# A Validation of Symmetric 2D + T Model Based on Single-Stepped Planing Hull Towing Tank Tests

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

- The speed, V, was assumed to be constant for all two planing surfaces. In reality, the speed of the water would decrease aft of each step due to disturbances from the hull and turbulence. This would implicate that the lift from the middle and aft planing surface would be slightly exaggerated. By applying the effects of the transom and the steps, the forces will be calculated with a more accurate value.
- The planing surfaces are assumed to have triangular shapes.
- The sweep-back of the steps is not included in the model.
- The local deadrise angle ${\beta}_{L}$ has been assumed to be 2 degrees for each planing surface. This value depends on the ventilation length and has effects on the trim and resistance of the vessel because it affects the lift coefficients.
- The local trim angle, $\tau $, has also been assumed to be 2 degrees. This value is measured using the slope of the planing surface in relation to the horizon, which has a straightforward relationship with step height.

#### 2.1. Two Dimensional Forces

#### 2.2. Phase 1—The Dry-Chine Condition

#### 2.3. Phase 2—The Wet-Chine Condition

_{i}is calculated as follows.

_{i}is calculated as follows.

#### 2.4. Three Dimensional Forces

_{non}is the non-dimensional longitudinal position (from the transom and step) in which the reductions appear. Garme [25] proposed that the anon be set to 0.34 for the planing range.

#### 2.5. Frictional Forces

_{i}and α

_{i}are calculated separately for each planing surface, as follows,

#### 2.6. Resistance and Thrust

#### 2.7. Computational Procedure

## 3. Validation and Results

#### 3.1. Model Tested and Experimental Details

#### 3.2. Towing Tank vs. 2D + T Method Results

#### 3.3. Wetted Surfaces and Wetted Length Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Boat Characteristics | |

B | Beam of the boat (m) |

Cf_{i} | Frictional coefficient of the ith body |

$Fr$ | Froude number |

$F{r}_{B}$ | Beam Froude number |

${H}_{ste{p}_{1}}$ | Height of step (m) |

L | Length of the boat (m) |

$L{c}_{i}$ | Chine wetted length of the ith body (m) |

LCG | Longitudinal Center of Gravity (m) |

${L}_{{w}_{i}}$ | Wetted length of body ith body (m) |

Rn_{i} | Reynolds Number of the ith body |

${S}_{{P}_{i}}$ | Wetted area of the ith body |

V | Forward moving velocity of the boat (m s^{‒1}) |

α_{i} | stagnation line angle of the ith body |

β_{i} | Deadrise angle of the boat |

${\beta}_{{L}_{i}}$ | Local deadrise angle of the boat of the ith body |

Δ | Weight of boat (N) |

λ_{i} | Mean wetted length of the ith body |

τ_{i} | Local trim angle of the ith body |

θ | Dynamic trim angle of the hull |

Distance | |

a_{non} | Non-dimensional distance at which transom reduction appears |

${L}_{s}$ | Distance of step from the transom (m) |

${L}_{dry}$ | Dry length of step from the transom |

x, y, z | Longitudinal (positive forward), transverse (positive starboard), and vertical distances (positive downward) from CG (Oxyz) (m) |

ξ, η, ζ | Longitudinal (positive forward), transverse (positive starboard), and vertical distances (positive downward) (m) |

ξ_{i}′ | Distance of section from the step or transom just located behind the section (m) |

${\xi}_{{s}_{i}}$ | Distance of section from intersection of the keel and calm water of the ith body (m) |

Force and Moments | |

Df | Frictional drag on pressure area (N) |

F_{i} | Pressure force on ith body (N) |

${f}_{{s}_{i}}$ | Drag acting on the spray area (N) |

R | Total resistance of the vessel |

${R}_{spra{y}_{i}}$ | frictional drag of Whisker spray of the ith body |

Subscript x | Force component in surge direction (N) |

Subscript z | Force component in heave direction (N) |

Subscript θ | Force component in pitch direction (N) |

Physical Parameters | |

g | Gravitational constant |

P_{i} | Pressure of the ith body (Pa) |

ρ | Fluid density (kg m^{−3}) |

Sectional Parameters Related to 2.5D Theory | |

A_{i} | Submerged area of the ith body (N m^{−1}) |

c_{i} | Half beam of spray in transverse plane (m) |

${\dot{c}}_{i}$ | Time derivation of c (m s^{−2}) |

${C}_{t{r}_{i}}$ | Transom reduction at the section of the ith body (N m^{−1}) |

${f}_{H{D}_{i}}$ | Hydrodynamic force of each section of the ith body (N m^{−1}) |

${f}_{{B}_{i}}$ | Hydrostatic force of each section of the ith body (N m^{−1}) |

l | Distance from wedge apex in the direction of wedge wall (m) |

t | Time |

${t}_{c{w}_{i}}$ | Chine wetting time of the ith body (s) |

${t}_{{p}_{i}}$ | Solution time for water entry problem of the ith body |

w_{i} | Impact velocity of the ith body |

y_{i} | Lateral distance from wedge apex of the ith body |

Subscript H | component in horizontal direction |

Subscript V | component in vertical direction |

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**Figure 2.**The 2D + T theory for single-stepped planing hull; it passes through a fixed two-dimensional observation plane for each planing surface (

**left**); water entry problem for each planing surface from t = t

_{0}to t

_{p}(

**right**).

**Figure 6.**The computational workflow for determining the pressure distribution of the single-stepped planing hull.

**Figure 7.**The C03 model body plan (transversal section every 0.100 m) and profile (buttock line every 0.025 m).

Description | |
---|---|

Length overall: L_{OA} (m) | 0.935 |

Breadth max: B_{MAX} (m) | 0.335 |

Deadrise angle at transom (°) | 23 |

Step height (mm) | 6 |

Displacement (N) | 30.705 |

LCG (%L) | 33 |

Model scale | 1:10 |

Fr | RT_{M}/Δ | Trim | WS/∇^{2/3} | ||||||
---|---|---|---|---|---|---|---|---|---|

Exp. | 2D + T Approach | Error | Exp. | 2D + T Approach | Error | Exp. | 2D + T Approach | Error | |

(-) | (-) | (%) | (deg) | (deg) | (%) | (-) | (-) | (%) | |

0.866 | 0.182 | 0.181 | 0.2 | 3.550 | 4.500 | −26.8 | 6.63 | 3.43 | 48.3 |

1.151 | 0.208 | 0.201 | 3.1 | 4.420 | 3.755 | 15.0 | 4.85 | 2.99 | 38.4 |

1.702 | 0.261 | 0.255 | 2.3 | 3.270 | 2.880 | 11.9 | 3.88 | 2.47 | 36.2 |

1.973 | 0.318 | 0.288 | 9.5 | 2.870 | 2.605 | 9.2 | 3.54 | 2.31 | 34.7 |

2.330 | 0.415 | 0.336 | 19.1 | 2.690 | 2.326 | 13.5 | 3.32 | 2.15 | 35.2 |

2.683 | 0.501 | 0.389 | 22.3 | 2.520 | 2.113 | 16.2 | 3.23 | 2.03 | 37.3 |

2.958 | 0.566 | 0.434 | 23.5 | 2.580 | 1.976 | 23.4 | 2.85 | 1.94 | 31.7 |

Fr | RT_{M}/Δ | Trim | WS/∇^{2/3} | |||
---|---|---|---|---|---|---|

EXP–CFD | EXP–2D + T | EXP–CFD | EXP–2D + T | EXP–CFD | EXP–2D + T | |

(%) | (%) | (%) | (%) | (%) | (%) | |

0.866 | 5.46 | 0.20 | −9.01 | −26.76 | −8.14 | 48.33 |

1.702 | −1.90 | 2.32 | −1.22 | 11.93 | −31.20 | 36.24 |

2.330 | 9.33 | 19.07 | −0.37 | 13.53 | −31.15 | 35.19 |

2.958 | 5.26 | 23.46 | −3.10 | 23.41 | −36.13 | 31.71 |

**Table 4.**The detailed view of the dynamic wetted surface for the experimental test and analytical method.

Fr | Analytical Wetted Surface | Experimental Wetted Surface |
---|---|---|

0.866 | ||

1.151 | ||

1.702 | ||

1.973 | ||

2.330 | ||

2.683 | ||

2.958 |

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**MDPI and ACS Style**

Bilandi, R.N.; Mancini, S.; Vitiello, L.; Miranda, S.; De Carlini, M. A Validation of Symmetric 2D + T Model Based on Single-Stepped Planing Hull Towing Tank Tests. *J. Mar. Sci. Eng.* **2018**, *6*, 136.
https://doi.org/10.3390/jmse6040136

**AMA Style**

Bilandi RN, Mancini S, Vitiello L, Miranda S, De Carlini M. A Validation of Symmetric 2D + T Model Based on Single-Stepped Planing Hull Towing Tank Tests. *Journal of Marine Science and Engineering*. 2018; 6(4):136.
https://doi.org/10.3390/jmse6040136

**Chicago/Turabian Style**

Bilandi, Rasul Niazmand, Simone Mancini, Luigi Vitiello, Salvatore Miranda, and Maria De Carlini. 2018. "A Validation of Symmetric 2D + T Model Based on Single-Stepped Planing Hull Towing Tank Tests" *Journal of Marine Science and Engineering* 6, no. 4: 136.
https://doi.org/10.3390/jmse6040136