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# Journal Club February 2011: Challenges of Fluid-Structure Interaction Problems Involving Incompressible, Viscous Flow

Fluid-structure interaction (FSI) problems are common in nature, including biomechanics of birds, fishes, heart valves, arteries, etc. Understanding and accurate prediction of FSI responses are also important to many engineered structures, such as bridges, tall buildings, rotor blades, spars, sails, membranes, etc., in order to avoid potential aeroelastic/hydroelastic instability issues, or to improve performance by actively and/or passively tailoring the structural morphology. Much of the earlier work on FSI problems focused on aerospace structures, where the effects of fluid damping and fluid inertia could be approximated or ignored. Interested readers in aeroelasticity should consult classic introductory texts such as Fung (1955) and Bisplinghoff et al. (1955), or Dowell and Hall (2001) for a recent review article. Nevertheless, there exist broad classes of FSI problems that involve viscous, incompressible flows where the effects of fluid damping and fluid inertia are important. Examples include many problems in biomechanics, microfluidics, and advanced marine/naval structures. Hence, the focus of this issue is to identify some of the challenges related to modeling of FSI problems involving viscous, incompressible flows.

FSI problems involving viscous, incompressible flows are challenging because (1) questions regarding the appropriateness of standard turbulence/wall models because of FSI-induced modifications to the inner-layer structures, (2) the expense of constantly re-meshing the fluid domain to adapt to the deforming solid boundary, particularly for problems with a dense boundary layer mesh, and (3) numerical instability issues for partitioned approaches because of the artificial added mass effect caused by incorrect prediction of the coupling forces based on the predicted, instead of the true, interface displacements from the previous sub-iteration or time step (see van Brummelen, 2009 and references therein). Alternatively, fixed, non-body-conforming methodologies such as the Immersed Boundary (IB) method introduced by Peskin (1972) could be used to solve FSI problems on a fixed, Cartesian mesh that models both the fluid and (a portion or all of) the interior solid domain. IB methods modify the body forces or velocity fields near the body to account for the effect of the moving/deforming solid boundary. Interested readers of IB methods should refer to Mittal & Iaccarino (2005) for a recent review article. Nevertheless, IB methods still require significant mesh refinement near the moving/deforming solid boundary to accurately capture the FSI response. Moreover, the interpolation/extrapolation of the velocity and pressure fields near the interface is non-trivial for multi-dimension problems as the cells near the boundary switches between “solid” and fluid states. Another popular method for FSI problems is the Arbitrary Lagrangian-Eulerian (ALE) methods (e.g. Farhat et al., 2001; Tezduyar, 2003), where an arbitrary finite element mesh can be used inside the computational domains while the interface elements move along with the materials to explicitly tract the interface boundaries. However, for applications with thin material interfaces and large deformations, the frequency of re-meshing may become too high, and the accuracy of the ALE method would deteriorate due to increased anisotropy or uneven distribution of the grid points (van Loon et al., 2007). In summary, more research is needed to advance the state-of-the-art in FSI modeling, particularly for problems involving viscous, incompressible flows and complex material constitutive behaviors. References:

- Y.C. Fung, “An Introduction to the Theory of Aeroelasticity,” New York: John Wiley, 1955.
- R.L. Bisplinghoff, H. Ashley, and R.L. Halfman, “Aeroelasticity,” Cambridge, MA: Addison-Wesley, 1955.
- E.H. Dowell and K.C. Hall, “Modeling of Fluid-Structure Interaction,” Annual Reviews of Fluid Mechanics, 33:445-490, 2001.
- E.H. van Brummelen, “Added Mass Effects of Compressible and Incompressible Flows in Fluid-Structure Interaction,” Journal of Applied mechanics, 76: 021206, 2009.
- C.S. Peskin, “Flow Patterns around Heart Valves—Numerical Method,” Journal of Computational Physics, 10:252-271, 1972.
- R. Mittal and G. Iaccarino, “Immersed Boundary Methods,” Annual Reviews of Fluid Mechanics, 37:239-261, 2005.
- C. Farhat, P. Geuzaine, and C. Grandmont, “The Discrete Geometric Conservation Law and the Nonlinear Stability of ALE Schemes for the Solution of Flow Problems on Moving Grids,” Journal of Computational Physics, 174: 669-694, 2001.
- T.E. Tezduyar, “Computation of Moving Boundaries and Interfaces and Stabilization Parameters,” International Journal for Numerical Methods in Fluids, 43:555-575, 2003.
- R. van Loon, P.D. Anderson, F.N. van de Vosse, and S.J. Sherwin, “Comparison of Various Fluid-Structure Interaction Methods for Deformable Bodies,” Computers & Structures, 85:833-843, 2007.

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## Comments

## difficulties of FSI from my past experiences.

Here is just my personal opinion.

I had some experiences with FSI long time ago, and I had difficulties in several aspects. First of all, fluid behaviors were difficult to model. Then, people usually were good in either solid or fluid mechanics. It was difficult to have in-depth knowledge in both areas. Finally, computing cost was too high compared with testing.

Prof. Bathe and his students worked on FSI for a long time, and here are some interesting results.

http://www.adina.com/fluid-structure-interaction.shtml

## Re: difficulties of FSI

I would say fluid and solid each have their own challenges. In fluid, you have to deal with boundary layer and turbulence structures, and in some cases, changes in the phase of the fluid, waves, and compressibility. In solid, you may have to deal with nonlinear, inelastic constitutive material behavior with complex material failure mechanisms. A good understanding of both the fluid and the solid is necessary to properly model the FSI response and potential failure mechanisms.

Yin Lu (Julie) Young Associate Professor Dept. of Naval Architecture & Marine Engineering University of Michigan

## Discrete Approach for FSL

Hi Professor Young, Thank you for the insightful introduction. In addition to the continuum-based approach you mentioned, it is also worth-noting that problems which involves granular materials are something modeled by coupling discrete element method are with lattice Boltzmann to simulate particle suspensions or porous media [1]. What do you think of this approach? WaiChing

[1] K. Han, Y.T. Feng, D.J.R. Owen,Coupled lattice Boltzmann and discrete element modelling of fluid–particle interaction problems, Computers and Structures, 85:1080-1088 http://linkinghub.elsevier.com/retrieve/pii/S0045794906003907

## Re: DEM approaches

Yes, DEM approaches are promising, but the results depend on the coefficients used for the contact models, the shape of the particles, the background fluid model, as well as on enough realizations to generate proper statistics on the stress & displacement fields.

Yin Lu (Julie) Young Associate Professor Dept. of Naval Architecture & Marine Engineering University of Michigan