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Model order reduction and boundary control of incompressible Boussinesq flow Open Access


Other title
Linear quadratic regulator
Galerkin method
Proper orthogonal decomposition
boundary control
Reduced order modeling
Type of item
Degree grantor
University of Alberta
Author or creator
Wang, Zichuan
Supervisor and department
Koch, C R (Mechanical Engineering)
Flynn, Morris (Mechanical Engineering)
Examining committee member and department
Dubljevic, Stevan (Chemical and Materials Engineering)
Koch, C R (Mechanical Engineering)
Flynn, Morris (Mechanical Engineering)
Department of Mechanical Engineering

Date accepted
Graduation date
Master of Science
Degree level
The time evolution of a two dimensional incompressible, density stratified, Boussinesq flow in a rectangular cavity is numerically simulated for a range of parameters. Boundary control is then implemented along the upper boundary by adjusting the fluid density. More specifically, the top boundary condition of the cavity is a fixed function of space that is modulated by the control input. The resulting numerical simulation for the fluid density and velocity is computed using finite differences in the vertical direction and a spectral method in the horizontal direction. To develop the control strategy, the flow is simulated and a sequence of snapshots of the density and velocity fields are collected. Then, a reduced order modelling method suitable for a linear quadratic regulator (LQR) of the Boussinesq flow is developed using the proper orthogonal decomposition (POD)/Galerkin approach. The reduced order model is obtained by projecting the governing equations of the flow onto the sub space spanned by a finite number of basis functions obtained using the method of snapshots. For the flow in question, the POD method based on the snapshots yields 6 POD modes which capture 99\% of the flow energy. In turn, the boundary control is transferred to the governing equations using Duhamel's principle so that the resulting equations contain the control input. The feasibility of this method is assessed using a LQR boundary controller that is designed based on the reduced order model. The cost functional which is minimized in the LQR control design is defined to be the squared norm of the difference between the actual density field and the desired density field in the cavity. The weighting parameter of the cost functional is found to play a critical role in the process of controller design. To judge the effectiveness of the control, two metrics $\eta_a$ and $\eta_r$ are introduced. $\eta_a$ denotes the absolute effectiveness of the LQR controller and is a metric of the controller's ability to drive the system to the final desired state. Conversely $\eta_r$ denotes the relative effectiveness of the LQR controller and is a metric of the improvement of the LQR controller compared to an open-loop controller. For the control cases tested, the LQR controller is found to have $\eta_a =96.7\%$ and $\eta_r = 22.2\%$ in the best case for the steady state starting configurations, and $\eta_a =99.3\%$ and $\eta_r = 47.9\%$ for representative transient cases. $\eta_r$ is more than doubled in the comparison between the steady case and the transient case. This indicates that the LQR controller is able to reject complex transient flow much better than the open-loop controller. In conclusion, a relatively simple feedback control scheme applied on the boundary of a turbulent flow improves the performance in regulating the density field to its desired final state compared to open-loop control.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
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