2.1. Experimental set-up

The water-entry experiments were conducted at the State Key Laboratory of Ocean Engineering of Shanghai Jiao Tong University using a purpose-built drop facility, as shown in figure 1. The drop apparatus and its associated instrumentation are identical to those employed in an earlier work by the authors to study the influence of aeration on the impact pressure at one entry velocity (Wu et al. Reference Wu, Xiao, Kou and Liu2025). The present work provides a more complete picture of velocity and aeration influences and extends the analysis to additional response quantities, including impact duration, impulse and the expansion velocity of the wetted surface. The experimental set-up featured a transparent water tank with dimensions of 1.5 m in length, 1.25 m in width and 1.2 m in depth, maintaining a consistent water depth of 0.8 m. Structural support was provided by two high-strength steel frames, while a pair of guide rails ensured the wedge’s vertical trajectory. As displayed in figure 2, the impactor was an aluminium wedge with a length of 0.50 m, a width of 0.25 m and a deadrise angle of 20°. The total mass of the falling structure, including the wedge and associated support components, was 23.3 kg. An electromagnetic release system, mounted on an electric telescopic rod, allowed precise control of the wedge’s initial position.

Figure 1.

Schematic illustration of the equipment used for aerated water-entry experiments. The experimental set-up is housed within two rigid support frames and consists of a transparent water tank with dimensions of 1.5 m × 1.25 m × 1.2 m, filled to a depth of 0.8 m. A wedge-shaped body, positioned at the centre of the tank, is dropped vertically into the water to study the water-entry impact pressure. A bubble generator is placed at the bottom of the tank to create a controlled aerated environment. The entire water-entry process is captured by a high-speed camera for detailed flow visualisation.

The impact sequence was filmed using a high-speed camera (FASTCAM SA-Z, Photron) for qualitative flow visualisation. Impact pressures were measured with 24 strain-gauge pressure sensors (PGMC-A-200KPZ15, KYOWA) that were flush mounted with the wedge bottom surface. All the sensors have a sensing area of 8 mm in diameter and a maximum measurement capacity of 200 kPa. The locations and the identification number of pressure sensors are depicted in figure 2. Pressure data were recorded at a 20 kHz sampling rate. The wedge’s displacement during the drop was tracked using a laser displacement sensor (Y1TA100QXVT80, Wenglor) mounted on a vibration-isolated frame. These data enabled precise calculations of instantaneous velocity and acceleration. The repeatability of the wedge motion and the reliability of the water-entry apparatus are indicated by the consistent impact pressure profiles in figure 3.

Figure 2.

Photograph of the bubble generator and pressure sensor layout on the wedge model. The porous boxes are arranged in a grid-like pattern to ensure uniform bubble generation. Front view (top right) shows the wedge with deadrise angle $\beta =20^{\circ}$ , indicating the sensor distribution along the inclined surface. Bottom view (bottom right) provides the detailed arrangement of 24 pressure sensors (P01–P24). Sensors P01–P06 were installed along the centreline, and P07–P12 were aligned along the edge of the same side. Sensors P13–P23 were distributed at consistent vertical heights but varying longitudinal positions to capture pressure variation along the wedge face. Sensor P24 was symmetrically placed opposite P03 for validation of experimental symmetry. All dimensions are given in millimetres.

Figure 3.

Denoised impact pressure time histories obtained from five repeated tests and schematic diagram of parameter definition. The curves exhibit excellent repeatability of the pressure response during wedge entry. The inset illustrates the method for determining the peak pressure $P_{\textit{max}}$ , the duration $t_{d}$ and the corresponding pressure impulse $I$ , calculated as the area under the pressure curve between $t_{s}$ and $t_{e}$ .

A custom-designed bubble generator was employed to create controlled aerated water conditions for studying the effect of void fraction on wedge impact pressure. The system consisted of 48 quartz sand-based porous boxes, arranged in two rows at the bottom of the water tank, as depicted in figures 1 and 2. Each box, measuring 300 mm by 34 mm, featured a quartz top and allowed compressed air to escape through small gaps, forming finely dispersed bubbles. Compressed air was supplied by an air compressor, regulated by a digital mass flow controller (AS200, Aosong) with a range of 0–300 l min^–1. This system allowed precise control of the aeration level by adjusting the gas flow rate. This set-up ensured repeatable conditions for evaluating the impact of varying aeration on the impact pressure during wedge entry.

Prior to the impact tests, void fraction measurements were performed using a conductance-based WMS. The WMS functions by detecting variations in the electrical conductivity of the fluid between two orthogonally arranged planes of parallel wires. In this study, the purposely designed WMS comprised 64 parallel electrodes per plane with a pitch of 4 mm, forming a measurement matrix of 4096 measuring points (64 × 64 grid). The effective measuring area of the WMS was 256 × 256 mm, slightly larger than the pressure sensor array (less than 250 × 250 mm), thus providing full coverage. The sensor was positioned approximately 0.75 m above the tank bottom to measure the void fraction close to the water’s free surface. Data acquisition was carried out at 2500 Hz, with each recording lasting 20 s and repeated ten times for each air flow rate. For additional details on the WMS design and its accuracy validation, please refer to Wu et al. (Reference Wu, Xiao, Kou and Liu2025).

2.2. Test conditions

Pure water entry and aerated water entry were investigated in the present study. Seven different air flow rates were applied: $ Q=0, 60, 80, 100, 120, 140, 180$ l min^–1. Table 1 lists the average void fractions and corresponding standard deviations for each flow rate from ten repeated tests measured by the WMS. This experiment considered four distinct drop heights $H=330, 520, 750, 975$ mm, corresponding to initial water-entry velocities of $v_{i}=2.15, 2.70, 3.24, 3.69$ m s^–1, respectively. Each test was repeated at least five times for consistent and reliable results. As suggested by Mai et al. (Reference Mai, Mai, Raby and Greaves2019), a minimum rest period of 15 minutes was allowed between each test to ensure the system stabilised before initiating the next trial.

Table 1.

Aeration levels measured by WMS at seven air flow rates.

2.3. Impact pressure parametrisation

Given the unavoidable high-frequency noise, a filtering process is required before the raw data can be used to analyse the impact pressure characteristics. A wavelet denoising method was applied to eliminate noise interference from the measurements, which has been proven reliable for processing impact pressure data in previous studies (Guo et al. Reference Guo, Xiao, Teng, Kou and Liu2020; Zhang et al. Reference Zhang, Xiao, Cheng, Wei and Chen2024). The Daubechies-4 wavelet was chosen with 8 decomposition levels, following the universal threshold selection rule as described in Wu et al. (Reference Wu, Liu, Xiao, Shan, Kou, Li and Guo2024). All water-entry impact measurements were denoised using this approach.

Figure 3 illustrates the denoised impact pressure time histories obtained from five repeated tests. A distinct initial pressure peak is observed, followed by a secondary peak around $t=0.24 \mathrm{s}$ . This secondary peak is attributed to the formation of an air pocket at the bottom surface of the wedge at the time when the secondary peak occurs. The trapped air then undergoes multiple cycles of compression and expansion, inducing additional but significantly smaller pressure on the wedge. A detailed explanation can be found in Wu et al. (Reference Wu, Xiao, Kou and Liu2025). Before the secondary peak, the impact pressure curves exhibited excellent agreement, confirming the high repeatability and reliability of the experimental set-up. After the second peak, minor variations appeared due to the slight differences in the shape and behaviour of the air pocket between individual tests. However, given the relatively low magnitude of the secondary peak, only the first peak pressure and its associated impulse were considered in the subsequent analysis.

To quantitatively account for the aeration effect on impact pressure, this study identifies several parameters to describe the characteristics of the impact pressure, as shown in figure 3.

1. (i) The parameter $P_{max}$ : peak impact pressure. The maximum pressure measured by one pressure sensor in a water-entry event.

2. (ii) The parameter $t_{d}$ : duration, time interval between instants $t_{s}$ and $t_{e}$ .

3. (iii) The parameter $I$ : impact pressure impulse, the integral of impact pressure over the total duration.

The start time $t_{s}$ and the end time $t_{e}$ were identified using a threshold set at 1 % of the peak impact pressure $P_{\textit{max}}$ , minimising the influence of data oscillations. Impact pressure impulse $I$ was determined by integrating the impact pressure over the duration, as shown below

(2.1) \begin{align} I=\int _{t=t_{s}}^{t=t_{e}}P\mathrm{d}t .\end{align}

3.1. Classical wedge entry models for pure water

The Wagner model, one of the most classic models for water-entry impact, focused on the problem of blunt body entering water. It is widely regarded as a successful model, with numerous extensions and modifications developed from it. The basic assumptions of Wagner model are that the fluid is incompressible, inviscid and irrotational. Initially, the liquid is at rest, with its free surface being horizontal ( $z=0$ ) (Wagner Reference Wagner1932). The origin of the Cartesian coordinate system Oxz is defined at the initial contact point between the body tip and the free surface, as illustrated in figure 4. Additionally, the effects of gravity and surface tension are neglected. As depicted in figure 4, during the wedge entry process, the local water surface elevation near the body rises rapidly, causing premature wetting of the wedge surface. This water pile-up phenomenon represents the key distinction between the Wagner model and the earlier von Karman theory (von Karman Reference von Karman1929). Due to the water pile-up effect, the wetted portion of the wedge surface at a given time $t$ has a length denoted as $2c(t)$ , with a maximum rise height relative to the wedge bottom expressed as $\eta (x)$ . Wagner employed the flat-plate approximation, which equated the wedge water-entry problem to the impact of a flat plate of width $2c(t)$ subjected to a vertically upward flow, where the flow velocity is equal to the entry velocity $ v$ . The velocity potential $\phi$ in Wagner model satisfies the following linear boundary-value problem:

(3.1) \begin{align} \left\{\begin{array}{lllll}\! \boldsymbol{\nabla} ^{2}\phi =0,\\ \frac{\partial \phi }{\partial z}=-v, & z=0,|x|\lt c,\\ \phi (x,0)=0, & z=0,|x|\geq c,\\ \phi (x,z)\rightarrow 0, & \sqrt{x^{2}+z^{2}}\rightarrow \infty. \end{array}\right. \end{align}

For brevity, the repeated derivations are omitted. The velocity potential on the body surface is obtained as follows:

(3.2) \begin{align} \phi \left(x,0\right)=-v\sqrt{c^{2}-x^{2}},|x|\lt c, z=0. \end{align}

Here, $c$ represents the half-width of the wetted region, which depends on the geometry of the entering body. For a wedge characterised by a deadrise angle  $\beta$ , the wetted half-length $c$ is given by

(3.3) \begin{align} c\left(t\right)=\frac{\pi }{2\tan \beta }\int _{0}^{t}v\left(t\right)\mathrm{d}t. \end{align}

The variable $z$ describes the disturbed free-surface shape

(3.4) \begin{align} z(x,t)=f(x)-h(t). \end{align}

The shape function of a symmetric wedge is given by $f(x)=x\tan \beta$ , and $h(t)$ represents the penetration depth following initial contact with the water surface. The pressure $P$ at any point on the wedge surface is determined by employing the unsteady Bernoulli equation

(3.5) \begin{align} P\left(x,z,t\right)=-\rho \frac{\partial \phi }{\partial t}-\frac{1}{2}\rho \left| \boldsymbol{\nabla }\phi \right| ^{2}-\rho gz. \end{align}

Here, $\rho$ denotes the fluid density, and $g$ is the gravitational acceleration. Wagner indicated that the gravitational effect is negligible compared with the large impact forces generated during water-entry process. Moreover, $\boldsymbol{\nabla }\phi$ can be disregarded, as it is considerably smaller than $\partial \phi /\partial t$ . Thus, by linearising the Bernoulli equation, the following expression is obtained:

(3.6) \begin{align} P=\rho \frac{\mathrm{d}v}{\mathrm{d}t}\sqrt{c^{2}-x^{2}}+\rho \frac{vc}{\sqrt{c^{2}-x^{2}}}\frac{\mathrm{d}c}{\mathrm{d}t},|x|\lt c. \end{align}

The first term on the right-hand side is referred to as the added-mass term, which is related to the acceleration of the wedge $\mathrm{d}v/\mathrm{d}t\hspace{0pt}$ . The second term is known as the impact term, which is associated with the change rate of the wetted length $2\mathrm{d}c/\mathrm{d}t\hspace{0pt}$ . Integrating the pressure over the wetted width yields the impact force exerted on the wedge as

(3.7) \begin{align} F=\int _{-c}^{c}P\mathrm{d}x. \end{align}

In the Wagner solution, as indicated by (3.6), the impact pressure tends to infinity as  $x$ approaches $c$ . This singularity is physically unrealistic, hindering the calculation of peak impact pressure values that can be directly compared with experiments. A notable improvement to address this limitation was proposed by Zhao and Faltinsen. By matching asymptotic expansions, Zhao and Faltinsen derived a composite solution that avoids this singular behaviour, known as the ZFM (Zhao & Faltinsen Reference Zhao and Faltinsen1993). To avoid repetition of derivations already well documented, the resulting pressure calculation from their approach is directly provided below

(3.8) \begin{align} P_{\textit{ZFM}}= \left\{\begin{array}{llll} \! \rho vc\frac{1}{\sqrt{c^{2}-x^{2}}}\frac{\mathrm{d}c}{\mathrm{d}t}-\rho vc\frac{1}{\sqrt{2c(c-x)}}\frac{\mathrm{d}c}{\mathrm{d}t}+2\rho \frac{\sqrt{| \tau | }}{(1+\sqrt{| \tau | })^{2}}\left(\frac{\mathrm{d}c}{\mathrm{d}t}\right)^{2}\!, \quad |x| \leq c(t)\\ 2\rho \frac{\sqrt{| \tau | }}{(1+\sqrt{| \tau | })^{2}}\left(\frac{\mathrm{d}c}{\mathrm{d}t}\right)^{2}\!, \quad |x| \gt c (t) \end{array}\right. \end{align}

where $\delta$ is an asymptotic jet thickness

(3.9) \begin{align} \delta =2\pi v^{2}c\left(4\frac{\mathrm{d}c}{\mathrm{d}t}\right)^{-2}\! ,\end{align}

where $\tau$ is a parameter related to $x$

(3.10) \begin{align} x-c=\left(\frac{\delta }{\pi }\right)\big(-\ln | \tau | -4\sqrt{| \tau | }-| \tau | +5\big) .\end{align}

Another significant improvement to the Wagner model has been achieved through the inclusion of higher-order terms derived from a Taylor series expansion in the expression of the velocity potential, alongside the consideration of nonlinear terms present in the Bernoulli equation. This modification ensures that the flow velocity near the boundary of the wetted region remains finite. Prominent models developed based on this approach include the OLM proposed by Logvinovich (Reference Logvinovich1972), the MLM and the GWM introduced by Korobkin (Reference Korobkin2004). In comparison with the Wagner model, the Bernoulli equation in these extended models includes additional nonlinear terms, enabling the following expression to be formulated:

(3.11) \begin{align} P\left(x,t\right)&=\rho \dot{h}^{2}\left(t\right)\left[\frac{\dot{c}}{\dot{h}}\frac{c}{\sqrt{c^{2}-x^{2}}}-\frac{1}{2}\frac{c^{2}}{c^{2}-x^{2}}\frac{1}{1+f_{x}^{2}}-\frac{1}{2}\frac{f_{x}^{2}}{1+f_{x}^{2}}-\frac{\dot{d}}{\dot{h}}\right] \nonumber\\ &\quad+\rho \ddot{h}\big(\sqrt{c^{2}-x^{2}}+f\left(x\right)-h\left(t\right)-d\left(t\right)\big). \end{align}

In the equations, the dot notation represents the time derivative; $d(t)$ is a function with two general cases considered: one is $d(t)\equiv 0$ , which corresponds to the linearisation of the original Wagner problem at the initial liquid level $z=0$ . The second case is $d(t)=f[c(t)]-h(t)$ , which corresponds to the linearisation at the jet height. In the OLM, the shape of the object is not accounted for in (3.11). Specifically, the second-order terms related to the small deadrise angle $\beta$ is neglected and $d(t)\equiv 0$ is assumed, leading to the following expression for the pressure:

(3.12) \begin{align} P_{\textit{OLM}}\left(x,t\right)&=\frac{1}{2}\rho \dot{h}^{2}\left(\frac{\pi }{\tan \beta }\frac{c}{\sqrt{c^{2}-x^{2}}}-\frac{c^{2}}{c^{2}-x^{2}}\right) \nonumber\\ &\quad +\rho \ddot{h}\big(\sqrt{c^{2}-x^{2}}+f\left(x\right)-h\left(t\right)\big) .\end{align}

The velocity potential in the MLM, compared with the original Wagner model, adds a first-order term obtained by Taylor expansion at the surface of the wedge. This treatment allows the MLM to provide more accurate pressure distributions and impact force for objects with larger deadrise angles. The MLM considers the shape of the object by keeping the second-order term of the deadrise angle, while maintaining $d(t)=0$ , resulting in the following expression:

(3.13) \begin{align} P_{\textit{MLM}}\left(x,t\right)&=\frac{1}{2}\rho \dot{h}^{2}\left(\frac{\pi }{\tan \beta }\frac{c}{\sqrt{c^{2}-x^{2}}}-\cos ^{2} \beta \frac{c^{2}}{c^{2}-x^{2}}-\sin ^{2} \beta \right) \nonumber\\ &\quad +\rho \ddot{h}\big(\sqrt{c^{2}-x^{2}}+f\left(x\right)-h\left(t\right)\big). \end{align}

The GWM takes $d(t)=f[c(t)]-h(t)$ , and also accounts for the object shape by retaining the second-order term of the deadrise angle, yielding

(3.14) \begin{align} P_{\textit{GWM}}\left(x,t\right)&=\frac{1}{2}\rho \dot{h}^{2}\left(\frac{\pi }{\tan \beta }\frac{c}{\sqrt{c^{2}-x^{2}}}-\cos ^{2} \beta \frac{c^{2}}{c^{2}-x^{2}}-\sin ^{2} \beta +2-\pi \right) \nonumber\\ &\quad +\rho \ddot{h}\big(\sqrt{c^{2}-x^{2}}+f\left(x\right)-\frac{\pi }{2}h\left(t\right)\big) .\end{align}

For all three methods, there are regions where the pressure is negative, and the value approaches negative infinity as $x\rightarrow \pm c$ . This singularity is common to most modified Wagner models. Generally, to ensure that the pressure remains positive, the region considered for impact pressure is limited to a small distance before $x=c$ , denoted as point  $a$ , where $P [a(t),t]=0$ is the solution. The impact force is then obtained by integrating over the positive pressure regions

(3.15) \begin{align} F=\int _{-a}^{a}P\left(x,t\right)\mathrm{d}x .\end{align}

It has been demonstrated that the four models mentioned above are reasonably accurate in calculating the impact pressures of wedge-shaped bodies entering water (Tassin et al. Reference Tassin, Jacques, El Malki Alaoui, Nême and Leblé2010; Wang & Faltinsen Reference Wang and Faltinsen2017; Wen et al. Reference Wen, Qu, Liu and Hu2024). However, the existing formulations of these models are only applicable to pure water entry and are not suitable for aerated water conditions. Consequently, in the next subsection, modifications of these four theoretical models (ZFM, OLM, MLM and GWM) will be conducted to consider the aeration effect.

3.2. Modified wedge entry models with aeration effects

In the modified model, aerated water is represented as a homogeneous mixture of two immiscible phases, compressible gas (air) and compressible liquid (pure water). A representative volume element is considered, within which the volumes of liquid and gas phases are denoted by $V_{l}$ and $V_{g}$ , respectively. The total representative volume $V$ is thus expressed as

(3.16) \begin{align} V=V_{l}+V_{g} .\end{align}

The void fraction $\alpha$ , representing the volumetric fraction of gas within the representative volume, is defined as

(3.17) \begin{align} \alpha =\frac{V_{g}}{V} .\end{align}

In this study, surface tension effects are neglected. A no-slip condition is enforced between the water and gas phases, implying that both phases share the same velocity field. Furthermore, the representative volume is assumed to be in static equilibrium, and thus the detailed behaviour of individual bubbles and the nonlinear effects associated with air bubble oscillations are not considered. Accordingly, the pressures in the liquid and gas phases, denoted by $P_{l}$ and $P_{g}$ , are equal ( $P$ ). These simplifying assumptions align with those adopted in previous studies on aerated water impact (Peregrine & Thais Reference Peregrine and Thais1996; Bredmose et al. Reference Bredmose, Peregrine and Bullock2009). With this hypothesis, the gas–water mixture is described as a barotropic fluid governed by a single pressure–density relationship.

The gas phase is assumed to be compressible and undergoes isentropic compression. Accordingly, the relationship between density and pressure is expressed as follows (Miller et al. Reference Miller, Jasak, Boger, Paterson and Nedungadi2013):

(3.18) \begin{align} \frac{\partial \rho _{g}}{\partial P}=\frac{1}{c_{g}^{2}}=\frac{1}{a_{c}\gamma }\left(\frac{P}{a_{c}}\right)^{\frac{1-\gamma }{\gamma }}\! .\end{align}

Here, $\rho _{g}$ denotes the gas density, $c_{g}$ is the speed of sound in the gas and $a_{c}=P/\rho ^{\gamma }$ represents the isentropic constant, where $\gamma$ is the polytropic index. Specifically, $\gamma =1$ corresponds to isothermal conditions, while $\gamma =1.4$ indicates adiabatic conditions. Peregrine & Thais (Reference Peregrine and Thais1996) investigated the influence of the polytropic index and determined that its effect on the impact loads in aerated water is minor. In the present study, the equilibrium radius of bubbles generated in the aerated water was approximately $10^{-3}$ m. Given that compression occurs significantly faster than thermal exchange during impact, the process can be reasonably approximated as adiabatic. Consequently, a value of $\gamma =1.4$ was adopted. For validation, additional calculations with $\gamma =1.0$ (isothermal conditions) were also performed, yielding results closely aligned with those for $\gamma =1.4$ .

By integrating (3.18), the variation of gas density $\rho _{g}$ can be expressed as follows:

(3.19) \begin{align} \rho _{g}=\rho _{g0}\left(\frac{P}{P_{0}}\right)^{\frac{1}{\gamma }}\! ,\end{align}

where $\rho _{g0}$ denotes the reference ambient gas density, and $P_{0}$ is the atmospheric pressure. In this study, $\rho _{g0}$ is set at 1.29 $\,\mathrm{kg}\,\mathrm{m}^{-3}$ , and $P_{0}$ at $10^{5}$ Pa. The initial volumes of gas, liquid and mixture are denoted by $V_{g0}$ , $V_{l0}$ and $V_{0}$ , respectively, while $\alpha _{0}$ represents the initial gas void fraction. Upon compression, the volume of the gas phase, $V_{g}$ , adheres to the adiabatic relationship

(3.20) \begin{align} V_{g}=V_{g0}\left(\frac{P}{P_{0}}\right)^{-\frac{1}{\gamma }}\! .\end{align}

The liquid phase is regarded as weakly compressible, and the relationship between its density and pressure is provided by Miller et al. (Reference Miller, Jasak, Boger, Paterson and Nedungadi2013)

(3.21) \begin{align} \frac{\mathrm{D}\rho _{l}}{\mathrm{D}P}=\frac{1}{c_{l}^{2}} ,\end{align}

where $c_{l}$ denotes the speed of sound in water, which is taken as 1500 m s^–1 in this study. Integration of (3.21) yields a linear relationship between liquid density and pressure

(3.22) \begin{align} \rho _{l}=\rho _{l0}+\frac{1}{c_{l}^{2}}(P-P_{0}) .\end{align}

Here, $\rho _{l0}$ represents the density of water under atmospheric conditions, taken as 1000 $\,\mathrm{kg}\,\mathrm{m}^{-3}$ . The volumetric change in the liquid phase is linked to pressure variations through the linear equation of state

(3.23) \begin{align} \frac{V_{l}}{V_{l0}}=\left(1+\frac{P-P_{0}}{K_{l0}}\right)^{-1}\! ,\end{align}

where $K_{l0}$ denotes the bulk modulus of water, defined by $K_{l0}=\rho _{l0}c_{l}^{2}$ . Combining (3.19) and (3.22), the density of the gas–liquid mixture can be expressed as

(3.24) \begin{align} \rho =\alpha \rho _{g}+(1-\alpha )\rho _{l} .\end{align}

An increase in pressure leads to the compression of the gas–water mixture, and the volume ratio $V/V_{0}$ can be directly related to the volumetric changes of both gas and liquid phases

(3.25) \begin{align} \frac{V}{V_{0}}=\frac{V_{g}+V_{l}}{V_{0}}=\frac{V_{g0}}{V_{0}}\frac{V_{g}}{V_{g0}}+\frac{V_{l}}{V_{l0}}\frac{V_{l0}}{V_{0}}=\alpha _{0}\frac{V_{g}}{V_{g0}}+\left(1-\alpha _{0}\right)\frac{V_{l}}{V_{l0}} .\end{align}

Figure 5.

Snapshots of wedge entering aerated water process at different time instants. $t=0 \mathrm{ms}$ marks the moment when the wedge first made contact with the water surface. The bright white region highlights the distribution and accumulation of bubbles around the wedge as it progressively submerges.

By substituting (3.20) and (3.23) into (3.25), the relationship between the void fraction  $\alpha$ and pressure $P$ is obtained

(3.26) \begin{align} \alpha =\frac{V_{g}}{V}=\frac{\alpha _{0}\times \left(\frac{P}{P_{0}}\right)^{-\frac{1}{\gamma }}}{\alpha _{0}\times \left(\frac{P}{P_{0}}\right)^{-\frac{1}{\gamma }}+\left(1-{\unicode[Arial]{x03B1}} _{0}\right)\times \left(1+\frac{P-P_{0}}{K_{l0}}\right)^{-1}} .\end{align}

For a homogeneous air–water mixture, the mixture sound speed $c_{m}$ can be expressed as

(3.27) \begin{align} \frac{1}{\rho c_{m}^{2}}=\frac{\alpha }{\rho _{g}c_{g}^{2}}+\frac{1-\alpha }{\rho _{l}c_{l}^{2}} ,\end{align}

where $c_{g}$ is the speed of sound in pure air, and $c_{l}$ is the speed of sound in pure water. The sound speed $c_{m}$ of the air–water mixture drops dramatically with increasing void fraction. This reduction in sound speed has an immediate effect on the local Mach number, defined as

(3.28) \begin{align} \text{Mach}=\frac{v_{e}}{c_{m}} ,\end{align}

where $v_{e}$ is the expansion velocity of the wetted surface. A higher Mach number indicates that compressibility effects are more pronounced.

It is worth noting that, in the present study, the highest Mach number achieved is approximately 0.15. The compressibility of the liquid phase has a more negligible effect on the impact loads than that of the gas phase. Within the pressure range considered, the volume change of the gas phase in aerated water is significantly greater than that of the liquid phase. The dominant contribution to the volume change of the aerated water arises from the compression of the gas phase. However, the inclusion of liquid compressibility in the model is intended to enhance the model’s robustness and extend its applicability. This consideration may allow the proposed method to remain valid and accurate in scenarios involving higher impact pressures or alternative engineering conditions where liquid compressibility may become comparable to the gas compressibility.

Figure 5 illustrates high-speed snapshots capturing the process of wedge entering aerated water. It can be observed that the brightness distribution on the wetted surface of the wedge appeared relatively uniform and gradually intensified with increasing penetration depth. This observation indicates that bubbles near the wedge surface were approximately uniformly distributed and progressively accumulated as the wedge penetrated deeper. Evidently, the actual void fraction in the region surrounding the wedge during entry significantly exceeded the initial void fraction of the water prior to entry. Consequently, when developing a theoretical model for aerated water-entry impacts, it is essential to consider the temporal variation of the void fraction within the fluid regions close to the wedge surface.

The wedge entry process is governed by the principle of momentum conservation, expressed as

(3.29) \begin{align} Mv+M_{a}v=Mv_{0} ,\end{align}

where $M$ represents the mass of the wedge, $M_{a}$ is the added mass, $v$ is the instantaneous wedge velocity and $v_{0}$ is the initial entry velocity. According to the results reported by Yettou, Desrochers & Champoux (Reference Yettou, Desrochers and Champoux2007), the added mass for a three-dimensional wedge entry is given by

(3.30) \begin{align} M_{a}=C_{\!a}\rho 'c(t)^{2} ,\end{align}

where $\rho '$ denotes the density of the fluid, and the coefficient $C_{\!a}$ is defined as

(3.31) \begin{align} C_{\!a}=\frac{\pi }{2}\left(1-\frac{\beta }{2\pi }\right)^{2}\boldsymbol{\cdot }\varepsilon .\end{align}

In this expression, $\varepsilon$ represents a correction factor accounting for three-dimensional effects due to the finite length of the wedge. This correction factor $\varepsilon$ is dependent on the ratio of the wetted region width $2c$ to the wedge length $L$ , defined as $K=2c/L$ . Based on the Blagovenshchensky formula (Baarholm Reference Baarholm2009), the correction factor $\varepsilon$ can be expressed as

(3.32) \begin{align} \varepsilon =\frac{1}{\sqrt{1+K^{2}}}\left(1-\frac{0.425K}{1+K^{2}}\right)\! .\end{align}

The concept of added mass arises from the physical phenomenon in which a structure moving through a fluid must accelerate a portion of the surrounding fluid, thereby experiencing an additional inertial force. This effect becomes especially significant during rapid acceleration or deceleration, as in water-entry scenarios. As first proposed by von Karman (Reference von Karman1929), the predominant mechanism driving impact loads during water entry is an added-mass effect due to the fluid being displaced by the submerging wedge. Therefore, the added mass can be interpreted as a crucial parameter that characterises the inertial influence of fluid on structural dynamics.

Figure 6.

Schematic illustration depicting the determination of the equivalent depth and void fraction calculations during wedge entry into aerated water. Due to the upward motion of the bubbles at relative velocity $V=V_{\textit{wedge}}+V_{\textit{bubble}}$ , additional bubbles (shown in red dashed circles) enter the considered volume, increasing the gas content. This process updates the void fraction to a new value $\alpha _{1}$ , capturing the temporal variation of void fraction essential for accurately modelling aeration effects on impact loading.

In accordance with Wagner model (Wagner Reference Wagner1932) and associated models (Lin & Shieh Reference Lin and Shieh1997; Belden et al. Reference Belden, Speirs, Hellum, Jones, Paolero and Truscott2023, Reference Belden, Speirs, Hellum, Loubimov and Truscott2024), the fluid region directly beneath the structure is considered as the primary contributor to the impact load. Accordingly, the added mass is typically attributed to this localised region, which dominates the force response during water entry. Consistent with this approach, the present study adopts the same treatment for the added mass. As illustrated in figure 6, the dot-dashed rectangular area directly beneath the wedge denotes the control volume associated with the added mass. Since the void fraction is a volume-based physical quantity, it is meaningful only when defined within a clearly specified finite region, such as the representative volume defined previously. Consequently, it is reasonable to define the void fraction within the control volume associated with the added mass, shown in figure 6. Moreover, given the flat-plate approximation in these theoretical models, the void fraction is assumed to be spatially uniform along the wetted wedge surface at any given time step. An equivalent depth $Z_{a}$ of the region corresponding to the added mass is obtained as follows, enabling a more accurate quantification of void fraction variation during impact:

(3.33) \begin{align} Z_{a}=\frac{M_{a}}{\rho '\boldsymbol{\cdot }2c(t)}=\frac{C_{\!a}\boldsymbol{\cdot }c(t)}{2} .\end{align}

At time $t_{0}$ , let the void fraction be $\alpha _{0}$ , the equivalent depth of the added-mass region $Z_{a0}$ , and the wetted half-width $c_{0}$ . After a short interval $\mathrm{d}t$ , these become $\alpha _{1}$ , $Z_{a1}$ and $c_{1}$ at time $t_{1}$ . Given the known $\alpha _{0}$ at $t_{0}$ and the need to obtain $\alpha _{1}$ at $t_{1}$ , the gas volume balance is performed over the updated control region characterised by the new wetted half-width $c_{1}$ and equivalent depth $Z_{a1}$ . The gas volume in this region is $V_{0}=Z_{a1}\times 2c_{1}\times \alpha _{0}$ when evaluated at $t_{0}$ , and $V_{1}=Z_{a1}\times 2c_{1}\times \alpha _{1}$ at $t_{1}$ . The increase $V_{1}-V_{0}$ represents the additional gas introduced into this volume during $\mathrm{d}t$ by bubbles entering from below. The outer bubbles (shown in red dashed circles in figure 6) enter the considered fluid volume ( $Z_{a1}\times 2c_{1}$ ) with a relative velocity $V=V_{w\textit{edge}}+V_{\textit{bubble}}$ , contributing an additional gas volume ${\unicode[Arial]{x0394}} V=V\mathrm{d}t\times 2c_{1}\times \alpha _{0}$ . Within this short time step, the escape and collapse of bubbles are neglected. The void fraction is then updated by

(3.34) \begin{align} \alpha _{1}=\frac{(Z_{a1}+V\mathrm{d}t)\boldsymbol{\cdot }\alpha _{0}}{Z_{a1}} .\end{align}

In this study, bubbles are assumed to be uniformly distributed with a constant rising velocity $V_{\textit{bubble}}$ of 0.3 m s^–1, as determined experimentally by Wu et al. (Reference Wu, Xiao, Kou and Liu2025), to whom readers are referred for further methodological details.

Furthermore, since the previously described four models are based on two-dimensional theory, three-dimensional effects must be considered for practical wedge geometries of finite length, such as the one in the current study with a length-to-breadth ratio $L/B=2$ . Consequently, the following correction of the impact force for three-dimensional effects is applied:

(3.35) \begin{align} F=F_{2D}\boldsymbol{\cdot }\varepsilon ,\end{align}

where $F_{2D}$ is the two-dimensional impact force obtained directly from (3.15).

By applying numerical methods and iteratively solving for the void fraction and the effective density of the gas–liquid mixture at each time step, the local pressure at any given instant and location can be determined using (3.8), (3.12), (3.13) and (3.14), which correspond to the ZFM, OLM, MLM and GWM, respectively. Additionally, it should be noted that theoretical models typically yield pressure predictions at a single point on the wedge surface. To facilitate direct comparisons with experimental measurements, an area-averaged pressure corresponding to an 8 mm diameter sensor is calculated.

In summary, by incorporating the enhanced compressibility of the aerated water, the time-dependent variation in void fraction, the aeration effect on gas–liquid mixture density and added mass, along with three-dimensional effects and area-averaged pressure, modified models (M-ZFM, M-OLM, M-MLM, M-GWM) have been developed for wedge entry impact considering aeration effects. These models will subsequently be validated against experimental results to evaluate their accuracy and be further analysed to provide deeper insights into the pressure characteristics during wedge entry into aerated water.

4.1. Impact pressure time series

Water-entry impact is regarded as a typical fluid–structure impact phenomenon, with the resulting impact pressure serving as a critical parameter in such interactions. Figure 7 presents typical pressure time series in pure water recorded by sensors P01–P06 (located along the centreline of the wedge), as well as sensor P10 (positioned at wedge edge, aligned with P04 in height) and sensor P24 (placed symmetrically relative to P03 with respect to the keel).

Figure 7.

Pressure time series for pure water measured by sensors P01–P06, P10 and P24. The time series were aligned by setting the start time $t_{s}$ of the first sensor at the keel as $t=0$ , with the start time identified using a threshold set at 1 % of the peak impact pressure. The dashed horizontal black line represents zero pressure, and the dashed vertical grey line denotes the moment of the near-simultaneous pressure drop. Sensors P01–P06 were located along the centreline of one side of the wedge, with each sensor spaced 20 mm apart. Sensors P10 was placed at the same height as P04, but near the wedge edge. Sensors P24 was installed at symmetrical longitudinal and vertical positions relative to P03, but on the opposite side of the wedge.

Figure 8.

High-speed camera snapshot showing the jet flow development during wedge impact process. The dashed red line marks the symmetry axis of the jet flow, highlighting that the wedge entered the water in a symmetrical way.

Upon initial contact between the sensors and the water surface, a pronounced pressure spike was generated, lasting of the order of $10^{-3}$ seconds. As the water surface passed successively over the sensors, the peak pressures were recorded sequentially by sensors P01–P06. Then, a simultaneous decline was detected in the recordings from all the sensors, as illustrated by the grey dashed line in figure 7. Near-identical pressure profiles were observed between sensors P03 and P24, implying a perfectly vertical and symmetrical entry which was further confirmed by the symmetrical jet flow relative to the red dashed line displayed in figure 8. This agreement highlights the precision of the experimental set-up and further supports the reliability of the results. Comparing sensors located along the wedge’s keel with same height, the peak pressure recorded by sensor near edge (P10) is distinctly lower than that measured at centre (P04). Additionally, the timing of the peak pressure for the edge sensor, denoted by $t_{P10}$ , exhibited a measurable delay than the corresponding peak at the centreline $t_{P04}$ . Similar variations were identified when comparing other paired sensors along the keel (P01–P06 and P07–P12). This discrepancy is primarily attributed to the three-dimensional effect of the wedge’s interaction with the water, as the wedge behaved as a finite three-dimensional structure, rather than a two-dimensional one as assumed in theories such as Wagner (Reference Wagner1932). The water flow became more dispersed at edge, which induced momentum along the keel direction (as shown in figure 8), leading to a delay in the arrival at edge sensor and a subsequent reduction in impact pressure. Therefore, it is crucial to account for the three-dimensional flow effect in the modified analytical models.

Analysis of pressure distributions along the centreline (sensors P01–P06) revealed a nonlinear relationship between the magnitude of impact loads and the proximity to the wedge’s keel, characterised by an initial rise followed by subsequent decay. This trend aligned with experiments by Barjasteh, Zeraatgar & Javaherian (Reference Barjasteh, Zeraatgar and Javaherian2016), who investigated symmetric wedge entry (deadrise angles ranging from 5° to 30°) using 9 mm diameter pressure sensors. In contrast, Yettou, Desrochers & Champoux (Reference Yettou, Desrochers and Champoux2006) employed free-falling wedges with deadrise angles of 15° to 35° and reported that the impact pressures, measured by 19 mm diameter sensors, attained maximum values closest to the keel. Such conflicting spatial distributions emphasise the sensitivity of pressure measurements to sensor dimensions and positioning. Specifically, the finite sensing area inherently averages local pressure gradients, while variations in sensor size and installation position across studies amplify discrepancies through spatial averaging effects. Consequently, from an experimental perspective, the variation of impact pressure with respect to the distance from the keel can hardly be described by a unified deterministic model in the absence of point pressure sensors – a capability currently unattained due to technological limitations in sensor miniaturisation. These findings are consistent with earlier work by Chuang (Reference Chuang1966) and Yamamoto, Ohtsubo & Kohno (Reference Yamamoto, Ohtsubo and Kohno1984), which similarly emphasised the lack of a systematic trend in the variation of peak pressures from locations near the keel to those further away.

Figure 9.

Evolution of impact pressure coefficient $C_{P}$ for a range of water-entry velocities in pure water, obtained from both theoretical models and model tests. Panel (a) shows the results for sensor P01, while panel (b) displays the results for sensor P02. The theoretical predictions are based on five models: O-ZFM (grey solid curve), M-ZFM (blue solid curve), M-OLM (orange dotted curve), M-MLM (green dashed curve) and M-GWM (red dash-dotted curve). The phases of the data have been adjusted so that at $t=0$ , the pressure at P01 starts to rise.

To validate the applicability of the theoretical models presented in § 3, figure 9 displays the evolution of impact pressure coefficient $C_{P}$ at sensors P01 and P02 predicted by five analytical models, together with the experimental data for varying water-entry velocities in pure water. Here, $h_{\textit{max}}$ refers to the vertical distance from the keel to the knuckle of the wedge. The data were adjusted in time so that at dimensionless time ${tv}_{i}/h_{\textit{max}}=0$ the pressure at sensor P01 begins to rise, with the model calculations adjusted accordingly. Notably, the peak pressure was over-predicted by original Zhao & Faltinsen model (O-ZFM). A gradual reduction in peak impact pressure was observed along the centreline in the O-ZFM model, which contrasted with the experimental trend. Moreover, the impact pressure from O-ZFM exhibited an extremely rapid, near-singular rise, a phenomenon not matched by the pressure time history from the model test. This disagreement arises primarily from the point-based pressure prediction in O-ZFM, combined with its failure to account for three-dimensional flow effect.

From figure 9 it is evident that, until the attainment of the peak impact pressure, the temporal evolution of impact pressure was accurately captured by all four modified models, showing a more gradual pressure rise compared with the O-ZFM. In pure water, the four refined models, which take into account the area-averaged pressure corrections and three-dimensional flow effects, generally demonstrated improved predictive accuracy for peak pressure magnitudes, with the M-ZFM delivering superior agreement with experimental data. The predictive accuracy of the peak pressures in both pure water and aerated water will be discussed in further detail in § 4.3. However, as the immersion depth increased, more pronounced discrepancies in the pressure time series were observed, as illustrated in figure 9. Although the M-OLM and M-MLM models tended to overestimate the pressure time series after the peak, their temporal profiles were in closer alignment with experimental trends than those of the M-ZFM. Notably, the M-GWM model achieved the highest fidelity to experimental measurements during the post-peak period. This enhanced predictive capability may be attributed to the integration of body geometry into the Bernoulli equation and the linearisation of the original Wagner problem at the splash-up height, enabling more physically consistent modelling throughout the entire impact process. Further validation was provided by the temporal delay in pressure rise between sensors P01 and P02, which was successfully reproduced by all modified models.

Figure 10.

Dimensionless impact pressure $C_{P}$ time histories at sensor P01 predicted by modified theoretical models (M-ZFM, M-OLM, M-MLM, M-GWM) under aerated water conditions. Panels (a) and (b) correspond to moderate ( $\alpha =1.07\mathrm{\,\% }$ ) and high ( $\alpha =1.77\mathrm{\,\% }$ ) void fractions, respectively. The inset in (a) displays a zoomed-in view of the post-peak pressure region, highlighting the oscillations observed in the experimental pressure measurements. The experimental data are plotted for entry velocities ( $v_{i}$ ) ranging from 2.15 to 3.69 m/s. As plotted earlier, time series are shifted so that the point at which the reading on pressure sensor P01 starts to rise lies at ${tv}_{i}/h_{\textit{max}}=0$ .

Accurate prediction of impact loads in aerated water is also critical, as aeration of the water–air mixture is common in open seas. Figure 10 presents the dimensionless impact pressure at P01 calculated by different models and measured during impact tests under moderate ( $\alpha =1.07\,\%$ ) and high ( $\alpha =1.77\,\%$ ) void fraction conditions. The phases of the data had been appropriately adjusted to ensure that $t=0$ aligned with the start time  $t_{s}$ of the pressure rise at sensor P01. Consistent with the pure water case, the modified models closely replicated both the peak pressure and pre-peak pressure rise characteristics in aerated conditions. A reduction in peak pressure is evident in aerated water compared with pure water.

Figure 11 further illustrates the temporal evolution of errors between the model predictions and experimental measurements of the dimensionless pressure coefficient $C_{P}$ , plotted as a function of the normalised time ${tv}_{i}/h_{\textit{max}}$ . Notably, the post-peak pressure time series exhibited marked deviations: while the M-GWM maintained good agreement with experimental trends (relative errors generally below 10 %), the M-ZFM was again shown to overpredict the post-peak pressures, as clearly depicted in figure 11. The deviations may be due to the inherent limitations in the ZFM, specifically the neglect of the quadratic velocity term in Bernoulli’s equation within the outer solution, and the unsatisfactory treatment of the body boundary condition in the outer flow solution. Fortunately, the accuracy of the peak pressure was not significantly affected, as detailed in § 4.3. Unlike in pure water, the M-OLM and M-MLM achieved post-peak predictive accuracy comparable to the M-GWM in aerated water. This suggests that under aerated conditions, aeration effects play a more dominant role in governing post-impact pressure dynamics than both the location of velocity potential linearisation and the geometric fidelity of the wetted surface.

Figure 11.

The error between model predictions and experimental measurements of the dimensionless pressure coefficient $C_{P}$ as a function of normalised time ${tv}_{i}/h_{\textit{max}}$ for $\alpha =1.07\mathrm{\,\% }$ . The error is defined as $C_{P}^{\textit{model}}-C_{P}^{\textit{exp}}$ . Results are presented for four different impact velocities $v_{i}=2.15, 2.70, 3.24, 3.69$ m/s, shown from top to bottom. Obviously, the M-OLM, M-MLM and M-GWM demonstrate better agreement with the experimental data, particularly in the post-peak region. In the vicinity of the pressure peaks, noticeable fluctuations in the error are observed, primarily due to the oscillations present in the experimental pressure. A more detailed comparison between the experimental and predicted peak values is provided in § 4.3. It is also noted that within the peak region, the agreement could be further improved by applying a slight time shift.

In comparison with pure water cases, the pressure oscillations observed following the peak were more pronounced in aerated water, as highlighted in the inset of figure 10. The period of these oscillations corresponds closely to the natural pulsation frequency (Minnaert frequency) of the millimetric air bubbles in the aerated water, as demonstrated by Wu et al. (Reference Wu, Xiao, Kou and Liu2025). The oscillations can be interpreted as bubble pulsations: immediately after impact, compressed bubbles expand and contract, creating damped pressure fluctuations. Because the bubbles in the mixture have a distribution of sizes and are not oscillating in unison, the resulting pressure oscillations are largely stochastic and non-deterministic in nature, as evidenced by the marked reduction in amplitude after ensemble averaging of repeated tests (Wu et al. Reference Wu, Xiao, Kou and Liu2025). Such oscillations were not observed in the theoretical results, which explains the larger errors observed near the peak pressure regions depicted in figure 11. This discrepancy stems from the model simplifications employed in the theoretical approach, wherein the water was modelled as a homogeneous, uniformly aerated mixture without explicitly considering the detailed behaviour of individual bubbles. These simplifications were necessary, as modelling the unsynchronised expansion and compression of numerous bubbles, along with random effects, pose substantial challenges for accurate theoretical representation.

4.2. Expansion velocity of wetted surface

The expansion velocity of the wetted surface ( $v_{e}$ ) is a critical parameter governing hydrodynamic impact dynamics, as it directly quantifies the propagation rate of the liquid–solid interface. One of the key contributions of Wagner theory was the definition of the wetted surface half-length $c(t)$ for specific geometries. This foundational concept has been adopted in subsequent Wagner-based theoretical models, which establish a deterministic functional relationship between impact pressure and $c(t)$ , as shown in (3.8), (3.12), (3.13) and (3.14) in § 3.1. For a wedge with a deadrise angle $\beta$ , $c(t)$ is defined as

(4.1) \begin{align} c(t)=\frac{\pi v_{i}t}{2\tan \beta } ,\end{align}

where $v_{i}$ denotes the water-entry velocity and $\beta$ is the wedge deadrise angle. Considering the real geometry of the wedge and constant entry velocity $v_{i}$ , the expansion velocity of the wetted surface $v_{e}$ along the wedge can be obtained from Wagner’s definition of $c(t)$ as

(4.2) \begin{align} v_{e}=\frac{\mathrm{d}c(t)}{\mathrm{d}t}\boldsymbol{\cdot }\frac{1}{\cos \beta }=\frac{\pi v_{i}}{2\sin \beta } .\end{align}

For all the experimental results, the expansion velocity $v_{e}$ within the range of the first and last sensors was determined by tracking the position of the jet using the first pressure spike recorded at each sensor. Let $x_{01}$ and $x_{02}$ denoted the positions of sensors P01 and P02, respectively, with $t_{P01}$ and $t_{P02}$ representing the corresponding times to peak pressure. The local expansion velocity between sensors P01 and P02 ( $v_{e}^{12}$ ) was calculated as

(4.3) \begin{align} v_{e}^{12}=\frac{(x_{02}-x_{01})/\cos \beta }{t_{P02}-t_{P01}} .\end{align}

This methodology was applied to adjacent sensor pairs along the wedge centreline (P01–P06), yielding five independent expansion velocities ( $v_{e}^{12}$ , $v_{e}^{23}$ , $v_{e}^{34}$ , $v_{e}^{45}$ , $v_{e}^{56}$ ). Similar calculation was performed for the sensors at the edge (P07–P12). Figure 12(a) and 12(b) depict the measured expansion velocity $v_{e}$ along the wedge in the central (P01–P06) and edge (P07–P12) regions, respectively. Each column of hollow markers in figure 12 represents the $v_{e}$ values from five repeated trials at a specific void fraction ( $\alpha$ ) and entry velocity ( $v_{i}$ ), while solid markers indicate the corresponding mean values. Theoretical results from the Wagner model (4.2) and the M-ZFM (derived from (4.3) using predicted impact pressure histories) are also plotted for comparison.

Figure 12.

Relationship between the wet surface expansion velocity $v_{e}$ and water-entry velocity $v_{i}$ at varied void fractions. Panel (a) shows the data for sensors P01–P06 (central region), while panel (b) shows the data for sensors P07–P12 (edge region). Experimental results are represented by hollow markers (five repeated tests for each condition) and solid markers (mean values). The dashed black line represents theoretical predictions from Wagner model, and the dotted purple line corresponds to the results from the M-ZFM. The expansion velocity predicted by M-ZFM in the central region keeps constant across all void fractions. Notably, for each experiment at a specific void fraction and entry velocity, five expansion velocities ( $v_{e}^{12}$ , $v_{e}^{23}$ , $v_{e}^{34}$ , $v_{e}^{45}$ , $v_{e}^{56}$ ) in the central region are grouped under a single marker type, with same grouping also applied to edge region sensors aligned parallel to the centreline.

One observes that the Wagner theory overestimated the $v_{e}$ in both the central and edge regions. This discrepancy arose primarily from the assumption of a constant water-entry velocity, a simplification commonly adopted in many other experimental studies and theoretical models. The constant velocity assumption failed to account for the varying dynamics during impact, leading to a higher predicted expansion velocity and thereby the impact pressure. In contrast, the M-ZFM presented in this study aligned well with the measured expansion velocity in the central region, with the error being less than 4 % in pure water. As shown in figure 9, the time interval between the peak pressures at sensors P01 and P02 was also well captured by the theoretical model, which accounts for varying entry velocities.

The expansion velocity at the edge was generally lower than that at the centre. Near the edge, jet flow momentum dispersed laterally, inducing a horizontal velocity component that diverted energy from the primary wetting surface development along the wedge. As a result, the expansion velocity at the edge was significantly lower compared with the central region, highlighting the importance of considering the spatial variation in jet flow dynamics.

In the experiment, the introduction of aeration inevitably caused minor surface disturbances, potentially leading to slight variations in the initial water contact timing at the sensors, and thereby introducing small errors for calculating $v_{e}$ . However, as depicted in figure 12, the mean expansion velocities for all void fractions remained relatively consistent, with fluctuations kept within 10 %. Furthermore, the expansion velocity predicted by M-ZFM, developed under the assumption of a flat water surface, remained constant across all tested void fractions for a certain entry velocity, and compared well with the experiments in the central region with a maximum error of less than 10 %. This alignment suggests that the disturbances in the water surface due to aeration are within an acceptable range.

Figure 13.

Variation of peak impact pressure $P_{\textit{max}}$ recorded by sensor P01 relative to the square of the entry velocity $v_{i}^{2}$ at varied void fractions. For each test condition, the $P_{\textit{max}}$ was calculated by averaging the records at all the tests carried out. The linear relationship between $P_{\textit{max}}$ and $v_{i}^{2}$ remains evident at all tested void fractions, while the slope of this linear correlation decreases with increasing void fraction. The dashed lines represent the linear fit of the scatter data for each corresponding void fraction, with the fitting coefficient $R^{2}$ provided in the legend.

4.3. Peak impact pressure

An analysis of the peak impact pressure characteristics was conducted to examine the effects of impact velocity and initial void fraction on impact load during water entry. Figure 13 plots the peak impact pressures $P_{\textit{max}}$ , averaged from five repeated impact tests for each test condition (characterised by $v_{i}$ and $\alpha$ ), as recorded by sensor P01. From the figure, $P_{\textit{max}}$ was found to scale linearly with the square of the velocity $v_{i}^{2}$ in pure water, consistent with the expectations for an inertia impact process. A similar scaling behaviour was also reported by Han et al. (Reference Han, Li, Zhang, Zou and Zhao2024) using free-falling wedges with deadrise angles of 20°, 30° and 45° in pure water. Additionally, the linear relationship persisted in aerated water ( $\alpha =0.80\,\%{-}1.77\,\%$ ), with only a few points at high entry velocities deviating from the linear trend. This deviation may be due to the limited temporal resolution of the pressure sensor. The slopes of the linear fits of the $P_{\textit{max}}-v_{i}^{2}$ curve decreased with increasing void fraction. For pure water ( $\alpha =0.00\,\%$ ), the slope measured 7.08 whereas it decreased to 4.49 at $\alpha =1.77\,\%$ . In an inertia-dominated impact process, the slope of the $P_{\textit{max}}-v_{i}^{2}$ relationship for a given wedge geometry is primarily governed by the effective density of the impacting fluid. The observed reduction in slope demonstrated that the introduction of gas bubbles lowered the density of the air–water mixture, thereby reducing the peak pressures. This further validates the rationality and necessity of integrating aeration effects into the theoretical model proposed in § 3. Further analysis was conducted to quantify the influence of aeration on peak impact pressures. As demonstrated in figure 13, $P_{\textit{max}}$ exhibits a marked reduction with increasing void fraction, with the reduction becoming more pronounced at higher aeration level. For instance, at $v_{i}=3.69\,\mathrm{m}\,\mathrm{s}^{-1}$ , the $P_{\textit{max}}$ decreases from 95.35 kPa in pure water to 64.04 kPa at $\alpha =1.77\,\%$ , corresponding to a 32.8 % decrease.

This reduction can be attributed to the presence of entrained air, which provides an effective air cushioning effect. Upon impact, the entrained gas compresses and absorbs a portion of the impact energy, thereby acting as a compliant cushion between the wedge and the liquid. This mechanism reduces the instantaneous force on the wedge, resulting in a lower peak pressure. Meanwhile, in a water-entry event, the dominant mechanism of force generation is the fluid’s inertia (added mass) resisting the body’s motion. With a lower density fluid due to air voids, the effective added mass is reduced, meaning the fluid offers less resistance to the wedge’s penetration. This fundamentally leads to smaller impact pressures. Both effects contribute to a reduced peak impact pressure in aerated water.

In addition, the sound speed $c_{m}$ drops sharply as the void fraction increases in an aerated mixture. Upon impact of the wedge, the initial pressure disturbance propagates into the fluid at speed $c_{m}$ . Over a short time interval ${\unicode[Arial]{x0394}} t$ immediately after impact, the disturbance travels a distance $c_{m}{\unicode[Arial]{x0394}} t$ and accelerates the fluid in that region of mass $M_{f}=\rho c_{m}A{\unicode[Arial]{x0394}} t$ beneath the patch of area $A$ . Equating the momentum gained by this fluid ( $M_{f}v_{i}$ ) with the impulse delivered by the pressure ( $P_{a}A{\unicode[Arial]{x0394}} t$ , where $P_{a}$ is the average pressure), the average pressure can be obtained as

(4.4) \begin{align} P_{a}=\rho c_{m}v_{i} ,\end{align}

which is often referred to as the acoustic pressure for the impact. This analysis illustrates that a reduction in $c_{m}$ directly lowers the impedance $Z_{m}=\rho c_{m}$ and hence yields a lower attainable peak pressure $P_{a}$ for a given impact velocity $v_{i}$ . Although the acoustic pressure  $P_{a}$ is rarely observed in experiments due to the more complex real-world conditions, it links the sound speed to impact pressure and provides a mechanistic rationale for why entrained air leads to a reduction of the peak impact pressure $P_{\textit{max}}$ in aerated water.

In the experimental set-up employed in this study, the evolution of impact pressures was observed to follow the inertial scaling even under aerated conditions, which formed the basis for comparing experimental results with theoretical predictions. Figure 14 displays the non-dimensionalised peak pressure measured at sensors P01–P06 for three representative cases: pure water ( $\alpha =0.00\,\%$ ), low void fraction ( $\alpha =0.80\,\%$ ) and high void fraction ( $\alpha =1.77\,\%$ ), alongside standard deviations from repeated trials. Theoretical predictions from the O-ZFM (pure water only), M-ZFM, M-OLM, M-MLM and M-GWM are also presented. For pure water, the O-ZFM not only overestimated the impact loads at sensors P01 and P02 (as also shown in figure 9) but also overpredicted the load at P03 while underpredicting P06. This misalignment would be more pronounced under aerated conditions, where traditional models neglecting the aeration effect significantly overestimated impact pressures.

Figure 14.

Variation of the dimensionless maximum pressure measured by sensors P01–P06 in different void fractions and entry velocities, with error bars representing the standard deviations of the repeated tests. Sensors P01–P06 were evenly distributed along the centreline of the wedge, maintaining a 20 mm spacing between adjacent sensors. The theoretical predictions from the O-ZFM model (represented by solid black lines for pure water) and the modified models (M-ZFM, M-OLM, M-MLM and M-GWM) are shown with different coloured lines: red solid line for M-ZFM, orange dashed line for M-OLM, light blue dash-dotted line for M-MLM and blue dotted line for M-GWM.

A clear pattern emerged from the data of all six sensors, indicating that peak pressures initially increased with the vertical position of the sensor, reached a maximum at sensor P02 and subsequently decreased toward higher positions (P06). All modified models successfully captured this characteristic pressure distribution. One can observe that, in pure water, experimental peak pressures exhibited low variability and high repeatability. In contrast, greater variance in the peak pressure was introduced by aeration because the bubble field generated by the bubble generator cannot be perfectly consistent in each repeated test. However, despite this variability, the aerated impact pressure was generally lower than pure water, a trend reproduced by the proposed aeration-inclusive models. The modified models proposed in this study, which account for the aeration effect, performed well in predicting the peak impact pressures at P01–P06 in aerated conditions, thereby validating the effectiveness of the model. Among the analytical models, the M-MLM appeared to provide the most accurate results. This improved prediction of hydrodynamic pressure was achieved by considering the nonlinear effects associated with the quadratic term of the Bernoulli equation and the realistic geometry of the wetted surface.

4.4. Impact pressure impulse and duration

The impact pressure measured directly on the wedge is the most basic indicator to describe the impact event. However, in many cases, impact pressure can exhibit great randomness between tests conducted at the same conditions. As previously mentioned, this variability can stem from the limited temporal resolution of the pressure sensors or from the aeration effect. It has been suggested that the time-integrated pressure impulse is more critical for quantifying local damage, owing to its improved repeatability and predictability (Wu et al. Reference Wu, Liu, Xiao, Shan, Kou, Li and Guo2024). Additionally, Manjula & Sannasiraj (Reference Manjula and Sannasiraj2019) found from wave impacts on a slender cylindrical member that waves producing high pressures over short durations were less significant for structural damage than those producing lower pressures over longer durations. Therefore, the safety of structure is not only determined by the peak impact pressure but also by the impact duration. Since the pressure impulse is closely related to the impact duration, the characteristics of impact pressure duration are investigated first in this section, followed by an analysis of the pressure impulse.

The distribution of the impact pressure duration at a range of void fractions is presented in figure 15, denoted by circular markers in different colours. The vertical dashed lines in corresponding colours indicate the mean duration for each $\alpha$ , providing a clear reference for comparing the durations at varying aeration levels. A clear trend is observed that the mean duration gradually increased with higher void fraction. As shown in figure 15(a), the durations rose from 0.04 to 0.05 s in pure water to predominantly exceed 0.05 s at $\alpha =1.77\mathrm{\,\% }$ . This increase in duration can be explained by the reduced inertial resistance of the aerated water. With entrained air lowering the fluid’s effective density, the added mass is reduced, and the fluid provides less resistance to the wedge’s penetration. The wedge penetrates slightly further into the fluid before decelerating, thereby extending the duration of the impact event. In essence, aeration leads to a lower instantaneous impact force but allows the impact to last longer, distributing the energy over a longer time.

Figure 15.

Distributions of impact pressure duration under different void fractions. The vertical dashed lines in each panel correspond to the mean impact duration for each void fraction.

Additionally, the reduction in $c_{m}$ also slows the establishment of the pressure field. The characteristic time for pressure wave to communicate across a length $L$ is the acoustic time scale

(4.5) \begin{align} \tau _{a}=\frac{L}{c_{m}}, \end{align}

where $L$ is a characteristic length (e.g. the equivalent depth of the added-mass region). Thus, a smaller $c_{m}$ in more aerated water yields a larger $\tau _{a}$ , producing a more gradual rise and a broader pulse. This also explains why the aerated cases exhibit longer-lasting pressure pulses compared with the pure water cases. Furthermore, a comparison between figure 15(a) and 15(b) reveals that, as expected, impact durations decreased with increasing water-entry velocity.

The impact pressure impulse, as a parameter combining both impact pressure and duration, serves as a direct indicator of the energy exerted on the wedge during the water-entry process. Figure 16 illustrates the evolution of the dimensionless impulse as a function of dimensionless time for pure water and high-aeration water ( $\alpha =1.77\,\%$ ). Predictions from the M-MLM and M-GWM are also presented, as these models demonstrate better accuracy in predicting peak values and account for the realistic wedge geometry.

Figure 16.

Temporal evolution of normalised impact impulse ( $I/({1}/{2})\rho _{\mathrm{w}}v_{i}h_{\textit{max}}$ ) at sensor P01 comparing experimental measurements (Exp) with predictions from the M-MLM and M-GWM models under two aeration conditions: pure water ( $\alpha =0.00\,\%$ ) and aerated water ( $\alpha =1.77\,\%$ ). For clarity, symbols are shown every 10 samples.

For the experimental results, the impulse in aerated water remained lower than that in pure water in the pre-peak phase. After a certain point following the peak (approximately at ${tv}_{i}/h_{\textit{max}}=1$ ), the pressure impulse under aerated conditions surpassed that of pure water. This may arise from the prolonged impact duration caused by the presence of gas. In contrast, it is observed that the predicted impulse in aerated water was consistently lower than that in pure water for both the M-MLM and M-GWM. This discrepancy could be explained by the assumption in the models that the gas bubbles are uniformly distributed and rise at the same velocity, which is determined based on the experimentally measured average rise velocity of the bubbles near the surface. However, experimentally observed bubble volumetric expansion and inter-bubble collision processes may induce variations in terminal rise velocities. According to the momentum theorem, the individual momentum of the bubbles also affects the total impulse applied to the wedge, potentially leading to an underestimation of the impulse prediction for aerated water. Generally, in pure water, both M-MLM and M-GWM provided good agreement with the experimental evolution of the impact impulse. Comparative analysis revealed that under aerated conditions, the M-MLM model outperformed the M-GWM in predicting the impact impulse evolution.

Figure 17 depicts the dimensionless impact pressure impulse measured by sensors P01–P06 against the void fractions. Error bars representing the standard deviation for five repeated tests are provided at the top of the histogram columns. Spatially, the magnitude of the impulse declined with the location being away from the wedge’s keel, due to the progressive velocity reduction of the wedge during water entry. Comparing figures 14 and 17, it is evident that the variation of impact impulse from repeated experiments was much less than that of peak pressure, for both pure and aerated water entry. Furthermore, there is no reduction of the impact pressure impulses from pure water to aerated water entry. Indeed, a slight increase was even observed at certain sensor locations, such as P01. Introducing gas into pure water reduces the wedge’s peak impact pressure and prolongs the impact load duration. But, by integrating the pressure over time, these two opposing effects partially compensated for each other, resulting in an almost stable impulse. At higher aeration levels, the effect of extended duration became more pronounced, leading to an increased total load applied to the wedge, which may pose a greater risk to the structure. This finding aligns with the conclusions of Cuomo, Shimosako & Takahashi (Reference Cuomo, Shimosako and Takahashi2009) that less intense but longer-lasting wave loads may be more critical for the deck than more intense but shorter-lasting impacts, based on the research involving wave-induced impact loads on bridges.

Figure 17.

Variation of the dimensionless maximum impact impulse at different void fractions for sensors P01 to P06, with error bars at the top of the columns indicating the standard deviation from repeated measurements.