This is the 4th edition of a book originally published by Kluwer Academic Publishers. It is an exhaustive monograph on turbulence in fluids in its theoretical and applied aspects, with many advanced developments using mathematical spectral methods (two-point closures like the EDQNM theory), direct-numerical simulations, and large-eddy simulations. The book is still of great actuality on a topic of great importance for engineering and environmental applications, and presents a very detailed presentation of the field. The fourth edition incorporates new results coming from research work done since 1997. Many of these results come from direct and large-eddy simulations methods, which have provided significant advances in problems such as turbulent mixing or thermal exchanges (with and without gravity effects).

to Turbulence in Fluid Mechanics.- Basic Fluid Dynamics.- Transition to Turbulence.- Shear Flow Turbulence.- Fourier Analysis of Homogeneous Turbulence.- Isotropic Turbulence: Phenomenology and Simulations.- Analytical Theories and Stochastic Models.- Two-Dimensional Turbulence.- Beyond Two-Dimensional Turbulence in GFD.- Statistical Thermodynamics of Turbulence.- Statistical Predictability Theory.- Large-Eddy Simulations.- Towards "Real World Turbulence".

Turbulence is a dangerous topic which is often at the origin of serious ?ghts in the scienti?c meetings devoted to it since it represents extremely di?erent points of view, all of which have in common their complexity, as well as an inability to solve the problem. It is even di?cult to agree on what exactly is the problem to be solved. Extremely schematically, two opposing points of view had been adv- ated during these last thirty years: the ?rst one was "statistical", and tried to model the evolution of averaged quantities of the ?ow. This community, which had followed the glorious trail of Taylor and Kolmogorov, believed in the phenomenology of cascades, and strongly disputed the possibility of any coherence or order associated to turbulence. On the other bank of the river standed the "coherence among chaos" community, which considered turbulence from a purely deterministic point of view, by studying either the behaviour of dynamical systems, or the stability of ?ows in various situations. To this community were also associated the experimentalists and computer simulators who sought to identify coherent vortices in ?ows. Situation is more complex now, and the existence of these two camps is less clear. In fact a third point of view pushed by people from the physics community has emerged, with the concepts of renormalization group theory, multifractality, mixing, and Lagrangian approaches.