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 ad­ vocated during these last twenty years: the first one is "statistical", and tries to model the evolution of averaged quantities of the flow. This com­ munity, which has followed the glorious trail of Taylor and Kolmogorov, 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.

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 Potential vorticity and Rossby number.- 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 free-shear flows.- 4 Wall flows.- 5 Thermal convection.- 6 Transition, coherent structures and Kolmogorov spectra.- IV- Shear-flow turbulence.- 1 Reynolds equations.- 2 Coherent vortices in free-shear layers.- 3 Coherent structures in wall flows.- 4 Turbulence, order and chaos.- V- Fourier analysis of homogeneous turbulence.- 1 Fourier representation of a flow.- 2 Navier-Stokes equations in Fourier space.- 3 Boussinesq approximation in Fourier space.- 4 Craya decomposition.- 5 Complex helical waves decomposition.- 6 Utilization of random functions.- 7 Moments of the velocity field, homogeneity and stationarity.- 8 Isotropy.- 9 The spectral tensor of an isotropic turbulence.- 10 Energy, helicity, enstrophy and scalar spectra.- 11 Alternative expressions of the spectral tensor.- 12 Axisymmetric turbulence.- 13 Rapid-distorsion theory.- VI- Isotropic turbulence: phenomenology and simulations.- 1 Introduction.- 2Triad interactions and detailed conservation.- 3 Transfer and Flux.- 4 Kolmogorov's 1941 theory.- 5 The Richardson law.- 6 Characteristic scales of turbulence.- 6.6.1.1 The dimension of the attractor.- 7 Skewness factor and enstrophy divergence.- 8 Coherent structures in 3D isotropic turbulence.- 9 Pressure spectrum.- 10 Phenomenology of passive scalar diffusion.- 11 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 Decay of kinetic energy and backscatter.- 11 The Renormalization-Group techniques.- 12 The E.D.Q.N.M. isotropic passive scalar.- 13 The decay of temperature fluctuations.- 14 Lagrangian particle pair dispersion.- 15 Single-particle diffusion.- VIII- Two-dimensional turbulence.- 1 Introduction.- 2 Spectral tools for two-dimensional isotropic turbulence.- 3 Fjortoft's theorem.- 4 The enstrophy cascade.- 5 Coherent vortices.- 6 The inverse energy cascade.- 7 The two-dimensional E.D.Q.N.M. model.- 8 Diffusion of a passive scalar.- 9 2D turbulence in a temporal mixing layer.- IX- Geostrophic turbulence.- 1 Introduction.- 2 Geostrophic approximation.- 3 Quasi-geostrophic potential vorticity equation.- 4 Baroclinic instability.- 5 The n-layer quasi-geostrophic model.- 6 Ekman layer.- 7 2D barotropic and baroclinic waves.- 8 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.- 4Two-dimensional turbulence over topography.- 5 Inviscid statistical mechanics of 2D point vortices.- 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.- 5 Two-dimensional mixing-layer unpredictability.- XII- Large-eddy simulations.- 1 The direct-numerical simulation of turbulence.- 2 Formalism of large-eddy simulations.- 3 The Smagorinsky model.- 4 LES in spectral space.- 5 LES in physical space.- 6 LES of two-dimensional turbulence.- XIII- Towards real-world turbulence.- 1 Introduction.- 2 Stably-stratified turbulence.- 3 Rotating turbulence.- 4 Separated flows.- 5 Compressible flows.- 6 Conclusion.- References.