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

Thermal Convection in Porous Media at High Rayleigh Numbers

Film Cooling With Normal Injection Into a Supersonic Flow

An experimental system is designed, constructed, and operated to make local measurements of heat transfer from constant-temperature surfaces in a linear turbine cascade. The system includes a number of embedded heaters and a control system to maintain the turbine blades and end walls in the cascade at a uniform temperature. A five-axis measurement system is used to determine temperature profiles normal to the pressure and suction sides of the blades and to the end wall. Extrapolating these measurements close to the surface, the local heat transfer is calculated using Fourier's law. The system has been tested in the laboratory, and results are shown for the temperature distributions above the surfaces and for the local variations in the Nusselt number on the different surfaces in the cascade. The system can also be used to study the heat and mass transfer analogy as considerable data are available for mass transfer results with similar geometries.

Heat Transfer Study in a Linear Turbine Cascade Using a Thermal Boundary Layer Measurement Technique

Measurements of film cooling on a simulated turbine blade are conducted using a mass transfer technique. Under the influence of the endwall, dramatic changes of film cooling performance occur on the convex surface of the blade as compared to the region where the flow is two dimensional. The result is a triangular region, where coolant is swept away from the surface by the three-dimensional vortex-driven flow present between adjacent blades. In order to predict the area of this unprotected region, the influences of several parameters including density ratio, blowing rate, and number of rows of injection holes are studied. The presence of the endwall affects the film cooling performance on the concave surface only slightly.

Film Cooling of a Turbine Blade With Injection Through Two Rows of Holes in the Near-Endwall Region

Heat transfer is measured downstream of perpendicular injection of an air jet through a single hole into a turbulent mainstream boundary layer. The heat transfer coefficient, calculated from wall temperature measurements with a constant heat flux from the test surface, is determined with injection of both heated and unheated jets. The heat transfer coefficient near the hole is as much as 45 percent larger than the value without injection for a blowing rate (mass flux ratio) of 2.0. Even far downstream, the heat transfer coefficient is 10–15 percent greater than the flat plate value for blowing rates greater than 0.2. The increased value of the heat transfer coefficient near the point of injection is due to the high turbulence levels that arise from interaction between the jet and main flow near the point of injection. Significant variations of the heat transfer coefficient with Reynolds number or wall heat flux are not observed.

Richard J Goldstein

Papers

Thermal Convection in Porous Media at High Rayleigh Numbers

Film Cooling With Normal Injection Into a Supersonic Flow

Heat Transfer Study in a Linear Turbine Cascade Using a Thermal Boundary Layer Measurement Technique

Film Cooling of a Turbine Blade With Injection Through Two Rows of Holes in the Near-Endwall Region

Heat transfer and film cooling following normal injection through a round hole