Keywords: fluid-structure interaction, weakly coupled analysis, computational fluid dynamics, computational structural mechanics, finite element method, finite volume method.

The paper presents a procedure for computational analysis of weakly coupled fluid-structure interaction in a mixing vessel. In a mixing vessel two physical systems interact with each other, fluid flow and the vessel structure (container walls, mixing blades, shaft). The necessary condition for weakly coupled solution of fluid-structure interaction problems is that small deformations of the structure result only in small changes of the fluid flow and only approximate (weak) equilibrium between the fluid and the structure is sufficient.

For solving fluid dynamics problems the conservation laws, expressed in the form of Navier-Stokes equations, are predominantly computationally solved by the Finite Volume Method. The FVM is based on the integral form of the Navier-Stokes equations, which are discretised in space and time. For fluid dynamics problems the discretisation procedure with FVM results in the following discrete form of the Navier-Stokes equations:

The pressure field in the fluid and also at all surface nodes that represent the boundary between the fluid and the structure can be determined by computational solution of equation (10) together with continuity equation for a given set of boundary conditions. The pressure (force) of the fluid acting on the structure surface has to result in the same reaction force in the structure.

On the other hand, the elastic-static solid mechanics can be described with a system of equilibrium equations, kinematic equations and constitutive equations [2,3]. After the applied spatial discretisation with the finite element method, one can write the equilibrium equation for a single finite element as

where the external surface traction force in equation (11) can be substituted with the previously computed fluid pressure field at the boundary, i.e. , resulting in

As a test example, the influence of mixing of water and air in a mixing vessel with four blade turbine stirrer on blades deformation was analysed with weakly coupled fluid-structure analysis. Air is injected into the vessel through an inlet pipe located bellow the impeller blade at a speed of 5 m/s. The angular velocity of the shaft is 3 s. The stationary flow field in the mixing vessel was computed with the CFX code with Eulerian-Eulerian two-phase flow model and the standard wall function k- turbulence model. The computed pressure field at the surface boundary nodes was then used as external load for a structural analysis of a single blade with the MSC.visualNastran for Windows. The computed deformations and stresses of the blades are shown in Figure 1. Results show that the highest deformations appear at the upper free corner of the blade, while the highest stresses can be observed at the upper part of welding connection between the blade and the shaft.

Described procedure can be applied to any weakly coupled fluid-structure interaction problem, where small deformations of the structure result only in small changes of the fluid flow. However, for solving problems with large structural deformations, it is necessary to use fully coupled fluid-structure interaction analysis methods that consider also geometrical and even material nonlinearities.

1

Ferziger, H.J., Peric, M., Computational Methods for Fluid Dynamics, Springer Verlag, Berlin. 1997.

2

Zienkiewicz O.C., Taylor R.L.: The Finite Element Method - Volume 1 - The Basis. McGraw-Hill Ltd., London. 2000.

3

NAFEMS: A Finite Element Primer. NAFEMS. Glasgow, 1992.

4

CFX 5.6: User's Manual, AEA Technology, 2003.

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