I Introduction to turbulence in fluid mechanics --; 1 Is it possible to define turbulence? --; 2 Examples of turbulent flows --; 3 Fully developed turbulence --; 4 Fluid turbulence and 'chaos' --; 5 'Deterministic' and statistical approaches --; 6 Why study isotropic turbulence? --; II Basic fluid dynamics --; 1 Eulerian notation and Lagrangian derivatives --; 2 The continuity equation --; 3 The conservation of momentum --; 4 The thermodynamic equation --; 5 The incompressibility assumption --; 6 The dynamics of vorticity --; 7 The generalized Kelvin theorem --; 8 The Boussinesq equations --; 9 Internal inertial-gravity waves --; 10 Barré de Saint-Venant equations --; III Transition to turbulence --; 1 The Reynolds number --; 2 The Rayleigh number --; 3 The Rossby number --; 4 The Froude Number --; 5 Turbulence, order and chaos --; IV The Fourier space --; 1 Fourier representation of a flow --; 2 Navier-Stokes equations in Fourier space --; 3 Boussinesq equations in the Fourier space --; 4 Craya decomposition --; 5 Complex helical waves decomposition --; V Kinematics of homogeneous turbulence --; 1 Utilization of random functions --; 2 Moments of the velocity field, homogeneity and stationarity --; 3 Isotropy --; 4 The spectral tensor of an isotropic turbulence --; 5 Energy, helicity, enstrophy and scalar spectra --; 6 Alternative expressions of the spectral tensor --; 7 Axisymmetric turbulence --; VI Phenomenological theories --; 1 The closure problem of turbulence --; 2 Karman-Howarth equations in Fourier space --; 3 Transfer and Flux --; 4 The Kolmogorov theory --; 5 The Richardson law --; 6 Characteristic scales of turbulence --; 7 The skewness factor --; 8 The internal intermittency --; VII Analytical theories and stochastic models --; 1 Introduction --; 2 The Quasi-Normal approximation --; 3 The Eddy-Damped Quasi-Normal type theories --; 4 The stochastic models --; 5 Phenomenology of the closures --; 6 Numerical resolution of the closure equations --; 7 The enstrophy divergence and energy catastrophe --; 8 The Burgers-M.R.C.M. model --; 9 Isotropic helical turbulence --; 10 The decay of kinetic energy --; 11 E.D.Q.N.M. and R.N.G. techniques --; VIII Diffusion of passive scalars --; 1 Introduction --; 2 Phenomenology of the homogeneous passive scalar diffusion --; 3 The E.D.Q.N.M. isotropic passive scalar --; 4 The decay of temperature fluctuations --; 5 Lagrangian particle pair dispersion --; IX Two-dimensional and quasi-geostrophic turbulence --; 1 Introduction --; 2 The quasi-geostrophic theory --; 3 Two-dimensional isotropic turbulence --; 4 Diffusion of a passive scalar --; 5 Geostrophic turbulence --; X Absolute equilibrium ensembles --; 1 Truncated Euler Equations --; 2 Liouville's theorem in the phase space --; 3 The application to two-dimensional turbulence --; 4 Two-dimensional turbulence over topography --; XI The statistical predictability theory --; 1 Introduction --; 2 The E.D.Q.N.M. predictability equations --; 3 Predictability of three dimensional turbulence --; 4 Predictability of two-dimensional turbulence --; XII Large-eddy simulations --; 1 The direct numerical simulation of turbulence --; 2 The Large Eddy Simulations --; 3 L.E.S. of 3-D isotropic turbulence --; 4 L.E.S. of two-dimensional turbulence --; XIII Towards 'real world turbulence' --; 1 Introduction --; 2 Stably Stratified Turbulence --; 3 The Mixing Layer --; 4 Conclusion --; References.