A complex interface exists between water flow and solid particles during hydraulic soil erosion. In this study, the particle discrete element method (DEM) has been used to simulate the hydraulic erosion of a granular soil under moving bed conditions and surrounding terrain changes. Moreover, the weakly compressible smoothed particle hydrodynamics (WCSPH) approach has been exploited to simulate the instability process of the free surface fluid and its propagation characteristics at the solid–liquid interface. The influence of a suspended medium on the water flow dynamics has been characterized using the mixed viscosity concept accounting for the solid–liquid mixed particle volume ratio. Numerical simulations of wall-jet scouring and reservoir sediment flushing on a mobile bed were performed and validated with experiments. The results show that the proposed WCSPH–DEM coupling model is highly suitable for determining parameters, such as the local maximum scour depth, the scour pit width, and the sand bed profile. The effects on the hydraulic erosion process of two important parameters of the mixed viscosity coefficient (initial solid volume concentration and initial viscosity coefficient) are also discussed to a certain extent in this study.
Hydraulic soil erosionWCSPH-DEM coupling modelsuspended mediummixed viscosity coefficientNational Natural Science Foundation of China51568022Science and Technology Project of Education Department, Jiangxi Province, ChinaGJJ217404Introduction
Typical hydraulic soil erosion, such as runoff erosion and overtopping erosion, usually occur when water flows through loose rivers, subgrade slopes, earth-rock dams, and other structures. These occurrences have raised concerns about the mechanism of hydraulic soil erosion and the intensity of its activity, which are primarily studied through physical model tests. Numerous experimental results have shown that hydrodynamic scour damage is mainly caused by the vortex and submerged flow [1,2], and is broadly divided into three stages; initial scour, scour pit development, and scour depth balance [3]. The scale and intensity of scouring activity are predicted using empirical or semi-empirical formulas derived from relevant model tests. Some examples are the formula proposed by Xue et al. [4] for calculating the overtopping erosion depth of an earth-rock dam and the formula derived by Zhao et al. [5] for calculating the maximum scour depth of debris flow. However, the model test has a scale effect, and it struggles to accurately and comprehensively capture the process of development of the scoured bed as well as the flowfield information of the sand-carrying flow [6,7].
Numerical methods are increasingly being used for accurate refined analysis and to predict the trend of development of the water–soil coupled contact process. The most important aspect of the numerical simulation of hydraulic soil erosion is the treatment of the moving interface between the solid and liquid phases [8]. Coupled continuum–discrete methods are ideal for modeling hydraulic soil erosion behavior; particulate matter in a fluid is modeled by discrete elements, whereas the fluid is modeled as a continuum governed by constitutive laws [9]. The discrete element method (DEM) and element-free Galerkin (EFG) (EFG, improved DEM) method, which are used to determine general behavior through interaction laws in small-scale particle assembly, are ideal tools for granular systems [10,11]. Many continuum approaches such as the finite volumes method (FVM), finite difference method (FDM), and lattice Boltzmann method (LBM), have been coupled to the DEM or EFG to characterize the interaction of heterogeneous fluids with moving granular soils [12–15]. However, in mesh-based methods (e.g., FVM, FDM, and LBM), tedious meshing is required to track the position of the moving solid–liquid interface when dealing with the coupled contact problem of the solid–liquid two–phase flow. Moreover, on a fixed mesh element, the free surface, deformation boundary, and moving interface may struggle to define some defects accurately [16]. Although the DEM and EFG are highly effective for analyzing the deformation of contact between soil particles as well as between soil particles and structures, it is difficult to use these methods directly for obtaining the non-physical parameters (such as friction coefficient and elastic modulus) of the solid–liquid contact zone through experiments. Consequently, it is impossible to accurately describe the stress–strain behavior of solid particles in the liquid phase [17]. For the coupled contact analysis involving multi-phase flow, DEM and EFG methods based on geotechnical mechanics should be coupled with hydrodynamic methods.
Recently, the smoothed particle hydrodynamics (SPH) method proposed by Gingold et al. [18] in 1977 has rapidly advanced. The SPH method uses Lagrangian particles to discretize the continuous spatial computational domain. These particles carry physical quantities (such as mass, density, and velocity) that characterize the behavior of the computational domain. Thus, the method easily determines the contact interface between different materials and is suitable for simulating interface changes caused by hydraulic erosion [19–21]. Ma et al. [22] established the SPH solid–liquid two-phase flow model based on the Oldroyd-B constitutive governing equations and the modified Shields criterion, and they simulated the sediment transport and erosion on the free surface. Padova et al. [23] used the weakly compressible SPH and two-dimensional turbulence model to study the mechanism of the vortex formation of downstream hydraulic erosion. Ng et al. [24] proposed an SPH scheme, which is a coupled Newtonian viscosity equation and Hersch-Bulkley-Papanasasiou (HBP) non-Newtonian control equation. The SPH scheme was used to describe the solid–liquid contact of the two-phase flow, and the feasibility of the method was verified through a simulation of wave erosion on bank sediment. Although the SPH method can adequately characterize the flow pattern changes and soil deformation during hydraulic soil erosion, the hydrodynamics motion governing model inaccurately describes the stiffness degradation and strength reduction of solid soil particles [25]. Therefore, SPH has been coupled with DEM to investigate the behavior of scouring between fluids and solids and to fully characterize the nonlinear properties of interactions between soil particles. For example, Abbas et al. [26] used the one-way coupling of the weakly compressible SPH (WCSPH) method and the DEM to show the additional circulating sediment transport mechanism in front of the vertical breakwater, as well as its direct impact on the bed surface sediment scour scale. Iwamoto et al. [27] introduced the SPH-DEM coupled model based on the local average equation of motion to explore the breakwater instability and failure mechanism caused by the tsunami erosion of the soil behind the breakwater. To improve the computational efficiency and numerical stability of the conventional SPH-DEM model, Kim et al. [28] proposed an improved ISPH-CGDEM model and verified the applicability of the coupled model through three hydraulic soil erosion simulation cases.
However, these studies are limited to the clear water erosion of particles and do not fully consider the change in flow properties caused by suspended particles during erosion. Bed particles typically undergo different states from static to dynamic during erosion, and some particles remain suspended at the bottom of the flow, causing moving bed erosion. Hence, the single-phase SPH method considers the use of elastoplastic or viscoelastoplastic constitutive equations to describe the motion of solid–liquid two-phase flow [29,30]. However, existing control equations of the SPH-DEM model often only assume the viscosity coefficient of water flow as a constant and do not adequately characterize the dynamic change in water flow properties. Therefore, researchers should dynamically modify the coupling control variables of the SPH-DEM model and explore the influence of suspended bed particles on the erosion morphology of the moving bed.
This paper presents a WCSPH-DEM coupling model for modeling hydraulic soil erosion. First, the flow control equation and coupling mechanism of the WCSPH-DEM model are introduced. Unlike the traditional method, the mixed viscosity coefficient of water flow that characterizes the influence of suspended bed particles is combined with water-sediment contact calculation. Subsequently, two hydraulic soil erosion cases are considered for numerical simulation, and the simulation results are compared with the experimental results. The verified WCSPH-DEM model is then applied to explore the influence of the mixed viscosity coefficient on hydraulic erosion morphology. Finally, we present the conclusion and prospects of the coupling strategies adapted in the current research.
Numerical Model
The focus of this study is the complex interface interaction between water flow and soil particles in hydraulic soil erosion. The SPH method based on Lagrange description as well as the DEM method based on Newton’s second law interpolation is used to establish a coupling model. The model comprises three main parts: a water flow model based on the N-S equation; a soil deformation and failure model based on bond contact relationship; and a water-soil interface interaction and erosion model.
Governing Equations
Solid–liquid two-phase flow is often involved in hydraulic soil erosion. Owing to the large difference in the soil particle sizes, the contact flow around soil particles is prone to velocity difference, forming shape resistance; friction resistance also exists between solid particles. The control equation of a weakly compressible fluid under the Lagrangian framework does not consider the resistance term caused by differences in solid particles. Therefore, the combined drag force of shape resistance and friction resistance, and the buoyancy caused by particle volume drainage are introduced as the control components of momentum exchange and energy balance between continuous flow and discontinuous soil particles [28,31,32].
The governing equations (mass and momentum conservation equations) of a weakly compressible fluid, considering the coupling of two-phase flow, can be expressed as follows [28,32]:
dρfdt+ρf∇⋅uf=0,
ρfdufdt=∇⋅σ−fdf−fbf+ρfg,where ρ, u, σ, and g represent density, velocity component, total stress tensor, and gravity component, respectively. fdf and fbf denote the drag force and buoyancy of the fluid element, respectively. ∇ is the divergence symbol.
The motion characterization of a solid particle satisfies the following equation [32]:
ρsdusdt=−∇P+fds+fcs+ρsg+fbs,where P is pressure;fds, fcs, and fbs represent the drag force on the solid particle, interaction forces between solid-phase particles, and buoyancy exerted by the fluid on the solid particle, respectively. The subscripts f and s refer to fluid elements and solid particles, respectively.
Fluids Governed by SPH
Hydraulic soil erosion has a complex flow pattern, and the shallow-water equation does not accurately reflect scouring flow characteristics. The discrete modification of the mass and momentum conservation equations is often used to characterize the motion of particle flow. Thus, the governing equations of weakly compressible fluids in the Lagrangian framework can be written as follows [32–34]:
dρidt=∑j=1Nmjuijα∂Wij∂xiα,
duiαdt=gα+∑j=1Nmj(σiαβρi2+σjαβρj2)∂Wij∂xiβ−Fds→f,iα+Fbs→f,iαmi,where xiα and uijα=uiα−ujα represent the position vector and velocity vector, respectively; α and β denote the Cartesian components x, y, z; σiαβ is the total stress tensor of SPH particles; Fds→f,i and Fbs→f,i represent the drag force and buoyancy force exerted by DEM particles on SPH particle i, respectively; subscript i, j indicates an arbitrary SPH (fluid) particle.
The kernel function Wij determines the performance of the SPH simulation; it not only determines the interpolation accuracy of the SPH approximation, but is also related to the numerical stability. In this paper, the cubic spline kernel function is used as follows:
W(R,h)={αd(23−R2+12R3),0≤R<1αd6(2−R)3,1≤R<20,R≥2,where αd = 157πh2 and 32πh3 represent the 2D and 3D spatial distributions of SPH particles, respectively, |rij|=|xi−xj|=L0, R=|rij|/|rij|hh. As the smoothing length h of the particle decreases, its radius of action decreases, reducing the number of interpolation iterations and saving computer memory. However, Wu et al. [35] found that as the smooth length decreases, the constraint force in the SPH particle influence domain decreases, which easily causes numerical interruption. Accordingly, L0 < h ≤ 1.5 L0 should be satisfied between the initial particle spacing L0 and the smoothing length h for a sufficient number of correlated particles in the influence range, which ensures the convergence of the calculation results [35].
The total stress tensor can be decomposed into an isotropic normal stress P and a shear stress tensor ταβ:
σαβ=−Pδαβ+ταβ,where δαβ is the Kronecker delta. When α=β, δαβ = 1; otherwise, δαβ = 0.
When we consider the effect of suspended particles, they are generally regarded as a mixture of water and particles; their internal flow resistance is expressed by viscous shear stress and obtained from the material model [31]. The heterogeneous water is modeled using the Bingham model [36,37]:
ταβ=2μeff⋅ε˙αβ=2(τy|γ˙|+μ0)ε˙αβ,where μeff and μ0 represent effective viscosity and plastic viscosity coefficient, respectively; τy represents yield strength, which controls the flow state before fluid instability failure and satisfies the Mohr–Coulomb yield criterion.
For a weakly compressible fluid, ε˙αβ is the shear strain rate tensor and can be described by the following vector expression based on the velocity gradient [25,38]:
In the above expression, the discrete expression for the shear strain rate ε˙iαβ of arbitrary particle i in SPH can be written in the following form:
ε˙iαβ=∂uα∂xβ|i=12(∑j=1Nmjρj(ujα−uiα)∂Wij∂xiβ+∑j=1Nmjρj(ujβ−uiβ)∂Wij∂xiα),where uα is the velocity component, and the SPH discrete form of its velocity gradient satisfies the mass conservation criterion. |γ˙| is the shear rate which can be defined as
Among the commonly used rheological models, the Bingham model [39] reflects the linear relationship between the shear rate and shear stress of the fluid after yielding. In a stable state, when the shear rate |γ˙|→0, the shear stress is equal to the yield stress. However, according to Eq. (8), the shear rate cannot be zero in the Bingham model. To avoid the singularity of the flow model at the boundary of the numerical calculation domain, the regularized Bingham equation [40] is introduced as follows:
μeff=τy|γ˙|(1−e−m|γ˙|)+μ0=τyεΠ+μ0,where εΠ is the second invariant of the shear strain tensor, and m is a regularized constant. Eq. (15) shows that when |γ˙|→0, the fluid is about to enter the plastic deformation stage, and its initial viscosity is close to the viscosity coefficient when the fluid yields. At this time, the shear stress can be approximated to the yield stress. However, when m is infinite, the approximate solution mτy+μ0 of the effective viscosity is extremely large, easily causing a result overflow. Therefore, to ensure computational efficiency and accuracy, m is usually taken in a small order of magnitude (100–1000) [41].
Soil particles at a water–soil coupling interface are suspended at the bottom of the water flow by the scouring and buoyancy of water, increasing the viscosity of the bottom water flow. The increased viscosity directly affects the dynamic characteristics of the two-phase interface and influences the calculation results. Therefore, the effective viscosity coefficient should be modified to consider the influence of suspended particles on the flow properties. The most common method is to track the change in the solid–liquid volume concentration in the water flow containing suspended particles, after which the effective viscosity coefficient is amplified by an exponential function [38,42]. Therefore, the mixed viscosity coefficient of flow containing suspended particles satisfies:
μmix={μeffe2.5αs1−39αs/64,αs≤0.3μeff,αs>0.3where αs is the volume ratio of solid particles (sediment volume fraction) in the affected flow zone, and its discrete calculation formula is
αs,i=∑ksat∈2hNmkρk/mkρk∑k∈2hNmkρk∑k∈2hNmkρk,where ksat is a solid particle that satisfies the erosion initiation condition in the influence domain of particle i; h is the smooth length. The influence domain of SPH particles on DEM particles can be determined by referring to the water–soil coupling interface treatment method proposed by Manenti et al. [43] (as shown in Fig. 1). The 2h depth range at the coupling interface is the potential erosion area (exposed soil particles), and the DEM particles beyond the 2h depth range are regarded as part of the fixed boundary (hidden soil particles).
Water-sediment contact interface diagram
In this paper, we refer to the contact treatment between the suspended medium and the boundary in the mechanics of sediment motion [38,42]. When αs≤0.3, the water erosion occurs in the exposed soil particles layer, and the mixed viscosity is applied to calculating the relative motion of the mixed medium. When αs>0.3, the suspended medium is considered to have reached the interface between the exposed soil particles layer and the hidden soil particle layer. At this time, the contact between the suspended medium and the fixed boundary particles is calculated by the effective viscosity.
In the SPH method for weakly compressible fluid problems, the motion of water particles is produced by the pressure gradient. Therefore, the dynamic pressure change in the particle calculation domain can be corrected by the SPH state equation in weak compression form [44]:
Pi=B[(ρiρ0)λ−1],where B=c02ρ0/c02ρ0λλ; λ = 7. ρ0 and ρi represent the original density (base density) and liquid density of the fluid in the SPH particle calculation domain, respectively. c0 is the artificial sound velocity, and, generally, c0≥10umax, where umax represents the maximum velocity in the fluid flow process.
The SPH control equations (Eqs. (4) and (5)) are solved using the liquid phase coupling of the state equation (Eq. (18)) and the stress–strain equation (Eq. (7)), and the rheological model of the flow with suspended particles is established.
DEM Description of Soil Particles
The DEM is a molecular-dynamics-based numerical simulation method developed by Cundall et al. [10], and it was first applied to the problem of discontinuous media in rock mechanics. The fundamental concept of the DEM is the expression of discontinuous points through discrete particles; Newton’s second law and time step iteration are then used to solve the motion control equations of the discrete particles, obtaining the overall motion form of the discontinuous points.
We simulated the solid particles under hydraulic scouring using spherical DEM particles. The motion includes two kinds of six degrees of freedom; translational and rotational, which are mainly controlled by force and moment. The force of solid particles is usually dominated by the contact force between particles as well as the force between particles and the water flow medium.
Meanwhile, natural sand grains are not spherical. Hence, we used the rotational moment to reduce the inhomogeneity of the soil particles and simulate the tumbling of the particles by changing the angle [28]. Overall, the governing equations satisfying the translation and rotation of DEM particles are as follows:
∂udem,kα∂t=Fdf→s,kα+Fbf→s,kα+Fcklαmk+gα,
Ik∂ωk∂t=Tckl,where Fckl is the solid interparticle force; Fdf→s,k and Fbf→s,k represent the resistance and buoyancy of the liquid on the solid particle k, respectively, and are the main forces driving the movement of the solid phase particles caused by the fluid medium, as defined in Section 2.3. Ik=mkdk2/mkdk21010 represents the moment of inertia, dk is the particle diameter, ωk is the rotational speed, and Tckl is the interaction torque. Subscript k, l indicates an arbitrary DEM particle.
Fig. 2 illustrates the mechanical model of the interaction between DEM particles [10]. When two DEM particles collide, the contact force Fckl can be decomposed into the normal and tangential directions, represented by the local coordinates between the particles. The specific contact force equation is
Diagram of contact between solid particles in the DEM model
Fckl=Fckln+Fckls,where the contact force component in the normal direction Fckln and the contact force component in the tangential direction Fckls satisfy [45,46]
in which μ is the static friction coefficient between DEM particles. K, Δx, η, and Δu are the elastic coefficient, relative displacement, viscous damping coefficient, and relative velocity, respectively. When the tangential component of the contact force satisfies |Fckls|>μ|Fckln|, the DEM particles slide at the contact point.
Additionally, the displacement and rotation of particles can be tracked in the movement of DEM particles. Through the first-order Taylor expansion of the increment of velocity and angular velocity, we derive the iterative calculation equations for displacement and angle at each time step:
(ukα)t+Δt2=(ukα)t−Δt2+(∂ukα∂t)t⋅Δt,
(xkα)t+Δt=(xkα)t+(ukα)t+Δt2⋅Δt,
(ωk)t+Δt2=(ωk)t−Δt2+(∂ωk∂t)t⋅Δt,
(θk)t+Δt=(θk)t+(ωk)t+Δt2⋅Δt.
WCSPH-DEM Coupling Model
During hydraulic soil erosion, the interaction between the solid and liquid phases is related to the power transfer and energy dissipation between various media. Therefore, we propose a two-phase flow coupling model based on the contact forces (drag force and buoyancy) between the particle DEM component and SPH component (as shown in Fig. 3), and we applied the model to the solid–liquid SPH-DEM simulation.
Contact force and torque of fluid particles acting on solid particles
The drag force is related to the solid–liquid friction velocity, the density of distribution of DEM particles in the influence range of SPH particles, and the volume of DEM particles. In view of the change in the flow properties caused by suspended particles under water erosion, we replace the original clear water viscosity coefficient with the modified mixed viscosity coefficient in the SPH-DEM coupling model. In reference to the two-fluid calculation model [32,46], the drag force of SPH particles on DEM particles satisfies
εk=∑k∈demWikVk∑k∈dem,sphWikVk,u¯sph,kα=∑iuiαWikVi∑iWikVi,where u¯sph,kα is the average velocity of SPH particles around DEM particles, and Vk is the volume of DEM particles. εk is the local porosity, which depends on the unit volume density of DEM particles. To make the calculated local porosity continuous in space, the smooth length h should be at least 2 times larger than the median diameter of DEM particles [47]. The fluid resistance coefficient ζk is defined by the equations of Ergun [48] and Wen et al. [49] to meet different particle porosity conditions.μmix,f, dk, and ρf represent the mixed viscosity coefficient, DEM particle diameter, and fluid density, respectively. Cd is the drag coefficient of DEM particles, which satisfies the following expression [50]:
Cd={24Re(1+0.15Re0.681),Re≤10000.42,Re>1000,
Re=|u¯sph,kα−udem,kα|⋅εkρf⋅dkμmix,f,where Re is the DEM particle Reynolds number, and |u¯sph,kα−udem,kα| is the friction velocity.
In the case of an extremely low friction velocity, the variance can be neglected, and the drag is mainly caused by the friction of the liquid acting on the surface of the particle. When the friction velocity increases gradually, the streamlines on the DEM particle surface are separated, the pressure difference is generated before and after, and the shape resistance is finally developed [31]. Therefore, the general form of the drag force can be defined by the resultant force of surface friction F_{1} and shape resistance F_{2}.
Buoyancy is also a crucial contact force in the SPH-DEM coupling model and is usually characterized by fluid pressure gradient under the control of state equation [27,32]:
Fbf→s,k=Vdem,k∇Psph,i(xiα)=Vk∑imipiρi⋅∂Wik∂xiα.
However, for the analysis of the solid–liquid two-phase flow under water erosion, the influence of suspended particles on fluid properties cannot be ignored. Therefore, we introduce the mixed viscosity coefficient μmix,f and average velocity u¯sph,kα to increase the buoyancy calculation term under the action of suspended particles:
Because Newton’s third law governs the interaction forces, the drag force and buoyancy of the DEM particle on the SPH particle are of the same magnitude as that of the reaction of the DEM object on the SPH fluid, but in opposite directions.
Operation Process of the Coupled Model
A fixed time step is used in the DEM and SPH modules, and the calculation time step should be less than the critical time step; otherwise, the calculation will not converge. The critical time step of solid particles (Δts) in the DEM model can be controlled by the normal elastic coefficient Kn of DEM particles, and is expressed as [51]:
Δts=2π5mk/mkKnKn.
The critical time step for the SPH module (Δtf) is usually determined by the Courant-Friedrichs-Lewy (CFL) condition [52]:
Δtf=CFL⋅min(Δtw,Δtcv),
Δtw=h|ai|max,Δtcv=hmax(c0,10|umax|),where |ai|max and |ui|max denote the maximum acceleration and maximum velocity on SPH particles, respectively; CFL is the Courant number, and CFL = 0.1 was selected in the hydraulic erosion research.
Owing to the differences in material properties and algorithms, the values of critical time steps Δts and Δtf are largely different, and Δts is usually smaller than Δtf. In addition, the SPH method is an interpolation algorithm, and the value of Δtf is insufficiently large, which affects the calculation efficiency and the accuracy of the results. To synchronize the operation of the SPH and DEM models, we optimized and integrated the multi-time step operation program. Fig. 4 describes the computational process of the SPH-DEM program.
Simplified flow chart representation of the procedure for one time step of the coupled SPH-DEM modelModel Validation
We investigated the validity and accuracy of the SPH-DEM coupling model in simulating hydraulic soil erosion. A water storage depth h = 0.15 m (as illustrated in Fig. 5) was used for the simulation according to the wall-jet scouring and reservoir sediment flushing experimental model designed by Khanpour et al. [53]. First, for the wall-jet test, a 0.15 m high water flow was stored in a 0.2 m × 0.7 m flume, and the flume gate was designed to be lifted quickly by 0.05 m to form a water jet. The fine sediment riverbed (0.085 m × 0.45 m, d_{50} = 0.85 mm) was set in the downstream of the flume. The arrival of the discharge flow induced the hydraulic soil erosion. Second, for the reservoir sediment flushing test, a 0.15 m × 1 m dense sedimentary sand layer (d_{50} = 0.85 mm) was tiled at the bottom of the storage tank, which was 0.05 m higher than the downstream riverbed. As the water storage depth of the upper part of the sedimentary sand layer reached 0.15 m, the gate was quickly opened to initiate the erosion of the sediment layer by the burst water flow.
Two-dimensional schematic of the model tests for (a) sediment scouring and (b) reservoir sediment flushing
Considering the different aspects of the solid–liquid two-phase flow, we used SPH discrete particles to represent the stored water flow in the simulation process and DEM particles to define the sediment. NULL material was used for the liquid unit as the material shares similar parameters with water. Solid particles were assigned by the PLASTIC_KINEMATIC material to ensure that they have the same density, elastic modulus, and Poisson’s ratio as those of natural sand. The NULL material was also used to replace the test groove and gate made of smooth glass, whereas SPH particles with rigid constraints were used to characterize the boundary, satisfying the calculation criteria of the boundary particle method [54]. The weakly compressible SPH state equation was selected for the motion control and dynamic pressure balance of the water flow. The initial stress state, boundary conditions, and model size of the SPH-DEM numerical model established in the simulation were consistent with those of Khanpour’s experiment [53]. Table 1 presents the material properties of the reservoir flushing and wall-jet scouring caused by a dam break.
Material models and parameters for simulating hydraulic soil erosion
Parameters
Symbol
Value
Reference
Boundary
Young’s modulus (GPa)
200
[54]
Poisson ratio
0.25
Density (kg/m^{3})
7850
Water phase
Sound velocity c0 (m/s)
1480
[53]
Initial density ρ0 (kg/m^{3})
1000
[53]
The constant λ
7
[53]
Dynamic viscosity μ0 (Pa.s)
0.001
[53]
Sediment phase(natural sand)
Young’s modulus (MPa)
100
[28]
Poisson ratio ν
0.3
[28]
Grain density ρs (kg/m^{3})
2650
[53]
Initial porosity εk
0.66
[53]
Friction coefficient μ
0.6
[53]
Elasticity coefficient in both normal and tangential directions Kn, Ks (N/m)
10^{3}
[32]
Damping coefficient in both normal and tangential directions ηn, ηs
0
[32]
Test 1: Sediment Scouring Caused by Wall-Jet Flow
To simulate the behavior of the erosion of the downstream bed sand caused by a wall-jet flow, we established a coupled SPH-DEM simulation model (see Fig. 6). The liquid SPH particles were filled into a groove with a width of 0.7 m, length of 0.1 m in the paper direction, and height of 0.15 m. The solid DEM particles in the downstream sediment layer were distributed randomly. For the movable sediment layer, 85350 DEM particles (d_{50} = 0.85 mm) were used, satisfying the initial porosity of 0.66. Water particles and boundary particles were placed at the same particle spacing of 5.0 mm, yielding 40500 SPH particles and 8250 fixed SPH particles, respectively.
Two-dimensional schematic of the SPH-DEM coupling model
To verify the applicability of the SPH-DEM coupling model, we compared the numerical results with the experimental results (Fig. 7). The simulated solid–liquid two-phase flow pattern is highly consistent with that of the experiments. The profile of the bed sediment after erosion at different time nodes is consistent with the profile observed in the model experiment. Fig. 8 illustrates the velocity distribution when the dam-break flow erodes the downstream sand bed. As shown in the refined area of Figs. 8b and 8c, the shallow DEM particles have obvious erosion entrainment under the action of SPH particles, and according to the velocity distribution gradient, a layer of suspended DEM particles is formed at the solid–liquid interface. As shown in Fig. 8d, the wall-jet is squeezed and accumulated at the gate, and the free surface in the reservoir is turbulent, forming reverse waves. As the reverse waves develop, the flow velocity of SPH particles at the outlet gradually decreases. Consequently, the downstream liquid level hump turns reverses direction (see Figs. 8e and 8f), causing turbulence in the suspended DEM particle layer at the solid–liquid interface. The overlap between SPH particles and DEM particles is due to the visualization of SPH particles which has no effect on the simulation [46].
Experimental and numerical results of the wall-jet scouring experimentCloud map of velocity distribution at each time point
As shown in Fig. 8b, nonphysical splashing of SPH particles occurs in the refined area, and similar cases have been reported in other SPH-based calculations [35,55]. Bakti et al. [55] reported that changes in the density of SPH particles near the boundary are more obvious than those of other regions because of boundary constraints, which considerably enlarge the density discontinuity of boundary particles and the calculation complexity. At the moment of contact of some SPH particles, the interaction force exceeds the constraint force in the particle influence domain, resulting in nonphysical splashing of particles. In addition, the smooth length h is small (h = 1.05 L0 in this study), and the constraint force of the SPH particle influence domain is relatively small [35], which also facilitates particle splashing. However, with the gradual release of pressure, the nonphysical splash of the particles disappears (see Figs. 8e and 8f). Compared with the experimental results, the simulation results show that the slight nonphysical particle splash of the current particles has little effect on the calculation results.
In accordance with previous model test results [53], we took the maximum depth of the scour hole D_{final} = 0.02 m as the basis for regularizing the numerical results. Fig. 9 illustrates the dimensionless expression (D/D_{final}) of the scour pit depth D at each time point; the numerical simulation results were observed to be in good agreement with the experimental results. In the early stage of the dam break, the flow violently scours the downstream sand bed, and the scour depth increases rapidly. As water flow is discharged, the change in the scour pit tends to be stable after time = 2 s. Between time = 1 s and time = 2 s, the depth of the scour hole decreases. This decrease is likely because after time = 1 s, reverse free surface waves appear near the gate, driving some bed sand to backfill the hole. More sediment particles are carried by water flow with the intensification of hydraulic scour. The change in the suspended medium concentration affects the physical and mechanical properties of water flow, as well as the final formation of the scour pit [42].
Time history of dimensionless scour hole depth
For the comparison of the scour depth of the one-dimensional angle, the influence of the suspended medium was not obvious. Thus, we considered the erosion analysis of the two-dimensional angle. Fig. 10 shows the time distribution curve of the aspect ratio (D/L) of the scour hole; the dimensionless size distribution of the scour hole calculated by the WCSPH-DEM coupled model lies between the ISPH-CGDEM simulation results and the model experimental results. Hence, the improved strategy of the SPH-DEM coupling model, which uses the fluid mixing viscosity coefficient to characterize the dynamic change in the suspension medium, is effective and feasible. In addition, because of the different solid–liquid coupling control equations selected by each simulation model, some differences exist between the numerical calculations and the experimental observations, but the overall motion change trend is consistent.
Dimensionless expressions for depth and width scour hole with timeTest 2: Numerical Simulation of Reservoir Sediment Flushing
Another important case analysis in the study of hydraulic soil erosion behavior is the flushing of the bottom sedimentary sand layer by reservoir discharge. We referred to the small-scale flume discharge experiment conducted by Khanpour et al. [53] (see Fig. 5b) to construct an SPH-DEM simulation model for reproducing an idealized reservoir sediment flushing. The initial sketch of the numerical model is shown in Fig. 11. At the beginning, 380000 DEM particles were randomly distributed to fill the sediment layer at the bottom of the tank while 30000 SPH particles were uniformly distributed on the sediment layer with a spacing of 0.01 m. The initial values of the field variables, such as pressure and velocity, were set to zero.
Initial schematic representation of the SPH-DEM numerical model
Fig. 12 illustrates the numerical simulation of the sediment flushing test. As the gate opens, the sediment layer and upper water are released at the same time, discharging the sand-carrying water flow. By tracking and comparing the simulated sediment profile and water level in the flume at different times, we observed that the SPH-DEM numerical calculation results were consistent with the experimental observations.
Experimental and numerical results of reservoir sediment flushing test
The output of the velocity distribution nephogram (Fig. 13) indicates that the coupling model completely captures the peak value and corresponding position of the discharge solid–liquid velocity. The fine areas in Figs. 13a–13c show that the erosion effect of SPH particles on DEM particles is significant, and the exposed soil gradually forms a suspended medium layer. On the velocity curve, the difference in velocity between the particles and the liquid is also fully reflected in the bottom precipitation layer because of the damping contact between the DEM particles and SPH particles. Fig. 13d shows that the subsequent movement of the particles is essentially smooth when the suspended media layer disappears and the relative particle velocity approaches zero.
Time history of velocity contour; (a) T = 0.2 s, (b) T = 0.4 s, (c) T = 0.8 s, (d) T = 2.2 s
The time distribution curve of the free surface height and sediment thickness at the gate is shown in Fig. 14. When the gate is opened, the water and sand are discharged, and the liquid level peaks quickly. At time = 0.8 s, the sediment thickness fluctuates slightly. By also considering the flow velocity near the gate (Fig. 13c), we believe that the flow gradually develops from turbulent flow to laminar flow, weakening the relative motion of the solid–liquid interface and thus forming a short-term accumulation of sediment. With time, the height of the free surface gradually decreases, and the flushing effect of water flow on the sediment weakens. After time = 1 s, the change in sediment thickness gradually stabilizes.
Time history of the water level and sediment thickness at the gate
The scouring efficiency of the sand-carrying flow also needs to be analyzed. The area domain of the graph was determined using AutoCAD software, and the cross-sectional area of the material in the sink at different time nodes was calculated. The volume ratio (Vwater/Vsediment) of the cross-sectional unit was obtained from the area ratio, and the flushing efficiency of the water flow was characterized (as illustrated in Fig. 15). The fitting analysis of the volume ratio distribution of the cross-sectional unit shows that the sediment change during the water flushing process (0–3.4 s) satisfies the quadratic equation.
Dimensionless expression of flushing efficiency of the sediment with time
In summary, the two test cases were adequately simulated and reproduced by numerical calculations. The comparison of results shows that our WCSPH-DEM coupling model that considers the influence of a suspended medium is suitable for determining the physical and mechanical parameters in the coupling contact process of a solid–liquid two-phase flow. Therefore, the coupling model is effective for analyzing and solving the problem of hydraulic soil erosion.
Numerical Analysis and Discussion
As earlier mentioned, the suspended medium is an important factor that differentiates between moving bed erosion and clear water erosion, and the suspended mass concentration is the key physical parameter affecting the erosion morphology of the moving bed. We adjusted the sediment thickness at the inlet to analyze the erosion of downstream sediment by sand-carrying flow with different initial concentrations. This approach was selected because of the change in suspended sediment concentration caused by hydraulic soil erosion, as well as the simple and reliable acquisition of the initial viscosity coefficient and the initial sediment volume fraction in the mixed viscosity coefficient, which are the main characterization parameters of the influence of suspended media.
As derived from Eqs. (12) and (13), the sediment erosion is mainly determined by the initial viscosity coefficient of the fluid. When sand-laden flow is used, the influence of the initial sediment content and the initial viscosity coefficient of the flow should be considered [56]. Therefore, we used Case 1 (Section 3) for the simulation model. The relevant material parameters were unchanged, and the sand-carrying flow erosion and clear water flow erosion with different suspended sediment concentrations and flow viscosity coefficients were set, respectively. For the concentration distribution of suspended sediment at the entrance, we used the suspended sediment concentration range of 2456–2700 kg/m^{3} (d50 = 0.18 mm) [57] in Debnath’s test. The sediment density of 2650 kg/m^{3} was converted into the initial sediment volume fraction of 0.000927–0.001019. Therefore, 0.001 was selected as the sediment volume fraction of the sand-carrying flow for calculation, and 0.002, 0.003, 0.004, and 0.005 were set for an extended discussion. When the sediment volume fraction of the inlet water flow was 0.001 (the initial water flow depth is 15 cm), the sediment thickness in the storage tank satisfied one layer of DEM particle filling with a median particle size of 0.18 mm. When the gate was opened, the sediment at the bottom collapsed instantaneously, and the initial volume fraction of the sediment met the theoretical setting. By analogy, the settings of other simulation groups met the corresponding sand thickness requirements.
Fig. 16 illustrates the relationship between the initial sediment volume fraction and the local scour depth. As the concentration of suspended sediments in the water flow increases, the peak erosion depth of the water flow to the downstream sand bed gradually increases. However, when the initial sediment concentration reaches 0.004, the peak local scour depth of the downstream sand bed decreases. Chien et al. [31] reported that the increase in sediment thickness is accompanied by an increase in the energy dissipation during sediment transformation from bed load to suspended load, which affects the erosion of suspended sediment flow on the downstream sediment bed. In addition, sediment scours caused by water flow with different initial viscosity coefficients has a strong influence.
Scour depth peak <italic>vs.</italic> initial sediment volume fraction under different water conditions
Fig. 17 shows the time history curve of the depth of erosion of the downstream bed sand under different suspended sediment concentrations at the entrance of the flume. In the early stage of scouring, the higher the suspended sediment concentration, the smaller the scouring depth. In the later stage of erosion, the higher the suspended sediment concentration, the greater the erosion depth. This behavior occurs mainly because in the two-phase flow model, the existence of suspended load in the initial stage of scouring weakens the turbulent form of water flow. The higher the suspended load concentration, the less obvious the turbulent form of water flow, and the weaker the degree of local scour. In the later stage of erosion development, the presence of suspended sediment increases the viscosity of the sediment-laden flow, increasing the thickness of the laminar layer near the wall, occupying the hydraulic rough area of the hydraulic perimeter, smoothening the perimeter, reducing the reduction of flow velocity or energy loss, and enhancing its ability to scour the riverbed [31]. Comparing clear water erosion (Fig. 9) and sand-carrying flow erosion (Fig. 17), no obvious reverse wave is observed when the suspended sediment flow scours the downstream bed sand, and the change in the scour hole depth does not fluctuate significantly between time = 1 s and time = 2 s.
Time history of scour depth peak under different initial sediment volume fractions
We also used the relative error norm (εYi) of the scour sand surface elevation (Yi) to evaluate the applicability of each numerical method and model test. εYi is expressed as follows:
εYi=∑iN(δYi)2/∑iN(Yi)2,where Y represents the model test observation elevation in the scour hole, and δY represents the difference between the numerical calculation results and the experimental observation results in the scour hole. N represents the number of local scour hole depth observations. Table 2 presents the summary of relative error norm of the results under different numerical conditions.
<inline-formula id="ieqn-117">
<mml:math id="mml-ieqn-117"><mml:mi>Y</mml:mi><mml:mi>i</mml:mi></mml:math>
</inline-formula> relative error norm in calculated elevation of the sediment bed surface
Case
Fluid parameter μmix
Relative error norm εYi
Initial sediment volume fraction αs,0
Initial viscosity coefficient μ0
1
0
0.001
0.0901
IXSPH [47], μmix=μ0
2
0
0.001
0.1479
ISPH-CGDEM [21], μmix=μ0
3
0
0.001
0.1193
Present SPH-DEM, μmix≠μ0
4
0
0.0001
0.1335
Present SPH-DEM, μmix≠μ0
5
0
0.01
0.1456
Present SPH-DEM, μmix≠μ0
6
0
0.1
0.2581
Present SPH-DEM, μmix≠μ0
7
0.001
0.001
0.0777
Present SPH-DEM, μmix≠μ0
8
0.002
0.001
0.0909
Present SPH-DEM, μmix≠μ0
9
0.003
0.001
0.0886
Present SPH-DEM, μmix≠μ0
10
0.004
0.001
0.1002
Present SPH-DEM, μmix≠μ0
11
0.005
0.001
0.0924
Present SPH-DEM, μmix≠μ0
As presented in Table 2, for the case of clear water erosion (Cases 1–3), the relative error of the modified WCSPH-DEM coupling model is smaller than that of the ISPH-CGDEM model, indicating the feasibility of introducing the mixed viscosity coefficient to dynamically characterize the influence of suspended sediment on the flow pattern characteristics. For the influence of the initial viscosity coefficient (Cases 3–6), the error of the numerical calculation results gradually increases with the increase in the initial viscosity coefficient of the water flow. It is reasonable to select μ0 = 0.001 for the numerical calculation. For the case of scouring sand-carrying water flow (Cases 7–11), the deviation norm of the overall numerical calculation results is smaller than that of the scouring case of clear water flow (Case 3), indicating that the selection of mixed viscosity coefficients instead of effective viscosity coefficients improves the calculation accuracy of the coupling model.
Overall, the occurrence of hydraulic soil erosion depends on the input of initial flow parameters and affects the morphological development of the entire scouring process. The suspended load cannot be ignored in the whole scour analysis, and different concentrations of suspended load can change the characteristics of the scour process.
Conclusions
This study aimed to investigate the complex interface between water flow and soil particles in hydraulic soil erosion. A numerical model of hydraulic erosion simulation based on WCSPH-DEM coupling was developed. In the coupling model, the Mohr-Coulomb yield criterion is used for judging whether erosion occurs. The eroded water-sand mixed particles are described with reference to the non-Newtonian fluid characteristics of the Bingham rheological model, and the SPH constitutive control of weakly compressible fluid is established. For the solid part, the DEM particles based on the bond contact relationship adequately track the deformation and failure between the soils. Drag force and buoyancy are introduced to complete the solid-liquid interface processing, and the parallel computing of the WCSPH method and DEM method is realized. The reliability of the WCSPH-DEM model was verified by comparing its results with those of a wall-jet flushing test and reservoir flushing test [47]. The change in topography, the position of maximum scour depth, and the final topography at an equilibrium scour depth were successfully simulated.
The mixed viscosity coefficient is introduced into the WCSPH-DEM model to characterize the influence of suspended sediment on the dynamic characteristics of soil-water two-phase interface. We analyzed and discussed the effects of initial sediment concentration and initial viscosity coefficient on hydraulic erosion. The local scour depth is positively correlated with the initial sediment concentration and the initial viscosity coefficient, indicating that the suspended medium has a significant effect on the flow pattern and scour pattern of the sand bed. In addition, the comparison of the deviation norm between the calculation results of the different numerical methods and the experimental results further shows that it is feasible to introduce the mixed viscosity coefficient into the proposed model to characterize the influence of the suspended medium on the flow pattern.
However, several problems of the model need to be addressed. For example, the influence of the gap size between discrete particles in the model on the flow needs to be studied and improved. Because of computational resource limitations, the proposed WCSPH-DEM coupling model is two-dimensional. Subsequent research should expand the model to three dimensions using parallel computing or GPU graphics acceleration technology, and the model verification should be conducted through more comparison and quantitative analyses.
Funding Statement
This research was funded by the National Natural Science Foundation of China (No. 51568022) and the Science and Technology Project of Education Department, Jiangxi Province, China (No. GJJ217404).
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
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