Turbulence in Fluids (2^nd Ed., 2nd ed. 1990. Softcover reprint of the original 2nd ed. 1990) Stochastic and Numerical Modelling Fluid Mechanics and Its Applications Series, Vol. 1

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.

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.- 2.10.1 Derivation of the equations.- 2.10.2 The potential vorticity.- 2.10.3 Inertial-gravity waves.- 2.10.3.1 Analogy with two-dimensional compressible gas.- 11 Gravity waves in a fluid of arbitrary depth.- III Transition to turbulence.- 1 The Reynolds number.- 2 Linear-instability theory.- 3.2.1 The Orr-Sommerfeld equation.- 3.2.2 The Rayleigh equation.- 3.2.2.1 Kuo equation.- 3 Transition in shear flows.- 3.3.1 Free-shear flows.- 3.3.1.1 Mixing layers.- 3.3.1.2 Mixing layer with differential rotation.- 3.3.1.3 Plane jets and wakes.- 3.3.2 Wall flows.- 3.3.2.1 The boundary layer.- 3.3.2.2 Poiseuille flow.- 3.3.3 Transition, coherent structures and Kolmogorov spectra.- 3.3.3.1 Linear stability of a vortex filament within a shear.- 3.3.4 Compressible turbulence.- 3.3.4.1 Compressible mixing layer.- 3.3.4.2 Compressible wake.- 3.3.4.3 Compressible boundary layer.- 4 The Rayleigh number.- 5 The Rossby number.- 3.5.1 Quasi-two-dimensional flow submitted to rotation.- 3.5.1.1 Linear analysis.- 3.5.1.2 The straining of absolute vorticity.- 6 The Froude Number.- 7 Turbulence, order and chaos.- IV The Fourier space.- 1 Fourier representation of a flow.- 4.1.1 Flow “within a box”:.- 4.1.2 Integral Fourier representation.- 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.- 6.1.1 The mixing-length theory.- 6.1.2 Application of mixing-length to turbulent-shear flows.- 6.1.2.1 The plane jet.- 6.1.2.2 The round jet.- 6.1.2.3 The plane wake.- 6.1.2.4 The round wake.- 6.1.2.5 The plane mixing layer.- 6.1.2.6 The boundary layer.- 2 Triad interactions and detailed conservation.- 6.2.1 Quadratic invariants in physical space.- 6.2.1.1 Kinetic energy.- 6.2.1.2 Helicity.- 6.2.1.3 Passive scalar.- 3 Transfer and Flux.- 4 The Kolmogorov theory.- 6.4.1 Oboukhov’s theory.- 5 The Richardson law.- 6 Characteristic scales of turbulence.- 6.6.1 The degrees of freedom of turbulence.- 6.6.1.1 The dimension of the attractor.- 6.6.2 The Taylor microscale.- 6.6.3 Self-similar decay.- 7 Skewness factor and enstrophy divergence.- 6.7.1 The skewness factor.- 6.7.2 Does enstrophy blow up at a finite time?.- 6.7.2.1 The constant skewness model.- 6.7.2.2 Positiveness of the skewness.- 6.7.2.3 Enstrophy blow up theorem.- 6.7.2.4 A self-similar model.- 6.7.2.5 Oboukhov’s enstrophy blow up model.- 6.7.2.6 Discussion.- 6.7.3 The viscous case.- 8 The internal intermittency.- 6.8.1 The Kolmogorov-Oboukhov-Yaglom theory.- 6.8.2 The Novikov-Stewart (1964) model.- 6.8.3 Experimental and numerical results.- 6.8.4 Temperature and velocity 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.- 7.11.1 The R.N.G. algebra.- 7.11.2 Two-point closure and R.N.G. techniques.- 7.11.2.1 The k–5/3 range.- 7.11.2.2 The infrared spectrum.- VIII Diffusion of passive scalars.- 1 Introduction.- 2 Phenomenology of the homogeneous passive scalar diffusion.- 8.2.1 The inertial-convective range.- 8.2.2 The inertial-conductive range.- 8.2.3 The viscous-convective range.- 3 The E.D.Q.N.M. isotropic passive scalar.- 8.3.1 A simplified E.D.Q.N.M. model.- 8.3.2 E.D.Q.N.M. scalar-enstrophy blow up.- 4 The decay of temperature fluctuations.- 8.4.1 Phenomenology.- 8.4.1.1 Non-local interactions theory.- 8.4.1.2 Self-similar decay.- 8.4.1.3 Anomalous temperature decay.- 8.4.2 Experimental temperature decay data.- 8.4.3 Discussion of the L.E.S. results.- 8.4.4 Diffusion in stationary turbulence.- 5 Lagrangian particle pair dispersion.- 6 Single-particle diffusion.- 8.6.1 Taylor’s diffusion law.- 8.6.2 E.D.Q.N.M. approach to single-particle diffusion.- IX Two-dimensional and quasi-geostrophic turbulence.- 1 Introduction.- 2 The quasi-geostrophic theory.- 9.2.1 The geostrophic approximation.- 9.2.2 The quasi-geostrophic potential vorticity equation.- 9.2.3 The n-layer quasi-geostrophic model.- 9.2.4 Interaction with an Ekman layer.- 9.2.4.1 Geostrophic flow above an Ekman layer.- 9.2.4.2 The upper Ekman layer.- 9.2.5 Barotropic and baroclinic waves.- 3 Two-dimensional isotropic turbulence.- 9.3.1 Fjortoft’s theorem.- 9.3.2 The enstrophy cascade.- 9.3.3 The inverse energy cascade.- 9.3.4 The two-dimensional E.D.Q.N.M. model.- 9.3.5 Freely-decaying turbulence.- 4 Diffusion of a passive scalar.- 5 Geostrophic turbulence.- 9.5.1 Rapidly-rotating stratified fluid of arbitrary depth.- 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.- 12.2.1 Large and subgrid scales.- 12.2.2 L.E.S. and the predictability problem.- 3 The Smagorinsky model.- 4 L.E.S. of 3-D isotropic turbulence.- 12.4.1 Spectral eddy-viscosity and diffusivity.- 12.4.2 Spectral large-eddy simulations.- 12.4.3 The anomalous spectral eddy-diffusivity.- 12.4.4 Alternative approaches.- 12.4.5 A local formulation of the spectral eddy-viscosity.- 5 L.E.S. of two-dimensional turbulence.- XIII Towards real-world turbulence.- 1 Introduction.- 2 Stably-stratified turbulence.- 13.2.1 The so-called “collapse” problem.- 13.2.2 A numerical approach to the collapse.- 3 The two-dimensional mixing layer.- 13.3.1 Generalities.- 13.3.2 Two-dimensional turbidence in the mixing layer.- 13.3.3 Two-dimensional unpredictability.- 13.3.4 Two-dimensional unpredictability and 3D growth.- 4 3D numerical simulations of the mixing layer.- 13.4.1 Direct-numerical simulations.- 13.4.2 Large-eddy simulations of mixing layers.- 13.4.3 Recreation of the coherent structures.- 13.4.4 Rotating mixing layers.- 5 Conclusion.