A methodology proposed for actively controlling flows involves the use of proper orthogonal decomposition (POD) to derive computational models of reduced order. The methodology could be particularly useful for controlling flows of gases and liquids in real time.

The need for reduced-order modeling arises because of the inherent complexity of the computations for solving the full Navier-Stokes equations for the dynamics of fluids. One can reduce the time and cost of computation by use of reduced-order models. The problem is to derive suitable reduced-order models that approximate the essential dynamics well enough for purposes of control. The present POD-based methodology provides a systematic and optimal way to derive reduced-order models of relatively high accuracy while maintaining well-conditioned matrices in a matrix-vector form of the dynamical equations ("system matrices" for short). The methodology may not be effective in all cases and must be applied with care. In those cases in which it is effective, it can provide adequate control performance at significantly reduced computational cost.

The conventional approach to the discretization of the Navier-Stokes equations or other nonlinear partial differential equations by use of a finite-difference, finite-element, spectral method involves the use of basis functions (e.g., trigonometric functions, Legendre polynomials, or piecewise polynomials) that are mathematically convenient but that have very little connection with either the underlying physics of a specific case or the corresponding partial differential equations. In contrast, POD involves the use of basis functions generated from experiments or from numerical solutions of the partial differential equations. More specifically, it involves the extraction of an optimal set of basis functions (perhaps containing only a few basis functions) from a computational or experimental data base, by use of an eigenvalue analysis. Then by means of Galerkin projection, a solution is calculated as a linear combination of basis functions from the optimal set. This solution is the desired reduced-order model solution.

In a test case, the methodology was applied to a two-dimensional flow in a channel that includes a backward-facing step. At high Reynolds numbers, the flow separates and recirculation appears (see figure). The problem was formulated as one of blowing of fluid on part of the step surface to reduce the recirculation and thereby reduce the length of reattachment. Computational simulations with as few as 9 POD modes showed that optimal blowing control could effectively eliminate separation and significantly reduce the size of the recirculation bubble and the reattachment length.

This work was done by S. S. Ravindran of Langley Research Center. L-17846