# Fluid Structure Interaction Using Modal Superposition and Lagrangian CFD

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equation

#### 2.1. Incompressible Fluid Flow

#### 2.2. Mode Superposition

## 3. Methodology

- Calculate the mass-normalized mode shapes and corresponding natural frequencies, denoted as ${\omega}_{1},{\omega}_{2},...,{\omega}_{n}$.
- Define the initial and boundary conditions for the simulation, including the initial displacement of the structural system if any.
- Create Radial Basis Function (RBF) connections between fluid mesh and structural mesh to transfer the mode shape information if the structural mesh is not the same as the fluid mesh [40].
- At every time step of the simulation $\Delta t$:
- (a)
- Compute the forces exerted on the structural mesh due to fluid force.
- (b)
- Solve equation of motion (15) for each mode shapes and natural frequencies
- (c)
- Determine the overall deformation vector using Equation (12) and apply the resulting deformation to the mesh.
- (d)
- Solve the fluid equations for time $t+\Delta t$, taking into account of structural deformation.

## 4. Results and Discussion

#### 4.1. Validation of the Flow and Structural Solvers

#### 4.1.1. Sloshing in Cobiodal Tank

#### 4.1.2. Cantilever Beam Vibration

#### 4.2. Deformation of Cantilever Gate

^{3}and Young’s modulus $E=12$ MPa is employed. The simulation considers only the first mode shape and its associated natural frequency. Validation is undertaken for the tank filled with water, density $\rho =1000$ kg/m

^{3}and dynamic viscosity $\mu ={10}^{-3}$ Pa.s. Due to the pressure difference, the water flows through the rubber gate exerting the force to deflect it. The elastic deformation of the cantilever due to fluid loading is illustrated in Figure 7.

#### 4.3. Elastic Baffle in Sloshing Tank

^{3}, Young’s modulus of 38.4 MPa, and Poisson’s ratio of 0.3. The first five mode shapes were calculated from modal analysis and used for the simulation. Mode shapes are shown in Figure 9 and corresponding natural frequencies are shown in Table 3. By incorporating the elastic baffle model, the investigation aims to analyze and comprehend the effects of fluid–structure interaction within the tank system. Dimensions, material properties and displacement X at the location D1 (the top left corner of the baffle looking forward as shown in Figure 8) of the baffle are considered from the investigation conducted by Sampann [46]. The reference X-displacement is compared with the X-displacement obtained from the simulation and the results are presented graphically in Figure 10. The comparison shows that both results follow the same trend and also match closely at all the points where the most significant deformation takes place, thus verifying the precision of the modal coupling. At the peak, the deviation of deformation is minimal compared to the transition region where change in deformation takes place, due to the linear effect of mode superposition. The corresponding values are listed in Table 4. This provides an understanding of the baffle dynamic response under the specified sinusoidal excitation, which contributes added value to the present implementation of FSI.

#### 4.4. Elastic Beam in Shallow Oil

^{3}, a Young modulus of 60 MPa, and a Poisson ratio of 0.49. On the other hand, the oil has a density of 917 kg/m

^{3}and a kinematic viscosity of $5\times {10}^{-5}$ m²/s. The first five mode shapes obtained from modal analysis are used for the simulation. Mode shapes are presented in Figure 12 and corresponding natural frequencies are shown in Table 5.

#### 4.5. Flow Impact on a Cantilever Beam

#### 4.6. Water Entry of a Wedge

^{3}, the modulus of elasticity of 68 GPa, and the Poisson’s ratio of 0.33. The accuracy of the properties were important to consider, as they can have a significant effect on the behavior of the wedge during the free fall water entry [49]. The tank has dimensions of 5 × 5 × 1.75 m and is filled with water up to a height of 1.5 m in the z direction, as shown in Figure 17. For both structural modal analysis and fluid simulation, the surface mesh of the wedge was represented by 9200 triangles, which is shown in Figure 17. To conduct the numerical analysis of the impact loads, the wedge was dropped vertically from a height of 0.25 m (measured from the free surface up to the keel) into the water, with zero initial velocity. During the free fall, the velocity increased to $U=2.2$ m/s as the wedge made contact with the water. The numerical time step was taken as constant, $\delta t=2.5\times {10}^{-3}$. The modal analysis of the structural model resulted in obtaining the first five significant modes, which are shown in Figure 18 and corresponding natural frequencies are shown in Table 6. The same surface mesh and the modes obtained were used as input for the LDD-modal coupling solver.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dnamics |

LDD | Lagrangian Differencing Dynamics |

FSI | Fluid Structure Interaction |

RANS | Reynolds-Averaged Navier–Stokes |

URANS | Unsteady Reynolds-Averaged Navier–Stokes |

VOF | Voluem Of Fluid |

RBF | Radial Basis Function |

DES | Detached Eddy Simulation |

SPH | Smooth Particle Hydrodynamic |

FEM | Finite Element Method |

DEM | Discrete Element Method |

MPS | Moving Particle Semi-implicit |

CFPI | Complementary Function and Particular Integral |

MDOF | Multi-Degree-Of-Freedom |

SDOF | Single-Degree-Of-Freedom |

ODE | Ordinary Differential Equation |

DOF | Degree-Of-Freedom |

PBD | Particle-Based Dynamics |

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**Figure 1.**FSI with LDD workflow: This flowchart explains the process of integrating weak coupling of the modal solver.

**Figure 2.**Tank considered for sloshing to validate the simulation pressure with Rhee [41] Experiment pressure.

**Figure 4.**Cantilever beam subjected to the tip load is considered for comparative analysis of Mode Superposition: Theoretical vs. Numerical.

**Figure 7.**Comparison of cantilever beam deformation from the current study with Antoci et al. [42] at different time steps such as t = 0.08 s, 0.16 s, 0.24 s, 0.32 s and 0.40 s.

**Figure 8.**Rhee sloshing tank is considered and an elastic baffle is introduced at the bottom of the tank.

**Figure 9.**Modal results of an elastic baffle with reference to its undeformed wireframe, i.e., the first five mode-shapes.

**Figure 11.**Idelsohn sloshing tank is considered and an elastic beam is introduced at the bottom of the tank.

**Figure 12.**Modal results of an elastic beam with reference to its undeformed wireframe, i.e., the first five mode-shapes.

**Figure 15.**Dynamic response over a time period at the mid and tip section,

**Left**: X Displacement and

**Right**: Y Displacement.

**Figure 16.**Displacement at the mid and tip section until 1 s,

**Left**: X Displacement and

**Right**: Y Displacement.

**Figure 17.**Numerical setup to replicate the experiment of [49]. On the left, a tank filled with water up to 1.5 m and a wedge placed at a distance of 0.25 m above the free surface. On the right, a wedge with a surface mesh was used for the modal coupling solver.

**Figure 18.**Modal results of a wedge with reference to its undeformed wireframe, i.e., the first five mode shapes.

**Figure 20.**Deformation of Wedge in X, Y, and Z directions and deformation magnitude at maximum impact load condition, where wedge touches the water with 2.2 m/s velocity.

**Table 1.**Deviation of pressure from the simulation with respect to Rhee Experiment at peak location.

Region | 1st Peak | 2nd Peak |
---|---|---|

Deviation of pressure [kPa] | 0.1 | 0.35 |

Mode | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Natural Frequency (Hz) | 15.09 | 93.433 | 257.66 | 374.8 | 495.57 |

Mode | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Natural Frequency (Hz) | 100.12 | 270.05 | 301.33 | 392.05 | 505.44 |

Region | Peak | Transition |
---|---|---|

Max Deviation of deformation [m] | 0.005 | 0.01 |

Mode | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Natural Frequency (Hz) | 186.4 | 560.61 | 673.29 | 856.07 | 939.09 |

Mode | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Natural Frequency (Hz) | 213.8 | 376.6 | 440.95 | 529.12 | 535.23 |

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## Share and Cite

**MDPI and ACS Style**

Paneer, M.; Bašić, J.; Sedlar, D.; Lozina, Ž.; Degiuli, N.; Peng, C.
Fluid Structure Interaction Using Modal Superposition and Lagrangian CFD. *J. Mar. Sci. Eng.* **2024**, *12*, 318.
https://doi.org/10.3390/jmse12020318

**AMA Style**

Paneer M, Bašić J, Sedlar D, Lozina Ž, Degiuli N, Peng C.
Fluid Structure Interaction Using Modal Superposition and Lagrangian CFD. *Journal of Marine Science and Engineering*. 2024; 12(2):318.
https://doi.org/10.3390/jmse12020318

**Chicago/Turabian Style**

Paneer, Manigandan, Josip Bašić, Damir Sedlar, Željan Lozina, Nastia Degiuli, and Chong Peng.
2024. "Fluid Structure Interaction Using Modal Superposition and Lagrangian CFD" *Journal of Marine Science and Engineering* 12, no. 2: 318.
https://doi.org/10.3390/jmse12020318