Disturbance source identification for flow control: Problem formulation for infinitesimal disturbances

We introduce a method for identifying sources of disturbances in a mixture measured by sensors in shear flows. To recover all disturbance sources from the recorded signals, we consider blind source separation (BSS) techniques in a case where the sources and the mixing process are unknown and only the recordings of the mixtures are available. We adapt and use the independent component analysis (ICA) method for preforming BSS. In the theoretical framework we define the term 'source' in shear flows, and derive a model describing the mixtures measured by N_x sensors due to N_s various disturbances generated in shear boundary layer. We obtain a criterion for successful separation of 3D disturbances in on N_s x N_x source-sensor system by the ICA-BSS method. The criterion dictates the proper placement of sensors in the flow. We study, numerically and analytically, various sensor-actuator arrangements. In our numerical simulations we consider shear flows with two disturbance-generators and two sensors. Linear stability theory (LST) for plane Poiseuille flow (PPF) and Blasius flow with parallel flow assumption are employed to describe the downstream propagation of small disturbances. Results from application of the ICA-BSS method on simulated measurements in shear flow are validated against our theoretical analysis.