Proper orthogonal decomposition reconstruction of a transitional boundary layer with and without control

Proper orthogonal decomposition (POD) has been performed for controlled and uncontrolled transitional boundary layer data in an effort to reconstruct and possibly control the transitional boundary layer. Although the POD provides mathematically defined optimal basis functions for a given flow, they are only optimal for a given flow condition (e.g., specific Reynolds number, boundary conditions, etc.). In the context of flow control, one is usually forced to use the POD modes extracted from an uncontrolled flow as the controlled flow is not known a priori. The present investigation reveals that the most energetic POD modes for uncontrolled and controlled modes show a striking similarity, and unlike in turbulent flows, the present transitional boundary layer flow can be reliably captured by a few POD modes which contain almost all of the flow energy. It is then shown that it is possible to reconstruct the controlled flow using POD modes from the uncontrolled flow. Therefore, it can be conjectured that low-dimensional models based on the uncontrolled POD modes may be successfully used as online control tools. After this it is shown that "new" snapshots, i.e., the time evolution of the flow, can also be reconstructed by appropriately modifying the coefficients of the POD modes extracted from "previous" or earlier snapshots. This is accomplished using flow-field information at some "sensor positions" and a least-squares fit to the uncontrolled POD modes from former snapshots at these points. The approach introduced in this paper offers a simple, experimentally realizable approach to calculate the temporal coefficients, hence to reconstruct the unknown flow field without the need for a low-dimensional model based on the projection of the Navier-Stokes equations.