Introduction
The fluid-structure interaction (FSI) problem remains one of the most challenging in computational
mechanics and computational fluid dynamics. Researchers have conducted various studies on certain types
of FSI problems (such as the fluid-rigid body interaction [1, 2, 3, 4, 5], fluids with a non-rotational structure
[6, 7, 8, 9, 10, 11], and FSI with a stationary fluid domain [12, 13, 14, 15]). However, there is a dearth
of practical models on FSI problems involving a structure that rotates and deforms, i.e., an elastic rotor.
The development of mathematical models and numerical methodologies is critical in practice for large-scale
advanced FSI simulations involving an elastic rotor, in order to guide the design, evaluation, and prediction
of various applications, such as hydro-turbines, jet engines, and artificial heart pumps. Therefore, it is of
great significance to develop an efficient and accurate mathematical model and numerical method to handle
fluid-structure interactions involving a rotational and deformable structural motion.
However, numerical simulations of fluids interacting with a rotating structure can be challenging. The
difficulties associated with simulating the rotational structure in FSI stem from the fact that the simulations
rely on the coupling of two distinct descriptions: the Lagrangian description for the solid and the Eulerian
coordinate for the fluid. The arbitrary Lagrangian Eulerian (ALE) method [6, 16, 17, 18, 19] copes with this
difficulty by adapting the fluid mesh to accommodate the deformations of the solid on the interface. When
the ALE method is used, the meshes of the fluid and structure conform on the interface if the Lagrangian
structure mesh is moved to the Eulerian one following ALE mapping. This is important as the degrees
of freedom on the interface are naturally shared by both the fluid and the structure, which facilitates the
implementation of the discretization proposed herein. However, the ALE method has a severe drawback; i.e.,
when the structure has a large displacement or deformation, it is very likely that ALE mapping will distort
the fluid mesh. Even the most advanced and most finely tuned ALE-based scheme cannot perform well
without re-meshing. And, if re-meshing is used to produce the fluid mesh, then the number of mesh nodes
and/or elements over different time levels can no longer be guaranteed to be the same. Thus, interpolations
of variables between every two adjacent time steps are unavoidable, resulting in a time-consuming and even
unstable geometrical process, especially in high-dimension cases. In fact, it is difficult to directly apply the
ALE method to fluid-rotating structure interaction problems.
Many approaches have been proposed to deal with rotational structures in FSI problems. Several of these
approaches model the wind turbine rotor [20, 21, 22, 23, 24] by coupling the finite element method (FEM) for
fluid dynamics, isogeometric analysis (IGA) for structural mechanics, and the non-conforming discretization
on the interface between fluid and structure. In order to apply ALE, these approaches introduce an artificial
cylindrical buffer zone to enclose the rotor; let the mesh of the sliding cylindrical interface weakly enforce the
continuity of the solution fields; and introduce extra unknowns and/or penalties to reinforce the continuity
on the interface via, for example, the Lagrange multiplier or the discontinuous Galerkin (DG) method.
The shear-slip method [25, 26, 20] was introduced to locally reconnect the mesh in order to maintain its
the quality when the structure is undergoing translation or rotation. There are also some interesting ALE
remeshing techniques that are able to deal with large structure displacement. For example, the fixed-mesh
ALE method [27] and the universal mesh method [28] locally rebuild the mesh in order to accommodate the
arbitrary motion of the interface.
In our work, we develop a new ALE method in order to produce a body-fitted moving fluid mesh that
conforms with the rotational and deformable structure mesh on the interface. And, to make our ALE
method work for an elastic rotor immersed in fluid, we first derive a linear structure equation that involves
the rotational matrix from the nonlinear structure model based on decomposing the structure displacement
into two components: rotation and deformation. In addition, we define an artificial cylindrical buffer zone
in the fluid domain to enclose the elastic rotor and rotate together on the same axis of rotation with the
same angular velocity. If the fluid channel is non-axisymmetric, we need to determine the relative motion
information between the rotational fluid subdomain (the cylindrical buffer zone) and the stationary fluid
subdomain (the rest of fluid domain) by matching the grid on the sliding interface and defining them in
our new ALE mapping. Finally, we develop a very stable and easily attainable ALE method designed to
generate a rotational and deformable fluid mesh that matches the structure mesh on the interface.
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