ABSTRACT
The robustness of control is a requirement to maintain a fluid layer at conductive equilibrium heated to a highly supercritical condition. Robustness determines how much uncertainties, or design parameter mismatches, can be tolerated. Both linear stability analysis and three-dimensional fully nonlinear simulations are used for the study of the linear quadratic Gaussian (LQG) controller. The parameter mismatches from the nominal conditions are introduced into the plant model, while the LQG compensator assumes nominal conditions. The mismatches arise from boundary properties, actuator lag, sensor level uncertainty, and wall thickness, as well as from the major parameters such as Prandtl number, Rayleigh number, wave number, and truncation number in the reduced-order model. The results suggest that the LQG compensator action can preserve closed-loop stability at over ten times the critical Rayleigh number, provided that the mismatches in the sensor level and wall thickness are small. Mismatches in the Prandtl number and wall material properties have little impact. Mismatches in Rayleigh number and wave number are relatively benign compared with the sensor and thickness parameters. Techniques for measuring the plant output temperature at multiple levels with sufficient accuracy may be an implementation challenge.
ABSTRACT
Recent studies in the feedback control of Rayleigh-Bénard convection indicate that one can sustain the no-motion state at a moderate supercritical Rayleigh number (Ra) using only proportional compensation. However, stabilization occurs at a much higher Rayleigh number using linear-quadratic-Gaussian (LQG) control synthesis. The restriction is that the convection model is linear. In this paper, we show that a comparable degree of stabilization is achievable for a fully nonlinear convection state. The process is demonstrated in two stages using a fully nonlinear, 3D Boussinseq model, compensated by a reduced-order LOQ controller and a gain-schedule table. In the first stage a fully-developed convective state is suppressed through the control action at a moderate supercritical Ra. After the residual convection decays to a sufficiently small amplitude, in the second stage, we increase the Ra by a large step and switch the compensator gains using the gain-schedule table. During this change the control action is in place. Our nonlinear simulation results suggest that the nonlinear system can be stabilized to the limit predicted by the linear analysis. The simulation shows that the large Ra jump induces a large transient temperature in the conductive component, which appears to have very small impact on the stabilization.
ABSTRACT
The onset condition of convection in a layer of fluid bounded by isothermal walls, with lower temperature varying sinusoidally in time at very low nondimensional modulation frequency, is derived in closed form, based on the Floquet theory and using a matched-asymptotic WKB method.
ABSTRACT
We study numerically the onset of temporally modulated Rayleigh-Bénard convection with zero mean gradient for cases of antisymmetric and asymmetric boundary temperatures over a continuous range of nondimensional frequencies omega, from omega approximately O(10(-1)) to omega approximately O(10(3)). For omega below 1, the neutral curves for Pr=7 in both cases alternate between synchronous and subharmonic responses, with increasingly shorter intervals as omega becomes small. At large omega, the critical wave number k(c) asymptotes to omega(1/2) and the critical Rayleigh number R(c) asymptotes to omega(3/2), via a subharmonic response in both cases. A comparison with the experimental results of Niemela and Donnelly [Phys. Rev. Lett. 57, 583 (1986)] shows fairly reasonable agreement.
