Analysis of Wave Load Characteristics of Hovercraft Based on Model Test

Abstract

The prediction of the wave load on a hovercraft is essential for the design of the hull structure and safety. However, theoretical methods for the prediction of wave loads are still not mature enough due to the unique and complex nature of the air cushion structure, and numerical modeling and simulation are challenging due to the complexity of the gas-solid-liquid three-phase coupling, so the study of wave loads on hovercrafts still relies on experimentation. In this study, we aim to analyze the wave load response characteristics of a four-chamber hovercraft by conducting a wave load model test under medium/low sea states. The load components and amplitude-frequency response characteristics were thoroughly analyzed based on the acquired data of the cushion pressure, acceleration, and bending moment. The main characteristics of the wave-induced response of the hovercraft were described in detail, and an analytical relationship between the cushion pressure and hull acceleration was derived. The reliability of the experimental results was confirmed through a comparison with the derived results. The relationship between the cushion pressure and cushion volume was investigated in terms of the observed geometric volume of the air chamber, and the relationship between the cushion pressure and flow rate was analyzed to validate the derivation of the theory of wave loads on hovercrafts.

1. Introduction

A fully cushioned hovercraft (hereinafter referred to as “hovercraft”) is a water surface vehicle that uses air cushions as the support to hover over water. Owing to their high speed and low resistance, hovercrafts have become an important type of vessel for ocean exploration, maritime rescue, and military operations [1]. It is an irreplaceable and important piece of equipment for water surface transportation. Its working principle is to use the lift fan on the vessel to generate airflow that passes through the air ducts and flexible annular nozzles (referred to as “fingers”) around the bottom of the flexible skirt. This airflow is then directed into the air chambers enclosed by the lower deck, the skirt, and the water surface. The presence of the skirt structure means that the motion and wave load of the hovercraft depend significantly on the response properties of the craft, which differ markedly from traditional surface displacement vessels and require further study.

The prediction of wave loads on hovercrafts is critical for structure design and safety, but the complexities arising from air-solid-liquid interactions make developing numerical methods for estimating motion and wave loads challenging. Even when using scale models for testing, the number of sensors available for measuring physical quantities is severely limited due to the weight and distribution restrictions of the models. Furthermore, the continuous variations in load and response within this complex system make it difficult to observe key physical quantities such as air chamber flow rates and flow coefficients [2]. Given the limited number of sensors available, appropriate methodologies must be employed to ensure the accuracy and reliability of the measurements.

Significant advancements have been made in the study of wave loads on hovercrafts, particularly in analytical formulations and numerical simulations. In the 1970s, Reynolds [3,4] paved the foundation for the dynamic modeling of a hovercraft with linear motion equations. Based on Reynolds’ work, Doctors et al. [5] extended the motion theory from the frequency domain to the time domain, which significantly facilitated the derivation of nonlinear differential equations describing hovercraft motion in waves. This advancement enabled the description of hovercraft motion at any given time and, for the first time, accounted for the nonlinear effects of the fan, air duct, and skirt.

In recent decades, numerical simulation techniques, particularly Computational Fluid Dynamics (CFD), have significantly enhanced the accuracy in predicting hovercraft behavior in complex marine environments. Milewski et al. [6,7] developed the ACVSIM program based on the Immersed Boundary Method (IBM) to provide numerical support for the U.S. Navy’s LCAC hovercraft or landing craft. Xu et al. [8] conducted a numerical study on the pressure response characteristics of hovercraft skirt cushion systems, particularly those with finger-type skirts, proposing a method for analyzing these characteristics and the influence of the skirt’s geometric parameters and cushion parameters on pressure response. Jiang et al. [9,10], building upon the numerical model of automotive safety airbags, introduced a numerical simulation method by combining the Control Volume (CV) method and the Arbitrary Lagrangian-Eulerian (ALE) method to analyze the response of airbags under various factors, such as entry speed, loading mass, material properties, and inclination angle. Gu et al. [11] performed an in-depth analysis of the air cushion wave-making phenomenon of Air Cushion Vehicles (ACVs) under different sailing conditions and its impact on the motion characteristics of the vessel, including investigations into the air cushion wave-making phenomenon, numerical calculation methods, and dynamic coupling models. Petoft et al. [12] studied the influence of geometric shapes on the lift force of hovercraft and proposed a new formula for estimating air cushion lift based on parameters such as air gap, side gap, intake velocity, and scaling factor.

In almost all existing methods for predicting wave loads on fully cushioned hovercrafts, simplified models are employed for the air cushion structure to reduce complexity. For instance, the fan characteristic formula used to describe the relationship between the cushion pressure and flow rate within the air chamber ignores the dynamic response characteristics of the air cushion interacting with the water and the flexible skirt, making it valid only under certain low-sea-state conditions. Numerical simulations face significant challenges in accurately modeling the gas-solid-liquid coupling dynamics, and strong assumptions and simplifications are typically made in existing numerical studies.

In the past two decades, several experimental studies have been conducted to investigate the behavior of hovercrafts. Pinkster et al. [13] used a 1:200 scale model to examine the behavior of an air-supported mobile offshore base and found that the mid-ship bending moment was reduced due to the air cushion effect. Ricci et al. [14] performed an experimental comparative study on the seakeeping performance of four types of skirts on the US LCAC hovercraft; they obtained and analyzed the amplitude-frequency responses of the LCAC in regular waves [6,7]. Zhang et al. [15] conducted segmented ship model tests on a four-chamber hovercraft to determine the wave load characteristics and cushion pressure response characteristics. Liu et al. [16] performed segmented seakeeping model tests on a similar hovercraft, measuring the motion and wave load response in both regular and irregular waves, and conducted a structural strength assessment based on the wave loads. Xu et al. [17] analyzed the calculation method of skirt configuration and its impact on the static hovering characteristics on land through segmented skirt configuration tests and static hovering tests with a hovercraft scale model. These studies primarily focused on the motion of the hovercraft in waves and the wave load response. However, since the dynamic flow in the air chamber is difficult to observe, the relationship between the cushion pressure and flow rate in the air chamber remains understudied and unclear.

Model-test-based methods remain the most reliable approach to study the wave load characteristics of hovercrafts. In this study, a comprehensive analysis of the hovercraft cushion pressure, acceleration, motion, and cross-sectional bending moment responses is carried out based on model test data. A kinematic model of a hovercraft is established, and physical quantities are deduced from the test data to validate the accuracy and reliability of the measurements. Through an analysis of the experimental data and numerical calculations, the relationship between the cushion pressure and cushion volume is investigated.

2. Hovercraft Test and Kinematic Model

In actual waves, the motion of a hovercraft is influenced by external factors such as waves and wind, as well as its own weight distribution, geometric dimensions, and control system. The primary motion characteristics of a hovercraft in waves include the following:

(1)

Heave: The hovercraft experiences vertical motion due to fluctuating cushion pressure, which forces it to rise and fall vertically.

(2)

Pitch: The hovercraft undergoes pitching motion in waves, which is caused by changes in the front and rear air chambers and the impact force when the skirt contacts the water.

Waves induce pressure fluctuations within the hovercraft’s chambers, as illustrated in Figure 1. These fluctuations lead to vertical motions and changes in the leakage flow rate and cushion volume.

2.1. Seakeeping Model Test

2.1.1. Scale Model

To study the wave load response characteristics of a hovercraft in a wave environment, a four-chamber hovercraft test model was fabricated. The key parameters of the hovercraft model are shown in

Table 1. The key parameters of the hovercraft test model.

Parameter	Value
Cushion length LC (front chamber length and rear chamber length: l1 + l2)	2.83 m (1.44 m + 1.39 m)
Width of hovercraft BC (same as cushion beam)	1.22 m
Flexible skirt height	0.14 m
Mass M	92.9 kg
Pitch moment of inertia Iy	56.9 kg × m2
Longitudinal position of center of gravity (after mid-ship)	0.114 m

.

The test model features a longitudinal segmented structure design, as shown in Figure 2. Taking into account the stringent requirements on the strength and weight of the materials for fabricating the model, composite carbon fiber plates were used in most of the structure members of the model. Three-dimensional printing technology was used to fabricate the model to reduce the total weight.

Special attention was paid so as to ensure the segmentation cross sections were not located at the positions of fans and air ducts. Based on the weight distribution of the entire hovercraft as well as the structural integrity of the model, four longitudinal segments were used to constitute the entire model. The segmentation cross sections are located at L_C/4 (stern), 2L_C/4 (mid-ship), and 3L_C/4 (bow). A 10 mm gap was ensured between the neighboring segments. The segments were connected using a backbone to form the scale model. Strain gauges were installed on the backbone to measure the strains at the segmentation cross sections. Thin elastic rubber strips were used to seal the gaps between segments for the purpose of water tightness.

For the hovercraft tank test, the determination of the scale ratio of the hovercraft is constrained by factors such as the wave-making capacity of the tank, the towing speed, the deployment of sensors, etc. An overly small model size may not ensure the rationality of the hovercraft’s mass distribution, as the hull beam, air channel, fans, skirt, and various sensors are arranged on the hull, while an overly large model size puts higher demands on the wave-making and towing speed capabilities of the test water tank, which is particularly important for high-speed hovercrafts.

The skirt, shown in Figure 3, is made of flexible fabric with a weight-to-area ratio of $0.2128 {\mathrm{kg}/\mathrm{m}}^2$. Vent holes are fabricated on the bag of the skirt, and two partitioning skirts are attached to form four chambers. Four axial flow lift fans are used to pump air into the air cushion, as shown in Figure 4, and each of them weighs 3.0 kg. The formula used to fit the fan’s theoretical characteristic curve takes the following form:

$$P=-4275.430Q^2+476.649Q+540.595,$$

(1)

where P is the fan pressure head (Pa), and Q is the fan flow rate (m³/s).

2.1.2. Test Cases

After the calibration of the sensors, a series of wave model tests for the hovercraft are carried out under medium/low wave conditions. The wave height-to-cushion length ratio is 2ζ_w/L_C = 0.025, and the other parameters of the model test are shown in

Table 2. Test conditions.

Quantity	Value
Towing speed (m/s)	4.8
Wave amplitude ζw (mm)	81
Wave frequency (Hz)	0.385~1.00
Wave direction (deg)	0 (head wave)

. The vessel motion, cushion pressure, acceleration, and cross-section bending moment time series are measured and analyzed. Among these parameters, acceleration does not only have an immediate influence on the passenger comfort, but is also a significant contributing factor to the wave load (inertial load). However, only the influence of the latter is addressed in this study as the wave load is the main concern in the design of the hull structures.

2.1.3. Possible Error Sources and Experimental Uncertainty

In this section, the possible error sources and uncertainties associated with the experimental procedures and results are discussed.

(1)

Environmental conditions

Environmental conditions, such as the water temperature and wave loads, may affect the results of the experiments. The tests were conducted at room temperature, and its influence is negligible in wave load tests. The wave height is monitored by wave gauges, and the measurements of the wave height are used for dimensionless calculation. Because the hovercraft is a type of high-speed vessel, the influence of the banks of the towing tank does not exit [18].

(2)

Data acquisition and processing

All of the sensors were calibrated before the test, and the data acquisition methods including the sampling rate were determined appropriately to capture all of the extrema. The algorithms used for data filtering and analysis, such as the Fast Fourier Transform tool, have been extensively validated. Detailed information can be found in [16].

(3)

Reproducibility of experimental results

In order to obtain trust-worthy experimental results, model tests were carried out using the same model at two tanks. A comparison of the results was carried out for the selected number of cases, and it was found that the results obtained at the two tanks are consistent for low and medium sea states. For medium and higher sea states, discrepancies were found between the results, which might have been caused by differences in the large wave amplitude that the wave makers can generate. Due to the significant nonlinear dynamic behavior under high sea states and the limited capability of the wave makers in the towing tanks, this study will focus on medium and low sea states.

2.2. Theoretical Model of Kinematics

In waves of small heights, the hovercraft motion response is relatively mild, and the nonlinear motion and wave load components are insignificant [19]. The pressure in the air chamber can be regarded as uniformly distributed, and the water contact force on the skirt is small compared to the cushion pressure. According to the three-dimensional perturbation theory of hovercrafts proposed by Reynolds in 1972 [3,4], the air chamber can be simplified into a rectangular box, and the static cushion pressure in the air chamber can then be solved using the static equilibrium equation of the hovercraft. According to the dynamic equilibrium equation of the hovercraft, the (angular) acceleration of the hull can be obtained from the known air chamber’s pulsating pressure (see Equation (5)), and the motion can be obtained from the time integration. By using the d’Alembert principle, the cross-section load can be obtained. Similarly, if the (angular) acceleration of the hull is known, the air chamber’s pulsating pressure and the cross-section load of the two- and three-chamber hovercrafts can also be obtained (see Figure 5).

In this study, the static and dynamic equilibrium equations are derived for a four-chamber hovercraft.

2.2.1. Static Equilibrium Equation

By placing the origin at the center of gravity of the hovercraft, a body-fixed coordinate system, G-X_GY_GZ_G, is defined, as shown in Figure 6. The static equilibrium equation [3,4] of the hovercraft is used to solve the static cushion pressure in the air chambers as follows:

$$A_N{P}_{\mathrm{static}}=W_M,$$

(2)

where $A_{{\rm N}}$ is the air cushion geometry matrix, $P_{\mathrm{static}}$ is the static cushion pressure vector of the air chambers, and $W_M$ is the initial weight vector. For a four-chamber hovercraft, the specific forms of the above matrices and vectors are as follows:

$$A_N=\left[\begin{array}{cccc}{A}_{C11}& A_{C12}& A_{C21}& A_{C22}\\ -A_{C11}{x}_{c11}& -A_{C12}{x}_{c12}& -A_{C21}{x}_{c21}& -A_{C22}{x}_{c22}\\ A_{C11}{y}_{c11}& A_{C12}{y}_{c12}& A_{C21}{y}_{c21}& A_{C22}{y}_{c22}\end{array}\right],$$

(3)

$$P_{\mathrm{static}}=\left[\begin{array}{l}{P}_{s11}\\ P_{s12}\\ P_{s21}\\ P_{s22}\end{array}\right], W_M=\left[\begin{array}{c}Mg\\ -M_y\\ -M_x\end{array}\right],$$

(4)

where $M$ is the mass of the hovercraft; $M_y, M_x$ are the initial pitch moment and the initial roll moment, respectively; $g$ is the gravity acceleration; $A_{Cij}$ is the area of the air chamber; $P_{sij}$ is the static cushion pressure, and $x_{cij}, y_{cij}$ are the abscissa and ordinate of centroid of each air chamber (the definitions of the subscripts $ij$ are given in Figure 6).

2.2.2. Dynamic Equilibrium Equation

According to the (angular) acceleration, the total external force (moment) of the three degrees of freedom can be obtained, and thus, the pulsating pressure in the air chamber and the (angular) acceleration satisfy the following hovercraft dynamic equilibrium equation [3,4]:

$$A_N{P}_{\mathrm{dynamic}}=W_I\left[\begin{array}{l}{\alpha }_{\mathrm{heave}}\\ {\alpha }_{\mathrm{pitch}}\\ {\alpha }_{\mathrm{roll}}\end{array}\right],$$

(5)

where $W_I$ is the mass matrix; $P_{\mathrm{dynamic}}$ is the pulsating cushion pressure vector of the air chambers; and ${\alpha }_{\mathrm{heave}}, {\alpha }_{\mathrm{pitch}}, {\alpha }_{\mathrm{roll}}$ are the (angular) acceleration of the heave, pitch, and roll motions, respectively. The specific forms of the above matrices and vectors for a four-chamber hovercraft are as follows:

$$W_I=\left[\begin{array}{ccc}M& 0& 0\\ 0& I_y& 0\\ 0& 0& I_x\end{array}\right], P_{\mathrm{dynamic}}=\left[\begin{array}{l}{P}_{d11}\\ P_{d12}\\ P_{d21}\\ P_{d22}\end{array}\right],$$

(6)

where $I_y, I_x$ are the pitch and roll moments of inertia of the hovercraft, respectively, and $P_{dij}$ is the pulsating cushion pressure in each air chamber. By solving the linear algebra equations in Equation (5), the pulsating pressure in the air chambers and the (angular) acceleration can be calculated from each other.

2.2.3. Air Chamber Volume and Leakage Formula

The volume of the hovercraft air chamber can be calculated as follows:

$$V_C=A_C{h}_c, h_c=z_0+\mathrm{d}z-x_c\left({\varphi }_0+\mathrm{d}\varphi \right)+y_c\mathrm{d}\theta ,$$

(7)

where $V_C$ is the cushion volume of the air chamber; $A_C$ is the area of the air chamber; $h_c$ is the height of the air chamber at the center; $z_0$ is the height of the center of gravity of the hovercraft when static hovering; ${\varphi }_0$ is the trim angle; $x_c, y_c$. are the coordinates of the air chamber centroid; and $z, \varphi , \theta$ are the heave, pitch, and roll motions, respectively.

The leakage flow rate of a hovercraft can be calculated as follows:

$$Q_e=C_e{v}_e{A}_e=C_e{v}_{e}L{h}_e,$$

(8)

where $Q_e$ is the leakage flow rate (m³/s); $C_e$ is the flow coefficient; $v_e$ is the leakage velocity (m/s); $A_e$ is the leakage area (m²); and $L, h_e$ are the cushion periphery and hover gap height (m), respectively.

3. Test Results and Analysis

3.1. Amplitude-Frequency Response Analysis

3.1.1. Model Test Results for the Amplitude-Frequency Response

Fast Fourier Transform (FFT) was used to filter the experimental data, including the front and rear air chamber pressures, heave/pitch motion, acceleration at Lc/2 and 3L_C/4, and the vertical bending moment M_v (hereinafter referred to as the “bending moment”) at the mid-ship and 3L_C/4 cross section. The experimental results presented hereafter are non-dimensionalized (

Table 3. Dimensionless expressions for physical quantities.

Quantity	Dimensionless Expression
Cushion pressure	$P_C/\left({\rho }_{a}g{\zeta }_w\right)$
Cushion leakage flow rate	$Q_e/\left(A_e\sqrt{2\Delta P_C/{\rho }_a}\right)$
Acceleration	$a/\left(g{\zeta }_w/L_C\right)$
Heave	$z/{\zeta }_w$
Pitch	$\theta /\left(k{\zeta }_w\right)$
Cross-sectional bending moment	$M_v/\left({\rho }_{w}g{B}_C{L}_C^2{\zeta }_w\right)$

where ${\rho }_a$ is the air density, $g$ is the acceleration of gravity, ${\zeta }_w$ is the wave amplitude, $A_e$ is the air chamber leakage area, $L_C$ is the overall length of the air cushion, $k$ is the wave number, ${\rho }_w$ is the density of fresh water, and $B_C$ is the width of the hovercraft.

and

Table 4. Correspondence between wave frequency and wavelength-to-cushion length ratio.

Wave Frequency (Hz)	0.385	0.455	0.500	0.588	0.625	0.667	0.714	0.769	0.833	0.909	1.00
λ/LC	3.72	2.67	2.21	1.60	1.41	1.24	1.08	0.93	0.80	0.67	0.55

), and the characteristics of the wave load and motion response of the hovercraft are analyzed.

As shown in Figure 7, the hovercraft tank towing test is roughly divided into a wave-making stage, an acceleration stage, a stationary stage, and a braking stage. The stationary stage has an effective window for measuring the responses (motion, load, pressure, etc.), and it is obvious that the sagging cushion pressure is greater than the hogging cushion pressure. The measurements in the stationary stage (50~70 s), as shown in Figure 7, were used for the FFT, and the results are shown in Figure 8, where it can be seen that the major frequency component is the same as the encounter frequency. In the low-pass filter, a cutoff frequency 2.5 times the encounter frequency was used. In this study, P_C is the cushion pressure obtained by deducting the standard atmospheric pressure P₀ from P_a, and P_a in the P_aV is the cushion pressure including the standard atmospheric pressure, that is, P_a = P₀ + P_C.

As shown in Figure 9, there is a local peak in the cushion pressure response when λ/L_C is between 1.5 and 2.0. The amplitude of the cushion pressure in the front air chamber is greater than that of the rear air chamber, which coincides with the observation that the rear body of the hovercraft motion in waves is small than that of the fore body.

As shown in Figure 10, since the acceleration is closely related to the cushion pressure pulsation, the acceleration has a maximum peak value when λ/L_C is between 1.5 and 2.0. Compared with the dimensionless acceleration in [15], where the model has similar main dimensions, the acceleration of the model in this study is generally smaller, which is a result of the difference in mass distribution. Generally, the acceleration response at the 3L_C/4 cross section is greater than that at the mid-ship cross section.

As shown in Figure 11, as λ/L_C increases, the pitch response approaches 1.3 and exceeds the wave steepness. This is very different from that of surface displacement ships because the hovercraft is supported by the air cushion rather than buoyancy. The skirt is flexible, and it continues to deform after touching the water surface, under the action of slamming force during the vertical bow-down motion, and under the action of horizontal force from the wave’s impact on the skirt due to the high forward speed of waves.

As shown in Figure 12, the hovercraft differs from traditional surface displacement ships. Owing to the pulsating cushion pressure in the front and rear air chambers, it exhibits certain independent response characteristics. Its maximum vertical bending moment does not necessarily occur near the mid-ship cross section. For example, the bending moment response at the 3L_C/4 cross section, as shown in Figure 12, is marginally larger than that at the mid-ship cross section. Unlike the cushion pressure and acceleration, the bending moment response at the mid-ship cross section reaches the maximum λ/L_C near 1.0. The magnitudes of the dimensionless mid-ship and 3L_C/4 bending moments are, in general, in the order of 10⁻³. Compared with the hovercraft in [15], the magnitude of 10⁻² of the dimensionless mid-ship bending moment is one order of magnitude smaller, which is likely related to the mass distribution.

3.1.2. Numerical Calculation and Results for the Amplitude-Frequency Response

The mass and geometric parameters of the hovercraft model were used to establish the mass and geometry matrices for the dynamic model of the hovercraft, as described in the Section 2.2.

The hovercraft’s static equilibrium equation is satisfied during the static hovering process. The static pressures of the front and rear air chambers are calculated using Equation (2), and they are 264.6 Pa and 267.9 Pa, which is generally consistent with the measured cushion pressure (260~270 Pa) in the static cushion test of the hovercraft. Figure 13 shows the calculated vertical shear force F_z and vertical bending moment M_v distribution of the cross sections along the vessel’s length.

The main external load of the hovercraft is the air chamber pressure. Through Equation (5), the acceleration and cross-section load response can be derived based on the air chamber pressure. For hovercrafts with two or three air chambers, it is theoretically feasible to derive the cushion pressure and cross-section load response based on acceleration; for four-chamber hovercrafts, since the geometry matrix of Equation (3) is not a square matrix, only the least-squares-based solution can be obtained by relying on the pseudo-inverse matrix.

In Figure 14, Figure 15, Figure 16 and Figure 17, based on the front and rear air chamber pressure time series obtained from the seakeeping test, the acceleration, cross-section load time series, and their amplitude were solved using the simplified kinematic model of the hovercraft (see Equation (5)) and the d’Alembert principle and compared with the test values.

As shown in Figure 14 and Figure 15, the correlation coefficients between the test value and the calculated value of the mid-ship and 3L_C/4 cross-section accelerations are 0.7157 and 0.9612, respectively. According to the dynamic equilibrium equation, the acceleration calculated from the front and rear air chamber pressure is in good agreement with the experimental value. To solve the dynamic equilibrium equations for the solution to the acceleration, the accuracy is affected by the mass and geometry parameters.

As shown in Figure 16 and Figure 17, the correlation coefficients between the experimental result and the solution obtained from the calculation for the bending moments at the mid-ship and 3L_C/4 cross section are 0.1976 and 0.8622, respectively. According to the d’Alembert principle, the bending moment at the mid-ship cross section is affected by the accuracy of the parameters, and the agreement is poor; the bending moment at the 3L_C/4 cross section obtained from the front and rear cushion pressures is in a good agreement with the experimental results.

In the equations of motion, the hovercraft reaches a state of dynamic equilibrium under the action of the cushion pressure, gravity, and the inertial force in low sea conditions, where the slamming force on the skirt can be neglected. In Figure 14, Figure 15, Figure 16 and Figure 17, apart from the bending moment at the cross sections, the solutions of the calculation are generally consistent with the experimental results, which indicate that the model’s test results are reliable and accurate.

3.2. Analysis of the Relationship between the Air Chamber Pressure and Volume

The hovercraft is supported by the air cushion structure, and the relationships between the cushion pressure, the cushion volume, and the flow rate are some of the core mechanisms that govern the dynamics of fully cushioned hovercrafts.

3.2.1. Model Test results for the Air Chamber Pressure and Volume

Figure 18 shows a flowchart of the data processing procedures used to acquire the cushion pressure P_C and motion response x through the towing test of a hovercraft in regular waves, to calculate the motion response $x$ to the cushion volume V_C (the volume between the top of the air chamber and the still water surface, excluding the surface elevation) through Equation (7), and to obtain the P_C-V_C fitting relationships.

The front-left air chamber has a volume of 0.3014 m³, and the solution of the calculation for the cushion pressure is 264.6 Pa; the rear-left air chamber has a volume of 0.2722 m³, and the cushion pressure is 267.9 Pa.

A total of 500 data records from each wave frequency’s stationary stage were used for the correlation analysis of P_C-V_C. A quadratic polynomial fit was adopted to address the nonlinear characteristics of the P_C-V_C curves, and a linear fit was adopted to describe the linear characteristics of the P_aV_C-V_C curves (see Figure A1 and Figure A2 in Appendix A). The fitting functions are presented in Table 5 and Table 6. The fitting function curves of P_C-V_C and P_aV_C-V_C of the front and rear air chambers for a range of wave frequencies are plotted in the same figure for comparison purposes, as shown in Figure 19 and Figure 20.

As shown in the P_aV_C-V_C curves in Figure 19 and Figure 20, and according to the ideal gas law, P_aV = nRT, it can be seen that the value of P_aV_C inside the chamber changes with the wave motion and the dynamic inflow and outflow rate of the chamber, indicating that there also exists a strong correlation between the cushion pressure of the chamber and the inflow/outflow.

As can be seen in Equations (7) and (8), the cushion volume is related to the cushion height, while the leakage flow rate is related to the leakage height. For small oscillations, both the air cushion volume and the leakage flow rate are directly related to the heights and can be calculated from each other. The cushion pressure and flow rate equations in the hovercraft motion theory [3,4,5,6,7] are expressed through the fan duct characteristic equation. Under the small perturbation assumption, the curve of this equation, as shown in Figure 21, exhibits a convex shape similar to the P_C-V_C curves in Figure 19 and Figure 20.

3.2.2. Numerical Calculation and Results for the Air Chamber Pressure and Volume

Based on the P_C-V_C fitting curve equations listed in Table 5 that are linearized at the static cushion volume, the average P_C-V_C linear fitting equations were obtained by means of linear superposition.

Table 5. Equations for the fitted curves of P_C-V_C of the front and rear chambers and the Coefficient of Determination R².

Frequency (Hz)	Front-Left Air Chamber
	PC-VC Fitting Function	Domain	R2
0.385	$y=-1663.7x^2+101.52x+436.22$	[0.255, 0.348]	0.9460
0.455	$\begin{array}{l}y=-16065x^2+8173.7x-682.86\hfill \end{array}$	[0.250, 0.359]	0.9708
0.500	$\begin{array}{l}y=-19252x^2+9482.4x-792.53\hfill \end{array}$	[0.240, 0.373]	0.9587
0.588	$\begin{array}{l}y=-54465x^2+28889x-3444.9\hfill \end{array}$	[0.272, 0.336]	0.9809
0.625	$y=-63943x^2+33808x-4065.2$	[0.283, 0.329]	0.9729
0.667	$y=-206026x^2+117581x-16412$	[0.287, 0.320]	0.9238
0.714	$\begin{array}{l}y=-345283x^2+200402x-28716\hfill \end{array}$	[0.293, 0.314]	0.7632
0.769	$y=377253x^2-233847x+36516$	[0.295, 0.311]	0.4233
0.833	$y=136193x^2-85375x+13660$	[0.293, 0.311]	0.2091
0.909	$\begin{array}{l}y=97612x^2-61719x+10047\hfill \end{array}$	[0.293, 0.316]	0.1846
1.00	$y=-92792x^2+54147x-7577.5$	[0.292, 0.320]	0.1114
Frequency (Hz)	Rear-Left Air Chamber
	PC-VC Fitting Function	Domain	R2
0.385	$\begin{array}{l}y=-1539x^2+83.025x+421.4\hfill \end{array}$	[0.228, 0.315]	0.8899
0.455	$y=-9591.1x^2+3886.5x-16.22$	[0.237, 0.307]	0.8040
0.500	$\begin{array}{l}y=2612.3x^2-3375.7x+1048.1\hfill \end{array}$	[0.239, 0.299]	0.7922
0.588	$\begin{array}{l}y=-76153x^2+37269x-4174.6\hfill \end{array}$	[0.257, 0.283]	0.8205
0.625	$y=-39208x^2+17156x-1437.5$	[0.260, 0.284]	0.7279
0.667	$y=-48098x^2+20606x-1719.9$	[0.264, 0.280]	0.7012
0.714	$\begin{array}{l}y=-60324x^2+27477x-2681.7\hfill \end{array}$	[0.265, 0.280]	0.6135
0.769	$y=103327x^2-64070x+10075$	[0.266, 0.282]	0.6816
0.833	$y=-89760x^2+40744x-4154.9$	[0.264, 0.279]	0.5730
0.909	$y=91532x^2-54232x+8273.2$	[0.265, 0.287]	0.2568
1.00	$y=-2166.5x^2-2643.7x+1176.5$	[0.265, 0.283]	0.1990

Table 5. Equations for the fitted curves of P_C-V_C of the front and rear chambers and the Coefficient of Determination R².

Frequency (Hz)	Front-Left Air Chamber
	PC-VC Fitting Function	Domain	R2
0.385	$y=-1663.7x^2+101.52x+436.22$	[0.255, 0.348]	0.9460
0.455	$\begin{array}{l}y=-16065x^2+8173.7x-682.86\hfill \end{array}$	[0.250, 0.359]	0.9708
0.500	$\begin{array}{l}y=-19252x^2+9482.4x-792.53\hfill \end{array}$	[0.240, 0.373]	0.9587
0.588	$\begin{array}{l}y=-54465x^2+28889x-3444.9\hfill \end{array}$	[0.272, 0.336]	0.9809
0.625	$y=-63943x^2+33808x-4065.2$	[0.283, 0.329]	0.9729
0.667	$y=-206026x^2+117581x-16412$	[0.287, 0.320]	0.9238
0.714	$\begin{array}{l}y=-345283x^2+200402x-28716\hfill \end{array}$	[0.293, 0.314]	0.7632
0.769	$y=377253x^2-233847x+36516$	[0.295, 0.311]	0.4233
0.833	$y=136193x^2-85375x+13660$	[0.293, 0.311]	0.2091
0.909	$\begin{array}{l}y=97612x^2-61719x+10047\hfill \end{array}$	[0.293, 0.316]	0.1846
1.00	$y=-92792x^2+54147x-7577.5$	[0.292, 0.320]	0.1114
Frequency (Hz)	Rear-Left Air Chamber
	PC-VC Fitting Function	Domain	R2
0.385	$\begin{array}{l}y=-1539x^2+83.025x+421.4\hfill \end{array}$	[0.228, 0.315]	0.8899
0.455	$y=-9591.1x^2+3886.5x-16.22$	[0.237, 0.307]	0.8040
0.500	$\begin{array}{l}y=2612.3x^2-3375.7x+1048.1\hfill \end{array}$	[0.239, 0.299]	0.7922
0.588	$\begin{array}{l}y=-76153x^2+37269x-4174.6\hfill \end{array}$	[0.257, 0.283]	0.8205
0.625	$y=-39208x^2+17156x-1437.5$	[0.260, 0.284]	0.7279
0.667	$y=-48098x^2+20606x-1719.9$	[0.264, 0.280]	0.7012
0.714	$\begin{array}{l}y=-60324x^2+27477x-2681.7\hfill \end{array}$	[0.265, 0.280]	0.6135
0.769	$y=103327x^2-64070x+10075$	[0.266, 0.282]	0.6816
0.833	$y=-89760x^2+40744x-4154.9$	[0.264, 0.279]	0.5730
0.909	$y=91532x^2-54232x+8273.2$	[0.265, 0.287]	0.2568
1.00	$y=-2166.5x^2-2643.7x+1176.5$	[0.265, 0.283]	0.1990

Table 6. Equations for the fitted curves of P_aV_C-V_C of the front and rear chambers and the Coefficient of Determination R².

Frequency (Hz)	Front-Left Air Chamber			Rear-Left Air Chamber
	PaVC-VC Fitting Function	Domain	R2	PaVC-VC Fitting Function	Domain	R2
0.385	$y=101365x+81.817$	[0.255, 0.348]	1	$y=101446x+56.03$	[0.228, 0.315]	1
0.455	$\begin{array}{l}y=101112x+153.47\hfill \end{array}$	[0.250, 0.359]	1	$y=101281x+100.39$	[0.237, 0.307]	1
0.500	$\begin{array}{l}y=100882x+215.04\hfill \end{array}$	[0.240, 0.373]	1	$y=101126x+141.65$	[0.239, 0.299]	1
0.588	$\begin{array}{l}y=100336x+384.43\hfill \end{array}$	[0.272, 0.336]	1	$y=100600x+285.54$	[0.257, 0.283]	1
0.625	$y=100019x+484.11$	[0.283, 0.329]	1	$y=100527x+306.02$	[0.260, 0.284]	1
0.667	$\begin{array}{l}y=99563x+618.73\hfill \end{array}$	[0.287, 0.320]	1	$y=100135x+412.14$	[0.264, 0.280]	0.9999
0.714	$\begin{array}{l}y=98999x+795.18\hfill \end{array}$	[0.293, 0.314]	1	$y=100170x+403.39$	[0.265, 0.280]	0.9999
0.769	$\begin{array}{l}y=99991x+495.76\hfill \end{array}$	[0.295, 0.311]	1	$y=99516x+571.82$	[0.266, 0.282]	0.9998
0.833	$\begin{array}{l}y=100619x+303.87\hfill \end{array}$	[0.293, 0.311]	0.9996	$y=99443x+589.35$	[0.264, 0.279]	0.9996
0.909	$\begin{array}{l}y=100929x+214.14\hfill \end{array}$	[0.293, 0.316]	0.9998	$y=100493x+306.43$	[0.265, 0.287]	0.9996
1.00	$y=100965x+201.84$	[0.292, 0.320]	0.9996	$y=100567x+286.86$	[0.265, 0.283]	0.9996

Table 6. Equations for the fitted curves of P_aV_C-V_C of the front and rear chambers and the Coefficient of Determination R².

Frequency (Hz)	Front-Left Air Chamber			Rear-Left Air Chamber
	PaVC-VC Fitting Function	Domain	R2	PaVC-VC Fitting Function	Domain	R2
0.385	$y=101365x+81.817$	[0.255, 0.348]	1	$y=101446x+56.03$	[0.228, 0.315]	1
0.455	$\begin{array}{l}y=101112x+153.47\hfill \end{array}$	[0.250, 0.359]	1	$y=101281x+100.39$	[0.237, 0.307]	1
0.500	$\begin{array}{l}y=100882x+215.04\hfill \end{array}$	[0.240, 0.373]	1	$y=101126x+141.65$	[0.239, 0.299]	1
0.588	$\begin{array}{l}y=100336x+384.43\hfill \end{array}$	[0.272, 0.336]	1	$y=100600x+285.54$	[0.257, 0.283]	1
0.625	$y=100019x+484.11$	[0.283, 0.329]	1	$y=100527x+306.02$	[0.260, 0.284]	1
0.667	$\begin{array}{l}y=99563x+618.73\hfill \end{array}$	[0.287, 0.320]	1	$y=100135x+412.14$	[0.264, 0.280]	0.9999
0.714	$\begin{array}{l}y=98999x+795.18\hfill \end{array}$	[0.293, 0.314]	1	$y=100170x+403.39$	[0.265, 0.280]	0.9999
0.769	$\begin{array}{l}y=99991x+495.76\hfill \end{array}$	[0.295, 0.311]	1	$y=99516x+571.82$	[0.266, 0.282]	0.9998
0.833	$\begin{array}{l}y=100619x+303.87\hfill \end{array}$	[0.293, 0.311]	0.9996	$y=99443x+589.35$	[0.264, 0.279]	0.9996
0.909	$\begin{array}{l}y=100929x+214.14\hfill \end{array}$	[0.293, 0.316]	0.9998	$y=100493x+306.43$	[0.265, 0.287]	0.9996
1.00	$y=100965x+201.84$	[0.292, 0.320]	0.9996	$y=100567x+286.86$	[0.265, 0.283]	0.9996

The average fitting equation for P_C-V_C of the front air chamber is as follows:

$$P_C=-3813.00\left(V_C-0.3014\right)+313.1011;$$

(9)

The average fitting equation for P_C-V_C of the rear air chamber is as follows:

$$P_C=-4320.75\left(V_C-0.2722\right)+317.2457.$$

(10)

To investigate the influence of surface elevation on the cushion pressure based on the experimental data, the P_C-V_C fitting relationship of the front and rear air chambers above and the static and dynamic equilibrium equations in Section 2.2 were used to calculate the static hovering cushion height. Then, the equations of hovercraft motion in waves and the air chamber response model were established to solve for the hovercraft motion in the time domain. Since the cushion volume is affected by the surface elevation, the cushion pressure was solved by using Equations (9) and (10). The combined force of gravity and cushion pressure causes the hull to heave and pitch. Based on the average fitting equations, a series of numerical calculations were carried out to obtain the hovercraft motion in the test conditions.

Since the surface elevation under the cushion could not be measured in the experiment, the cushion volume was calculated from the still water surface and the instantaneous position; however, in the calculation model, the surface elevation was accounted for in the cushion volume. This is the reason why there are some differences between the two curves in Figure 22.

Since the influence of surface elevation cannot be considered in the cushion volume of the experimental model, while the surface elevation will affect the cushion volume in the calculation model, the pressure-volume results calculated from the numerical calculation model deviate from the experimental results. As shown in Figure 22, the surface elevation likely has a significant impact on the cushion pressure in the wave frequency range of [0.588, 0.714] (corresponding wavelength-to-cushion length ratio of [1.6, 1.08]), resulting in a large gap between the calculated cushion pressures and the actual measured pressures. This phenomenon suggests a strong correlation between the surface elevation and the cushion pressure. Specifically, changes in surface elevation directly affect the pressure values within the air chamber, which should be a crucial factor for consideration in future research.

4. Conclusions

Based on the analysis of the results, the following conclusions are drawn. The data indicate that the cushion pressure and mid-ship acceleration reach a local maximum when the wavelength-to-cushion length ratio is approximately 1.6. Additionally, the wave appears to have a significant impact on the wave load when the wavelength-to-cushion length ratio is around 1.2.

When a hovercraft encounters waves, the airflow within the air chamber experiences dynamic inflow and outflow rather than remaining in a static state as it would in a fully enclosed chamber. The cushion pressure is shown to be not only closely related to the chamber volume but also directly influenced by the inlet and outlet airflow. An analysis of the results reveals a complex interdependence among the cushion pressure, cushion volume, and the inflow/outflow dynamics. This suggests that the inherent correlation between the cushion pressure and leakage flow rate can be indirectly inferred through monitoring the changing patterns of cushion pressure and volume. The fitting and calculation methods used in this study demonstrate significant feasibility and practicality for analyzing hovercraft wave loads.

In this study, the variation in volume within the air chamber is estimated based on the vessel’s motion, which might introduce errors. However, due to the technical challenges associated with monitoring both the spatial and temporal variations in the volume and flow rate within the air chamber, the experimental validation and quantification of these errors appear unlikely. The analysis and conclusions presented here are applicable to medium and low sea states, and further studies are required for higher sea states.