A Mixed-Dimensional Formulation for the Simulation of Slender Structures Immersed in an Incompressible Flow
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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 problem can be modeled by mixed-dimensional coupled equations (3D for the fluid and 1D for the solid) [1, 3, 5]. Several challenges must be faced when dealing with this type of problems. From a mathematical point of view, these include defining well posed trace operators of codimension 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 [2, 4]. We define the continuous formulation using 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. One of the fundamental advantages of our approach is that it enables the approximation of the problem within the framework of the Finite Element Method (FEM). We establish the energy stability of the discrete formulation and provide extensive numerical evidence of the accuracy of the mixed-dimensional model, notably with respect to a fully resolved (ALE based) model.
