Control of a Simplified Unsteady Film-Cooling Flow Using Gradient-Based Optimization

Abstract

Maximization of film-cooling effectiveness in an idealized film-cooling flow is accomplished using a gradient-based control strategy. The state equations are the two-dimensional unsteady incompressible Navier-Stokes equations and the temperature equation. The control is a spatially and temporally varying normal-velocity boundary condition subject to a constraint of zero mass and energy flux. The control that maximizes the objective function is found using a conjugate gradient method, in which the gradient of the objective function with respect to the control variables is obtained from solving a set of adjoint equations. The effectiveness of choices for the objective function are examined, along with the effect of the event horizon over which the control is obtained by the optimizer. Drastic improvements in the film-cooling effectiveness are obtained for certain choices of the objective function and event horizon. Although not a specific target of the optimizer, one result of the control is that the wall jet remains attached to the wall. The numerical issues involved with finding the control are discussed and the features of the resulting control are analyzed, with the goal of understanding the mechanisms that affect the film-cooling effectiveness.