New estimations of the added mass and damping of two cylinders vibrating in a viscous fluid, from theoretical and numerical approaches

TL;DR

This paper introduces a new theoretical and numerical approach to estimate fluid forces, including added mass and damping, on two cylinders vibrating in a viscous fluid, highlighting the effects of viscosity, separation distance, and Stokes number.

Contribution

It presents a novel Helmholtz expansion and bipolar coordinate method for calculating fluid forces, and introduces a penalization-based numerical approach for small Stokes numbers in fluid-structure interactions.

Findings

Added mass and damping decrease with Stokes number and separation distance.

Cross-added coefficients increase with Stokes number and separation distance.

Viscous effects add a correction term scaling as Sk^{-1/2}.

Abstract

This paper deals with the small oscillations of two circular cylinders immersed in a viscous stagnant fluid. A new theoretical approach based on an Helmholtz expansion and a bipolar coordinate system is presented to estimate the fluid forces acting on the two bodies. We show that these forces are linear combinations of the {\textcolor{black}{cylinder accelerations}} and velocities, through viscous fluid added coefficients. {\textcolor{black}{To assess the validity of this theory, we consider the case of two equal size cylinders, one of them being stationary while the other one is forced sinusoidally}}. The self-added mass and damping coefficients are shown to decrease with both the Stokes number and the separation distance. The cross-added mass and damping coefficients tend to increase with the Stokes number and the separation distance. Compared to the inviscid results, the effect of viscosity is to add a correction term which scales as $Sk^{-1/2}$. When the separation distance is sufficiently large, the two cylinders behave as if they were independent and the Stokes predictions for an isolated cylinder are recovered. Compared to previous works, the present theory offers a simple and flexible alternative for an easy determination of the fluid forces and related added coefficients. To our knowledge, this is also the first time that a numerical approach based on a penalization method is presented in the context of fluid-structure interactions for relatively small Stokes numbers, and successfully compared to theoretical predictions.