Feedback Stabilization of a Reduced-Order Model of a Jet in Crossflow

Airfoils in gas turbine engines used for aeropropulsion are actively cooled by film cooling; and, to cost-effectively optimize or control the film-cooling behavior, reduced-order models combined with a suitable control strategy are needed. Motivated by this film-cooling application, a linear state feedback controller for stabilization of a proper orthogonal decomposition-based reduced-order model of a film-cooling jet injected over the airfoil surface into a hot crossflow is presented. A full-order direct numerical simulation of the jet in crossflow is used to generate the data required to build the reduced-order model. The control objective is to prevent the mixing of the coolant jet and hot gases from the combustor so that the coolant remains close to the blade surface. The kinetic energy of the turbulent flow is used as a measure of mixing, which in turn is used to guide the design of a desired steady-state condition with low kinetic energy. The linear controller guarantees that the nonlinear reduced-order model will be driven to the desired steady state if the system is operating inside a region of quadratic attraction. The controller gain and region of quadratic attraction are easily calculated by solving a set of linear matrix inequalities. In addition, the suggestion is made to utilize a switching strategy to deal with the case in which the state of the reduced-order model is outside of the calculated region of quadratic attraction. Results show that in closed-loop, the reduced-order model is completely stabilized, whereas the direct numerical simulation is only stabilized for a period of time.