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.
We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, perhaps, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are also concentrated only near this perimeter. Perhaps as important, the measured quantities can refer to the total acoustic field rather than only to its unwanted component, and the methodology can automatically distinguish between the two. In the paper, we construct a general solution to the aforementioned noise control problem. The apparatus used for deriving the general solution is closely connected to the concepts of generalized potentials and boundary projections of Calderon's type. For a given total wave field, the application of Calderon's projections allows us to definitively split its incoming and outgoing components with respect to a particular domain of interest, which may have arbitrary shape. Then the controls are designed so that they suppress the incoming component for the domain to be shielded or alternatively, the outgoing component for the domain, which is complementary to the one to be shielded. To demonstrate that the new noise control technique is appropriate, we thoroughly work out a twodimensional model example that allows full analytical consideration. To conclude, we very briefly discuss the numerical (finite-difference) framework for active noise control that has, in fact, already been worked out, as well as some forthcoming extensions of the current work: optimization of the solution according to different criteria that would fit different practical requirements, applicability of the new technique to quasi-stationary problems, and active shielding in the case of the broad-band spectra of disturbances. In the future, the aforementioned finitedifference framework for active noise control is going to be used for analyzing complex configurations that originate from practical designs.
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