1   Introduction

Water exit are common phenomena in marine engineering, encompassing events such as slamming of ships and offshore structures (Faltinsen, 1993; Kaplan, 1987; Baarholm and Faltinsen, 2004), marine lifting operation (De Andrade et al., 2023), sea landings and take-offs of aircraft (Tassin et al., 2013) and sloshing (Faltinsen and Timokha, 2009; Lugni et al., 2014). Research on the water exit problem is critically important in marine engineering. By investigating the phenomena and mechanisms associated with water exit, valuable guidance may be provided for engineering applications, such as: (1) Guiding lifting operation of shipwreck salvage, equipment recovery etc. to prevent sudden structural damage; (2) Instructing the operation of ships and offshore platforms during the water entry and exit process; (3) Optimizing structural design and improving the safety of the structure; (4) Developing cross-media vehicle for ensuring the stability during the water-to-air transition; (5) Enhancing weapon system performance to improve the launch stability and accuracy of submarine-launched weapons.

The research progress on the water exit problems was systematically summarized by Zheng et al. (2024). Water exit problems can be generally divided into two categories based on whether the surface of the object is completely in contact with water during the water exit process: (1) Cavitated water exit; (2) Fully wetted water exit. Cavitated water exit refers to the phenomenon wherein an underwater object passes through the water-air interface enveloped by cavitation bubble. The bubble is generated either by high-speed motion-induced cavitation or through active ventilation. The study can be applied to underwater vehicle ejection and the design optimization of underwater vehicles. Currently, cavity formation and evolution, hydrodynamic characteristics and motion stability are paid much attention to (Shi et al., 2021; Nguyen et al., 2022; Zhang et al., 2024; Zhao et al., 2024; Gao et al., 2025). Fully wetted water exit describes an object remaining completely in contact with water, which moves at low speed without cavitation throughout the entire water exit stage. This phenomenon is relevant to marine lifting operations, aircraft take-off, and seakeeping performance of ships and offshore structures. Research on the fully wetted water exit will help deepen the understanding of the fluid-body interaction mechanism, especially for the free surface deformation and hydrodynamic loads acting on the object during the water exit process, therefore providing valuable guidance on the performance and operation of offshore structures such as offshore platforms, ships, seaplanes and trans-media vehicle etc. To be specific, feasible strategies summarized by the hydrodynamic characteristics of marine vehicle lifting may be employed in the marine lifting operation to reduce the influence of sudden load acting on the object and improve the safety of the operation.

Experimental and numerical methods have been extensively utilized to investigate fully wetted water exit problems, which can be categorized into two aspects considering whether there is external force acting on the body: 1) Free water exit; 2) Forced water exit. Free water exit describes the phenomenon where low-density objects, subject to buoyant force exceeding gravity, rise upward and ultimately exit the water surface. Forced water exit occurs when externally propelled objects traverse the water-air interface under specific kinematic conditions, such as constant velocity, constant acceleration, or defined displacement.

Concerning the experimental study, free surface deformation is attached importance to especially for the free water exit tests (Greenhow and Lin, 1983; Wu et al., 2017). Besides, the pop-up phenomenon and trajectory of a rising buoyant sphere during the free water exit process was investigated by Truscott et al. (2016). With respect to the forced water exit, several objects with typical shape such as cylinder, wedge, disc, sphere, cone, flat plate etc. were adopted to explore the free surface elevation and hydrodynamic force (Miao, 1989; Tveitnes et al., 2008; Vega-Martínez et al., 2019; Breton et al., 2020; Li and Zheng, 2022; Yun et al., 2024). Reis et al. (2010) conducted a physical water exit experiment and applied theoretical analysis to investigate the micro-mechanism of the drinking water process in domestic cats. Tassin et al. (2017) investigated the evolution of the wetted surface of a floating circular disc and a square flat plate during the water exit stage. Han et al. (2024) established an experimental platform to assess the hydrodynamic effects on a Foldable-wing Unmanned Aerial-Underwater Vehicle during the water exit process under varying angles and speeds. And an enhanced formula for predicting the maximum resistance was proposed. However, few of studies compared the experimental results with verified dataset. Moreover, experimental investigation mainly focused on water exit problems with constant velocity, lacking systematical analysis on the hydrodynamic force at low constant accelerations.

Regarding the numerical study, two-dimensional Boundary Element Method (BEM) has been employed to study the free surface deformation and force during forced water exit of horizontal cylinders, as demonstrated by Greenhow (1988), Greenhow and Moyo (1997), Liju et al. (2001), Vinayan and Kinnas (2010), Rajavaheinthan and Greenhow (2015). Ni et al. (2015) adopted a three-dimensional BEM to study the free water exit of a fully-submerged spheroid, focusing on free surface deformation and breakup, as well as force and pressure distributions. Based on the numerical method proposed by Ni et al. (2015), Wu et al. (2018) investigated the free surface breakup and reformation of a fully-submerged spheroid during the forced water exit process. Zhu et al. (2007) employed a Constrained Interpolation Profile (CIP) method coupled with a Finite Difference Method (FDM) to simulate the free surface deformation and hydrodynamic force of a horizontal circular cylinder under the free water exit condition. Volume of Fluid (VOF) method was employed by Zhang et al. (2014) and Huang et al. (2024) to investigate the characteristic of the flow field of a horizontal circular cylinder and a deep-sea mining vehicle respectively. Hao et al. (2019) and Xiao et al. (2022) utilized Lattice Boltzmann Methods (LBM) to investigate the free surface deformation and hydrodynamic force during the forced water exit process. An improved diffuse interface-immersed boundary method was developed by Yan et al (2024) to capture the contact interfaces and free surface deformation of the fluid–structure interaction problems, which was then employed to investigate flow phenomena of a moving sphere during the forced and free water exit process respectively (Yan et al., 2025).

Besides experimental and numerical methods, several theoretical models have been developed for predicting water exit process. Kaplan (1987) used a modified Wagner approach to analyze the water exit stage during two-dimensional flat bottom slamming. This approach was later applied to the study of sea landing of an airplane by Bensch et al. (2001) and to the investigation of wet deck slamming by Baarholm and Faltinsen (2004). Tassin et al. (2013) employed a modified von Karman approach to estimate the hydrodynamic force during the water exit of a two-dimensional body. Korobkin (2013) proposed a linearized model for evaluating the hydrodynamic force during the water exit of a two-dimensional object with a constant acceleration significantly greater than the gravitational acceleration. Korobkin et al. (2017) presented an improved model that accounts for time-varying shapes, arbitrary motions, and nonlinear effects. The current analytical models are primarily derived based on two-dimensional framework, which may not be applicable for asymmetric bodies. In addition, current analytical models may be unable to predict hydrodynamic force of an object at low constant accelerations based on the proposed model hypothesis and its limitation.

Most experimental, numerical, and theoretical studies mentioned above primarily focus on free surface deformation during water exit at constant velocities. In marine engineering applications, hydrodynamic loads associated with water exit at low accelerations are of particular interest, as they relate to marine lifting operations, aircraft takeoff (e.g., seaplanes), and the seakeeping performance of ships. For example, the Nordic Cooperative Project (1987) specifies that the vertical acceleration of ships must be restricted to below 0.2 g to ensure vessel safety and operational performance (where g denotes gravitational acceleration). While several publicly available studies address the water-exit problem under extremely large constant accelerations (nearly several times gravitational acceleration), few investigations have focused on hydrodynamic forces during water exit under low constant accelerations. This research gap motivates the present study.

In this work, the water exit of a floating sphere under low constant accelerations is investigated, with a focus on the vertical hydrodynamic loads. For the water exit of a floating sphere, viscous effects can be safely neglected; however, they play a critical role in the water exit of a fully submerged sphere (Huang et al., 2025). This feature enables the analysis of the floating sphere’s water exit using potential flow theory. We integrate experimental investigations and a theoretical model to examine the vertical hydrodynamic forces acting on the floating sphere during its water exit under low constant accelerations. The results obtained are validated through numerical simulations, which are conducted in Star-CCM+ using the Finite Volume Method (FVM) and Volume of Fluid (VOF) technique. Furthermore, the dynamics of free surface evolution are discussed in detail.

The originality of this work concerns three main aspects: (1) The study of the water-exit of a surface piercing half sphere has been done through a certified experimental dataset, i.e. including an uncertainty assessment analysis; (2) A wide range of prescribed constant accelerations (from low to medium values) has been used to understand the role of the physical phenomena acting on the body and on the variation of the hydrodynamic force as function of the prescribed acceleration, providing a reliable reference for the water exit research, especially for studies related to slamming of ships and offshore structures, seaplane take-off, underwater vehicle ejection and marine equipment recovery. Within the scope of the author’s knowledge, relevant studies are seldom found in the existing literature; (3) A numerical and theoretical model for the hydrodynamic force have been proposed and compared with this particular shape, complementing the physical investigation.

The paper is structured as follows: Section 2 presents the experimental set-up and uncertainty analysis, Section 3 briefly outlines the numerical method, Section 4 describes the theoretical model, Section 5 provides and discusses the results, and Section 6 draws the conclusions.

2   Experimental setup

The experiments were conducted in a rectangular tank at Harbin Engineering University, with dimensions of 2.4 m (length) × 1.0 m (width) × 1.0 m (height). The tank walls are constructed of transparent plexiglass to facilitate observation of the water exit process. The schematic and photographic experimental setup is depicted in Figure 1.

Figure  1  Sketch of the experimental set-up

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In the experiment, the sphere was fabricated by using 3D printing technology with photosensitive resin. It is waterproof and possesses a highly smooth surface achieved through grinding. The sphere has a diameter (D) of 0.280 m. Initially, the draught (d) of the sphere measures 0.140 m, which is equivalent to the radius of the sphere. The corresponding mass of the displaced water (M_d) is 5.747 kg. The mass of the equipment connected to the sphere, M_e, measures 2.827 kg. The mass of the sphere (M_s) is adjustable by changing the weight placed inside the sphere. M_s is set to be 2.920 kg such that the mass of the whole system is equal to the mass of the displaced water. These properties are summarized in Table 1.

During the water exit process, a draw-wire potentiometer (MILONT, MT80-600 mm) is used to measure the vertical displacement of the sphere. Its full-scale range is 600 mm, with the linearity error being 0.15% of the full scale. A capacitive MEMS, triaxial accelerometer (KISTLER 8396A2D0) is used to measure the vertical acceleration of the sphere. Additionally, it is capable of measuring horizontal accelerations, which serves to verify the reliability of the mechanism. The triaxial accelerometer has a range of ±2 g, where g represents the gravitational acceleration. The linearity error of the accelerometer is within 0.3% of the full scale. To measure the vertical loads on the sphere, a force transducer (LZCG, LFS‒35A‒100N) is used. This transducer has a full-scale range of 100 N and an accuracy of 0.05% of the full scale. The positions of the accelerometer and force transducer are illustrated in Figure 2. The triaxial accelerometer is mounted on the side of the vertical beam, enabling it to measure the accelerations of the mechanism system in both vertical and horizontal directions. The force transducer is fixed to an aluminum plate located beneath the vertical beam, positioned close to the sphere to minimize the influence of the entire system. These physical quantities are recorded using a data acquisition device (HOPE, DJ8425) with a maximum sampling frequency of 200 kHz.

Figure  2  Detail of the installation of transducers

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The evolution of the free surface is captured by a high-speed camera (Mega speed, MS90K), boasting a maximum resolution of 1 280 × 850 pixels. At this maximum resolution, the camera is capable of recording at 1 000 frames per second. The high-speed camera is seamlessly triggered by the data acquisition device, facilitating precise control over the timing of image capture and ensuring consistency in the experimental data.

A series of low constant accelerations were selected to evaluate the hydrodynamic force acting on the sphere, with the value being as follows: 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00 m/s². The sampling frequency of the data acquisition device was set to be 10 000 Hz in the water exit experiments. Meanwhile the frame rate of the high-speed camera is set to be f_ps = 500 frames per second.

Before the experiment stage, preliminary tests should be conducted to verify the reliability of the mechanism system. For tests involving constant acceleration, it is crucial that the movement throughout the entire process is smooth enough to prevent any sharp jumps in the measured data. One effective approach is to divide the process into four distinct stages. Initially, in Stage Ⅰ, the body is at rest. During Stage Ⅱ, it moves upward at a constant acceleration. At the conclusion of Stage Ⅱ, the body reaches its maximum speed. In Stage Ⅲ, the speed is gradually reduced to zero through constant deceleration. Lastly, in Stage Ⅳ, the mechanism system stops and the body returns to a resting state.

Based on the experimental conditions, the maximum motion distance allowed by the belt module has been determined to be approximately 400 mm. Taking the diameter of the sphere and the entire movement process into account, the constant acceleration distance was set at 300 mm, while the constant deceleration distance was set at 80 mm.

The motion of the mechanism can be expressed as follows

where, Z denotes the vertical displacement, Z₁ represents the acceleration distance, Z₂ is the total distance, a₁ is the acceleration, a₂ is the deceleration, v₁ is the maximum speed at the ending time of Stage Ⅱ, t_i (i = Ⅰ, Ⅱ, Ⅲ) represents the ending time of different stages of the whole process. a₁, Z₁, Z₂ are given values, with Z₁ = 0.30 m and Z₂ = 0.38 m. t_Ⅰ is obtained by the data acquisition device. a₂ can be calculated by $a_{\mathrm{2}}=-Z_{\mathrm{1}}\cdot a_{\mathrm{1}}/(Z_{\mathrm{2}}-Z_{\mathrm{1}})$ and the maximum velocity $v_{\mathrm{1}}=\sqrt[]{\mathrm{2}{a}_{\mathrm{1}}{Z}_{\mathrm{1}}}$. The durations of stage Ⅱ and stage Ⅲ can be calculated using t_Ⅱ = $v_{\mathrm{1}}/a_{\mathrm{1}}$ and t_Ⅲ = $-v_{\mathrm{1}}/a_{\mathrm{2}}$.

Repetitive tests were conducted to validate the experiment. Each series of these repetitive experiments consisted of 20 trials conducted under identical conditions. The measurements of displacement, acceleration, and force obtained from the transducers, along with other relevant variables, were analyzed for uncertainty using the root-sum-square model (Abernethy et al., 1985), which satisfies,

where $U_{\mathrm{R}\mathrm{S}\mathrm{S}}$ is the measurement uncertainty; B_R refers to the total bias uncertainty, t represents the t-statistic from Student’s t-distribution corresponding to a 95 percent confidence level (denoted by the subscript), N denotes the number of repeated tests, S indicates the biased standard deviation, which is calculated using the following equation

Here, X_k represents the individual data and $\overline{X}$ indicates the average value of the measurements.

The total bias uncertainty is determined by taking the root-sum-square of the individual elemental bias uncertainties, based on the functional relationship between the measurement result and its independent parameters

Repetitive tests were carried out to verify the vertical displacement of the sphere. As an illustrative example, we will consider a test with a constant acceleration of a = 2.00 m/s². A comparison of the vertical displacement is shown in Figure 3. The error bar of the raw experimental data displayed in Figure 3 represents only a portion of the processed data, with a gap of 300 data points. The displacement measured by the potentiometer is in good agreement with the theoretical value, indicating that the actual motion of the body closely aligns with the expected motion.

Figure  3  Comparison of the vertical displacement with a constant acceleration of a = 2.00 m/s²

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Figure 4 presents the vertical force component acting on the model. $F_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}$ represents the force acting on the force transducer. By applying Newton’s third law, we can deduce that the force acting on the sphere, $F_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}^{’}$, is equal to -$F_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}$. During the water exit stage, the sphere is influenced by the hydrodynamic force, $F_H$, which is assumed to be positive and acting upward. Additionally, the sphere is also subject to the forces of gravity and inertia.

Figure  4  Schematic diagram of vertical force components on the sphere

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For a sphere of mass M, the equilibrium equation can be expressed as

Then the hydrodynamic force on the sphere is

where a in Eq. (6) adopts the acceleration measured by the experiment.

It is noted that the computation of the hydrodynamic force was based on the raw data. Specifically, only the accelerometer signals underwent filtering, where high frequency components were eliminated using Empirical Mode Decomposition technology first proposed by Huang et al. (1998). Furthermore, the resultant hydrodynamic force curve was subjected to post-processing through the application of a Convolution filter and a Gaussian-weighted moving average filter. For a comprehensive understanding of the process, details are provided in Appendix 1.

The high-speed camera was synchronized with the data acquisition device using a trigger signal. This signal initiated the camera at time t_c and was also recorded by the data acquisition device. The frame number is defined as $N_i(i=\mathrm{1},\mathrm{2},\cdots ,m)$ and the real-time corresponding to each frame is given by $t_i=t_c+(N_i-N_{\mathrm{1}})/f_{\mathrm{p}\mathrm{s}}$.

According to Figure 5, the body started to move upward vertically. The moving distance of the body is described as Z and c(t) denotes the minimum radius of the water column. R is the radius of the sphere and the blue solid line indicates the instantaneous free surface.

Figure  5  Sketch of the water exit

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3   Numerical method

The CFD method is used to investigate the water exit problem by employing the software Star-CCM+. In this study, Reynolds-Averaged Navier-Stokes (RANS) method is adopted to solve the Navier-Stokes equation. The SST k-ω model, proposed by Menter (1994), is selected as the eddy viscosity model for determining the turbulent viscosity. The VOF method is utilized to capture the interface between the water and the air. Additionally, the Finite Volume Method (FVM) is implemented in Star-CCM+ to discretize the governing equations in integral form. By utilizing the SIMPLE algorithm, the pressure-velocity coupling equations can be decoupled and solved iteratively.

The numerical settings are consistent with the experimental layout. The background domain has dimensions of 2.4 m (length) × 1.0 m (width) × 1.0 m (depth), with a liquid domain depth of 0.37 m. The geometric scene and boundary conditions are shown in Figure 6. The Overset Grids Method is applied in dealing with complex shape grid layout and motion of bodies, which can accelerate simulation and improve model precision. To improve the numerical results, fine meshes are adopted in the overset domain, free surface zone and motion region, which is illustrated in Figure 7.

Figure  6  Geometric scene and boundary conditions

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Figure  7  Sectional mesh scene (Grid number N = 2.6 million)

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In the numerical simulations, the vertical velocity of the body is prescribed according to the experimental data. This velocity is derived by differentiating the vertical displacement of the body, which is measured by the potentiometer, and any noise in the data is subsequently removed using the Savitzky-Golay filter. To ensure numerical reliability, a grid convergence study is conducted by employing four distinct grid resolutions: coarse (1.8 million cells), medium (2.6 million cells), fine (3.6 million cells) and very fine (5.1 million cells). The grid refinement ratio between successive resolutions is maintained at approximately $\sqrt[]{\mathrm{2}}$, following best practices for convergence analysis. For instance, when considering a constant acceleration of 1.00 m/s², all other variables are held constant except for the grid number. Figure 8 shows the numerical hydrodynamic force and the absolute error when compared with the results obtained under the finest grid (N = 5.1 million).

Figure  8  Grid convergence verification (a = 1.00 m/s²)

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The results demonstrate excellent agreement throughout the entire process of water exit, conclusively verifying the convergence of hydrodynamic force curves. With a negligible absolute error observed under 2.6 million grid resolution, an optimal balance between computational efficiency and solution accuracy was achieved. Consequently, the medium grid (N = 2.6 million) was selected as the baseline for subsequent simulations.

Figure 9 illustrates the components of the vertical hydrodynamic forces during the water exit process. The vertical force is primarily governed by the pressure force, with the shear force being so small that it can be safely ignored. Given the short duration of the water-exit process and the absence of sharp corners on the wetted surface of the body, it is reasonable to assume that viscous flow separation does not occur and that vorticity is negligible. Consequently, the potential theory is applicable for describing the water-exit process and predicting the vertical hydrodynamic force.

Figure  9  Hydrodynamic component during water exit (a = 1.00 m/s²)

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4   Theoretical model

Neglecting viscous effects and assuming that the flow is irrotational and the water is incompressible, the water flow can be well represented by the potential-flow model (Wang et al., 2019). The velocity potential satisfying Laplace’s equation

is introduced. Consider a three-dimensional fluid domain $\mathrm{\Omega }$ illustrated in Figure 10, where $S_B$ denotes the wetted body surface, $S_F$ the free surface, $S_{\mathrm{\infty }}$ the fixed vertical surface far away from the body, and $S_{\mathrm{0}}$ the sea bottom. On the body surface, the normal velocity of the fluid particles is equal to the normal velocity of the boundary. On the free surface, the water pressure becomes the atmospheric pressure by neglecting the surface tension, the normal velocity of the boundary is the normal velocity of the fluid particles, and the high-frequency free-surface boundary condition, $\varphi =\mathrm{0}$, is applied. Since $S_{\mathrm{\infty }}$ is fixed, the normal velocity of the boundary vanishes. On the sea bottom, both the normal velocity of the boundary and the normal velocity of the fluid particles are zero.

Figure  10  Sketch of the three-dimensional fluid domain and boundary conditions. U_n is the normal velocity of the boundary and V_n is the normal velocity of the fluid particles on the boundary.

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The momentum of the fluid inside $\Omega$ is written as

By using Gauss’s theorem, then we have

Applying the transport theorem, the rate of change of the fluid momentum can be expressed as

It satisfies $U_n=V_n$ on $S_B$, thus $\vec{U}\cdot \vec{n}=\partial \varphi /\partial n$. Based on Eq. (10) and the Bernoulli’s equation, the force acting on the body can be expressed as

It can be shown that the last term on the right-hand side of Eq. (11) is approximately zero, which can be safely neglected (see Appendix 2 for details). The force acting on the body becomes

Therefore, the vertical force F₃ is

In Eq. (13), the first integral represents the high-frequency added mass in the vertical direction multiplied by the velocity of the body, while the second integral represents the buoyancy of the body. They can be written as

and

Here, $A_{\mathrm{33}}$ denotes the high-frequency added mass in the vertical direction and $\nabla$ the displaced volume of the sphere. Then, the vertical force is expressed as

The first term on the right-hand side of Eq. (16) is the added mass force. The second term is related to the time change rate of the wetted body surface. In water-entry problems, it is called ‘slamming force’ (Faltinsen, 1993). For the water-exit problem, it may not be important and can be neglected. Then, the vertical force is simplified into

$\mathrm{d}V/\mathrm{d}t$ is exactly the acceleration of the sphere, which is prescribed in the present study. Referring to Figure 11, the displaced volume of the sphere can be approximated as

Figure  11  Estimation of the displaced volume of the sphere

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Here, the free-surface elevation is neglected, which corresponds to the von Karman’s assumption. Furthermore, the vertical added mass of the sphere is approximated by that of a disc with radius s(t), as discussed by Lamb (1906). This approximation yields

It should be noted that Eq. (19) is not valid for small value of Z/R where the vertical added mass satisfies the hemisphere approximation: $A_{\mathrm{33}}\approx \frac{\mathrm{\pi }}{\mathrm{3}}\rho s^{\mathrm{3}}(t)$. The nondimensional coefficient of the vertical added mass in this case is relatively close to that derived from the disc approximation. Thus, retaining the disc-based added mass formulae introduces negligible errors even for small Z/R.

5   Results

The water exit process was recorded by the high-speed camera, which is shown in Figure 12.

Figure  12  Images of water exit

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Compared with images featuring various accelerations, the disturbance of the free surface is small when Z/R < 1. Deformation of the free surface occurs in the vicinity of the contact line between the free surface and the body surface, as discussed in detail by Breton et al. (2020). When Z/R > 1, the water column gradually forms and ultimately collapses under the interaction of inertia force and gravity. With the non-dimensional moving distance fixed, Figure 13 compares images from tests conducted with various accelerations.

Figure  13  Comparison of images with different accelerations

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When Z/R > 1, water adhered to the sphere begins to form a water column due to the combined effects of inertia entrainment and gravity. As the acceleration increases under the same moving distance, the volume of the adhered water grows. Inertia entrainment drives the adhered water upwards, elongating the water column, while gravity pulls the water downwards, causing the column to thin and forming a distinct boundary between the water column and the free surface.

The experimental evolution of c(t)/R as a function of the non-dimensional elevation Z/R under varying acceleration conditions is depicted in Figure 14. It is observed that c(t)/R decreases more rapidly with lower accelerations when Z/R > 1.1. This behavior results from the interplay between inertia force and gravity. At lower accelerations, gravity becomes a more dominant factor, leading to a significantly smaller volume of adhered water. Due to the greater influence of gravity and the reduced effect of inertia force, the water column is more prone to collapse.

Figure  14  Evolution of c(t)/R as a function of the non-dimensional elevation Z/R

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Figure 15 presents a comparison of the vertical hydrodynamic forces for several accelerations, integrating experimental data, numerical results, and theoretical predictions. The remarkable agreement among these different approaches underscores the high level of accuracy and consistency in the analysis. The theoretical vertical force, calculated as the sum of buoyancy and the added mass force as described by Eq. (17), provides an accurate prediction of the hydrodynamic force during the water exit process. It is important to note that wave excitation forces must be accounted for in the theoretical formulae when waves are present.

Figure  15  Comparison of hydrodynamic force

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It is instructive to compare the individual components of buoyancy and the added mass force. Using a constant acceleration of a = 2.00 m/s² as an illustrative example, Figure 16 presents a detailed comparison of buoyancy, the added mass force, and the total force. Notably, during the early stages of the water exit process, buoyancy is the dominant force. It is because the acceleration is much less than the gravitational acceleration, and the vertical added mass is of the same order as the mass of the displaced water.

Figure  16  Comparison of force components (a = 2.00 m/s²)

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6   Conclusions

The water exit of a floating sphere under low, constant acceleration has been investigated. This study focuses on the evolution of the free surface and the vertical hydrodynamic force exerted on the sphere. A detailed and accurate mechatronic system was developed to impose the input motion, allowing for the execution of the experiments. These experiments utilized a validated experimental dataset that includes an uncertainty assessment analysis. Additionally, both numerical and theoretical models for the hydrodynamic forces associated with this particular spherical shape have been proposed and compared, thereby complementing the physical investigation.

Experiments were conducted in a transparent plexiglass tank, with a high-speed camera employed to capture the free surface deformation during the water exit process. Before the sphere’s bottom detaches from the mean water level, free surface deformation occurs primarily near the contact line between the sphere’s surface and the water. However, the overall deformation remains minimal during this phase. Once the sphere’s bottom lifts off the mean free surface, a layer of water adheres to the sphere and is carried upward due to inertia, potentially forming a water column beneath the sphere. Under the influence of gravity, this water column elongates and narrows, eventually leading to its collapse. A higher acceleration leads to an increase in the volume of the water column, consequently resulting in a larger minimum radius of the column. An accelerometer and a force transducer were used to measure the vertical acceleration and force, respectively. The hydrodynamic force was then calculated using Newton’s second law. To validate and verify the experimental hydrodynamic loads, numerical simulations were performed using the Finite Volume Method combined with the Volume of Fluid technique in the software Star-CCM+. The numerical results show good agreement with the experimental data throughout the entire water exit process. The results demonstrate that the vertical force is primarily governed by the pressure force, with the shear force being negligible and therefore safely ignored. Due to the short duration of the water exit process and the absence of sharp corners on the wetted surface of the sphere, it is assumed that viscous flow separation does not occur and that vorticity is negligible. Consequently, potential flow theory is employed to describe the water exit process and predict the vertical hydrodynamic force. Based on the conservation of fluid momentum, a theoretical expression for the vertical hydrodynamic force has been derived. The theoretical force, expressed as the sum of buoyancy and the added mass force, provides an accurate prediction of the hydrodynamic force during the water exit process. The buoyancy dominates during the early stages of the water exit process when the acceleration is significantly less than gravitational acceleration. The present study neglects viscous effects, which play an important role in the water exit of fully submerged bodies or bodies with wetted sharp corners. In such cases, viscous drag forces must be included in the theoretical force model. Furthermore, when waves are present, wave excitation forces should also be incorporated into the theoretical force formulation.

Appendix 1

In this appendix, several post-processing techniques for data denoising, such as Empirical Mode Decomposition (EMD) analysis and the Gaussian-weighted moving average filter, were applied to the measured data and hydrodynamic forces. The results are presented as follows.

Empirical Mode Decomposition (EMD), introduced by Huang et al. (1998), is a powerful tool for extracting useful information in the field of signal denoising. It adaptively decomposes a complex, multiscale signal into a series of Intrinsic Mode Functions (IMFs), expressed as:

where x(t) represents the original signal, m denotes the number of IMFs, and r_m(t) is the residual component. The instantaneous frequency and amplitude of the signal can then be determined using the Hilbert transform. By selectively removing certain IMF components, the associated frequencies and energy distributions can be identified, enabling effective noise reduction and signal analysis.

Regarding the water exit tests conducted with a constant acceleration of a = 2.00 m/s², the EMD method and Hilbert transform were applied to the mean value of the vertical acceleration obtained from repetitive tests. The instantaneous frequencies of the first five modes of the IMFs are illustrated in Figure 17.

Figure  17  Instantaneous frequency of the first five modes of IMFs (a = 2.00 m/s²)

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Based on the frequency response of the triaxial accelerometer, the minimum frequency range within which the measurement error remains below ±5% is between 0 and 250 Hz. The instantaneous frequencies of the first two modes of IMFs in Figure 17 exceed this range, suggesting that the results of these two IMF modes are unreliable. However, removing the first two IMFs, which contain high-frequency components, has minimal impact on the overall characteristics of the signal. After eliminating the first two IMFs from the raw data measured by the triaxial accelerometer, the acceleration curve still exhibits significant oscillations (represented by the blue dotted line), as shown in Figure 18. Therefore, it is necessary to apply additional filtering methods to remove high-frequency signals.

Figure  18  Comparison of raw and filtered data of vertical acceleration (a = 2.00 m/s²)

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Two filtering techniques were employed for denoising: the EMD filter and the Moving Average filter. The first two IMFs were identified as noise using Fast Fourier Transform (FFT) and were subsequently removed using the EMD filter (represented by the red dotted line). Additionally, the Moving Average filter was applied with a window length of 500, and the resulting filtered signal is depicted by the black solid line in Figure 18.

The measured horizontal accelerations in horizontal plane are plotted in Figure 19, with accelerations in the X direction and Y direction shown separately. The red dotted line represents the raw data after removing the first two IMFs, while the black solid line indicates the data filtered using the Moving Average filter.

Figure  19  Measured accelerations of the mechanism in horizontal plane

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The raw data exhibits strong oscillations near the zero value in both X and Y directions. After applying the Moving Average filter, the amplitude of the acceleration time-history curve becomes significantly smaller, almost negligible. This demonstrates the effectiveness of the mechanism system in minimizing accelerations in the X and Y directions.

For the vertical hydrodynamic force, both the Convolutional filter and the Gaussian filter were employed to remove noise from the signals. The Convolutional filter was applied with a window length of 100 for light filtering, while the Gaussian filter was set with a window length of 1 000 for more aggressive filtering. A comparison of the raw and filtered hydrodynamic force data is presented in Figure 20.

Figure  20  Comparison of raw and filtered data of hydrodynamic force (a = 2.00 m/s²)

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Appendix 2

Regarding the final term on the right-hand side of Eq. (11), it can be shown to vanish to zero through the following proof.

For the enclosed surface, the formulae is presented as follows (Newman, 2017).

Thus, the integration over S_B can be transformed into an integration over other control surfaces,

Regarding the integration over $S_{\mathrm{\infty }}$ in Eq. (22), the velocity induced by the body decays at least as fast as r^-3 at large distances. Since the integrand is proportional to r^-6 and the surface element dS is proportional to r², the contribution from the integration over $S_{\mathrm{\infty }}$ can be safely neglected. It can be shown that the integration over S₀ in Eq. (22) is estimated to decay with an order of h^-4 (h denotes the water depth) and can be safely ignored for the present case.

Because of $U_n=V_n$ on $S_F$, $\vec{U}\cdot \vec{n}=\partial \varphi /\partial n$. Based on Eq. (10), we have

Then the integration over S_B can be transformed into an integration over S_F

with the high-frequency free-surface boundary condition, $\varphi =\mathrm{0}$, applied on S_F, the integration of the first term in Eq. (23) approaches to zero. By using the Bernoulli’s equation,

the last two terms on the right-hand side of Eq. (24) can be expressed as

Note that it satisfies p = p_a on S_F, and the integration term representing buoyancy on S_F approaches to zero. Therefore, the integration in Eq. (22) vanishes, i.e.