A higher-order semi-Lagrangian mixed finite element methodology in an arbitrary Lagrangian-Eulerian framework for vortex-induced vibrations of rigid structures

Abstract

Vortex-induced vibrations (VIV) arise from the interaction between bluff structures and unsteady vortex shedding. This work presents a numerical framework for solving the two-dimensional incompressible Navier-Stokes equations in an arbitrary Lagrangian-Eulerian formulation. The equations are discretized using the Galerkin finite element method with an inf-sup stable second-order element, avoiding additional stabilization. Two semi-Lagrangian formulations are employed for the material derivative: a two-step scheme (SL2) and a back-and-forth error compensation and correction scheme (SL-BFECC). The framework is validated for flow past a stationary cylinder at Re = 100, with the predicted aerodynamic coefficients and Strouhal number differing by less than 7.1% from reference results. Forced-cylinder simulations reproduce the experimental synchronization region and the variation of the aerodynamic coefficients with forcing frequency. For an elastically supported cylinder, lock-in occurs roughly in between 80 < Re < 140, with a maximum cross-flow amplitude of approximately 0.6D. Three-cylinder triangular and four-cylinder square arrangements are subsequently investigated. The downstream cylinders exhibit up to 2.9 times larger lift fluctuations and approximately 39%-47% larger cross-flow amplitudes than the upstream cylinders. Overall, both semi-Lagrangian schemes capture the principal aerodynamic and structural characteristics of single- and multiple-cylinder VIV configurations.