Scaling Behavior of Sloshing Impact Pressures Based on Event Distribution and Regime Classification

Abstract

Sloshing in partially filled tanks generates significant impact pressures that threaten the structural integrity of LNG cargo containment systems, and accurate scaling of these impacts remains a critical issue. Although Froude-based scaling has been widely applied, its validity may be limited under conditions where multiple impact mechanisms coexist. In this study, sloshing impact pressures measured across different scales were analyzed based on individual impact events. Distribution-based representative metrics, including mean and upper-percentile values, were introduced, and scale dependency was quantified using a power-law relationship. The results show that under low filling conditions, impact responses exhibit relatively consistent distributions, and gravity-based scaling yields nearly scale-independent results. In contrast, high filling conditions lead to increased variability and a pronounced expansion of the upper tail, resulting in stronger scale dependency, particularly for high-intensity events. The increase in the power-law exponent indicates that extreme impacts are more sensitive to scale variation. These findings demonstrate that sloshing impact scaling is governed not by a uniform change in pressure magnitude, but by a redistribution of impact intensity across events. Consequently, reliable scaling requires consideration of both distribution characteristics and underlying impact mechanisms.

1. Introduction

Sloshing in partially filled tanks is a highly nonlinear free-surface flow phenomenon that can generate significant impact loads that threaten the structural integrity of LNG cargo containment systems [1,2]. These impacts may reach locally extreme magnitudes and contribute to structural fatigue and damage. Therefore, accurate prediction of sloshing impacts is critical for structural design and safety assessment.

Previous studies on sloshing impacts have primarily focused on free-surface motion and the associated impact mechanisms. The impact of a liquid surface on a rigid wall has been classically described by Wagner’s impact theory [3], while Ralph Alger Bagnold introduced the concept of air-pocket-induced pressure amplification [4]. These studies established that sloshing impacts can be categorized into different regimes depending on the free-surface configuration and impact conditions.

To account for scale effects in sloshing flows, most experimental and numerical studies have adopted gravity-based scaling using Froude similarity. The scaling ratio itself may influence sloshing impact loads, even when similarity conditions are applied [5]. This approach assumes that free-surface motion is governed by the balance between inertia and gravity, allowing dynamic similarity to be maintained across different scales. However, real sloshing impacts involve complex physical processes, including air entrapment, jet formation, and localized flow variations. Previous studies have pointed out that while Froude scaling is appropriate for describing global flow behavior, local impact pressures are significantly affected by additional factors such as gas–liquid interaction and density ratio effects, making the scaling problem more complex [6]. As a result, deviations from Froude scaling have been frequently reported, and its applicability remains a subject of ongoing debate. In particular, Froude scaling has been reported to provide conservative estimates at smaller scales [7]. These limitations indicate that the applicability of Froude scaling depends on the underlying impact conditions and mechanisms.

Extensive efforts have been made to address the scaling of sloshing impact pressures. Abramson et al. [8] and Cox et al. [9] derived governing dimensionless parameters based on Π-theorem and conducted early experimental investigations, suggesting that viscous and surface tension effects are relatively negligible and that Froude scaling can serve as the dominant similarity criterion. However, they also noted that gas entrapment and compressibility effects may introduce additional scaling complexities. Subsequent studies proposed multi-parameter scaling frameworks incorporating compressibility effects [10], while numerical investigations have further explored the influence of two-phase flow dynamics on scaling behavior [11,12,13]. In addition, experimental studies have shown that gas–liquid interactions and air entrapment significantly affect impact pressure magnitude and scaling characteristics [5,14]. Furthermore, it has been reported that local sloshing impact pressures cannot be fully described by Froude scaling alone, as they are strongly influenced by density ratio effects and gas–liquid interaction during impact events [6].

Despite these efforts, a clear scaling criterion for sloshing impacts involving mixed impact regimes—such as air-pocket-dominated impacts and pure impacts—has not yet been established. In particular, the role of impact mechanisms in determining scaling behavior has not been systematically clarified. This issue is closely related to the filling level, which serves as a key control parameter governing internal flow patterns and impact characteristics. Under high filling conditions, multiple impact mechanisms tend to coexist, leading to increased variability and difficulty in achieving consistent scaling behavior.

Previous studies have also highlighted significant uncertainties associated with sloshing experiments, including those arising from experimental setup, motion systems, and data acquisition processes [15], as well as the influence of geometric parameters such as tank width on impact pressure statistics [16].

Furthermore, most previous studies have evaluated scaling behavior based on mean or single representative values, with limited consideration of the statistical distribution of impact events or the behavior of extreme impacts. However, from a structural design perspective, the maximum loads induced by extreme events are of primary importance rather than average responses. Experimental studies have also shown that measured impact pressures can vary significantly even under nominally identical conditions due to microscopic flow differences and sensor-related effects, such as thermal shock and droplet formation, leading to substantial measurement uncertainty [17]. Therefore, conventional approaches may not adequately capture the actual characteristics of sloshing impact pressures.

In particular, the scale dependency of sloshing impact pressures is strongly influenced by the distribution of impact events, especially the behavior of the upper tail corresponding to extreme impacts. Changes in dominant impact mechanisms directly affect the distribution characteristics of impact pressures, which in turn influence the observed scaling behavior. Consequently, a proper interpretation of scaling behavior requires a combined consideration of both distribution characteristics and underlying impact mechanisms.

The importance of understanding the relationship between loading conditions and impact response characteristics has also been recognized in other engineering domains. For example, Yang et al. [18] demonstrated that in percussive rock drilling, the ratio of shear stress to hydrostatic pressure governs the rock-breaking efficiency in a nonlinear manner, highlighting that the dominant failure mechanism—rather than the absolute load magnitude—determines the system response. This observation is conceptually analogous to the present finding that the dominant impact mechanism governs the scaling behavior of sloshing pressures.

In this study, sloshing impact pressures are defined based on individual impact events, and distribution-based representative metrics, including mean and upper-percentile values, are introduced to analyze scaling behavior. A power-law framework is employed to quantitatively evaluate scale dependency, and the relationship between impact mechanisms and scaling behavior is systematically examined.

The results demonstrate that the scaling behavior of sloshing impact pressures cannot be adequately described by a single similarity law, but is governed by the combined effects of distribution characteristics and impact mechanisms. In particular, the expansion of the distribution tail and the increasing contribution of extreme events play a dominant role in determining scale dependency.

Based on these findings, this study proposes a distribution-based and mechanism-aware scaling framework for sloshing impact pressures, providing a more physically consistent basis for interpreting scaling behavior and enabling improved prediction of impact loads from model-scale experiments within typical scale ranges.

2. Experimental Setup

2.1. Experimental Facilities

2.1.1. Sloshing Motion Platform

A six-degree-of-freedom motion platform (Sloshing Motion Platform, SMP) was employed to accurately reproduce sloshing phenomena. The platform was designed to support dynamic loads of up to approximately 4000 kgf and is capable of generating six-degree-of-freedom motions, including surge, sway, heave, roll, pitch, and yaw, using six independent actuators. The overall configuration of the experimental system, including the model tank and sensor arrangement, is illustrated in Figure 1.

Each actuator converts the rotational motion of a servo motor into linear motion through a ball-screw mechanism, and the target motion is generated through the coordinated motion of the actuators. The platform is also equipped with an integrated control system consisting of a DSP-based controller and an interface control unit (ICU), enabling precise reproduction of the prescribed motion scenarios.

The kinematic specifications of the platform are summarized in

Table 1. Kinematic specifications of the six-degree-of-freedom sloshing motion platform (SMP).

DOF	Range	Max. Velocity
Surge	−1060~1030 mm	2000 mm/s
Sway	−970~970 mm	1900 mm/s
Heave	−540~540 mm	1000 mm/s
Roll	−34.5~34.5 deg	83 deg/s
Pitch	−36.5~34.9 deg	83 deg/s
Yaw	−59.8~59.8 deg	170 deg/s
Max. load	Static: 2500 kgf Dynamic: 4000 kgf	-

. In the present study, all experiments were conducted under unidirectional (sway) excitation, as this configuration is most representative of the dominant ship motion component governing sloshing behavior in LNG cargo tanks.

2.1.2. Model Tank

A two-dimensional model tank was used in the experiments. The model tanks were designed based on a No.2 cargo tank of a 174k LNG carrier (LNGC), which was selected as the reference configuration for scaling. The tanks were constructed at scale ratios of 1/70, 1/50, and 1/35, while maintaining a constant internal width of 0.2 m for all models to minimize geometric effects associated with scale variation. Among the tested configurations, the 1/35 scale was considered as the reference scale, as it represents the largest model with relatively reduced scale-induced distortions.

The adoption of a two-dimensional configuration allows the dominant sloshing behavior under uni-directional excitation to be captured while reducing the complexity associated with three-dimensional flow features. This simplification is particularly suitable for investigating the scaling characteristics of impact pressure and the statistical behavior of peak responses, which are the primary focus of this study. To facilitate a consistent comparison across different scales, selected results are presented in a normalized form with respect to the 1/35 scale, enabling clearer identification of scale-dependent trends in both pressure magnitude and distribution characteristics.

The tanks were fabricated from acrylic, with a uniform wall thickness of 20 mm. This design ensures sufficient structural rigidity while allowing visual observation of the internal flow. The geometry and principal dimensions of the model tanks are presented in Figure 2 and

Table 2. Geometrical dimensions of the model tank for different scale ratios.

Scale Ratio	Relative Scale	Length [mm]	Width [mm]	Height [mm]
1/70	0.5	702.72	200.00	402.72
1/50	0.7	1024.00	200.00	544.00
1/35	1	1405.43	200.00	805.43

.

2.1.3. Pressure Sensor and DAQ

A Kistler IEPE-type pressure sensor (model 601CBA, Kistler, Winterthur, Switzerland) was used to measure sloshing impact pressures. The sensor has a high natural frequency and a wide measurement range, enabling reliable acquisition of transient impact pressure signals.

Data acquisition was performed using a National Instruments PXI-4497 module (National Instruments, Austin, TX, USA), which provides a 24-bit resolution and a maximum sampling rate of 204.8 kS/s. In the present experiments, the sampling rate was set to 20 kHz to capture transient sloshing impact events with sufficient temporal resolution.

The measurement system provides a reliable basis for analyzing the temporal variability and extreme characteristics of sloshing impact pressures.

2.1.4. Flow Visualization System

A GoPro HERO9 camera (GoPro, San Mateo, CA, USA) was used to visualize sloshing flow patterns. The camera was installed outside the tank at a distance of approximately 500 mm from the tank wall to simultaneously record the free-surface motion and impact events. Illumination was provided by multiple halogen lamps installed in the experimental facility, ensuring adequate brightness for visual observation of the free-surface behavior.

The camera was operated at a frame rate of 24 fps with a resolution of 1920 × 1080 pixels. Synchronization between the pressure DAQ system and the camera was achieved through manual triggering: video recording was initiated simultaneously with the start of data acquisition using a camera remote control. The resulting temporal offset is estimated to be less than 1 s, which is sufficient for correlating observed flow patterns with impact event clusters for regime classification purposes, but insufficient for resolving the detailed dynamics of individual impact events occurring on a millisecond timescale.

This visualization enabled correlation analysis between pressure signals and flow patterns, allowing different impact mechanisms, such as air pocket formation and pure impact, to be identified.

2.2. Test Configuration

2.2.1. Sensor Arrangement

To quantitatively measure the spatial distribution of sloshing impact pressures, a total of 90 pressure sensors were installed on the internal walls and upper structure of the tank. The sensors were arranged over both the side walls and the upper region, considering areas where sloshing impacts are most likely to occur. The measurement domain was divided into six sections to enable location-dependent analysis of impact characteristics.

In the side-wall region, sensors were uniformly distributed in the vertical direction to capture impacts associated with free-surface rise and wall collision. In the upper region, sensors were installed with a higher spatial density to capture impacts related to air pocket collapse and direct impact events. This sensor configuration provides a basis for analyzing not only the spatial distribution of impact pressures but also variations in impact characteristics depending on the impact location. The overall sensor arrangement and section definition are shown in Figure 3a.

Each cluster consists of a 3 × 5 array of sensors (3 columns × 5 rows). Within each cluster, the center-to-center spacing between adjacent sensors is 10.0 mm in both the horizontal and vertical directions. The distance from the tank wall to the center of the nearest sensor is 9.0 mm. The sensor diameter is 5.5 mm. For clarity, the schematic in Figure 3b shows a partial view of the cluster; the full array follows the same uniform spacing throughout.

2.2.2. Test Conditions

The experiments were conducted under different scale conditions (1/70, 1/50, and 1/35) and filling levels. The experimental conditions, including sampling strategy, test duration, and repetition, were selected in accordance with widely accepted guidelines for sloshing model tests, such as those provided by ITTC [19]. To effectively induce sloshing impacts in each case, the excitation frequency was set to match the natural frequency of the free surface in the tank.

The natural frequency was determined based on the linear dispersion relation under finite-water-depth conditions, which can be expressed as follows.

$$f_n={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{\pi }{L}}g\tanh \left({\displaystyle \frac{\pi H}{L}}\right)}$$

(1)

Here, $L, H,$ and $g$ denote the tank length, the liquid depth, and the gravitational acceleration, respectively. This relation represents the natural sloshing frequency under finite-depth conditions and was used as the reference criterion to ensure dynamic similarity across different scales.

The excitation amplitude was adjusted for each scale to reproduce comparable sloshing flow patterns, thereby enabling consistent comparison between different scale conditions.

Repeated experiments were conducted under each test condition to obtain a sufficient number of impact events for statistical analysis. The main experimental conditions are summarized in

Table 3. Summary of experimental conditions. The test duration was fixed at 600 s for all cases, with a ramp function of 5 s applied at both the beginning and end of each run, resulting in an effective steady-state duration of 590 s per run.

Scale Ratio	Relative Scale	Filling Level [%]	Frequency [Hz]	Amplitude [mm]	Repetitions
1/70	0.5	18	0.767	38	10
		80	0.996	9
1/50	0.7	18	0.648	53
		80	0.843	13
1/35	1.0	18	0.542	75
		80	0.705	18

.

Based on the excitation frequencies and steady-state duration of 590 s, the total number of steady-state excitation cycles per condition across the 10 repetitions ranged from approximately 3200 to 5880, providing a sufficient pool of impact event candidates for statistical analysis.

In addition to the kinematic test conditions above, the thermophysical properties of the working fluids are provided in

Table 4. Thermophysical properties of water and air at 25 °C and standard atmospheric pressure (101.325 kPa).

Property	Unit	Water (25 °C)	Air (25 °C, 1 atm)
Density	kg/m3	997.0	1.184
Dynamic viscosity	Pa·s	8.900 × 10−4	1.849 × 10−5
Kinematic viscosity	m2/s	8.926 × 10−7	1.562 × 10−5
Surface tension	N/m	0.0720	-
Speed of sound	m/s	1497	346
Density ratio	-	842	-

, as these quantities directly affect the relevant dimensionless parameters governing sloshing scaling behavior, particularly under conditions where viscous and compressibility effects may deviate from the Froude similarity assumption.

The working fluid used in all experiments was water, and the free surface above the liquid was exposed to ambient air. All experiments were conducted at a controlled laboratory temperature of 25 ± 1 °C. At this temperature, the key physical properties of the fluids are summarized in Table 4. These values are consistent with standard reference data [ISO 5725/NIST] [20] and were used as the basis for computing the dimensionless parameters relevant to scaling analysis, including the Reynolds number, Weber number, and Cauchy number.

The dynamic viscosity of water at 25 °C is approximately 8.900 × 10⁻⁴ Pa·s, and its density is 997.0 kg/m³. The density of air at standard atmospheric pressure (101.325 kPa) and 25 °C is 1.184 kg/m³, with a dynamic viscosity of 1.849 × 10⁻⁵ Pa·s. The density ratio of water to air is approximately 842. The surface tension of water at 25 °C is 0.0720 N/m.

2.3. Scaling Analysis

The scaling behavior of sloshing impact pressure is not governed by a single physical mechanism, and the dominant physical effects may vary depending on the flow pattern and the location of impact. In particular, the two representative impact conditions considered in this study—low filling conditions dominated by side-wall impacts and high filling conditions dominated by top impacts—exhibit distinct flow structures.

Under low filling conditions, the free surface rises along the side wall, leading to the formation of strong velocity gradients near the wall, which may enhance localized shear effects. In contrast, under high filling conditions, the free surface interacts directly with the upper structure, and the overall flow behavior is primarily governed by inertia and gravity.

Based on these physical differences, it is hypothesized that the scaling behavior of sloshing impact pressure depends on the dominant impact mechanism. To examine this hypothesis, two normalization approaches representing different physical effects—gravity-based scaling and shear-stress-based scaling—were applied, and their scaling behaviors were systematically compared under different conditions.

2.3.1. Gravity-Based Scaling

Gravity-based scaling assumes that sloshing flows are governed by the interaction between inertia and gravity. In this framework, the characteristic velocity is defined using gravitational acceleration and a representative length scale, and the impact pressure can be normalized as

$$P^{\ast }={\displaystyle \frac{P}{\rho g{L}_c}}$$

(2)

where $P$ is the measured impact pressure, $\rho$ is the fluid density, $g$ is the gravitational acceleration, and $L_c$ is the characteristic length.

In the present study, the internal width of all model tanks was kept constant, resulting in geometric dissimilarity across scales. Under such conditions, using a single directional length as the characteristic scale may not adequately represent the overall flow scale. To address this limitation, an equivalent length scale based on the tank volume was adopted as the characteristic length.

The characteristic length is defined as

$$L_c=V^{1/3}$$

(3)

where $V$ denotes the volume of the model tank. This definition corresponds to the side length of an equivalent cube with the same volume and is used as a representative length scale for the overall flow.

This definition allows a consistent comparison of flow scales across different models, even under fixed-width conditions, and provides a more rational basis for evaluating the applicability of gravity-based scaling.

In general, the length and height of a tank influence the flow field in fundamentally different ways: the tank length governs the horizontal wavelength of the free-surface motion and determines the natural sloshing frequency, while the tank height controls the hydrostatic pressure head and the depth-dependent wave celerity. Therefore, selecting either dimension alone as the characteristic scale may introduce a directional bias, particularly when geometric similarity is not maintained across models. In the present study, since the tank width was held constant while the length and height were scaled proportionally, no single directional dimension remains consistent across all scale ratios. Under such conditions, the volume-equivalent cube length L_c = V^1/3 provides a scale-invariant measure that integrates all three spatial dimensions, enabling a consistent and physically meaningful comparison of the overall fluid domain across different scale models.

2.3.2. Shear-Stress-Based Scaling

Under low filling conditions, the free surface rises along the side wall, leading to the formation of strong velocity gradients in the near-wall region. This flow structure exhibits characteristics similar to a jet-like motion along the wall, where localized shear effects may become significant, particularly in the near-impact region.

In such conditions, the resulting impact pressure may not be solely governed by hydrostatic or gravity-driven responses but can also be influenced by local flow structures and velocity gradients near the wall. Based on this observation, it is hypothesized that shear stress may contribute to the formation of impact pressure under low filling conditions.

Based on this physical consideration, a shear-stress-based scaling is introduced using shear stress as the representative physical quantity. Under the assumption of a Newtonian fluid, the shear stress can be expressed as

$$\tau =\mu {\displaystyle \frac{\partial u}{\partial y}}$$

(4)

where $\mu$ is the dynamic viscosity and $\partial u/\partial y$ is the velocity gradient.

Since direct measurement of local velocity gradients in sloshing flows is challenging, the shear stress is approximated using a characteristic velocity and length scale as

$$\tau =\mu {\displaystyle \frac{U_t}{h}}$$

(5)

where $U_t$ is the characteristic velocity associated with sloshing motion, and $h$ is the characteristic length scale of the free-surface or local flow region.

Accordingly, the impact pressure is normalized as

$${P_{\tau }}^{\ast }={\displaystyle \frac{P}{\tau }}$$

(6)

where $P$ is the measured impact pressure.

This shear-stress-based scaling is introduced not as an expected superior alternative to gravity-based scaling, but as a hypothesis-driven comparison to examine whether localized viscous shear effects contribute meaningfully to impact pressure under low filling conditions. By systematically comparing its performance against gravity-based scaling, the relative importance of different physical mechanisms can be assessed, and the dominant scaling framework can be identified on an empirical basis.

2.4. Data Processing and Impact Definition

Sloshing impact pressure data are measured in the form of time histories and exhibit strong variability and non-stationary characteristics across individual impact events. Therefore, reliable scaling analysis requires a systematic procedure, including impact event definition, data preprocessing, and the determination of representative metrics.

2.4.1. Signal Preprocessing

The measured pressure signals contained sensor noise and low-frequency drift; therefore, a preprocessing procedure was applied to accurately identify impact events. First, baseline drift was corrected to remove low-frequency trends and to obtain signals suitable for subsequent analysis. When necessary, a second-order Butterworth high-pass filter was applied to further eliminate low-frequency components, with a cutoff frequency of 5 Hz.

The preprocessing procedure was designed to apply only minimal filtering so that the peak values of the impact pressures were not distorted. This approach ensured that the essential characteristics of the original impact signals were preserved.

2.4.2. Peak Detection

Peak detection was performed to identify impact events from the pressure time history data measured at each channel. A threshold pressure of 2.5 kPa was adopted, consistent with values widely used in sloshing model test practice [21], to exclude non-impact background signals while retaining physically meaningful impact events. The conceptual procedure of the peak detection process is illustrated in Figure 4a.

Each time history signal was segmented into fixed time intervals with a window length of 0.5 s. This window length is smaller than the excitation period across all tested conditions (minimum period: 1.0 s at 1/70 scale, high filling), ensuring that impact events from successive excitation cycles are assigned to separate windows while multiple pressure peaks arising from a single impact event are captured within the same window. Within each interval, the maximum value exceeding the threshold was defined as the representative impact pressure.

If multiple peaks were detected within the same interval, and the time difference between them was smaller than the interval length, they were considered to originate from a single impact event. In such cases, only the largest peak was selected as the representative value. This procedure eliminates multiple peaks arising from the same impact and ensures that independent impact events are identified.

Figure 4b–d illustrates the thresholding process, time window segmentation, and representative peak selection, respectively.

2.4.3. Event Filtering and Selection

Since sloshing impact data include both transient and steady-state regimes, only steady-state data were selected to ensure consistency in the analysis.

The initial and final stages of the sloshing motion were excluded because transient flow conditions may occur in these periods. For regular wave conditions, only data after the first five wave periods were considered. For irregular wave conditions, data obtained after 1 min from the start of the experiment were used for analysis.

In addition, impact events were screened for spatial consistency across adjacent sensor locations. Pressure peaks recorded at an isolated sensor that were disproportionately large relative to simultaneously measured signals at neighboring sensors were identified as spurious outliers, likely attributable to localized droplet impacts directly on the sensor surface rather than global sloshing-induced structural loads. Such events were excluded from the dataset to retain only physically meaningful impact events representative of the overall sloshing flow.

2.4.4. Definition of Representative Impact Pressure

Sloshing impact pressures exhibit large variability between individual events; therefore, defining a representative value based solely on a single maximum value is not appropriate. To account for the statistical characteristics of impact pressures, representative metrics were defined in this study. The definition of representative impact pressure is consistent with engineering practices, where sloshing loads are evaluated based on statistical and extreme values, as recommended in classification society guidelines [22,23].

The representative impact pressure was evaluated using the mean value of all detected events to characterize the overall impact level. In addition, the mean values of the top one-third and top one-tenth of the events were considered to capture the influence of high-intensity impacts.

In particular, the mean of the top 10% of events was used as an indicator of near-extreme impact behavior. This metric provides relevant information from a structural design perspective, where high-intensity impacts are of primary importance.

2.4.5. Spatial Classification of Impact Events

The impact events identified at each channel were classified according to the sensor locations to analyze the spatial distribution of sloshing impact pressures and to evaluate location-dependent impact characteristics.

In addition, temporally adjacent peaks were grouped into a single cluster to represent a unified impact event. In this study, peaks occurring within a time interval of 0.5 s were defined as belonging to the same cluster. For each cluster, the maximum pressure, corresponding sensor location, and occurrence time were extracted.

For irregular wave conditions, video analysis was conducted in parallel to record the timing of impact events and the associated flow patterns. This enabled correlation analysis between the pressure signals and the underlying impact mechanisms.

3. Analysis

This section presents a systematic methodology for analyzing the scaling behavior of sloshing impact pressures. The proposed approach incorporates spatial classification, definition of representative impact pressures, and scaling evaluation criteria.

Since sloshing impacts exhibit different characteristics depending on the impact location and underlying flow mechanisms, simple comparisons based on mean values alone are not sufficient to describe scaling behavior. Accordingly, spatial classification, representative pressure metrics, and scaling evaluation criteria are defined in a stepwise manner, enabling consistent comparison of impact pressure data obtained under different scale and filling conditions.

3.1. Spatial Analysis

Sloshing impact pressures vary significantly depending on their location within the tank, both in terms of magnitude and underlying impact mechanisms. Therefore, it is essential to account for spatial distribution in the analysis. In this study, based on the sensor arrangement described in Section 2.2, a total of 90 pressure sensors were grouped into six sections, and the impact pressures were analyzed on a section-wise basis.

For each pressure sensor $i$, the set of detected impact events is defined as

$$P_i=\left\{p_{i1},p_{i2},\dots ,p_{i{n}_i}\right\}$$

(7)

Here, $p_{ij}$ denotes the $j$-th detected impact peak at sensor $i$, and $n_i$ is the total number of impact events detected at that sensor.

Each sensor is assigned to a specific section $S_k$ based on its spatial location. The impact event sets from sensors belonging to the same section are combined to construct a section-level impact pressure set as

$$P_{S_k}=\bigcup _{i\in S_k}{P}_i$$

(8)

where $P_{S_k}$ represents the aggregated set of impact events within section $S_k$.

This aggregation reduces local variability associated with individual sensors and enables the extraction of representative impact characteristics for each spatial region. Based on $P_{S_k}$, representative impact pressures were evaluated for each section, allowing comparison of spatial distributions and scaling behavior across different locations.

In particular, in sections corresponding to the side-wall region, impacts are predominantly associated with free-surface rise and wall collision. In contrast, in the upper region, impacts are governed by a combination of air pocket formation and collapse, as well as direct free-surface impacts. Therefore, section-based analysis serves not only as a spatial classification but also as an indirect means of distinguishing dominant impact mechanisms.

Furthermore, since a single scaling approach may not be uniformly applicable across all spatial locations, comparison of section-wise results provides insight into the spatial dependency of scaling behavior.

3.2. Definition of Comparative Metric

Sloshing impact pressures exhibit large variability between individual events and show a wide distribution even under identical experimental conditions. Therefore, evaluating scaling behavior based solely on a single maximum or mean value is insufficient. To account for the statistical characteristics of the impact pressures, representative metrics were defined in this study.

Based on the section-level impact event set $P_{S_k}$ defined in Section 3.1, the impact pressures were sorted in descending order to construct an ordered set as

$$P_{S_k}^{sorted}=p_1,p_2,p_3,\dots ,p_N,\hspace{1em} p_1\ge p_2\ge \dots \ge p_N$$

(9)

where $N$ is the total number of impact events in section $S_k$.

Based on this ordered set, the representative impact pressures were defined as follows. The mean impact pressure is given by

$${\overline{P}}_{\mathrm{mean}}={\displaystyle \frac{1}{N}}\sum _{j=1}^N{p}_j$$

(10)

which represents the overall impact level.

To account for higher-intensity events, the mean values of the upper subsets were additionally defined. The mean of the top one-third of events is expressed as

$${\overline{P}}_{\mathrm{top} 1/3}={\displaystyle \frac{1}{\lfloor N/3\rfloor }}\sum _{j=1}^{\lfloor N/3\rfloor }{p}_j$$

(11)

which captures medium-to-high intensity impact characteristics.

Similarly, the mean of the top one-tenth of events is defined as

$${\overline{P}}_{\mathrm{top} 1/10}={\displaystyle \frac{1}{\lfloor N/10\rfloor }}\sum _{j=1}^{\lfloor N/10\rfloor }{p}_j$$

(12)

which represents near-extreme impact behavior and provides a relevant indicator for characterizing high-intensity impact events, though it is not intended as a direct structural design metric (see Section 5 for further discussion).

By defining multiple representative metrics, both average and extreme impact behaviors can be analyzed separately. This approach enables a more detailed evaluation of how the distribution of impact pressures changes with scale. In particular, the upper-percentile-based metrics play a key role in capturing extreme-event characteristics and are therefore essential for assessing scaling behavior.

3.3. Scaling Evaluation Method

To evaluate the scaling behavior of sloshing impact pressures, the normalization methods defined in Section 2.3 were applied to the representative impact pressures obtained at different scales. The representative pressures were converted into dimensionless form, enabling direct comparison across different scale conditions.

In the case of ideal scaling, the dimensionless pressure should remain invariant with respect to scale, indicating that scale dependency has been eliminated. To quantitatively assess this dependency, the relationship between representative impact pressure and scale ratio was modeled using a power law expression as

$$P=\alpha {\lambda }^p$$

(13)

where $\lambda$ is the scale ratio, $\alpha$ is a coefficient, and $p$ is the exponent representing scale dependency.

For ideal scaling, the dimensionless pressure remains constant across scales, and therefore the exponent $p$ approaches zero. Accordingly, $p$ was used as the primary indicator of scaling validity in this study. A value of $p$ close to zero indicates that the corresponding normalization method provides scale-independent results, whereas larger values of $p$ imply that the scaling method does not fully capture similarity across scales.

The power law analysis was performed for each representative metric defined in Section 3.2 (mean, top 1/3, and top 1/10). This allows the scale dependency to be evaluated across different impact intensity levels.

In particular, an increase in the exponent $p$ for upper-percentile-based metrics indicates that extreme impact events are more sensitive to scale variation. This behavior reflects the influence of distribution tails on scaling characteristics.

Furthermore, gravity-based scaling and shear-stress-based scaling were evaluated under the same framework to compare their effectiveness. If a given scaling method yields a value of $p$ closer to zero under specific conditions, it suggests that the corresponding physical mechanism is more dominant in governing the impact behavior.

3.4. Impact Regime Classification

Sloshing impact pressures can arise from different flow mechanisms even under identical conditions, and these mechanisms directly influence not only the magnitude and temporal characteristics of the impact pressures but also their scaling behavior. Therefore, it is necessary to classify impact events according to the dominant impact mechanism rather than treating all events as a single dataset.

In this study, impact events were classified into two categories based on the underlying flow mechanism: air-pocket-dominated impacts and pure impacts. This classification was performed by considering both the characteristics of the pressure signals and the corresponding flow visualization results.

Air-pocket-dominated impacts occur when an air layer is entrapped between the free surface and the upper structure, and the impact is generated through the compression and collapse of the air pocket. These impacts typically exhibit a relatively gradual pressure rise, multiple peaks with oscillatory components, longer impact duration, and comparatively lower peak pressure. Such behavior is attributed to the cushioning effect of the entrapped air layer and is consistent with the characteristics of Bagnold-type impacts. In particular, Pressure oscillations observed near the peak are associated with the compression dynamics of the air pocket.

In contrast, pure impacts occur when the free surface directly collides with the structure with little or no air entrainment. These impacts are characterized by a rapid pressure rise over a very short duration, typically exhibiting a single sharp peak with high pressure magnitude. This behavior is associated with direct fluid–structure interaction without air cushioning and is consistent with Wagner-type impact characteristics.

The classification of impact events was carried out through a combined analysis of pressure signals and flow visualization. First, the time history signals were analyzed to evaluate the rise time, duration, and peak shape of each event. Subsequently, camera recordings were examined to identify the presence of air entrainment and the corresponding free-surface behavior. Based on these combined observations, each event was assigned to one of the two impact regimes.

For air-pocket-dominated impacts, the characteristic signal features include a relatively long rise time, multiple oscillatory peaks, and a gradual pressure decay. For pure impacts, the signal is characterized by a short rise time, a single sharp peak, and rapid pressure decay. These qualitative criteria were used as the basis for regime classification, and a quantitative characterization of representative signal parameters is presented in Section 4.1.

For quantitative characterization of impact signal shape, the rise time and decay time were evaluated for each representative event. Following the peak modeling approach widely adopted in sloshing analysis [22,23,24], these parameters are defined as

$$T_{rise}={\displaystyle \frac{T_p-T_{\alpha P_{max},up-crossing}}{1-\alpha }}$$

(14)

$$T_{decay}={\displaystyle \frac{T_{\alpha P_{max},down-crossing}-T_p}{1-\alpha }}$$

(15)

$$T_{duration}=T_{rise}+T_{decay}$$

(16)

where $T_p$ is the time at which the peak pressure $P_{max}$ occurs, $T_{\alpha P_{max},up-crossing}$ and $T_{\alpha P_{max},down-crossing}$ are the times at which the pressure crosses the level $\alpha P_{max}$ on the rising and falling sides of the signal, respectively, and $\alpha$ is a fractional coefficient. In the present study, $\alpha$ = 0.5 was adopted for both rise and decay, consistent with the definition widely used by classification societies including DNV, BV, ABS, and GTT, as individually documented in [22,23] and comparatively summarized in [24]. The total impact duration is defined as $T_{duration}=T_{rise}+T_{decay}$.

Furthermore, impact events were grouped using the cluster definition described in Section 2.4, and a representative impact mechanism was assigned to each cluster. This enables classification at the level of spatially extended impact events rather than at individual sensor measurements.

The classified impact events were then used as a key basis for subsequent scaling analysis. In particular, the proportion of different impact mechanisms may vary with scale even under the same filling condition, which directly affects the distribution of normalized pressures. Therefore, the interpretation of scaling behavior in this study considers not only experimental conditions but also the variation in dominant impact mechanisms.

Accordingly, the scaling behavior of sloshing impact pressures is analyzed under the premise that it is governed not only by scale and filling conditions but also by the dominant impact mechanism. This perspective provides a fundamental basis for interpreting the results presented in Section 4.

4. Results

4.1. Impact Regime Identification

Sloshing impact pressures arise from different flow mechanisms even under identical experimental conditions, and these mechanisms directly influence not only the magnitude but also the temporal characteristics of the pressure signals. Therefore, accurate interpretation of scaling behavior requires classification of impact events according to the dominant flow mechanism.

As defined in Section 3.4, the impact events were classified into two categories: air-pocket-dominated impacts and pure impacts. The representative flow patterns and corresponding pressure signals for these two impact mechanisms are presented in Figure 5, Figure 6 and Figure 7.

Under low filling conditions, an air layer is frequently entrapped between the free surface and the side wall, and most of the observed impacts were identified as air-pocket-dominated events. The corresponding pressure signals exhibit a gradual rise with multiple peaks containing oscillatory components. These characteristics are attributed to the compression and cushioning effects of the entrapped air layer and are consistent with Bagnold-type impact behavior, as discussed in Section 3.4.

In contrast, under high filling conditions, both air-pocket-dominated impacts and pure impacts were observed, indicating a coexistence of different impact mechanisms. Pure impacts are characterized by a sharp pressure rise and a single distinct peak, corresponding to direct fluid–structure interaction without significant air cushioning. This behavior is consistent with Wagner-type impact characteristics.

In addition, under high filling conditions, the geometric effect of the upper tank structure was found to influence the impact behavior. In the corner region where the side wall and top structure meet, the rising free surface tends to trap air locally, leading to frequent formation of compressed air pockets. This configuration sustains the occurrence of air-pocket-dominated impacts while enhancing oscillatory features in the pressure signals due to repeated compression and expansion of the entrapped air.

To further quantify the temporal differences between these two impact regimes, the representative pressure signals for Bagnold-type and Wagner-type impacts are compared in Figure 8, and their characteristic temporal parameters are summarized in

Table 5. Temporal parameters of representative pressure signals for each impact regime, evaluated with α = 0.5.

Parameter	Air Pocket Dominant (Ch.85)	Pure Impact (Ch.62)
T_rise [ms]	69.853	8.346
T_decay [ms]	266.650	9.350
T_duration [ms]	336.503	17.696

.

To provide a quantitative basis for regime classification, the temporal parameters of representative pressure signals were evaluated for each impact regime, as summarized in Table 5. For the Bagnold-type impact (Ch.85), the rise time and decay time were $T_{rise}= 69.9 \mathrm{m}\mathrm{s}$ and $T_{decay}=266.7 \mathrm{m}\mathrm{s}$, respectively, yielding a total impact duration of $T_{duration}=336.5 \mathrm{m}\mathrm{s}$. In contrast, the Wagner-type impact (Ch.62) exhibited significantly shorter characteristic times, with $T_{rise}= 8.3 \mathrm{m}\mathrm{s}$ and $T_{decay}= 9.4 \mathrm{m}\mathrm{s}$, corresponding to a total duration of $T_{duration}= 17.7 \mathrm{m}\mathrm{s}$. The ratio of impact durations between the two regimes is approximately 19:1, and the decay time ratio is approximately 28:1, reflecting the sustained pressure oscillation associated with air pocket compression and expansion in Bagnold-type impacts. These quantitative differences confirm that the temporal characteristics of the pressure signal provide a clear and objective basis for regime classification.

4.2. Spatial Distribution of Impact Events

To analyze the spatial distribution of impact mechanisms, the occurrence of impact events in each section was classified under low and high filling conditions, as shown in Figure 9 and Figure 10. The analysis was conducted based on the spatial classification defined in Section 3.1.

Figure 9a presents the occurrence frequency of impact events in each section for different scales under low filling conditions. The detailed occurrence ratios are summarized in Table 6. As shown in the figure, impact events are not uniformly distributed but tend to concentrate in specific sections. For the 1/70 and 1/35 scales, the highest occurrence is observed in Section 1, followed by Sections 6 and 5. For the 1/50 scale, Sections 1 and 5 exhibit comparable occurrence frequencies. In all scales, the occurrence in Section 2 is relatively low, while almost no impact events are observed in Sections 3 and 4.

To provide a clearer interpretation, the sections were further grouped into side-wall and upper regions, as shown in Figure 9b. At the 1/70 scale, impact events are predominantly concentrated in the side-wall region. In contrast, at the 1/50 and 1/35 scales, the occurrence frequencies in the side-wall and upper regions become comparable, indicating a transition in the dominant impact locations with increasing scale.

Figure 10a shows the spatial distribution of impact events under high filling conditions. The detailed occurrence ratios are summarized in Table 7. Due to the elevated free-surface level, impact events are observed only in Sections 4–6, with a pronounced concentration in Section 5. This distribution indicates that, once the free surface reaches the upper corner region, impact events occur more frequently beneath the top structure rather than along the side wall.

Table 6. Distribution of peak occurrences across sections under low filling conditions [%].

Section No.	1/70	1/50	1/35
1	66.1	41.4	41.9
2	6.0	2.6	4.9
3	0.2	0.0	0.0
4	0.0	0.0	0.0
5	7.4	44.2	21.5
6	11.8	15.8	31.1

Table 6. Distribution of peak occurrences across sections under low filling conditions [%].

Section No.	1/70	1/50	1/35
1	66.1	41.4	41.9
2	6.0	2.6	4.9
3	0.2	0.0	0.0
4	0.0	0.0	0.0
5	7.4	44.2	21.5
6	11.8	15.8	31.1

Table 7. Distribution of peak occurrences across sections under high filling conditions [%].

Section No.	1/70	1/50	1/35
1	0.0	0.0	0.0
2	0.0	0.0	0.0
3	0.0	0.0	0.0
4	17.6	18.8	15.0
5	72.4	67.6	73.5
6	9.9	13.6	11.5

Table 7. Distribution of peak occurrences across sections under high filling conditions [%].

Section No.	1/70	1/50	1/35
1	0.0	0.0	0.0
2	0.0	0.0	0.0
3	0.0	0.0	0.0
4	17.6	18.8	15.0
5	72.4	67.6	73.5
6	9.9	13.6	11.5

This behavior can be explained by the geometric constraint under high filling conditions. The reduced clearance between the free surface and the upper structure facilitates local air entrapment near the corner region, leading to a higher frequency of impact events in the upper region.

Figure 10b presents the overall occurrence ratio of impact mechanisms for different scales under high filling conditions. The occurrence frequency is normalized as

$$N^{\ast }={\displaystyle \frac{N}{d\times f_n}}$$

(17)

where N* is the normalized occurrence frequency, $N$ is the number of events corresponding to a given impact mechanism, $d$ is the duration of the scenario, and $f_n$ is the excitation frequency.

As shown in the figure, air-pocket-dominated impacts remain the dominant mechanism across all scales. However, the proportion of pure impacts increases with increasing scale. In particular, the occurrence of pure impacts increases from approximately 0.3% at the 1/70 scale to about 7.5% at the 1/35 scale. In contrast, the proportion of air-pocket-dominated impacts shows a slight decrease with scale, although it still accounts for the majority of events.

4.3. Distribution of Peak Impact Pressure

Sloshing impact pressures exhibit significant variability between individual events even under identical excitation conditions, making it difficult to adequately characterize the overall response using a single representative value. Therefore, in this section, individual impact events were defined using time-based clustering, and the maximum peak pressure within each event was selected as the representative value to analyze the event-level pressure distribution.

This distribution-based approach is essential for interpreting scale dependency, particularly for high intensity events, as it enables direct assessment of how the distribution structure varies with scale.

Figure 11 show the sorted event-level peak pressures in descending order. Figure 11a presents the pooled distribution across all scales, while Figure 11b–d corresponds to the results for the 1/70, 1/50, and 1/35 scales, respectively. Across all scales and in the pooled results, the low filling condition consistently exhibits higher peak pressure levels than the high filling condition over the entire range. This difference is observed not only in the extreme-value region but also across the entire distribution, indicating that the effect of filling condition is robust and not limited to a specific scale.

Under high filling conditions, although the overall pressure level is lower, the decay of the distribution becomes more gradual and the tail extends toward higher pressure values, indicating a broader spread of impact intensities. This suggests that the variability of impact pressure increases under high filling conditions, despite the reduction in absolute magnitude.

Comparing the distributions across different scales, the 1/50 and 1/35 cases exhibit similar decay trends in the sorted curves. However, the 1/70 scale shows a relatively flatter slope in the intermediate rank region, indicating a more uniform distribution of impact pressures.

This behavior may be partially attributed to the geometric characteristics of the model configuration. Since the tank width was kept constant across all scales, the relative width-to-length ratio increases at smaller scales. As a result, lateral confinement effects are reduced, and the flow becomes more spatially distributed, preventing strong localization of impact energy. Consequently, the distribution of event-level peak pressures becomes more flattened at smaller scales.

To further investigate the scaling behavior, the event-based sorted peak pressures were compared across different scales by separating the low and high filling conditions, as shown in Figure 12.

Under low filling conditions, the distributions exhibit partial overlap across scales depending on the rank range. In particular, the 1/50 and 1/35 scales show similar distributions in the intermediate rank region, whereas the 1/70 scale consistently maintains a lower pressure level and remains relatively separated from the other two scales. This behavior indicates that the scale effect under low filling conditions is not uniformly manifested over the entire distribution, but rather varies depending on the rank region.

This non-uniform behavior across the distribution directly explains why different representative metrics (e.g., mean, top 1/3, and top 1/10) exhibit different scaling trends, as discussed in Section 4.4.

It should be noted that, in the present experiments, the tank width was fixed at 200 mm for all scales, resulting in geometric non-similarity in terms of aspect ratio. In particular, the 1/70-scale model becomes relatively wider compared to the larger-scale models, which may influence the flow structure, especially under low filling conditions where the fluid motion is primarily confined near the side-wall region. This geometric difference can affect the degree of flow localization and impact formation, potentially contributing to the overall lower pressure levels observed for the 1/70 scale.

Specifically, as the scale ratio decreases from 1/35 to 1/70, the width-to-length ratio increases from 0.142 to 0.285, and the width-to-height ratio increases from 0.248 to 0.497, as summarized in Table 2. These changes in aspect ratio indicate that the degree of lateral confinement and free-surface confinement varies substantially across the tested scales, which may affect three-dimensional flow features such as corner effects, wave breaking, and impact localization.

Therefore, the scale-dependent behavior observed under low filling conditions should be interpreted as a combined effect of both dynamic scaling and geometric distortion, rather than as a purely scale-driven phenomenon.

In contrast, under high filling conditions, a more pronounced scale dependency is observed compared to the low filling case. While the distributions under low filling conditions exhibit rank-dependent similarity and partial overlap, the high filling condition shows a reduction in such overlap, with the 1/35 scale consistently maintaining higher peak pressure levels over almost the entire rank range. This indicates that the differences are not limited to extreme events but rather reflect a systematic shift in the entire distribution with scale.

Furthermore, while the 1/50 and 1/70 scales exhibit relatively similar distributions, the 1/35 scale remains distinctly separated, indicating that the scale effect under high filling conditions is characterized primarily by the upward shift in the 1/35-scale distribution rather than a uniform separation across all scales. These results demonstrate that the pressure response under high filling conditions exhibits increased sensitivity to scale, and that this sensitivity extends beyond the extreme-value region to the overall distribution of impact events.

4.4. Scaling Behavior of Impact Pressure

To quantitatively evaluate the scaling behavior of sloshing impact pressures, the power law relationship $P=\alpha {\lambda }^p$ was applied to the representative impact pressures obtained at different scales. Prior to normalization, the representative impact pressures under low and high filling conditions are compared in Figure 13, illustrating the scale-dependent variation in pressure levels in absolute terms. Here, the exponent $p$ serves as a key indicator of scale dependency; under ideal scaling conditions, $p\approx 0$.

As shown in Figure 14 and

Table 8. Power law fitting parameters for representative peak impact pressures under different filling conditions and normalization methods. The fitting follows $P=\alpha {\lambda }^p$, where $\lambda$ is the scale ratio.

Condition	Metric	$\mathit{\alpha }$	$\mathit{p}$
Low filling–Gravity-based (Figure 14a)	$\mu$	7.079	0.024
	${\mu }_{1/3}$	11.450	−0.043
	${\mu }_{1/10}$	17.267	−0.123
High filling–Gravity-based (Figure 14b)	$\mu$	1.457	0.039
	${\mu }_{1/3}$	2.389	0.263
	${\mu }_{1/10}$	3.620	0.476
Low filling–Shear-based (Figure 15a)	$\mu$	0.587	1.211
	${\mu }_{1/3}$	0.949	1.145
	${\mu }_{1/10}$	1.431	1.065
High filling–Shear-based (Figure 15b)	$\mu$	2.841	1.206
	${\mu }_{1/3}$	4.658	1.430
	${\mu }_{1/10}$	7.058	1.643

, under low filling conditions, gravity-based scaling yields values of $p$ close to zero for all representative metrics (mean, top 1/3, and top 10%), with $p\approx 0.02,-0.04,$ and $-0.12$, respectively (based on three scale points; see limitation discussion below). This indicates that the dimensionless impact pressures remain nearly invariant with scale, suggesting that gravity-based scaling is reasonably valid under these conditions. This behavior is closely related to the relatively uniform impact mechanisms and stable distribution characteristics observed under low filling conditions, as further discussed in Section 4.5.

In contrast, under high filling conditions, the exponent $p$ increases with increasing impact intensity level. In particular, the top 10% mean exhibits the largest value $p\approx 0.48, \mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{ }\mathrm{o}\mathrm{n}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{ }\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{ }\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}$), indicating that the contribution of extreme events becomes more significant as the scale increases. This behavior suggests that scale similarity is not preserved under high filling conditions and that the distribution structure of impact pressures changes with scale. In particular, the increased spread of the distribution and the enhanced contribution of upper-tail events play a dominant role in determining the observed scale dependency.

For shear-stress-based scaling, the exponent $p$ exceeds unity even under low filling conditions ($p\approx 1.21,1.15,1.07$), and becomes even larger under high filling conditions ($p\approx 1.21,1.43,1.64$), as shown in Figure 15 and Table 8. This indicates that shear-stress-based normalization does not promote convergence across scales and does not adequately represent the dominant physics governing sloshing impact pressures under the present experimental conditions.

As hypothesized in Section 2.3, the dominant physical mechanisms may vary depending on the impact conditions. However, the present results show that shear-stress-based scaling fails to capture the scaling behavior even under low filling conditions. This suggests that sloshing impact pressures are governed primarily by inertia and gravity rather than localized shear effects.

Overall, gravity-based scaling is found to be reasonably valid under low filling conditions, whereas its applicability deteriorates under high filling conditions due to the coexistence of multiple impact mechanisms and the increased spread of the pressure distribution.

These results indicate that the scaling behavior of sloshing impact pressure cannot be fully described by a single physical parameter or similarity law, but is strongly influenced by both the statistical characteristics of impact events—particularly the distribution tail—and the underlying flow mechanisms.

The relatively good agreement observed under low filling conditions can be attributed to the more uniform flow structure and consistent impact mechanisms, which result in stable pressure distributions across scales. In addition, the excitation conditions, determined based on the linear dispersion relation, inherently follow gravity-dominated scaling, further contributing to the effectiveness of gravity-based normalization.

In contrast, under high filling conditions, the interaction between air-pocket-dominated and pure impact mechanisms leads to increased variability and a broader distribution, particularly in the upper tail, where extreme events play a dominant role.

These results indicate that the scaling behavior of sloshing impact pressures cannot be fully described by a single physical parameter and is strongly influenced by both the statistical characteristics of impact events and the underlying flow mechanisms.

It should be noted that the power law fitting in the present study is based on three scale points (1/70, 1/50, and 1/35), which limits the statistical robustness of the fitted exponents. With only three data points, confidence intervals for the exponent p cannot be rigorously estimated, and the observed differences in p across filling conditions and intensity levels should be interpreted with caution. Increasing the number of tested scales would be necessary to provide statistically well-supported power law exponents, and this is identified as a priority for future experimental work.

Furthermore, detailed per-case statistical summaries—including event counts, standard deviations, and confidence intervals for each scale, filling level, and representative metric—are not reported in the present study. Providing such information would allow a more complete assessment of the repeatability and statistical reliability of the observed scaling trends, and is identified as a further area for improvement in future work.

4.5. Regime-Dependent Scaling Anaylsis

To interpret the differences in scaling behavior identified in Section 4.4, the relationship between scaling characteristics and impact mechanisms was examined. As shown in Figure 14 and Table 8, the power law exponent $p$ remains close to zero under low filling conditions, whereas under high filling conditions, $p$ increases with increasing impact intensity level. This contrast is closely associated with differences in the dominant impact mechanisms as well as the resulting distribution characteristics of impact pressures.

It should be noted that the excitation conditions were determined based on the linear dispersion relation, which inherently follows gravity-dominated scaling. This provides a consistent baseline for gravity-based normalization and contributes to the relatively good agreement observed under low filling conditions.

As defined in Section 3.4, sloshing impacts can be classified into air-pocket-dominated impacts, governed by air compression and collapse, and pure impacts, resulting from direct fluid–structure interaction. Considering the distribution characteristics presented in Section 4.3, most impact events under low filling conditions are dominated by air-pocket mechanisms, which exhibit relatively gradual pressure rise and limited variability.

In addition, under sway excitation, the flow is primarily governed by side-wall interactions, which exhibit relatively consistent kinematics across scales. This leads to a stable distribution of impact pressures with limited spread, and as a result, the representative impact pressures remain nearly invariant across scales, consistent with the near-zero values of $p$ observed in Table 8.

In contrast, under high filling conditions, impact events occur predominantly in the upper region, where multiple impact mechanisms—including air-pocket-dominated and pure impacts—coexist. Although the overall pressure magnitude is lower than that in low filling conditions, the distribution exhibits a significantly wider spread, particularly in the upper tail. This indicates that a small number of high intensity events contribute disproportionately to the overall distribution.

The pronounced increase of $p$ for the top 10% metric suggests that scaling behavior is governed not by the absolute magnitude of impact pressure, but by the expansion of the distribution and the increasing contribution of extreme events. In other words, the relative importance of extreme events increases with scale, leading to a scale-dependent increase in dimensionless impact pressure.

This behavior is closely related to the coexistence of different impact mechanisms. In the upper region, the combined effects of air entrapment due to structural geometry and direct free-surface impacts result in increased variability between events. Consequently, the scaling behavior is influenced more by the variability in impact events than by a uniform increase in impact intensity.

Therefore, the scaling behavior is governed not by a uniform shift in pressure magnitude, but by a redistribution of impact intensity across events. In particular, the expansion of the upper tail of the sorted peak distribution plays a dominant role in determining the scale dependency of extreme impact events.

Furthermore, as observed in Section 4.2, impact events under high filling conditions are concentrated near the upper corner region (Section 5), where geometric effects promote air entrapment. This condition facilitates the coexistence of multiple impact mechanisms, leading to a broader distribution rather than localized pressure amplification. As a result, the observed scaling behavior reflects changes in the distribution structure—such as the increased spread and variability of impact events—rather than a simple increase in peak pressure.

For shear-stress-based scaling, the exponent $p$ exceeds unity under both low and high filling conditions, indicating a persistent increase in normalized impact pressure with scale. This suggests that shear-based normalization does not adequately capture the dominant physics of the observed impact mechanisms under the present experimental conditions.

This further supports that sloshing impact pressures are governed primarily by inertia–gravity interactions rather than localized shear effects.

It should also be noted that the present experiments were conducted using a two-dimensional tank configuration, in which the internal width was fixed at 200 mm across all scales. While this simplification reduces the complexity associated with three-dimensional flow features and is widely adopted in sloshing scaling studies [16], it introduces inherent limitations. In particular, out-of-plane flow effects—including sidewall friction, lateral confinement, and three-dimensional air leakage during impact—are not fully captured by the present setup. These effects may influence local impact pressure magnitudes and regime classification, particularly under high filling conditions where air entrapment and roof impact dynamics are dominant. The present results should therefore be interpreted with the understanding that the two-dimensional assumption represents an idealization, and that three-dimensional effects may contribute to residual scaling deviations not fully explained by the present analysis.

From a physical perspective, deviations of p from zero can be interpreted as evidence of unresolved scaling imbalances associated with physical quantities not captured by the applied normalization. When p deviates positively from zero under gravity-based scaling, it suggests that additional physical effects beyond inertia–gravity interaction contribute to the impact pressure at larger scales. In particular, under high filling conditions where air-pocket-dominated impacts are prevalent, gas compressibility effects—characterized by the Cauchy number—become increasingly significant, as the dynamics of air pocket compression and collapse are not fully represented by Froude-based scaling alone. Furthermore, under conditions where the free surface interacts with structural boundaries, surface tension effects—characterized by the Weber number—may also contribute to scaling deviations, particularly at smaller scales where the Bond number is relatively large. Therefore, a positive deviation of p from zero under high filling conditions is consistent with the interpretation that gas compressibility and, to a lesser extent, surface tension effects introduce additional scale dependencies that are not accounted for by gravity-based normalization alone.

5. Conclusions

In this study, the distribution characteristics and scaling behavior of sloshing impact pressures were experimentally investigated across different scale conditions, and their relationship with underlying impact mechanisms was systematically analyzed. By adopting an event-based distribution approach and applying power law analysis, the scale dependency of sloshing impact pressures was quantitatively evaluated.

Under low filling conditions, impact events exhibit relatively consistent behavior, and gravity-based scaling yields nearly scale-independent results, with dimensionless impact pressures remaining almost invariant across scales. This indicates that gravity-dominated scaling is reasonably valid under these conditions.

In contrast, under high filling conditions, mixed impact mechanisms involving both air-pocket-dominated impacts and pure impacts were observed. Although the overall pressure magnitude is lower, the distribution becomes more dispersed, with a pronounced expansion in the upper tail. The increase in the power law exponent $p$ for upper-percentile-based metrics demonstrates that extreme events become more influential in determining scaling behavior. This indicates that the scaling behavior is governed not by a uniform increase in pressure magnitude, but by a redistribution of impact intensity across events, particularly through the expansion of the distribution tail.

Shear-stress-based scaling does not produce convergence across scales under any condition, indicating that sloshing impact pressures are governed primarily by inertia and gravity rather than localized shear effects.

Overall, the results demonstrate that the scaling behavior of sloshing impact pressures cannot be adequately described by a single similarity law. Instead, it is strongly influenced by both the statistical distribution of impact events and the dominant impact mechanisms. In particular, the variability of the distribution and the contribution of extreme events play a critical role in determining scale dependency.

From a practical perspective, the present results suggest that, within the commonly adopted model-scale range (1/70–1/35), the scaling behavior derived from model tests can be used to provide approximate predictions of impact pressure levels within the localized regions where the tests were conducted.

However, such predictions should be interpreted with caution, particularly under conditions where multiple impact mechanisms coexist and distribution variability becomes significant.

Based on the power law exponents obtained in this study, a preliminary assessment of scaling reliability can be made. Under low filling conditions, the exponent p for gravity-based scaling ranges from −0.123 to 0.024 across all representative metrics (mean, top 1/3, and top 10%), indicating that the associated scaling error remains well within acceptable engineering limits across the tested scale range (1/70–1/35). Under high filling conditions, p increases progressively with impact intensity level, from 0.039 (mean) to 0.263 (top 1/3) and 0.476 (top 10%), indicating a more pronounced scale dependency for extreme events. For mean-level predictions under high filling conditions (p = 0.039), the scaling error remains within acceptable limits across the full tested range. However, for the top 10% metric (p = 0.476), the deviation relative to the 1/35 reference scale reaches approximately 15% at the 1/50 scale and 28% at the 1/70 scale, both of which exceed the commonly adopted engineering tolerance of 10%. Based on power law extrapolation, the 10% deviation threshold is estimated to occur at a relative scale of approximately 0.79, corresponding to a model scale of approximately 1/44. Therefore, Froude-based scaling for extreme impact events under high filling conditions should be applied with caution at scales smaller than this threshold.

These findings highlight that reliable prediction of sloshing impact loads requires an integrated scaling approach that explicitly accounts for distribution characteristics, especially the upper tail, as well as regime-dependent flow mechanisms.

From an engineering design perspective, the present findings suggest that modifications to tank geometry—such as adjusting the chamfer angle or introducing curved wall profiles in the upper region—may help reduce the coexistence of mixed impact mechanisms under high filling conditions, thereby improving scaling consistency. In addition, the installation of internal damping baffles could suppress the expansion of the impact pressure distribution tail by attenuating extreme events, contributing to more reliable scaling behavior across model scales.

It is also noted that the representative metrics adopted in this study—mean, top one-third, and top one-tenth—are intended for comparative characterization of distribution-level scaling behavior across scales, rather than for direct structural design application. For design purposes, a more rigorous extreme-value analysis, such as fitting the impact pressure distribution to a parametric extreme-value distribution (e.g., Weibull) and estimating return-period-based load levels with confidence intervals, would be required. Such an analysis is identified as an important direction for future work.