Viscous hydrodynamic coupling of parallel cylinders: A comprehensive theoretical and numerical approach

This study investigates, both theoretically and numerically, the hydrodynamic interaction between two parallel circular cylinders undergoing small-amplitude forced harmonic oscillations in an initially quiescent viscous two-dimensional fluid. The theoretical framework builds on a Helmholtz decomposition of the fluid velocity field combined with a bipolar coordinate system. This decomposition leads to a coupled system comprising a Laplace equation and a Helmholtz equation with a variable Lamé coefficient. While similar in spirit to our previous work on the same configuration, the present approach fundamentally differs in the resolution of the Helmholtz equation. In the earlier study, the Helmholtz equation was replaced by an ad hoc version with a constant Lamé coefficient, introducing a residual in the linearized Navier–Stokes equations and providing only an approximate description of the fluid forces. In contrast, the present formulation solves both the Laplace and Helmholtz equations in their full form, thereby preserving the complete structure of the viscous flow problem. This refined model now makes it possible to analyze both axial and transverse oscillations of the cylinders, whereas the previous study was limited to axial motion (i.e., motion aligned with the centers of the cylinders). The fluid forces remain expressible as linear combinations of the cylinder velocities and accelerations, with the coefficients of these combinations corresponding to viscous self- and cross-added mass and damping terms. The variations of these coefficients with the dimensionless separation distance, the Stokes number, and the oscillation direction are explicitly quantified through parametric studies. The accuracy of the improved theory is assessed by direct comparisons with numerical simulations performed using the Arbitrary Lagrangian–Eulerian method implemented in our open-source TrioCFD software, as well as with reference results from the literature. The new theoretical approach yields more accurate estimates of the fluid-added coefficients over a wide range of Stokes numbers, although some deviations between theory and numerics persist for very small Stokes numbers. To foster reproducibility and further investigations, a Python implementation of the present theory is made available for computing the fluid-added coefficients.