Métodos basados en área

Although stabilization of structural vibrations is not directly a concern par se, these studies can illustrate problems that also appear in other systems involving control of structural motion. For example, both cases are concerned with characteristic values that oscillate at a given period (at least in the case of the fixed plate).

Most of the literature on the subject is dedicated to the control of structural defor- mation due to the aerodynamic forces or to the control of the flow through various devices like a dynamically deforming airfoil (deforming leading edge to reduce drag for instance), or boundary layer transpiration to delay separation.

Meirovitch and Silverberg (1984) [39] present a method based on the modal control (based on the mode of excitation of the free structure), and a displacement and velocity feedback. They then provide a linear model for the relation between the aerodynamic forces and the structural displacement. Although the formulation is very close to the problem tackled here, its input is limited to a flap angle on a wing and they have no interest in the flow characteristics. Nevertheless, it emphasizes the possibility of real- time control on this type of structure and the fact that the aerodynamic forces in the case of flutter (coupling between the aerodynamic forces and the vibration mode of the

plate ) can be modeled through a linear model.

Ballas (1978) [4], proposes an active control method for a distributed parameter sys- tem through modal control. However, he also provides an elegant solution to reduce the effect of the spillover (instability due to uncontrolled elastic modes of a structure) and an example with a flexible beam. To apply this approach to the rigid body mo- tion case would require a rewriting of the body displacement equation. Nevertheless, it could prove to be useful if the emphasis was put on the damping ability of the control.

Fromme and Golberg [19], discuss the solution of the equations of motion with struc- tural damping, aerodynamic stiffness and nonlinear generalized forces provided. Then a complete method of resolution is provided but without aerodynamic forces. Although it has a different focus considering the project need, this paper nonetheless supplies an interesting point of view on how to implement and solve the equations of motion.

Ribeiro (1998) [47] models a vibrating structure through a hierarchical method, in- volving a finite element method with a high degree polynomial as an approximation for the element. This is beyond the scope of the project.

Monaco and Normand-Cyrot (1997) [42] provide a common framework for the study of nonlinear discrete-time and sampled dynamics. Although the simulation in the work described in section 5.3 could be considered as giving some discrete time input, this study is of limited use for the project purposes. Newman (1994) [44] used a distributed controller for the control of structural vibration. Although giving an interesting point of view, it too is not directly relevant to this project.

The discrete vortex method

2.1

Approach

The vortex element method enables one to recreate the physically relevant dy- namics of two-dimensional incompressible flows through the use of the Lagrangian or the Lagrangian-Eulerian description of the evolution of discretized vorticity fields. Helmholtz was the first to show that, in what is now regarded as one of the most im- portant contribution to fluid mechanics, that in an inviscid fluid vortex lines remain continually composed of the same fluid elements and flows with vorticity can be mod- eled with vortices of appropriate circulation.

As a Lagrangian technique, its advantages lies in the grid-free nature of the simula- tion, the exact treatment of the far-field boundary condition (if the Vortex In Cell scheme is excluded), and the concentration of the computational power where it is nec- essary (i.e. vorticity at specific points). On the other hand, the use of vortex methods introduces some error in the convection, and special treatments are needed to take into account viscous diffusion (random walk scheme for example). The method also requires explicit treatment of the turbulent diffusion, otherwise it is limited to preturbulent, almost laminar flow.

There are additional problems associated with use of the vortex method and the as-

sumption of a purely two-dimensional (2D) flow. As stated by Sarpkaya [53], the 2D

method is unable to capture three-dimensional effects such as vortex filament tilting or stretching, whatever the specific scheme considered. This means that without the use of an ad-hoc circulation reduction scheme, it is difficult to predict accurately the

dynamics of 3D flows, in particular the lift and drag forces and the pressure distribu- tion. Nevertheless, the Strouhal number is generally correctly predicted. Note, though, that circulation reduction is strictly an ad-hoc scheme for 2D methods and does not enable to properly emulate the physical 3D effects in the flow thus it is a purely artificial mathematical model. As such, it requires to be adjusted depending on the type of the body and flow. This kind of model is thus somewhat awkward due to the use of purely ad-hoc parameters with no physical means.

Finally, other problems arise due to the fact that for stability reasons the vortices must have a finite core. Unfortunately however, these finite-core or ’blob’ vortices violate Helmholtz’s law that vorticity is a material quantity. This introduces a formal incon- sistency in the dynamics. Furthermore, the nonlinearity in the Navier-Stokes equations does not allow the superposition of finite-core vortices. However, there are some ad- vantages to be gained through the use of blob vortices, in particular a smoothing of the velocity and vorticity distributions, provided that one increases the number of blobs, by forcing them to overlap, and by judiciously choosing the core radius and the shape of the velocity cutoff function. Besides, increasing the number of vortices also mitigates the effect of the Navier-Stokes nonlinearities.

For this study, two different models are considered for separated flow, both of which are modifiable to take into account the effect of a moving plate. The first is based on the discrete introduction of vorticity at a fixed pre-specified separation point using blob vortices to represent both the flow and the plate. This scheme was coded in the C language, with the aim of coupling it with the Matlab utility Simulink. The second was adapted from the algorithm (and Fortran code) of Spalart [57], which represents the boundary layer by blob vortices that are eventually shed into the flow.

Both methods have been used previously to model a moving body in a flow using the vortex method. For the first method, a similar scheme was proposed by Ham [23] to model a flat plate during dynamic stall, as well as by Sarpkaya [51] to model a transversely oscillating cylinder. As for the second method, Spalart [57] used his code to model an airfoil during dynamic stall, Blevins [7] characterized a transversely oscil- lating cylinder, and Shiels [56] used a similar method with blob vortices with growing cores to account for the viscous diffusion to model a free-falling flat plate as well as the transversely oscillating cylinder.

Both schemes are used for two different roles; the first method is used to further develop and validate the control, while the second method serves as a benchmark for validating the code based on the first method, particularly for moving-body flows. Indeed, the Fortran code could not be directly coupled to a Matlab utility without first translating it into C. However, the second method is much more accurate than the discrete intro- duction of vorticity at predetermined locations, due to the higher count of vortices and less arbitrary parameters. On the other hand, it is also more difficult to use correctly as there can be significant discrepancies from one simulation to another simply by chang- ing one parameter, for example the timestep, within the same flow condition. In that respect, the first method is more consistent as it does not have such discrepancies; that is why trial were made to couple it with the control.

The main complication with both methods when introducing a rotation is the extrac- tion of the force and moment, as some terms are added, due not only to the rotation itself but also to the frame of reference used when modeling the flow around the plate.