Finding an optimal flow control with multi-point penalty method*

Finding an optimal control of turbulent flows is inherently difficult due to its chaotic dynamics. The objective functional becomes extremely non-convex, having many local extrema that impede search strategies for finding effective controls. A multi-step penalty method, which temporarily introduces discontinuities at intermediate time points before minimizing them along the optimization, is demonstrating to be a promising approach to this challenge. However, while it is shown to find an effective control for turbulence, it can be still challenging for the flows in free space such as jets, where the controllable subspace is confined to a spatially limited region, near the actuator. We further develop additional techniques to extend the multi-step method to such flows in a spatially extensive domain.  A compact support for intermediate state is introduced, where the multi-step method can be selectively applied only to the region that exhibits chaos. A modified L₂-norm is suggested as a better penalty for flow optimization problem. This approach is demonstrated in a model system then demonstrated on a two-dimensional shear-layer noise-control application.