamentary plasma flows
I.1.3 Dissipation
I.2 Statistical models
I.2.1 (Un-)Predictability
I.2.2 The K41 and KBL67 theories
I.2.3 Turbulence models
I.3 Mathematical view
I.3.1 Weak solutions and well-posedness
I.3.2 Dynamical systems and attractors
I.3.3 Singular limits
I.4 Numerical approach
I.4.1 Foundations
I.4.2 Technological requirements
I.4.3 Examples of current achievements
II Wavelet tools for flow analysis
II.1 Mathematical theory
II.1.1 Multiresolution analysis
II.1.2 Fast wavelet transform
II.1.3 Wavelet families
II.1.4 Representation of differential operators
II.1.5 Denoising
II.2 Implementation
II.2.1 Review of some existing implementations
II.2.2 General structure of our approach
II.2.3 Adaptive wavelet transform
II.2.4 Parallelization
II.3 Verification and benchmarking
II.3.1 Parallel efficiency
II.3.2 Representation of translation operators
II.3.3 Power spectrum estimation
II.3.4 Denoising correlated noises
II.4 Application: edge plasma tomography
II.4.1 Introduction
II.4.2 Reconstruction method
II.4.3 Validation
II.4.4 Application to Tore Supra movies
II.4.5 Conclusion
III Particle-in-Wavelets approach for the Vlasov equation
III.1 Wavelet-based density estimation
III.1.1 Introduction
III.1.2 Methods
III.1.3 Applications
III.1.4 Summary and Conclusion
III.2 Particle-in-Wavelets scheme for the 1D Vlasov-Poisson equations
III.2.1 Background
III.2.2 Description of the PIW scheme
III.2.3 Numerical results
III.2.4 Discussion
IV Regularization of inviscid equations
IV.1 1D Burgers equation
IV.1.1 Introduction
IV.1.2 Numerical method
IV.1.3 Deterministic initial condition
IV.1.4 Random initial condition
IV.1.5 Conclusion
IV.2 Incompressible 2D Euler equations
IV.2.1 Introduction
IV.2.2 Numerical method
IV.2.3 Results
IV.2.4 Conclusion and Perspectives
IV.3 Remarks on Galerkin discretizations
V Dissipation at vanishing viscosity
V.1 Volume penalization
V.2 Molecular dissipation in the presence of walls
V.2.1 Introduction
V.2.2 Model and numerical method
V.2.3 Results
V.2.4 Conclusion
V.3 Turbulent dissipation in 2D homogeneous turbulence
V.3.1 Introduction
V.3.2 Conditional statistical modelling
V.3.3 Mathematical framework and numerical method
V.3.4 Statistical analysis
V.3.5 Scale-wise coherent vorticity extraction
V.3.6 Interscale enstrophy transfers and production of incoherent enstrophy
V.3.7 Dynamical influence of the incoherent part
V.3.8 Conclusion
V.4 Analysis of 3D turbulent boundary layers
V.4.1 Introduction
V.4.2 Flow configuration and parameters
V.4.3 Orthogonal wavelet decomposition of the turbulent boundary layer flow
V.4.4 Numerical results
V.4.5 Conclusions and perspectives
Conclusion
