A mixed-dimensional formulation for the simulation of slender structures immersed in an incompressible flow

Abstract

We consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixed-dimensional coupled equations. Taking advantage of the slenderness of the structure and thus considering 3D/1D coupled problems raise several challenges and difficulties. From a mathematical point of view, these include defining well-posed trace operators of co-dimension two. On the computational standpoint, the non-standard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixed-dimensional discrete formulation as compared to a fully resolved one. Here we proposed to circumvent theses issues by imposing the fluid–structure coupling conditions on the 2D fluid–structure interface but in a reduced way still taking advantage of the 1D dynamic of the slender structure. We consider the Navier–Stokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixed-dimensional version of the fluid–structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional Fourier space on each beam cross section. Furthermore, we develop a discrete fictitious domain formulation within the framework of the finite element method, establish the energy stability of the scheme, provide extensive numerical evidence of the accuracy of the discrete formulation, notably with respect to a fully resolved (ALE based) model and a standard reduced modeling approach.