Proper Orthogonal Decomposition Modeling of a Controlled Ginzburg-Landau Cylinder Wake Model

Abstract

A short computational program was undertaken to evaluate the effectiveness of a closed-loop control strategy for the stabilization of an unstable bluff-body flow. In this effort, the nonlinear one-dimensional GinzburgLandau wake model at 20% above the critical Reynolds number was studied. The numerical model, which is a nonlinear partial differential equation with complex coefficients, was solved using the FEMLAB/MATLAB package and validated by comparison with published literature. Based on computationally generated data obtained from solving the unforced wake, a low-dimensional model of the wake was developed and evaluated. The lowdimensional model of the unforced Ginzburg-Landau equation captures more than 99.8% of the kinetic energy using just two modes. Two sensors, placed in the absolutely unstable region of the wake, are used for real-time estimation of the first two modes. The estimator was developed using the linear stochastic estimation scheme. Finally, the loop is closed using an PID controller that provides the command input to the variable boundary conditions of the model. This method is relatively simple and easy to implement in a real-time scenario. The control approach, applied to the 300 node FEMLAB model at 20% above the unforced critical Reynolds number stabilizes the entire wake for a proportional gain of 0.06. While the controller uses only the estimated temporal amplitude of the first mode of Im(A(x,t)), all higher modes are stabilized. This suggests that the higher order modes are caused by a secondary instability that is suppressed once the primary instability is controlled.