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

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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.

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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

Second-order pressure structure functions, estimated from direct numerical simulations of homogeneous and nonhomogeneous turbulent flows, exhibit a significant Reynolds number dependence in the dissipative range. This dependence mainly reflects the contribution from low wave numbers to the instantaneous pressure gradient.

Reynolds number dependence of the second-order turbulent pressure structure function

The frequency dependence of a turbulent Prandtl number, defined using the Reynolds shear stress and normal heat flux co‐spectra, in various turbulent shear flows accentuates the inadequacy of the Reynolds analogy from a spectral point of view. For the same flows, the analogy between spectral distributions of the kinetic energy and the temperature variance is validated over a significant frequency domain. This analogy suggests a spectral parameter which is very close to unity in this frequency range for all the flows considered.

Spectral relationships between velocity and temperature fluctuations in turbulent shear flows

Hot-wire anemometry measurements are made in a rough wall turbulent boundary layer to assess the dependence of the mean turbulent kinetic energy dissipation coefficient, $$C_\varepsilon$$
 , on the distance from the wall and the Taylor microscale Reynolds number, $$Re_\lambda$$
 . The locally isotropic expression $$\langle \varepsilon _\mathrm{{iso}}\rangle$$
 is used as a surrogate for the mean turbulent kinetic energy dissipation rate, $$\langle \varepsilon \rangle$$
 , for calculating $$C_\varepsilon =\langle \varepsilon \rangle L/u^{\prime 3}$$
 ; L and $$u'$$
 are the integral length scale and the rms of the longitudinal velocity fluctuation, respectively. The measurements show that $$C_\varepsilon$$
 varies significantly near the wall, but becomes almost constant in the region $$0.2 \le y/\delta \le 0.6$$
 of the boundary layer, a region where $$Re_\lambda$$
 is also practically constant.

Behaviour of the energy dissipation coefficient in a rough wall turbulent boundary layer

Starting with the Navier-Stokes equations, a transport equation is written for the sum of the squared vorticity increments in homogeneous isotropic turbulence. This equation is compared with that for the sum of the squared velocity increments; whereas the latter equation exhibits a linear dependence on separation, the former does not. In the limit of a negligibly small separation, the new equation expresses a balance between the production and dissipation of the mean square vorticity gradient. All terms in the equation have been measured using a three-component vorticity probe in the self-preserving region of a low Reynolds number turbulent wake.

Transport of turbulent vorticity increments

The dominant view in the theory of fluid turbulence assumes that, once the effect of the Reynolds number is negligible, moments of order n of the longitudinal velocity increment, ( δ u), can be described by a simple power-law r ζ n, where the scaling exponent ζn depends on n and, except for ζ 3 ( = 1 ), needs to be determined. In this Letter, we show that applying Holder's inequality to the power-law form ( δ u ) n ¯ ∼ ( r L ) ζ n (with r / L ⪡ 1; L is an integral length scale) leads to the following mathematical constraint: ζ 2 p = p ζ 2. When we further apply the Cauchy–Schwarz inequality, a particular case of Holder's inequality, to | ( δ u ) 3 ¯ | with ζ 3 = 1, we obtain the following constraint: ζ 2 ≤ 2 / 3. Finally, when Holder's inequality is also applied to the power-law form ( | δ u | ) n ¯ ∼ ( r L ) ζ n (this form is often used in the extended self-similarity analysis) while assuming ζ 3 = 1, it leads to ζ 2 = 2 / 3. The present results show that the scaling exponents predicted by the 1941 theory of Kolmogorov in the limit of infinitely large Reynolds number comply with Holder's inequality. On the other hand, scaling exponents, except for ζ3, predicted by current small-scale intermittency models do not comply with Holder's inequality, most probably because they were estimated in finite Reynolds number turbulence. The results reported in this Letter should guide the development of new theoretical and modeling approaches so that they are consistent with the constraints imposed by Holder's inequality.

R. A. Antonia

Papers

Reynolds number dependence of the second-order turbulent pressure structure function

Spectral relationships between velocity and temperature fluctuations in turbulent shear flows

Behaviour of the energy dissipation coefficient in a rough wall turbulent boundary layer

Transport of turbulent vorticity increments

Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence