Numerical simulation of planing motion and hydrodynamic performance of a seaplane in calm water and waves

ABSTRACT

The high-speed motion of a seaplane involves the coupled hydrodynamic and aerodynamic effects. The suction force, pressure, free surface and motion of the seaplane model were numerically investigated to understand the characteristics of the seaplane's planing motion. The study utilized the SST-DDES turbulence model to analyse the coupled hydrodynamic and aerodynamic effects. Overset mesh method and rigid body motion were employed to simulate the high-speed and substantial motion of seaplane. The volume of fluid method (VOF) was used to capture and sharpen the interface between water and air. First, verification and validation (V&V) were performed by comparing the results with those of the towing tank experiments. Second, the air-water entrainment in calm water and free surfaces were presented, and the pressure distribution on the seaplane was analysed and discussed. Numerical simulations were performed while considering the wave parameters of different velocities, wavelengths, and wave heights. The accelerations of the fore, aft, and centre of gravity of the seaplane demonstrated the presence of the suction effect. The evolution of the air-water entrainment at the bottom of the fuselage was observed. The investigation of suction characteristics revealed that the aerodynamic force in waves plays a substantial role in influencing motions of the seaplane.

Highlights

• The study of hydrodynamics and aerodynamics of seaplanes is interdisciplinary. The numerical schemes, including a SST-DDES turbulent model, overset mesh method, and volume of fluid (VOF) method, have proven to be effective and accurate for simulating the motion of the seaplane and flow field characteristics.

• The greater the speed and wave height, the faster is the motion of the seaplane and the greater are peak and trough values of the pitch. As the wavelength increased, the peak value of the motion decreased gradually.

• The accelerations of the aft, fore, and centre of gravity of the seaplane in the waves exhibited significant periodicity. The peak value of the acceleration at the aft was the largest, whereas that at the centre of gravity was mild, and the peak value of the acceleration at the centre of gravity was greater than that at the fore. The acceleration indicates that suction at the aft of the seaplane hinders the takeoff during the planing motion.

KEYWORDS:

• Seaplane
• multiphase flow
• regular waves
• suction
• drag

1. Introduction

The aerodynamic and hydrodynamic prediction of seaplanes during the planing motion is challenging, especially the study of mechanical properties in waves, which is not only challenging but also has not achieved good prediction results, and further research is therefore required.

The application of computational fluid dynamics (CFD) technology provides a reliable method for accurately predicting the hydrodynamic and aerodynamic performance of seaplanes (Hu et al., 2021; J. Zhang et al., 2022). Theory and empirical data in the fields of aerodynamics (Zhao et al., 2021), hydrodynamics, and flight mechanics directly affect and help improve the safety and reliability of seaplanes. For example, based on different research purposes, Müller et al. (2018) studied the flow-structure interaction by combining the finite volume method (FVM) and the finite element method (FEM) and analysed the slamming motion and the surface pressure of the structure. Kontogiannis et al. (2016) conducted a numerical study of the remotely piloted aircraft ATLAS IV to summarize its variation patterns such as lift and efficiency and to optimize the aerodynamic design of its shape based on RANS. There are also detailed studies on the aerodynamic and the hydrodynamic in vortex (Lasher & Flaherty, 2009), free surface (Jagadeesh & Murali, 2010), etc. However, previous studies did not study the contribution of the coupled aerodynamics and hydrodynamics systematically. Since most aerodynamic and hydrodynamic studies do not interact with each other (Liu et al., 2022; Sajedi & Ghadimi, 2020; Wang et al., 2014), there are few coupled research results.

In aerodynamic studies, there are effects of high Reynolds numbers and separation of airflow within an aircraft. In recent years, more and more scholars have begun to combine LES and RANS for aerodynamic research (Misaka & Obayashi, 2017; Nelson et al., 2017). For instance, Larsson et al. (2016) introduced the near-wall problem of Large eddy simulation (LES), analysed the near-wall turbulence of the LES model, and discussed its advantages over methods such as RANS for aerodynamic numerical simulations. Misaka and Obayashi (2017) used the RANS/LES hybrid method to study the interaction between the aircraft exhaust nozzle and the generated wake at the boundary of the fluid domain or in a small area. However, there is a difference in the fuselage flow characteristics for seaplanes between the water and air, which plays an important role in the selection of turbulence models. For the study of hydrodynamics, the motion of a planing boat in the water is similar to the take-off process of a seaplane; therefore, scholars have successfully carried out research on planing boats. For example, Mansoori and Fernandes (2016) used RANS to simulate the stability of a planing boat in high-speed motion and found that a reasonable installation of baffles can significantly improve the stability of the planing boat. Brizzolara and Serra (2007) carried out numerical simulation of the planing boat model with a low lift angle of ${20}^{\circ }$ based on RANS and k-epsilon models and compared the obtained lift resistance and hull pressure with the experimental results (Chambliss & Boyd, 1953). This verified the reliability of the numerical simulations in Savitsky (1964) and Shuford (1958), and the error was within 10%. In contrast, unlike the resistance prediction of a boat in the planing stage, the coupled effect of air and water must be fully considered when the structure moves at a high speed.

Additionally, the structural problems based on the finite volume method are mature (Guo et al., 2013; Zheng et al., 2019). Computational fluid dynamics (CFD) simulations conducted for various hull tests (Duan et al., 2019; Xiao et al., 2019) have shown that the difference in hydrodynamics between different simulated conditions is quite obvious. Wu et al. (2004) and Xu et al. (2010) successfully analysed the coupled motion of structures and fluids. Because the planing of a seaplane is a strongly nonlinear fluid–solid coupling phenomenon (Hughes et al., 2013), it is particularly difficult to simulate the effects of splash, atomization, and waves. The accuracy of the air–water phase interface during the numerical simulation must be improved. Seaplane planing in waves involves significant multi-degree-of-freedom motion, which is not conducive to flying safety (Karman, 1929; Spinosa et al., 2022). The numerical solution of the air-water coupling of seaplane is helpful in reducing the difference between the numerical results and the actual motion of the seaplane (Bisagni & Pigazzini, 2018; Qi et al., 2016; Siemann et al., 2018). However, they did not study the contribution of the coupled aerodynamics and full geometry of the seaplane model. The interactions of the seaplane motion between effects of aerodynamics and hydrodynamics are not recognized, which is useful in improving the seaplane safety. The details of pressure, free surface, drag at various speed and various waves are not revealed. Based on experimental results, we carried out numerical simulation to investigate the coupled high-speed motion of this problem.

In this paper, a turbulence model coupled with aerodynamics and hydrodynamics is used to analyse the suction characteristics of a seaplane. The influence of velocity, wave height and wavelength are concerned. The experimental setup, seaplane geometry and numerical method used in the study are introduced in Sections 2 and 3. The experimental results with regard to resistance are presented and discussed. The verification and validation are also introduced. In Section 4, we present the results of our numerical investigation based on SST-DDES to characterize the flow fields around the seaplane. The cases of the velocities, wave heights and wavelengths are discussed in three parts. The typical pressure distributions, free surfaces, motions and suctions (accelerations) are analysed and discussed in detail.

2. Numerical method

2.1. Governing equations

Owing to the high speed of the seaplane in an air-water system, multiphase flow in commercial software Star-CCM+ (Version 16.02) was adopted to simulate the interactions of the seaplane and turbulent free-surface flow. Assuming an isothermal and immiscible fluid, the governing equations of Newtonian fluid consisted of continuity and the well-known Navier-Stokes equations: (1) $\begin{array}{rl}& \mathrm{\nabla }\cdot \mathrm{U}=0\end{array}$(1) (2) $\begin{array}{rl}& \frac{\mathrm{\partial }\rho \mathrm{U}}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \rho \mathrm{U}\mathrm{U}-(\mu +{\mu }_t)\mathrm{\nabla }\cdot (\mathrm{\nabla }\mathrm{U}+{\mathrm{\nabla }\mathrm{U}}^T)\\ & =-\mathrm{\nabla }p+\rho g+f\end{array}$(2) where the U, ρ , p, g and f is the velocity, density, pressure, gravitational acceleration, and the stress tensor, respectively. μ and ${\mu }_t$ represent the mixture dynamic viscosity coefficient and viscosity coefficient, respectively. (3) $\begin{array}{rl}\rho & ={\rho }_{\mathrm{a}\mathrm{i}\mathrm{r}}{\alpha }_{\mathrm{a}\mathrm{i}\mathrm{r}}+{\rho }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}{\alpha }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}\end{array}$(3) (4) $\begin{array}{rl}\mu & ={\mu }_{\mathrm{a}\mathrm{i}\mathrm{r}}{\alpha }_{\mathrm{a}\mathrm{i}\mathrm{r}}+{\mu }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}{\alpha }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}\end{array}$(4) The VOF schemes can be divided into two main categories: algebraic and geometric VOF. The algebraic VOF model was adopted to capture the air-water interface with a liquid volume fraction 0.5. High-resolution interface capture (HRIC) is one of the most widely used algorithms for interface calculations. In the modified HRIC, it is easy to develop a stable interface and retain its boundness. The value of the normalized variable at the face of the control volume can be evaluated as follows: (5) ${\alpha }_f=\{\begin{array}{ll}{\alpha }_c,& \mathrm{i}\mathrm{f}{\alpha }_c<0\\ {\displaystyle (\frac{20}{3}-\frac{100}{9}{\alpha }_c){\alpha }_c,}& \mathrm{i}\mathrm{f}0\le {\alpha }_c<0.3\\ 1,& \mathrm{i}\mathrm{f}0.3\le {\alpha }_c<1\\ {\alpha }_c,& \mathrm{i}\mathrm{f}1\le {\alpha }_c\end{array}$(5) where ${\alpha }_c$ denotes the cell value of the normalised control volume. Additionally, the value of ${\alpha }_c$ is further corrected according to the local Courant number. Finally, the modified HRIC can be calculated as: (6) ${\alpha }_f^{HRIC}={\alpha }_f({\alpha }_D-{\alpha }_U)+{\alpha }_U$(6) where ${\alpha }_D$ denotes the values in the central and downwind cells. ${\alpha }_U$ is extrapolated using the volume fraction gradient and the centroid-centroid distance vector. These modified HRIC operations make algebraic VOF accurate enough and faster than the geometric alternative simultaneously.

In this study, the effect on the coupled motion and hydrodynamics of the seaplane at high speeds was emphasized. A high-resolution MDCD scheme was used to reconstruct the left and right states of the Riemann problem. The fifth-order WENO interpolation technique was used for the spatial discretization of the finite volume method. An implicit unsteady algorithm was used for velocity-pressure coupling, and a second-order scheme was used for temporal discretization.

2.2. Turbulence model SST-DDES

In this study, the turbulence model is SST-DDES. The high-speed motion of seaplane are often accompanied by a large separation vortex, and the standard turbulence model eg SST k-omega, overestimates the vortex viscosity in the turbulence flow separation area, resulting in excessive turbulent stress. This significantly affects the simulation accuracy of vortex shedding, such as cavitation. As a hybrid RANS-LES method, the SST-DDES method significantly improves the ability to simulate separation vortices and has been verified in engineering applications. The formula for SST-DDES is as follows: (7) $\begin{array}{rl}& \frac{\mathrm{\partial }\rho \mathrm{k}}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \rho \mathrm{U}\mathrm{k}-(\mu +\sigma {\mu }_t)\mathrm{\nabla }\cdot \left(\mathrm{\nabla }\mathrm{k}\right)\\ & =P_k-\rho \sqrt{(}{k}^3)/l_{DDES}\\ & \frac{\mathrm{\partial }\rho \omega }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \rho \mathrm{U}\omega -(\mu +\sigma {\mu }_t)\mathrm{\nabla }\cdot \left(\mathrm{\nabla }\omega \right)\\ & =-2(1-F_1)\rho {\sigma }_{\omega 2}\frac{\mathrm{\nabla }\mathrm{k}\cdot \mathrm{\nabla }\omega }{\omega }-\alpha \frac{\rho }{{\mu }_t}{P}_k+\text{β}\rho {\omega }^2\end{array}$(7) (8) $\begin{array}{rl}& {\mu }_t=\frac{\rho a_{1}k}{max(a_1\cdot \omega ,F_2\cdot S)}\end{array}$(8) where μ is the dynamics viscosity, ${\mu }_t$ is the turbulence eddy viscosity, $P_k$ is the turbulence production term, and $F1$ ($F2$) is the bounding function. The SST-DDES is obtained by modifying the k equation in the SST and coupled DES models. The turbulence production term $P_k$ is defined as follows: (9) $P_k=min({\mu }_t{S}^2,10C_{\mu }\rho k\omega )$(9) DDES turbulenee length scale can be expressed as (10) $\begin{array}{rl}{l}_{LES}& =C_{DES}{h}_{max}\\ l_{RANS}& =\frac{\sqrt{k}}{C_{\mu }\omega }\\ C_{DES}& =C_{DES1}\cdot F_1+C_{DES2}\cdot (1-F_1)\\ l_{DDES}& =l_{RANS}-f_{d}max(0,l_{RANS}-l_{LES})\end{array}$(10) The details of the parameters can be obtained from the literature (Menter & Kuntz, 2004; Shur et al., 2008).

2.3. Rigid body motion and seaplane model

Considering the coupled complex motion of the seaplane, overset mesh and 2-DOF (two Degrees of Freedom) method were used. Due to the use of a symmetrical half model, the motion of the seaplane in the numerical analysis is limited to the two degrees of freedom: heave and pitch. A background mesh block with an air-water interface and an overset mesh block that moves along the seaplane were used. The data exchange between the background mesh and overset mesh was realized by least-squares interpolation. As shown in Figures 1 and 2, two coordinate systems in the fluid region are assigned to solve the relevant motion in the 2-DOF (two Degree of Freedom). One is called the initial coordinate system (geodetic coordinate system), and the other is called the non-initial coordinate system (seaplane coordinate system). The origin coordinate system of the seaplane is always at the centre of gravity, based on the Earth's coordinate system.

Figure 1. Definition of the coordinate system.

Figure 2. Illustration of the seaplane. (a) top view, (b) back view, (c) front view, (d) axial side view and (e) side view.

2.4. Computational domain

The computational domain of the numerical simulation is illustrated in Figure 3. The computational domain firstly referred to the flow field of the seaplane observed in towing experiments and drew upon the computational experience from ship and aircraft engineering studies (Spinosa et al., 2022; T. Zhang et al., 2012). A CFD model was developed based on the experimental model. Secondly, since the seaplane possesses axisymmetric geometry and symmetrical flow characteristics, a semi-model computational domain was employed to conserve computational resources. To ensure the complete development of the wake and free surface waveform when the seaplane is planing on the free surface, the computational domain was designed to provide sufficient interaction between the seaplane and the free surface. Additionally, ample space was allotted for fluid development during the establishment of the computational domain.

Figure 3. (a) The towing tank (T. Zhang et al., 2012), (b) schematic of experimental configuration, (c) boundary condition of simulation and (d) computational domain.

Figure 3(c) shows the sketch of the computational domain: $-2.15L\le x\le 7.3L$, $0.0L\le y\le 2.7L$, and $-2.7L\le z\le 2.7L$. L is the total length of the seaplane. The inlet boundary was the velocity inlet, the outlet boundary was the pressure outlet, the upper and bottom boundaries were consistent with the velocity inlet, the sides and symmetry planes of the computational domain had symmetric boundary conditions, and the seaplane had no slipping walls.

The computational domain was discretized into meshes using volume rectangular meshes, while the fuselage's boundary layer was divided using prismatic boundary layer meshes. To accurately capture the physical phenomena around the fuselage and rear area of the seaplane, appropriate mesh refinement was applied. Details of the mesh are shown in Figure 4. The mesh distribution of the seaplane at a specific time is shown in Figure 5. Different mesh details were employed for the water and air parts around the seaplane, considering their distinct media. Surface remesher and prism layer mesher models were used to obtain high-quality meshes. The Two-Layer All $y^{+}$ wall surface in Star-CCM+ was used for the boundary layer. In the model-scale aerodynamic and hydrodynamic calculations, the non-dimensional distance $y^{+}$ values for the water part ranged from 30 to 60, while for the air part, the $y^{+}$ values ranged from 0 to 1. These $y^{+}$ values satisfy the requirements for the calculations. To capture the wave height shape effectively, at least 20 meshes were used within the range of the wave height. Similarly, for wavelength shape capture, a minimum of 120 meshes were employed within the wavelength range. Under-relaxation factors of 0.7 and 0.2 were used for the velocity and pressure, respectively. Each simulation case was calculated using a two-way 56-core 2.5 GHz Intel(R) Xeon(R) Platinum 8180 processor.

Figure 4. Mesh topology, (a) the free surface and Kelvin waves area, (b) the boundary layer mesh distribution around the step, (c) the boundary layer mesh distribution at the front of seaplane, and (d) the boundary layer mesh distribution at the tail of the fuselage.

Figure 5. The overset meshes, (a) overset domain, (b) side view, (c) back view and (d) top view.

3. Experimental setup

3.1. Towing tank specifications

The schematic of the test device is shown in Figure 3(a,b). The test conditions were a free-drag test with no power and two degrees of freedom. To simulate the dynamic influence of the wing lift and moment on the fuselage at the scale of a real seaplane, the method of unloading the fuselage weight with an equivalent load was adopted in the test, that is, adding the equivalent unloading lift at the centre of gravity. The effective torque was unloaded to match the actual machine state as much as possible.

The effect of the calm water and of waves on the motion of the complete seaplane model was experimental investigated in China Special Type Flier Research Institute, whose high-speed hydrodynamic laboratory is an Aeronautic Scientific & Technologic Laboratory. The test tank in experimental sections is 510 m long, 6.5 m wide, 6.8 m deep, 5 m deep. A high speed trailer, whose maximum speed is 22 m/s, is installed in the test section, and the stability precision is 0.1% to 0.2%. A rocker-flap wave maker driven by an AC servo motor is installed at the end of the towing tank. The short-crested regular waveslength can be set to values between 0.05 and 15 m/s, with long-crested irregular waves with significant wave height values of less than 0.3 m.

In this paper, the towing test was carried out in calm water and short-crested regular waves. The test equipment consists of trailers, seaplane models, tanks, and transducers. The seaplane model is attached to the trailer using a hook. Once the trailer reaches a predetermined position at a specific speed, the seaplane is released into the water. The parameters of seaplane for experiments and numerical data analysis is shown in Figure 6. The rear of the seaplane will planing in contact with the free surface. The density of the air-water mixtures is significantly lower than that of gas, with the mixtures of air and water below the rear and air above it. According to Bernoulli's principle, Downward force will generate at the rear of the seaplane. This force (downward) is the source of suction for seaplanes. In this paper, we quantify the effect of suction by comparing differences in accelerations of the seaplane using transducers on its surface.

Figure 6. The parameters of seaplane for experiments and numerical data analysis.

3.2. Model geometry

The shapes of the fuselage, wings, and empennage in this paper are similar to those in the literature (Duan et al., 2019). There is a difference in the geometry of the model, as the literature includes a turbofan engine while it is omitted in this paper. The actuator disk model was used to simulate the aerodynamic flow field characteristics of the turbofan engine and a 1:15 scaled model with six blades and one nacelle has been employed in the literature in the literature (Duan et al., 2019). This study primarily focuses on the planing performance of the seaplane. The scale ratio is 1:10, and the parameters such as the size, mass, speed, moment of inertia and test speed of the seaplane model satisfy the corresponding proportional relationship based on the similarity of the Froude number (Fr). The non-dimensional speed coefficient $C_v$ is widely used and identified as follow: (11) $C_v=V/\sqrt{g{B}_{\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{e}}}$(11) where V is the speed, g is the acceleration of gravity, $B_{\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{e}}$ is the width of the fuselage. The speed coefficient $C_v$ is similar to the Froude number Fr based on ship length, but it is still difficult to identify the states of the seaplane by the coefficient. Therefore, we adopted both $C_v$ and displacement Froude number $F{r}_{\mathrm{\nabla }}$, which is shown as follow, to identify the state of the seaplane. The corresponding speed coefficient $C_v$ and displacement Froude numbers $F{r}_{\mathrm{\nabla }}$ are presented in Table 1 (12) $F{r}_{\mathrm{\nabla }}=V/\sqrt{g{\mathrm{\nabla }}^{\frac{1}{3}}}$(12)

Table 1. Conversion relationship of experimental seaplane model at various speeds.

Planing speed V(m/s)	3	4	5	5.5	6	6.5	8	9	10	11	12	13
Speed coefficient $C_V$	0.498	0.664	0.831	0.914	0.997	1.08	1.329	1.495	1.661	1.827	1.993	2.159
Displacement Froude number $F{r}_{\mathrm{\nabla }}$	1.272	1.701	2.183	2.417	2.649	2.898	3.9	4.425	4.95	5.525	6.09	6.691
Initial angle of inclination (°)	1.1	1.93	2.77	2.91	2.96	3.48	5.46	5.43	5.55	5.62	5.53	5.5

We used %MAC to calculate the weight and balance with respect to the aircraft datum and The Mean Aerodynamic Chord (MAC) is 26% in this paper. As shown in Figure 2, the seaplane model components mainly include the fuselage, wings, flaps, elevator and some parts. The parameters of seaplane model are listed in Table 2. To monitor the motion and suction of the seaplane, we have selected three representative measurement points. The transducers records the pressure, speed, acceleration, and other response information of the seaplane. These points are located at the fore, near the centre of gravity, and the aft of the seaplane. Transducers were installed at these measurement points to collect data, which was then utilized for experimental analysis and subsequent numerical comparisons. The horizontal distances from the transducers to the frontmost point of the seaplane model are 555, 1610 and 2930 mm respectively. The PCB acceleration transducers used in this experiment have a measuring range of 2 g, with an accuracy of 5%. The displacement transducer has a measuring range of 1 m, with an accuracy of 1%. The angle transducer has a measuring range of ${60}^{\circ }$, with an accuracy of ${0.01}^{\circ }$. The force transducer has a measuring range of 40 kg, with an accuracy of 0.01 kg.

Table 2. Parameters of seaplane model.

Parameters	Description	Value	Unit
M	Seaplane mass	49	kg
CG	Center of gravity position (%MAC)	26	–
L	Length over all	3.694	m
H	Height over all	1.210	m
$L_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}}$	Wing span	3.880	m
$S_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}}$	Wing area	1.672	${\mathrm{m}}^2$
AR	Aspect ratio	9	–
$L_{\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{e}}$	Fuselage length	3.636	m
$B_{\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{e}}$	Fuselage width	0.34	m
$H_{\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{e}}$	Fuselage height	0.472	m
$L_{\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l}}$	Horizontal Tail Span	1.258	m
$L_{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}$	Vertical Tail Span	0.660	m
${\alpha }_{\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{p}}$	leading edge sweep angle	25	°
${\alpha }_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{p}}$	flap angle	20	°
${\alpha }_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}}$	elevator angle	-12	°

4. Verification and validation

For the computational domain, the mesh sizes dx were 0.07, 0.06, 0.05 and 0.04 m, respectively, whereas the Courant Friedrichs Lewy (CFL) number was less than 1. The detailed meshes and time resolutions are listed in the Table 3. The background area was divided into three layers, gradually decreasing the mesh size. Taking the fine mesh condition as an example, the background mesh size in the X, Y, Z direction is set to 0.06 m, and the mesh size in the Z direction is changed to 0.03, 0.015 and 0.0075 m near the free surface. In the Kelvin wave area, the mesh size was modified in the X, Y, Z directions, with densification values of 0.03, 0.015, and 0.0075 m. The splash phenomenon during seaplane takeoff occurs across a wide range of takeoff speeds. Splashing and atomization typically involve droplets smaller than 1 mm in size. To capture this phenomenon accurately, the mesh size needs to be approximately 1/8 to 1/16 of the droplet size, resulting in a considerable number of meshes. In this study, we employed various numerical strategies such as overset mesh method, SST-DDES turbulent model, and 2-DOF (two Degree of Freedom) to minimize computational instability. The refinement rate between different meshes (i.e. dxmedium / dxcoarse) was 0.86, 0.83, and 0.8, respectively; The time steps were 0.005, 0.004, 0.003 and 0.002 s; the implicit unsteady algorithm was adopted for the velocity–pressure coupling, and the second-order scheme with 10 inner iterations per time step was used for the temporal discretization. Figure 7 shows the time history of each monitored physical quantity. The numerical results obtained using different meshes and time-step intervals were in good agreement with each other. At 2 s, the pitch, heave and drag exhibited stable periodic changes, and the calculation stabilized after 2 s. The obtained results were used to validate the numerical simulation.

Figure 7. Time history of (a) Drag, (b) Heave and (c) Pitch at the planing speed V = 8 m/s in calm water.

Table 3. Details of meshes and time resolutions of seaplane.

Mesh type	Number of all meshes	Number of background meshes	Number of overset meshes	Time step $\mathrm{\Delta }t\left(s\right)$	Mesh size $dx\left(m\right)$	Mean drag (N)	Mean heave (mm)	Mean pitch (°)
Coarse	8,818,234	1,821,724	6,994,510	0.005	0.07	36.169	108.037	4.440
Medium	12,303,387	2,889,426	9,413,981	0.004	0.06	35.515	111.123	4.635
Fine	18,304,810	4,887,525	13,437,285	0.003	0.05	34.944	113.363	4.767
Tiny	30,082,389	9,283,024	20,819,345	0.002	0.04	34.499	114.978	4.856

The planing motion of seaplane (half model) at different speed are simulated, and the drag, pitch and heave of the seaplane are analysed in Figure 8. The corresponding drag, heave and pitch are the mean values. Compared with the experimental data, the average drag error was 6.38% in the simulations; the drag and heave are in good agreement with the experimental data in Figure 8(a,b); In general, the heaves were also in good agreement with the experimental data at several planing speeds; However, the values between the experiment and numerical simulation demonstrated large errors at the drag peak area, e.g. 5 and 6.5 m/s. As shown in Figure 8(c), in the planing speeds range of $5-6.5$ m/s, the error between the experimental and numerical simulation results increased, indicating that the drag and motion of the seaplane undergo strong dynamic instabilities.

Figure 8. Comparison between numerical results and experimental data at the planing speed $V=8$ m/s in calm water. (a) Drag, (b) Heave and (c) Pitch.

Specifically, before takeoff, seaplanes undergo a transition from a displacement state to a planing state. Owing to the lifting of the front body of the fuselage, the drag increased rapidly. A fluctuation in the pitch motion was noted in the transition stage of the medium speed, but the seaplane continued to rise. When the speed reached 8 m/s, the seaplane entered the planing stage, and the pitch angle gradually stabilized. In the speed range of $5-6.5$ m/s, the drag is closely related to the pitch of the seaplane. Peak drag was observed at the beginning of the planing motion and declined as the pitch decreased. The dynamic instability of the pitch was caused by a change in the state of the seaplane. In other words, owing to the coupled effects of pitch and speed, the drag decreased after increasing at various velocities, as shown in Figure 8(a).

The calculated drag, heave and pitch of seaplane are in good agreement with those of the experimental data in Figure 8. In the following simulations, the dx = 0.005 m and dt = 0.0005 s were adopted to ensure the accuracy and the rationalization of computer resources. Snapshots of the free-surface wave elevation at various instants are shown in Figure 9. The free-surface wave height was evident, and a Kelvin wake was evident at the tail of the seaplane.

Figure 9. Typical snapshot of the wave elevation at the planing speed $V=12$ m/s in calm water, (a) front view, (b) side view, (c) Partial enlarged view (top) and (d) Partial enlarged view (bottom).

We considered the drag, heave, and pitch after a simulation time of 10 seconds, which corresponds to the stabilized motion of the seaplane. We averaged the data and compared it with the experimental values. The values of mean drag, mean heave and mean pitch are in Table 3.

For estimating the uncertainty in the numerical solution, The numerical results with coarse, medium, fine and tiny mesh type are denoted by $s_1$, $s_2$, $s_3$ and $s_4$, respectively. ${ϵ}_{12}$, ${ϵ}_{23}$ and ${ϵ}_{34}$ denote the difference between the results using the medium mesh and coarse mesh, that between the fine mesh and medium mesh, and that between the tiny mesh and fine mesh, respectively: (13) $\begin{array}{rl}{ϵ}_{12}& =s_2-s_1\end{array}$(13) (14) $\begin{array}{rl}{ϵ}_{23}& =s_3-s_2\end{array}$(14) (15) $\begin{array}{rl}{ϵ}_{34}& =s_4-s_3\end{array}$(15) The convergence rate $R_k$ is calculated as: (16) $\begin{array}{r}{R}_{ki}=\frac{{ϵ}_{i,i+1}}{{ϵ}_{i-1,i}}\end{array}$(16) The convergence rate distinguishes three types of numerical solutions as follows: $0<R_k<1$: indicates that the solution converges monotonically, and $R_k\le 0$: indicates that the solution is of oscillatory convergence $R_k\ge 1$: indicates that the solution is divergent. The values of the parameters are shown in Table 4. They show that the numerical results are monotonically convergent with the size of the meshes, which indicates that the fine and tiny mesh used in this study are effective for simulating the flow field around the seaplane model. Considering computational economy, fine mesh is selected as the mesh settup for numerical simulation.

Table 4. Verification of numerical results.

Numerical results	$s_1$	$s_2$	$s_3$	$s_4$	${ϵ}_{12}$	${ϵ}_{23}$	${ϵ}_{34}$	$R_{k1}$	$R_{k2}$
Mean drag (N)	36.169	35.515	34.944	34.499	−0.654	−0.571	−0.445	0.873	0.779
Mean heave (mm)	108.037	111.123	113.363	114.978	3.086	2.240	1.615	0.726	0.721
Mean pitch (°)	4.440	4.635	4.767	4.856	0.195	0.132	0.089	0.677	0.674

In this study, regular waves were provided directly at the boundary as incident conditions (elevation and speed). The wave heights were $0.05$ and 0.1 m. The wavelengths were 8, 12 and 16 m. The numerical wave absorption method is an artificial attenuation method (sponge damping wave absorption) that adds a damping term to the damping zone prior to the boundary. A regular wave validation was also performed. The wave height and wavelength were 0.05 m and 8 m, respectively. The time history of the wave elevation between the numerical simulation and theoretical data is shown in Figure 10. The regular wave remained stable such that the maximum crest elevation predicted by the probe monitor was the same as the theoretical value. These curves are in good agreement with the theoretical results.

Figure 10. Time history of wave elevation in regular wave, H = 0.05 m, $L=8$ m.

Figure 11 shows the typical snapshot of the dimensionless wave horizontal and vertical speed profile. These components are generally the cosine and sinusoidal components of the axial speed. It can be observed that numerical simulation works well and the orbital motion of the regular wave is well reproduced.

Figure 11. The dimensionless wave velocity profile in regular wave, H= 0.05 m, L = 8 m. (a) Horizontal velocity and (b) Vertical velocity.

5. Numerical results and discussion

5.1. Pressure distribution around the step on seaplane in calm water

The pressure around the step of seaplane are investigated in Figure 12. It can be observed that high-pressure area at each speed is concentrated in the front area of the step. As the speed increases, the pressure amplitude in the pressure zone varies from a wide range of low pressures to a small range of high pressures and then to a small range of low pressures. As shown in Figure 12(a), at 5.5 m/s, the seaplane is in the transitional displacement stage, as mentioned in Section 3.1. The pressure distribution over a large area from the step to the head of the seaplane is relatively uniform, and there is no evident high-pressure area. After entering the planing state, as shown in Figure 12(b–d), a triangular high-pressure area appears in the front area of the step, and the triangular high-pressure area moves backward as the speed increases. When the speed is 11 m/s, as shown in Figure 12(e), the pressure on both sides of the triangular area near the step increases obviously. Before takeoff, the high pressure in front of the step decreases, as shown in Figure 12(f).

Figure 12. The pressure profiles around the step at various planing speeds in calm water. (a) V = 5 m/s, (b) V = 8 m/s, (c) V = 10 m/s, (d) V = 11 m/s, (e) V = 12.0 m/s and (f) V = 13.0 m/s.

Figure 8(c) shows that the corresponding centre of lift force of the seaplane moves backward to the front of the step as the seaplane speed increases. At this time, in Figure 12(b–d), the pressure area is also moved backward closer to the centre of gravity. When the centre of pressure is the same as the centre of gravity, the head-up moment of the seaplane is converted into a head-down moment.

Figure 13 shows the pressure distribution along the bottom surface of the fuselage in the direction of the seaplane length, with the vertical axis being the pressure coefficient and the direction of the seaplane length being represented by the $(1,0,0)$ on the horizontal axis of Figure 13. Along the seaplane length, the pressure changes significantly in the areas of the step and tail. Particularly, there is a large difference in pressure before and after this step. The pressure before the step is significantly higher than that after the step. As shown in Figure 13(a), there are differences in the pressure before the step at different speeds. The faster the speed, the greater the is the pressure before the step, and the differences in pressure after the step are not significant. This step effectively reduces the pressure at the tail of the seaplane at different speeds. At a speed of 5.5 m/s, it can be observed that, in the unstable speed range, the pressure before and after the step fluctuates; furthermore, the pressure on the fuselage surface far from the step also changes. Similarly, As shown in Figure 13(b,c), the pressure fluctuation at the step and tail of seaplane was caused by different waves. When the wave height and wavelength were 0.1 m and 8 m, the pressure at the tail of seaplane is larger, but the high pressure area at the step is smaller.

Figure 13. The pressure at the bottom surface around the seaplane, (a) various speed in calm water, (b) wavelengths and (c) wave heights.

Typical free-surface wave elevations behind the seaplane from the top view are shown in Figure 14. Evidently, with an increase in speed, the Kelvin wake extends in the backward direction. The Kelvin wake at various speeds was well captured during planing in water. The angle of the Kelvin wake gradually decreases with the increasing seaplane speed. In the transitional displacement stage, the Kelvin wake clearly forms two elevated ‘long stripes’, as shown in Figure 14(a,b). After entering the planing stage, the Kelvin wake rises and extends in the backward direction, exhibiting a horseshoe shape, as shown in Figure 14(c). When the speed is close to the takeoff from water, the Kelvin wake becomes smooth and flat without an obvious wave elevation, as shown in Figure 14(d). The wake does not exhibit a complete Kelvin wave as the speed changes, which is significantly different from that of a planing boat.

Figure 14. Free-surface wave elevation from front views in calm water.

5.2. Free-surface air-water entrainment on seaplane

Compared with this paper, the treatment of the mesh in the literature (Duan et al., 2019) is relatively rough, and the simulation method based on RANS (Reynolds-Averaged Navier-Stokes) sacrifices simulation accuracy. In terms of numerical simulation, the literature (Duan et al., 2019) uses the motion simulation method of moving mesh method, while this paper adopts the motion simulation method of overset mesh method. The VOF method employed in the literature (Duan et al., 2019) is MULES, whereas we use a modified HRIC method. Regarding the hydrodynamic conditions, the numerical simulation conditions in this paper are similar to those in the literature, and a similar mesh strategy is implemented to ensure a reasonable number of Courant Friedrichs Lewy (CFL) for stability.

We conducted a comparison of the pressure and free surface with the literature (Duan et al., 2019) and simultaneously demonstrated the vortex and velocity of the airflow field. Figure 15 shows comparisons of the numerical flow fields around the seaplane. Comparing the numerical flow field of the literature (Duan et al., 2019) in Figure 15(a) that the surface pressure of seaplane can be better presented. The free-surface wave elevation and seaplane motion in Figure 15(b) are effectively and precisely simulated based on VOF (HRIC) method. Figure 16 illustrates the vortex and velocity distribution of the airflow field. From Figures 15 and 16, it can be observed that the effect on the aerodynamic and hydrodynamic flow field of the seaplane works well. Overall, the numerical results can better capture the flow fields by SST-DDES turbulence model.

Figure 15. Comparisons of the flow fields around the seaplane in calm water, (a) the pressure and (b) free-surface. (a) Pressure distribution: Reference (Duan et al., 2019) (left) and this paper (right) and (b) Free-surface: Reference (Duan et al., 2019) (left) and this paper (right).

Figure 16. Comparisons of the air flow fields around the seaplane, the vorticity(left) and velocity(right).

The free-surface water entrainment characteristics of the underwater part (underwater fuselage) of the seaplane are shown in Figure 17. The top and bottom parts of Figure 17 show the air-water distribution and the air volume fraction on the bottom surfaces of the fuselage, respectively. The draft at the seaplane bottom was slightly deeper, with a speed of 3 m/s, and the lower surfaces of the fuselage did not have free-surface water entrainment, as shown in Figure 17(a). At a speed of 4 m/s in Figure 17(b), the phenomenon of free-surface water entrainment occurred at the edge of the step, and the entrained water of free surface appeared as a trachea, which reached the tail of the fuselage.

Figure 17. Typical profiles of free-surface water entrainment in calm water. (a) V = 3 m/s, (b) V = 4 m/s, (c) V = 5 m/s, (d) V = 6.5 m/s, (e) V = 12.0 m/s and (f) V = 13.0 m/s.

When the speed of seaplane is 5 m/s, a large air cavity area is observed behind the step in Figure 17(c). The volume fraction of air is concentrated around 0.9–1.0. The volume fraction of air is concentrated around 0.9–1.0. This step can reintroduce gas around the fuselage surface to the bottom of the fuselage, thus reducing the contact area between the fuselage surface and free-surface water. Then, the upper part of Figure 17(d) shows that, at the speed of 6.5 m/s, the phenomenon of air-water entrainment occurs at the front of the fuselage surface. The entrained water at the front of the fuselage takes the shape of a water pipeline and extends from the front of the fuselage to the step. The edge of the air-water interface on the fuselage surface extends to the rear of the fuselage. After the transition to the high-speed stage of 12 m/s, Figure 17(e) shows that a large air cavity develops around the bottom areas of seaplane, and there are two air-water mixture area at the step and the tail of seaplane. Finally, at a speed of 13 m/s, the volume fraction of air in the air cavity separation area is approximately 0.5–0.9. Figure 17(f) presents the flow instability behind the step. That is, a part of the air cavity merges with the interface near the fuselage surface, and another part merges with the interface at the tail of the seaplane.

5.3. Effect of planing velocities on seaplane in regular wave

In this section, the influence of planing speed on the stability of seaplane in regular waves is discussed. The wave height was 0.05 m, the wavelength is 8 m, and the corresponding speed were 4, 5, 6, 8, and 9 m/s respectively. The attack angle of elevator was $-{12}^{\circ }$. The seaplane's position is set far away from the wave generation zone to ensure stable regular waves. The numerical data is collected until the wave generation reaches a stable state. We monitored the wave height at the front of the seaplane. The wave periods corresponding to wavelengths of 8, 12, and 16 m are 2.264, 2.772, and 3.201 s, respectively. When the wavelength is 8 m and the wave height is 0.05 m, the wave encountering periods at planing speeds of 4, 5, 6, 8, and 9 m/s are 1.06, 0.94, 0.84, 0.69 and 0.64 s, respectively.

Figure 18 presents the time histories of drag (half model) at various velocities. The curves at different speeds show similar periodicities during the planing motion of the seaplane. The peak and mean values of the drag increase with speed. Simultaneously, with an increase in speed, the number of waves encountered by the seaplane increases gradually, and the interaction with the waves becomes more violent. Significantly, for the lower speed of 4 and 5 m/s, as shown in Figure 18(a,b) that the drag period is similar, and the peak and trough values of the drag do not change significantly. The drag amplitude of the seaplane is positively related to speed. Additionally, the mean value of the seaplane drag in waves appears to increase progressively with increasing speed.

Figure 18. Drag of seaplane with various velocities in waves, H = 0.05 m, L = 8 m. (a) V = 4 m/s, (b) V = 5 m/s, (c) V = 6 m/s, (d) V = 8 m/s and (e) V = 9 m/s.

From Figure 19, it can be observed that the heave change of the seaplane is consistent with the trend of the drag change. An increase in speed results in an increase in the number of waves encountered by the seaplane per unit of time. Consequently, the heave amplitude in the waves increases when the speed of the seaplane increases, indicating that the interaction between the seaplane and waves becomes increasingly intense. Particularly, when the speed is 6 m/s, as shown in Figure 19(c), the motion of the seaplane is more frequent than other speeds, and the seaplane is in the unstable motion stage.

Figure 19. Heave of seaplane with various velocities in waves, H = 0.05 m, L = 8 m. (a) V = 4 m/s, (b) V = 5 m/s, (c) V = 6 m/s, (d) V = 8 m/s and (e) V = 9 m/s.

We further compared the pitch of the seaplane in waves at various velocities, as shown in Figure 20. Similar to the variation of drag or heave in waves of seaplane, in Figure 20, it can be observed that the pitch motion of seaplane in waves also has significant periodicity. Pitch is directly affected by the encountering frequency. Therefore, the greater the speed, the faster the frequency of the pitch motion, and the greater the amplitude of the peak and trough of the pitch.

Figure 20. Pitch of seaplane wit h various velocities in waves, H = 0.05 m, L = 8 m. (a) V = 4 m/s, (b) V = 5 m/s, (c) V = 6 m/s, (d) V = 8 m/s and (e) V = 9 m/s.

Suction is a negative (downward) force acting on the seaplane. The occurrence of suction forces at the rear and their effects on the aircraft dynamics were highlighted in literature (Climent et al., 2006; Del Buono et al., 2021; T. Zhang et al., 2012). A region of negative pressure (i.e. below the atmospheric value) develops at the rear of seaplane (Spinosa et al., 2022). Figure 21 shows the accelerations induced by the waves with different encountering frequency. Additionally, the accelerations at the fore, aft, and centre of gravity have significant periodicity. With an increase in speed, the peak and amplitude of acceleration increase. The peak value of the acceleration at the aft is the largest, while that of the acceleration at the centre of gravity and aft is mild. Furthermore, the acceleration indicates that the suction at the CG position and fore of the ship is smaller than that at the aft.

Figure 21. The acceleration of seaplane with various velocities in waves, H= 0.05 m, L = 8 m. (a) V = 4 m/s, (b) V = 5 m/s, (c) V = 6 m/s, (d) V = 8 m/s and (e) V = 9 m/s.

The air-water interface is not conducive to takeoff, as shown in Figure 21. Furthermore, the profiles of the wave elevation of the seaplane at different speeds are shown in Figure 22. The breaking wave can be clearly observed after the seaplane encounters the wave, and the wake appears discontinuous with peaks and troughs. With an increase in the speed, the Kelvin angle of the waves on the free surface decreases gradually. When the planing speed of the seaplane increases, the suction effect of the seaplane can induce unstable motion, which hinders the seaplane's take-off process. Therefore, it is advisable to avoid premature aircraft control or adjust the angle of attack to ensure a safe takeoff.

Figure 22. The profile of the wave elevation at typical velocities, H= 0.05 m, L = 8 m. (a) V = 4 m/s, (b) V = 5 m/s, (c) V = 6 m/s, (d) V = 8 m/s and (e) V = 9 m/s.

5.4. Effect of regular wavelengths on seaplane in regular wave

In the study of the influence of wavelength on the planing stability of the seaplane, the wave height and speed were 0.05 m and 8 m/s, respectively, and the wavelengths are 8, 12, and 16 m. When the planning speed is 8 m/s and the wave height is 0.05 m, the wave encountering periods corresponding to wavelengths of 8, 12, and 16 m are 0.69, 0.97, and 1.3 s, respectively. Figure 23 shows the drag of seaplane at different wavelengths. The heave and pitch of the seaplane at different wavelengths are shown in Figures 25 and 26.

Figure 23. The drag of seaplane at various wavelengths, H = 0.05 m, V = 8 m/s. (a) L = 8 m, (b) L = 12 m and (c) L= 16 m.

Figure 24. The heave of seaplane at various wavelengths (half mode), H= 0.05 m, V = 8 m/s. (a) L = 8 m, (b) L = 12 m and (c) L = 16 m.

Figure 25. The pitch of seaplane at various wavelengths (half mode), H= 0.05 m, V = 8 m/s. (a) L = 8 m, (b) L = 12 m and (c) L = 16 m.

Figure 26. The acceleration of seaplane in various wavelengths, H = 0.05 m, V = 8 m/s. (a) L = 8 m, (b) L = 12 m and (c) L= 16 m.

As shown in Figures 24–26, the drag and motion under different wavelengths have obvious periodicities. At the same wave height, as the wavelength increases, the peak value of the drag gradually decreases. Additionally, the mean value of the drag appears to decrease gradually with an increase in wavelength. The periodicity of the heave and pitch is more significant, and both the amplitude and mean value decrease with an increase in wavelength. The accelerations of seaplane at different wavelengths are shown in Figure 27.

Figure 27. Contour distribution of wave profiles around the seaplane at different wavelengths, $H=0.05$ m, V = 8 m/s. (a) L = 8 m, (b) L= 12 m and (c) L = 16 m.

The acceleration of the fore, aft and centre of gravity of the seaplane in waves also have significant periodicity. The peak value of the acceleration at the aft is greater than that at the centre of gravity, and the peak value of the acceleration at the centre of gravity is greater than the that at the fore. The peak value and range of acceleration decrease with increasing wavelength. The numerical results for the acceleration are in agreement with the previous ones mentioned in section 4.3, as shown in Figure 27. The acceleration indicates that suction at the aft of the seaplane hinders the takeoff during the planing motion.

The contour distributions of the wave profiles around the seaplane at different wavelengths are shown in Figure 27. With an increase in the wavelength, the encountering period between the seaplane and the wave is different, which is further confirmed by the relative position between the wave crests and seaplane. Because the speed of the seaplane is the same for the three wavelengths, the Kelvin angle of the free surface is essentially the same. Owing to the mutual interference of the waves, the discontinuous effect of periodic wave crests and troughs is more pronounced at the free surface behind the fuselage. Additionally, for the shorter wavelengths shown in Figure 27, the interaction of between seaplane and regular wave is more significant.

5.5. Effect of regular wave heights on seaplane in regular wave

In this section, we studied the effect of wave height on the planing stability of a seaplane. The wave heights were 0.05, 0.075 and 0.1 m, respectively. The wavelength and speed are 8 m and 8 m/s, respectively. The wave height does not affect the wave period in deep water. The wave encountering period at a wavelength of 8 m and a planing speed of 8 m/s is 0.69 s.

As shown in Figures 28–30, the motion and force under different wave heights show obvious periodicity. The peak drag of the seaplane in waves increases with wave height, and the periodicity of the heave and pitch of the seaplane at various wave heights is more pronounced. With an increase in the wave height, the amplitude and mean value of the heave and pitch increase. At the same wavelength, the wave steepness increases with increasing wave height, and the wave strength encountered by the seaplane increases.

Figure 28. Drag of seaplane at various wave heights, attack angle of elevator $-{12}^{\circ }$, $L=8$ m, V = 8 m/s. (a) H = 0.05 m, (b) H= 0.075 m and (c) H = 0.1 m.

Figure 29. Heave of seaplane at various wave heights, attack angle of elevator $-{12}^{\circ }$, $L=8$ m, V = 8 m/s. (a) H = 0.05 m, (b) H= 0.075 m and (c) H = 0.1 m.

Figure 30. Pitch of seaplane at various wave heights, attack angle of elevator $-{12}^{\circ }$, $L=8$ m, V = 8 m/s. (a) H = 0.05 m, (b) H= 0.075 m and (c) H = 0.1 m.

The accelerations of the fore, aft and centre of gravity of seaplane in waves at different wave heights are shown in Figure 31. Additionally, the acceleration of the fore, aft, and centre of gravity of the seaplane in waves has significant periodicity. The peak value of acceleration at the aft is greater than that at the centre of gravity, and the peak value of acceleration at the centre of gravity is greater than that at the fore. As the wave height increases, both the peak value and amplitude of the acceleration increase. To enhance the comfort of passengers and pilots, it is advisable to minimize planing in waves, as this helps reduce the range of pitchs.

Figure 31. The acceleration of seaplane at various wave heights, attack angle of elevator $-{12}^{\circ }$, L = 8 m, V = 8 m/s. (a) H = 0.05 m, (b) H = 0.075 m and (c) H = 0.1 m.

The wave profile at different wave heights are shown in Figure 32. It can be observed in Figure 32 that the free surface distribution at different wave heights is obvious. Since the wavelength does not change, the seaplane has the same encountering frenquency in waves. The angles of the Kelvin wake at different wave heights are the same. However, the larger the wave height, the more pronounced the discontinuous effect of the periodic crests and troughs on the free surface. As seen in Figure 32, the larger the wave height, the larger the pitch and heave.

Figure 32. The profile of the wave elevation at various wave heights, attack angle of elevator $-{12}^{\circ }$, L = 8 m, V = 8 m/s. (a) H = 0.05 m, (b) H = 0.075 m and (c) H = 0.1 m.

It is evident that once the seaplane achieves a stable motion relative to the waves, the period of its steady movement at different speeds aligns closely with the encountering period of the waves. The heave and pitch motions of the seaplane remain coordinated with the waves.

6. Conclusions

The high-speed motion of a seaplane involves the coupled hydrodynamic and aerodynamic effects. The motions, pressures, free surfaces, and suctions of the seaplane in waves were numerically investigated. The main conclusions are as follows:

1. The numerical schemes, including a SST-DDES turbulent model, overset mesh method, and volume of fluid (VOF) method, have proven to be effective and accurate for simulating the motion of the seaplane and flow field characteristics.

2. As the planing speed increases, the area of high pressure in front of the step decreases. The drag, heave, and pitch amplitudes of the seaplane gradually increase with increasing speed and wave height. However, it decreases with an increase in wavelength. The air-water entrainment at the bottom of the fuselage appeared behind the step. Futhermore, as the speed increased, the air-water entrainment occurred in front of the step.

3. The greater the speed and wave height, the faster is the motion of the seaplane and the greater are peak and trough values of the pitch. As the wavelength increased, the peak value of the motion decreased gradually.

4. The accelerations of the aft, fore, and centre of gravity of the seaplane in the waves exhibited significant periodicity. The peak value of the acceleration at the aft was the largest, whereas that at the centre of gravity was mild, and the peak value of the acceleration at the centre of gravity was greater than that at the fore. The acceleration indicates that suction at the aft of the seaplane hinders the takeoff during the planing motion.

7. Inadequate and prospects

Although the force and pressure of the seaplane can be accurately predicted, the numerical simulation and experimental motion errors of the seaplane increase at unstable planing speeds such as 5.5 m/s. This imposes more stringent requirements on numerical calculations. From the above research, the HRIC method can be observed to have certain limitations, and the air-water interface capture is still not sufficiently fine. Precise free-surface capture and interaction between water and air are beyond the scope of present Computational Fluid Dynamics (CFD) numerical methods for multiphase flow on seaplane in this study. Further development of new modelling of the numerical method is required for high-fidelity simulations of the free-surface around the seaplane. In the future, geometric VOF methods, such as PLIC and IsoAdvector will be required to simulate the high-speed hydrodynamic motion of the seaplane. Considering the influence of the water interface capture on the seaplane's pitch and heave degree-of-freedom motion, this will be carried out in future work.

Acknowledgments

We would like to express our sincere gratitude to the designers of the seaplane model. Thanks to the seaplane model you have designed, we were able to conduct numerical simulations and experiments, which have provided valuable insights for the optimization and development of seaplanes.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Financial support was provided by the Project of Research and Development Plan in Key Areas of Guangdong Province [grant number 2020B1111010002]; the National Key Research and Development Program of China [grant number 2021 YFC2800700]; the National Natural Science Foundation of China [grant numbers 52171330; 52101379; 52101380; 516790 53]; the Foundation of Key Laboratory of Marine Environmental Survey Technology and Application, Ministry of Natural Resources [grant number MESTA-2021-B010]; the Innovation Group Project of Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) [grant number 311020011], the Guangdong Basic and Applied Basic Research Foundation [grant number 2019A1515110721]; the China Postdoctoral Science Foundation [grant number 2019M663243]; and the Natural Science Foundation of Guangdong Province, China [grant number 2021A1515012134].