# Numerical Prediction of Hydrodynamic Performance of Planing Trimaran with a Wave-Piercing Bow

^{*}

## Abstract

**:**

_{Δ}) ranging from 1.31 to 4.98. Dynamic sinkage and trim in the Dynamic Fluid Body Interaction (DFBI) six-degree-of-freedom model were considered. A validation study carried out by comparison of experimental test results with numerical results showed good consistency. To analyze the process of tunnel penetration and pressure change at the bottom of the boat, numerical simulation results for free surface, bottom streamline, and pressure distribution around the hull are given. A large triangular high-pressure area was observed in the front of the main hull for all volume Froude numbers. Consequently, the central drainage body, in reference to the profile of single planing craft with distinctive resistance performance, was redesigned into a wave-piercing shape. Total resistance, sinkage, and trim angle of the new model were then predicted by numerical method. The results show that the central drainage body has a significant impact on the hydrodynamic performance of the planing trimaran. Furthermore, the wave-piercing shaped main hull has a drag reduction effect.

## 1. Introduction

## 2. Experimental Tests

#### 2.1. Geometrical Description of Planing Trimaran Model

#### 2.2. Experimental Set-Up and Measurement

#### 2.3. Experimental Results and Uncertainty Analysis

_{T}to the overall weight in Newtons $\Delta $. Dimensionless dynamic sinkage $Z/{\nabla}^{1/3}$ is defined as the ratio of sinkage value Z to one-third power of volumetric displacement $\nabla $.

_{Δ}= 1.57, then appears to go up as Fr

_{Δ}increases. The dynamic sinkage curve records a rapid increase at lower Froude numbers and then seams to stabilize for Fr

_{Δ}> 3.0. In relation to the dynamic trim angle curve, a sharply increases to a large maximum value is noted, and then decreases gradually. When Fr

_{Δ}< 1.57, the tunnel of the planing trimaran model did not penetrate, with a large amount of air inflow turned back at the bell mouth of the tunnel. This caused the model′s dynamic trim angle to increase rapidly and the hull to be quickly lifted. With the combination of air resistance and hydrodynamic force, total resistance also rises rapidly. The largest trim angle is realized at Fr

_{Δ}= 1.83. At this point, the resistance of the water flow to the hull decreased due to the rise of the hull; therefore, the total resistance leveled off and did not rise rapidly. After Fr

_{Δ}= 1.83, as the model is lifted, there is a little space in the roof of the tunnel, which penetrates forward and backward, which could be attributed to the gradual decrease in dynamic trim angle. The air flow passing through the tunnel creates lift and further rises the hull leading to a much slower increase in total resistance. At Fr

_{Δ}= 2.89, the model is lifted to a certain height, the roof of the tunnel is penetrated, the lift force generated by the air tends to be the maximum, and dynamic sinkage of the model will also be flattened. As a result, dynamic trim angle slowly decreases. As the volume Froude number increases, the lift on the hull is insignificant and the total resistance of the model mainly due to the resistance of the water inflow.

## 3. Numerical Simulation and Verification

#### 3.1. Governing Equations

#### 3.2. Governing Equations of Free Motion

#### 3.3. Free Surface Treatment

#### 3.4. Coordinate System and Computational Domain Analysis

_{Δ}= 4.72 is shown in Figure 8.

#### 3.5. Sensitivity Analysis of Mesh near the Free Surface

_{Δ}= 4.72. In the computational fluid dynamics software STAR-CCM+, it is necessary to refine the free surface mesh. Since the free surface is a projection surface of the entire computational domain, its range is large, and the mesh size directly determines the total mesh amount. However, a fine mesh around the model is also needed to capture the details of the flow field around the hull. Generally, for the sensitivity verification of the mesh, several mesh schemes are designed in order from sparse to dense and calculated separately. The calculation error is correlated to changes in mesh size where smaller size of the mesh yields smaller calculation error. Finally, a mesh scheme in which the calculation error is within the range of engineering error and the amount of mesh can be bore by the computer is selected.

_{∇}= 4.72, while “Num” represents the numerical simulation value at same speed. $Error=\frac{\left|Num-Exp\right|}{Exp}$.

#### 3.6. Results and Discussions

_{∇}= 2.1, 3.15, 4.2, 4.72, in terms of free surface, streamline and pressure distribution. These are crucial in the analysis of the state of the planing trimaran model. Finally, suggestions are given to further improve the hydrodynamic performance of planing trimaran.

#### 3.6.1. Total resistance, dynamic sinkage, and trim angle

_{∇}= 1.31, 1.57, 1.84, 2.1, 2.36, 2.62, 2.89, 3.15, 3.41, 3.67, 3.94, 4.2, 4.46, 4.72, 4.98 (conditions corresponding to the experimental test) are obtained. The numerical simulation and experimental test results of the dimensionless total resistance, the dimensionless dynamic sinkage, and trim angle against Fr

_{∇}are shown in Figure 10.

_{∇}< 4.5, the trend of the calculated value and the experimental test value remains highly consistent, and the error between the two methods decreases uniformly as the speed increases. In this interval, the largest error is 6.35% at the position of the resistance peak. However, for volume Froude number is greater than 4.46, compared with the experimental tests, the numerical results show significant distortion. Although the maximum error in this interval is 12.09% at Fr

_{∇}= 4.98, it can be considered that the calculated results are not accurate. In view of this phenomenon, the subsequent section will analyze the free surface and the pressure distribution on the bottom of the hull.

_{∇}in numerical simulation. When Fr

_{∇}is greater than 2.5, the numerical value increases at a slower rate, and the numerical simulation results are almost the same as the experimental value (Figure 10b). The calculated values of dynamic trim angle are in the same trend as the experimental test values, in which they increase sharply and then decrease slowly. However, the numerical simulation results are smaller than the experimental test values (Figure 10c).

#### 3.6.2. Analysis of Free Surface and Bottom Pressure

_{∇}= 2.1, 3.15, 4.2, 4.72. This is done to explore the intrinsic mechanism of the hydrodynamic performance of the planing trimaran. Further, suggestions are given to improve the hydrodynamic performance of planing trimaran.

_{∇}= 2.1 at the front end of the tunnel around the bell mouth, the free surface rolls along the bottom surface of the hull and a nonlinear phenomenon occurs. The current hits the main hull resulting in a triangular high-pressure area H1 at the front end of the bottom of the main hull. At the same time, due to the large trim angle of the model, the flow directly hits the front end of the straight section of the tunnel after passing through the bell mouth. This causes a high-pressure area H2, as shown in Figure 11a. There is a little streamline at the roof of the tunnel and a little air passes through the tunnel. However, most of the air flow enters the bell mouth of the tunnel and then turns back, as shown by streamline A in Figure 11a. A small amount of air passing through the tunnel will provide aerodynamic lift causing the hull to rise further.

_{∇}= 3.15, the trim angle is further reduced. As can be seen from Figure 11b, the high-pressure area H1 of the main hull moves forward, and the high-pressure area H2 of the straight section in the tunnel goes backward temporarily. At this time, the roof of the tunnel has been completely penetrated. Compared with Fr

_{∇}= 2.1, more streamlines pass through the tunnel and the roof of the tunnel is filled with air. However, air aerodynamic lift is no longer significant for lifting the hull. Under the action of gravity, the front structure of the model approaches the water surface causing the trim angle to decrease further.

_{∇}= 4.2, the trim angle is about 5 degrees currently. The triangle high-pressure area H1 of the main hull does not change significantly, but the high-pressure area H2 of the straight section of the tunnel completely disappears. The streamlines in the tunnel further increases and the air can pass smoothly. The model has been lifted to a certain height and would not rise any further. The navigation attitude of the model becomes stable. The air resistance of the hull tends to a fixed value. The resistance of the model is mainly derived from the resistance of the water flow, so the dimensionless resistance increases linearly with the volume Froude number.

_{∇}= 4.72, the free surface around the planing trimaran, the bottom streamline and pressure distribution are consistent with Fr

_{∇}= 4.2. However, the pressure value of the model is further increased. The triangular high-pressure area H1 of the main hull still exists. Under this volume Froude number, the numerical dynamic sinkage is consistent with the experimental test value. However, the trim angle value is quite different and the numerical result is smaller than the experimental test value. In the process of numerical simulation, the interaction between the bottom of the main hull and the water flow decreases. The result indicates that the pressure value is less than the actual value. Finally, the total resistance value differs from the experimental test value.

_{∇}. If the shape of the bottom of the model is changed, the resistance performance of the model may be improved. Next, the shape of the bottom of the model was changed according to the above analysis. Numerical simulation scheme introduced in this earlier was used to predict the hydrodynamic performance of the improved model.

## 4. Improved Hull and Hydrodynamic Performance Prediction

#### 4.1. Improved Hull

#### 4.2. Mesh and Time-Step Sensitivity Analysis

_{∇}= 4.72. Mesh and time-step sensitivity analysis results in the Table 7.

#### 4.3. Numerical Results and Analysis of Hydrostatic Performance of Improved Hull Type

_{∇}= 1.57, 2.1, 2.62, 3.15, 3.67, 4.2, and 4.72.

_{∇}. Model b and Model a in this section represents the improved hull and the original hull, respectively. It can be seen from Figure 13 that the trend of Model a and Model b in regards to total resistance dimensionless number, dynamic sinkage dimensionless number, and dynamic trim angle change with Fr

_{∇}is consistent. The resistance peak of Model b is more obvious in the dimensionless number of total resistance with Fr

_{∇}. To volume Froude numbers ranging from 1.57 to 4.72, the total resistance of Model b decreases in comparison to Model a. The minimum value is 6.33%, and the maximum value is 22.78%. The total resistance reduction is small at low speed. When Froude numbers are greater than 2.62, the total resistance reduction effect is obvious. In the navigation attitude, the dynamic sinkage and trim angle have also been reduced. At low speeds, the maximum reduction is 32.21%. However, the dynamic sinkage is reduced generally by around 5% at high speeds. The reduction of dynamic trim angle is about 50%.

_{∇}> 2.1, especially in the high-speed sailing state, the hydrodynamic performance is significantly improved. In Model a, due to the existence of two tunnels, the air cushion for planing trimaran is formed in advance. This reduces its total resistance. The central drainage body of Model b adopts wave-piercing shape, further reducing its resistance and improving the navigation attitude. This could be attributed to the reduction in the interaction between the current and the main hull.

#### 4.4. Bottom Wetted Area and Pressure Distribution Analysis

_{∇}= 2.1, 3.15, 4.2, 4.72.

#### 4.5. Relationship between Main Hull Shape and Hydrostatic Performance

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Acronyms | |

DFBI | Dynamic Fluid Body Interaction |

FVM | Finite Volume Method |

NUMELS | Numerical Marine Engineering Laboratory Software |

UDF | User Define Function |

VOF | Volume of Fluid |

ANNs | Artificial Neural Networks |

UA | Uncertainty Analysis |

## Appendix A. Experimental Uncertainty Analysis

U [m/s] | Fr_{∇} | R_{T} [N] | Z [mm] | τ [deg] |

2.5 ± 0.1% | 1.31 ± 0.1% | 81.34 ± 1.45% | −6.5 ± 26.63% | 5.21 ± 0.56% |

3 ± 0.1% | 1.57 ± 0.1% | 91.92 ± 1.29% | 18.6 ± 9.31% | 7.3 ± 0.40% |

3.5 ± 0.1% | 1.84 ± 0.1% | 92.90 ± 1.27% | 41 ± 4.22% | 7.95 ± 0.36% |

4 ± 0.1% | 2.10 ± 0.1% | 93.79 ± 1.26% | 54.3 ± 3.19% | 7.78 ± 0.37% |

4.5 ± 0.1% | 2.36 ± 0.1% | 94.77 ± 1.25% | 65.3 ± 2.65% | 7.58 ± 0.38% |

5 ± 0.1% | 2.62 ± 0.1% | 97.41 ± 1.22% | 70.2 ± 2.47% | 7.09 ± 0.41% |

5.5 ± 0.1% | 2.89 ± 0.1% | 100.74 ± 1.18% | 73.5 ± 2.35% | 6.8 ± 0.43% |

6 ± 0.1% | 3.15 ± 0.1% | 105.15 ± 1.13% | 76.3 ± 2.27% | 6.4 ± 0.45% |

6.5 ± 0.1% | 3.41 ± 0.1% | 110.54 ± 1.07% | 75.8 ± 2.28% | 6.2 ± 0.47% |

7 ± 0.1% | 3.67 ± 0.1% | 114.66 ± 1.04% | 79 ± 2.19% | 5.93 ± 0.49% |

7.5 ± 0.1% | 3.94 ± 0.1% | 120.74 ± 0.98% | 79.2 ± 2.19% | 5.84 ± 0.50% |

8 ± 0.1% | 4.20 ± 0.1% | 127.60 ± 0.93% | 79.7 ± 2.17% | 5.6 ± 0.52% |

8.5 ± 0.1% | 4.46 ± 0.1% | 136.22 ± 0.87% | 82 ± 2.11% | 5.51 ± 0.53% |

9 ± 0.1% | 4.72 ± 0.1% | 144.06 ± 0.83% | 82.5 ± 2.10% | 5.31 ± 0.55% |

9.5 ± 0.1% | 4.98 ± 0.1% | 151.61 ± 0.79% | 83.2 ± 2.08% | 5.24 ± 0.55% |

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**Figure 7.**Mesh encryption around the hull: Volumetric Controls 1 and 2; Prismatic mesh around the hull.

**Figure 10.**Comparisons of numerical and experimental results for volume Froude numbers ranging from 1.31 to 4.98.

**Figure 13.**Comparisons of the numerical results of Model a and Model b for volume Froude numbers ranging from 1.57 to 4.72.

Main Feature | Symbol | Value |
---|---|---|

Length overall (m) | L | 1.8333 |

Beam overall (m) | B | 0.7 |

Model depth (m) | H | 0.3667 |

Main hull beam (m) | B_{M} | 0.4 |

Tunnel beam (m) | B_{T} | 0.141 |

Tunnel height (m) | H_{T} | 0.067 |

Draft (m) | T | 0.1067 |

Displacement (kg) | Δ | 50.9 |

Initial trim Angle (deg) | τ | 3.74 |

Longitudinal center of gravity from stern (m) | L_{CG} | 0.75 |

Deadrise angle at the transom (deg) | β | 13 |

The length of the sheet (m) | L_{d} | 1.63 |

Length | 510 m |

Breadth | 6.5 m |

Depth | 6.8 m |

Water depth | 5.0 m |

Carriage speed | 0.1–22 m/s |

No | Name of Instrument | Model Type | Amount of Routine | Accuracy |
---|---|---|---|---|

1 | Data collection system | TDEC-PCI20016 | ±10 V | 16 bit 0.3% |

2 | Position sensor | FWP-1.2 | 1.2 m | 0.2% |

3 | Gyroscope | MTC-1 | 0~80° | ±0.25° |

4 | Wave height gauge | NH-1 | 0~500 mm | 1 mm |

5 | Acceleration sensor | FML-A | ±5 g | <0.5% |

Mesh | A | B | C | D |
---|---|---|---|---|

X, Y, Z (%L) | 16.36, 16.36, 0.55 | 8.18, 8.18, 0.55 | 8.18, 8.18, 1.1 | 8.18, 8.18, 2.2 |

TOTAL (million) | 0.75 | 1.03 | 0.43 | 0.23 |

Mesh | R_{T}/Δ
| Error (%) | Z/∇
^{1/3} | Error (%) | τ (deg) | Error (%) | |||
---|---|---|---|---|---|---|---|---|---|

Exp | Num | Exp | Num | Exp | Num | ||||

A | 0.2888 | 0.2664 | 7.75 | 0.2226 | 0.2081 | 6.52 | 5.31 | 4.86 | 8.48 |

B | 0.2888 | 0.2687 | 6.96 | 0.2226 | 0.2065 | 7.24 | 5.31 | 4.81 | 9.42 |

C | 0.2888 | 0.2055 | 28.83 | 0.2226 | 0.2186 | 1.81 | 5.31 | 5.00 | 5.77 |

D | 0.2888 | 0.2160 | 25.21 | 0.2226 | 0.2297 | −3.19 | 5.31 | 4.99 | 6.08 |

Main Feature | Symbol | Model a Value | Model b Value |
---|---|---|---|

Length overall (m) | L | 1.8333 | 1.8333 |

Beam overall (m) | B | 0.7 | 0.6 |

Model depth (m) | H | 0.3667 | 0.31 |

Main hull beam (m) | B_{M} | 0.4 | 0.4 |

Tunnel beam (m) | B_{T} | 0.141 | 0.095 |

Tunnel height (m) | H_{T} | 0.067 | 0.072 |

Draft (m) | T | 0.1067 | 0.115 |

Displacement (kg) | Δ | 50.9 | 50.9 |

Initial trim Angle (deg) | τ | 3.74 | 0 |

Longitudinal center of gravity from stern (m) | L_{CG} | 0.75 | 0.75 |

Deadrise angle at the transom (deg) | β | 13 | 21 |

The length of the sheet (m) | L_{d} | 1.63 | 1.63 |

Hull Surface Mesh Size | |||

R_{T}/Δ
| Z/∇^{1/3} | τ(deg) | |

Coarse Mesh (2%L) | 0.2155 | 0.1986 | 1.81 |

Medium Mesh (1%L) | 0.2240 | 0.1972 | 1.74 |

Fine Mesh (0.5%L) | 0.2311 | 0.1999 | 1.74 |

Time-Step | |||

R_{T}/Δ
| Z/∇^{1/3} | τ(deg) | |

0.002 s | 0.2140 | 0.1978 | 1.79 |

0.001 s | 0.2240 | 0.1972 | 1.74 |

0.0005 s | 0.2329 | 0.1938 | 1.70 |

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## Share and Cite

**MDPI and ACS Style**

Su, G.; Shen, H.; Su, Y. Numerical Prediction of Hydrodynamic Performance of Planing Trimaran with a Wave-Piercing Bow. *J. Mar. Sci. Eng.* **2020**, *8*, 897.
https://doi.org/10.3390/jmse8110897

**AMA Style**

Su G, Shen H, Su Y. Numerical Prediction of Hydrodynamic Performance of Planing Trimaran with a Wave-Piercing Bow. *Journal of Marine Science and Engineering*. 2020; 8(11):897.
https://doi.org/10.3390/jmse8110897

**Chicago/Turabian Style**

Su, Guangsheng, Hailong Shen, and Yumin Su. 2020. "Numerical Prediction of Hydrodynamic Performance of Planing Trimaran with a Wave-Piercing Bow" *Journal of Marine Science and Engineering* 8, no. 11: 897.
https://doi.org/10.3390/jmse8110897