Experimental Study on the Stabilization of the No-Motion State in the Rayleigh-Benard Convection Problem

Abstract

We demonstrate experimentally that through the use of proportional-differential control, it is possible to stabilize the no-motion state of a fluid layer heated from below, cooled from above, and confined in an upright, circular cylinder (the Rayleigh-Be´nard problem). An array of 24 independently controlled heaters (thermal actuators), microfabricated on a silicon wafer, constitutes the bottom boundary of the test cell. A cooling system maintains the top boundary at a constant temperature. Silicon diodes located at the mid-height of the cell, above the actuators, measure the fluid's temperature. The multi-input, multi-output controller adjusts the heaters' power in proportion to the deviation of the fluid's temperatures, as recorded by the diodes, from preset values associated with the no-motion, conductive state. First, a set of experiments was conducted in the absence of a controller to determine the uncontrolled, reference state. Advantage is taken of the linear dependence of the mid-height temperature on the power input in the no-motion state. The preset temperatures are determined by extrapolating the mid-height temperatures to the desired input power values. A proportional controller is then engaged. We show that as the controller's gain increases so does the critical Rayleigh number for the onset of convection. The proportional controller allows us to increase the critical Rayleigh number by as much as a factor of 1.4. When the controller's gain is larger than a critical value, the system becomes time-wise oscillatory (Hopf bifurcation) and the controller's performance deteriorates. The oscillatory convection can be significantly damped out by engaging a proportional-differential (PD) controller. The PD controller allows us to further increase the critical Rayleigh number for the onset of convection to as much as a factor or 1.7 compared to the uncontrolled case. Further increases in the critical Rayleigh number were not possible due to the actuators' saturation. We also compared the supercritical flow patterns at the mid-height of the test cell in the presence of the controller with the flow patterns in the absence of a controller. The proportional controller modified the flow pattern from a single convective cell with ascending fluid in one half of the cell and descending in the other half, to fluid ascending at the center of the cell and descending at near the lateral wall. Our work represents an improvement over previous experimental investigations on the stabilization of Rayleigh-Be´nard convection in which the critical Rayleigh number was increased by only a factor of 1.2. Almost uniform temperature distribution at the mid-height is obtained through the combined action of proportional and derivative controllers. The Rayleigh-Be´nard convection is suppressed under conditions when, in the absence of a controller, flow would persist.