Turbulence in Fluids

Turbulence is a dangerous topic which is often at the origin of serious fights in the scientific meetings devoted to it since it represents extremely different points of view, all of which have in common their complexity, as well as an inability to solve the problem. It is even difficult to agree on what exactly is the problem to be solved. Extremely schematically, two opposing points of view have been advocated during these last ten years: the first one is "statistical", and tries to model the evolution of averaged quantities of the flow. This com­ has followed the glorious trail of Taylor and Kolmogorov, munity, which believes in the phenomenology of cascades, and strongly disputes the possibility of any coherence or order associated to turbulence. On the other bank of the river stands the "coherence among chaos" community, which considers turbulence from a purely deterministic po­ int of view, by studying either the behaviour of dynamical systems, or the stability of flows in various situations. To this community are also associated the experimentalists who seek to identify coherent structures in shear flows.

Springer Book Archives

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?.- 7 One-point closure modelling.- 8 Outline of the following chapters.- 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 approximation.- 9 Internal inertial-gravity waves.- 10 Barré de Saint-Venant equations.- 11 Gravity waves in a fluid of arbitrary depth.- III Transition to turbulence.- 1 The Reynolds number.- 2 Linear-instability theory.- 3 Transition in shear flows.- 4 The Rayleigh number.- 5 The Rossby number.- 6 The Froude Number.- 7 Turbulence, order and chaos.- IV The Fourier space.- 1 Fourier representation of a flow.- 2 Navier-Stokes equations in Fourier space.- 3 Boussinesq approximation 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 Inhomogeneous turbulence.- 2 Triad interactions and detailed conservation.- 3 Transfer and Flux.- 4 The Kolmogorov theory.- 6.4.1 Oboukhov's theory.- 5 The Richardson law.- 6 Characteristic scales of turbulence.- 7 Skewnessfactor and enstrophy divergence.- 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 The Renormalization-Group 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.- 6 Single-particle diffusion.- 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 The Smagorinsky model.- 4 L.E.S. of 3-D isotropic turbulence.- 5 L.E.S. of two-dimensional turbulence.- XIII Towards real-world turbulence.- 1 Introduction.- 2 Stably-stratified turbulence.- 3 The two-dimensional mixing layer.- 4 3D numerical simulations of the mixing layer.- 5Conclusion.