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

http://dns2.asia.edu.tw/~ysho/YSHO-English/2000%20Engineering/PDF/Int%20J%20Hea%20Mas%20Tra15,%201787.pdf

A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows

Review on thermal energy storage with phase change: materials, heat transfer analysis and applications

Laser-Doppler measurements of velocity distribution and reattachment length are reported downstream of a single backward-facing step mounted in a two-dimensional channel. Results are presented for laminar, transitional and turbulent flow of air in a Reynolds-number range of 70 < Re < 8000. The experimental results show that the various flow regimes are characterized by typical variations of the separation length with Reynolds number. The reported laser-Doppler measurements do not only yield the expected primary zone of recirculating flow attached to the backward-facing step but also show additional regions of flow separation downstream of the step and on both sides of the channel test section. These additional separation regions have not been previously reported in the literature.Although the high aspect ratio of the test section (1:36) ensured that the oncoming flow was fully developed and two-dimensional, the experiments showed that the flow downstream of the step only remained two-dimensional at low and high Reynolds numbers.The present study also included numerical predictions of backward-facing step flow. The two-dimensional steady differential equations for conservation of mass and momentum were solved. Results are reported and are compared with experiments for those Reynolds numbers for which the flow maintained its two-dimensionality in the experiments. Under these circumstances, good agreement between experimental and numerical results is obtained.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112083002839

Experimental and Theoretical Investigation of Backward-Facing Step Flow

▪ AbstractWe review the experimental evidence on turbulent flows over rough walls. Two parameters are important: the roughness Reynolds number ks+, which measures the effect of the roughness on the buffer layer, and the ratio of the boundary layer thickness to the roughness height, which determines whether a logarithmic layer survives. The behavior of transitionally rough surfaces with low ks+ depends a lot on their geometry. Riblets and other drag-reducing cases belong to this regime. In flows with δ/k ≲ 50, the effect of the roughness extends across the boundary layer, and is also variable. There is little left of the original wall-flow dynamics in these flows, which can perhaps be better described as flows over obstacles. We also review the evidence for the phenomenon of d-roughness. The theoretical arguments are sound, but the experimental evidence is inconclusive. Finally, we discuss some ideas on how rough walls can be modeled without the detailed computation of the flow around the roughness element...

Turbulent flows over rough walls

The complex three-dimensional flow in the endwall region near the base of a turbine blade has an important impact on the local heat transfer. The initial horseshoe vortex, the passage vortex, and resulting corner vortices cause large variations in heat transfer over the entire endwall region. Due to these large surface gradients in heat transfer, conventional measurement techniques generally do not provide in accurate determination of the local heat transfer coefficients. In the present study the heat/mass transfer analogy is used to examine the local transport coefficients for two different endwall boundary layer thicknesses and two free-stream Reynolds numbers. A linear turbine blade cascade is used in conjunction with a removable endwall plate. Napthalene (C{sub 10}H{sub 8}) is cast into a mold on the plate and the rate of naphthalene sublimation is determined at 6,000+ locations on the simulated endwall by employing a computer-aided data acquisition system. This technique allows one to obtain detailed contour plots of the local convection coefficient over the entire endwall. By examining the mass transfer contours, it is possible to infer information on three-dimensional flow in the passage between the blades. Extremely high transport coefficients on the endwall indicate locations of potential overheating and failuremore » in actual turbine.« less

Turbulent transport on the endwall in the region between adjacent turbine blades.

Overall heat transfer and mean temperature distribution measurements have been made of turbulent thermal convection in horizontal water layers heated from below. The Nusselt number is found to be proportional to Ra0·278 in the range 2·76 × 105 < Ra < 1·05 × 108. Eight discrete heat flux transitions are found in this Rayleigh number range. An interferometric method is used to measure the mean temperature distribution for Rayleigh numbers between 3·11 × 105 and 1·86 × 107. Direct visual and photographic observations of the fluctuating interferogram patterns show that the main heat transfer mechanism is the release of thermals from the boundary layers. For relatively low Rayleigh numbers (up to 5 × 105) many of the thermals reach the opposite surface and coalesce to form large masses of relatively warm fluid near the cold surface and masses of cold fluid near the warm surface, resulting in a temperature-gradient reversal. With increasing Rayleigh numbers, fewer and fewer thermals reach the opposite bounding surface and the thermals show persistent horizontal movements near the bounding surfaces. The central region of the layer becomes an isothermal core. The mean temperature distributions for the high Rayleigh number range are found to follow a Z−2 power law over a considerable range, where Z is the distance from the bounding surface. A very limited agreement with the theoretically predicted Z−1 power law is also found.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112073000091

Turbulent convection in a horizontal layer of water

Experiments have been performed to explore the qualitative and quantitative characteristics of thermals which ascend through the fluid environment above a heated horizontal surface. With water as the participating fluid, an electrochemical technique was employed which made the flow field visible and facilitated the direct observation of thermals. Measurements were also made of the fluid temperature above an active site of thermal generation.As seen in flow field photographs, a thermal has a mushroom-like appearance, with a blunted nearly hemispherical cap. At a given heating rate, thermals are generated at fixed sites which are spaced more or less regularly along the span of the heated surface. At these sites, the generation of thermals is periodic in time, thereby validating a prediction of Howard. Both the spatial frequency of the sites and the rate of thermal production increase with increases in heating rate. The break-up Rayleigh number of the conduction layer is shown to be a constant (within the uncertainties of the experiment), which is in accord with Howard's phenomenological model.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112070000927

Observations and other characteristics of thermals

Experiments are conducted to determine the heat transfer to a jet impinging at different oblique angles to a plane surface The main portion of the test plate contains a composite sheet of temperature-sensitive liquid crystal, which is sandwiched between a thin metallic-foil heater and a specially designed liquid bath The results indicate a displacement of the peak heat transfer from the geometric center of the jet origin, the displacement being a function primarily of impingement angle Contours of constant heat transfer coefficient are obtained and correlated with an empirical equation that permits determination of average Nusselt numbers over areas of interest

Heat transfer from a flat surface to an oblique impinging jet

Film cooling effectiveness measured following air injection through discrete holes into turbulent boundary layer of air on flat plate

/pdf/film-cooling-following-injection-through-inclined-circular-3vsguei14v.pdf

Richard J Goldstein

Papers

Turbulent transport on the endwall in the region between adjacent turbine blades.

Turbulent convection in a horizontal layer of water

Observations and other characteristics of thermals

Heat transfer from a flat surface to an oblique impinging jet

Film cooling following injection through inclined circular tubes