# A Numerical Study on the Response of a Very Large Floating Airport to Airplane Movement

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Calculation Method

#### 2.1. Governing Equations

_{i}(ρ

_{1}< ρ

_{2}< … < ρ

_{I}) of the i-layer is spatially uniform and temporally constant in each layer. The thickness of the i-layer is h

_{i}(

**x**) in still water, where

**x**is the coordinate in the horizontal plane, namely (x, y). The origin of the vertical axis z is located at the top surface of the system in the stationary state, and the positive direction of z is vertically upward. The elevations of the lower and upper interfaces of the i-layer are expressed by z = η

_{i}

_{,0}(

**x**, t) and z = η

_{i}

_{,1}(

**x**, t), respectively, and the pressures at the lower and upper interfaces of the i-layer are defined as p

_{i}

_{,0}(

**x**, t) and p

_{i}

_{,1}(

**x**, t), respectively.

_{i}and δ

_{i}, respectively. When m

_{i}, δ

_{i}, and the flexural rigidity of the i-plate are zero, the plate yields no resistance to fluid motion, where two immiscible fluids touch each other directly without any plate. Both surface tension and capillary action are ignored, and friction is also ignored for simplicity. Moreover, the energy attenuation inside the thin plates is not considered.

_{i}in the i-layer, and ϕ

_{i}is expanded into a power series of z with weightings f

_{i}

_{,α}as

_{i}is the number of terms for an expanded velocity potential in the i-layer.

_{i}

_{,1−j}(

**x**, t) (j = 0 or 1), and the pressure on the other interface, p

_{i}

_{,j}(

**x**, t), are known, the unknown variables are the velocity potential ϕ

_{i}(

**x**, z, t) and interface displacement η

_{i}

_{,j}(

**x**, t). Then, the definition of the functional for the variational problem in the i-layer, F

_{i}, is as follows [29]:

^{2}. The plane A, which is the orthogonal projection of the object domain on to the x-y plane, is assumed to be independent of time.

_{i}

_{,j}+ P

_{i}+ W

_{i})/ρ

_{i}as an interfacial pressure term, without the terms related to vorticity. Using the functional of [30], after omitting the vorticity terms, the set of nonlinear equations for one-layer problems without thin plates was derived by [31].

_{i}and W

_{i}in Equation (2) are expressed by

_{i}expanded in Equation (1) into Equation (2), the Euler–Lagrange equations on η

_{i}

_{,j}and f

_{i}

_{,α}are derived as

_{2,3}is the weighting of z

^{3}in the 2-layer.

^{4n+2}, where σ is the representative ratio of water depth to wavelength. Conversely, the order of error in the extended Green–Naghdi equation [33] is σ

^{2n+2}. Therefore, especially when O(σ) ≪ 1, the accuracy of the former is significantly higher than that of the latter for n ≥ 1.

_{i}is the flexural rigidity of the i-plate between the (i − 1)- and i-layers. Although both the plate density m

_{i}and vertical width δ

_{i}are assumed to be constant throughout the i-plate for simplicity, the flexural rigidity B

_{i}can be distributed along the thin plate.

^{2}) ≪ 1. Thus, the first term on the left-hand side of Equation (9) can be ignored. Without this term, we obtain the i-plate equation for the dimensional quantities as

**x**, z, t) = f

_{α}z

^{α}. Thus, the unknown values are the weighting factors f

_{α}and the surface displacement η

_{1,1}(

**x**, t), which is simply described as ζ(

**x**, t) for the horizontally two-dimensional cases and η(x, t) for the one-dimensional cases.

#### 2.2. Numerical Method

_{i,α}(

**x**, 0 s) of the expanded velocity potential in Equation (1) were all zero, so the initial velocity was zero everywhere. In this paper, the values are written without considering significant digits, although the calculations were conducted using 64-bit floating-point numbers. In the present study, the number of terms for the velocity potential expanded as in Equation (1), i.e., N

_{1}= N was one, so the governing equations were reduced to nonlinear shallow water equations for velocity potential considering the flexural rigidity of a floating thin plate. The numerical calculation method described above is also applicable in this case.

^{2}was installed in a wave channel [19], as sketched in Figure 2.

^{−2}m and Δt = 2.5 × 10

^{−5}s, respectively. When a solitary wave is incident, Figure 5 presents the experimental and numerical displacements ζ of the floating thin plate or water surface at x = 7.0 m and 14.5 m. Based on the results, it is confirmed that the surface displacements in the almost one-dimensional wave propagation were simulated successfully.

## 3. Calculation Conditions

#### 3.1. 1D Calculations

#### 3.1.1. Common Conditions

#### 3.1.2. Conditions for Touch-and-Go

#### 3.1.3. Conditions for Landing

#### 3.1.4. Conditions for Takeoff

#### 3.2. 2D Calculations

#### 3.2.1. Common Conditions

^{11}N·m

^{2}, was given in the area covered by the airport, and it was possible to consider both the reflection and transmission of waves at the edges of the floating airport, including the side edges. The Sommerfeld open boundary condition was adopted at the lateral boundaries other than the x-axis. The grid sizes and time interval were Δx = Δy = 20 m and Δt = 2.0 × 10

^{−4}s, respectively.

^{2}ω/(3g).

#### 3.2.2. Conditions for Landing

^{2}, so that the running distance and run time are 0.938 km and 25 s, respectively. The conditions of Case LC are listed in Table 4.

#### 3.2.3. Conditions for Takeoff

^{2}, so the running distance and run time are 0.938 km and 25 s, respectively, which are relatively short values. The conditions of Case TC are listed in Table 4.

## 4. 1D Response of a Floating Airport to Airplane Movement

#### 4.1. Touch-and-Go

^{11}N·m, m = 1000 kg/m

^{3}, and h = 20 m. As depicted in Figure 10, we also obtained the corresponding numerical result of the linearized present model when the number of expansion terms for the velocity potential, N, is one, by considering the wave dispersion only due to the flexural rigidity of the floating body. For comparison, the phase velocity of linear water waves is depicted in the figure, which indicates that the difference in traveling velocity C between floating-body waves and water surface waves increases as the wavelength λ decreases. In Case GA-S1, the wavelength of the floating-body waves generated by the running airplane was less than 120 m based on Figure 8 and Figure 9. Moreover, the phase velocity of water waves decreases as the still water depth decreases in shallow water. Therefore, in Case GA-S1, the traveling velocity ratio between the floating-body waves and water waves was large, and the wave reflectance at the airport edges was large.

^{3}, the wavelength of the floating-body waves is 100 m, and the still water depth h is 10 m. Based on the figure, as B is increased, C increases remarkably when B > 10

^{9}N·m. When the flexural rigidity B decreases, the difference between the decreased traveling velocity of floating-body waves and the phase velocity of water waves decreases, so the wave reflectance is reduced. Thus, the reflectance of floating-body waves depends on both the flexural rigidity of an airport and the wavelength-to-water-depth ratio.

^{11}N·m, m = 1000 kg/m

^{3}, and the wavelength of the floating-body waves is 100 m. Figure 12 indicates that C decreases as h is increased for the same wavelength.

_{max}, and the still water depth h in Cases GA-L2 and GB-L2, when B747 and B737 left the long-enough airport in touch-and-go, respectively. These relationships are linearly approximated by η

_{max}= 0.0025h + 0.013 and η

_{max}= 0.0009h + 0.0087 (unit length in meter), respectively. When installing a floating airport in deeper water, it should be noted that the wave height of floating-body waves may increase because of airplane movement. The reason why larger water depths produce larger floating-body waves due to a running airplane will be discussed in Section 4.2.

#### 4.2. Landing

#### 4.3. Takeoff

^{11}N·m throughout the airport for 0.5 km ≤ x ≤ 5.5 km, R was approximately 0.67. Conversely, when B was 1 × 10

^{11}N·m for 0.5 km ≤ x < 5 km, and 1 × 10

^{10}N·m for 5 km ≤ x ≤ 5.5 km, R was approximately 0.55. This reduction in wave reflectance was due to two-stage reflection, which suppressed the wave height of the reflected waves. Furthermore, when B was 1 × 10

^{11}N·m for 0.5 km ≤ x < 5 km, and B decreased linearly from 1 × 10

^{11}N·m at x = 5 km to 1 × 10

^{9}N·m at x = 5.5 km, R was approximately 0.27 because of successive reflection near the airport edge. Consequently, the wave reflectance was reduced by lowering B near the airport edge. Even if the flexural rigidity is structurally or economically fixed for most of a floating airport, the wave reflectance can be reduced by modifying the structure or installing accessories to lower the flexural rigidity only near the airport edges, leading to an increase in the calmness of the floating airport.

## 5. 2D Response of a Floating Airport to Airplane Movement

#### 5.1. Landing

#### 5.2. Takeoff

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Wave flume used in the hydraulic experiments [19].

**Figure 4.**Distribution of the flexural rigidity at the surface of the computational domain depicted in Figure 3.

**Figure 5.**Surface displacements ζ at x = 7.0 m (

**a**) and 14.5 m (

**b**). The still water depth was 0.4 m and the incident wave height was 0.02 m.

**Figure 6.**Photographs of the airplane models for size comparison. The upper model is B747-400 and the lower model is B737-800, which were produced by SkyMarks Models and Solaseed Air Inc., respectively. The length scale is 1/200.

**Figure 7.**Computational domain for the 2D calculations. The floating airport is located in the marble-colored area. The perfect reflection condition is adopted along the x-axis, so the x-axis is the neutral axis of the domain, assuming that the length and width of the airport are 5 km and 1 km, respectively. At Point S, located at x = 1 km and y = 0 km, an airplane touches down for landing and starts running for takeoff.

**Figure 8.**Profiles of the floating airport and water surface at every 4 s when B747 performed touch-and-go in Case GA-L1, the conditions of which are described in Table 1. The still water depth h was 10 m, the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m, and the airport length L was 15 km. The black dotted line indicates the location of the airplane running on the floating airport. The red and green dotted lines indicate the waves generated by the touchdown and leaving of the airplane, respectively.

**Figure 9.**Profiles of the floating airport and water surface at every 4 s when B747 performed touch-and-go in Case GA-S1, the conditions of which are described in Table 1. The still water depth h was 10 m, the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport. The two blue dotted ellipses indicate examples of wave height reduction.

**Figure 10.**Traveling velocities of floating-body waves from Equation (12) and the linearized present model when the number of expansion terms for the velocity potential, N, is one. The flexural rigidity B and density m are 1.0 × 10

^{11}N·m and 1000 kg/m

^{3}, respectively. The still water depth h is 20 m. The phase velocity of linear water waves is also depicted for comparison.

**Figure 11.**Relationship between the traveling velocity of floating-body waves, C, and the flexural rigidity of the floating body, B, using Equation (12). The density of the floating body, m, is 1000 kg/m

^{3}, the wavelength of the floating-body waves is 100 m, and the still water depth h is 10 m.

**Figure 12.**Relationship between the traveling velocity of floating-body waves, C, and the still water depth h, using Equation (12). The flexural rigidity B and density m of the floating body are 1.0 × 10

^{11}N·m and 1000 kg/m

^{3}, respectively. The wavelength of the floating-body waves is 100 m.

**Figure 13.**Profiles of the floating airport and water surface at every 4 s when B747 performed touch-and-go in Case GA-S2, the conditions of which are described in Table 1. The still water depth h was 10 m, the flexural rigidity of the airport, B, was 1 × 10

^{10}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 14.**Profiles of the floating airport and water surface at every 4 s when B737 performed touch-and-go in Case GB-L1, the conditions of which are described in Table 1. The still water depth h was 50 m, the flexural rigidity of the airport, B, was 1 × 10

^{10}N·m, and the airport length L was 15 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 15.**Profiles of the floating airport and water surface at every 4 s when B737 performed touch-and-go in Case GB-S1, the conditions of which are described in Table 1. The still water depth h was 50 m, the flexural rigidity of the airport, B, was 1 × 10

^{10}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 16.**Profiles of the floating airport and water surface at every 4 s when B747 performed touch-and-go in Case GA-S1, the conditions of which are described in Table 1. The still water depth h was 20 m, the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 17.**Relationships between the maximum displacement of the floating airport, η

_{max}, and the still water depth h in Cases GA-L2 (red) and GB-L2 (blue), when B747 and B737 left the airport in touch-and-go, respectively. The conditions of these cases are described in Table 1. The flexural rigidity of the airport, B, was 1 × 10

^{11}N·m in Case GA-L2, whereas B was 1 × 10

^{10}N·m in Case GB-L2. The airport length L was 15 km in both cases.

**Figure 18.**Profiles of the floating airport and water surface at every 8 s when B747 landed in Case LA-L, the conditions of which are described in Table 2. The still water depth h was 10 m, the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m, and the airport length L was 15 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 19.**Profiles of the floating airport and water surface at every 8 s when B747 landed in Case LA-S, the conditions of which are described in Table 2. The still water depth h was 10 m, the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 20.**Profiles of the floating airport and water surface at every 4 s when B737 landed in Case LB-L, the conditions of which are described in Table 2. The still water depth h was 50 m, the flexural rigidity of the airport, B, was 1 × 10

^{10}N·m, and the airport length L was 15 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 21.**Profiles of the floating airport and water surface at every 4 s when B737 landed in Case LB-S, the conditions of which are described in Table 2. The still water depth h was 50 m, the flexural rigidity of the airport, B, was 1 × 10

^{10}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 22.**Profiles of the floating airport and water surface at every 8 s when B747 took off in Case TA-L, the conditions of which are described in Table 3. The still water depth h was 10 m, the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m, and the airport length L was 15 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 23.**Profiles of the floating airport and water surface at every 8 s when B747 took off in Case TA-S, the conditions of which are described in Table 3. The still water depth h was 10 m, the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 24.**Profiles of the floating airport and water surface at every 4 s when B737 took off in Case TB-L, the conditions of which are described in Table 3. The still water depth h was 50 m, the flexural rigidity of the airport, B, was 1 × 10

^{10}N·m, and the airport length L was 15 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 25.**Profiles of the floating airport and water surface at every 4 s when B737 took off in Case TB-S, the conditions of which are described in Table 3. The still water depth h was 50 m, the flexural rigidity of the airport, B, was 1 × 10

^{10}N·m, and the airport length L was 5 km. The black dotted line indicates the location of the airplane running on the floating airport.

**Figure 26.**Surface-level distributions at t = 25 s, at which B787 stopped after landing on the floating airport in Case LC. The still water depth h values were 10 m, 20 m, and 100 m for the figures on the left (

**a**), middle (

**b**), and right (

**c**), respectively. The flexural rigidity of the airport, B, was 1 × 10

^{11}N·m

^{2}.

**Figure 27.**Surface profiles along the x-axis at t = 25 s, at which B787 stopped after landing in Case LC. The still water depth h values were 10 m, 20 m, and 100 m, and the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m

^{2}.

**Figure 28.**Time variations of the vertical positions of B787, ζ

_{p}, (

**a**), and the surface gradient ∂ζ/∂x at the location of B787 (

**b**), in Case LC, in which B787 stopped at t = 25 s after landing. The still water depth h values were 10 m, 20 m, and 100 m, and the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m

^{2}.

**Figure 29.**Surface-level distributions at t = 25 s, at which B787 left the floating airport in Case TC. The still water depth h values were 10 m, 20 m, and 100 m for the figures on the left (

**a**), middle (

**b**), and right (

**c**), respectively. The flexural rigidity of the airport, B, was 1 × 10

^{11}N·m

^{2}.

**Figure 30.**Surface profiles along the x-axis at t = 25 s, at which B787 left the airport in Case TC. The still water depth h values were 10 m, 20 m, and 100 m, and the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m

^{2}.

**Figure 31.**Time variations of the vertical positions of B787, ζ

_{p}, (

**a**), and the surface gradient ∂ζ/∂x at the location of B787 (

**b**), in Case TC, in which B787 left the airport at t = 25 s. The still water depth h values were 10 m, 20 m, and 100 m, and the flexural rigidity of the airport, B, was 1 × 10

^{11}N·m

^{2}.

Case * | Airplane | Airport | Water Depth | ||||
---|---|---|---|---|---|---|---|

Type (Mass) | Running Speed | Running Distance | Run Time | Length L | Flexural Rigidity B | h | |

GA-L1 | B747-400 | 83 m/s | 3 km | 36.1 s | 15 km | 1 × 10^{11} N·m | 10 m |

GA-L2 | (397,000 kgs) | 10 m to 50 m | |||||

GA-S1 | 5 km | 10 m, 20 m | |||||

GA-S2 | 1 × 10^{10} N·m | 10 m | |||||

GB-L1 | B737-800 | 78 m/s | 2 km | 25.6 s | 15 km | 50 m | |

GB-L2 | (79,000 kgs) | 10 m to 50 m | |||||

GB-S1 | 5 km | 50 m |

Case * | Airplane | Airport | Water Depth | |||||
---|---|---|---|---|---|---|---|---|

Type (Mass) | Landing Speed ** | Running Deceleration | Running Distance | Run Time | Length L | Flexural Rigidity B | h | |

LA-L | B747-400 | 72 m/s | 0.86 m/s^{2} | 3 km | 83.7 s | 15 km | 1 × 10^{11} N·m | 10 m |

LA-S | (397,000 kgs) | 5 km | ||||||

LB-L | B737-800 | 1.3 m/s^{2} | 2 km | 55.4 s | 15 km | 1 × 10^{10} N·m | 50 m | |

LB-S | (79,000 kgs) | 5 km |

Case * | Airplane | Airport | Water Depth | |||||
---|---|---|---|---|---|---|---|---|

Type (Mass) | Takeoff Speed ** | Running Acceleration | Running Distance | Run Time | Length L | Flexural Rigidity B | h | |

TA-L | B747-400 | 83 m/s | 1.15 m/s^{2} | 3 km | 72.2 s | 15 km | 1 × 10^{11} N·m | 10 m |

TA-S | (397,000 kgs) | 5 km | ||||||

TA-S-B | 1 × 10^{9} N·m to | 50 m | ||||||

1 × 10^{11} N·m | ||||||||

TB-L | B737-800 | 78 m/s | 1.52 m/s^{2} | 2 km | 51.3 s | 15 km | 1 × 10^{10} N·m | |

TB-S | (79,000 kgs) | 5 km |

Case * | Airplane | Airport | Water Depth | |||||
---|---|---|---|---|---|---|---|---|

Type (Mass) | Landing/Takeoff Speed ** | Running Acceleration | Running Distance | Run Time | Length L | Flexural Rigidity B | h | |

LC | B787 | 75 m/s | −3 m/s^{2} | 0.938 km | 25 s | 5 km | 1 × 10^{11} N·m^{2} | 10 m, 20 m, |

TC | (228,400 kgs) | 3 m/s^{2} | or 100 m |

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## Share and Cite

**MDPI and ACS Style**

Kakinuma, T.; Hisada, M.
A Numerical Study on the Response of a Very Large Floating Airport to Airplane Movement. *Eng* **2023**, *4*, 1236-1264.
https://doi.org/10.3390/eng4020073

**AMA Style**

Kakinuma T, Hisada M.
A Numerical Study on the Response of a Very Large Floating Airport to Airplane Movement. *Eng*. 2023; 4(2):1236-1264.
https://doi.org/10.3390/eng4020073

**Chicago/Turabian Style**

Kakinuma, Taro, and Masaki Hisada.
2023. "A Numerical Study on the Response of a Very Large Floating Airport to Airplane Movement" *Eng* 4, no. 2: 1236-1264.
https://doi.org/10.3390/eng4020073