# Optimizing coupled fluid-structure simulations for nuclear relevant geometries

Daniele Vivaldi, Guillaume Ricciardi

► To cite this version:

Daniele Vivaldi, Guillaume Ricciardi. Optimizing coupled fluid-structure simulations for nuclear relevant geometries. Journal of Pressure Vessel Technology, 2024, 146 (4), pp.041405. 10.1115/1.4065584. hal-04664780

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Submitted on 30 Jul 2024

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# Optimizing coupled fluid-structure
simulations for nuclear relevant geometries.

**Daniele Vivaldi**

Institut de Radioprotection et de Sûreté

Nucléaire
(IRSN)

Cadarache, St Paul-lez-Durance, BP3, 13115,
France

Email: daniele.vivaldi@irsn.fr

**Guillaume Ricciardi**

Commissariat à l'Energie Atomique et
aux Energies Alternatives

CEA/DES/IRESNE, Dep. of Nuclear Technology
Cadarache F-13108 Saint-Paul-Lez-Durance,
France

Email: guillaume.ricciardi@cea.fr

## ABSTRACT

The numerical simulation of fluid-structure interactions has gained interest to study flow-induced vibrations. Nevertheless, the high computational resources required by such simulations can represent a significant limitation for their application to industrial configurations. Therefore, simplified modelling approaches, when physically applicable, can represent an interesting compromise. This can be the case of slender structures (tubes, rods) often encountered in nuclear power plants. In this paper, an Euler-Bernoulli beam finite element model is implemented inside the CFD code Saturne. With the goal of finding CFD methods less expensive than Large Eddy Simulations (LES), Unsteady Reynolds Navier-Stokes (URANS) and hybrid URANS/LES approaches are considered. The resulting fluid-structure model is able to calculate the vibration response of cantilever beams under a fluid flow, avoiding the necessity of CFD-FEM code coupling. The first part of the paper describes the model and its implementation: it allows to perform 2-way explicit fluid-structure coupling, using the Arbitrary Lagrangian-Eulerian approach to account for the structure deformations. Validation test cases are presented in the second part: first, the model is validated in terms of frequency, added mass and damping for a cylinder vibrating in static air and water; then, the model is validated towards the vortex-induced resonance and lock-in mechanisms for a cylinder subjected to water cross-flow. The model is then applied to a real experimental configuration of two in-line cylinders in water cross-flow: the calculated vibrations are found to be in good agreement with the experimental measurements.

## NOMENCLATURE

* *E* Young modulus [$N \cdot m^{-2}$].

* *I* Second moment of area [$m^4$].

* *ρ* Material density [$kg \cdot m^{-3}$].

* *A* Beam section [$m^2$].

* *u* Displacement [$m$].

* *F*<sub>ext</sub> External force applied to the beam [$N$].

* *v* Fluid velocity [$m \cdot s^{-1}$].

* *P* Fluid pressure [$N \cdot m^{-2}$].

* *μ* Dynamic viscosity [$Ns \cdot m^{-2}$].

* *μ*<sub>t</sub> Turbulent viscosity [$Ns \cdot m^{-2}$].

* *ρ*<sub>f</sub> Fluid density [$kg \cdot m^{-3}$].

* *φ* Rotation [*rad*].
L Length of the beam element [m].

p Load [N].

$f_n$ Natural frequency of the cylinder [Hz].

$f_s$ Shedding frequency [Hz].

$\delta$ Logarithmic decrement [-].

$C_a$ Added mass coefficient [-].

m Material density per unit length [kg·m⁻⁴].

St Strouhal number.

D Cylinder diameter [m].

U Free stream velocity [m·s⁻¹].

S Flow strain rate magnitude [s⁻¹].

$\Omega$ Flow vorticity rate magnitude [s⁻¹].

k Turbulent kinetic energy [m²·s⁻²].

# 1 INTRODUCTION

Fluid-structure interaction (FSI) of slender structures is a concern in nuclear power plants (NPPs). Two main examples are the vibrations of fuel rods and fuel assemblies due to the single-phase water flow inside the reactor pressure vessel, and the vibrations of U-tube steam generator tubes due to the two-phase steam-water flow. Thanks to the increasing computational performances, numerical simulation of FSI phenomena has become more and more feasible in the recent years and simulations accounting for the structure displacement and the induced deformation of the fluid domain are now affordable.

One solution to account for the displacement of the structure is to adopt an Arbitrary Lagrangian Eulerian (ALE) formulation for the fluid. This yields to challenging numerical problems. Ahn and Kallinderis [1] proposed a conservative ALE scheme to account for hybrid meshes. Hay et al. [2] developed a stable adaptive algorithm for time integration. Couture-Peck et al. [3] presented a monolithic approach to accurately solve the ALE problem. Sonawane et al. [4] proposed a function-based interpolation to move the mesh. One of the first applications of ALE to simulate cylinder vibrations in cross-flow was presented by Longatte et al. [5]; ALE has then been adopted by several other authors to simulate cross-flow vibrations, as will be discussed later in this section.

A different technique to take into account FSI in CFD simulations is the immersed boundary method. In this method, the structure is meshed inside the calculation domain, a time and space dependent porosity (with the fluid-structure interface lying in cells featuring a porosity between 0 and 1) is employed to track the position of the structure. Therefore, the mesh is stationary and does not deform to follow the structure displacement. This method was used, for example, by Benguigui et al. [6]. Another technique, referred to as Chimera [7], superimposes several grids and obtains the solution by interpolation in overlapping sections to establish boundary conditions for the solution on each separate grid. The major advantage of this technique is that a stationary mesh can be used for the flow (background mesh) and a deforming mesh can be superimposed on top of that which moves along with the deforming structure (component mesh). Delcour et al. [8] employed this method to simulate flutter instability of a clamped-free cylinder. Immersed boundary and Chimera techniques allow to cope with larger deformations of the structure, compared to ALE, which tends to bad quality meshes in case of large structure deformations. However, the strength of ALE lies in the fact of implicitly maintaining a consistent near-wall refinement along the fluid-structure interface; immersed boundary and Chimera techniques, on the other hand, require to adopt a near-wall-like refinement in a larger region of the domain, i.e., the region where the structure is supposed to move.

Mittal and Kumar [9] showed that 2D laminar simulations could reproduce *lock-in* phenomenon observed in vortex-induced vibration of a cylinder in cross-flow. Longatte et al. [10] even simulated fluid-elastic instabilities in a cylinder array. Papaioannou et al. [11] studied numerically the effect of spacing between two cylinder using high order spectral elements, they emphasized the reattachment of vortices on the downstream cylinder. Zhao and Cheng [12] showed the effect of attack angle on a square lattice of four cylinders reaching Reynolds number of 20,000 with a $k-\omega$ STT model for turbulence. Gohel et al. [13] proposed similar study with a triangular arrangement performing Direct Numerical Simulations (DNS) for a Reynolds number of 10,000. Tang et al. [14] simulated a large number of cylinders but for laminar flows. Athar Khan et al. [15] performed laminar simulations for a square cylinder. Xiong et al. [16] studied the effect of viscoelastic fluid on vortex-induced vibrations. 2D URANS simulations were performed
by Sadek et al. [17] to study streamwise fluidelastic instability in tube bundles under two-phase flow. Zhang et al. [18] also performed 2D URANS simulations, to study unsteady fluidelastic forces in tube arrays and associated viscous damping coefficient conventions. All the studies cited above only account for rigid body translation motion of the cylinder mostly with 2D simulation, even though Chen and Wu [19] also account for rotation of the cylinder.

Other authors performed 3D simulations of FIV of cylinders in cross-flow, but with the assumption of the structure moving as a rigid body, i.e., with zero rotation and pure translation (that is, without deformation). This allowed to solve the structure displacement through a simple dynamic equation of its center of mass, with no space discretization required, analogously to what is done in 2D simulations. This approach was retained to simulate FIV in tube bundles at high Reynolds numbers (up to about 60,000) by Shinde et al. [20, 21], Berland et al. [22, 23], Benhamadouche and Benguigui [24] and Vivaldi and Baccou [25]. All these simulations employed the ALE method.

Some authors proposed 3D simulations accounting for structural deformations. The classical case of a cylinder with attached flag has been simulated by Hofstede et al. [26] and Xu et al. [27]. Hofstede et al. [26] showed that a strongly coupled method is necessary to couple a fluid and a structural solver, and that a wrong choice of time step can lead to artificial damping. Fully 3D coupled simulation have been performed accounting cylinders subjected to axial flow. Xu et al. [28] compared different URANS models for turbulence. De Santis and Shams [29] used the simulation to extract added mass and damping induced by the flow and showed good agreement with literature. In a similar way, Lagrange et al. [30] used this type of approach, imposing a deformation to the cylinder to identify added mass forces. Feng et al. [31] proposed a 3D fully coupled simulation to study cross-flow vortex-induced vibration for a flexible cylinder with a LES model for turbulence. Brockmeyer et al. [32] performed similar simulations for a cylinder array.

Involving a 3D structural solver accounting for a strong coupling with a CFD solver can be time consuming. Therefore, taking advantage of the slenderness of a cylinder, Liu et al. [33, 34] have implemented a in-house Euler-Bernoulli beam solver to account for the structural dynamic. A similar approach was retained by Kumar and Sriram [35]. Baraglia et al. [36] extended this type of reduced-order approach to treat also large deformations (non-linear beams).

The purpose of the present study is to propose a high fidelity simulation of vortex and turbulent induced vibration of cylinders with optimized computational time. To reach that goal, we combine a deformable mesh imposed by the kinematic of an Euler-Bernoulli beam model with a hybrid URANS/LES approach for the turbulence modelling, which represents the originality of the study. The geometries studied are slender structures with relatively high length over diameter ratio, typically encountered in nuclear reactors. We consider a validation test case (the test case of section 4.2) with Reynolds number up to 30,000. However, the considered experimental test case (presented in section 5) features a Reynolds number significantly lower than the prototypical ones. Nevertheless, the goal of the present work is to establish a methodology that can be later used to consistently simulate more representative configurations, such as portions of tube bundles or spans of fuel assemblies.

The paper is organized as follows. First, the structural model and its implementation are described. Then, two validations tests are proposed. And finally, simulations are compared to experimental results.

## 2 BEAM MODEL

Instead of using 3D FEM, we aim at representing the structure through a reduced order modeling approach based on the Euler-Bernoulli beam model. The Euler-Bernoulli beam fundamental dynamic bending equation is:

$$ \rho A \frac{\partial^2 u}{\partial t^2} + EI \frac{\partial^4 u}{\partial x^4} = F_{ext} \quad (1) $$

Where it is assumed that the beam is homogeneous along its spanwise abscissa (x), i.e., the Young modulus (E) and the moment of inertia (I) are constant. One assumption of the Euler-Bernoulli theory consists in plane sections to remain plane after deformation: this assumption is generally valid for bending beams, unless the beam experiences significant shear or torsional stresses relative to the bending stresses. This can be the case for beams with large diameters and short lengths, which is not the case of the geometries considered here. Eq. 1 is obtained assuming that $sin\phi = \phi$ and $cos\phi = 1$ ($\phi$ being the rotation): this assumption is valid only for small displacements (up to 5° the approximation is almost exact). The slender structures we want to study are not expected to feature displacements higher than this value.
We consider the beam element of Fig. 1, characterized by 4 degrees of freedom (DOF): the displacement of each node and the rotation of each node. The axial displacement can be decoupled from the vertical displacement-rotation (u-φ) solution, and is anyway not considered in the present model, since axial stresses are neglected.

The array of the 4 nodal unknowns of the beam element is:

$$ \{d\}^T = \{u_1, \phi_1, u_2, \phi_2\} \qquad (2) $$

The dynamic solution of the motion of the Euler-Bernoulli beam can be obtained by the following monodimensional finite element equation:

$$ [M] \{\ddot{d}_n\} + [C] \{\dot{d}_n\} + [K] \{d_n\} = \{F\} \qquad (3) $$

Where the subscript n indicates that displacements refer to the displacements of the nodes that replace the continuum solid, once the space discretization is applied. The elementary (i.e., of the beam element of Fig. 1) stiffness matrix ($K_e$) can be obtained by applying the principle of virtual works, considering the Hooke law from Eq. 1 and the shape functions applying to the 4 DOF beam element considered. The resulting matrix is (see, for example, Logan [37]):

$$ [K_e] = \int_L \{B\}^T EI \{B\} dz = EI \begin{bmatrix} \frac{12}{L^3} & -\frac{6}{L^2} & -\frac{12}{L^3} & -\frac{6}{L^2} \\ -\frac{6}{L^2} & \frac{4}{L} & \frac{6}{L^2} & \frac{2}{L} \\ -\frac{12}{L^3} & \frac{6}{L^2} & \frac{12}{L^3} & \frac{6}{L^2} \\ -\frac{6}{L^2} & \frac{2}{L} & \frac{6}{L^2} & \frac{4}{L} \end{bmatrix} \qquad (4) $$

Where B is the matrix containing the second derivative of the shape functions, and L is the length of the element. A lumped mass matrix is retained, i.e., a diagonal mass matrix resulting from the assumption of lumping the mass of the element on its nodes. For a homogeneous Euler-Bernoulli beam, the displacement nodal masses correspond to the total mass of the element divided by 2 (two equal parts are assigned to each node); a simple and consistent assumption is to assume a zero inertial effect associated with rotational degrees (see, for example, Logan [37]), which results in the following singular elementary lumped mass matrix:

$$ [M_e] = \frac{1}{2} \rho A L \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \qquad (5) $$

In the present model, the damping matrix (C) is obtained by multiplying the stiffness matrix by a “damping coefficient”, the value of which depends on the specific structure configuration and generally ranges from $1 \cdot 10^{-7}$ to $1 \cdot 10^{-5}$: this value represents the purely structural damping that, even if generally much lower than the fluid damping (at least for the configurations studied here), provides additional numerical stability to the fluid-structure coupling solution.

F represents the equivalent nodal load array. Assuming an arbitrary vertical constant load p applying on the element length (this assumption is consistent provided that the space discretization of the beam is fine enough to avoid significant gradients of the distributed load along the beam element), the elementary equivalent nodal load array ($F_e$) will be:
$$ \{F_e\} = p \begin{bmatrix} \frac{L}{2} \\ \frac{L^2}{12} \\ \frac{L}{2} \\ \frac{-L^2}{12} \end{bmatrix} \qquad (6) $$

The elementary mass ($M_e$), damping ($C_e$) and stiffness ($K_e$) matrices and the elementary nodal force array ($F_e$) are then assembled in order to obtain the global mass ($M$), damping ($C$) and stiffness ($K$) matrices and the global nodal force array ($F$), used to solve the global beam Eq. 3.

# 3 IMPLEMENTATION INSIDE CODE_SATURNE

The Euler-Bernoulli beam model presented in section 2 was implemented inside the CFD open source code `code_Saturne v7`. The details of the implementation are provided in this section.

## 3.1 Space discretization approach

We refer to the 3D CFD mesh shown in Fig. 2, showing an arbitrary cylindrical cantilever beam inside a fluid domain ($z=0$ in Fig. 2) represents the fixed boundary condition of the cantilever beam). The present model is meant to be applied to hexahedral meshing of the beam surface (the CFD wall boundary condition), with all vertices of a specific layer $j$ located at the same spanwise position ($z_j$). Provided that this condition is satisfied, any other type of mesh can be retained in the rest of the computational domain. Generally speaking, the CFD space discretization required along the beam spanwise direction is finer than the discretization required by the beam model: several CFD mesh layers are comprised between two beam nodes, as represented in Fig. 2.

Once the number of nodes ($N_{nodes}$) retained for the discretization of the beam is fixed, a reference distance equal to $L_{ref} = L_{beam}/N_{nodes}$ (where $L_{beam}$ is the total length of the beam) is used to loop on all $z_j$ positions: the position of each beam node $i$ is fixed in correspondence of the mesh layer $j$ closest to $L_{ref} \cdot i$. It follows that all beam nodes are located in correspondence of the nodes of one CFD layer mesh spanwise position, as shown in Fig. 2(c). Once the beam node positions are defined, the mass, damping and stiffness elementary and, then, global matrices can be built. In order to build the equivalent load array, it is assumed that each node sees half the force of the upstream beam element plus half the force of the downstream beam element (except for the node $i = N_{nodes}$, the free end, that receives only half the force of the upstream element).

## 3.2 Dynamic solution

The algorithm proposed by Hilber, Hughes and Taylor [38], which represents a slight modification of the standard Newmark approach, is retained to solve Eq. 3 and calculate the time and space evolution of the beam node displacement. Compared to the standard Newmark algorithm, this algorithm proved to provide a more robust behavior for the configurations studied and presented in this article.

Giving an initial value of $\dot{d}$, $\dot{d}$ and $d$ (indicated with subscript $t$), the system that allows to solve for their value after a time step equal to $\Delta t$ (indicated with subscript $t+1$) is the following:

$$ M \ddot{d}_{t+1} + C \dot{d}_{t+1} + (1 + \alpha) K d_{t+1} - \alpha K d_t = F_{t+1} \qquad (7) $$

$$ d_{t+1} = d_t + \Delta t \dot{d}_t + \Delta t^2 [(0.5 - \beta) \ddot{d}_t + \beta \ddot{d}_{t+1}] \qquad (8) $$

$$ \dot{d}_{t+1} = \dot{d}_t + \Delta t [(1 - \gamma) \ddot{d}_t + \gamma \ddot{d}_{t+1}] \qquad (9) $$
$\alpha$, $\beta$ and $\gamma$ are parameters that govern the stability and the numerical dissipation of the algorithm. Hilber, Hughes and Taylor [38] showed that an optimized combination of these parameters is $\alpha = -0.1$, $\beta = (1 - \alpha)^2/4$ and $\theta = 0.5 - \alpha$. These values were therefore retained for the present model.

The term $F_{t+1}$ in Eq. 7 represents the load applied by the fluid, converted into equivalent nodal load array (Eq. 6). Fluid forces consist in pressure forces and viscous forces. They can be expressed in terms of stress tensor $\vec{\sigma}$:

$$ \vec{\sigma} = -P\vec{I} + \mu (\nabla\vec{v} + \nabla^T\vec{v}) \quad (10) $$

For a general beam moving both in the x and the y directions, two independent (uncoupled) equations (Eq. 3) are solved, one for each direction: considering the small deformation assumption, non-linear effects are negligible. The vector representing the fluid forces applying on the 2 directions considered, on a specific near-wall cell face is:

$$ \vec{p} = \begin{bmatrix} p_x \\ p_y \end{bmatrix} = \vec{\sigma} \cdot \vec{n} \quad (11) $$

Considering a given direction, the algorithm of Eqs. 7, 8 and 9 can be applied to the present beam model, using the mass, damping and stiffness matrices and the displacement and load arrays detailed in paragraph 2. Substituting Eqs. 8 and 9 into Eq. 7, one obtains:

$$ \begin{aligned} & [M] \{\ddot{d}_{t+1}\} \\ & + [C] \left[ \{\dot{d}_t\} + \Delta t \left( (1-\gamma) \{\ddot{d}_t\} + \gamma \{\ddot{d}_{t+1}\} \right) \right] \\ & + (1+\alpha) [K] \left[ \{d_t\} + \Delta t \{\dot{d}_t\} \right] \\ & + (1+\alpha) [K] \left[ \Delta t^2 \left( (0.5-\beta) \{\ddot{d}_t\} + \beta \{\ddot{d}_{t+1}\} \right) \right] \\ & - \alpha [K] \{d_t\} = \{F_{t+1}\} \end{aligned} \quad (12) $$

where the subscript n, used in Eq. 3, was omitted for the vector $\{d\}$ and its derivatives for the sake of simplicity. One can solve for the unknown $\ddot{d}_{t+1}$ rewriting Eq. 12 in the following way:

$$ [A] \{\ddot{d}_{t+1}\} = \{F_{t+1}\} - [C] \{B\} - [K] \{D\} \quad (13) $$

Where:

$$ [A] = [M] + [C] \gamma \Delta t + (1 + \alpha) [K] \beta \Delta t^2 \quad (14) $$

$$ \{B\} = \{\dot{d}_t\} + (1 - \gamma) \Delta t \ddot{d}_t \quad (15) $$

$$ \begin{aligned} \{D\} = (1 + \alpha) & \left[ \{d_t\} + \Delta t \{\dot{d}_t\} + (0.5 - \beta) \Delta t^2 \{\ddot{d}_t\} \right] \\ & - \alpha \{d_t\} \end{aligned} \quad (16) $$

Eq. 13 represents a linear system and allows to calculate the acceleration array $\{\ddot{d}_{t+1}\}$; from $\{\ddot{d}_{t+1}\}$, the velocity and displacement arrays $\{\dot{d}_{t+1}\}$ and $\{d_{t+1}\}$ can be obtained by Eqs. 9 and 8, respectively.
## 3.3 Mesh displacement

The FSI simulation is based on the Arbitrary Lagrangian Eulerian approach (available in code_Saturne) allowing the mesh deformation in order to take into account the fluid-solid interface motion. The mesh deformation is calculated by the following Laplace diffusion equation:

$$ \nabla \cdot (\lambda \nabla \mathbf{u}_c) = 0 \qquad (17) $$

the boundary condition of which is the moving wall velocity. The mesh motion diffusivity coefficient (λ) specifies the rate of spreading of the boundary mesh motion to the interior mesh and, therefore, drives the cell deformation (u_c is the mesh velocity): in the simulations presented in this paper, the value of λ was set very high in the region near the fluid-solid interface, in order to prevent cell deformation in the boundary layer region.

The beam node displacements calculated by Eq. 3 are used to impose the corresponding displacements of the CFD wall (the fluid-solid interface) boundary condition nodes. For the CFD mesh layers positioned between two beam nodes, the imposed displacement is calculated as a linear interpolation between the upstream and the downstream beam node displacements (as shown in Fig. 3).

## 3.4 FSI algorithm

The fluid forces calculated by the CFD are transferred to the beam finite element model at the end of each time step. These forces are used to calculate the beam node displacements, and the mesh deformation is imposed consistently. The updated CFD mesh configuration is used for the following time step, and so on. No inner fluid-solid sub-iterations are realized within each time step. The resulting FSI coupling approach is 2-way explicit. This type of algorithms is stable and accurate provided that the numerical time step is low enough, compared to the specific FSI mechanism. In other words, 2-way explicit approaches can provide accurate results only if the evolution of the fluid-solid configuration (for example, the evolution of the structure deformation) along one time step is small enough to be consistently predicted by one single fluid-solid iteration. In spite of this limitation, previous FSI simulations performed with a 2-way explicit approach have proven to provide consistent results [39]: this can be explained by the fact that the value of the numerical time step imposed by CFD-driven criteria is generally much smaller than first, the time scale of the simulated FSI mechanism and second, the natural vibration period of the simulated structure. This allows to retain an explicit approach, without losing noticeable precision and stability in terms of FSI results. The 2-way explicit algorithm employed for the present FSI model is presented in Fig. 4.

# 4 VALIDATION TESTS

## 4.1 First test: natural frequency and damping

The first mandatory test of the here proposed FSI model consists in the validation of the calculated beam natural frequency and damping, together with the added mass coefficient, along a free vibration test in static air and in static water.

### 4.1.1 Geometrical configuration

The configuration presented in Fig. 5 was employed for this analysis: a cantilever cylinder, with a length of 250 mm and a diameter of 10 mm, inside a fluid domain. The fluid domain boundary conditions are placed far enough from the cylinder to consider it totally non-confined.

### 4.1.2 CFD model

FSI simulations were performed imposing an insignificant flow velocity (equal to 1 · 10⁻⁴ m/s) at one boundary of the domain, in order to simulate a static flow condition (a CFD simulation with strictly zero velocity boundary condition is generally numerically not straightforward to run). All other boundaries were modeled as constant pressure conditions, with the exception of the cylinder wall where a no-slip condition was applied.
4.1.3 Solid model

In terms of solid mechanical properties, the Young modulus was fixed equal to 2 · 10¹¹ Pa and the density equal to 7850 kg/m³. At the beginning of the simulation, a constant force was imposed to the cylinder free end in order to generate an initial displacement; the force was then removed and only the fluid forces were applied to the cylinder. For the purposes of the present tests, only the cylinder motion along one direction (the x-direction) was solved. Generally speaking, considering the mode shapes of a fixed-free cantilever beam, a discretization into 20 nodes is expected to be able to describe at least the first 4 natural modes. To confirm this assumption, a convergence analysis on the number of nodes is presented in the following section.

4.1.4 Results

The calculated vibration amplitude of the cylinder free end (tip), as a function of the time, is shown in Fig. 6. Fig. 7 shows a zoom of the cylinder displacement along 0.1 seconds: one can observe that the motion in air features some high (i.e., higher than the first) natural modes, whereas only the main sinusoidal mode is present when the cylinder vibrates in water. The corresponding results in terms of natural frequency, damping and added mass are presented in Table 1. $f_n$ in air can be compared to the analytical solution of a cantilever beam: the value calculated by the present FSI simulation is almost identical to the one obtained by the analytical solution. The calculated logarithmic decrement in water is in good agreement with experimental data found in King [40] and in Bolshukhin et al. [41]. Also the added mass coefficient is consistent with experimental and analytical values, as the ones found in Gao et al. [42] and Bolshukhin et al. [41].

Fig. 8 shows the Power Spectral Density (PSD) of the displacement of the free end of the cylinder, along a free vibration test in static air (the one presented in Figs. 6 and 7). Results are presented for four different beam discretizations: 5, 10, 20 and 25 nodes. No visible discrepancies are observed starting from a discretization of 10 nodes. For the discretization with 10 or more nodes, the PSDs contain 6 clear peaks. It can be easily verified that the first 4 calculated mode frequencies (peaks) agree very well with the analytical solution of a fixed-free cantilever beam, whereas a slight discrepancy is obtained for the modes 5 and 6.

4.2 Second test: vortex-induced resonance and lock-in

The second configuration tested is a cylinder in water cross-flow, in order to study the vortex-induced vibrations and the associated resonance and lock-in mechanisms. Vortex-induced vibrations are an interesting FSI problem, since the fluid-structure system response is the result of a tightly coupled fluid-structure mechanism. It is, therefore, a relevant test for validation purpose of the present FSI model.

4.2.1 Geometrical configuration

The configuration simulated is presented in Fig. 9. The cylinder has a diameter of 10 mm and is 350 mm long. This specific geometrical configuration with high length-to-diameter ratio was retained in order to obtain a natural frequency low enough (about 57Hz, see section 4.2.3) to achieve vortex-induced resonance (see section 4.2.4) for reasonable flow velocities. The inlet boundary condition surface is located 20·D upstream of the cylinder, the outlet boundary condition surface is located 30·D downstream of the cylinder. Lateral walls are placed at a distance of 12·D from the cylinder axis, the distance between the cylinder free end and the top wall of the domain corresponds to 15·D.

4.2.2 CFD model

The CFD mesh is shown in Fig. 9. Constant velocity was imposed at the inlet boundary, a constant pressure outlet boundary condition was employed. With the exception of the cylinder wall, all walls were treated as slip walls. The cylinder near-wall cell size respected the $y^+ < 1$ condition, in order to solve the entire boundary layer. The mesh is fully hexahedral and features 2.4 million cells. The refinement was based on the simulation with the highest free stream velocity (3.0m/s), the same mesh was kept for the other simulations at lower velocities. The $k-\omega$-SST turbulence model [43] was used. Second-order upwind scheme was retained for the convective terms of equations. The size of the numerical time step was set based on a maximum CFL value of about 3. The goal of this numerical campaign is to assess the behavior of the FSI model, and not to validate the CFD results: from this point of view, the CFD results obtained through the $k-\omega$-SST model were assumed consistent enough to
obtain relevant results from the FSI simulations (this assumption will prove to be correct, considering the simulation results presented in the following section 4.2.4).

### 4.2.3 Solid model

The cylinder Young modulus and density were set equal to $2 \cdot 10^{11}$ Pa and 7850 kg/m$^3$, respectively, typical values of stainless steel. The resulting natural frequency in air is 57Hz. The beam (the cylinder) was discretized into 25 nodes. Both the axial (drag) and cross (lift) directions were simulated: two beam motion equations (Eq. 3) were therefore solved, one for the direction x and one for the direction y (see Fig. 9). The time step used for the beam solution corresponds to the CFD time step (this is not a choice, but a necessary condition). In order to keep the maximum CFL value equal to around 3 (as discussed in section 4.2.2), the time step was gradually decreased as a function of the simulated fluid velocity: the resulting time steps allowed to perform between 90 (for the lowest velocity simulated) to 260 (for the highest velocity simulated) time steps per cylinder vibration period. This natural period time discretization was assumed to be fine enough to allow using a 2-way explicit approach.

### 4.2.4 Results

A water flow crossing a cylinder is known to generate a vortex shedding mechanism behind it. The shedding frequency ($f_s$) depends on the water free stream velocity ($U$) and the cylinder diameter ($D$): the Strouhal number, defined as $St = f_s D/U$, is commonly used to describe the correlation between $f_s$, $D$ and $U$. For a wide range of Reynolds numbers, $St$ is experimentally found to be constant and equal to 0.2. From these considerations, one would expect vortex-induced cylinder resonance to occur for a water velocity of around 2.6m/s: for this velocity, the vortex shedding frequency estimated through $St=0.2$ would correspond to the cylinder natural frequency expected in water. Different numerical simulations were performed: the imposed water velocity was gradually increased, the details of the simulated test conditions are provided in Table 2. The calculated vortex shedding frequency and the calculated cylinder vibrations were monitored in order to study the consistency of the cylinder response with the flow behavior.

In order to obtain the calculated shedding frequency, the velocity behind the cylinder was monitored at the two different positions shown in Fig. 10: the velocity as a function of the time can be used to calculate the PSD, which provides the frequency content of the signal.

**Shedding frequency and vibration amplitude** Fig. 11 sums up the results, in terms of shedding frequency and amplitude of vibrations, for the seven values (see Table 2) of the free stream velocity simulated. For each velocity, the shedding frequency is provided for both a CFD stand-alone simulation and a FSI simulation. The two values are identical for $U \le 2.4$ m/s, and correspond to $St=0.2$, as expected. For higher flow velocities, the CFD stand-alone simulations still predict a shedding frequency corresponding to $St=0.2$, whereas the FSI simulations modify the shedding frequency, which locks into the cylinder natural frequency (equal to 53-54Hz, in water): this is a well-known FSI mechanism referred to as lock-in [40]. Lock-in occurs for flow velocities in the vortex-induced resonance range: the high amplitudes of vibration induced by the resonance cause the vortex shedding frequency to coincide with the cylinder natural frequency. For $U \ge 2.4$ m/s, the resonance is clearly visible by the drastic increasing of the amplitude of vibrations, as it can be observed in Fig. 11.

The values of the shedding frequency reported in Fig. 11 refer to both the vertical positions monitored and indicated in Fig. 10: no visible difference was found between the two positions.

**Insight into resonance and lock-in** A detailed insight into the lock-in mechanism is provided in Figs. 12 and 13, which present the PSD of the calculated velocity behind the cylinder (at the position of $z=180$ mm shown in Fig. 10) and the PSD of the calculated cylinder displacement along the lift direction, for different values of $U$. At $U=2.4$ m/s, the shedding frequency is very close to the cylinder natural frequency, and the amplitudes of vibrations start to increase significantly, as shown in Fig. 11. The cylinder vibrates mainly at the shedding frequency, slightly lower than its natural frequency; the shedding frequency of the FSI simulation keeps unchanged, compared to the CFD stand-alone simulation. At $U=2.6$ m/s, resonance is triggered, as the drastic increase of the amplitudes of vibrations indicates (see Fig. 11): the FSI simulation predicts the lock-in of the shedding frequency to the cylinder vibration frequency, as the slight difference between the shedding frequency calculated by the FSI and the CFD stand-alone simulations suggests. The lock-in becomes more evident for the flow velocities of 2.8 and 3.0 m/s: the CFD stand-alone simulations calculate
a shedding frequency corresponding to the expected St=0.2 condition; on the other hand, the FSI simulations predict a shedding frequency identical to the cylinder vibration frequency, which is lower than the St=0.2 condition.

The signals of the displacement and of the total fluid force on the cylinder wall as a function of time are presented in Figs. 14-17, for the flow velocities of 0.6-2.4m/s. Simulations were initially run without FSI, for a physical time ranging from 1.5 s to 2.0 s: depending on the specific U, this transient corresponds to 25 to 100 vortex shedding periods. This allows to have an insight of the influence of the FSI on the fluid forces, once the FSI is activated. The impact of the FSI on the fluid force is evident looking, for example, at the drag force for U=1.2 m/s and at the drag and lift forces for U=2.4 m/s: once the cylinder starts to vibrate, the fluid force fluctuations increase significantly, compared to the CFD stand-alone simulation.

It is worth discussing the significant amplitude of vibrations in the drag direction for U=1.2 m/s, noticeable in Fig. 15. In the drag direction, the frequency of the fluid force is twice the value in the lift direction, consistently with the fact that two vortices of equal strength and opposite sign are shed at each shedding period behind the cylinder, but the drag force is insensitive to the sign of the force. Looking at the shedding frequency at U=1.2 m/s (Fig. 11, which represents the shedding frequency of the vortices on one side of the cylinder, consistently with the monitoring position shown in Fig. 10), one can notice that the double of its value is close to the natural frequency of the cylinder: the FSI simulation consistently calculates a resonance-like behavior in the drag direction.

**Cylinder mean deflection in the drag direction** A quantitative validation of the cylinder displacement can be assessed in terms of maximum deflection along the drag direction. In this direction, the mean value of the force represents the pressure loss across the cylinder, and the fluctuating force is generated by the vortex shedding mechanism. Discarding the negligible (at least for flow velocities low enough not to trigger resonance) fluctuating component, the mean value of the force can be retained and supposed to represent a uniformly distributed load on the cylinder. For a cantilever beam submitted to a uniform constant load, an analytical solution is available. Therefore, the time-averaged deflection of the cylinder tip calculated by the FSI model can be compared to the analytical solution obtained imposing the time-averaged fluid force along the drag direction. The analytical solution of the maximum (the tip) deflection ($v_{TIP}$) of a cantilever beam submitted to a uniformly distributed vertical load W (in $[N]$) is (see, for example, [44]):

$$v_{TIP} = \frac{WL^3}{8EI} \qquad (18)$$

Applying to Eq. 18 the calculated (mean) fluid force in drag direction shown in Figs. 14-17, the results listed in Table 3 are obtained. One can notice that the time-averaged tip deflection calculated by the FSI model agrees with the analytical tip deflection. This allows to state that the calculated deflection for a given force is consistently calculated by the beam model implemented in the present FSI approach.

# 5 SIMULATION OF THE EXPERIMENTAL CONFIGURATION BY BOLSHUKHIN ET AL.

The two preliminary validation test cases presented in the previous sections 4.1 and 4.2 allowed to show that the proposed FSI model is able to calculate the relevant fluid-structure mechanisms in cross-flow configuration. The second step in the validation process consists in a quantitative assessment of the model versus experimental data of fluid-induced vibrations: this was realized by applying the FSI model to the experimental configuration of Bolshukhin et al. [41].

## 5.1 Geometrical configuration

The experimental configuration used by Bolshukhin et al. [41, 45] is presented in Fig. 18 and consists of two in-line cylinders subjected to a water cross-flow. The cylinders have a cantilever attachment on one end, and a free end inside the channel. The length of the cylinders is 19 8mm, the test section height is 200 mm, which means that only 2 millimeters separate the cylinder free end from the test section top wall. The cylinder diameter is 7.0 mm, they are made of metal thin-walled tube, the wall thickness is 0.3 mm. The distance between the two cylinders is 45 mm. The material of the cylinders is stainless steel. The natural frequency of the first cylinder is 107 Hz in air and 98 Hz in
water, the one of the second cylinder is 98 Hz in air and 90 Hz in water.

The dimensions of the channel are provided in Fig. 18. The width of the test section is only 30 mm, which results in a non-negligible confinement of the cylinders: this aspect adds complexity to the actual FSI mechanisms.

The instantaneous flow velocity is monitored at different positions behind the two cylinders (see Fig. 19) and the velocity PSD is obtained from these signals. Accelerometers are mounted inside the cylinders to measure flow-induced vibrations.

This experimental campaign was the object of a CFD benchmark in the frame of the OECD-WGAMA [45]. The experimental test featuring the lowest (among the ones tested experimentally) water flow rate was retained for the assessment of the present FSI model.

## 5.2 CFD model

Two CFD approaches were attempted: a URANS simulation based on the $k-\omega$-SST two-equation linear-eddy-viscosity turbulence model, and the Delayed Detached Eddy Simulation (DDES) implemented in the $k-\omega$-SST model [46]. DDES is a hybrid URANS/LES approach: a URANS-like behavior is obtained in the near-wall region, whereas the model tends to a LES-like behavior far from the wall. This is obtained by modifying the dissipation source term of the turbulent kinetic energy ($k$) transport equation of the $k-\omega$-SST model, that becomes [46]:

$$D_k = -\frac{\rho_f k^{\frac{3}{2}}}{l_{DDES}} \qquad (19)$$

with $l_{DDES}$ defined as:

$$l_{DDES} = l_0 - f_d \max(0, l_0 - C_{DES} \Delta_{\max}) \qquad (20)$$

where $l_0$ is the turbulent length scale:

$$l_0 = \frac{k^{\frac{1}{2}}}{C_{\mu}\omega} \qquad (21)$$

and $\Delta_{\max}$ the length of the longest face of the cell. The function $f_d$ is defined as:

$$f_d = 1 - \tanh\left[(C_{d1}r_d)^{C_{d2}}\right] \qquad (22)$$

with

$$r_d = \frac{\mu + \mu_t}{\rho_f \kappa^2 d_w^2 [0.5 (S^2 + \Omega^2)]^{\frac{1}{2}}} \qquad (23)$$

where $\kappa$ the von Karman constant, $d_w$ the distance from the wall. The constants $C_{DES}$, $C_\mu$, $C_{d1}$ and $C_{d2}$ are equal to 0.61, 0.09, 20 and 3, respectively [46].

A low-Reynolds wall-resolving approach was employed, i.e., the value of $y^+$ was kept lower than or equal to one for all near-wall cells. The time step of the simulation was fixed equal to $1.8 \cdot 10^{-4}$s and to $9 \cdot 10^{-5}$s for the URANS and DDES simulations, respectively. The value of the time step in the case of DDES guaranteed to keep the maximum CFL number lower than 1 in the region where a LES solution was expected. This value of time step resulted in around 50 and around 100 time steps per cylinder vibration period for URANS and DDES, respectively. First order time scheme discretization was used. Second order upwind and second order central scheme were retained for space discretization
of the momentum equations for URANS and DDES, respectively.

Based on the turbulence length scales estimated by the URANS simulation, the retained grid size was estimated to be able to solve at least 80% of the turbulent spectrum, in the DDES.

The simulated numerical domain and meshing are shown in Fig. 20. The CFD mesh consists of about 10 million hexahedral cells. The inlet boundary conditions were fixed based on the experimental measurements of mean and rms water velocity available 50 mm upstream of the first cylinder.

## 5.3 Solid model

The beam model was applied to the cylinders through a discretization into 30 nodes. This space discretization proved to represent a converged structural mesh (see also section 4.1.4), since results did not change by further increasing the number of nodes. The Young modulus and density (available in [45]) are equal to 1.8·10¹¹ Pa and 7850 kg/m³; the value of the density was locally increased for the beam elements lying in the region where the accelerometers are installed, consistently with their higher material density in this zone. It has to be noted that the weight of the accelerometers was not identical between the two cylinders, and this explains the slight different natural frequencies between the two cylinders.

## 5.4 Results

### 5.4.1 Flow velocity PSD

The velocity PSD behind the first (position v1 of Fig. 19) and second (position v3 of Fig. 19) cylinder are presented in Fig. 21. The PSD refer to the vertical measurement point located at the middle of the cylinder height (100 mm from the fixed end). For both cylinders, the vortex shedding harmonic is well calculated by DDES, in terms of both frequency and amplitude. The $k-\omega$-SST slightly underestimates the shedding frequency. However, the main difference between DDES and $k-\omega$-SST is represented by the energy content of the spectrum: the $k-\omega$-SST largely underestimates it and calculates a harmonic cascade not observed in the experimental spectrum; on the other hand, DDES calculates a spectrum very close to the experimental one from the lowest frequencies to frequencies as high as 150 Hz (the numerical PSD content starts to decrease more than the experimental one beyond this frequency).

Numerical results obtained in the frame of the OECD benchmark, available in [45], suggest that non-linear eddy-viscosity models and Reynolds-stress models can in some way improve the predictions, compared to linear eddy-viscosity models. However, simulations adopting scale-resolving approaches (LES, hybrid URANS/LES) were found to predict overall superior results. This shows the need for scale-resolving simulations for the correct prediction of fluid forces for such configurations and, in turn, the interest of using reduced order structure models for such approaches requiring fine grid size and low numerical time steps.

### 5.4.2 Cylinder acceleration PSD

The cylinder acceleration PSD is reported in Figs. 22 and 23 for the drag and lift directions, respectively. Generally speaking, all experimental PSD show two main peaks: the first one corresponds to the cylinder response at the force generated by the vortex shedding mechanism, that excites the cylinder at the main shedding frequency (twice the shedding frequency in the drag direction); the second peak corresponds to the cylinder response at its natural frequency, excited by random (turbulent) forces, i.e., forces that act on the cylinder with no dominant frequency.

In Figs. 22 and 23, the numerical results obtained by Vivaldi [39] for the same experimental test, using Ansys Fluent coupled to Ansys Mechanical are reported: these simulations retained the Scale Adaptive Simulation [47] hybrid URANS/LES approach for the CFD part, a 3D FEM discretization of the cylinders, and a two-way explicit approach to couple the CFD simulation to the FEM simulation. This allows to compare the present results, obtained by a reduced order model for the structure directly implemented inside a CFD code, to a full order simulation result obtained through code-to-code coupling.

In the drag direction (Fig. 22), the DDES results are overall in good agreement with the experimental ones for both cylinders: the shedding harmonic at $2f_s$ is well calculated in terms of both frequency and amplitude, the natural frequency harmonic ($f_n^{1cyl}$ and $f_n^{2cyl}$) is well calculated in terms of frequency, nevertheless slightly underestimated in terms of amplitude. Consistently with the poor results in terms of flow velocity PSD, the $k-\omega$-SST is unable to predict any trend of the vibrational experimental results. The DDES results obtained with the present FSI model are
almost identical to the results obtained with a full order structural model. The experimental PSD shows a peak at $f_s$ (this is clearly visible for the second cylinder), which is not present in the numerical results: this experimental peak is unexpected and would suggest a small misalignment of the accelerometers with respect to the drag direction, that would cause the accelerometers to capture a contribution of the signal along the lift direction.

In the lift direction (Fig. 23), the DDES results well calculate (both in frequency and amplitude) the vibration peak corresponding to the shedding frequency harmonic ($f_s$), whereas the amplitude of the peak corresponding to the natural frequency harmonic ($f_n^{1cyl}$ and $f_n^{2cyl}$) is quite significantly underestimated. These results are very similar to the ones obtained with a full order structural model: the underestimation suggests that, despite the good results shown in Fig. 21, the fluid force spectrum calculated by the CFD model needs to keep closer to the experimental one even for frequencies beyond 150Hz, if one wants to capture the necessary exciting force.

The calculated $f_n$ of both cylinders is slightly overestimated, as the numerical peaks at $f_n^{1cyl}$ and $f_n^{2cyl}$ show: this is probably the result of the mass distribution (the mass per unit length along the cylinder length is not constant, due to the presence of the accelerometers) that is not exactly the same as the actual one (approximation linked to the space discretization of the cylinders, which cannot guarantee to have the local correct mass in case of mass discontinuities, such as the presence of an accelerometer). However, it can be noticed that a similar overestimation of the vibrating frequency for the first cylinder was obtained also by the full order structural model, which suggests that the mass distribution is not the only source of discrepancy.

### 5.4.3 CPU time considerations

The main goal of developing reduced-order structure models and hybrid URANS/LES approaches is to reduce computational time, keeping a degree of precision of the calculated results comparable to 3D FEM and LES.

In terms of CPU time, the FSI simulations run with the present model required 220 hours to simulate 10 seconds of physical time, with a parallelization on 80 CPU. Table 4 provides a quantitative comparison of the CPU time and mesh size necessary to run an FSI simulation, as a function of different modeling approaches for the fluid flow (URANS, hybrid URANS/LES and LES) and the structure (reduced-order beam model and 3D FEM). The values provided in the table refer to wall-resolved (WR) CFD, which is the approach retained for the simulations presented in this article. The number of mesh cells required by a hypothetical WR-LES was estimated, based on a grid size similar to the one retained for DDES far from the wall, but a WR LES mesh in the near-wall region and a LES-like refinement upstream of the first cylinder, necessary to properly apply LES to the entire domain.

The CPU time ratio between a given CFD approach coupled to 3D FEM and to the beam model was found to be in the range of 5 to 10. Switching from URANS to hybrid URANS/LES and to LES, the increasing necessary mesh size results in an increasing CPU time<fnref n="1" />, because of the CFD solving time but also because of the mesh deformation process performed at each time step: the corresponding increasing CPU-time is difficult to be exactly evaluated, the values provided in Table 4 for LES must be considered as approximated. Anyway, the goal of Table 4 is to show that, for the simulation of fluid-induced vibrations of slender structures, hybrid URANS/LES CFD approaches coupled to a beam model can represent an optimized compromise between solution precision and CPU time.

## 6 CONCLUSIONS

This article deals with the use of coupled fluid-structure simulations for predicting vibrations of slender structures (beams) subjected to a water flow. Considering the final goal of adopting such simulations for industrial configurations, the goal of the article is to study the use of optimized simulation approaches, in terms of compromise between precision and CPU time. To do that, a reduced-order beam approach coupled to URANS and hybrid URANS/LES CFD simulations was considered. The 1D beam model features a drastically reduced number of discretization nodes, compared to a general 3D FEM approach. Furthermore, it was implemented inside the CFD code, thus avoiding the necessity of code-to-code coupling.

The proposed FSI model was first applied to validation tests of a cylinder in water cross-flow, which allowed to confirm its capability to predict consistently damping, added mass, vortex-induced resonance and lock-in.

<footnote n="1">¹CPU time estimated for URANS in Table 4 refers to eddy-viscosity models (linear or non-linear). URANS based on Reynolds-stress models imply the solution of 5 additional transport equations, compared to 2-equation eddy-viscosity models and to hybrid approaches based on 2-equation eddy-viscosity models (such as SAS and DDES discussed here): considering the higher number of transport equations to solve, but at the same time the theoretically slightly less strict requirements in terms of mesh refinement and CFL, the resulting CPU time of wall-resolved Reynolds-stress models is comparable to the one of hybrid approaches.</footnote>
Then, an experimental campaign accessing local flow behavior and vibration response of two in-line cylinders in water cross-flow was simulated: numerical results are found to be in good agreement with the experimental measurements, both in terms of fluid flow and cylinder vibrations, despite some underestimations of the cylinder displacement amplitudes. URANS was found to significantly underestimate the energy content of the fluid force spectrum and does not appear (at least the linear eddy-viscosity approach tested here) as a proper approach for such applications. On the other hand, DDES (a hybrid URANS/LES model) was found to potentially represent a powerful option to calculate a consistent spectrum of fluid forces and, in turn, a consistent structure vibration response.

In terms of structure simulation, the proposed FSI approach based on a beam model was found to require a CPU time significantly lower (by a factor 5 to 10) than a corresponding approach based on a 3D FEM for the structure resolution. Compared to LES, DDES allows to significantly reduce the grid size in the near-wall region (wall-parallel directions) and in regions where a LES-like solution is not strictly required for the consistent calculation of the relevant phenomena. It can be concluded that, for applications similar to the ones studied here, FSI simulations based on hybrid URANS/LES approaches coupled to beam models represent an interesting compromise between CPU-time feasibility and result precision.

## REFERENCES

[1] Ahn, H. T., and Kallinderis, Y., 2006, “Strongly coupled flow/structure interactions with a geometrically conservative ale scheme on general hybrid meshes,” *Journal of Computational Physics*, **219**(2), pp. 671–696.

[2] Hay, A., Etienne, S., Garon, A., and Pelletier, D., 2015, “Time-integration for ale simulations of fluid-structure interaction problems: Stepsize and order selection based on the BDF,” *Computer Methods in Applied Mechanics and Engineering*, **295**, pp. 172–195.

[3] Couture-Peck, D., Garon, A., and Delfour, M. C., 2020, “A new k-TMI/ALE fluid-structure formulation to study the low mass ratio dynamics of an elliptical cylinder,” *Journal of Computational Physics*, **422**, p. 109734.

[4] Sonawane, C., Praharaj, P., Pandey, A., Kulkarni, A., Kotecha, K., and Panchal, H., 2022, “Case studies on simulations of flow-induced vibrations of a cooled circular cylinder: Incompressible flow solver for moving mesh problem,” *Case Studies in Thermal Engineering*, **34**, p. 102030.

[5] Longatte, E., Bendjeddou, Z., and Souli, M., 2003, “Application of Arbitrary Lagrange Euler formulations to flow-induced vibration problems,” *J. Pressure Vessel Technol.*, **125**(4), pp. 411–417.

[6] Benguigui, W., Laviéville, J., and Merigoux, N., 2019, “Fluid-structure interaction in two-phase flow using a discrete forcing method,” *Int. J. Numer. Meth. Fluids*, **91**, pp. 247–261.

[7] Benek JA., Steger JL., D. F. “A flexible grid embedding technique with application to the Euler equations,” *6th Computational Fluid Dynamics Conference Danvers, MA, U.S.A. 13 July - 15 July (1983)*.

[8] Delcour, L. e. a., 2022, “Investigating the influence of compressibility on the second mode flutter instability of a clamped-free cylinder in axial flow using fluid-structure interaction simulations with the Chimera technique,” *Journal of Fluids and Structures*, **109**.

[9] Mittal, S., and Kumar, V., 1999, “Finite element study of vortex-induced cross-flow and in-line oscillations of a circular cylinder at low Reynolds numbers,” *International Journal for Numerical Methods in Fluids*, **31**(7), pp. 1087–1120.

[10] Longatte, E., Bendjeddou, Z., and Souli, M., 2003, “Methods for numerical study of tube bundle vibrations in cross-flows,” *Journal of Fluids and Structures*, **18**(5), pp. 513–528 Arrays of Cylinders in Cross-Flow.

[11] Papaioannou, G., Yue, D., Triantafyllou, M., and Karniadakis, G., 2008, “On the effect of spacing on the vortex-induced vibrations of two tandem cylinders,” *Journal of Fluids and Structures*, **24**(6), pp. 833–854.

[12] Zhao, M., and Cheng, L., 2012, “Numerical simulation of vortex-induced vibration of four circular cylinders in a square configuration,” *Journal of Fluids and Structures*, **31**, pp. 125–140.

[13] Gohel, H. R., Shah, B. A., and Lakdawala, A. M., 2013, “Numerical investigation of flow induced vibration for the triangular array of circular cylinder,” *Procedia Engineering*, **51**, pp. 644–649 Chemical, Civil and Mechanical Engineering Tracks of 3rd Nirma University International Conference on Engineering (NUiCONE2012).

[14] Tang, D., Bao, S., Luo, L., Zhu, H., and Cui, H., 2019, “A CFD/CSD coupled method with high order and its applications in flow induced vibrations of tube arrays in cross flow,” *Annals of Nuclear Energy*, **130**, pp. 347–356.
[15] Athar Khan, M., Masood, S., Anwer, S. F., Khan, S. A., and Arif, M. R., 2023, "Vortex induced vibration for mixed convective flow past a square cylinder," *International Journal of Heat and Mass Transfer*, **202**, p. 123722.

[16] Xiong, Y., Peng, S., Zhang, M., and Yang, D., 2019, "Numerical study on the vortex-induced vibration of a circular cylinder in viscoelastic fluids," *Journal of Non-Newtonian Fluid Mechanics*, **272**, p. 104170.

[17] Sadek, O., Mohany, A., and M., H., 2020, "Numerical simulation of streamwise fluidelastic instability of tube bundles subjected to two-phase cross flow," *Journal of Fluids and Structures*, **92**.

[18] Zhang, M., Wang, X., and Oiseth, O., 2022, "Numerical simulation and modeling convention of unsteady fluidelastic forces of tube arrays," *J. Pressure Vessel Technol.*, **144(2)**.

[19] Chen, L., and Wu, G., 2022, "Vortex-induced three-degree-of-freedom vibration of a piggyback circular cylinder system," *Applied Ocean Research*, **123**, p. 103145.

[20] Shinde, V., et al., 2014, "Numerical simulation of the fluid-structure interaction in a tube array under cross flow at moderate and high Reynolds number," *Journal of Fluids and Structures*, **47**.

[21] Shinde, V., Longatte, E., and Baj, F., 2018, "Large eddy simulation of fluid-elastic instability in square normal cylinder array," *J. Pressure Vessel Technol.*, **140(4)**.

[22] Berland, J., Deri, E., and Adobes, A. "Large-Eddy Simulation of cross-flow induced vibrations of a single flexible tube in a normal square tube array," *Proceedings of the ASME 2014 Pressure Vessels & Piping Division Conference, July 20-24, 2014, Anaheim, USA*.

[23] Berland, J., Deri, E., and Adobes, A. "Investigation of cross-flow induced vibrations in a normal square tube array by means of Large-Eddy Simulations for tube damage risk assessment," *Proceedings of Computational Fluid Dynamics for Nuclear Reactor Safety Applications-6 (CFD4NRS-6), 13-15 September 2016 Cambridge, United States*.

[24] Benhamadouche, S., and Benguigui, W. "Wall-resolved LES and low-Reynolds number URANS combined to an arbitrary lagrangian eulerian approach for predicting water cross-flow induced vibrations of a single flexible tube in a normal square tube array," *Proceedings of NURETH-20, 20-25 August 2023, Washington (USA)*.

[25] Vivaldi, D., and Baccou, J. "URANS and hybrid URANS/LES coupled fluid-structure simulations of fluid-induced vibrations in a square-pitch tube bundle subjected to water cross-flows," *Proceedings of NURETH-20, 20-25 August 2023, Washington (USA)*.

[26] Ter Hofstede, E., Shams, A., and van Zuijlen, A., 2015, "Numerical prediction of flow induced vibrations in nuclear reactor applications," *Conference: 16th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-16)At: Chicago, USA*, 09.

[27] Xu, B., Wang, H., Deng, Y., Shen, X., Geng, L., Liu, K., and Zhang, D., 2022, "Nonlinear vortex dynamic analysis of flow-induced vibration of a flexible splitter plate attached to a square cylinder," *Ocean Engineering*, **264**, p. 112433.

[28] Ter Hofstede, E., Kottapalli, S., and Shams, A., 2017, "Numerical prediction of flow induced vibrations in nuclear reactor applications," *Nuclear Engineering and Design*, **319**, pp. 81–90.

[29] De Santis, D., and Shams, A., 2017, "Numerical modeling of flow induced vibration of nuclear fuel rods," *Nuclear Engineering and Design*, **320**, pp. 44–56.

[30] Lagrange, R., Puscas, M. A., Piteau, P., Delaune, X., and Antunes, J., 2022, "Modal added-mass matrix of an elongated flexible cylinder immersed in a narrow annular fluid, considering various boundary conditions. new theoretical results and numerical validation," *Journal of Fluids and Structures*, **114**, p. 103754.

[31] Feng, Z., Jiang, N., Zang, F., Zhang, Y., Huang, X., and Wu, W., 2016, "Nonlinear characteristics analysis of vortex-induced vibration for a three-dimensional flexible tube," *Communications in Nonlinear Science and Numerical Simulation*, **34**, pp. 1–11.

[32] Brockmeyer, L., Merzari, E., Solberg, J., Karazis, K., and Hassan, Y., 2019, "High fidelity simulation and validation of crossflow through a tube bundle and the onset of vibration," *International Journal of Non-Linear Mechanics*, **117**, p. 103231.

[33] Liu, Z., Liu, Y., and Lu, J., 2012, "Fluid-structure interaction of single flexible cylinder in axial flow," *Computers & Fluids*, **56**, pp. 143–151.

[34] Liu, Z., Liu, Y., and Lu, J., 2013, "Numerical simulation of the fluid-structure interaction for two simple fuel assemblies," *Nuclear Engineering and Design*, **258**, pp. 1–12.

[35] Kumar, J., and Sriram, V., 2020, "Development of a hybrid model based on mesh and meshfree methods and its application to fluid-elastic structure interaction for free surface waves," *Journal of Fluids and Structures*, **99**.
[36] Baraglia, F., Benguigui, W., and Denefle, R., 2020, “Development of a hybrid model based on mesh and meshfree methods and its application to fluid-elastic structure interaction for free surface waves,” *Journal of Fluids and Structures*, **99**.

[37] Logan, D. L., 2018, *A first course in the Finite Element Method, Fourth Edition* Springer.

[38] Hilber, H. M., Hughes, T. J. R., and Taylor, R., 1977, “Improved numerical dissipation for time integration algorithms in structural dynamics,” *Earthquake engineering and structural dynamics*, **5**, pp. 283–292.

[39] Vivaldi, D. “Fluid-structure simulations of vortex shedding induced vibrations on two in-line cylinders in water cross-flow,” *Proceedings of the ASME Pressure Vessels Piping Conferences, July 17-22, 2022, Las Vegas, Nevada, USA*.

[40] King, R., 1977, “A review of vortex shedding research and its application,” *Ocean Engineering*, **4**, pp. 141–171.

[41] Bolshukhin, M. A., et al., 2021, “Dynamic measurements of the flow and structure oscillations to validate FSI calculations,” *Nuclear Engineering and Design*, **381**.

[42] Gao, Z., et al., 2020, “Hydrodynamic damping of a circular cylinder at low KC: Experiments and an associated model,” *Marine structures*, **72**.

[43] Menter, F. R., 1994, “Two-equation eddy-viscosity turbulence models for engineering applications,” *AIAA Journal*, **32**(8)(1598-1605).

[44] Megson, T. H. G., 2005, *Structural and Stress Analysis, Second Edition*. Elsevier Butterworth-Heinemann.

[45] “The fluid-structure interaction (FSI) benchmark based on OKBM experiments to validate coupled computational fluid dynamics (CFD) and computational structural dynamics (CSD) calculations,” *Nuclear Safety NEA/CSNI/R(2023)2*.

[46] Gritskevich, M., Garbaruk, A., Schutze, J., and Menter, F., 2012, “Development of DDES and IDDES formulations for the $k-\omega$ shear stress transport model,” *Flow Turbulence Combust*, **88**, pp. 431–449.

[47] Menter, F. R., and Egorov, Y., 2010, “The Scale-Adaptive Simulation method for unsteady turbulent flow predictions. Part 1: Theory and model description,” *Journal of Flow, Turbulence and Combustion*, **85**, pp. 113–138.

[48] Chen, S., and Wambsganss, M., 1974, “Design guide for calculating natural frequencies of straight and curved beams on multiple supports,” *Argonne National Laboratory report ANL-CT-74-06*.
LIST OF FIGURES

<table>
  <tbody>
    <tr>
      <td>1</td>
      <td>The 4-DOF Euler-Bernoulli beam element considered.</td>
      <td>19</td>
    </tr>
    <tr>
      <td>2</td>
      <td>3D CFD mesh and representation of the beam discretization: (a) CFD mesh cut on a x-z plane, showing the presence of the beam and its discretization into several nodes, (b) CFD mesh cut on a x-y plane, showing the presence of the beam, (c) CFD mesh faces at the fluid-solid interface (i.e., the CFD wall boundary corresponding to the cylindrical beam surface) showing the correspondence between the beam node position and the CFD mesh layer in the spanwise (z) direction (the hexahedral mesh is built such that all CFD nodes of a given spanwise layer are placed at the same z position).</td>
      <td>20</td>
    </tr>
    <tr>
      <td>3</td>
      <td>(a) displacement imposed to the nodes of the CFD mesh located at the fluid-solid interface; (b) example of resulting mesh deformation (an artificial high displacement is imposed for the sake of clear visualization).</td>
      <td>21</td>
    </tr>
    <tr>
      <td>4</td>
      <td>FSI algorithm employed. U stands for the fluid velocity, P for the fluid static pressure, {F} for the equivalent nodal array of the fluid force, t is the time and dt the time step.</td>
      <td>22</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Free vibration test numerical domain and CFD meshing.</td>
      <td>23</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Displacement of the cylinder tip during a free vibration test in static air and in static water.</td>
      <td>24</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Displacement of the cylinder tip during a free vibration test: zoom between 0.1 and 0.2 seconds. the markers correspond to the actual numerical solution points (time step equal to 2 · 10<sup>-4</sup>s).</td>
      <td>25</td>
    </tr>
    <tr>
      <td>8</td>
      <td>PSD (normalized by its maximum) of the cylinder tip displacement, along a free vibration test in static air, as a function of the number of nodes employed for the discretization of the beam.</td>
      <td>26</td>
    </tr>
    <tr>
      <td>9</td>
      <td>Numerical domain and meshing details of the vortex-induced vibration simulations.</td>
      <td>27</td>
    </tr>
    <tr>
      <td>10</td>
      <td>Positions behind the cylinder where the velocity was monitored.</td>
      <td>28</td>
    </tr>
    <tr>
      <td>11</td>
      <td>Calculated shedding frequency (blue) and cylinder rms vibration amplitude (red) (referring to the displacement of the cylinder free end), as a function of the water free stream velocity.</td>
      <td>29</td>
    </tr>
    <tr>
      <td>12</td>
      <td>PSD of the flow velocity behind the cylinder, for U=2.4m/s and U=2.6m/s. U-CFD and U-FSI refer to the velocity PSD calculated by the CFD stand-alone and FSI simulation, respectively. displ. refers to the PSD of the calculated cylinder displacement along the lift direction.</td>
      <td>30</td>
    </tr>
    <tr>
      <td>13</td>
      <td>PSD of the flow velocity behind the cylinder, for U=2.8m/s and U=3.0m/s. U-CFD and U-FSI refer to the velocity PSD calculated by the CFD stand-alone and FSI simulation, respectively. displ. refers to the PSD of the calculated cylinder displacement along the lift direction.</td>
      <td>31</td>
    </tr>
    <tr>
      <td>14</td>
      <td>Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=0.6m/s simulation. The structure motion is activated at 2.0 seconds, in order to let the water flow and the shedding mechanism develop.</td>
      <td>32</td>
    </tr>
    <tr>
      <td>15</td>
      <td>Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=1.2m/s simulation. The structure motion is activated at 2.0 seconds, in order to let the water flow and the shedding mechanism develop.</td>
      <td>33</td>
    </tr>
    <tr>
      <td>16</td>
      <td>Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=1.8m/s simulation. The structure motion is activated at 2.0 seconds, in order to let the water flow and the shedding mechanism develop.</td>
      <td>34</td>
    </tr>
    <tr>
      <td>17</td>
      <td>Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=2.4m/s simulation. The structure motion is activated at 1.5 seconds, in order to let the water flow and the shedding mechanism develop.</td>
      <td>35</td>
    </tr>
    <tr>
      <td>18</td>
      <td>Geometrical configuration of the channel with cylinders of the considered experimental campaign (dimensions given in mm) [41].</td>
      <td>36</td>
    </tr>
    <tr>
      <td>19</td>
      <td>Horizontal locations of the measurement points behind the cylinders. These measurements are realized at a vertical position of 25mm, 100m and 175mm from the fixed end of the cylinders.</td>
      <td>37</td>
    </tr>
    <tr>
      <td>20</td>
      <td>CFD domain: (a) region of the test section simulated, (b) meshing details along a horizontal plane and (c) meshing details in the cylinder free end region along a vertical plane.</td>
      <td>38</td>
    </tr>
    <tr>
      <td>21</td>
      <td>PSD of the velocity behind the first and second cylinders (position v1 and v3 of Fig. 19, respectively, both at a vertical position of 100mm, i.e., at half of the cylinder height).</td>
      <td>39</td>
    </tr>
    <tr>
      <td>22</td>
      <td>PSD of the cylinder acceleration in the drag direction.</td>
      <td>40</td>
    </tr>
    <tr>
      <td>23</td>
      <td>PSD of the cylinder acceleration in the lift direction.</td>
      <td>41</td>
    </tr>
  </tbody>
</table>
LIST OF TABLES

<table><tr><td>1</td><td>Results of the free vibration simulations.</td><td>42</td></tr><tr><td>2</td><td>Details of the different conditions simulated.</td><td>43</td></tr><tr><td>3</td><td>Cylinder free end deflection.</td><td>44</td></tr><tr><td>4</td><td>CPU time and mesh size as a function of the CFD approach and the structure approach.</td><td>45</td></tr></table>
<figure><img src="image_2.png" /><figcaption>Fig. 1: The 4-DOF Euler-Bernoulli beam element considered.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2: 3D CFD mesh and representation of the beam discretization: (a) CFD mesh cut on a x-z plane, showing the presence of the beam and its discretization into several nodes, (b) CFD mesh cut on a x-y plane, showing the presence of the beam, (c) CFD mesh faces at the fluid-solid interface (i.e., the CFD wall boundary corresponding to the cylindrical beam surface) showing the correspondence between the beam node position and the CFD mesh layer in the spanwise (z) direction (the hexahedral mesh is built such that all CFD nodes of a given spanwise layer are placed at the same z position).</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3: (a) displacement imposed to the nodes of the CFD mesh located at the fluid-solid interface; (b) example of resulting mesh deformation (an artificial high displacement is imposed for the sake of clear visualization).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig 4: FSI algorithm employed. *U* stands for the fluid velocity, *P* for the fluid static pressure, {*F*} for the equivalent nodal array of the fluid force, *t* is the time and *dt* the time step.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5: Free vibration test numerical domain and CFD meshing.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6: Displacement of the cylinder tip during a free vibration test in static air and in static water.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7: Displacement of the cylinder tip during a free vibration test: zoom between 0.1 and 0.2 seconds. the markers correspond to the actual numerical solution points (time step equal to 2 · 10⁻⁴ s).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8: PSD (normalized by its maximum) of the cylinder tip displacement, along a free vibration test in static air, as a function of the number of nodes employed for the discretization of the beam.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 9: Numerical domain and meshing details of the vortex-induced vibration simulations.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 10: Positions behind the cylinder where the velocity was monitored.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 11: Calculated shedding frequency (blue) and cylinder rms vibration amplitude (red) (referring to the displacement of the cylinder free end), as a function of the water free stream velocity.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 12: PSD of the flow velocity behind the cylinder, for $U=2.4\text{m/s}$ and $U=2.6\text{m/s}$. *U-CFD* and *U-FSI* refer to the velocity PSD calculated by the CFD stand-alone and FSI simulation, respectively. *displ.* refers to the PSD of the calculated cylinder displacement along the lift direction.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 13: PSD of the flow velocity behind the cylinder, for $U=2.8$m/s and $U=3.0$m/s. *U-CFD* and *U-FSI* refer to the velocity PSD calculated by the CFD stand-alone and FSI simulation, respectively. *displ.* refers to the PSD of the calculated cylinder displacement along the lift direction.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Fig. 14: Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=0.6m/s simulation. The structure motion is activated at 2.0 seconds, in order to let the water flow and the shedding mechanism develop.</figcaption></figure>

<figure><img src="image_16.png" /></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 15: Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=1.2m/s simulation. The structure motion is activated at 2.0 seconds, in order to let the water flow and the shedding mechanism develop.</figcaption></figure>

<figure><img src="image_18.png" /></figure>
<figure><img src="image_19.png" /><figcaption>Fig. 16: Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=1.8m/s simulation. The structure motion is activated at 2.0 seconds, in order to let the water flow and the shedding mechanism develop.</figcaption></figure>

<figure><img src="image_20.png" /></figure>
<figure><img src="image_21.png" /><figcaption>Fig. 17: Displacement of the cylinder tip and total force on the cylinder wall, as a function of time, along the drag and the lift directions, for the U=2.4m/s simulation. The structure motion is activated at 1.5 seconds, in order to let the water flow and the shedding mechanism develop.</figcaption></figure>

<figure><img src="image_22.png" /></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 18: Geometrical configuration of the channel with cylinders of the considered experimental campaign (dimensions given in mm) [41].</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 19: Horizontal locations of the measurement points behind the cylinders. These measurements are realized at a vertical position of 25mm, 100m and 175mm from the fixed end of the cylinders.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. 20: CFD domain: (a) region of the test section simulated, (b) meshing details along a horizontal plane and (c) meshing details in the cylinder free end region along a vertical plane.</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Fig. 21: PSD of the velocity behind the first and second cylinders (position v1 and v3 of Fig. 19, respectively, both at a vertical position of 100mm, i.e., at half of the cylinder height).</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>Fig. 22: PSD of the cylinder acceleration in the drag direction.</figcaption></figure>
<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /><figcaption>Fig. 23: PSD of the cylinder acceleration in the lift direction.</figcaption></figure>
<table><caption>Table 1: Results of the free vibration simulations.</caption><thead><tr><th></th><th>Present simulations</th><th>Reference</th></tr></thead><tbody><tr><td>f<sub>n</sub> in static air [Hz]</td><td>112.5</td><td>113.0<br><sup>1</sup>&frasl;<sub>2&pi;</sub> 1.875<sup>2</sup> (<sup>EI</sup>&frasl;<sub> mL<sup>4</sup></sub>)<sup>0.5</sup> [48]</td></tr><tr><td>C<sub>a</sub> in water [-]</td><td>1.05</td><td>1.00-1.10 [42, 41]</td></tr><tr><td>&delta; in water [-]</td><td>0.08</td><td>0.06-0.09 [40, 41]</td></tr></tbody></table>
<table><caption>Table 2: Details of the different conditions simulated.</caption><thead><tr><td>Case no.</td><td>Free stream vel. [m/s]</td><td>Reynolds number</td></tr></thead><tbody><tr><td>1</td><td>0.6</td><td>6,000</td></tr><tr><td>2</td><td>1.2</td><td>12,000</td></tr><tr><td>3</td><td>1.8</td><td>18,000</td></tr><tr><td>4</td><td>2.4</td><td>24,000</td></tr><tr><td>5</td><td>2.6</td><td>26,000</td></tr><tr><td>6</td><td>2.8</td><td>28,000</td></tr><tr><td>7</td><td>3.0</td><td>30,000</td></tr></tbody></table>
<table><caption>Table 3: Cylinder free end deflection.</caption><thead><tr><th rowspan="2">Force [N]</th><th colspan="2">Tip deflection [m]</th></tr><tr><th>Present simulations</th><th>Analytical solution</th></tr></thead><tbody><tr><td>0.65 (U=0.6m/s)<br>(see Fig. 14)</td><td>3.40x10<sup>-5</sup><br>(see Fig. 14)</td><td>3.55x10<sup>-5</sup></td></tr><tr><td>2.65 (U=1.2m/s)<br>(see Fig. 15)</td><td>1.40x10<sup>-4</sup><br>(see Fig. 15)</td><td>1.45x10<sup>-4</sup></td></tr><tr><td>5.50 (U=1.8m/s)<br>(see Fig. 16)</td><td>2.90x10<sup>-4</sup><br>(see Fig. 16)</td><td>3.00x10<sup>-4</sup></td></tr><tr><td>11.00 (U=2.4m/s)<br>(see Fig. 17)</td><td>6.65x10<sup>-4</sup><br>(see Fig. 17)</td><td>6.00x10<sup>-4</sup></td></tr></tbody></table>
<table><caption>Table 4: CPU time and mesh size as a function of the CFD approach and the structure approach.</caption><thead><tr><th rowspan="2"></th><th colspan="2">WR-URANS</th><th colspan="2">WR-Hybrid</th><th colspan="2">WR-LES</th></tr><tr><th>Beam</th><th>3D FEM</th><th>Beam</th><th>3D FEM</th><th>Beam</th><th>3D FEM</th></tr></thead><tbody><tr><td>Number of mesh cells</td><td colspan="2">~5M</td><td colspan="2">10M</td><td colspan="2">~40M</td></tr><tr><td>CPU time ratio with reference to URANS-Beam</td><td>1</td><td>~5-10</td><td>~3</td><td>~15-30</td><td>~12</td><td>~75-150</td></tr></tbody></table>