Keywords: fluid-structure interaction, flutter, structural dynamics, Navier-Stokes equations, fictitious domain algorithm.

This work presents a numerical study of fluid-structure interaction of a filament in a viscous flow. The filament structure is modelled as a cantilever beam using finite elements. The flow of Newtonian fluid is considered laminar and 2D case. Viscous fluid flow is governed by Navier-Stokes equations expressing the conversion of mass and momentum. The momentum equations are solved using a commercial finite element code FIDAP (Fluent. Inc.) that enables imposition of a moving boundary constraint. The boundary conditions are no slip condition on the walls and on the surface of filament.

To resolve the fluid-structure coupling, a fictitious domain algorithm is used to describe the interaction of the filament and the fluid. In this method the Navier-Stokes equations using the outer boundary conditions of the walls and imposing the prescribed motion of the filament are iteratively solved. The pressure distribution around the moving boundary is applied to the structure of filament in a weakly coupled algorithm [2]. Initially the position and velocity of the filament is assumed known. The Navier-Stokes equations are solved primarily at each time level to obtain the hydrodynamic force acting on the body, the pressure forces are used as an input in the structural solver to predict the displacement and velocity of the structure at the next time level. This displacement and velocity is updated then as an internal boundary condition on the mesh in the Navier-Stokes solver.

In this weak coupling solution technique, the algorithm is sometimes "unstable". To cover this kind of problem iterative procedures have been developed. In unsteady fluid dynamic part, a ramp function of the hydrodynamic force is applied to the structure based on the pressure calculation to describe the gradual change of the external load. In structural dynamic part, to avoid possible the numerical instability, time-dependent problem is solved by applying a Newmark-Wilson approach [8]. Assuming a linear acceleration within a time step, this implicit scheme is unconditionally stable and second-order accurate in time for linear and first-order accurate for geometrically large deflection structure.

Experiments concerning the dynamic behaviour of aluminium film in airflow have been carried out in a wind tunnel by Huang [3]. We note that particular interesting filament properties were observed in unstable flutter boundaries such as shown in Figure 1.

Validation of the method and by comparing with wind tunnel test results have been carried out through similarity [6] where the numerical results should be interpreted by multiplying with frequency scaling factor. The numerical solution shows good relationship with wind tunnel test especially at the flutter boundary.

1

J. Zhang et al, "Flexible Filaments in a Flowing Soap Film as a Model for One-Dimensional Flags in a Two Dimensional Wind", Nature, 408: 835-839, 2000. doi:10.1038/35048530

2

D.J.J. Farnell et al., "Numerical Simulations of a Filament in a Flowing Soap Film", Int. J. Numer. Meth. Fluids, 44: 313-330, 2004. doi:10.1002/fld.640

3

L. Huang, "Flutter of Cantilevered Plates in Axial Flow", Journal of Fluids and Structures, 9, 127-147, 1995. doi:10.1006/jfls.1995.1007

4

Y. Yadykin et al, "The Added mass of a Flexible Plate Oscillating in a Fluid", Journal of Fluids and Structures, 17, 115-123, 2003. doi:10.1016/S0889-9746(02)00100-7

5

J. De Hart et al, "A Three-Dimensional Computational Analysis of Fluid-Structure Interaction in the Aortic Valve", Journal of Biomechanics, 36, 103-112, 2003. doi:10.1016/S0021-9290(02)00244-0

6

J. Glasser, "Seminar on Aeroelasticity", Seminar to the Airworthiness Branch Transport Canada Aviation Group Ottawa Canada, Boeing of Canada, October 26-30, 1987.

7

R.H. Scanlan, R. Rosenbaum, "Aircraft Vibration and Flutter", Macmillan, 1951.

8

G. Dhatt, G. Touzot, "The Finite Element Method Displayed", John Wiley & Sons, 1985.

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