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

Redistribution of the total energy of a fluid in motion, which is called "energy separation," has been observed in various flow situations. Understanding the underlying mechanism of this interesting phenomenon has been limited due to lack of the temporal information on flow and temperature fields. In the present study, numerical simulation of a viscous circular jet was performed to provide detailed temporal information on pressure, vorticity, and total temperature fields. Nondimensionalized governing equations, including mass, momentum, and total energy conservation equations, were simultaneously solved by an equal-order linear finite element and fractional four-step method. The results show that the formation and transport of vortices induce a pressure fluctuation in the flow field. The fluid, which flows through the disturbed pressure field, exchanges pressure work with the surroundings, and gains or loses total energy. This work exchange leads to higher and lower total temperature regions than the surroundings. In addition to the presence and movement of the vortices, the results indicate that the vortex-pairing process significantly intensifies the pressure fluctuation and corresponding total temperature difference. This implies that the vortex-pairing process is a very important process in intensifying energy separation and might explain the enhancement of energy separation in a jet using acoustic excitation.

A Numerical Study of Energy Separation in a Jet Flow

Temperature distribution and heat transfer in a transitional separated shear later

Heat/Mass Transfer and Leakage Flow on the Tip of a Simulated Gas Turbine Blade

Adiabatic wall temperature and heat transfer downstream of injection through two rows of holes.

A parametric study investigates the effects of wall shear on a two-dimensional turbulent boundary layer. A belt translating along the direction of the flow imparts the shear. Velocity measurements are performed at 12 streamwise locations with four surface-to-freestream velocity ratios (0, 0.38, 0.52, and 0.65) and a momentum-based Reynolds number between 770 and 1776. The velocity data indicate that the location of the "virtual origin" of the turbulent boundary layer "moves" downstream toward the trailing edge of the belt with increasing surface velocity. The highest belt velocity ratio (0.65) results in the removal of the "inner" region of the boundary layer. Measurements of the streamwise turbulent kinetic energy show an inner scaling at locations upstream and downstream of the belt, and the formation of a new self-similar structure on the moving surface itself. Good agreement is observed for the variation in the shape factor (H) and the skin friction coefficient (c f ) with the previous studies. The distribution of the energy spectrum downstream of the belt indicates peak values concentrated around 1 kHz for the stationary belt case in the near wall region (30 < y + < 50). However, with increasing belt velocity, this central peak plateaus over a wide frequency range (0.9―4 kHz).

Richard J Goldstein

Papers

A Numerical Study of Energy Separation in a Jet Flow

Temperature distribution and heat transfer in a transitional separated shear later

Heat/Mass Transfer and Leakage Flow on the Tip of a Simulated Gas Turbine Blade

Adiabatic wall temperature and heat transfer downstream of injection through two rows of holes.

Effect of Upstream Shear on Flow and Heat (Mass) Transfer Over a Flat Plate—Part I: Velocity Measurements