1. General presentation

1.1. Reminders

The dynamic fluid forces exerted on a moving structure can be classified into two categories:

the forces**independent* of the movement of the structure, at least in the range of small displacements; these are mainly random forces generated by turbulence or the two-phase nature of the flow,

the fluid forces**dependent**on the movement of the structure, called « **fluid-elastic forces* », responsible for fluid-structure coupling.

In this document, we are interested in the four fluid-elastic force models integrated into the CALC_FLUI_STRU operator. The computer aspects related to the integration of these models have been the subject of specification notes [bib. 1], [bib. 2].

1.2. Modeling

The dependence of fluid-elastic forces on the movement of the structure results, for low amplitudes, in a transfer matrix between the fluid-elastic force and the displacement vector. The projection of the equation of motion of the coupled fluid-structure system on the modal basis of the structure alone is written, in the Laplace domain:

\(\{[{M}_{\mathit{ii}}]{s}^{2}+[{C}_{\mathit{ii}}]s+[{K}_{\mathit{ii}}]-[{B}_{\mathit{ij}}(U,s)]\}(q)=({Q}_{t})\) eq. 1.2- 1

where	\([{M}_{\mathit{ii}}],[{C}_{\mathit{ii}}]\mathit{et}[{K}_{\mathit{ii}}]\) refer respectively to the diagonal matrices of structural mass, damping and stiffness in air;
	\((q)\) refers to the vector of generalized movements in air;
	\(({Q}_{t})\) is the vector of generalized random excitations (independent forces);
and	\([{B}_{\mathit{ij}}(U,s)]\) represents the fluid-elastic force transfer matrix, projected on the modal basis of the structure alone. This matrix depends in particular on , the characteristic speed of the flow, as well as on the frequency of the movement through the Laplace variable .

In principle, \([{B}_{\mathit{ij}}(U,s)]\) is any matrix whose extra-diagonal terms, if they are not zero, introduce a coupling between modes. On the other hand, the terms of \([{B}_{\mathit{ij}}(U,s)]\) evolve in a non-linear manner with the complex frequency \(s\).

Each fluid-elastic force model is associated with a specific transfer matrix. In all cases, the formulation of the modal problem under flow can be characterized by the relationship [éq. 1.2-1].

For the various types of configurations that can be simulated using the operator CALC_FLUI_STRU, the representations of fluid-elastic force transfer matrices are explained in the rest of this document.