Order and disorder in two- and three-dimensional Bénard convection

Abstract

The character of transition from laminar to chaotic Rayleigh–Bénard convection in a fluid layer bounded by free-slip walls is studied numerically in two and three space dimensions. While the behaviour of finite-mode, limited-spatial-resolution dynamical systems may indicate the existence of two-dimensional chaotic solutions, we find that, this chaos is a product of inadequate spatial resolution. It is shown that as the order of a finite-mode model increases from three (the Lorenz model) to the full Boussinesq system, the degree of chaos increases irregularly at first and then abruptly decreases; no strong chaos is observed with sufficiently high resolution.In high-Prandtl-number σ two-dimensional Boussinesq convection, it is found that there are finite critical Rayleigh numbers Ra for the onset of single- and two-frequency oscillatory motion, Ra [gsim ] 60 Rac and Ra [gsim ] 290 Rac respectively, for σ = 6.8. These critical Rayleigh numbers are much higher than those at which three-dimensional convection achieves multifrequency oscillatory states. However, in two dimensions no additional complicating fluctuations are found, and the system seems to revert to periodic, single-frequency convection at high Rayleigh number, e.g. when Ra [gsim ] 800Rac at σ = 6.8.In three dimensions with σ = 10 and aspect ratio 1/√2, single-frequency convection begins at Ra ≈ 40Rac and two-frequency convection starts at Ra ≈ 50Rac. The onset of chaos seems coincident with the appearance of a third discrete frequency when Ra [gsim ] 65Rac. This three-dimensional transition process may be consistent with the scenario of Ruelle, Takens & Newhouse (1978).As Ra increases through the chaotic regime, various measures of chaos show an increasing degree of small-scale structure, horizontal mixing and other characteristics of thermal turbulence. While the three-dimensional energy in these flows is still quite small, it is evidently sufficient to overcome the strong dynamical constraints imposed by two dimensions.Gollub & Benson (1980) found experimentally that frequency modulation of lower boundary temperature Ra(t) = Ra(0) [1 + ε sin ωt] induces chaotic behaviour in a quasi-periodic flow close to transition. We investigate numerically the effects of finite modulation of Ra on the flow far below natural transition (R = 50Rac). By choosing ε = 0.1 and the Rayleigh-number oscillation frequency ω incommensurate with the frequencies of the quasi-periodic motion, transition to chaos is induced early. This result also seems consistent with the Ruelle et al. scenario and leads to the conjecture that periodic modulation of the Rayleigh number of the above form in a two-frequency flow may provide the third frequency necessary for chaotic flow.For moderate Prandtl number, σ = 1, our results show that two-dimensional flow seems free of oscillation, while three-dimensional flow is vigorously turbulent for Ra [gsim ] 70Rac.