Introduction

Two classic benchmarks are provided to validate the capability of SPH model for FSI problems with different density ratios. The two test cases correspond to the FSI2 and FSI3 cases proposed by Turek and Hron (2006). As shown in Fig.1, an elastic plate is attached to a rigid cylinder. The elastic plate oscillates under the forces induced by the vortex shedding. Case FSI2 is more widely used in the literature than FSI3 because in FSI3 the density ratio between the structure and the fluid is only 1, which can be regarded as a small density ratio, and therefore makes FSI3 more challenging and requires a stronger FSI coupling algorithm in the numerical solver.

Fig. 1. Sketch for the problem of FSI between an elastic object and laminar incompressible flow

Flow phenomena

Incompressible Flow

Viscous Flow

Fluid-structure Interaction

Geometry

As shown in Fig.1, an elastic plate is attached to a rigid cylinder of diameter 𝐷=0.1 m and they are immersed in a viscous current with density 𝜌=1000 kg/m3 and kinematic viscosity 𝜈=1E−3 m2/s. The length of the fluid domain is 𝐿2=2.5 m and the width is 𝐻2=0.41 m.

The cylinder center marked by 𝐶 is located at (0,0) which is the origin of the reference frame, and the tip point 𝐴 of the plate is located at (0.25 m, 0). The bottom of the flow is located at 𝑦=−0.2 m, and therefore the axis of symmetry of the immersed body is located slightly below the axis of symmetry of the fluid domain with a shifted distance of 0.005 m.

Boundary conditions

• Wall boundary

On all the solid walls, including the lateral walls, the cylinder and plate surfaces, no-slip boundary conditions are imposed.

• Left-side boundary

The current enters from the left side and freely exits from the right side. A parabolic velocity profile is prescribed at the left channel inflow:

𝑢(−0.25,𝑦) = 1.5 𝑈 (𝑦+0.2)(𝐻2−𝑦−0.2) (0.5𝐻2)^(-2),

where the mean inflow velocity is 𝑈 and the maximum of the inflow velocity profile is 1.5 𝑈.

• Right-side boundary

The right-side boundary is treated as a free outlet boundary.

Initial conditions

• Test case FSI2

In the test case FSI2, the density of the elastic plate is 𝜌0𝑠=10^4 kg/m3 which leads to a density ratio between the structure and fluid of 𝜌0𝑠/𝜌0𝑓=10. The Young modulus and Poisson ratio are 𝐸𝑠=1.4x1E6 Pa and 𝜈𝑠=0.4, respectively. The mean inflow velocity at the in-flow boundary is 𝑈=1 m/s (which leads to a larger Reynolds number ℜ=𝑈𝐷/𝜈=100). Due to the lift force exerted by the viscous fluid on the elastic plate, the plate deforms and vibrates under the periodic vortex shedding from the structure surface and, finally, a steady VIV state is achieved.

• Test case FSI3

The challenging test case named as FSI3 proposed by Turek and Hron (2006) is considered by increasing the mean inflow velocity to 𝑈=2 m/s (which leads to a larger Reynolds number ℜ=𝑈𝐷/𝜈=200) and increasing the Young modulus of the elastic plate to be 𝐸𝑠=5.6x1E6 Pa (which will lead to a higher vibrating frequency). In addition, the density ratio is reduced to be 𝜌0𝑠/𝜌0𝑓=1.

Results specifications

To validate the numerical results, the displacements of the tip of the elastic plate (point 𝐴 in Fig. 1) is compared with other reference solutions. For FSI2, solutions from FEM, LBM and SPH models are available (see the folder named “FSI2”). For FSI3, solutions from SPH and IB-RLB (immersed boundary - regularized lattice Boltzmann) are available (see the folder named “FSI3”).

Results format

1. For all provided ASCII files, the first column is time (unit: s) and the second column is the displacement of point A (unit: m).

2. In the folder named “FSI2”, four numerical results related to BEM (Turek and Hron, 2006), LBM (Li et al., 2017), IBFEM (Bhardwaj and Mittal, 2012) and SPH (Sun et al., 2021) are provided with an ASCII format.

3. In the folder named “FSI3”, two numerical results related to IB-RLB (Li et al., 2019) and SPH (Sun et al., 2021) are provided with an ASCII format.

Benchmark results of FSI2

Fig.2. Comparison of the vertical displacement of point A between the results of δ+-SPH, Bhardwaj and Mittal (2012), Li and Favier (2017) and Turek and Hron (2006)

Fig. 3. δ+-SPH results of the case FSI2: distributions of the stress component 𝜎11in the elastic plate and the vorticity in the flow field at four time instants

Benchmark results of FSI3

Fig.4. Time evolution of the vertical displacement of point A at the end of the elastic plate, compared to the reference result by Turek and Hron (2006).

Fig.5. Time evolution of the vertical displacement of point A at the end of the elastic plate, compared to the LBM result by Li et al. (2019)

Fig. 6. δ+-SPH results of the case FSI3: distributions of the stress component 𝜎11in the elastic plate and the vorticity in the flow field at four time instants

Download

You can download the full test case below:

Publications using this test case as a benchmark

   O'Connor, J., & Rogers, B.D. (2021). A fluid–structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU. Journal of Fluids and Structures, 104, 103312.

References

Turek, S., & Hron, J. (2006). Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow. In Fluid-structure interaction (pp. 371-385). Springer, Berlin, Heidelberg.

Li, Z., Cao, W., & Le Touzé, D. (2019). On the coupling of a direct-forcing immersed boundary method and the regularized lattice Boltzmann method for fluid-structure interaction. Computers & Fluids, 190, 470-484.

Li, Z., Wang, K., Wu, W., Leo, C. J., & Wang, N. (2017). Vertical vibration of a large-diameter pipe pile considering the radial inhomogeneity of soil caused by the construction disturbance effect. Computers and Geotechnics, 85, 90-102.

Bhardwaj, R., & Mittal, R. (2012). Benchmarking a coupled immersed-boundary-finite-element solver for large-scale flow-induced deformation. AIAA journal, 50(7), 1638-1642.

Sun, P. N., Le Touze, D., Oger, G., & Zhang, A. M. (2021). An accurate FSI-SPH modeling of challenging fluid-structure interaction problems in two and three dimensions. Ocean Engineering, 221, 108552.

Sun, P. N., Le Touzé, D., & Zhang, A. M. (2019). Study of a complex fluid-structure dam-breaking benchmark problem using a multi-phase SPH method with APR. Engineering Analysis with Boundary Elements, 104, 240-258.

Zhang, C., Rezavand, M., & Hu, X. (2021). A multi-resolution SPH method for fluid-structure interactions. Journal of Computational Physics, 429, 110028.