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.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications

This review covers Verification, Validation, Confirmation and related subjects for computational fluid dynamics (CFD), including error taxonomies, error estimation and banding, convergence rates, surrogate estimators, nonlinear dynamics, and error estimation for grid adaptation vs Quantification of Uncertainty.

/pdf/quantification-of-uncertainty-in-computational-fluid-3imbkhwf41.pdf

Quantification of uncertainty in computational fluid dynamics

AIJ guidelines for practical applications of CFD to pedestrian wind environment around buildings

Keywords: visualisation de l'ecoulement ; tourbillon ; dynamique des : fluides ; aubes ; cylindre ; instabilite ; ecoulement : secondaire Note: moult photos Reference Record created on 2005-11-18, modified on 2016-08-08

An Album of Fluid Motion

A professional problem exists in the computational fluid dynamics community and also in the broader area of com­ putational physics. Namely, there is a need for higher stan­ dards on the control of numerical accuracy. The numerical fluid dynamics community is aware of this problem but, although individual researchers strive to control accuracy, the issue has not to our knowledge been addressed

Editorial Policy Statement on the Control of Numerical Accuracy

The three-dimensional motion of a freely descending parachute is studied with a five-degreeoffreedom analysis (the roll motion is neglected). The equations of motion are nondimensionalized and the resulting parameters discussed. Exact expressions are given for the longitudinal and lateral small-disturbance stability of the familiar gliding motion of parachutes. The breakdown of these small-disturbance expressions is illustrated by exact large-disturbance studies. A large longitudinal disturbance of most parachutes will result in a large pitching motion, whereas a large lateral (out of the glide plane) disturbance will usually cause a large angle vertical coning motion. Exact algebraic expressions are given for the coning mode, which is a stable rotation, and a small amount of available coning data is included for comparison. Some parametric computer studies of the various motions are also shown.

A theory of three-dimensional parachute dynamic stability

A finite difference model is presented for viscous two dimensional flow of a uniform stream past an oscillating cylinder. A noninertial coordinate transformation is used so that the grid mesh remains fixed relative to the accelerating cylinder. Three types of cylinder motion are considered: oscillation in a still fluid, oscillation parallel to a moving stream, and oscillation transverse to a moving stream. Computations are made for Reynolds numbers between 1 and 100 and amplitude-to-diameter ratios from 0.1 to 2.0. The computed results correctly predict the lock-in or wake-capture phenomenon which occurs when cylinder oscillation is near the natural vortex shedding frequency. Drag, lift, and inertia effects are extracted from the numerical results. Detailed computations at a Reynolds number of 80 are shown to be in quantitative agreement with available experimental data for oscillating cylinders.

Numerical Solution for Laminar Two Dimensional Flow About a Cylinder Oscillating in a Uniform Stream

Experiments were conducted to study transition to turbulence in pipe flows started from rest with a linear increase in mean velocity. The data were taken at the Unsteady Flow Loop Facility at the Naval Underwater System Center, using a 5-cm diameter pipe 30 meters long. Instrumentation included static pressure, wall pressure, and wall shear stress sensors, as well as a laser Doppler velocimeter and a transient flowmeter. A downstream control valve was programmed to produce nearly constant mean flow accelerations, a from 2 to 12 m/s 2

Frank M. White

Papers

An Album of Fluid Motion

Editorial Policy Statement on the Control of Numerical Accuracy

A theory of three-dimensional parachute dynamic stability

Numerical Solution for Laminar Two Dimensional Flow About a Cylinder Oscillating in a Uniform Stream

Experiments on Transition to Turbulence in a Constant-Acceleration Pipe Flow