Proceedings in Applied Mathematics and Mechanics, 31 May 2016

3D Fluid-Structure Interaction Experiment and Benchmark Results Andreas Hessenthaler1,⇤ , Stephanie Friedhoff2 , Oliver Röhrle1 , and David A. Nordsletten3 1 2 3

Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany Mathematical Institute, University of Cologne, Weyertal 86–90, 50931 Cologne, Germany Division of Imaging Sciences and Biomedical Engineering, King’s College London, 4th Floor, Lambeth Wing St. Thomas Hospital London, SE1 7EH, UK

Testing and comparison of well-known and recently developed numerical methods and algorithms are the key for their thorough assessment and continued improvement. We will present a short overview of a new 3D FSI experiment that proposes steady and dynamic test cases (referred to as Phase I and Phase II), complementing well-known standard benchmark problems. In this context, we discuss a mathematical FSI model, that has previously proofed capable of good predictions, and demonstrate that the periodic steady-state solution for Phase II is insensitive to the initial deformation of the solid. Copyright line will be provided by the publisher

1

3D FSI Experiment

Numerical and experimental fluid-structure interaction (FSI) benchmark tests, such as [1–8] and others, have become standard in assessing and comparing various FSI methods and algorithms and a recently developed 3D FSI experiment [9] aims at extending the current test bed. The new test case has been developed using computer-aided design (CAD) tools to ensure precise definition of the underlying fluid and solid domains of the 3D FSI experiment. The fluid domain consists of two inlets with inflow being constant (Phase I) or periodic (Phase II) and an elastic solid is attached to the rigid fluid domain wall in the merging region of the two inflow jets, see Figure 1. Phase I and Phase II inflow conditions result in steady-state and periodic steady-state test cases (period time of 6 s), respectively. Material parameters of the fluid and solid domains were determined and mechanical behavior of the solid was characterized using a uniaxial tensile load-displacement test. Phase-contrast (PC) magnetic resonance imaging (MRI) techniques were used to acquire (time-dependent) data including the flow velocity (in each coordinate direction) at the inlets and in the fluid domain interior and the motion of the solid. Employed sequences were optimized for improved accuracy, for example, the velocity encoding (VENC) sensitivity was tuned to reduce measurement errors. Parabolic profiles were fit to acquired PC MRI data at the inlets for each velocity component at each time point to provide easy to apply boundary condition data for the numerical setting. For more details on the experimental setup, see [9].

2

Numerical Simulation

To model the FSI phenomena observed in the 3D FSI experiment, we employed the arbitrary Lagrangian-Eulerian (ALE) Navier-Stokes equations for incompressible Newtonian flow, the equations governing transient finite elasticity (with an incompressible Neo-Hookean solid model) and used a Lagrange multiplier [10] to constrain the system at the fluid-solid interface and to enforce dynamic and kinematic interface constraints. Further, the fluid domain deformation was modeled using an anisotropic diffusion equation, such that the set of equations to be solved for t 2 [0, T ] were, ⇢f @t v f + ⇢f (v f ⇢s @ t v s

rx ·

w f ) · rx v f s

= (⇢s f

rx ·

= 0 and

in ⌦s

⇢f ) g

· nf +

f

and

rx · v f = 0 in ⌦f ,

Js (us )

1=0

in ⌦0s ⇥ [0, T ],

@t wf + rX · ( rX wf ) = 0 in ⌦0f ⇥ [0, T ], s

· ns = 0 and v f

(1)

v s = 0 in ⌦ ,

with fluid and solid density ⇢f and ⇢s , fluid and solid velocity v f and v s , fluid domain velocity w⇣f , Cauchy stress tensor ⌘ for

F s :F s s the fluid f := µf rx v f 'f I (with dynamic viscosity of the fluid µf ) and the solid s := Jµ5/3 F s F Ts I 's I 3 (with Neo-Hookean parameter µs , deformation gradient tensor F s ), outward boundary normal of the fluid and solid domain nf and ns , fluid and solid pressure 'f = pf ⇢f x · g Po and 's = ps ⇢f x · g Po (denoting a change-of-variables to remove the hydrostatic pressure contribution from the fluid equations), Jacobian mapping Js := det F s of the solid domain displacement us , the Eulerian and Lagrangian gradient operator rx and rX , fluid and solid domain ⌦f and ⌦s , fluid and solid reference domain ⌦0f and ⌦0s and the Lagrange multiplier domain ⌦ . ⇤

Corresponding author: e-mail [email protected], phone +49 711 685 66215 Copyright line will be provided by the publisher

2

PAMM header will be provided by the publisher

Fig. 1: Fluid and solid domain.

Fig. 2: Velocity component vy at t ⇡ 1.2 s: Experimental data and numerical results show good qualitative agreement in observed flow patterns.

dy

The FSI model given in Equation 1 was solved in CHeart [11]. A more detailed description is provided in [12], where it has been shown, 0 that the model in Equation 1 can accurately govern the FSI phenomena observed in the 3D FSI experiment. For the steady-state test case Phase I, numerical results are in exceptional agreement with experimental data -5 and the required simulated time is short, i.e., fluctuations in all velocity components vanish rather quickly after a transition phase due to the inidyref -10 tial conditions. Good qualitative agreement is observed for the periodic dy, different initial position steady-state test case Phase II, for example, observed and predicted ve0 1 2 3 4 5 6 locity component vy at t ⇡ 1.2 s are shown in Figure 2. Furthermore, t [s] important flow features observed in the experiment, such as a double-⌦ shape (see [9]), are governed if the spatial domains are resolved approFig. 3: Relative displacement of solid’s tip during first priately, see [12]. simulated period starting from hydrostatic equilibrium Due to a difference in fluid and solid density (see [9]), the solid in or partially deflected state, dyref and dy. its hydrostatic equilibrium naturally bends upwards under the influence of gravitational forces. Mimicking this initial state (corresponding to the initial state in the experiment) proofed valuable (see [12]), however, different initial states were investigated: (i) undeformed state as shown in Figure 1 and (ii) a partially deflected state. The final periodic steady-state solution obtained from the numerical solution did not show dependence on the initial state. For example, the relative displacement at the end of the first simulated period did not differ significantly, whether the initial state corresponded to the hydrostatic equilibrium or a partially deflected state, see Figure 3. This can be related to the fact, that no inflow occurs during the latter part of each cycle (see [9]) yielding decelerating flow and slowed solid motion.

References [1] [2] [3] [4] [5]

W. Wall, and E. Ramm, CIMNE, Barcelona (1998). W. Wall, Ph.D. Thesis, Institut für Baustatik, Universität Stuttgart (1999). D. P. Mok, Ph.D. Thesis, Institut für Baustatik, Universität Stuttgart (2001). S. Turek, J. Hron, Springer Berlin Heidelberg (2006). Gomes JP, and Lienhart H, in: Fluid-structure interaction I – modelling, simulation, optimization, edited by H. J. Bungartz, and M. Schäfer, Lecture notes in computational science and engineering (Springer, 2006). [6] Gomes JP, and Lienhart H. in: Fluid-structure interaction II – modelling, simulation, optimization, edited by H. J. Bungartz, M. Mehl, M. Schäfer, Lecture notes in computational science and engineering (Springer, 2010). [7] Kalmbach A, De Nayer G, Breuer M. A new turbulent three-dimensional FSI Benchmark FSI-PFS-3A: Definition and measurements. V International Conference on Computational Methods in Marine Engineering 2013. [8] A. Kalmbach, Ph.D. Thesis, Professur für Strömungsmechanik, Helmut-Schmidt Universität Hamburg (2015). [9] A. Hessenthaler, N. R. Gaddum, O. Holub, R. Sinkus, O. Röhrle, and D. A. Nordsletten, Int J Numer Method Biomed Eng (under review). [10] D. Nordsletten, D. Kay, and N. Smith, J Comput Phys 229, pp. 7571–7593 (2010). [11] J. Lee, A. Cookson, I. Roy, E. Kerfoot, L. Asner, G. Vigueras, T. Sochi, S. Deparis, C. Michler, N. P. Smith, and D. Nordsletten, SIAM J Sci Comput 38, pp. C150–C178 (2016). [12] A. Hessenthaler, O. Röhrle, and D. A. Nordsletten, Int J Numer Method Biomed Eng (under review).

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