Numerical solution of fluid–structure interaction in piping systems by Glimm's method

This work presents a numerical procedure for obtaining approximated solutions for one-dimensional fluid–structure interaction (FSI) models, which are used in transient analyses of liquid-filled piping systems. The FSI model considered herein is formed by a system of hyperbolic partial differential equations and describes, simultaneously, pressure waves propagating in the liquid as well as axial, shear and bending waves traveling in the pipe walls. By taking advantage of an operator splitting technique, the flux term is split away from the source one, giving rise to a sequence of simpler problems formed by a set of homogeneous hyperbolic differential equations and by a set of ordinary differential equations in time. The numerical procedure is constructed by advancing in time sequentially through these sets of equations by employing Glimm's method and Gear's stiff method, respectively. To implement Glimm's method, analytical solutions for the associated Riemann problems are presented. The boundary conditions are properly accounted for in Glimm's method by formulating and analytically solving suitable (non-classical) Riemann problems for the pipe's ends. The proposed numerical procedure is used to obtain numerical approximations for the well-known eight-equation FSI model for two closed piping systems, in which transients are generated by the impact of a rod onto one of the ends. The obtained numerical results are compared with experimental data available in the literature and very good agreement is found.