Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed

Robust control of linearized Poiseuille flow

An approach to feedback control of linearized planar Poiseuille flow using H\infty control is developed. Surface transpiration is used to control the flow and point measurements of the wall shear stress are assumed to monitor its state. A high-but-finite dimensional model is obtained via a Galerkin procedure, and this model is approximated by a low dimensional one using Hankel-optimal model reduction. For the purposes of control design the flow is modeled as an interconnection of this low dimensional system and a perturbation, reflecting the uncertainty in the model. The goal of control design is to achieve robust stability (i.e. to stabilize any combination of the nominal plant and a feasible perturbation), and to satisfy certain performance requirements. Two different types of surface actuation are considered -- harmonic transpiration and a model of a pair of suction/blowing panels. It is found that the latter is more efficient in suppressing disturbances in terms of the control effort required

Parallel discrete vortex methods on commodity supercomputers; an investigation into bluff body far wake behaviour

Parallel discrete vortex methods are ideally suited for studying the behaviour of bluff body wakes due to their ability to capture the motion of vortex structures and lack of downstream grid boundaries. However, the availability of suitable parallel computers to run long simulations is always an issue. The convergence of the high-end workstation and commodity PC markets means that it is now possible to build cheap, powerful supercomputer-level machines at a fraction of the cost of proprietary systems. The characteristics, programming methodology and performance of such systems are discussed in this paper. We also present results showing vortex merging behaviour in the far-wake of a circular cylinder. Previous simulations using the random walk method have shown a doubling of the shedding wavelength in the far wake compared with the near wake. New results using a deterministic vortex method are presented. The parallel vortex method used incorporates an O(NlogN) fast multipole method and vortex panel method to satisfy solid body boundary conditions. In order to account for viscous effects, both near the body and in the far wake, the vorticity redistribution method is used. This is considerably more accurate than stochastic methods The aim of this paper is twofold. To present our findings on the behaviour of fully viscous, far wakes behind bluff bodies and to demonstrate that sufficient resource can be obtained on a cost-effective commodity supercomputer

The automatic control of boundary-layer transition: Experiments and computation

In recent years there has been an increasing interest in the control of boundary-layer transition through the use of wall suction. In the current work suction is provided through one or more suction panels situated close to the leading edge of a plate. Experiments show that boundary-layer pressure fluctuation measurements can be used to identify the position of transition. Transition can be maintained at a desired location with minimum power consumption by employing an automatic adaptive feedback control loop which regulates the suction flow rates of two independent suction panels. This can be expressed as a constrained optimization problem. To allow the suction flow rates to be updated, a modified least mean squares algorithm is used within the control loop. Experimental measurements show that the control algorithm allows fast and stable convergence towards the optimum suction distribution for a double suction panel configuration. Numerical simulations have also been performed. The two-dimensional boundary layer was calculated allowing the viscous boundary layer to interact with the inviscid outer flow. Following linear stability theory the spatial growth rates are calculated by solving an Orr-Sommerfeld type eigenvalue problem, with the streamwise location of transition predicted via the eN -method. Applying the same optimization strategy as in the experiments, good qualitative agreement between computations and experiments was found. The optimization algorithm has been applied to computer models where the relation between suction flow rates and transition location is described by an empirical analytical function. This shows that the controller can in principle be applied to systems with more than two suction panels.</p

Reducing aerodynamic load fluctuation in wind turbines using iterative learning control laws designed using reduced order models of the flow

Developments in actuators and sensors have led to considerable interest in their use for aerodynamic load control for wind turbines, thereby increasing power extraction efficiency, including economic competitiveness against other sources of alternative energy. In particular, the route is to embed smart devices into the rotor blades and use them in combination with active control to modify the blade section aerodynamics, aiming to minimize lift fluctuations due to disturbances. Previous research has shown that iterative learning control can be used in this area, starting with model-free designs and proceeding to model-based designs. This paper uses proper orthogonal decomposition to construct finite-dimensional models from the computational fluid dynamics-based representations of the defining partial differential equations. The performance of the resulting control laws is examined using the computational fluid dynamics representation of the dynamics

Parallel Discrete Vortex Methods on Commodity Supercomputers; an Investigation into Bluff Body Far Wake Behaviour

Parallel discrete vortex methods are ideally suited for studying the behaviour of bluff body wakes due to their ability to capture the motion of vortex structures and lack of downstream grid boundaries. However, the availability of suitable parallel computers to run long simulations is always an issue. The convergence of the high-end workstation and commodity PC markets means that it is now possible to build cheap, powerful supercomputer-level machines at a fraction of the cost of proprietary systems. The characteristics, programming methodology and performance of such systems is discussed in this paper. We also present results showing vortex merging behaviour in the far-wake of a circular cylinder. Previous simulations using the random walk method have shown a doubling of the shedding wavelength in the far wake compared with the near wake. New results using a deterministic vortex method are presented. The parallel vortex method used incorporates an O(NlogN) fast multipole method and v..