Coupling of non-hydrostatic model with unresolved point-particle model for simulating particle-laden free surface flows

Abstract

Sediment-laden flow is a common phenomenon in nature and the deposition of sediments can make a great difference in landscape formation or marine systems. The complexity of this issue can be further increased with temporal variations in the free surface elevation. This paper aims to present a two-phase flow model that effectively integrates the non-hydrostatic free surface model with the Lagrangian point-particle model. The free surface elevation is conceptualized as a height function and is tracked using a Lagrangian-Eulerian method. This new model is validated by five test cases, showing a good agreement with analytical or experimental results. This demonstrates the model's proficiency in handling sediment-laden flow under various free surface flow conditions, particularly with surface waves. Consequently, the proposed model holds promise for investigating sediment-laden flow issues in coastal regions.

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Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
Coupling of non-hydrostatic model with unresolved point-particle
model for simulating particle-laden free surface flows
Yuhang Chen a,b,c,, Yongping Chen a,b,∗, Zhenshan Xu a,b, Pengzhi Lin d,
Zhihua Xie c, ,∗
aThe National Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing, 210098, Jiangsu, China
bCollege of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing, 210098, Jiangsu, China
cSchool of Engineering, Cardiff University, Queen’s Buildings, Cardiff, CF24 3AA, UK
dState Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, 610065, Sichuan, China
A R T I C L E I N F O A B S T R A C T
Keywords:
Non-hydrostatic model
Point-particle
Particle-laden free surface flows
Sediment-laden flow is a common phenomenon in nature and the deposition of sediments can
make a great difference in landscape formation or marine systems. The complexity of this issue
can be further increased with temporal variations in the free surface elevation. This paper aims
to present a two-phase flow model that effectively integrates the non-hydrostatic free surface
model with the Lagrangian point-particle model. The free surface elevation is conceptualized
as a height function and is tracked using a Lagrangian-Eulerian method. This new model is
validated by five test cases, showing a good agreement with analytical or experimental results.
This demonstrates the model’s proficiency in handling sediment-laden flow under various free
surface flow conditions, particularly with surface waves. Consequently, the proposed model holds
promise for investigating sediment-laden flow issues in coastal regions.
1. Introduction
Sediment-laden free surface flow is a prevalent phenomenon in nature. It is vital for comprehending a range of environmental and
geological processes. This interaction, characterized by the movement of water containing sediment particles, significantly influences
landscape formation [1,2], river [3,4] and marine ecosystems [5,6]. Understanding the intricate dynamics in sediment-laden flow
is essential for both appreciating natural mechanisms and devising practical applications in the fields of environmental engineering
and hydrodynamics.
To effectively capture such phenomena, numerical simulation has emerged as an efficient method. Over the past decade, extensive
research has been conducted on sediment transport patterns and various numerical models have been utilized. They can be divided
mainly into three categories, namely Euler–Euler model [7–12], Euler–Lagrange model [13–16] and Lagrange-Lagrange model [17–
22]. The Euler–Euler model treats sediment phase as a continuum and sediment transport is modeled using an advection–diffusion
equation. It effectively captures the sediment transport process by solving the mass and momentum equations for both fluid and
sediment phases, incorporating closures for interphase momentum transfer, turbulence, and intergranular stresses. However, for
polydisperse particles or those particles with greater inertia (Stokes number (𝑆𝑡) ≥1), such kind of model encounters challenges [23].
*Corresponding authors.
E-mail addresses: ypchen@hhu.edu.cn (Y. Chen), zxie@cardiff.ac.uk (Z. Xie).
https://doi.org/10.1016/j.apm.2025.115962
Received 9 May 2024; Received in revised form 9 January 2025; Accepted 20 January 2025
Applied Mathematical Modelling 142 (2025) 115962
Available online 22 January 2025
0307-904X/© 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license
( http://creativecommons.org/licenses/by/4.0/ ).

Y. Chen, Y. Chen, Z. Xu et al.
In such circumstances, the Lagrangian approach, where sediment particles are individually tracked via the Maxey–Riley equations
[24], offers a more suitable alternative. Apart from the Lagrange-Lagrange model, which treats both fluid and sediment phases
as particles, the Euler–Lagrange model can achieve relatively good results with reasonable computational cost. Hence, the Euler–
Lagrange model is becoming more and more popular in simulating two-phase flow problems.
The Euler–Lagrange model can be categorized into two main approaches: the fully resolved particle model and the unresolved
point-particle approach. As for the fully resolved particle model, the immersed boundary method [25–27] is adopted and no-slip
boundary condition is directly enforced on the surfaces of individual particles [28], necessitating a grid size smaller than the sediment
diameters to resolve both the particles and the detailed flow fields around them. In that case, the fully resolved particle model has
the potential to substantially enhance our understanding of microscale sediment transport processes and foster the development
of improved closures for averaged equations [29]. However, the limitations on the computational resources are always a problem
for simulating sediment transport at different spatial and temporal scales [23]. By contrast, in the unresolved point-particle model,
particles are treated as point sources without resolving the particle–fluid interfaces by the continuous phase grid, allowing for a grid
size significantly larger than the sediment diameter. Hence, the grid size in this model can be several times larger than the sediment
diameter. Considering its much smaller computational cost, this approach is now well established as a research tool in a number of
fields [30–33], which is more suitable for application when a large number of small particles are involved [34].
When simulating sediment-laden flows with free surface elevation changes using the Euler–Lagrange model, the volume-of-fluid
(VOF) method [35] is commonly used and has been successfully applied in many conditions [36–39]. This interface capturing method
can capture the water surface well although it is constrained by substantial computational demands. In contrast, non-hydrostatic
models, which conceptualize the free-surface elevation as a single-valued function dependent on horizontal coordinates, offer an
efficient solution for tracking surface movements with reduced computational demands [40]. The key advantage of non-hydrostatic
models lies in their ability to accurately predict wave dispersion with relatively few vertical grid points. For example, studies by Lin
and Li [41] and Ma et al. [42] have shown that 10–20 vertical layers are sufficient to describe wave dispersion to an acceptable level,
even with simplified pressure boundary conditions at the top layer. Further advancements [43,44] enable even more computational
efficiency by positioning pressure at the cell faces rather than at the cell centers. Hence, these non-hydrostatic models have been
widely used for simulating free-surface flows in both ocean and coastal environments [41–49]. Ma et al. [50] used the non-hydrostatic
model -NHWAVE to simulate both turbidity currents and the tsunami wave generation by a landslide. Berard et al. [51] simulated
the bathymetry change of a steep sand dune under waves using the XBeach model. Zhang et al. [52] proposed a two-layer coupled
model to investigate submarine landslides and resulting tsunami generation over irregular bathymetry. However, in these models,
the sediment phase is considered as a continuum and in fact, they are all Euler-Euler models.
It can be seen from the above literature review that contemporary Euler–Lagrange models predominantly utilize a combination of
the Navier-Stokes equation and an interface capturing method when applied to sediment-laden flow problems with free surfaces. How-
ever, these models generally incur higher computational costs than non-hydrostatic models. Most non-hydrostatic models are based
on the Euler-Euler method when simulating particle-laden flows. The integration of non-hydrostatic models with the point-particle
model remains relatively rare. It is evident that the fusion of non-hydrostatic models with point-particle models holds significant
potential. This approach capitalizes on the relatively lower computational demands of non-hydrostatic models while enhancing the
accuracy of transport patterns for particles with greater inertia, as facilitated by the point-particle model.
In this paper, the main novelty is to develop a two-phase flow model which couples the non-hydrostatic free surface model with a
Lagrangian point-particle model to simulate dilute sediment-laden free surface flow problems. A series of numerical simulation cases,
encompassing regular waves, wave-structure interactions, a single spherical particle settling in stationary and oscillation environment,
and sediment-laden jet in stationary and wave environments are conducted to verify the model’s applicability. The paper is organized
as follows: the governing equation and boundary conditions for the two-phase flow model are introduced in Section 2. The numerical
methods are presented in Section 3. Five test cases are validated in Section 4. Finally, the conclusions are given in Section 5.
2. Governing equations
2.1. Continuous phase
A three-dimensional large eddy simulation model developed by Chen et al. [46] was adopted. The governing equations of fluid
phase in the non-hydrostatic model are the spatially filtered Navier-Stokes equations, which can be written as:
𝜕𝑢1
𝜕𝑥∗+𝜕𝑢2
𝜕𝑦∗+𝜕𝑢3
𝜕𝑧∗=0 (1)
𝜕𝑢1
𝜕𝑡∗+𝑢1
𝜕𝑢1
𝜕𝑥∗+𝑢2
𝜕𝑢1
𝜕𝑦∗+𝑢3
𝜕𝑢1
𝜕𝑧∗=− 1
𝜌𝑓
𝜕𝑝
𝜕𝑥∗+𝑔𝑥+𝜕𝜏𝑥𝑥
𝜕𝑥∗+𝜕𝜏𝑥𝑦
𝜕𝑦∗+𝜕𝜏𝑥𝑧
𝜕𝑧∗−𝜕𝜏𝑆𝐺𝑆
𝑥𝑥
𝜕𝑥∗−
𝜕𝜏𝑆𝐺𝑆
𝑥𝑦
𝜕𝑦∗−𝜕𝜏𝑆𝐺𝑆
𝑥𝑧
𝜕𝑧∗(2)
𝜕𝑢2
𝜕𝑡∗+𝑢1
𝜕𝑢2
𝜕𝑥∗+𝑢2
𝜕𝑢2
𝜕𝑦∗+𝑢3
𝜕𝑢2
𝜕𝑧∗=− 1
𝜌𝑓
𝜕𝑝
𝜕𝑦∗+𝑔𝑦+𝜕𝜏𝑦𝑥
𝜕𝑥∗+𝜕𝜏𝑦𝑦
𝜕𝑦∗+𝜕𝜏𝑦𝑧
𝜕𝑧∗−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑥
𝜕𝑥∗−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑦
𝜕𝑦∗−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑧
𝜕𝑧∗(3)
𝜕𝑢3
𝜕𝑡∗+𝑢1
𝜕𝑢3
𝜕𝑥∗+𝑢2
𝜕𝑢3
𝜕𝑦∗+𝑢3
𝜕𝑢3
𝜕𝑧∗=− 1
𝜌𝑓
𝜕𝑝
𝜕𝑧∗+𝑔𝑧+𝜕𝜏𝑧𝑥
𝜕𝑥∗+𝜕𝜏𝑧𝑦
𝜕𝑦∗+𝜕𝜏𝑧𝑧
𝜕𝑧∗−𝜕𝜏𝑆𝐺𝑆
𝑧𝑥
𝜕𝑥∗−
𝜕𝜏𝑆𝐺𝑆
𝑧𝑦
𝜕𝑦∗−𝜕𝜏𝑆𝐺𝑆
𝑧𝑧
𝜕𝑧∗(4)
Applied Mathematical Modelling 142 (2025) 115962
2

Y. Chen, Y. Chen, Z. Xu et al.
where 𝑢𝑖(𝑖= 1, 2, 3) are the velocity components in horizontal, transverse and vertical directions, respectively;
(𝑥∗,𝑦
∗,𝑧
∗,𝑡
∗)are
the spatial and time coordinates in the physical domain; 𝑡is the time; 𝑝is the pressure; 𝜌𝑓is the water density; 𝑔𝑖(𝑖= 1, 2, 3 or
𝑥, 𝑦, 𝑧) are the acceleration due to gravity; 𝜏𝑖𝑗 and 𝜏𝑆𝐺𝑆
𝑖𝑗 (𝑖, 𝑗 = 1, 2, 3 or 𝑥,𝑦, 𝑧 ) are shear stress and sub-grid scale (SGS) stress.
This model employs the 𝜎coordinate in the vertical direction, as is shown below:
𝑡=𝑡∗,𝑥=𝑥∗,𝑦 =𝑦∗,𝜎 =𝑧∗+ℎ
𝜂+ℎ
(5)
where
(𝑥, 𝑦, 𝜎, 𝑡)are the spatial and time coordinates in the 𝜎coordinate system. 𝜂is the free surface displacement and ℎis the still
water level. After the 𝜎transformation, the spatially filtered Navier-Stokes equations can be transformed to:
𝜕𝑢1
𝜕𝑥
+𝜕𝑢1
𝜕𝜎
𝜕𝜎
𝜕𝑥∗+𝜕𝑢2
𝜕𝑦
+𝜕𝑢2
𝜕𝜎
𝜕𝜎
𝜕𝑦∗+𝜕𝑢3
𝜕𝜎
𝜕𝜎
𝜕𝑧∗=0 (6)
𝜕𝑢1
𝜕𝑡
+𝑢1
𝜕𝑢1
𝜕𝑥
+𝑢2
𝜕𝑢1
𝜕𝑦
+𝜔𝜕𝑢1
𝜕𝜎 =− 1
𝜌𝑓𝜕𝑝
𝜕𝑥 +𝜕𝑝
𝜕𝜎
𝜕𝜎
𝜕𝑥∗+𝑔𝑥+𝜕𝜏𝑥𝑥
𝜕𝑥
+𝜕𝜏𝑥𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗+𝜕𝜏𝑥𝑦
𝜕𝑦
+𝜕𝜏𝑥𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗+𝜕𝜏𝑥𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗−
𝜕𝜏𝑆𝐺𝑆
𝑥𝑥
𝜕𝑥
−𝜕𝜏𝑆𝐺𝑆
𝑥𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗−
𝜕𝜏𝑆𝐺𝑆
𝑥𝑦
𝜕𝑦
−
𝜕𝜏𝑆𝐺𝑆
𝑥𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗−𝜕𝜏𝑆𝐺𝑆
𝑥𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗
(7)
𝜕𝑢2
𝜕𝑡
+𝑢1
𝜕𝑢2
𝜕𝑥
+𝑢2
𝜕𝑢2
𝜕𝑦
+𝜔𝜕𝑢2
𝜕𝜎 =− 1
𝜌𝑓𝜕𝑝
𝜕𝑦 +𝜕𝑝
𝜕𝜎
𝜕𝜎
𝜕𝑦∗+𝑔𝑦+𝜕𝜏𝑦𝑥
𝜕𝑥
+𝜕𝜏𝑦𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗+𝜕𝜏𝑦𝑦
𝜕𝑦
+𝜕𝜏𝑦𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗+𝜕𝜏𝑦𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑥
𝜕𝑥
−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑦
𝜕𝑦
−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗−
𝜕𝜏𝑆𝐺𝑆
𝑦𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗
(8)
𝜕𝑢3
𝜕𝑡
+𝑢1
𝜕𝑢3
𝜕𝑥
+𝑢2
𝜕𝑢3
𝜕𝑦
+𝜔𝜕𝑢3
𝜕𝜎 =− 1
𝜌𝑓
𝜕𝑝
𝜕𝜎
𝜕𝜎
𝜕𝑧∗+𝑔𝑧+𝜕𝜏𝑧𝑥
𝜕𝑥
+𝜕𝜏𝑧𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗+𝜕𝜏𝑧𝑦
𝜕𝑦
+𝜕𝜏𝑧𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗+𝜕𝜏𝑧𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗−
𝜕𝜏𝑆𝐺𝑆
𝑧𝑥
𝜕𝑥
−𝜕𝜏𝑆𝐺𝑆
𝑧𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗−
𝜕𝜏𝑆𝐺𝑆
𝑧𝑦
𝜕𝑦
−
𝜕𝜏𝑆𝐺𝑆
𝑧𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗−𝜕𝜏𝑆𝐺𝑆
𝑧𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗
(9)
where
𝜔=D𝜎
D𝑡∗=𝜕𝜎
𝜕𝑡∗+𝑢1
𝜕𝜎
𝜕𝑥∗+𝑢2
𝜕𝜎
𝜕𝑦∗+𝑢3
𝜕𝜎
𝜕𝑧∗(10)
As for the shear stress 𝜏𝑖𝑗 and sub-grid shear stress 𝜏𝑆𝐺𝑆
𝑖𝑗 (𝑖, 𝑗 = 1, 2, 3), they can be calculated as:
𝜏𝑖𝑗 =𝜈𝜕𝑢𝑖
𝜕𝑥𝑗
+𝜕𝑢𝑖
𝜕𝜎
𝜕𝜎
𝜕𝑥∗
𝑗
+𝜕𝑢𝑗
𝜕𝑥𝑖
+𝜕𝑢𝑗
𝜕𝜎
𝜕𝜎
𝜕𝑥∗
𝑖(11)
𝜏𝑆𝐺𝑆
𝑖𝑗 =−2𝜈𝑡𝑆𝑖𝑗 =−𝜈𝑡𝜕𝑢𝑖
𝜕𝑥𝑗
+𝜕𝑢𝑖
𝜕𝜎
𝜕𝜎
𝜕𝑥∗
𝑗
+𝜕𝑢𝑗
𝜕𝑥𝑖
+𝜕𝑢𝑗
𝜕𝜎
𝜕𝜎
𝜕𝑥∗
𝑖(12)
where 𝜈is the kinematic viscosity; 𝜈𝑡is the eddy viscosity, which can be obtained from the Smagorinsky model [53] as:
𝜈𝑡=𝐶𝑠Δ22𝑆𝑖𝑗 𝑆𝑖𝑗 (13)
Δ=Δ𝑥1Δ𝑥2Δ𝑥31∕3 (14)
where 𝐶𝑠is the Smagorinsky constant and should be calibrated and chosen based on the type of flow. In this study, the value is set
to 0.2; Δis a representative grid spacing and Δ𝑥1, Δ𝑥2, Δ𝑥3are the grid sizes in the coordinates of 𝑥∗, 𝑦∗, 𝑧∗, respectively.
2.2. Dispersed phase
The motion equation for a spherical particle within an unsteady and non-uniform fluid field is expressed as [24]:
𝜌𝑝𝑉𝑝
d𝐮𝐩
d𝑡
=𝐅𝐛+𝐅𝐝+𝐅𝐩+𝐅𝐚+𝐅𝐁𝐚𝐬𝐬𝐞𝐭 (15)
where 𝐅𝐛, 𝐅𝐝, 𝐅𝐩, 𝐅𝐚, 𝐅𝐁𝐚𝐬𝐬𝐞𝐭 are body force, drag force, fluid acceleration due to local pressure gradient, added mass force, Basset
history force, respectively.
The body force 𝐅𝐛acting on a single particle is calculated by
𝐅𝐛=𝜌𝑓−𝜌𝑝𝑉𝑝𝐠(16)
where 𝜌𝑝is the particle density; 𝑉𝑝is the volume of particle and 𝐠is the gravitational acceleration.
Applied Mathematical Modelling 142 (2025) 115962
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Y. Chen, Y. Chen, Z. Xu et al.
Fig. 1. Computational domain and boundary conditions.
The drag force 𝐅𝐝is expressed as follows:
𝐅𝐝=−1
2𝜌𝑓𝐶𝑑𝐴𝑝𝐮𝐩−𝐮𝐟𝐮𝐩−𝐮𝐟(17)
where 𝐴𝑝is the projected area of a particle; 𝐮𝐩is the particle velocity in three different directions; 𝐮𝐟is the fluid velocity in three
different directions; 𝐶𝑑is the drag coefficient, it is related with particle Reynolds number 𝑅𝑒𝑝as follows:
𝐶𝑑=𝑓(𝑅𝑒𝑝)=






24
𝑅𝑒𝑝
, 𝑅𝑒𝑝<0.4
24
𝑅𝑒𝑝1+0.15𝑅𝑒0.687
𝑝+0.42
1+42500𝑅𝑒−1.16
𝑝
, 0.4≤𝑅𝑒𝑝≤1000
0.45 , 𝑅𝑒𝑝>1000
(18)
𝑅𝑒𝑝=𝐮𝐩−𝐮𝐟𝑑
𝜈(19)
where 𝑑is the particle diameter.
The fluid acceleration due to local pressure gradient force 𝐅𝐩is formulated by:
𝐅𝐩=𝜌𝑓𝑉𝑝d𝐮𝐟
d𝑡 𝑓
(20)
The added mass force 𝐅𝐚follows:
𝐅𝐚=𝜌𝑓𝐶𝑀𝑉𝑝d𝐮𝐩
d𝑡
−d𝐮𝐟
d𝑡 𝑓(21)
where 𝐶𝑀is the added mass coefficient and equals 0.5.
The Basset history force 𝐅𝐁𝐚𝐬𝐬𝐞𝐭 here follows:
𝐅𝐁𝐚𝐬𝐬𝐞𝐭 =−3
2𝑑2𝜋𝜌𝑓𝜇
𝑡
∫
0
d𝐮𝐫
d𝜏
𝑡−𝜏
𝑑𝜏 (22)
where 𝜇is the dynamic viscosity of fluid and 𝐮𝐫=𝐮𝐩−𝐮𝐟is the relative velocity between particle and fluid.
2.3. Problem setup and boundary conditions
The setup of the computational domain is shown in Fig. 1. The dimensions of the domain are 𝐿𝑥in length, 𝐿𝑦in width and
𝐿𝑧in depth. For wave cases, a numerical damping zone with a length of 𝐿𝑥3is switched on to reduce wave reflection. For cases
involving a jet, a jet outlet boundary is switched on. Additionally, the jet horizontal position is placed several meters upstream of the
damping zone to maintain relatively stable flow conditions under wave conditions. A detailed description of the setup can be found
in respective benchmark sections.
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Six distinct boundary conditions, each corresponding to specific physical scenarios, are applied in this study. For the inflow bound-
ary, both horizontal velocity 𝑢1(𝑧,𝑡), vertical velocity 𝑢3(𝑧, 𝑡)and free surface displacement 𝜂(𝑡)are given by analytical expressions
[54] for wave cases as:
𝜂(𝑡)= 𝐻
2
cos( 𝜋
2
−2𝜋
𝑇𝑡)+ 𝐻2𝑘
16
cosh 𝑘ℎ
sinh3𝑘ℎ
(2 + cosh 2𝑘ℎ) cos 2( 𝜋
2
−2𝜋
𝑇𝑡),−ℎ≤𝑧≤0
𝑢1(𝑧, 𝑡)= 𝐻𝑔𝑘𝑇
4𝜋
cosh 𝑘(ℎ+𝑧)
cosh 𝑘ℎ
cos( 𝜋
2
−2𝜋
𝑇𝑡)+ 3𝜋𝐻2𝑘
8𝑇
cosh 2𝑘(ℎ+𝑧)
sinh4𝑘ℎ
cos 2( 𝜋
2
−2𝜋
𝑇𝑡),−ℎ≤𝑧≤0
𝑢3(𝑧, 𝑡)= 𝐻𝑔𝑘𝑇
4𝜋
sinh 𝑘(ℎ+𝑧)
cosh 𝑘ℎ
sin( 𝜋
2
−2𝜋
𝑇𝑡)+ 3𝜋𝐻2𝑘
8𝑇
sinh 2𝑘(ℎ+𝑧)
sinh4𝑘ℎ
sin 2( 𝜋
2
−2𝜋
𝑇𝑡),−ℎ≤𝑧≤0
(23)
where 𝐻is the wave height from the mean water level, 𝑘is the wave number, 𝑇is the period of oscillation and ℎis the initial water
depth. The pressure at the inflow boundary is derived based on the assumption of negligible vertical acceleration at the free surface
and is expressed as follows [41]:
𝜕𝑝
𝜕𝑥 +𝜕𝑝
𝜕𝜎
𝜕𝜎
𝜕𝑥∗=−𝜌𝑓𝑔𝑧
𝜕𝜂
𝜕𝑥,𝜕𝑝
𝜕𝑦 +𝜕𝑝
𝜕𝜎
𝜕𝜎
𝜕𝑦∗=−𝜌𝑓𝑔𝑧
𝜕𝜂
𝜕𝑦
(24)
For the bottom boundary, velocity gradients at the first interior node are estimated using the free-slip boundary condition. These
gradients are then utilized in advection calculations. Meanwhile, the log-law wall function is used to calculate wall shear stress for
the diffusion step for the bottom boundary. The pressure gradient in the normal direction follows a hydrostatic assumption:
𝜕𝑝
𝜕𝜎 =𝜌𝑓(𝜂+ℎ)𝑔𝑧(25)
This approach yields satisfactory results with relatively coarse meshes, as evidenced by Lin and Liu [55]. At the front/back wall
boundary, conditions similar to those for the bottom boundary are applied. For the outflow boundary, a zero-gradient condition is
imposed on the velocity in the normal direction, while the vertical pressure gradient follows the hydrostatic assumption.
For wave related cases, a numerical damping zone is applied at the outflow boundary to minimize wave reflection. This zone
replicates experimental setups where breakwaters [56,57] or porous structures [58] are commonly placed near the end of the flume
to dissipate waves. By employing this approach, surface waves are confined within the numerical channel, preventing artificial
interactions with the boundary. The damping method utilized in this study can be expressed as [59]:
𝜙𝑅=𝜙+Δ𝑡⋅𝛼⋅𝜎⋅𝑥−𝑥𝑠
𝑥𝑒−𝑥𝑠2
⋅𝜙−𝜙0(26)
where 𝜙is the variable to be solved, such as 𝑢𝑖and 𝜂. 𝜙𝑅is the resulting variable after the numerical damping; The empirical
parameter 𝛼is assigned a value of -1.0 in this study. The subscripts ‘𝑠’ and ‘𝑒’ represent the start and end points of the damping zone
in the 𝑥-direction, respectively.
Finally, at the moving free surface 𝜂, the pressure is assumed to be equal to the air pressure (i.e., equal to zero). And the kinematic
boundary condition is set as:
𝜕𝜂
𝜕𝑡
+𝑢1
𝜕𝜂
𝜕𝑥 +𝑢2
𝜕𝜂
𝜕𝑦
−𝑢3=0 (27)
The jet velocity boundary is generated by the Synthetic-Eddy-Method (SEM) outlined by Jarrin et al. [60]. The instantaneous
velocities at the jet outlet are divided into the sum of a time-averaged part and a fluctuating part. The time-averaged velocity at each
node matches the experiment performed by Lu and Yuan [61]. The fluctuating part of the velocity field at each grid point is given as:
𝑢𝑖′=1
𝑁
𝑁

𝑖=1
𝜀𝑖𝑓𝜎(𝑥𝑖−𝑥)𝑓𝜎(𝑦𝑖)(28)
where
𝑓𝜎𝑥 −𝑥𝑘(𝑡)=𝑉𝐵
𝜎𝑥𝜎𝑦𝜎𝑧
⋅𝑓𝑥−𝑥𝑘(𝑡)
𝜎𝑥⋅𝑓𝑦−𝑦𝑘(𝑡)
𝜎𝑦⋅𝑓𝑧−𝑧𝑘(𝑡)
𝜎𝑧(29)
The triangular function is written as
𝑓(𝜁)=1.5(1−𝜁)𝜁≤1
0𝜁>1(30)
Detailed description on the method can be found in Jarrin et al. [60]. In this study, the fluctuation velocity and Reynolds stress
distribution align with the DNS results conducted by Wu and Moin [62].
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3. Numerical methods
3.1. Fluid phase
3.1.1. Numerical schemes for Navier-Stokes equations
An operator splitting technique [63,64] is utilized for the numerical solution of the governing equations. Within each time interval,
the momentum equations are segmented into three separate steps, namely advection, diffusion, and pressure propagation [41]. For
brevity, in this part, only governing equations in the horizontal direction will be presented below, equations in other directions can
be solved in a similar way.
The finite difference form for the advection step is shown as:
(𝑢1)𝑛+1∕3
𝑖,𝑗,𝑘 −(𝑢1)𝑛
𝑖,𝑗,𝑘
Δ𝑡
=−𝑢1
𝜕𝑢1
𝜕𝑥
+𝑢2
𝜕𝑢1
𝜕𝑦
+𝜔𝜕𝑢1
𝜕𝜎 𝑛
𝑖,𝑗,𝑘
(31)
Equation (31) can be further split into following three sub-steps:
(𝑢1)𝑛+1∕9
𝑖,𝑗,𝑘 −(𝑢1)𝑛
𝑖,𝑗,𝑘
Δ𝑡
=−𝑢1
𝜕𝑢1
𝜕𝑥
𝑛
𝑖,𝑗,𝑘
(𝑢1)𝑛+2∕9
𝑖,𝑗,𝑘 −(𝑢1)𝑛+1∕9
𝑖,𝑗,𝑘
Δ𝑡
=−𝑢2
𝜕𝑢1
𝜕𝑦 𝑛+1∕9
𝑖,𝑗,𝑘
(𝑢1)𝑛+3∕9
𝑖,𝑗,𝑘 −(𝑢1)𝑛+2∕9
𝑖,𝑗,𝑘
Δ𝑡
=−𝜔𝜕𝑢1
𝜕𝜎 𝑛+2∕9
𝑖,𝑗,𝑘
(32)
To solve these above equations, the quadratic backward characteristic method [65] and the Lax-Wendroff method are employed
at the same time. Final numerical results are calculated by using the average value on the above two methods to ensure stability and
accuracy, which can be written as:
(𝑢1)𝑛+1∕9
𝑖,𝑗,𝑘 =(𝑢1)𝑛+1∕9
𝑖,𝑗,𝑘 𝑄𝐶 +(𝑢1)𝑛+1∕9
𝑖,𝑗,𝑘 𝐿𝑊
2
(33)
where
(𝑢1)𝑛+1∕9
𝑖,𝑗,𝑘 𝑄𝐶 =Δ𝑥𝑖−1 −Δ𝑥𝑎−Δ𝑥𝑎
Δ𝑥𝑖−2 Δ𝑥𝑖−2 +Δ𝑥𝑖−1(𝑢1)𝑛
𝑖−2,𝑗,𝑘 +Δ𝑥𝑖−2 +Δ𝑥𝑖−1 −Δ𝑥𝑎−Δ𝑥𝑎
Δ𝑥𝑖−2−Δ𝑥𝑖−1(𝑢1)𝑛
𝑖−1,𝑗,𝑘
+Δ𝑥𝑖−2 +Δ𝑥𝑖−1 −Δ𝑥𝑎Δ𝑥𝑖−1 −Δ𝑥𝑎
Δ𝑥𝑖−2 +Δ𝑥𝑖−1Δ𝑥𝑖−1
(𝑢1)𝑛
𝑖,𝑗,𝑘
(34)
(𝑢1)𝑛+1∕9
𝑖,𝑗,𝑘 𝐿𝑊 =Δ𝑥𝑎Δ𝑥𝑖+Δ𝑥𝑎
Δ𝑥𝑖−1 Δ𝑥𝑖−1 +Δ𝑥𝑖(𝑢1)𝑛
𝑖−1,𝑗,𝑘 +Δ𝑥𝑖−1 −Δ𝑥𝑎−Δ𝑥𝑖−Δ𝑥𝑎
Δ𝑥𝑖−1 −Δ𝑥𝑖(𝑢1)𝑛
𝑖,𝑗,𝑘
+Δ𝑥𝑖−1 −Δ𝑥𝑎−Δ𝑥𝑎
Δ𝑥𝑖−1 +Δ𝑥𝑖Δ𝑥𝑖
(𝑢1)𝑛
𝑖+1,𝑗,𝑘
(35)
In the above equations, the Δ𝑥𝑎is the advection distance and is defined as Δ𝑥𝑎=(𝑢1)𝑛
𝑖,𝑗,𝑘 Δ𝑡.
In the diffusion step, the equation to be solved is as follows:
(𝑢1)𝑛+2∕3
𝑖,𝑗,𝑘 −(𝑢1)𝑛+1∕3
𝑖,𝑗,𝑘
Δ𝑡
=𝜕𝜏𝑥𝑥
𝜕𝑥
+𝜕𝜏𝑥𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗+𝜕𝜏𝑥𝑦
𝜕𝑦
+𝜕𝜏𝑥𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗+𝜕𝜏𝑥𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗−𝜕𝜏𝑆𝐺𝑆
𝑥𝑥
𝜕𝑥
−
𝜕𝜏𝑆𝐺𝑆
𝑥𝑥
𝜕𝜎
𝜕𝜎
𝜕𝑥∗−
𝜕𝜏𝑆𝐺𝑆
𝑥𝑦
𝜕𝑦
−
𝜕𝜏𝑆𝐺𝑆
𝑥𝑦
𝜕𝜎
𝜕𝜎
𝜕𝑦∗−𝜕𝜏𝑆𝐺𝑆
𝑥𝑧
𝜕𝜎
𝜕𝜎
𝜕𝑧∗𝑛+1∕3
𝑖,𝑗,𝑘
(36)
The stress term can be calculated according to Equation (11) and Equation (12). The central difference is used to discretize all
partial differentiation terms in the above equation as:
𝜕𝜏𝑥𝑥
𝜕𝑥 𝑛+1∕3
𝑖,𝑗,𝑘
=𝜏𝑥𝑥𝑛+1∕3
𝑖+1∕2,𝑗,𝑘 −𝜏𝑥𝑥 𝑛+1∕3
𝑖−1∕2,𝑗,𝑘
Δ𝑥𝑖−1 +Δ𝑥𝑖∕2
(37)
where
Applied Mathematical Modelling 142 (2025) 115962
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Y. Chen, Y. Chen, Z. Xu et al.
𝜏𝑥𝑥𝑛+1∕3
𝑖+1∕2,𝑗,𝑘 =𝜈+𝜈𝑡
(𝑢1)𝑛+1∕3
𝑖+1,𝑗,𝑘 −(𝑢1)𝑛+1∕3
𝑖,𝑗,𝑘
Δ𝑥𝑖
+
(𝑢1)𝑛+1∕3
𝑖+1∕2,𝑗,𝑘+1 −(𝑢1)𝑛+1∕3
𝑖+1∕2,𝑗,𝑘−1
Δ𝜎𝑘−1 +Δ𝜎𝑘𝜕𝜎
𝜕𝑥∗𝑛+1∕3
𝑖+1∕2,𝑗,𝑘 
(38)
𝜏𝑥𝑥𝑛+1∕3
𝑖−1∕2,𝑗,𝑘 =𝜈+𝜈𝑡
(𝑢1)𝑛+1∕3
𝑖,𝑗,𝑘 −(𝑢1)𝑛+1∕3
𝑖−1,𝑗,𝑘
Δ𝑥𝑖−1
+
(𝑢1)𝑛+1∕3
𝑖−1∕2,𝑗,𝑘+1 −(𝑢1)𝑛+1∕3
𝑖−1∕2,𝑗,𝑘−1
Δ𝜎𝑘−1 +Δ𝜎𝑘𝜕𝜎
𝜕𝑥∗𝑛+1∕3
𝑖−1∕2,𝑗,𝑘 
(39)
In the above equations, (𝑢1)𝑛+1∕3
𝑖+1∕2,𝑗,𝑘+1 means the velocity between nodes and can be obtained by the linear interpolation.
In the pressure propagation step, the following equation is to be solved:
(𝑢1)𝑛+1
𝑖,𝑗,𝑘 −(𝑢1)𝑛+2∕3
𝑖,𝑗,𝑘
Δ𝑡
=− 1
𝜌𝑓𝜕𝑝
𝜕𝑥 +𝜕𝑝
𝜕𝜎
𝜕𝜎
𝜕𝑥∗𝑛+1
𝑖,𝑗,𝑘
+𝑔𝑥(40)
The central difference scheme in space is used to discretize the above two equations. To satisfy the continuity requirement, the
resultant equation is incorporated into the continuity equation, leading to the derivation of the modified Poisson equation as follows:
𝜕2𝑝
𝜕𝑥2+𝜕2𝑝
𝜕𝑦2+𝜕𝜎
𝜕𝑥∗2
+𝜕𝜎
𝜕𝑦∗2
+𝜕𝜎
𝜕𝑧∗2𝜕2𝑝
𝜕𝜎2+2𝜕𝜎
𝜕𝑥∗
𝜕2𝑝
𝜕𝑥𝜕𝜎 +𝜕𝜎
𝜕𝑦∗
𝜕2𝑝
𝜕𝑦𝜕𝜎 +𝜕2𝜎
𝜕𝑥∗𝜕𝑥 +𝜕2𝜎
𝜕𝑦∗𝜕𝑦𝜕𝑝
𝜕𝜎 𝑛+1
𝑖,𝑗,𝑘
=𝜌𝑓
Δ𝑡
𝜕𝑢1
𝜕𝑥
+𝜕𝑢1
𝜕𝜎
𝜕𝜎
𝜕𝑥∗+𝜕𝑢2
𝜕𝑦
+𝜕𝑢2
𝜕𝜎
𝜕𝜎
𝜕𝑦∗+𝜕𝑢3
𝜕𝜎
𝜕𝜎
𝜕𝑧∗𝑛+2∕3
𝑖,𝑗,𝑘
(41)
The CGSTAB method is used to solve the pressure Equation (41) discretized using central differences [66].
3.1.2. Lagrangian-Eulerian method for tracking free surface
In order to get the surface elevation, the Lagrange–Euler method [66] is adopted in this study. Assume we have a particle which
is at the free surface, the location of this particle at 𝑡𝑛+1 is marked as
𝑥𝑡𝑛+1 ,𝑦
𝑡𝑛+1 and we can also mark the corresponding particle
position at 𝑡𝑛as
𝑥𝑡𝑛,𝑦
𝑡𝑛. By the Lagrange displacement equation and Taylor expansion, we can get the following equations:
𝑥𝑡𝑛+1 −𝑥𝑡𝑛=
𝑡𝑛+1
∫
𝑡𝑛
𝑢1(𝑥(𝑡),𝑦(𝑡),𝑡
)𝑑𝑡 =𝑢1𝑥𝑡𝜃,𝑦𝑡𝜃,𝑡
𝜃Δ𝑡
=𝑢1𝑛+1
𝑖,𝑗 −𝜃𝜕𝑢1
𝜕𝑥
𝑛+1
𝑖,𝑗 𝑥𝑡𝑛+1 −𝑥𝑡𝑛−𝜃𝜕𝑢1
𝜕𝑦 𝑛+1
𝑖,𝑗 𝑦𝑡𝑛+1 −𝑦𝑡𝑛−𝜃𝜕𝑢1
𝜕𝑡 𝑛+1
𝑖,𝑗
Δ𝑡Δ𝑡
(42)
𝑦𝑡𝑛+1 −𝑦𝑡𝑛=
𝑡𝑛+1
∫
𝑡𝑛
𝑢2(𝑥(𝑡),𝑦(𝑡),𝑡
)𝑑𝑡 =𝑢2𝑥𝑡𝜃,𝑦𝑡𝜃,𝑡
𝜃Δ𝑡
=𝑢2𝑛+1
𝑖,𝑗 −𝜃𝜕𝑢2
𝜕𝑥
𝑛+1
𝑖,𝑗 𝑥𝑡𝑛+1 −𝑥𝑡𝑛−𝜃𝜕𝑢2
𝜕𝑦 𝑛+1
𝑖,𝑗 𝑦𝑡𝑛+1 −𝑦𝑡𝑛−𝜃𝜕𝑢2
𝜕𝑡 𝑛+1
𝑖,𝑗
Δ𝑡Δ𝑡
(43)
The central difference method for spatial derivatives and the forward difference method for temporal derivatives are used to
determine the variables (𝑥𝑡𝑛, 𝑦𝑡𝑛) by solving Equation (42) and Equation (43). Given the particle’s location within element (𝑖, 𝑗) at
time 𝑡𝑛, the initial surface elevation 𝜂𝑛
𝑥,𝑦 is interpolated from the grid node values within the element. Subsequently, the Lagrange
displacement equation is applied to update the surface elevation at the subsequent time step 𝑛+1.
𝜂𝑛+1
𝑖,𝑗 =𝜂𝑛
𝑖,𝑗 +𝑢3𝑛+1
𝑖,𝑗 −𝜃𝜕𝑢3
𝜕𝑥
𝑛+1
𝑖,𝑗 𝑥𝑡𝑛+1 −𝑥𝑡𝑛−𝜃𝜕𝑢3
𝜕𝑦 𝑛+1
𝑖,𝑗 𝑦𝑡𝑛+1 −𝑦𝑡𝑛−𝜃𝜕𝑢3
𝜕𝑡 𝑛+1
𝑖,𝑗
Δ𝑡Δ𝑡(44)
3.1.3. Stability criteria
To ensure the stability and computational efficiency of this model, two criteria must be met. The first criterion is related to the
Courant-Friedrichs-Lewy (CFL) condition, which is expressed as:
Δ𝑡≤𝛽⋅max( Δ𝑥𝑖
(𝑢𝑖)max
)(45)
Here, 𝑖=1,2,3refers to the spatial dimensions (horizontal, transverse, and vertical), with Δ𝑥𝑖representing the grid sizes and (𝑢𝑖)max
denoting the maximum particle velocity in each respective direction. Although the theoretical upper limit for 𝛽is 1.0, in practice, it
is generally assigned a more conservative value below 0.2 to maintain both stability and computational accuracy. In this model, the
value is set to 0.1.
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Y. Chen, Y. Chen, Z. Xu et al.
Another criterion applies to the diffusion term, which requires the following condition to be met:
Δ𝑡≤𝛾⋅
(Δ𝑥𝑖)2
𝜈(46)
where 𝛾is typically assigned a value of 0.2. However, in most practical simulations, the time step is primarily constrained by the more
restrictive CFL condition in Equation (45), as it imposes a tighter limit on the time step size due to the higher velocities encountered
in the advection process.
3.2. Dispersed phase
When it comes to the dispersed phase, one main difficulty lies in how to get numerical solution of the Basset force, as the integrand
is ill-behaved and would become an infinity when 𝜏⟶𝑡. To address this issue, the integral is divided into a series of small integrals,
each calculated over a brief time step Δ𝑡. Assuming that the variation in relative velocity can be determined using the central
difference method and that acceleration remains constant during Δ𝑡, the Basset integral can be computed as the cumulative total of
these smaller integrals [67]. Hence,
d𝐮𝐫
d𝑡
is evaluated at the middle of the time step.
𝐁=
𝑡+Δ𝑡
∫
0
d𝐮𝐫
d𝜏
𝑡+Δ𝑡−𝜏
d𝜏
=
Δ𝑡
∫
0
d𝐮𝟎
𝐫
d𝜏
𝑡+Δ𝑡−𝜏
d𝜏+
2Δ𝑡
∫
Δ𝑡
d𝐮𝚫𝐭
𝐫
d𝜏
𝑡+Δ𝑡−𝜏
d𝜏+... +
𝑀Δ𝑡+Δ𝑡
∫
𝑀Δ𝑡
d𝐮𝐌𝚫𝐭
𝐫
d𝜏
𝑡+Δ𝑡−𝜏
d𝜏
=d𝐮𝟎
𝐫
d𝜏
Δ𝑡
∫
0
1
𝑡+Δ𝑡−𝜏
d𝜏+d𝐮𝚫𝐭
𝐫
d𝜏
2Δ𝑡
∫
Δ𝑡
1
𝑡+Δ𝑡−𝜏
d𝜏+... +d(𝐮𝐩−𝐮𝐟)
d𝜏
𝑀Δ𝑡+Δ𝑡
∫
𝑀Δ𝑡
1
𝑡+Δ𝑡−𝜏
d𝜏
=𝐮𝐫,𝟏−𝐮𝐫,𝟎
Δ𝑡
⋅2(𝑀Δ𝑡+Δ𝑡−𝑀Δ𝑡−Δ𝑡+Δ𝑡)+... +𝐮𝐫,𝐌−𝐮𝐫,𝐌−𝟏
Δ𝑡
⋅
2(𝑀Δ𝑡−(𝑀−1)Δ𝑡+Δ𝑡−𝑀Δ𝑡−𝑀Δ𝑡+Δ𝑡)+2
Δ𝑡
𝑑(𝐮𝐩−𝐮𝐟)
d𝑡
=2
Δ𝑡
𝑀−1

𝑛=0 𝐮𝐫,𝐧+𝟏−𝐮𝐫,𝐧𝑀−𝑛+1−𝑀−𝑛−1+1
+2
Δ𝑡d(𝐮𝐩−𝐮𝐟)
d𝑡
=𝐁𝐭
𝟎+2
Δ𝑡d(𝐮𝐩−𝐮𝐟)
d𝑡
(47)
where 𝐁𝐭
𝟎is the Basset integral between 𝜏=0 to 𝑡.
At 𝜏=𝑡, the simplified version of Equation (15) can be written as
d𝐮𝐩
d𝑡
=(1 − 𝑠)𝑔+(1+𝐶𝑀)( d𝐮𝐟
d𝑡
)𝑓−3𝐶𝐷
4𝑑𝐮𝐩−𝐮𝐟(𝐮𝐩−𝐮𝐟)− 9
𝑑𝜈
𝜋𝐁𝐭
𝟎
(𝑠+𝐶𝑀)
=𝐅(48)
While when 𝜏=𝑡+Δ𝑡, after applying the Equation (47), the governing equation can be written as
d𝐮𝐩
d𝑡
=(1 − 𝑠)𝑔+(1+𝐶𝑀+18
𝑑𝜈
𝜋Δ𝑡)( d𝐮𝐟
d𝑡
)𝑓−3𝐶𝐷
4𝑑𝐮𝐩−𝐮𝐟(𝐮𝐩−𝐮𝐟)− 9
𝑑𝜈
𝜋𝐁𝐭
𝟎
(𝑠+𝐶𝑀+18
𝑑𝜈
𝜋Δ𝑡)
(49)
At the 𝑁th time step, the particle position 𝐱𝐩,𝐍, particle velocity 𝐮𝐩,𝐍, and flow velocity 𝐮𝐟,𝐍are known. Initially, a preliminary
estimate for the particle velocity 𝐮𝐩,𝐍+𝟏=𝐮𝐩,𝐍+Δ𝑡×𝐅(𝐮𝐩,𝐍,𝐱𝐩,𝐍)is made at 𝜏=𝑡+Δ𝑡, enabling the calculation of the corresponding
particle position 𝐱𝐩,𝐍. Subsequently, a second-order implicit iteration method is employed to determine the particle’s velocity and
position at the (𝑁+1)th time step. This iterative process continues until the relative deviation between two successive iterations
is less than 0.1%. Ultimately, the final particle velocity and position are obtained. It is noteworthy that this numerical method is
applicable beyond situations where the Basset force is considered. In scenarios where the Basset force is excluded, the term 𝐁𝐭
𝟎can be
disregarded, and the equations in Equation (48) can be further simplified. The detailed numerical scheme for solving the governing
equation of the dispersed phase is outlined in Algorithm 1.
3.3. Solution procedure
In order to make the solution procedure of the model a more direct and easier to understand, a flow chart is presented in Fig. 2.
1. Calculate 𝑢𝑛+1∕3
𝑖(𝑖=1,2,3) at advection step from Equations (31)-(35).
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Algorithm 1 Numerical method for dispersed phase.
1: 𝐮𝐩,𝐍+𝟏(0)←𝐮𝐩,𝐍+Δ𝑡∗𝐅(𝐮𝐩,𝐍,𝐱𝐩,𝐍), 𝐱𝐩,𝐍+𝟏(0)←𝐱𝐩,𝐍+Δ𝑡∗𝐮𝐩,𝐍+𝟏(0)
2: while 𝐮𝐩,𝐍+𝟏(𝑘+1)−𝐮𝐩,𝐍+𝟏(𝑘)∕𝐮𝐩,𝐍+𝟏(𝑘)≥10−3 do
3: 𝐮𝐩,𝐍+𝟏(𝑘+1)←𝐮𝐩,𝐍+Δ𝑡
2
∗𝐅𝐮𝐩,𝐍,𝐱𝐩,𝐍+𝐅𝐮𝐩,𝐍+𝟏(𝑘),𝐱𝐩,𝐍+𝟏(𝑘); 𝐱𝐩,𝐍+𝟏(𝑘+1)←𝐱𝐩,𝐍+𝐮𝐩,𝐍+𝟏(𝑘+1)∗Δ𝑡, 𝑘=0,1,2, ...
4: end while
5: 𝐮𝐩,𝐍+𝟏←𝐮𝐩,𝐍+𝟏(𝑘+1), 𝐱𝐩,𝐍+𝟏←𝐱𝐩,𝐍+𝟏(𝑘+1)
Fig. 2. Solution procedure of the model.
2. Solve Equations (36) to obtain 𝑢𝑛+2∕3
𝑖(𝑖=1,2,3) in the diffusion step.
3. Use the Conjugate Gradient Squared (CGSTAB) method to solve Equation (41) for the pressure field 𝑝𝑛+1 and velocity 𝑢𝑛+1
𝑖.
4. Calculate the free surface displacement 𝜂𝑛+1 using Equation (44).
5. Get the particle positions and velocities from Equation (49), based on the flow velocities derived in Steps 1-3.
4. Results and discussion
4.1. Benchmark: non-hydrostatic wave flume
4.1.1. Regular waves
A regular progressive wave is generated in the non-hydrostatic wave flume, as depicted in Fig. 1. The dimensions of the flume are
𝐿𝑥=15m in length, 𝐿𝑦=0.5m in width, and 𝐿𝑧=0.5m in depth. The damping zone has a length of 𝐿𝑥3=5m and is positioned
at the end of the flume to dissipate wave energy. The amplitude of the regular wave is 0.02 m and the wave period is 1 s.
Eight different grid systems, as detailed in Table 1, are performed to check the spatial convergence of the model. Grid1 to Grid4
focus on spatial convergence in the horizontal direction, while Grid2 and Grid5 to Grid8 are employed for the convergence in
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Table 1
Grid systems of spatial convergence study on regular
waves.
Grid Grid points min Δ𝑥min Δ𝜎
Grid-1 151x11x101 0.1 m 0.01
Grid-2 301x11x101 0.05 m 0.01
Grid-3 601x11x101 0.025 m 0.01
Grid-4 1201x11x101 0.0125 m 0.01
Grid-5 301x11x51 0.05 m 0.02
Grid-6 301x11x41 0.05 m 0.025
Grid-7 301x11x21 0.05 m 0.05
Grid-8 301x11x11 0.05 m 0.1
Fig. 3. Spatial convergence study of regular waves along the whole flume at 𝑡=25 s: (a) horizontal direction; (b) vertical direction.
the vertical direction. For the transverse direction, the inflow condition of the water level along this direction is the same and as
discussed in Section 2.3, a zero-gradient condition is applied to the wall’s normal direction. The discretization in the 𝑦-direction
does not influence changes in the water level 𝜂. Hence, fixed grids are set in the transverse direction for all the cases. The grids are
uniformly distributed in both horizontal and vertical directions. The normalized 𝐿1error 𝐸𝐿1, 𝐿2error 𝐸𝐿2, and maximum error
𝐸𝐿∞were used in this test, which were calculated as follows:
𝐸𝐿1=𝜂𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 −𝜂𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 
𝑛
(50)
𝐸𝐿2=(𝜂𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 −𝜂𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐 𝑎𝑙)2
𝑛
(51)
𝐸𝐿∞=𝑚𝑎𝑥 𝜂𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 −𝜂𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 (52)
In Grid systems 1–4, the water depth is discretized using 100 uniform grids, while the horizontal direction is discretized with
varying uniform grids. As shown in Fig. 3(a), the convergence order in the horizontal direction approaches first-order accuracy.
Similarly, Fig. 3(b) demonstrates that the convergence order in the vertical direction is also close to the first order. Finally, Grid-3
was selected for the simulation in this case.
Fig. 4shows the spatial profiles of water level throughout the entire flume at 𝑡= 25 s, as well as the temporal profile of water
level at 𝑥= 6 m. Evidently, the numerical results match well with analytical solutions. Additionally, Fig. 4(a) demonstrates that
wave reflection is negligible within the effective zone (0 m-10 m), which proves that wave energy was dissipated effectively in the
damping zone (10 m-15 m). Fig. 5shows the relative volume of water over time for regular waves. It can be seen that the volume of
water preserves well and stabilizes after 30 seconds, which confirms the validity of the model.
Fig. 6shows the maximum horizontal velocity distribution and maximum vertical velocity distribution along water depth in
one wave period and their comparison with analytical solutions at 𝑥= 6 m. The comparison of horizontal and vertical velocity
distributions with analytical results reveals negligible differences, indicating the flow field is well simulated. It is worth mentioning
that a slight difference in the horizontal velocity can be seen in Fig. 6close to the bottom, and this can be attributed to the wave
boundary layer which exists at a very narrow range close to the bottom. Due to the existence of the boundary layer, the real velocity
distribution would deviate from the analytical solution close to the bottom. In addition, the velocity should equal zero at the bottom,
which is consistent with our numerical results. In summary, the numerical model successfully replicates both the water surface
changes and the velocity distribution of regular waves.
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Fig. 4. Free surface verification for the regular waves: (a) Spatial distribution of water level along the whole flume at 𝑡= 25 s; (b) Temporal distribution of water
level at 𝑥= 6 m.
Fig. 5. Relative volume of water over time for the regular waves.
Fig. 6. Verification of the velocity distribution of a regular wave: (a) the horizontal velocity distribution along water depth in wave peak phase; (b) the vertical velocity
distribution along water depth in up-zero crossing phase.
4.1.2. Wave-structure interaction
When a regular wave encounters a submerged bar, it frequently experiences significant changes in the waveform, resulting in
noteworthy nonlinear energy interactions among various wave modes [41]. The simulation of such case is imperative, as it serves as
a foundational test case for numerical tanks and can yield critical insights into the dynamics of wave interactions with submerged
structures.
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Fig. 7. Sketch of the geometry for a regular wave passing the submerged bar. PS: G stands for wave gauges.
Table 2
Mesh convergence study on wave-structure interaction.
Grid Grid points min Δ𝑥min Δ𝑦min Δ𝜎
Grid-1 1201x9x47 0.025 m 0.05 m 0.011
Grid-2 2401x5x47 0.0125 m 0.10 m 0.011
Grid-3 2401x9x24 0.0125 m 0.05 m 0.022
Grid-4 2401x9x47 0.0125 m 0.05 m 0.011
Grid-5 2401x9x93 0.0125 m 0.05 m 0.0055
Grid-6 2401x17x47 0.0125 m 0.025 m 0.011
Grid-7 4801x9x47 0.00625 m 0.05 m 0.011
Our numerical simulations are based on experiments conducted by Beji and Battjes [68]. The geometry of our numerical com-
putations is depicted in Fig. 7. The length of the flume is 30 m, the width is 0.5 m and the water depth is 0.4 m. A regular wave
with a wave height of 0.02 m and a period of 2 s is generated from the inflow boundary and the damping zone 𝐿𝑥3is set as 10 m
to dissipate the wave energy. Seven wave gauges are positioned at various locations within the flume. To ensure grid convergence,
seven different grid configurations are employed in the model (as shown in Table 2), with a time step of Δ𝑡=0.005 s. The simulations
were carried out on a desktop computer with an AMD Ryzen(TM) 5 5600X CPU and 16 GB internal memory. The base frequency of
this CPU is 3.7 GHz. The total CPU time per time step required for the present model was about 1.8 s.
The comparisons between our numerical findings and the gauge data for free surface elevation at six designated locations are
depicted in Fig. 8. The results from seven different grid systems are also presented in the figure. For most grid systems, the numerical
simulation results match well with the experimental data, except for Grid-3, where a distinct phase lag is observed. This discrepancy
can be attributed to the relatively low grid resolution at the free surface. Ultimately, Grid-4 is selected for the simulation in this case.
It was found that at wave gauge 2 (𝑥= 10.5 m), the wave retains its sinusoidal feature, displaying strong concurrence between the
numerical results and experimental data. Moving from 𝑥= 10.5 m to 𝑥= 12.5 m, we observe the wave deformation as it climbs
the slope. From wave gauge 4 (𝑥= 13.5 m) to 7 (𝑥= 17.5 m), where the wave surmounts the breakwater, exhibit the emergence of
secondary wave growth.
Fig. 9shows the top and side view plots of the 3D free surface elevation at a representative time. The red, yellow, black, and
blue rectangle represents the upslope part, horizontal part, downslope part, and damping zone, respectively. It can be seen that the
wavelength gradually decreases while the wave height increases during the wave shoaling process. On the contrary, a decrease in wave
height is observed when the wave moves downhill, and more small waves can be seen. It can be attributed to the dissipation in wave
energy which in return lead to the generation of higher-order secondary waves. In summary, the numerical model accurately captures
this progression, although minor disparities in the variation of wave height exist between the numerical results and experimental
data. These distinctions may arise from numerical dissipation and the 𝜎transformation, stemming from the abrupt change in water
depth.
4.2. Benchmark: point particle
To validate the point-particle model, the accelerated process of a spherical particle settling in a stationary fluid is simulated.
When the particle Reynolds number 𝑅𝑒𝑝is relatively small (i.e., 𝑅𝑒𝑝<0.4), the analytical solution of the acceleration process can be
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Fig. 8. Comparison of the wave elevation between numerical results and experiment results at six different gauge points.
derived from linear (Stokes) drag law. By neglecting the Basset history term and assuming an initial velocity of zero, we can derive
the temporal evolution of particle velocity as follows:
𝑤𝑝(𝑡)=(𝑠−1
)𝑔𝑑2
18𝜈1−𝑒−18𝜈𝑡
𝑑2𝑠+𝐶𝑀(53)
When the Basset force is considered, the analytical solution can be found in Brush et al. [69] in a closed-form solution:
𝑤𝑝(𝑡)=(𝑠−1
)𝑔𝑑2
18𝜈1+ 𝑐2+ℎ2
ℎ
exp −ℎ2𝑡exp 𝑐2𝑡sin (2𝑐ℎ𝑡 −𝛼)erf c 𝑐𝑡
−2
𝑡
𝜋
ℎ
∫
0
exp 𝑦2𝑡cos [2𝑐(ℎ−𝑦)𝑡−𝛼]𝑑𝑦(54)
where
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Fig. 9. The surface elevation of regular wave interacts with a submerged bar. (a) 3D view (b) Top view (c) Front view of free surface elevation changes at representative
time (Red rectangle: 1:20 upslope; Yellow rectangle: horizontal crest; Black rectangle: 1:10 downslope; Blue rectangle: damping zone).
𝑐=9𝜈
2𝑑𝑠+𝐶𝑀(55)
ℎ=3
2𝑑𝑠+𝐶𝑀𝜈8𝑠+𝐶𝑀−9
(56)
𝛼=tan
−1 ℎ
𝑐(57)
and erf c (𝑡)is the complementary error function, which writes as:
erf c (𝑡)=2
𝜋
∞
∫
𝜋
exp −𝑡2𝑑𝑡 (58)
Fig. 10(a) shows the comparison of analytical and numerical solutions of the motion of a 50 μm diameter sphere falling in water
with the parameters 𝜌𝑝= 2500 kg/𝑚3, 𝑠=𝜌𝑝∕𝜌𝑓= 2.5, kinematic viscosity 𝜈=10
−6 𝑚2∕𝑠and particle Reynolds number 𝑅𝑒𝑝=0.1.
As it takes very short time for a single particle to reach its terminal settling velocity, the time step is set as Δ𝑡=0.0001 s. Significant
differences can be observed when Basset force was considered or not. Without the Basset force, the particle reaches its settling velocity
fast; while the particle accelerates at a much slower rate when Basset force is considered. It can also be clearly seen that the numerical
results match well with the analytical solution.
As mentioned before, the analytical solution of the settling process of a round particle is only applicable to the linear (Stokes) drag
law, experiments performed by Mordant et al. [70] are further adopted to validate the applicability of the point particle model in
non-linear drag range. The sphere diameter used in the experiment is 500 μm, 𝑠=𝜌𝑝∕𝜌𝑓= 2.565, 𝜈=9.0366 × 10−7 𝑚2∕𝑠, particle
Reynolds number 𝑅𝑒𝑝=41and the time step is set as Δ𝑡=0.00005 s. It can be found from Fig. 10(b) that the numerical results match
better with experimental data when the Basset force is considered. Both the early stage of particle settling and the terminal settling
velocity correspond well to the experiments.
4.3. Benchmark: single particle in an oscillating liquid
Given the good agreement between our simulated settling process of a single particle in a stationary liquid, our attention now
shifts to simulating settling process of a single particle in vertically oscillating liquid. The data are adapted from experiments done by
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Fig. 10. Accelerated process of a spherical particle settle in a stationary liquid: (a) 𝑅𝑒𝑝=0.1; (b) 𝑅𝑒𝑝=41.
Table 3
Summary of different cases on sphere settling in oscillation field.
Case 𝑅𝑒𝑝Diameter
[μm]
𝜌𝑝∕𝜌𝑓Viscosity
[10−6 m2/s]
Oscillation
period [s]
𝜈∕𝑑2𝜔𝑠
1 0.18 1587 6.27 250.80 0.15 2.3
2 0.18 1587 6.27 250.80 0.19 3.0
3 1.1 3175 6.16 250.80 0.15 0.59
4 1.1 3175 6.16 250.80 0.19 0.74
5 6 1587 6.53 35.77 0.15 0.330
6 6 1587 6.53 35.77 0.19 0.430
7 28 3175 6.41 35.77 0.15 0.085
8 28 3175 6.41 35.77 0.19 0.110
Ho [71]. The experiments were conducted in a cylinder which was filled with fluids, and it would oscillate vertically with different
amplitudes and frequencies as follows:
𝑤𝑓(𝑡)=𝑤𝑓0sin (𝜔𝑡)(59)
where 𝑤𝑓0is the velocity oscillation amplitude; 𝜔=2𝜋∕𝑇is the angular frequency and 𝑇is the oscillation period. The particle motion
process is computed for several oscillation cycles until the velocity deviation between two consecutive periods is less than 0.1%. The
average settling velocity 𝜔𝑠𝑎 is computed by averaging the velocities in an oscillation cycle. The time step is set as Δ𝑡=0.00005 s.
Four different 𝑅𝑒𝑝and two different oscillation periods are chosen. In total 8 cases are simulated. Detailed parameters used in the 8
cases are summarized in Table 3.
Fig. 11 shows the comparison between the simulated non-dimensional averaged settling velocity and oscillation acceleration ratio
with experiments. The results depicted by the solid lines in Fig. 11 are obtained through a sequence of simulations, all maintaining
the constant frequency while the wave amplitudes are systematically varied. Following these simulations, the oscillation acceleration
ratios for each amplitude are computed and the relation between the wave amplitudes and the oscillation acceleration ratios are drawn
into solid lines in this figure. It can be seen from these figures that with the increase of the oscillation amplitude, the non-dimensional
averaged settling velocity would decrease. This reduction in settling velocity can be attributed to the enhanced particle inertia during
the oscillating motion of the fluid, which, when increased, tends to overpower the effects of viscosity around the particle and can
lead to a more distinct decrease in averaged settling velocity. Another interesting finding is that when the Basset force is considered
in the settling process, the numerical simulation results deviate more than when the Basset force is not considered. One of the reasons
lies in that the Basset force is applicable in relatively small particle Reynolds number and is inapplicable to high particle Reynolds
number conditions [24]; Also the Basset force can be considered as a kind of viscous force and would resist the movement of the
particle. As a result, the average settling velocity in one oscillation period would decrease when Basset force is considered. Finally,
we can conclude that the numerical simulation results match better with experiments without Basset force. As the wave field can be
seen as a specific oscillation flow, we would not consider the Basset force in our later simulation on sediment-laden jet in flow with
waves.
4.4. Benchmark: sediment-laden jet in a stationary environment
To check the capability of the coupled model between the non-hydrostatic model and the point-particle model, we choose to
simulate the sediment-laden jet in a stationary environment. The setup of the experiment is the same as the experiments performed
by Chen et al. [72]. The tank has a length of 1.5 m, a width of 0.5 m, and a height of 0.6 m. The tank keeps a depth of 0.5 m.
The diameter of the jet orifice is 0.01 m and is located 0.15 m above the collection tray. The computational domain is discretized
into 131×41×139 grids. A non-uniform grid is implemented, and refined at the jet orifice, with minimum grid sizes in the 𝑥, 𝑦, and
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Fig. 11. Comparison of the numerical model and the experimental results of Ho [71] of accelerated process of a spherical particle settle in oscillation field.
𝜎directions of 0.001 m. The time step is set to 0.002 s and the total CPU time per time step required for the present model was
approximately 4.3 s. The jet outlet velocity is set as 0.76 m/s and the velocity boundary is specified using the SEM (Synthetic-Eddy-
Method) method. Given that the volume concentration of the simulated sediment-laden jet is below 0.1%, only one-way coupling is
considered [73], where both the reaction of particles to the flow field and interactions among particles are neglected.
Fig. 12 shows the comparison of centerline velocity decay and axial velocity distribution between experimental and numerical
results. To check the mesh convergence at the jet outlet boundary, the jet outlet was discretized by 8×8, 10×10 and 12×12 grids,
respectively. It is found that the centerline velocity decay is similar when the jet outlet boundary is discretized by finer grids like Grid
10-10 and Grid 12-12. Therefore, the 10×10 grid was selected for the jet outlet boundary. Additionally, it is found that the simulation
results match well with the experiments by Kwon and Seo [74]. When it comes to velocity distribution in cross-sections, it is found
that the distribution follows the Gaussian distribution at cross-sections in self-similar zone. In conclusion, the accuracy of the flow
field was verified.
We present 3D visualizations of the instantaneous and time-averaged isosurfaces of a jet in a stationary environment (see Fig. 13).
The isosurfaces represent velocities ranging from 0.2 m/s to 0.8 m/s, with intervals of 0.2 m/s. Fig. 13(a) shows the velocity distri-
bution of the jet at a specific time, where the isosurface appears irregular, reflecting the fluctuating structures caused by the inherent
unsteadiness and turbulence of the flow. In contrast, the time-averaged velocity isosurface (see Fig. 13(b)) appears smoother than
the instantaneous one. The shapes and behavior of both instantaneous and time-averaged isosurfaces are consistent with previous
studies on turbulent jets in stationary environments, thereby validating the accuracy of the numerical simulation model.
The particles used in the numerical simulation have a median size 𝐷50 = 200 μm. As shown in Fig. 14(a), the actual diameter
distribution follows the distributions used in experiments, which was measured using the Laser Diffraction Particle Size Analyzer
(Malvern Mastersizer 3000). A total of 8 different kinds of particle diameters are used in the simulation. Particles were put into
different grid cells on the jet outlet boundary. A Gaussian white noise is applied to determine the precise time step for particle release
from the jet boundary as particles do not exit each grid cell at every time step. This initial artificial asymmetry in particle distribution
can be quickly dissipated by turbulence as the jet flow develops downstream, having minimal impact on the final results [75].
The 1-D deposition pattern is shown in Fig. 14(b). To ensure the repeatability and avoid the effect of noise in particle numbers
on the final deposition pattern, two different particle numbers namely 81000 and 162000 particles are adopted in the numerical
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Fig. 12. Comparison of (a) centerline velocity decay with different grids; (b) axial velocity distribution (𝑏𝑣means jet Gaussian half-width) between experiments and
numerical simulation.
Fig. 13. Comparison of (a) instantaneous and (b) time-averaged velocity isosurfaces of the jet in a stationary environment, with isosurfaces shown at velocity magnitudes
of 0.2 m/s to 0.8 m/s, in 0.2 m/s increments.
simulation. It can be seen from the
figure that the simulated 1-D deposition pattern matches well with the experiments. The deposition
pattern shares great similarity when using 81000 or 162000 particles. Therefore, the number of sediment particles is set to 81000
hereafter to reduce computational cost.
The two-dimensional deposition pattern is shown in Fig. 15. The black contour lines are the 2-D deposition concentration which
equals the deposition rate divided by the bottom grid size. The profiles are consistent with the previously measured 1-D deposition
profiles, with a peak deposition at about 0.2 m from the jet orifice for this case. It can also be seen that the sediment deposition would
expand, and deposition patterns are almost symmetric.
Fig. 16 shows the three-dimensional visualization of a horizontal sediment-laden jet in a stationary environment. The positions of
the sediment particles are marked as spheres, while their instantaneous velocities are distinguished by various colors. The visualization
clearly shows that sediments initially follow the flow movement in the near field. As the flow velocity decreases downstream, the
sediments would gradually deviate from the jet centerline (i.e., 𝑧=0.18 m) and move downward due to gravity. Consequently, fewer
sediments are observed in the upper part of the jet cross-sections (i.e., 𝑧>0.18 m) and hardly can sediment be seen in the upper part
of jet far field (i.e., 𝑥>0.6m).
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Fig. 14. Verification of sediment-laden jet in an initially stationary environment: (a) particle diameter distribution; (b) 1-D deposition pattern.
Fig. 15. 2-D deposition pattern of sediment-laden jet in an initially stationary environment (Top View).
Fig. 16. 3-D visualization of sediment-laden jet in an initially stationary environment.
4.5. Benchmark: sediment-laden jet in flow with waves
Finally, we further use this model to simulate a sediment-laden jet in flow with waves. The experiments were carried out within
a wave flume located at the College of Harbor, Coastal and Offshore Engineering, Hohai University. Fig. 17 shows a sketch of the
experimental setup. The flume dimensions were 46 m in length, 0.5 m in width, and 1.0 m in depth. All experiments consisted of four
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Fig. 17. Experimental set up.
Fig. 18. Comparison of the numerical model and the experimental results of sediment-laden jet in flow with waves, (a) flow field; (b) 1-D deposition pattern.
systems, namely the wave generating system, the sediment-laden jet generating system, the measuring system, and the collection
system. The wave generating system consisted of a wave paddle and an absorber. This system was used to make regular waves.
The sediment-laden jet generating system mainly consisted of a constant head tank which was full of feeding particles and settling
equipment to feed sediment into the flow. The measuring system consisted of a continuous laser machine and two high-speed cameras,
which made it into a PIV system. Finally, the collection tray was used to collect all the settled sediment. A horizontal jet was introduced
through an acrylic nozzle with a diameter of 0.01 m. The nozzle was positioned at the flume’s midpoint, maintaining a 0.18 m clearance
above the bottom. To maintain a consistent exit velocity, water was continuously pumped at a constant head, while the discharge
rate was controlled by a rotameter. The wave height was 0.022 m and the wave period was set as 1.5 s. To fix the jet position in
flow with waves, a double-layer 𝜎coordinate [76] is adopted in the simulation. The computational domain is 15 m in length, 0.5
m in width, 0.5 m in depth and is discretized by 368×41×139 grids. A non-uniform grid is employed and refined at the jet orifice,
with minimum grid sizes in the 𝑥, 𝑦, and 𝜎directions of 0.1 m. The time step is set to 0.002 s and the total CPU time per time step
required for the present model was about 10.2 s. The jet outlet velocity is set as 0.85 m/s. As it was found in the former section the
number of sediment particles has no impact on the final deposition outcome once it exceeds 81000, the number of particles is set to
be 81000 and these particles are injected at each grid on the jet outflow boundary.
Fig. 18(a) shows the comparison of the velocity distribution of horizontal jet under wave conditions between experimental data
and numerical results at 𝑥∕𝑑=15 and 𝑥∕𝑑=30. It suggests that the simulated velocity profiles have the same trend as that in the
experiments but the values are over-predicted. The main reason for this outcome is that, in the experiments, the elevation inside the
sediment-feeding system would change in accordance with the outside elevation changes due to the existence of waves. As a result,
the outflow velocity near the jet orifice varies within a wave cycle. Therefore, an extra oscillation velocity should be added to the
jet outlet velocity. However, this simulation does not consider such specific phenomena. Nonetheless, the velocity remains consistent
with the experimental data.
Fig. 18(b) shows the 1-D deposition pattern of the horizontal sediment-laden jet in flow with waves and initially stationary
water. The results indicate that the simulated deposition aligns closely with experimental data, except in the relatively far field
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Fig. 19. Comparison of instantaneous sediment distribution between the numerical model and the experimental results of sediment-laden jet at wave peak phase.
where increased sediment deposition is observed when compared to that in the experiments. This discrepancy may be due to the
comparatively high outflow velocity used in our simulation. In addition, compared with the sediment-laden jet in initially stationary
water, it is found that the deposition rate decreases in the near field while a higher deposition rate is observed in the middle and
far-field when the sediment-laden jet is injected under wavy conditions, which are well captured by the model.
The comparison of instantaneous sediment distribution between the numerical model and the experimental results of the sediment-
laden jet at wave peak phase is shown in Fig. 19. Fig. 19(a) shows the original experimental image, while Fig. 19(b) presents the
same image with an inverted grayscale for better comparison with numerical simulation results. Obvious discontinuities in sediment
deposition which are marked in red ellipses can be seen in Fig. 19(b). The discontinuities also show periodic changes, which are likely
to be attributed to the impact of the wave. The numerical results capture the overall pattern well. In positions closer to the bottom,
discontinuities remain clearly visible in our model, whereas they are hardly to be observed in the experiments. The main reason is
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that the sediment particles adopted in our model are chosen to have the same diameter, which is different from the experiments.
Consequently, once the particles leave the jet body, they tend to move at the same speed due to uniform settling velocities. As a
result, discontinuities remain clearly visible in our model even in areas close to the bottom.
A set of three-dimensional visualizations of the horizontal sediment-laden jet at different wave phases are shown in Fig. 20. It can
be seen that particles would sway upward and downward under wave conditions, which is the most distinctive deposition pattern
compared to the initially stationary case. Additionally, the visualizations also reveal that particles are transported further under the
effect of waves. Fig. 21 shows the two-dimensional (2-D) deposition pattern of the sediment-laden jet under wave conditions. The
2-D sediment concentration contour lines cover a confined area with higher deposition rates, specifically when the deposition rate in
a grid is larger than 16 g/m2/s; whereas they cover a broader area in smaller values like when the deposition rate in a grid is less
than 1 g/m2/s. In summary, the model captures the behavior of the horizontal sediment-laden jet under wave conditions.
5. Conclusion
This paper introduces a two-phase flow model that couples the non-hydrostatic model with the point-particle model to simulate
sediment-laden flow problems associated with temporal changes in the free surface. A Lagrangian-Eulerian method is utilized to
track the free surface, and the movement of sediment particles is tracked by a point-particle model. The model’s accuracy is validated
through comparison with five distinct cases.
First, the propagation of regular waves at a constant depth is simulated using this model. It is found that both the free surface and
flow fields match well with analytical solutions. The model is then extended to address varying water depth scenarios, specifically
the interaction of regular waves with a submerged bar. The numerical results also correspond well with experimental findings. This
underscores the model’s effectiveness in simulating behavior related to changes in free surface. After that, the verification of the
point-particle model is conducted. The deposition processes of a single particle, both in stationary and oscillating environments are
simulated, yielding results that closely match with analytical solutions and experimental data. Finally, the two-phase flow model,
integrated with the point-particle model, is employed to simulate a dilute horizontal sediment-laden jet in both stationary and wave
environments. The observed deposition patterns are found to be in line with experimental outcomes, affirming the model’s capability
to accurately represent sediment transport processes.
The new model shows significant potential as a numerical tool for simulating sediment-laden free surface flows. Currently, the
point-particle model is limited to one-way coupling with the flow, where the reaction force of particles on the flow and the collisions
between particles are not considered. This limitation makes it primarily suitable for low-concentration sediment-laden flows. In the
future, the reaction force of particles on the flow will be incorporated into the governing equation as a source term, and either
the soft-sphere [77] model or hard-sphere model [78] will be employed for the dispersed phase to simulate problems with higher
sediment concentrations.
CRediT authorship contribution statement
Yuhang Chen: Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data
curation. Yongping Chen: Writing – review & editing, Supervision, Software, Project administration, Methodology, Investigation,
Funding acquisition, Formal analysis, Conceptualization. Zhenshan Xu: Supervision, Software, Investigation, Funding acquisition.
Pengzhi Lin: Writing – review & editing, Software, Methodology, Investigation, Formal analysis. Zhihua Xie: Writing – review &
editing, Supervision, Resources, Investigation, Funding acquisition, Formal analysis, Conceptualization.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgements
This work was supported by the National Key Research and Development Program of China (2023YFC3008100), the National
Natural Science Foundation of China (51979076, 52211530103) and United Kingdom Engineering and Physical Sciences Research
Council (EPSRC) grant (EP/V040235/1). The first author also would like to acknowledge the financial support from the China Schol-
arship Council (CSC) under PhD exchange program at Cardiff University [202206710100]. Constructive comments from anonymous
reviewers have helped to improve the manuscript and these are gratefully acknowledged. Special thanks are also given to editors
for their meticulous handling of the manuscript and dedication throughout the review process, which has significantly enhanced the
readability and overall quality of this work.
Data availability
Data will be made available on request.
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Fig. 20. 3-D visualization of initial stage of sediment-laden jet in flow with waves at four different wave phases (a) Wave up-crossing, (b) Wave peak, (c) Wave
down-crossing and (d) Wave trough.
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Fig. 20. (continued)
Fig. 21. 2-D deposition pattern of sediment-laden jet in flow with waves.
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