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Louisiana State University & A&M College

Dynamics of fluid flows are governed by the well-known Navier-Stokes equation which is nonlinear and distributed. Flows that are important from an engineering point of view require the discretization of the Navier-Stokes equation so that the equation can be solved approximately by computer simulation. The computer simulation consists of numerically solving thousands or millions of equations repeatedly. While the numerical methods required for these simulations are mature, control methods to deal with problems of this size are not. As such, flow control presents a challenge to scientists and engineers working in this field. Our control strategy proceeds in two steps: Model Reduction and Controller Design. In the first step we use the well known technique of Proper Orthogonal Decomposition (POD) to reduce the number of equations that describe the flow to ten or fifteen equations, without losing too much model fidelity. This new set of equations is called the Reduced Order Model (ROM). In the second step we design a linear feedback controller for the nonlinear ROM which will drive the flow to an operating point, in our case, a prescribed time invariant velocity field that has some desirable film cooling property. Due to the nonlinear dynamics, our controller will drive the flow to the operating point as long as the initial velocity is ‘close enough’, i.e., the initial velocity is in the domain of attraction of the operating point. Our controller design method provides an estimate for the domain of attraction and most importantly, maximizes it via a quadratic stability argument.

Dynamics of fluid flows are governed by the well-known Navier-Stokes equation which is nonlinear and distributed. Flows that are important from an engineering point of view require the discretization of the Navier-Stokes equation so that the equation can be solved approximately by computer simulation. The computer simulation consists of numerically solving thousands or millions of equations repeatedly. While the numerical methods required for these simulations are mature, control methods to deal with problems of this size are not. As such, flow control presents a challenge to scientists and engineers working in this field. Our control strategy proceeds in two steps: Model Reduction and Controller Design. In the first step we use the well known technique of Proper Orthogonal Decomposition (POD) to reduce the number of equations that describe the flow to ten or fifteen equations, without losing too much model fidelity. This new set of equations is called the Reduced Order Model (ROM). In the second step we design a linear feedback controller for the nonlinear ROM which will drive the flow to an operating point, in our case, a prescribed time invariant velocity field that has some desirable film cooling property. Due to the nonlinear dynamics, our controller will drive the flow to the operating point as long as the initial velocity is ‘close enough’, i.e., the initial velocity is in the domain of attraction of the operating point. Our controller design method provides an estimate for the domain of attraction and most importantly, maximizes it via a quadratic stability argument.

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