Theoretical and computational research has yielded some advances in the art of designing active feedforward and feedback controllers to suppress thermal convection and reduce drag (by suppressing turbulence) in boundary-layer flows. The advances include (1) improved means for designing reduced-order (and, hence, computationally more efficient) controllers and (2) discovery of a previously unknown phenomenon that could be exploited for feedforward control to reduce drag.
The mathematical model used in this research is that of three-dimensional flow of a viscous, incompressible liquid in a channel bounded by flat, parallel upper and lower walls, with three parallel sensor planes embedded in the channel and actuator planes coinciding with the walls (see figure). The actuation consists of a combination of blowing or suction normal to the upper wall and an equal amount of suction or blowing, respectively, normal to the lower wall at the same streamwise (horizontal) location so that there is no net transfer of mass into or out of the channel. Both sensing and actuation are assumed to be temporally continuous and spatially continuous on the affected planes.
In this research, the Navier-Stokes equations, linearized about laminar mean flow and about the no-motion state of a heated fluid, were used as a basis for designing reduced-order controllers for suppressing turbulence and convection. The linearized equations were cast into a state-space form by means of a two-dimensional Fourier expansion in the streamwise and spanwise directions and a Galerkin projection in the wall-normal direction. The resulting state-space equations were decoupled into independent Fourier-wave-number subsystems. Control laws were formulated by applying linear quadratic Gaussian (LQG) synthesis to these subsystems. The size of the controller was reduced by both limiting the number of subsystems to which LQG synthesis was applied (in effect, applying control to suppress perturbations only at certain Fourier wave numbers) and applying system-theoretic model-reduction techniques to each subsystem (in effect, by selecting only the most controllable and observable modes). Thus, the system of controller equations was reduced to about 2 percent of its original size.
In computational simulations, the controllers designed following the approach described in the preceding paragraph were found to be highly effective in suppressing convection in a heated fluid layer but only moderately effective in reducing drag in channel flow. The primary obstacle to improved drag-reduction performance appeared to be lack, in the linearized equations, of a cost function relating directly to skin-friction drag: because of this lack, feedback control laws for channel flow could affect viscous drag only indirectly.
In an attempt to overcome this obstacle, a more direct approach in which perturbations in each wave-number pair are linked to drag through nonlinearities in the governing equations was investigated. It was found the controllers synthesized as described above were found to transiently induce drag less than that of laminar flow when used with certain initial flow fields, prompting an inquiry into whether there could be a periodic process in which average drag could be sustained below that of laminar flow. This inquiry led to such a process: open-loop actuation in the form of upstream-traveling two-dimensional waves of wall-bounded, wall-normal blowing and suction have been found to force sustained reductions in drag to below laminar values in simulations initialized with both laminar and turbulent flows. The exact mechanism of drag reduction is not yet fully understood.
This work was done by Jason L. Speyer and J. John Kim of the University of California for the Naval Research Laboratory. For more information, download the Technical Support Package (free white paper) at www.defensetechbriefs.com/tsp under the Mechanics/ Machinery category. NRL-0004
This Brief includes a Technical Support Package (TSP).
Some Advances in Reducing Drag and Suppressing Convection
(reference NRL-0004) is currently available for download from the TSP library.
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