I will 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). 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. I define the continuous formulation using the Navier-Stokes equations for the fluid and a Timoshenko beam model for the structure. I 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 this approach is that it enables the approximation of the problem within the framework of the Finite Element Method (FEM). I 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.