Transient energy analysis of a spatially interconnected model for 3D Poiseuille flow

In this paper a new model for 3D Poiseuille flow is presented. The model is based on applying a combined spectral-finite difference approach on the velocity-vorticity formulation of the Navier-Stokes equations. In 3D the dominating feature of the problem is non-normality of the eigenvectors. One measure to assess the non-normality is the transient energy response. The model is validated by comparing the maximum transient energy for two different cases; for both cases the Reynolds number is fixed at 5000, in the first case the span-wise spatial frequency is set to zero and the stream-wise spatial frequency is varied from 0.1 to 2, while for the second case the streamwise frequency is set to zero and the span-wise frequency is varied from 0.1 to 3. It is known that for the first case the system is stable, while the second case is highly non-normal with maximum transient energy reaching 4500. It is observed that the model predicts both of these characteristics. Next, as a first step towards controller synthesis, a stabilizing state feedback controller is designed to minimize the transient energy response, by minimizing the induced ℒ
₂
-norm. The closed loop response shows that the energy is reduced by the factor of 30.