Nonlinear Estimation of Transient Flow Field Low Dimensional States Using Double Proper Orthogonal Decomposition

Feedback flow control of the wake of a circular cylinder at a Reynolds number of 100 is an interesting and challenging benchmark for controlling absolute instabilities associated with bluff body wakes. A two dimensional computational fluid dynamics simulation is used to develop low-dimensional models for estimator design. Actuation is implemented as displacement of the cylinder normal to the flow. The estimation approach uses a low dimensional model based on a truncated 6 mode Double Proper Orthogonal Decomposition (DPOD) applied to the streamwise velocity component of the flow field. Sensor placement is based on the intensity of the resulting spatial modes. Accurate estimation of the lowdimensional states of a wake in the presence of transient forcing is a challenging problem and there are few successful attempts to model off-design cases where the frequency and amplitude of the excitation differs from the design condition. A non-linear Artificial Neural Network Estimator (ANNE) was employed to map the velocity data to the mode amplitudes of the DPOD model. For a given four sensor configuration, developed using a previously validated strategy, ANNE was compared to two state-of-theart approaches, namely, a Quadratic Stochastic Estimator (QSE) and a Linear Stochastic Estimator with time delays (DSE). For the estimation of the first six DPOD modes, we show that a four sensor configuration using ANNE provides lower estimation errors (one order of magnitude for Mode 1,1 and about 50% improvement for the higher modes) when compared to conventional state-of-the-art techniques appearing in literature.