A new formulation for fluid–structure interaction in pipes conveying fluids using Mindlin shell element and 3-D acoustic fluid element

Abstract

This paper explores the vibratory behavior of fluid-conveying flexible shells using a new generic finite element formulation employing the first-order shear deformation theory. The flexible tube conveying fluid is modeled using eight-noded curved Mindlin shell elements, which incorporate the effects such as shearing deformations and rotary inertia. The fluid is modeled using twenty noded isoparametric acoustic fluid elements. Solving the wave equation for an abstract scalar field velocity potential, we get the equations of motion for the fluid element. The energy transfer within the fluid and the shell is idealized with the pressure and velocity boundary conditions, which guarantees proper contact between the fluid and structure. The flexible tubes find various applications in medical as well as pharmaceutical industries. Flexible tubes demand minimal energy to excite. Hence, they can find applications in the flow measuring devices, which use vibration techniques. There is a difference in the fundamental frequencies of silicone tubes measured in the horizontal and vertical planes. This difference is due to the sagging of flexible pipes, which causes a beat phenomenon. A novel laser scanning technique is proposed to obtain the actual dimensions of flexible tubes when it sags due to gravity. This actual dimension is analyzed using the new formulation developed. The numerical results, with the actual dimensions measured using the scanning technique, give a good match with the experimental results. Keywords Fluid-structure interaction • Flow through flexible pipes • Natural frequency • Beat phenomenon • Sagging of tubes • Pre-stretch • Thick shells List of symbols β ξ , β η Rotation vectors along ξ and η directions ε ξ , ε η , ε z Normal strain along ξ, η and z directions γ ξη , γ ηz , γ zξ Shear strains γ 0ξη , γ 0ηz , γ 0zξ Shell mid-plane shear strains K ξ , K η Radius of curvatures along ξ and η directions σ ξ , σ η Normal stresses , z , z Shear stresses A, B Lame's parameters E Young's modulus D s Constitutive matrix for shell G Shear modulus υ Poisson's ratio u 0 , v 0 , w 0 Mid-surface displacements of the shell d ei Generalized global displacement vector N i Shape function for the ith node ξ, η Local coordinates for the shell B s Strain displacement matrix of the shell K s Shell stiffness matrix K g Geometric stiffness matrix K e Kinetic energy of the shell m Mass of the shell U Displacement vector of the shell H Transformation matrix Φ Velocity potential c Velocity of sound U z Mean axial flow velocity M Mach number V z Velocity of the shell along the radial direction V r Radial velocity of fluid P s Stagnation pressure Technical Editor: Thiago Ritto.