The physical modeling for wave amplitude and runup induced by rotational cliff collapse was carried out in a three-dimensional water flume made up of plexiglass, as shown in Figs. 2 and 3. One end of the flume is vertical at 90°, whereas the other end is inclined and fixed at 30° (Fig. 3a and b). The flume is 0.55 m high, 0.5 m wide, and 1.4 m long along the base and 2.35 m long at the top. Furthermore, to measure the runup of induced water waves at various slope angles, two sliding rails were installed towards the inclined end at 45 and 60°. So, upon insertion of the gate at 45 and 60°, the top length of the flume was further reduced according to the Pythagoras theorem. To induce the rotational cliff collapse, a 0.55 m wide and 0.6 m high movable platform was designed, which can move in the vertical direction and can also rotate about its axis. The rotational motion was induced by pulling the hinge; the release ensured a pure rotational motion, which was visually verified by video analysis. The flume was marked with a vertical scale to measure the water depth. The wave amplitude was measured using capacitance-type wave gauges with an accuracy of ±0.5 mm, placed along the centerline at specified intervals. The runup height was measured using a graduated paper attached to the inclined surface. The entire process was recorded using a digital camera (240 fps, 720p resolution) placed perpendicular to the experimental flume; the velocity of the falling cliff was verified by frame-by-frame video analysis using Particle Image Velocitymeter (PIV).

Physical experiments were carried out by varying the water depth, fall height, number of fragments, bank slope angle, mass of falling rock, cliff height, and impact velocity. The tests were carried out for three water depths, i.e., 0.34, 0.27, and 0.20 m, and three fall heights, i.e., 0.64, 0.44, and 0.245 m from the surface of the water. Furthermore, the number of blocks was also varied, i.e., 6, 10, and 12 blocks having combined weights of 1.445, 2.29, and 2.82 kg, respectively. At the same time, the impact velocity changed by changing the fall height. The wave runup was measured by varying the bank slope angle, i.e., 30, 45, and 60°.

To replicate the field density of the rocks, red gutka bricks having a density of around 2000 kgm-3 were used. A singular block had a dimension of 0.055m×0.05m×0.042m. The mass volume and dimension of all the blocks were unchanged to ensure consistency in the experiments. A combination of 6, 10, and 12 blocks of red gutka bricks was used to form a cliff and measure the wave amplitude and runup of induced waves. The blocks were joined together with the help of cement paste having a water–cement ratio W/C 0.8 and cured for 2 h in front of an electric heater at 150 °C. To ensure the weak bond strength, several trials for bond strength were carried out after a curing period of 2 h, and it was found to be in the range of 0.42–0.5 MPa. In contrast, the inertial stresses at the time of impact were several times higher, such that they caused the fragmentation of the cliff. This condition was purposely designed to imitate naturally fractured cliff materials, confirming that the structure fragmented primarily along the joints upon impact with the water surface, consistent with field observations of rotational cliff collapses. The bonded blocks were placed on the rotational platform at specific heights, i.e., 0.64, 0.44, and 0.245 m from the water level, and were allowed to rotate under their own weight by pulling the hinge, such that the placed block falls in the water following rotation motion along its base Fig. 3b. To avoid the slippage of blocks and to ensure that it had sufficient frictional resistance needed for pure rotational motion of the simulated cliff, finely-grounded bricks of the same cliff material were pasted on the rotational platform, thereby preventing translational motion or vertical free fall into the water.

Furthermore, to reduce the impact of falling blocks on the base of the flume, a wooden plank weighing 2.69 kg and dimensions 0.65m×0.37m×0.01m was placed at the point of impact inside the flume. Due to its large surface area and lighter density, it tends to float in the flume, so two blocks of concrete weighing 3.58 kg were placed on it, Fig. 3a. Since the fall height was small, no considerable local breakage was observed in the blocks, and the brief water contact minimised the water absorption effect.

The induced wave amplitude was measured by placing the wave measuring gauges at a distance of 0.65 and 0.135 m from the vertical face; the gauges were wired and connected to the laptop. At the same time, the runup was measured manually with the help of a scale by pasting a scaled paper on the slope. Furthermore, the experiments were also recorded with the help of a high-resolution camera for verification purposes.

Simulating multi-phase flows is challenging due to the constant deformation of the liquid-gas interface. Various numerical methods have been developed to model these flows, each offering unique advantages depending on the specific flow regime and characteristics of interest. In this study, the Volume of Fluid (VOF) method is utilized for its effectiveness in handling significant interface distortions and topological changes. The VOF method offers superior mass conservation, which is critical in high velocity impact conditions where liquid fragmentation and wave generation are significant (Backbill et al., 1992; Hirt and Nichols, 1981). Alternative numerical schemes, such as the Front Tracking approach, are generally limited in handling complex topological changes (Tryggvason et al., 2001; Liu and Liu, 2010; Monaghan, 1994; Yang and Kong, 2018). Another approach is the Level Set method, but it suffers from mass conservation and convergence issues. The Lattice Boltzmann Method (LBM) is also common; however, its applicability to high velocity impact is rather limited (Aidun and Clausen, 2010). Given these trade-offs, the Volume of Fluid (VOF) method finds an optimal balance of computational efficiency, interface tracking capability, and proven reliability for modeling multiphase flow in the moderate-to-high velocity range relevant to this study. Therefore, a two-dimensional numerical model of a cliff, having the same properties as the experimental cliff mentioned previously, hitting the water surface is developed using the VOF method to accurately capture the liquid-gas interface.

In this approach, a volume fraction (α), ranging between 0 and 1, is applied across the entire computational domain. A value of α=1 indicates a control volume filled with liquid, while α=0 denotes a control volume filled with gas. The interface is represented by values where 0<α<1. In the Volume of Fluid (VOF) method, the momentum equation is solved across the entire computational domain, with the resulting velocity field shared by all phases. To account for surface tension effects, a continuum surface force (CSF) model is employed (Backbill et al., 1992). The normal vector n and interface mean curvature k are as follows, respectively: 1n=∇α∇α and 2k=∇⋅∇α∇α

The interface is maintained as sharp through the use of geometric reconstruction to ensure its clarity. The volume fraction (α) is discretised with the geo-reconstruct scheme, while the convective terms in the momentum equation are handled using a second-order upwind method. The PISO (Pressure-Implicit with Splitting of Operators) algorithm was employed for pressure-velocity coupling, which is well-suited for transient flows. Temporal discretisation employs a second-order implicit scheme, and spatial gradients are calculated using the Least Squares Cell-Based method.

The boundary conditions were defined as follows: the bottom boundary was modeled as a no-slip wall, while the top boundary was set as a pressure outlet at atmospheric conditions, and the lateral sides were modeled as stationary walls to confine the liquid film within the domain. For accurate simulation of the rotational motion of the cliff through the air-water interface in a multi-phase flow environment, dynamic meshing was implemented within the ANSYS Fluent framework. This approach facilitated the adaptation of the computational mesh to accommodate the cliff's movement while maintaining the integrity of the liquid-gas interface captured by the Volume of Fluid (VOF) method. Dynamic meshing was critical for modeling the complex interactions between the falling cliff and the surrounding air and water phases, allowing the mesh to deform and adapt in response to the cliff's trajectory. In ANSYS Fluent, the dynamic meshing strategy employed a combination of mesh deformation and local remeshing techniques to handle the cliff's motion. Mesh deformation was applied to adjust the existing mesh nodes smoothly as the cliff moved, preserving mesh quality in regions experiencing moderate displacement. For areas near the cliff where significant deformation could lead to poor mesh quality, local remeshing was utilized to regenerate mesh elements for better numerical stability and accuracy. The smoothing and remeshing algorithms were configured to maintain high mesh quality, with a skewness threshold set to prevent excessive element distortion.

The rotational cliff collapse was simulated using an in-house user-defined function (UDF). This UDF interfaced with ANSYS Fluent to dynamically update the rock's position and velocity. To enhance computational efficiency, a dynamic mesh zone was defined around the cliff, with a finer mesh resolution near its surface to capture the sharp gradients in the flow field and interface dynamics. The mesh was gradually coarsened away from the rock to reduce computational cost while maintaining sufficient resolution in the far-field regions. The dynamic meshing process was synchronised with the transient flow solver, using a time step size determined through a time step independence study to balance accuracy and computational efficiency. It is also worth mentioning that the numerical simulations were performed considering the rock as a unified mass. This approach describes the slight differences between the experimental and numerical results, which are nonetheless within the acceptable range.

The MEP model was developed for predicting wave amplitude and runup using experimental data, as shown in Table 2. A dataset of 81 experimental results was used as an input to a machine learning model. Furthermore, the data was divided into 70/30 ratios for training and validation purposes before developing the model. The model starts working by generating a random chromosome population, and it continues to generate the chromosomes until a terminal condition is achieved, generating an optimal expression from the data having input and output pairs over a certain number of generations, as shown in Fig. 4.

Based on a binary tournament process, parents are selected and then undergo a recombination process through a consistent crossover probability. This recombination produces two more offspring. These offspring go through mutation, and if these offspring perform better than the least fitting offspring in the current population, then the better offspring replace them. The illustrations used by MEP are similar to the ones used by C++ and Pascal compilers. The MEP chromosomes are comprised of numerous genes combined using various mathematical operators such as addition (+), subtraction (-), multiplication (×), and division (/), and these genes create expression trees (ETs) (Cheng et al., 2020). Moreover, there are several hyperparameters such as code length, sub-population size and number, crossover probability, and other sets of various functions involved in in generation of MEP code, and they also govern the overall performance of the code. Among these parameters, the size of the population tells us about the number of programs being generated, whereas an increase or decrease in subpopulation size directly affects the complexity and computation time of the model. Moreover, the length of the developed model is controlled by varying the code length parameter. During model development, prerequisite tuning procedures were applied to optimize these hyperparameters. This careful selection minimized the risk of premature convergence or underfitting while ensuring computational efficiency.

The experimental results of the wave amplitude and runup, induced by rotational cliff collapse, reveal complex hydrodynamic processes. As shown in Fig. 5, the failure is initiated by the rotational fall of the cliff, leading to a significant amount of impact energy upon hitting the water surface. The impact induced a huge splash, which is evident from Fig. 5b, e and h. It was observed that the shape of the splash also varies with water depth for all the cases; higher water depths resulted in a mushroom-shaped splash, i.e., broader on the top, as the momentum dissipates before interacting with the bottom surface, resulting in a vertical jet and the formation of a mushroom-shaped splash. as can be seen in Fig. 5h. The observed phenomena perfectly align with the basic concepts of fluid dynamics, which state that greater depths absorb more impact energy compared to shallow waters. Shallow waters produced a vertically elongated splash as can be seen in Fig. 5b and e. It can be observed that as the depth decreases, the splash becomes more elongated, as shallower depths intensify the upward momentum transfer, thus resulting in a more elongated shape (Kubota and Mochizuki, 2009).

The numerical simulations conducted in this study successfully captured key dynamic characteristics of the wave generated by the rotational cliff collapse, specifically the wave amplitude and wave runup, across a range of test cases. Moreover, the front velocity of the incident wave was also measured. The simulations were also focused on verifying the results obtained from the rotational cliff collapse in the experiments. To quantify the wave amplitude, runup, and velocity, a post-processing technique was employed. To establish the reliability of the wave front velocity measurements, the velocity was calculated at 5–7 distinct locations along the wave's propagation path and at multiple time steps during the simulation. This multi-point sampling approach minimized errors due to spatial and temporal variations. Figure 12 shows a representative case of wave formation and propagation in a water tank at a depth of d=0.2 m at various time frames.

The wave amplitude was defined as the peak vertical displacement of the liquid surface relative to the undisturbed free surface level. Figure 13 illustrates a representative case, depicting the wave front propagation.

To validate the results of simulations, we compared the results of the runup height with the experimental values. Table 3 presents the runup values for various runup slope angles, i.e., 30, 45, and 60°, for a water depth of 0.27 m. The comparison of simulated values was performed at this depth, as it lies in the middle of the experimental test range of water depths. Numerical modeling results indicate that for a fixed water depth, the runup values consistently decrease as the runup slope angle increases from 30–60°. At a water depth of 0.27 m, the runup decreases from 0.2 m at 30–0.17 m at 45°, and further to 0.11 m at 60°. This reduction is attributed to the changing momentum transfer dynamics with increasing slope angle. At less steep angles (closer to horizontal, e.g., 30°), the rock's momentum generates a stronger radial splash and greater upslope displacement of the liquid along the cliff. As the angle increases toward 60°, a larger component of the momentum is directed parallel to the cliff, reducing the vertical impulse. The experimental and numerical results agree well, and the difference lies within the acceptable range of 4 %–5 %. The experimental results for the other two water depths also indicate similar behavior.

Next, we measured the wave velocity through the numerical results, as it wasn't captured accurately through experimental images. Figure 14 illustrates the simulated wave fronts at a time instant of t=1 s following the impact of the solid rock on the liquid pool, for various water depths and a fixed slope angle of 30°. These visualizations highlight the propagation of the waves from the impact zone. The slope angle was varied across simulations to assess its influence on wave characteristics. It was observed that changes in the slope angle induced only minor variations in both the wave front velocity and wave amplitude for a given pool depth. These perturbations were typically within 1 %–2 % of the mean values. Consequently, to streamline the analysis and focus on dominant trends, the wave front velocity and height were averaged over the range of slope angles for each specific water depth.

However, variations in water depth exerted a pronounced effect on the wave dynamics, leading to significant alterations in both the propagation velocity and amplitude of the generated waves. This depth-dependent behavior is quantified in Table 4, which presents the averaged results from the numerical simulations. For a shallow water depth of d=0.2 m, the average wave front velocity was computed as 1.48 ms-1, with a corresponding average wave height of 0.11 m. As the pool depth increased to d=0.27 m, the velocity rose to 1.58 ms-1, while the wave height decreased to 0.07 m. Further deepening to 0.34 m yielded a velocity of 1.74 ms-1 and a reduced wave amplitude of 0.06 m. These trends indicate an approximately linear increase in velocity with depth, accompanied by an inverse relationship for wave amplitude.

The observed depth dependence can be rationalized through fundamental principles of wave propagation in gravity-dominated, multi-phase flows. In the shallow water regime, given that the pool depths (0.2–0.34 m) are comparable to or smaller than the wavelengths of the generated waves, the phase velocity c of long gravity waves approximates c≈gh, where g is the gravitational acceleration (9.81 ms-2), and h is the undisturbed water depth. This relation arises from the shallow water equations, where hydrostatic pressure balance and negligible vertical acceleration dominate, leading to a dispersionless incident wave speed that scales with the square root of depth. Substituting the water depths yields theoretical velocities of approximately 1.40 ms-1 for d=0.2 m, 1.63 ms-1 for d=0.27 m, and 1.83 ms-1 for d=0.34 m, which align closely with the simulated values (discrepancies of 7 %–10 % may stem from viscous dissipation, non-hydrostatic effects near the impact zone, or spreading of the wave front). A comparative analysis of the results is shown in Table 4.

Conversely, the decrease in wave amplitude with increasing water depth aligns with energy conservation and volume displacement considerations in impact-generated waves. The impact of rotational cliff collapse imparts a fixed kinetic energy and displaces a finite volume of liquid, creating an initial cavity and subsequent outflow that evolves into a propagating wave. In shallower pools, the displaced volume is confined to a smaller cross-sectional area, resulting in greater vertical amplification to accommodate the same mass redistribution. For deeper water depths, the energy is distributed over a larger water column, diluting the surface perturbation and yielding lower amplitudes. The trends observed in the numerical simulations for water waves induced by rotational cliff collapse are in good agreement with theoretical and experimental results, indicating that water depth has a direct effect on the wave velocity and an inverse effect on the wave amplitude and runup.

The purpose was to develop a precise model for wave amplitude and runup induced by rotational cliff collapse. The predicted model is a function of seven variables, i.e., water depth, fall height, cliff mass, impact velocity, cliff height, runup slope angle, and number of fragments, and can be described as follows, 3Wave amplitude and runup=f(d,H,m,v,h,α,Nf)

The relation among the parameters was evaluated using Pearson's correlation to analyze the multicollinearity and interdependency between the parameters, as they can obscure the interpretation of the developed model. The model was developed by splitting the data into two subsets, i.e., training (70 %) and testing (30 %). The randomization was done by MEP itself. Following the criteria, 70 % of the data, i.e., 57 data points, were taken as training data, whereas 30 % of the data, i.e., 24 data points, were considered for validation of the model. The mathematical expression for MEP is obtained by solving the C++ code and representing it as per optimized hyperparameter settings, as shown in Table 5. The prediction model for wave amplitude and runup was developed by analyzing the MEP code in MATLAB, as shown in Eqs. (4) and (5).

4Wave amplitudeA=dαd+Nf+m+2vh2m+Nf+(d+Nf+m)+2vhdα(d+Nf+m)5Wave runupR=Ah+A⋅-BαB/αA⋅BαA=v+hdB=v+m+hd Whereas d is the water depth (m), m is the mass of the cliff (kg), v is the impact velocity (ms-1), h is the cliff height (m), α is the runup slope angle, and Nf is the number of fragments.

The validation of the proposed model is an important feature in predictive modeling. It has been observed that sometimes the model performs very well for training data sets, but fails to perform during the validation stage for unseen data. So, the developed prediction model was further validated by conducting the sensitivity and parametric analysis for both the wave amplitude and runup.