Study on the Active Wave Absorption Methods in Lattice Boltzmann Numerical Wave Tank

Abstract

The active wave absorption method has been widely employed in numerical wave tanks. The wave absorption performance of active wave absorption methods is investigated within a numerical wave tank based on a lattice Boltzmann method. Specifically, two active wave absorption methods—the classical shallow water method and the extended range method—are compared. By analyzing the contributions of free and bound components in the harmonics of the reflected wave to the reflection coefficient, we found that the extended-range method is more effective than the shallow-water method in absorbing the reflection of the primary harmonic. Moreover, a wave absorption performance index is proposed to carry out rapid evaluation of active wave absorption method performance without resorting to numerical simulations. Our findings indicate that the performance index ratio of two active wave absorption methods closely mirrored their reflection coefficient ratio. Notably, the extended-range method significantly reduces the performance index in both shallow and deep waters, exhibiting superior active absorption performance within the lattice Boltzmann method-based numerical wave tank context compared to the shallow-water method.

1. Introduction

Coastal engineering is vital for protecting coastal communities and ecosystems from waves and storm surges. Recent advancements in numerical wave tanks (NWTs) have been essential for improving our understanding of wave–structure interactions, sediment and pollutant transport, and the enhancement of coastal protection designs. Numerical wave models track free surfaces and employ various methodologies, including Reynolds-averaged Navier–Stokes (RANS) equations [1,2,3], smoothed-particle hydrodynamics (SPH) [4,5,6,7], and the lattice Boltzmann method (LBM) [8,9,10,11]. These models are increasingly prevalent in research endeavors.

Accurate reproduction of target waves in numerical wave simulations is crucial and relies heavily on the performance of wave generation and absorption methods used in the simulation. Absorption methods are typically classified into passive wave absorption and active wave absorption (AWA) [12]. The AWA method, often referred to as the open boundary method in numerical models [12], absorbs outgoing waves by adjusting the boundary condition based on the measured incident and reflected waves [13]. Since AWA methods do not require additional area for absorption, their implementation costs are significantly reduced, making them favored choices. The earliest AWA method, proposed by Schäffer and Klopman [12], is based on linear shallow-water wave theory and is known as the shallow water AWA (SW-AWA) method. This method has been widely adopted by various numerical wave models, achieving satisfactory wave absorption performance across different models, including RANS models [14], SPH models [15], and LBM models [16]. Recently, an AWA method considering velocity distribution along the vertical direction has been proposed [13], demonstrating superior performance in a RANS-based NWT established by OpenFOAM.

In recent years, the LBM has gained widespread usage in engineering applications [17,18,19] due to its notable computational efficiency [20]. However, the extended-range AWA (ER-AWA) [13] method has yet to be implemented in LBM-based numerical wave tanks. This disparity arises due to the fact that while the lattice Boltzmann equation solves the mesoscopic particle density distribution function, the AWA method operates by adjusting velocities at open boundaries. Consequently, validating and assessing the wave absorption performance of the ER-AWA method within the LBM constitutes a primary objective of this study.

Furthermore, the performance of the ER-AWA method necessitates reevaluation, despite previous analysis conducted in RANS-based numerical wave tanks using a three-point wave separation method as proposed by Higuera [13]. However, the three-point wave separation method lacks the ability to distinguish the high-order bound and free harmonics [21], potentially introducing errors into the separation results of reflected waves and leading to inaccurate conclusions regarding the absorption performance of AWA methods. Hence, another aspect of this study involves utilizing a more reliable wave separation method [21,22,23,24] for wave reflection analysis and evaluating the wave absorption performance of the ER-AWA method.

Therefore, the main purpose of this paper is to implement the ER-AWA method in an LBM-based numerical wave tank (LBM-NWT) and utilize a more reliable wave separation method to evaluate and compare the wave absorption performance of both the SW-AWA method and the ER-AWA method within the LBM-NWT. This investigation aims to uncover the underlying mechanism driving the enhancement of wave absorption performance by the ER-AWA method. Additionally, it provides tools for future applications of the LBM-NWT in deep-water conditions, including wave–structure interactions and wave refraction over complex terrain.

The paper is organized as follows. Section 2 introduces the numerical model, including the LBM-NWT, the theory of the SW-AWA and ER-AWA methods and the wave separation method, followed by the two-dimensional model setup. In Section 3, a comprehensive analysis of reflection coefficients of the SW-AWA method and ER-AWA method in the 2D LBM-NWT is provided. Section 4 presents the disparities in wave absorption performance between RANS-NWT, physical flumes, and LBM-NWT, along with an analysis of the mechanism driving the enhancement of wave absorption performance by the ER-AWA method. Finally, conclusions are drawn in Section 5.

2. Numerical Model

2.1. Numerical Wave Tank Based on LBM

The numerical wave tank is developed based on a two-dimensional LBM model, which solves a discretized modified lattice Boltzmann equation with the D2Q9 velocity set [25] and the multi-relaxation-time collision operator [26], represented by:

$$f_{\alpha }\left(x+e_{\alpha }{\delta }_t,t+{\delta }_t\right)-f_{\alpha }\left(x,t\right)=-M^{-1}\left[\widehat{S}\left(m-m^{eq}\right)\right]$$

(1)

where $f_{\alpha }\left(x,t\right)$ denotes the fluid particle density distribution at lattice position x and time t, with velocity $e_{\alpha }$ and α ranging from 0 to 8. The particle speed is $c={\delta }_x/{\delta }_t$, ${\delta }_x$ is the lattice constant, and ${\delta }_t$ is the time step. M is the transformation matrix. $\widehat{S}$ is the diagonal collision matrix in moment space. m is the vector of moments; and m^eq is the vector of equilibrium moments. A detailed description of the multi-relaxation-time collision operator used in this paper is given by Lallemand and Luo [27]. The incompressible Navier–Stokes equations derived from the modified lattice Boltzmann equation using the Maxwell iteration method [19] are written as:

$$\nabla \cdot u=O\left({\delta }_x^2\right)$$

(2)

$$\frac{\partial u}{\partial t}+\nabla \cdot uu=-\frac{\nabla p^{*}}{{\rho }_0}+\nu {\nabla }^{2}u+O\left({\delta }_x^2\right)$$

(3)

where $u={\displaystyle {\sum }_{\alpha }{e}_{\alpha }{f}_{\alpha }}/{\rho }_0$ is the vector of fluid velocity, $p^{*}=\frac{c^2}{3}{\displaystyle {\sum }_{\alpha }{f}_{\alpha }}$ is the modified pressure [9] calculated by the distribution functions directly, $p=p^{*}+{\rho }_{0}g\cdot \left(x-x_{\mathrm{ref}}\right)$ is the pressure of fluid calculated from the modified pressure, x_ref = (0, d) is the vector of the reference position, d is the mean water depth, ${\rho }_0$ is the reference density of the fluid, $g=\left(0,-g\right)$ is the vector of gravity acceleration and g = 9.81 m/s², and ν is the kinetic viscosity of fluid.

The wall, slip, and open boundary conditions in the LBM model are implemented by the reconstruction of the density distribution functions streaming from the boundaries. The second-order accurate non-equilibrium extrapolation velocity scheme [28] is used to perform the reconstruction, and the scheme is given as:

$$f_{\alpha }\left(x_f,t+{\delta }_t\right)=f_{\alpha }\left(x_f+e_{\alpha }{\delta }_t,t+{\delta }_t\right)+f_{\alpha }^{eq}\left(p_f^{*},u_b\right)-f_{\alpha }^{eq}\left(p_f^{*},u_f\right)$$

(4)

where $x_f=x_B+e_{\alpha }{\delta }_t$ is the coordinate vector of the bulk lattice, which is located inside the simulation domain, x_B is the coordinate vector of boundary lattices, $p_f^{*}$ is the modified pressure of lattice located at x_f, u_b is the moving velocity vector of the boundary, f^eq is the equilibrium density distribution function, and u_f is the velocity vector of lattice located at x_f.

The single-phase VOF method proposed by [29] is adopted to capture the free surface of waves. The free-surface Körner scheme [30,31] is introduced to close the LBM model at the free surface, and the scheme is given as:

$$f_{\alpha }\left(x_f,t+{\delta }_t\right)=-f_{\beta }\left(x_f+e_{\beta }{\delta }_t,t+{\delta }_t\right)+f_{\alpha }^{eq}\left(p_b^{*},u_b\right)+f_{\beta }^{eq}\left(p_b^{*},u_b\right)$$

(5)

where f_β represents the density distribution function of particles with discrete velocity vector e_β = −e_α, $p_b^{*}={\rho }_{0}g\cdot \left(x_{\mathrm{ref}}-x_b\right)$ represents the modified pressure at the free surface, x_b is the coordinate vector of the intersection point between the line connecting x_f and $x_f+e_{\beta }{\delta }_t$ and the free surface, and u_b is the fluid velocity vector at the intersection point, extrapolated from interior lattices.

The modified pressure p_b^* used in the free-surface Körner scheme couples the influence of atmosphere boundary condition and gravity on wave motion [9]. Before executing the collision step in Equation (1) at the lattices near the “surface” region, it is necessary to reconstruct the density distribution functions that streaming from the atmosphere using the free-surface Körner scheme, thus ensuring the closure of the model at the free surface.

The static open boundary wave generator [14] and AWA method [12] are adopted. The velocity at the open boundary is given as:

$$u\left(x_B,t\right)=\left\{\begin{array}{l}r\left(t\right)u_T\left(x_B,t\right)+u_R\left(x_B,t\right)n, y\le {\eta }_M+\frac{{\delta }_x}{2}\hfill \\ r\left(t\right)u_T\left(x_B,t\right), y>{\eta }_M+\frac{{\delta }_x}{2}\hfill \end{array}\right.$$

(6)

where u_T is the velocity vector of the incoming wave, u_R is the velocity of the outgoing wave, n is the normal vector of the open boundary, r(t) is the ramp coefficient, which is used to ramp the model from the initial static field, y is the vertical coordinate, and η_M(t) is the surface elevation of one grid size away from the open boundary at time t. The details of evaluating the velocity of the outgoing wave and implementations of AWA methods in the LBM model are given in Section 2.2. More details about implementations of the active wave generator in the LBM model can found in [16]. The meanings of physical variables, mathematical symbols, and acronyms appearing in this paper can be found in Appendix A.

2.2. The SW-AWA and the ER-AWA

For wave absorption purposes, the velocity induced by the outgoing wave is provided by the AWA method. For the SW-AWA method [12], the velocity induced by the outgoing wave is given as:

$$u_R=\frac{{\eta }_R}{{\eta }_M+d}C$$

(7)

where C is the wave celerity, ${\eta }_R$ is the surface elevation of the outgoing wave at the open boundary, and d is the mean water depth. When there is an incoming wave at the open boundary, the surface elevation of the outgoing wave can be approximated by ${\eta }_R\approx {\eta }_M-{\eta }_I$, where ${\eta }_I$ is the surface elevation of the incoming wave; otherwise, ${\eta }_R={\eta }_M$.

According to the ER-AWA method, the vertical distribution of the horizontal velocity of the airy wave theory is introduced into Equation(7), and the velocity induced by the outgoing wave is given as:

$$u_R=\frac{{\eta }_R}{{\eta }_M+d}kd\frac{\cosh k\left(y+d\right)}{\sinh kd}C$$

(8)

where k is the wave number. Equation (8) is the same as Equation (5) of Higuera [13] when η_M is negligible compared with d. The use of ${\eta }_M+d$ in Equation (8) instead of d in Equation (5) of Higuera [13] makes the velocity expression consistent with Equation (7) in shallow water and makes sense in the whole domain. This change is trivial for practical application of wave absorption and could be neglected.

In the present LBM model, the surface elevation of one grid size away from the open boundary η_M(t) is given as:

$${\eta }_M\left(t\right)={\displaystyle \underset{y_{\min }}{\overset{y_{\max }}{\int }}\epsilon \left(x_B-n{\delta }_x,t\right)dy}-d$$

(9)

where y_min is the height of the bottom, y_max is the height of the top, and x_B represents the position of the open boundary lattice.

The volume fraction functions ε(x_B, t) and phase states s(x_B, t) of the open boundary lattices are evaluated according to the type of the open boundary. When there is a wave generation open boundary, the volume fraction function in lattices of the open boundary is calculated as follows:

$$\epsilon \left(x_B,t\right)=\left\{\begin{array}{l}0,y-\frac{{\delta }_x}{2}>{\eta }_I+d\hfill \\ \left(\frac{{\eta }_I+d-y}{{\delta }_x}+\frac{1}{2}\right),y+\frac{{\delta }_x}{2}>{\eta }_I+d\hfill \\ 1,y+\frac{{\delta }_x}{2}\le {\eta }_I+d\hfill \end{array}\right.$$

(10)

and the expression of the phase state in lattices of the open boundary is given as:

$$s\left(x_B,t\right)=\left\{\begin{array}{l}Gas,y-\frac{{\delta }_x}{2}>{\eta }_I+d\hfill \\ Surface,y+\frac{{\delta }_x}{2}>{\eta }_I+d\hfill \\ Liquid,y+\frac{{\delta }_x}{2}\le {\eta }_I+d\hfill \end{array}\right.$$

(11)

where y is the vertical coordinate. When there is a wave absorption open boundary, the volume fraction function and the phase state in lattices of the open boundary are given as:

$$\begin{array}{l}\epsilon \left(x_B,t\right)=\epsilon \left(x_B-n{\delta }_x,t\right)\\ s\left(x_B,t\right)=s\left(x_B-n{\delta }_x,t\right)\end{array}$$

(12)

A flowchart of the AWA method implemented in the present LBM-based NWT is presented in Figure 1.

2.3. Wave Separation Method

In order to distinguish between high-order bound (characterized by frequencies and wave numbers both being multiples of that of the main component) and free harmonics (where the frequency is a multiple of that of the main component while wave number is determined by the dispersion relation), we employ the 4-point method proposed by Andersen et al. [23].

The implementation of the 4-point method is tested using a group of elevation series generated by the Stokes wave theory without any reflected wave, as shown in

Table 1. Wave parameters for wave separation test.

Case	Wave Height (m)	Wave Period (s)	Depth (m)	Wave Theory
1	0.004	1.00	1.472	Stokes I
2	0.016	1.00	1.472	Stokes II
3	0.039	1.00	1.472	Stokes III
4	0.108	1.00	1.472	Stokes IV
5	0.542	2.00	5.886	Stokes V

. Subsequently,

Table 2. Wave separation results (in millimeters) of elevation time series created by the Stokes wave theory without reflected wave. Amplitude results by separation method listed as follows: subscripts I and R represent incident and reflected, respectively, subscripts B and F represent the bound component and the free component, respectively, and the digital superscripts 1, 2, 3, 4 represent frequencies of 1/T, 2/T, 3/T, and 4/T, respectively).

Case	${\mathit{a}}_{\mathit{I}}^{(1)}$	${\mathit{a}}_{\mathit{I},\mathit{B}}^{(2)}$	${\mathit{a}}_{\mathit{I},\mathit{F}}^{(2)}$	${\mathit{a}}_{\mathit{I},\mathit{B}}^{(3)}$	${\mathit{a}}_{\mathit{I},\mathit{F}}^{(3)}$	${\mathit{a}}_{\mathit{I},\mathit{B}}^{(4)}$	${\mathit{a}}_{\mathit{I},\mathit{F}}^{(4)}$	${\mathit{a}}_{\mathit{R}}^{(1)}$	${\mathit{a}}_{\mathit{R},\mathit{F}}^{(2)}$	${\mathit{a}}_{\mathit{R},\mathit{F}}^{(3)}$	${\mathit{a}}_{\mathit{R},\mathit{F}}^{(4)}$
1	2.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2	8.00	0.14	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
3	19.50	0.81	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
4	53.07	5.81	0.00	0.93	0.00	0.00	0.00	0.00	0.00	0.00	0.00
5	262.45	38.11	0.00	8.16	0.00	1.56	0.00	0.00	0.00	0.00	0.00

presents the separation results obtained using both methods. Upon examination of Table 2, it becomes evident that the separation results obtained by the 4-point method accurately reproduce the expectations set by the Stokes wave theory, thereby affirming the correctness of the present work’s implementation of the separation method. Specifically, no reflected waves are detected in the separated harmonics of each order, the bound component of super-harmonics is appropriately isolated, and the amplitude results of the free components are all zero.

2.4. Numerical Model Setup

The two-dimensional numerical wave tank is depicted in Figure 2, featuring dimensions of 6λ in length and d + 2H in height, where d is the water depth, λ represents the wavelength and H denotes the wave height. The lattice constant is set to H/8, while the Mach number is established at 0.064. The characteristic velocity of the flow is selected as the wave celerity, the time step is ${\delta }_t=\mathrm{Ma}{\delta }_x/C$, and the relaxation time is denoted as $\tau =3\nu {\delta }_t/{\delta }_x^2+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. Waves are generated at the leftmost open boundary and absorbed at the rightmost open boundary, with a no-slip boundary condition enforced at the bottom boundary. The initial condition of the model is a quiescent flow field, with the initial condition for the modified pressure set to p* = 0.

Surface elevation data from five gauges (G1–G5) are sampled to conduct wave separation analysis for benchmark cases. Gauge G1 is positioned at x₁ = 5λ, while the remaining gauges are positioned at intervals of 0.05λ, 0.12λ, 0.21λ, and 0.31λ away from G1, respectively. For the 3-point wave separation method [32] (3-point method), gauges G1, G2, and G4 are utilized. Conversely, for the 4-point wave separation method [23] (4-point method), all gauges are employed. The reflection coefficients (RCs) are computed according to the definition proposed by Lin and Huang [21], which involves determining the ratio of the square root of the sum of energies of the reflected and incident wave harmonics.

For detailed validation of the numerical wave tank used in this study, including its capacity to simulate various wave theories and wave deformation and breaking over submerged breakwaters and slopes, please refer to [9] and [16].

3. Numerical Results

The ER-AWA method has previously been investigated in a RANS-based numerical wave tank by Higuera [13], with its wave absorption performance analyzed and evaluated primarily using the 3-point method. However, it is imperative to reevaluate pertinent conclusions using the 4-point method, particularly focusing on the following aspects.

Firstly, it is noted that the velocity distribution induced by the outgoing wave, as assumed by the ER-AWA method based on linearized wave theory, may result in larger RCs as nonlinearity increases when applied to nonlinear waves.

Secondly, despite demonstrating significantly improved wave absorption performance compared to the SW-AWA method in deep water, the RC remains relatively high at 18.6%.

Therefore, this section aims to reassess the wave absorption performance of AWA methods for both shallow-water waves with strong nonlinearity and deep-water waves, utilizing the 4-point method.

The numerical wave tank setup remains consistent with that provided in Section 2. To mitigate interference from a re-reflected wave generated by the left open boundary on simulated results at wave gauges, the analysis duration is selected from the arrival of the reflected wave at the wave gauges to the arrival of the re-reflected wave at the wave gauges, spanning approximately 10 wave periods in total. For solitary waves, stability is typically achieved after 5 wave periods ($T=2d\mathrm{acosh}\sqrt{20}/\left(C\sqrt{3H/4d}\right)$), and thus the total simulation time is set to 17 wave periods. Conversely, for regular waves, stability usually requires 14 wave periods, necessitating a total simulation time of 27 wave periods.

Figure 3 illustrates the wave parameters utilized in the numerical simulation alongside their corresponding applicable wave theories. For shallow-water conditions, three cases (a, b, and c) are examined, all of which are suitable for solitary wave theory. For deep-water conditions, three groups (A, B, and C) of cases are considered, with applicable wave theories ranging from Stokes second to fifth order. Each group comprises three cases with varying relative water depths. Across all aforementioned cases, the mean number of grids employed in the simulation is set to 0.5 million.

3.1. Shallow-Water Benchmark Cases

In this section, the wave absorption performance of both the SW-AWA method and the ER-AWA method implemented in the LBM-NWT is tested using three solitary waves with varying wave heights and water depths. The separation of incident and reflected waves for solitary waves is conducted at the wave gauge located at x = 5λ. The relevant definitions used in this section are as follows, with their schematic diagram depicted in Figure 4. Incident wave height (h_I) represents the difference between the crest of the incident wave and the mean water level for the solitary wave, and reflected wave height (h_R) represents the wave height of the largest reflected wave observed for the solitary wave.

Table 3. The wave separation results of the shallow-water cases including the wave height and RC.

	H/d	SW-AWA		ER-AWA
		hI (m)	RC	hI (m)	RC
Case a	0.70	0.70	6.58%	0.70	5.61%
Case b	0.63	0.25	6.03%	0.25	4.31%
Case c	0.60	0.12	6.08%	0.12	4.76%

presents the wave separation results obtained from the simulated elevations. The RC observed at the wave gauge generated by the SW-AWA method amounts to 6%, whereas that generated by the ER-AWA method ranges from 4% to 5%. Both methods, when applied to the LBM-NWT, demonstrate satisfactory wave simulation outcomes (Figure 5).

Based on the RC results, the following conclusions can be drawn. Both the SW-AWA method and the ER-AWA method implemented in the present LBM-based NWT exhibit robust performance under shallow-water conditions. The ER-AWA method proves to be preferable to the SW-AWA method based on the observed RC values.

3.2. Deep-Water Benchmark Cases

In this section, the wave absorption performance of both the SW-AWA method and the ER-AWA method is evaluated using nine cases of deep-water waves, as depicted in Figure 3.

Table 4. Incident wave heights and RCs.

	H/d	SW-AWA		ER-AWA
		HI (m)	RC	HI (m)	RC
Case A1	0.027	0.04	61.15%	0.04	1.70%
Case A2	0.013	0.04	96.14%	0.04	2.33%
Case A3	0.006	0.05	61.29%	0.04	1.98%
Case B1	0.073	0.10	54.99%	0.11	2.50%
Case B2	0.034	0.10	77.09%	0.11	3.96%
Case B3	0.017	0.14	77.47%	0.11	3.51%
Case C1	0.090	0.53	47.69%	0.56	3.16%
Case C2	0.042	0.49	50.25%	0.56	4.49%
Case C3	0.021	0.67	70.43%	0.55	5.80%

displays the separation results obtained utilizing the 4-point method. It is evident that the RC values for waves generated by the SW-AWA method range from no less than 47% to as high as 96%. In contrast, the RC values for waves generated by the ER-AWA method do not exceed 5.80%. Representative simulated water elevations for Case A2, Case B3, and Case C3 are illustrated in Figure 6.

Based on the aforementioned results, the following conclusions can be drawn. The SW-AWA method proves ineffective in absorbing outgoing waves under deep-water conditions. Conversely, the ER-AWA method demonstrates satisfactory wave absorption performance in the present LBM-based numerical wave tank. Unlike the high RC values (up to 18.6%) for the ER-AWA method in deep-water conditions reported by Higuera [13] using the 3-point method, the RC values obtained in this study using the 4-point method do not exceed 5.8%. This discrepancy in RC values is attributed to the overestimation resulting from errors associated with the 3-point method.

4. Discussion

The wave absorption performance of the ER-AWA method significantly surpasses that of the SW-AWA method. This section will delve into three key points: 1. investigation of potential disparities in ER-AWA performance between LBM, RANS-NWTs, and physical wave flumes; 2. identification of the areas where the ER-AWA method outperforms the SW-AWA method; and 3. exploring the underlying mechanism responsible for enhancing the performance of the ER-AWA method.

4.1. Comparison between RANS-NWT and LBM-NWT

We compare the reflection analysis of Higuera’s Case b (with wave height 0.1 m, water depth 0.4 m, and wave period 1.0 s), which represents the largest reflection coefficient case reported in Higuera [13], based on the wave results obtained from the present LBM-NWT using both the 3-point method and the 4-point method. In Higuera’s [13] RANS-based NWT employing the ER-AWA method, the RC for Higuera Case b calculated using the 3-point method is 18.6%.

The wave separation results based on the present LBM-NWT employing the ER-AWA method are listed in

Table 5. Wave separation results (in millimeters) of Higuera Case b by LBM-NWT utilizing the ER-AWA method.

Method	${\mathit{a}}_{\mathit{I}}^{\left(1\right)}$	${\mathit{a}}_{\mathit{I},\mathit{B}}^{\left(2\right)}$	${\mathit{a}}_{\mathit{I},\mathit{F}}^{\left(2\right)}$	${\mathit{a}}_{\mathit{I},\mathit{B}}^{\left(3\right)}$	${\mathit{a}}_{\mathit{I},\mathit{F}}^{\left(3\right)}$	${\mathit{a}}_{\mathit{R}}^{\left(1\right)}$	${\mathit{a}}_{\mathit{R},\mathit{B}}^{\left(2\right)}$	${\mathit{a}}_{\mathit{R},\mathit{F}}^{\left(2\right)}$	${\mathit{a}}_{\mathit{R},\mathit{B}}^{\left(3\right)}$	${\mathit{a}}_{\mathit{R},\mathit{F}}^{\left(3\right)}$	RC
3-point	49.70	0.00	4.50	0.00	0.62	2.50	0.00	4.10	0.00	0.58	9.69%
4-point	49.60	6.80	1.30	1.90	0.579	1.50	0.88	0.00	0.54	0.53	3.78%

. The RC obtained using the 3-point method stands at 9.69%, exceeding the RC (18.6%) obtained from Higuera’s NWT. Conversely, the RC obtained using the 4-point method is 3.78%. This discrepancy in RC primarily stems from the differences between the two NWTs, possibly attributable to variations in the VOF methods employed. The VOF method utilized in the present LBM-NWT is geometrically based, requiring only one grid size thickness to track the free surface. In contrast, Higuera’s RANS-NWT employs an algebraic-based VOF method, necessitating at least three grid size thicknesses for free-surface tracking. Additionally, the grid size used by Higuera’s NWT is 1 cm, equivalent to only eight grids in a wave height, resulting in less precise tracking of free-surface motions [34]. To address this concern, a geometrically based VOF method was integrated into OpenFOAM by Gamet, Scala [35]. Comparative investigations on wave absorption performance based on this VOF method in NWT will be pursued in future research.

4.2. Comparison between Physical and Numerical Flumes

The RCs obtained from the numerical simulations in this study are compared with those measured in two different physical flumes using various active absorption methods. The M70 flume [36] utilized a paddle-type wavemaker with near-field feedback based on linear theory and pre-filter, while the A16 flume [37] employed a piston-type wavemaker with near-field feedback based on linear theory and FIR filtering. Figure 7 presents the RCs from both the numerical simulations and the physical flume tests. As illustrated in the figure, measured RCs obtained by [37] are higher than those obtained by [36]. The numerical model using SW-AWA achieved RCs significantly higher than those in the physical flumes. In contrast, the numerical model using ER-AWA yielded RCs consistent with [36] and superior to [37].

The difference between the two experimental datasets is likely due to the different types of wave absorption equips used. The figure also indicates that AWA methods in physical flumes are less effective for low-frequency waves, specifically waves with angular frequencies below 1 Hz. However, the AWA in the numerical wave tank performed well in eliminating low-frequency waves. In the shallow-water cases, even the RCs simulated using the SW-AWA did not exceed 7%. This is primarily due to the small size of the wavemakers used in physical flumes, which are unable to absorb long-period waves within the operational range.

Additionally, [37] noted that the proportion of RCs contributed by higher-order harmonics in the physical flume is significant and mainly composed of free component, with wave amplitudes 4–10 times larger than the bound component. We observed a similar phenomenon in our numerical simulations. However, regarding the second phenomenon, in Section 4.3 we observed that the contributions of the free and bound component to the RCs of higher-order harmonics are almost identical. This discrepancy may be attributed to differences in the principles of the AWA method between numerical and physical flumes. Future research can analyze these differences to reduce the contribution of bound higher-order harmonics in NWT, thereby further reducing RCs.

4.3. Reflection Analysis of Deep-Water Cases

Using the 4-point method enables the separation of incident and the reflected wave harmonics, as well as the distinction between free and bound components. By comparing their reflected components, the reason that the ER-AWA method outperforms the SW-AWA method can be found.

Table 6. The reflection coefficient (RC) contributed by the first three harmonics (represented by subscripts 1, 2, and 3) measured in simulation utilizing the SW-AWA method.

	CR1	CR2, B	CR2, F	CR3, B	CR3, F	Total
Case A1	61.09%	0.00%	0.06%	0.00%	0.00%	61.15%
Case A2	96.04%	0.00%	0.00%	0.00%	0.00%	96.14%
Case A3	61.17%	0.00%	0.12%	0.00%	0.00%	61.29%
Case B1	54.39%	0.05%	0.16%	0.22%	0.16%	54.99%
Case B2	76.78%	0.08%	0.08%	0.08%	0.08%	77.09%
Case B3	76.77%	0.23%	0.23%	0.08%	0.08%	77.47%
Case C1	41.49%	0.14%	0.10%	3.96%	2.00%	47.69%
Case C2	49.35%	0.15%	0.25%	0.30%	0.15%	50.25%
Case C3	68.95%	0.70%	0.14%	0.42%	0.14%	70.43%

and

Table 7. The reflection coefficient (RC) contributed by the first three harmonics (represented by the subscripts 1, 2, and 3) measured in simulation utilizing the ER-AWA method.

	CR1	CR2, B	CR2, F	CR3, B	CR3, F	Total
Case A1	0.05%	1.40%	0.14%	0.09%	0.01%	1.70%
Case A2	0.04%	1.74%	0.15%	0.10%	0.01%	2.33%
Case A3	0.08%	1.39%	0.08%	0.01%	0.00%	1.98%
Case B1	1.92%	0.08%	0.02%	0.28%	0.20%	2.50%
Case B2	1.54%	0.08%	0.01%	1.31%	1.00%	3.96%
Case B3	1.63%	0.03%	0.00%	1.02%	0.79%	3.51%
Case C1	1.98%	0.03%	0.00%	0.67%	0.44%	3.16%
Case C2	3.36%	0.05%	0.06%	0.70%	0.31%	4.49%
Case C3	3.46%	0.05%	0.01%	1.41%	0.86%	5.80%

present the proportion of the first three order harmonics in the reflected wave measured at the wave gauges using the SW-AWA method and the ER-AWA, respectively. When the SW-AWA method is employed (Table 6), the reflected wave is predominantly composed of the main harmonic wave (CR₁). The CR₁ contributes to a reflection coefficient ranging from 41.49% to 96.04%, whereas the combined reflection coefficient contributed by the second and third harmonic waves is less than 6.50%.

In contrast, when the ER-AWA method is utilized (Table 7), the components of the reflected wave exhibit notable differences. In Group A, the reflected wave primarily comprises the bound component of the second harmonic wave (CR₂, _B), contributing to a reflection coefficient ranging from 1.40% to 1.74%. In Groups B and C, the reflected waves are predominantly composed of the main (CR₁) and third-order (CR_{3, B} and CR_{3, F}) harmonic waves, with the main harmonic wave accounting for a larger proportion, resulting in a reflection coefficient ranging from 1.54% to 3.46%.

Higuera [13], based on their RANS-NWT simulation results and utilizing the 3-point method, concluded that the ER-AWA method would result in full reflection of the second-order super-harmonic. However, according to the analysis conducted using the 4-point method, the ER-AWA method effectively eliminates the generation of reflected waves. This stark disparity in conclusions arises from the fact that the 4-point method can differentiate between free and bound components, whereas the 3-point method cannot distinguish the bound component.

From these separation results, it becomes apparent that the superiority of the ER-AWA method over the SW-AWA method is primarily attributable to its ability to limit the contribution of the main harmonic wave to the reflected waves.

4.4. Comparison between SW-AWA and ER-AWA on Wave Field Continuities

Conventionally, assessing wave absorption performance involves computing the reflection coefficient of the simulated wave. However, active wave absorption methods uphold wave field continuity across the open boundary by presuming a predicted outgoing velocity distribution that approximates the analytical solution of the target wave. Consequently, the unabsorbed reflected wave primarily emerges from the discontinuity between the predicted wave field and the actual situation. Given this, it is worth investigating whether a quantity based on the velocity differential between the predicted and analytical solutions can be defined to evaluate active wave absorption method while accounting for wave field continuity.

To address this, we propose an index for indirectly evaluating the wave absorption performance (WAPI) of AWA methods:

$$\mathrm{WAPI}=\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}\frac{1}{d+\eta \left(t\right)}{\displaystyle \underset{0}{\overset{d+\eta \left(t\right)}{\int }}\frac{\left|u_x^{AWA}\left(y,t\right)-u_x^{analytical}\left(y,t\right)\right|}{C}dy}dt}$$

(13)

where T is the period of the wave, d is the water depth, $u_x^{AWA}$ is the horizontal velocity predicted by the AWA method, $u_x^{analytical}$ is the horizontal velocity calculated by the analytical solution, and C is the wave celerity.

The WAPI describes the degree of discontinuity between the predicted and actual velocities, normalized by the wave celerity C. For shallow-water cases, we compare the horizontal velocities of a solitary wave at crest time (using Case b as an example) calculated by the analytical solution, SW-AWA method, and ER-AWA method in Figure 8. It is evident that the velocity predicted by the SW-AWA method remains constant across the water depth and deviates significantly from the analytical solution. Conversely, the velocity predicted by the ER-AWA method closely aligns with the analytical solution, exhibiting a deviation of less than 25%. Figure 9 illustrates the depth-averaged deviation of horizontal velocity between the two AWA methods and the analytical solution within one wave period for Case-b wave conditions.

Table 8. WAPI indexes of the two AWA methods in shallow-water cases. SW in the superscript represents results calculated using the SW-AWA method, while ER represents results calculated using the ER-AWA method.

	WAPISW	WAPIER	WAPIER/WAPISW	RCER/RCSW
Case a	0.125	0.099	79.5%	85.3%
Case b	0.096	0.076	78.9%	71.5%
Case c	0.086	0.071	83.3%	78.3%

presents the WAPI values of the two AWA methods for the three shallow-water cases. The average WAPI ratio of the ER-AWA method to the SW-AWA method stands at 80.59%, while the average RC ratio of these two methods is 78.37%. The close alignment of these two ratios indicates that the WAPI, serving as a pre-calculated index, can be effectively employed to preliminarily estimate the wave absorption performance of an AWA method in shallow water by comparing it with another AWA method of known reflectivity.

For the deep-water cases, as discussed in Section 3.2, the ER-AWA method demonstrates significantly superior absorption performance compared to the SW-AWA method. This disparity can also be elucidated from the perspective of velocity deviation between the two AWA methods and the analytical solution. Take Case C3 as an illustration (refer to Figure 10): the depth-averaged deviation of the horizontal velocity predicted by the ER-AWA method is notably smaller than that by the SW-AWA method when compared with the analytical solution. Consequently, the WAPI of the ER-AWA method (0.096) is substantially smaller than that of the SW-AWA method (1.153). The ratio of WAPI from ER-AWA to SW-AWA stands at 8.33%, which roughly aligns with their corresponding RC ratio (8.24%). This further underscores the close correlation between WAPI and RC, reaffirming the utility of WAPI as an indicator for assessing wave absorption performance.

Figure 11 illustrates the WAPIs of the two AWA methods across the nine deep-water cases. A strong correlation is observed between the WAPI ratio and the RC ratio. On average, the WAPI of the ER-AWA method is approximately 6.6% that of the SW-AWA method, while their average RC ratio is approximately 5.2%.

In summary, the RC ratios of the ER-AWA method to the SW-AWA method closely mirror their respective WAPI ratios (WAPI^ER/WAPI^SW ≈ RC^ER/RC^SW). This indicates that the relative performance of the two AWA methods can be effectively judged based on their WAPIs. Fundamentally, the core wave absorption mechanism of the AWA method revolves around preserving the continuity of the wave field at the open boundary, thereby enabling the outward propagation of waves. In contrast to the SW-AWA method, the ER-AWA method notably reinforces the continuity of the wave field at the open boundary, thereby resulting in a significant improvement in wave absorption performance.

5. Conclusions

In this paper, we implemented the ER-AWA method in the LBM-NWT and extensively discussed its wave absorption performance. Utilizing the improved 4-point wave separation method, we evaluated the reflection coefficient at the wave gauges. Through a comprehensive analysis of wave propagation and reflection in both shallow and deep water, we arrived at the following conclusions.

The ER-AWA method demonstrates excellent applicability in the LBM-NWT, exhibiting a remarkable 95% improvement in wave absorption performance compared to the SW-AWA method in deep-water conditions.

We observed that the performance of the ER-AWA method implemented in the present LBM-NWT surpasses that of the RANS-NWT, a difference that may arise from variations in the VOF methods adopted in the NWTs.

We observed that the superiority of the ER-AWA method over the SW-AWA method primarily stems from its capability to restrict the contribution of the main harmonic wave to the reflected waves.

We established a roughly equivalent relationship between the WAPI ratio and the RC ratio of the two AWA methods. Analysis of the WAPI results suggests that the ER-AWA method notably bolsters the continuity of the wave field at the open boundary, consequently leading to a substantial enhancement in wave absorption performance.