An analysis is presented of the steady states of two-dimensional convection near threshold in a laterally finite container with aspect ratio 2L [Gt ] 1. It is shown that the allowed wavevectors which can occur in the bulk of the container are reduced from a band |q| ∼ [(R − R0)/R0]½ in the laterally infinite system to a band |q| ∼ (R − R0)/R0 in a system with sidewalls (R is the Rayleigh number and R0 its critical value in the infinite system). The analysis involves an expansion of the hydrodynamic equations in the small parameter [(R − R0)/R0]½, and leads to amplitude equations with boundary conditions, which generalize to higher order those previously obtained by Newell & Whitehead and Segel. The precise values of the allowed wavevectors depend on the Prandtl number of the fluid and the thermal properties of the sidewalls. For certain values of these parameters all the allowed wavevectors are less than the critical value q0. The applicability of the results to convection in a rectangular container is briefly discussed.

http://uqbar.rockefeller.edu/siggia/Publications/1980-9_files/crossJFluMec1981.pdf

Phase-winding solutions in a finite container above the convective threshold

Non-Oberbeck–Boussinesq (NOB) effects on the flow organization in two-dimensional Rayleigh–Benard turbulence are numerically analysed. The working fluid is water. We focus on the temperature profiles, the centre temperature, the Nusselt number and on the analysis of the velocity field. Several velocity amplitudes (or Reynolds numbers) and several kinetic profiles are introduced and studied; these together describe the various features of the rather complex flow organization. The results are presented both as functions of the Rayleigh number Ra (with Ra up to 108) for fixed temperature difference Δ between top and bottom plates and as functions of Δ (‘non-Oberbeck–Boussinesqness’) for fixed Ra with Δ up to 60K. All results are consistent with the available experimental NOB data for the centre temperature Tc and the Nusselt number ratio NuNOB/NuOB (the label OB meaning that the Oberbeck–Boussinesq conditions are valid). For the temperature profiles we find – due to plume emission from the boundary layers – increasing deviations from the extended Prandtl–Blasius boundary layer theory presented in Ahlers et al. (J. Fluid Mech., vol. 569, 2006, p. 409) with increasing Ra, while the centre temperature itself is surprisingly well predicted by that theory. For given non-Oberbeck–Boussinesqness Δ, both the centre temperature Tc and the Nusselt number ratio NuNOB/NuOB only weakly depend on Ra in the Ra range considered here.Beyond Ra ≈ 106 the flow consists of a large diagonal centre convection roll and two smaller rolls in the upper and lower corners, respectively (‘corner flows’). Also in the NOB case the centre convection roll is still characterized by only one velocity scale. In contrast, the top and bottom corner flows are then of different strengths, the bottom one being a factor 1.3 faster (for Δ = 40K) than the top one, due to the lower viscosity in the hotter bottom boundary layer. Under NOB conditions the enhanced lower corner flow as well as the enhanced centre roll lead to an enhancement of the volume averaged energy based Reynolds number of about 4% to 5% for Δ = 60K. Moreover, we find , with β the thermal expansion coefficient and Tm the arithmetic mean temperature between top and bottom plate temperatures. This corresponds to the ratio of the free fall velocities at the respective temperatures. By artificially switching off the temperature dependence of β in the numerics, the NOB modifications of ReE is less than 1% even at Δ = 60K, revealing the temperature dependence of the thermal expansion coefficient as the main origin of the NOB effects on the global Reynolds number in water.

Flow organization in two-dimensional non-Oberbeck-Boussinesq Rayleigh-Bénard convection in water

On the existence of thermoconvective rolls, transverse to a superimposed mean poiseuille flow

https://www.eng.tau.ac.il/~gelfgat/documents/Gelfgat_Ref13.pdf

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Linear stability of mixed convection flows in horizontal rectangular channels of finite transversal extension heated from below

We study the onset of convection in an infinite channel of rectangular cross section and heated from below; the four boundaries are supposed to be rigid. Three-dimensional transverse rolls are preferred to longitudinal rolls and to finite transverse rolls. We also describe the shape of the convective motion at the critical point. Introduction The study of the onset of convection inside a completely confined region began with Davis in 1967, [1]; this author, considering a fluid maintained in a rectangular parallelepiped heated from below, performed a linear analysis and predicted for a critical thermal gradient the appearance of \"convective finite rolls (rolls with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter sides of the box\". This earlier work, already criticized by Catton [2], regarding the validity of Davis' trial functions, was severely corrected by Davies-Jones [3], who demonstrated that the Davis' finite rolls could never be a solution of the equations that describe the movement of the fluid: the true structure at the onset of convection is purely threedimensional (no component of the fluid velocity vanishes identically). Yet, DaviesJones showed that, although the Davis' finite rolls could not exist, they represent a good approximation of the true, three-dimensional solution. To show the validity of this latest assertion, he calculated tKe stability of the movement of a fluid confined in an infinite channel of rectangular cross section, heated from below, and maintained between two \"free\" horizontal boundaries, which allows very simple — and exact — solutions of the equations but does not permit the comparison with experiments. The aim of this paper is a complete inquiry of the onset of convection in a \"reasonable\" apparatus (i.e. allowing comparison with experiments) — with rigid horizontal boundaries — and especially the investigation of the differences between the \"finite rolls\" approach and the real, three-dimensional convective structure. 0340-0204/81/0006-0141 $02.00 © Copyright by Walter de Gruyter & Co. Berlin New York 142 J. M. Luijkx, J. K. Platten 1. Basic equations A fluid is confined in an infinite channel of rectangular cross section (see Fig. 1). The horizontal boundaries (z = ± a) supposed to be rigid and perfectly heat-conducting are maintained at different temperatures. The lateral boundaries (y = ± b) are also rigid and perfectly heat-insulating. Fig. 1: System of coordinates. The boundary conditions are: = v(z=.±a) = and dy y=± = 0. (1) (2)

J.M. Luijkx

Papers

On the Onset of Free Convection in a Rectangular Channel

Critical Reynolds number in pipe Poiseuille flow