Keywords: fluid-structure interaction, coupled simulation, Coriolis flowmeter.

A fluid-conveying measuring tube that is maintained vibrating at its first natural frequency is a primary sensing element in a Coriolis flowmeter. As a result of the fluid forces acting on a straight clamped tube, the tube's first mode shape, which is otherwise symmetric with respect to the tube length, is distorted in an anti-symmetric way. This effect, which is actually caused by the appearance of the Coriolis forces, is exploited as the basic measuring principle. At early design stage, however, dynamic behaviour of such a coupled physical system should be carefully investigated in order to find the corresponding measuring sensitivity with respect to different design parameters [1]. Such an investigation can be done efficiently by means of a corresponding numerical modelling and subsequent computer simulation. In this contribution a slender-straight-tube flowmeter design is considered.

The first task in the dynamic characterization of the coupled fluid-structure system is the determination of its first natural frequency. Though in practice, with the Coriolis flowmeter being exposed to forced vibrations, the first natural frequency can be found by adaptive regulation of the respective imposed frequency, it is of great significance that a corresponding numerical model is built, which can be used for the analysis purposes. For trustful and reliable simulations such a model must be characterized by a satisfactory degree of physical objectivity. In this regard it can be stated that numerical response of a free vibrating fluid-structure coupled system is definitely, comparing to the forced vibrating one, more sensitive to deficiencies in the numerical model. It is therefore reasonable to base a development and calibration of the model on the study of the free vibrating system.

In this paper a simulation of fluid-structure interaction in a measuring tube of the Coriolis flowmeter is performed by the coupling of CFD (Computational Fluid Dynamics) and FE (Finite Element) code, using commercially available Comet 2.1 (Finite Volume Method) and Abaqus v6.3 (Finite Element Method), respectively. In order to determine initial conditions for the coupled analysis of the free vibrating system, first, fluid flow simulation with the tube assumed at rest is performed, yielding the pressure and velocity field in the fluid, as well as the fully developed inlet velocity profile. With the computed pressure applied to the tube, the stress-strain-displacement state in the resting tube is then obtained, which is followed by pushing the tube in oscilation by prescribing respective initial velocity field proper to the first vibrating mode. In principle, the respective coupled computation is performed, considering fluid-structure interaction and mutual fulfilment of energy balance at the common interface, iteratively. The iteration loop starts with structural computation, performed by Abaqus with given initial conditions and yielding in the first time step of the coupled simulation the displacement and velocity vectors of the tube. With the new tube geometry and velocity vectors used as the boundary conditions the fluid flow simulation is then performed by Comet to yield within the same time step the respective pressure and velocity response of the fluid to the induced vibration of the tube. In the next time step, considering the prescribed displacement conditions due to the imposed vibration, the calculated fluid forces, as obtained by the respective pressure distribution, are applied to the tube, and displacement and velocity of the tube are determined anew by Abaqus. The cyclic procedure of alternative exchanging the data between the two codes follows the following pattern: upon given initial and boundary conditions data are created within one code to be used in the other code as initial and boundary conditions, and yield there data which will be used in the subsequent time step as updated initial and boundary conditions for the former code. The just described procedure in principle works, but in its elementary version it needs rather long time to converge. Further numerical investigations have proven that rather fast convergence of the response can be obtained only under special boundary conditions application regime. Actually, "soft" application of pressure to a structure in the initial stage of computation proves to be favorable in reaching dynamic equilibrium, both for the structure and the fluid. A converged vibration is thus obtained in approximately six cycles.

1

G. Bobovnik, J. Kutin, I. Bajsic, "The effect of flow conditions on the sensitivity of the Coriolis flowmeter", Flow Measurement and Instrumentation, 15, 69-76, 2004. doi:10.1016/j.flowmeasinst.2003.12.001

2

C.A. Felippa, K.C. Park, C. Farhat, "Partitioned analysis of coupled mechanical systems", Computer Methods in Applied Mechanics and Engineering, 190, 3247-3270, 2001. doi:10.1016/S0045-7825(00)00391-1

3

C. Farhat, M. Lesoinne, "Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems", Computer Methods in Applied Mechanics and Engineering, 182, 499-515, 2000. doi:10.1016/S0045-7825(99)00206-6

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