Three-dimensional thermal cellular convection in rectangular boxes

Abstract

The extension of the classic Rayleigh–Bénard problem of a horizontal layer heated from below to the three-dimensional convection in rectangular boxes is dealt with in this paper both numerically and experimentally. Also discussed is the influence of shear flows in tilted boxes and the transition to time-dependent oscillatory convection. Three-dimensional numerical simulations allow the calculation of stationary solutions and the direct simulation of oscillatory instabilities. We limited ourselves to laminar and transcritical flows. For studying the particular characteristics of three-dimensional convection in horizontal containers, we carefully selected two container geometries with aspect ratios of 10:4:1 and 4:2:1. The onset of steady cellular convection in tilted boxes is calculated by an iterative application of a combined finite-difference method and a Galerkin method. The appearance of longitudinal and transverse convection rolls is determined by means of inter-ferometrical measuring techniques and is compared with the results of the linear stability theory. The spatial flow structure and the transition to oscillatory convection is calculated for selected examples in the range of supercritical Rayleigh numbers. Experimental investigations concerning the stability behaviour of the steady solutions with regard to time-dependent disturbances show a distinct influence of the Prandtl number and confirm the importance of nonlinear effects.

Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box

“…4(b) shows the vertical component w through the midline of the cell. The occurrence of two square-shaped convection cells of opposite vorticity is in good agreement with experiment [36]. collaborators [26] and with experiment.…”

“…We then calculate the critical Rayleigh number R c and plot the Nusselt number N(R) as a function of the Rayleigh number R, and obtain good agreement with an analytical expression [34] and with a spectral code [35]. Finally, we show the spatial structure of the fields near onset, to allow comparison with experiment [36] and with other codes.…”

Non-linear wavelength selection for pattern-forming systems in channels

“…Rayleigh-Benard experiments in long boxes have identified significantly different and as yet unexplained behaviours for high and low Prandtl number fluids. At high Prandtl numbers little change in wavelength is observed as the Rayleigh number increases to as much as ten times its critical value whereas at low Prandtl numbers the wavelength is observed to increase steadily through the same range (Kirchartz & Oertel 1988). It is hoped that the present approach will provide new insight into this behaviour and also a way of predicting the behaviour of other similar pattern-forming systems.…”

“…In many technological applications, such as crystal growth processes, solar collectors and heat exchangers, motion occurs within a channel-like geometry and is strongly influenced by the sidewalls and end walls of the channel. Carefully controlled Rayleigh-Benard experiments in long channels (Kirchartz & Oertel 1988) show that cellular patterns consisting of convection rolls parallel to the end walls can be subject to significant variations in wavelength as the Rayleigh number increases above its threshold value. Such variations affect the heat transfer characteristics of the system and are therefore of interest in applications of the kind mentioned above.…”

Numerical simulation of three‐dimensional thermal convection on the array processor DAP 510

“…All these methods have in common that during the time stepping process the computation of the hydrostatic pressure is decoupled from the computation of the velocity. We shall concentrate here on pressure-correction methods (Harlow- (2)- (4) we use a modification of the pressure-correction approach due to Chorin[lS]. For simplicity we drop the index h in the following.…”