Low-dimensional structure and online tracking of POD subspaces on the Grassmann manifold: application to flow around an airfoil
Representing flow states over a wide range of flow parameters and control inputs in a low-dimensional state space is a central challenge in fluid mechanics. Rather than representing the instantaneous flow field in a fixed subspace spanned by the leading proper orthogonal decomposition (POD) modes, this study regards the POD subspace itself as the flow state at each flow condition. The set of POD subspaces associated with flow conditions defines a state space on the Grassmann manifold. Diffusion maps identify the intrinsic low-dimensional structure of the family of POD subspaces, while Grassmannian rank-one update subspace estimation tracks the temporal evolution of a POD subspace online. The framework is experimentally demonstrated for flow around an airfoil. POD subspaces are extracted from wall-pressure fluctuations measured by a microphone array over a range of angles of attack and under different control inputs. The subspaces are found to lie on a one-dimensional submanifold of the Grassmann manifold. Moreover, the transition from separated to attached flow, induced by a plasma actuator, follows a reproducible trajectory along the same submanifold identified from statistically stationary flow data. The temporal evolution of the subspace is consistent with the transient evolution of the flow field observed using particle image velocimetry. These results show that POD subspaces can serve as representative flow states, enabling their temporal evolution across a wide range of flow conditions to be tracked online in a low-dimensional space based on wall-pressure fluctuations. This low-dimensional representation provides a basis for real-time flow-state estimation and feedback control.