An ALE formulation of embedded boundary methods for tracking boundary layers in turbulent fluid–structure interaction problems
Introduction
Turbulent viscous Fluid–Structure Interaction (FSI) problems arise in many scientific and engineering applications. Examples include limit cycle oscillation, buffet, dynamic loads analysis at high angles of attack, parachute dynamics, weapon bay acoustics, store separation trajectory predictions, boom refueling and egress operations, aeroelastic tailoring of aircraft and automotive systems, flapping wings, gate sliding, wind turbine and tire noise analysis, and hemodynamics and cardiovascular technology. All three Lagrangian, Arbitrary Lagrangian Eulerian (ALE), and Eulerian computational frameworks have been developed and explored for the solution of such problems. The Lagrangian and ALE computational frameworks move the Computational Fluid Dynamics (CFD) mesh, distort it with the fluid–structure interface, and transport this interface with the local fluid velocity. Unfortunately, large mesh distortions induced by large displacements, rotations, or deformations of the fluid–structure interface can reduce the accuracy and numerical stability of a Lagrangian method to the point where it becomes unpractical. Similarly, large structural motions and/or deformations challenge most if not all mesh motion schemes [1], [2], [3], [4] on which the ALE computational framework [5], [6], [7] relies. For this reason, topological changes such as those induced, for example, by topology optimization [8] or crack propagation typically make the ALE approach unfeasible. The Eulerian computational framework avoids all of the aforementioned issues associated with complex or large transformations of the fluid–structure interface by embedding the wet boundary surface of the structure of interest in a fixed CFD mesh, and relying on computational geometry tools [9], [10] for capturing or tracking the evolution of the position, shape, and topology of this dynamic boundary surface representing the fluid–structure interface.
Adopting the Eulerian computational framework for FSI problems and embedding the boundary surface of a rigid or flexible dynamic structure in a fixed computational fluid domain addresses in most cases the aforementioned limitations of the Lagrangian and ALE approaches. Furthermore, it leads to the concept of CFD computations on non-body-fitted meshes and therefore also simplifies mesh generation. For all these reasons, Eulerian methods for computing flows on embedding CFD meshes have gained popularity, albeit under different names such as “immersed boundary”, “embedded boundary”, “fictitious domain”, and “Cartesian” methods (for example, see Refs. [11], [12], [13]). Here, all of these methods and related computational approaches are collectively referred to as Embedded Boundary Methods (EBMs) for CFD.
Recent developments in EBMs for CFD have focused on various aspects of the treatment of wall boundary conditions [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. However, a remaining drawback of these methods is that they do not track the boundary layers around dynamic bodies, whether these are flexible or even rigid. Essentially, this is because EBMs for CFD are usually Eulerian methods. Consequently, their application to viscous FSI problems requires either high mesh resolutions in large parts of the computational fluid domain, or adaptive mesh refinement. The first proposition is computationally inefficient. The second one is labor intensive, particularly on massively parallel processors. Therefore, the main objective of this paper is to present an alternative approach for maintaining the boundary layers resolved at all times during the numerical simulation of viscous FSI problems characterized by large motions and/or deformations of the structure.
The proposed approach is simple and computationally reasonable. More importantly, it is applicable to any EBM for CFD of interest. Essentially, each computational step is preprocessed by a (rigid) translation and/or rotation of the underlying non-body-fitted CFD mesh that are designed to track the rigid component of the motion of the dynamic structure. Then, away from the embedded surface, the flow is computed using any ALE method, and the wall boundary conditions are treated by any preferred (Eulerian) EBM for CFD. This new computational approach is motivated, explained, and demonstrated in this work in the context of a single flexible or rigid body. To this effect, the remainder of this paper is organized as follows.
In Section 2, the main idea behind the computational approach proposed in this paper is outlined for a simple example. In Section 3, an algorithm for capturing the rigid component of the motion of a dynamic rigid or flexible obstacle of interest is described. This algorithm is proposed for updating in time the position of the embedding CFD mesh without deforming it. The purpose of the mesh position update is to track the boundary layer around the moving and/or deforming obstacle and keep it resolved at all times. The emphasis on a rigid mesh motion is to eliminate the problem of mesh crossovers that plagues ALE methods. In Section 4, the mathematical underpinnings of the resulting ALE-embedded computational framework are simply described. In Section 5, the rigid aspect of the mesh transformation is relaxed to allow special CFD treatments of practical significance. The proposed ALE-embedded computational framework is demonstrated in Section 6 with one turbulent FSI problem whose solution is amenable for verification. Its potential for the simulation of challenging FSI problems at reasonable computational costs is also highlighted in this section with its application to the simulation of turbulent flows past a family of highly flexible flapping wings for which experimental data is available. Finally, conclusions are offered in Section 7.
Section snippets
The main idea using a simple example
Fig. 1(a) shows a computational fluid domain discretized by an Eulerian CFD mesh in which at time , a discrete cylindrical surface is embedded. For example, this surface represents here the wet surface of a flexible cable whose computational domain is discretized by finite elements. Typically, a node of is said to be “real” at a given time t if it lies at this time inside the physical fluid domain; it is said to be “ghost”, if it lies at time t outside the physical fluid
Boundary layer tracking with the corotational method
Let denote the number of nodes on the embedded discrete surface , and C their barycenter. If the obstacle or structure of interest represented by is flexible, typically . If it is rigid, may be equal to 1, in which case the single point of interest is not on but rather the point C. Let denote the displacement vector of the embedded discrete surface at time , measured with respect to a global frame of reference — for example, that used for defining the coordinates of the
Computational framework
To implement the ALE-embedded computational strategy proposed in this paper, the governing Navier–Stokes equations need to be written in ALE form. For this purpose, is viewed as , where denotes the coordinates of a point in space. The configuration is chosen as the reference configuration for which the coordinates of a point in space are denoted by .
Let define a mapping function between and . Then, the ALE conservative
Variant with deformable ALE meshes
It has already been stated that the rigid aspect of the proposed ALE mesh transformation is motivated by the desire to completely eliminate the possibility of a mesh crossover. However, there are situations where this requirement must be relaxed, and the ALE mesh can be allowed to deform with little if any risk of incurring a mesh crossover. In general, such situations are not dictated by the wet surface of the structure , because in the proposed computational approach, the flow in the
Applications
The main objective of this section is to illustrate the basic features of the ALE-embedded computational framework for EBMs described above. For this purpose, this computational framework was implemented in the AERO Suite of Codes [35], [36]. Here the results of its application to the simulation of two different types of FSI problems are presented.
The first considered FSI problem focuses on the heaving of an airfoil in the low Mach number regime. In this case, turbulence is modeled using the
Conclusions
The arbitrarily large displacement of a boundary layer within a fixed computational fluid domain is the Achilles' heel of a conventional Embedded Boundary Method (EBM) for Computational Fluid Dynamics (CFD). This issue was addressed in this paper by reformulating an Eulerian EBM for CFD in an Arbitrary Lagrangian Eulerian (ALE) framework in which the underlying non-body-fitted mesh is (rigidly) translated and/or rotated to effectively track the position of the boundary layer. This proposed
Acknowledgements
The authors acknowledge partial support by the Army Research Laboratory through the Army High Performance Computing Research Center under Cooperative Agreement W911NF-07-2-0027, and partial support by the Office of Naval Research under Grant N00014-11-1-0538.
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