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

Abstract

In this paper a new model for 3D Poiseuille flow is presented. The model is based on applying a combined spectralfinite 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 L2-norm. The closed loop response shows that the energy is reduced by the factor of 30.

Flow estimation of boundary layers using DNS-based wall shear information

“…A particular case that has received much attention is that of channel flow, e.g. Hoepffner et al (2005), Hogberg et al (2003), Baramov et al (2004), McKernan et al (2007), Chughtai and Werner (2010), where the mean (time-averaged) flow is parallel to the walls and fully developed in the sense that it is invariant in the streamwise direction. A convenient consequence of this fact is that it enables a relatively straightforward analytic reformulation of (1) into an equivalent system expressed in terms of so-called 'divergence-free' variables of wall-normal velocity and vorticity.…”