Special Section: Computational Fluid Mechanics and Fluid–Structure Interaction

A Review of Full Eulerian Methods for Fluid Structure Interaction Problems

[+] Author and Article Information
Shu Takagi1

Kazuyasu Sugiyama, Satoshi Ii

Yoichiro Matsumoto

Department of Mechanical Engineering,  The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japanymats@fel.t.u-tokyo.ac.jp


Corresponding author.

J. Appl. Mech 79(1), 010911 (Dec 13, 2011) (7 pages) doi:10.1115/1.4005184 History: Received April 27, 2011; Accepted September 13, 2011; Published December 13, 2011; Online December 13, 2011

We have recently developed a novel numerical method for fluid–solid and fluid–membrane interaction problems. The method is based on a finite difference fractional step technique, corresponding to a standard numerical approach for simulating incompressible fluid flows, and applicable to treating nonlinear constitutive laws of solid/membrane and large deformations. The temporal change of the solid deformation is described in the Eulerian frame by updating the advection equations for a left Cauchy-Green deformation tensor, which is used to express the constitutive equations for materials and membranes. This method is reviewed in detail with some numerical results.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Schematic figure explaining the difference between the solid deformation descriptions of the Lagrangian and Eulerian approaches. In the Lagrangian method, the relative displacement between adjacent material points from the reference to current configurations quantifies the deformation level. In the present Eulerian method, to quantify the deformation, the left Cauchy-Green deformation tensor B is introduced in the Eulerian frame, and temporally updated.

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Figure 2

Comparison of the solid deformation in the lid-driven flow with the simulation result [22]. The dashed outline represents the result of Zhao [22], in which the Lagrangian tracking approach was employed to describe the solid deformation. The solid outline, the dotted material points, and the streamlines correspond to the present simulation results based on the full Eulerian approach with a mesh 1024 × 1024.

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Figure 3

Trajectories of the solid centroid in the lid-driven flow in a time range t ∈ [0,20] for various number of grid points

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Figure 4

The errors of the particle centroid in L2 norm and in L∞ norm versus the number Nx of grid points in the lid-driven flow

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Figure 5

Material point distribution in the imposing-releasing shear flow involving a circular particle between two parallel plates with the 1024 × 256 mesh. The solid obeys an incompressible Saint Venant-Kirchhoff law.

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Figure 6

Development of the material points with an extremely hard elastic property (Ca = 10−5 )

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Figure 7

The fully developed deformed membranes subjected to the linear shear flow for various capillary numbers

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Figure 8

The deformation parameter D versus time t for various capillary numbers. The lines are the present solutions, the filled circles indicate the solutions by Eggleton and Popel [23], and open triangles show those by Pozrikidis [48].

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Figure 9

Developments of the multiple membranes and flexible bodies in an elastic tube




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