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

Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112095000462

On the identification of a vortex

It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

/pdf/the-proper-orthogonal-decomposition-in-the-analysis-of-44vs42u998.pdf

The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

Progress in urban climatology over the two decades since the first publication of the International Journal of Climatology is reviewed. It is emphasized that urban climatology during this period has benefited from conceptual advances made in microclimatology and boundary-layer climatology in general. The role of scale, heterogeneity, dynamic source areas for turbulent fluxes and the complexity introduced by the roughness sublayer over the tall, rigid roughness elements of cities is described. The diversity of urban heat islands, depending on the medium sensed and the sensing technique, is explained. The review focuses on two areas within urban climatology. First, it assesses advances in the study of selected urban climatic processes relating to urban atmospheric turbulence (including surface roughness) and exchange processes for energy and water, at scales of consideration ranging from individual facets of the urban environment, through streets and city blocks to neighbourhoods. Second, it explores the literature on the urban temperature field. The state of knowledge about urban heat islands around 1980 is described and work since then is assessed in terms of similarities to and contrasts with that situation. Finally, the main advances are summarized and recommendations for urban climate work in the future are made. Copyright © 2003 Royal Meteorological Society.

/pdf/two-decades-of-urban-climate-research-a-review-of-turbulence-12p43b6trj.pdf

Two decades of urban climate research: a review of turbulence, exchanges of energy and water, and the urban heat island

Numerical simulations of fully developed turbulent channel flow at three Reynolds numbers up to Reτ=590 are reported. It is noted that the higher Reynolds number simulations exhibit fewer low Reynolds number effects than previous simulations at Reτ=180. A comprehensive set of statistics gathered from the simulations is available on the web at http://www.tam.uiuc.edu/Faculty/Moser/channel.

Direct numerical simulation of turbulent channel flow up to Reτ=590

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Rough-Wall Turbulent Boundary Layers

Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

/pdf/the-phenomenology-of-small-scale-turbulence-3a5uzrupts.pdf

The phenomenology of small-scale turbulence

Measurements are presented of the velocity structure function on the axis of a turbulent jet at Reynolds numbers Rλ ≤ 852 and in a turbulent duct flow at Rλ = 515. Moments of the structure function up to the eighteenth order were calculated, primarily with a view to establish accurately the dependence on the order of the inertial range power-law exponent and to draw conclusions about the distribution of energy transfer in the inertial range. Adequate definition of the probability density of the structure function was achieved only for moments of order n ≤ 10. It is shown, however, that, although the values of moments of n > 10 diverges from their true values, the dependence of the moment of the structure function on the separation r is still given to a fair accuracy for moments up to n ≈ 18. The results demonstrate that the inertial-range power-law exponent is closely approximated by a quadratic dependence on the power which for lower-order moments (n [lsim ] 12) would be consistent with a lognormal distribution. Higher-order moments diverge, however, from a lognormal distribution, which gives weight to Mandelbrot's (1971) conjecture that ‘Kolmogorov's third hypothesis’ is untenable in the strict sense. The intermittency parameter μ, appearing in the power-law exponent, has been determined from sixth-order moments 〈(δμ)6〉 ∼ r2−μ to be μ = 0.2 ± 0.05. This value coincides with that determined from non-centred dissipation correlations measured in identical conditions.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112084000513

High-order velocity structure functions in turbulent shear flows

The effects of surface roughness on a turbulent boundary layer are investigated by comparing measurements over two rough walls with measurements from a smooth wall boundary layer. The two rough surfaces have very different surface geometries although designed to produce the same roughness function, i.e. to have nominally the same effect on the mean velocity profile. Different turbulent transport characteristics are observed for the rough surfaces. Substantial effects on the stresses occur throughout the layer showing that the roughness effects are not confined to the wall region. The turbulent energy production and the turbulent diffusion are significantly different between the two rough surfaces, the diffusion having opposite sign in the region γ/δ < 0.5. Although velocity spectra exhibit differences between the three surfaces, the mean energy dissipation rate does not appear to be significantly affected by the roughness.

Surface roughness effects in turbulent boundary layers

Measurements in a zero-pressure-gradient turbulent boundary layer over a mesh-screen rough wall indicate several differences, in both inner and outer regions, in comparison to a smooth-wall boundary layer. The mean velocity distribution indicates that, apart from the expected k-type roughness function shift in the inner region, the strength of the rough-wall outer region ‘wake’ is larger than on a smooth wall. Normalizing on the wall shear stress, there is a significant increase in the normal turbulence intensity and a moderate increase in the Reynolds shear stress over the rough wall. The longitudinal turbulence intensity distribution is essentially the same for both surfaces. Normalized contributions to the Reynolds shear stress from the second (Q2) and fourth (Q4) quadrants are greater over the rough wall. The data indicate that not only are Q2 and Q4 events stronger on the rough wall but their frequency of occurrence is nearly twice as large for the rough wall as for the smooth wall. Comparison between smooth- and rough-wall spectra of the normal velocity fluctuation suggests that the strength of the active motion may depend on the nature of the surface.

R. A. Antonia

Papers

Rough-Wall Turbulent Boundary Layers

The phenomenology of small-scale turbulence

High-order velocity structure functions in turbulent shear flows

Surface roughness effects in turbulent boundary layers

Comparison between rough- and smooth-wall turbulent boundary layers