Fluid Structure Interactions for Tube Bundles: Physical Meaning of a Rayleigh Damping

It is well known that a fluid may strongly influence the dynamic behaviour of a structure. Many different physical phenomena may take place, depending on the conditions: fluid flow, fluid at rest, small or high displacements of the structure. Inertial effects can take place, with lower vibration frequencies, dissipative effects also, with damping, instabilities due to the fluid flow (Fluid Induced Vibration). In this last case the structure is excited by the fluid. The paper deals with the vibration of tube bundles under a seismic excitation or an impact. In this case the structure moves under an external excitation, and the movement is influenced by the fluid. The main point in such system is that the geometry is complex, and could lead to very huge sizes for a numerical analysis. Important developments have been made in the last years to develop homogenization methods for the dynamic behaviour of tube bundles. The numerical size of the problem is reduced, and it is possible to make numerical simulations on large tube bundles with reasonable computer times. These methods consider that the fluid movement is governed by the Euler equations for the fluid. They are based on an analysis on an elementary cell, corresponding to one tube, and on an expression of the forces applied by the fluid to the structure. This force only depends on the fluid’s and tube’s acceleration. Only “inertial effects” will theoretically take place, with globally lower frequencies. A research program is under progress to take into account dissipative effects also, with a homogenization of the Navier-Stokes equations in the tube bundle. It is common, in numerical simulations, to add a damping for the structures by using a global Rayleigh damping. The paper deals with the physical meaning of this Rayleigh damping in the Euler homogenized equations. It can be demonstrated that this damping corresponds to a force applied by the fluid to the structure depending not only on the acceleration, but on the fluid and structure velocity also. This Rayleigh damping is a first step to take into account the dissipative effects for FSI in tube bundles.