TU CE GSC CE People at the Graduate School CE Alumni Jane Ghiglieri Research Project of Jane Ghiglieri

Optimal flow control based on reduced models

The idea of controlling a fluid flow in order to optimize its characteristics in some way is highly attractive and has a wide range of potential applications.

Unfortunately, the optimal control of complex flows is not easy to handle. The fluid flow is governed by the Navier-Stokes equations and due to their nonlinearity the accurate numerical simulation of most practical flows is computationally expensive. As a consequence the optimization of flows based on the full Navier-Stokes system is very costly.

However, there exist suboptimal control algorithms which create reduced problems of tractable dimensions like receding horizon control in connection with reduced order models.

We plan to analyze and extend these flow control strategies and to implement them into the CFD code FASTEST using a large eddy simulation (LES). The goal is to control two-dimensional Tollmien-Schlichting waves in a flow and to delay the transition. To influence the flow, a body force is induced by plasma-actuators. The developed flow optimization methods will be validated by experiments in a wind tunnel.

A plasma-actuator consists of two electrodes separated by an insulating layer. One of the two electrodes is exposed to the flow, the other electrode is below the insulation layer and grounded. To the electrodes an AC voltage is applied. As a result an electric field is generated and the air above the actuator is ionized. This means that the neutral air molecules are broken down into their loaded components: electrons and positive ions. This gaseous state, where free charge carriers are present, is known as plasma. The electric field between the two electrodes exerts a body force on the charged carriers.

The experimental setup consists of a planar solid body, a plate, which is installed parallel to the flow in the wind tunnel. A plasma-actuator 400 mm downstream of the flat plate's leading edge is operated in pulsed mode to artificially induce TS-waves into the boundary layer. The second plasma-actuator is located 100 mm further downstream. Two velocity sensors measure each 50 mm behind the actuator the flow velocities 1mm above the plate.

The second actuator produces a body force, which will reduce the Tollmien-Schlichting waves. The quality of the cancellation of TS-waves depends on the body force. The size of the body force is controlled by the applied high voltage to the plasma-actuator. The right time for the control pulse relative to the Tollmien-Schlichting waves is controlled by the phase shift. Further control parameters are possible.

In a first step, derivative-free optimization methods have been applied to optimize the parameters of the plasma-actuator. The approach has the advantage that it can be based on the sensor signal itself without using a model for the Navier-Stokes equations. To account for the transient behaviour of the flow, several modifications of derivative-free optimization methods have been developed. The following optimization methods without derivatives have been used for the optimal control of the plasma-actuator:

- Parameter-wise closed-loop control: optimization of phase shift and maximum voltage is carried out iteratively in a bisection type manor and each parameter is adjusted separately
- The Nelder-Mead simplex method, which compares function values at the vertices of a simplex and transforms the simplex in an iterative way
- The NEWUOA method proposed by M. Powell, which uses a quadratic model based on the function evaluations in connection with a trust region method to find the minimum value of the objective function.

Proper Orthogonal Decomposition (POD) is a method for deriving reduced order models of dynamical systems. The construction is based on the information contained in so-called snapshots, which provide the spatial distribution of the dynamical system at pre-specified time instances. By solving an eigenvalue problem with these snapshots an optimal orthogonal basis is found - the POD basis - which represents the snapshots in an optimal way for a given weighted norm. The number of the basis elements can be small in comparison with the number of snapshots, while it is still possible to obtain a satisfactory level of accuracy, which is sufficient to compute effective optimal controls based on the resulting reduced order model.

The idea of Model Predictive Control (MPC) is to replace the open-loop optimal control problem on the full time horizon by a sequence of optimal control problems on short control horizons that move forward in time. To stabilize the method, the neglected costs-to-go can be replaced by a suitable control Ljapunov function. Instead of using the discretization of the full Navier-Stokes equations in each time step a Galerkin discretization with respect to the POD basis or another suitable reduced order model can be used. Thus, the computational cost can be reduced significantly. The control obtained is used to steer the system to the next time step, where the MPC procedure is repeated.

The aim of the project is to develop online optimal control methods based on MPC in connection with model reduction to cancel TS-waves by plasma actuators. The complexity of the considered flow will we increased during the project. The numerical studies of this project will be accompanied and validated by experiments in the wind tunnel.

Simulation Based Optimization: Optimization of Coupled and Multi-Physics Problems

Jane Ghiglieri

Dipl.-Math.

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