Elsevier

Journal of Computational Physics

Volume 280, 1 January 2015, Pages 529-546
Journal of Computational Physics

A discrete-forcing immersed boundary method for the fluid–structure interaction of an elastic slender body

https://doi.org/10.1016/j.jcp.2014.09.028Get rights and content

Abstract

We present an immersed boundary (IB) method for the simulation of flow around an elastic slender body. The present method is based on the discrete-forcing IB method for a stationary, rigid body proposed by Kim, Kim and Choi (2001) [25]. The discrete-forcing approach is used to relieve the limitation on the computational time step size. The incompressible Navier–Stokes equations are implicitly coupled with the dynamic equation for an elastic slender body motion. The first is solved in the Eulerian coordinate and the latter is described in the Lagrangian coordinate. The elastic slender body is modeled as a thin and flexible solid and is segmented by finite number of thin blocks. Each block is moved by external and internal forces such as the hydrodynamic, elastic and buoyancy forces, where the hydrodynamic force is obtained directly from the discrete forcing used in the IB method. All the spatial derivative terms are discretized with the second-order central difference scheme. The present method is applied to three different fluid–structure interaction problems: flows around a flexible filament, a flapping flag in a free stream, and a flexible flapping wing in normal hovering, respectively. Computations are performed at maximum CFL numbers of 0.75–1. The results obtained agree very well with those from previous studies.

Introduction

There are many elastic slender bodies interacting with ambient fluid such as flags fluttering in the wind and flapping insect wings [1], [2], [3]. Motions of these elastic slender bodies are of interest because they may provide us new ideas for bio-mimetic engineering applications such as an energy harvesting eel or wings of micro air vehicles [4], [5]. However, understanding the dynamics of fluid–structure interaction is quite complicated, because an elastic slender body, unlike rigid one, is significantly deformed by the fluid. Thus, for the movement and deformation of an elastic slender body, an efficient and accurate numerical method is required.

There have been quite a few numerical studies about elastic slender bodies using a conventional body-fitted or unstructured mesh to handle fluid–structure interaction (see, for example, [6], [7], [8], [9]). However, they suffer from complex mesh generation and frequent remeshing that may lead to losing accuracy during the procedure of projecting a solution from old mesh to new mesh. On the other hand, the immersed boundary (called IB hereafter) method initially developed by Peskin [10] uses Cartesian or cylindrical mesh and does not require mesh regeneration even for a deforming body. Therefore, the IB method is appropriate for the fluid–structure interaction of an elastic slender body in terms of its efficiency and convenience.

Over the last few decades, many researchers have conducted simulations of flows around elastic slender bodies by developing various versions of IB methods. Among them, Peskin [10] firstly simulated the blood flow in human heart assuming a low Reynolds number and two-dimensional flow. For the fluid–structure interaction inside human heart, the IB was made of moving Lagrangian points linked by springs, and the discrete delta function was used to interpolate the fluid velocity for Lagrangian points and to spread out the momentum forcing on Lagrangian points to fluid grid points (so is called continuous-forcing IB method). It is known that the discrete delta function has a role of attenuating the spurious force oscillations [11], [12]. This original IB method has been widely used to deal with various problems of elastic slender bodies such as animal locomotion, valveless pumping, and parachute [13], [14], [15]. Furthermore, there have been improved versions of this IB method for elastic slender bodies [16], [17], [18], [19] by considering the mass of a solid body, but they bring about a smearing interface problem from mass handling owing to the use of additional discrete delta function or virtual boundary, which decreases the order of accuracy near the body. These IB methods [13], [14], [15], [16], [17], [18], [19] based on the original one [10] have a clear advantage of their straightforward implementation into existing solvers [20], and satisfy the strong coupling between fluid and structure inherently because they use the same numerical method for both fluid and structure equations (monolithic algorithm). However, they have a fundamental difficulty in handling the structure with mass.

To handle the mass of structure more strictly, researchers employed fluid and structure solvers independently (partitioned algorithm). Huang et al. [21] adopted a feedback forcing approach [22] for the interaction between the fluid and flexible filament, where the inextensibility condition of the filament was satisfied by treating the tension force as the constraint of inextensibility. In this study, the discrete delta function was also used for interaction as done in the original IB method [10]. The two additional constants, α and β, used in the feedback forcing approach [22] had to be very large to obtain results especially for unsteady flow problems. These large values of constants provided a severe limitation on the size of computational time step. In another method [23], the momentum forcing was formulated from the inertial term of the structure equation to reduce the number of constants, but there still remained a free constant to be tuned. It is remarkable to note that these IB methods explicitly/semi-implicitly integrate the structure equation in time, which can cause the numerical instability. Nevertheless, this explicit/semi-implicit treatment of the structure equation may be acceptable because the time-step constraint from the IB method is less than or equal to that from the structure equation. In case stiff material is considered, an implicit treatment of the structure equation should be adopted.

Depending on how to implement the momentum forcing, the IB method is classified as two groups, continuous forcing and discrete forcing [24]. An important advantage of the discrete-forcing IB method over the continuous-forcing IB method is that the first has moderate limitation on the size of the computational time step [24], [25], [26]. Although previous studies for the fluid–structure interaction of an elastic body are mainly based on the continuous-forcing IB method, there have been some studies using the discrete-forcing approach [27], [28], [29], [30], [31], [32], [33]. The vocal fold vibration during phonation was studied as an example of biological flows [27], and the vortex-induced vibration of a flexible splitter behind a circular/rectangular cylinder was studied by varying the material properties and Reynolds numbers [28], [29]. Luo and co-workers [30], [31], [32], [33] studied flapping flights with a larger size of the computational time step than those used in the continuous-forcing IB methods [15], [16], [17], [18], [23], [34]. However, these discrete-forcing IB methods need additional interpolation schemes on the IB to obtain the hydrodynamic force or diminish the sharpness of the interface owing to the flow reconstruction remedy to reduce spurious force oscillations [12].

As mentioned above, there are many application areas associated with elastic slender bodies interacting with ambient fluid. For this type of bodies, there is a need to develop a numerical method that is more efficient than those developed for general bodies. Therefore, in the present study, we develop an IB method using the discrete-forcing approach [25] for the fluid–structure interaction of an elastic slender body and to relieve the limitation on the size of the computational time step. We obtain the hydrodynamic force directly from the Navier–Stokes equations without any interpolation on the IB and maintain the sharpness of the IB by avoiding any treatment on the fluid node. We test our method to three different fluid–structure interaction problems: flows around a flexible filament, a flapping flag in a free stream, and a flexible flapping wing in normal hovering. We show that the present numerical method accurately predicts these flows at moderate CFL numbers of 0.75–1.

Section snippets

Numerical method

The present numerical method for the fluid–structure interaction of an elastic slender body is constructed by coupling fluid and structure systems. The incompressible Navier–Stokes equations are solved in the Eulerian coordinate, whereas the motion of elastic slender body is described in the Lagrangian coordinate. Construction of the elastic slender body surface and coupling of the fluid and elastic slender body are given in the framework of the discrete-forcing IB method.

Numerical examples

The present numerical method is applied to three different fluid–structure interaction problems of elastic slender bodies: flows around a flexible filament, a flapping flag in a free stream, and a flexible flapping wing in normal hovering, respectively. All the computations are carried at maximum CFL=0.751. For the first flow problem, we discuss the convergence and stability of the present numerical method. The results obtained agree very well with those from previous studies, indicating the

Summary

In the present study, an IB method was developed for the simulation of flow around an elastic slender body. The present method was based on the discrete-forcing IB method for a stationary body proposed by Kim, Kim and Choi [25] and was fully coupled with the dynamic equation for an elastic slender body motion. The discrete-forcing approach was used to relieve the limitation on the size of the computational time step. The incompressible Navier–Stokes equations were solved in the Eulerian

Acknowledgements

This research is supported by the NRF Programs (NRF-2011-0028032, NRF-2012M2A8A4055647) of MSIP, Korea.

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