Models for controlling airfoil lift and drag

A reduced-order model (ROM) is constructed to measure and control aerodynamic forces on au airfoil. Proper orthogonal decomposition (POD) yields a finite series representing the flow velocity. The series is comprised of products of time-invariant basis functions and time-dependent coefficients. Projection of the weak form of the momentum-conservation equation for incompressible flow onto the set of basis functions produces a low-dimensional set of ordinary differential equations, which is the plant with time-dependent POD coefficients as the state variables. Control input consists of surface jet velocity and is represented explicitly in the plant. Control output is aerodynamic force - both pressure-based and viscous. System output is a direct function of the state variables in the measurement equation. Tests of the plant and output measurement compare data directly from CFD simulations with results from the ROM. The test problem consists of flow past a NACA 4412 airfoil at Mach 0.1, eight degrees angle of attack, and a Reynolds number of 1000 (based on chord length). At these conditions, a separation bubble above the downstream one-third of the airfoil produces periodic vortex shedding into the wake and oscillatory aerodynamic forces. The ROM accurately reflects system dynamics with as few as two POD modes - with and without surface jets. Computation of pressure-based force from the state variables involves second derivatives that produce errors in the mean value of the output measurement. The errors do not exist in the viscous component of force, which accurately reflects the evolution of system output.