# A Body-Nonlinear Green’s Function Method with Viscous Dissipation Effects for Large-Amplitude Roll of Floating Bodies

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Definite Problem of the $TGF\_N\_V$

#### 2.2. Green’s Function for the $TGF\_N\_V$

**is**the field point, $\mathit{q}\left(\xi ,\eta \right)$ is the source point, $\xi ,\text{}\eta $ is the coordinate of $\mathit{q}$ at $ox$ and $oy$ axis, respectively, $\tau $ is the time when $\mathit{q}$ pulses, and $\u03f5$ is the non-dimensional viscosity coefficient.

#### 2.3. Boundary Integral Equation for the $TGF\_N\_V$

#### 2.4. Equivalent Roll Damping Using the $TGF\_N\_V$

## 3. Application of the $\mathbf{TGF}\mathbf{\_}\mathbf{N}\mathbf{\_}\mathbf{V}$ for Solving Roll Damping of Hull Sections with Bilge Keels

#### 3.1. Hull Section Under Small Amplitude Roll

#### 3.2. Hull Section Under Large Amplitude Roll

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The coordinate system $o-xy$ and a hull section with bilge keels. The hull section harmonically rolls on the water surface. $A$ and $B$ are water surface—hull wetted surface intersections. The wetted surface of the hull section is discretized into $s$ segments.

**Figure 2.**A rectangular hull section with bilge keels harmonically rolls on the water surface, with beam $b=0.4\text{}\mathrm{m}$, mean draught $d=0.2\text{}\mathrm{m}$, bilge radius ${R}_{\mathrm{b}}=0.02b$, bilge keel depth ${K}_{\mathrm{d}}=0.04b$, and roll amplitude $\theta =0.05\text{}\mathrm{rad}$. The rotation axis is located at the center of waterplane.

**Figure 3.**Two viscosity distribution approaches for $\mathrm{TGF}\_\mathrm{N}\_\mathrm{V}$. (a) $\u03f5=0.25$ for segments $16\le i\le 60$ and $101\le i\le 145$, while $\u03f5=0$ for the rest segments; (b) $\u03f5=0.17$ for all segments.

**Figure 4.**Equivalent roll damping using different methods ($\mathrm{TGF}$ [25], $\mathrm{TGF}\_\mathrm{N}$ [25], $\mathrm{TGF}\_\mathrm{N}\_\mathrm{V}$ ) as compared with experimental fluid dynamic (EFD) data [13,26]. (

**a**) Equivalent roll damping related to the first viscosity distribution approach; (

**b**) Equivalent roll damping related to the second viscosity distribution approach.

**Figure 5.**A semi-submerged S60 midship section with bilge keels harmonically rolls in the presence of a free surface; beam $b=0.237\text{}\mathrm{m}$, mean draught $d=0.057\text{}\mathrm{m}$, bilge radius ${R}_{\mathrm{b}}=0.035\text{}\mathrm{m}$, bilge keel depth ${K}_{\mathrm{d}}=0.01\text{}\mathrm{m}$, roll amplitudes $\theta =0.15,\text{}0.20,\text{}0.25\text{}\left(\mathrm{rad}\right)$. The rotation axis is located at the center of the waterplane.

**Figure 6.**Equivalent roll damping using different methods ($\mathrm{TGF}$ [25], $\mathrm{TGF}\_\mathrm{N}$ [25], $\mathrm{TGF}\_\mathrm{V}\_\mathrm{N}$) as compared with results from CFD [15] and EFD [15]. (

**a**) Equivalent roll damping under roll amplitude $\theta =0.15\text{}\mathrm{rad}$; (

**b**) Equivalent roll damping under roll amplitude $\theta =0.20\text{}\mathrm{rad}$; (

**c**) Equivalent roll damping under roll amplitude $\theta =0.25\text{}\mathrm{rad}$.

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**MDPI and ACS Style**

Guo, Z.; Ma, Q.; Yu, S.; Qin, H.
A Body-Nonlinear Green’s Function Method with Viscous Dissipation Effects for Large-Amplitude Roll of Floating Bodies. *Appl. Sci.* **2018**, *8*, 517.
https://doi.org/10.3390/app8040517

**AMA Style**

Guo Z, Ma Q, Yu S, Qin H.
A Body-Nonlinear Green’s Function Method with Viscous Dissipation Effects for Large-Amplitude Roll of Floating Bodies. *Applied Sciences*. 2018; 8(4):517.
https://doi.org/10.3390/app8040517

**Chicago/Turabian Style**

Guo, Zhiqun, Qingwei Ma, Shuangrui Yu, and Hongde Qin.
2018. "A Body-Nonlinear Green’s Function Method with Viscous Dissipation Effects for Large-Amplitude Roll of Floating Bodies" *Applied Sciences* 8, no. 4: 517.
https://doi.org/10.3390/app8040517