"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.

Lattice boltzmann method for fluid flows

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

http://home.eps.hw.ac.uk/~ab226/tmp/implicit/lele_92.pdf

Compact finite difference schemes with spectral-like resolution

We develop the dynamic renormalization group (RNG) method for hydrodynamic turbulence. This procedure, which uses dynamic scaling and invariance together with iterated perturbation methods, allows us to evaluate transport coefficients and transport equations for the large-scale (slow) modes. The RNG theory, which does not include any experimentally adjustable parameters, gives the following numerical values for important constants of turbulent flows: Kolmogorov constant for the inertial-range spectrumCK=1.617; turbulent Prandtl number for high-Reynolds-number heat transferPt=0.7179; Batchelor constantBa=1.161; and skewness factor¯S3=0.4878. A differentialK-\(\bar \varepsilon \) model is derived, which, in the high-Reynolds-number regions of the flow, gives the algebraic relationv=0.0837 K2/\(\bar \varepsilon \), decay of isotropic turbulence asK=O(t−1.3307), and the von Karman constantκ=0.372. A differential transport model, based on differential relations betweenK,\(\bar \varepsilon \), andν, is derived that is not divergent whenK→ 0 and\(\bar \varepsilon \) is finite. This latter model is particularly useful near walls.

Renormalization group analysis of turbulence I. Basic theory

Two‐dimensional turbulence has both kinetic energy and mean‐square vorticity as inviscid constants of motion. Consequently it admits two formal inertial ranges, E(k)∼e2/3k−5/3 and E(k)∼η2/3k−3, where e is the rate of cascade of kinetic energy per unit mass, η is the rate of cascade of mean‐square vorticity, and the kinetic energy per unit mass is ∫0∞E(k) dk. The −53 range is found to entail backward energy cascade, from higher to lower wavenumbers k, together with zero‐vorticity flow. The −3 range gives an upward vorticity flow and zero‐energy flow. The paradox in these results is resolved by the irreducibly triangular nature of the elementary wavenumber interactions. The formal −3 range gives a nonlocal cascade and consequently must be modified by logarithmic factors. If energy is fed in at a constant rate to a band of wavenumbers ∼ki and the Reynolds number is large, it is conjectured that a quasi‐steady‐state results with a −53 range for k « ki and a −3 range for k » ki, up to the viscous cutoff. The t...

Inertial Ranges in Two‐Dimensional Turbulence

Inertial‐Range Spectrum of Hydromagnetic Turbulence

The theory of two-dimensional turbulence is reviewed and unified, and some hydrodynamic and plasma applications are considered. The topics covered include some equations of incompressible hydrodynamics, absolute statistical equilibrium, spectral transport of energy and enstrophy, turbulence on the surface of a rotating sphere, turbulent diffusion, MHD turbulence, and two-dimensional superflow. Finally, an attempt is made to assess the status and future of the principal research topics which have been discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/43/5/001/pdf

Two-dimensional turbulence

Single‐particle diffusion in a multivariate‐normal, incompressible, stationary, isotropic velocity field is calculated in two and three dimensions by computer simulation and by the direct‐interaction approximation. The computer simulations are carried out by storing the velocity field as a set of Fourier components and synthesizing in physical space only along the particle trajectories. The spectra taken for the velocity field are of the form E(k) ∝ δ(k − k0) and E(k) ∝ k4 exp (−2k2/k02) in three dimensions, and E(k) ∝ δ(k − k0) and E(k) ∝ k3 exp (−3k2/2k02) in two dimensions. Both frozen Eulerian fields and fields with Gaussian time correlation are treated. The simulation results agree with Taylor's picture of a classical diffusion process for times long compared with the eddy circulation time, except for the frozen‐Eulerian‐field runs in two dimensions, where strong trapping effects are found. The direct‐interaction approximations for Lagrangian velocity correlation, eddy diffusivity, dispersion, and mo...

Diffusion by a Random Velocity Field

Batchelor's theory of the turbulent straining of small‐spatial‐scale amplitude variations of a convected scalar field is re‐examined to see the effects of fluctuation of the rates of strain in space and time. The k−1 viscous‐convective‐range spectrum is unaltered, except for the constant of proportionality, but spectrum level in the viscous‐diffusive range displays a sensitivity to fluctuations which increases with wavenumber. The Gaussian cutoff found by Batchelor is replaced by more gently decreasing dependences of spectrum level on wavenumber. The scalar spectrum is also treated by the Lagrangian‐history direct‐interaction approximation. The k−5/3 inertial‐convective range of Obukhov and Corrsin, the k−1 viscous‐convective range, and the k−17/3 inertial‐diffusive range of Batchelor, Howells, and Townsend all are recovered. For a given rate of spectral transport of scalar variance, the predicted spectrum levels in the k−5/3 and k−1 ranges are too small, in comparison with experiment, by numerical factor...

Robert H. Kraichnan

Papers

Inertial Ranges in Two‐Dimensional Turbulence

Inertial‐Range Spectrum of Hydromagnetic Turbulence

Two-dimensional turbulence

Diffusion by a Random Velocity Field

Small‐Scale Structure of a Scalar Field Convected by Turbulence