Prediction of Scouring Hole Morphology Induced by Underwater Jets Using CFD–DEM Simulation

Abstract

Underwater jet scouring is an efficient, flexible underwater dredging technique, yet its complex physical mechanisms and dynamic evolution hinder dredging effectiveness evaluation. Existing studies mostly use empirical formulas and neglect the sediment properties’ influence on scour holes. This study integrates numerical simulation, theoretical derivation, and sediment characteristics to develop a universal model for efficiently predicting underwater jet scour hole morphology, overcoming existing models’ limitations of over-simplifying complex physics and insufficient experimental data alignment. Using CFD–DEM coupling to simulate scouring, it correlates key physical parameters (average/maximum shear rate, average/maximum shear velocity) with jet characteristics (nozzle diameter, velocity, distance) via theoretical derivation and simplifications, validated using multi-condition simulation data. Comparative analysis shows maximum relative errors of 13% for depth and 7% for width, confirming the engineering applicability in scour hole prediction.

1. Introduction

Sediment deposition has increasingly impacted the efficiency of hydraulic engineering infrastructure, particularly in pumped-storage power station reservoirs, port channels, and other critical scenarios, where siltation-induced reservoir capacity loss and navigation obstruction have become bottlenecks to sustainable operations [1,2]. Traditional dredging techniques, characterized by low efficiency, high costs, and significant environmental disturbances, struggle to meet the dynamic requirements of complex water environments. Submerged jet scouring technology, which utilizes high-pressure water flow to precisely disturb sediment and leverages natural currents for transportation, demonstrates advantages in efficiency and flexibility. However, its core challenge lies in the high uncertainty of dredging effectiveness, specifically that the accurate prediction of scour hole morphology and the optimization of operational parameters remain unresolved [3].

In recent decades, international research on jet scouring technology has achieved notable progress. The United States, as a pioneer in disturbance-based dredging, systematically explored jet scouring mechanisms starting in the mid-20th century. The U.S. Army Corps of Engineers validated the efficacy of bubble, jet, and propeller disturbance techniques in over ten projects, including the Mare Island Naval Shipyard and Louisiana Bay, establishing core operational criteria such as a sediment grain size of <0.1 mm and a tidal current velocity of >0.5 m/s [4,5,6]. The Netherlands, represented by the heavily silted Port of Rotterdam, implemented a towed bubble disturbance system in the last century, achieving routine dredging by synchronizing with ebb tides. This approach reduced costs by 50% compared to conventional dredgers, while environmental monitoring confirmed minimal impacts from the dispersion of suspended solids [7]. The UK further optimized disturbance equipment at Harwich Port, proposing a “more nature-based dredging methodology” that cut costs by 50% and carbon emissions by 65%, highlighting the low-carbon potential of disturbance techniques. Additionally, German and Japanese scholars contributed to jet parameter optimization. Taştan et al. [8] experimentally demonstrated that thin non-cohesive sediment layers are more susceptible to jet entrainment, whereas thicker layers require higher jet energy for effective scouring.

Experimental research on underwater jet dredging continues to progress. Scholars, employing diverse research methods and perspectives, are continuously deepening their exploration of this complex yet highly promising field. In experiments, researchers generally focus on the impact of key parameters such as jet velocity, target distance, nozzle configuration, sediment particle size, and sediment layer thickness on dredging efficiency. By quantitatively analyzing indicators such as scour depth, scour range, and sediment concentration distribution, dredging effectiveness can be evaluated under different working conditions, thereby providing a basis for optimizing jet parameters and operational strategies. The relevant research progress is summarized as follows. In experimental research, investigators have developed a variety of experimental setups to systematically examine the influence of jet parameters (such as jet velocity, nozzle diameter, and impingement distance) and sediment characteristics (such as particle size, density, and shape) on dredging effectiveness. Aderibigbe and Rajaratnam [9] conducted experiments on vertical cylindrical underwater jets with a scour intensity coefficient of less than 0.5. These researchers investigated the effect of target distance on maximum scour depth. In 2009, Yeh et al. [10] performed experimental studies on scour induced by large-scale moving cylindrical jets. They proposed functions (see Equations (1) and (2)) to describe the maximum scour depth and scour radius to characterize the morphological features of scour holes. However, most formulas in this study are empirical and not linked to sediment particles. Subsequently, Sutherland and Dalziel [11] experimentally studied the scouring effect of vertical jets on sediment beds and introduced a Rouse number model to describe the scouring process.

$$\mathrm{Scour} \mathrm{depth}: {\displaystyle \frac{{\epsilon }_m}{h}}={\displaystyle \frac{s-4.0}{305.4+3.00\left(s-4.0\right)}}$$

(1)

$$\mathrm{Scour} \mathrm{radius}: {\displaystyle \frac{r_1}{h}}={\displaystyle \frac{s}{1930.5+1.07s}}+0.48$$

(2)

Furthermore, Fan et al. [12] investigated the critical conditions for sediment resuspension and the scouring effect of water flow on sediment. These researchers validated their model in a flume by adjusting parameters including submerged flow velocity, the relative position of the outlet to the bottom, and sediment particle size. Their findings indicate that the submerged pipe flow velocity and outlet position must be adjusted to ensure that the Froude number remains below 0.5. If the Froude number exceeds this threshold, a portion of the sediment will be entrained into the water column. The release rate can be predicted using a formula based on Shields theory.

In the field of numerical simulation, researchers have developed various models to simulate the underwater jet dredging process. Among these, two-phase flow models have garnered significant attention for their ability to concurrently account for the motion of both the fluid and solid particles. Pham Van et al. [13] proposed a two-phase flow numerical model based on the Eulerian–Eulerian approach, successfully simulating the erosion of a horizontal granular bed by a plane jet and validating the model’s accuracy. Lee et al. [14] developed a three-dimensional two-phase model to simulate sediment transport, demonstrating applicability across a wide range of particle Reynolds numbers. Taştan et al. [8] investigated the influence of the thickness of non-cohesive sediment layers on riverbed scour results. Boroomand et al. [15] employed the Flow-3D method to simulate particle motion and deformation during both conventional and dispersed jet operations (see

Table 1. Jet scour hole morphology prediction model.

Research	Approach	Adaptation Conditions	Prediction Formula
Aderibigbe et al. (1996) [9]	Theoretical analysis	Scouring depth of jet in air	${\displaystyle \frac{r_{m\infty }}{h}}=11E_c^{0.65}{\left({\displaystyle \frac{\Delta \rho }{\rho }}+1\right)}^{6.6}/{\left({\displaystyle \frac{\Delta \rho }{\rho }}+1\right)}^{6.6}$
Yeh et al. (2009) [10]	Experimental data fitting	Depth and width of the scour hole	${\displaystyle \frac{r_1}{h}}={\displaystyle \frac{s}{1930.5+1.07s}}+0.48$ ${\displaystyle \frac{\Delta }{h}}={\displaystyle \frac{s-4.0}{1200.0+12.10(s-4.0)}}$
Taştan et al. (2016) [8]	Theoretical analysis and model verification	Scour depth having angles between 30° and 60°	${\displaystyle \frac{h_m}{H}}=0.73{\displaystyle \frac{F_0}{H/D_{0J}}}-0.1$
Fan et al. (2020) [12]	Theoretical analysis and model verification	Depth and width of the sediment resuspension	${\displaystyle \frac{V}{H^3}}=C_7{\displaystyle \frac{u_b^3}{\sqrt{{\left(g\frac{{\rho }_s-{\rho }_f}{{\rho }_f}D\right)}^3}}}-C_8{\displaystyle \frac{u_b^2}{g\frac{{\rho }_s-{\rho }_f}{{\rho }_f}D}}$
Pham et al. (2022) [13]	Experimental data fitting	Static scour depth	${\displaystyle \frac{{\epsilon }_{m{\infty }^{\prime }}}{h}}=0.87\left(1.26E_c^{0.11}-1\right)$
Liu et al. (2025) [3]	Regression from experimental data	Scour depth	$d^{*}=\alpha {\left(\ln \left(1+{\displaystyle \frac{t\mu }{\rho D^2}}\right)\times {\left({\displaystyle \frac{\lambda p_{\mathrm{w}}}{s_u}}\right)}^3\right)}^{\beta }\times {\left({\displaystyle \frac{\pi \rho D^2{v}_0^2}{4h^2{s}_u}}\right)}^{\gamma }$

). Their results indicated that the minimum scour distance exhibits an increasing trend with increasing density Froude number and jet velocity. An increase in tailwater depth leads to a reduction in scour distance until the maximum critical depth of the scour hole is reached; beyond this depth, the distance increases again. The optimal jetting angle depends on the jet velocity: 20° for velocities of 60 and 80 L/min, increasing to 40° for velocities of 100 and 120 L/min.

Despite these advancements, certain limitations remain in existing studies. Traditional numerical models fail to adequately resolve the complex interactions involved in water–sediment two-phase flows, such as turbulent dynamics and particle collisions, thereby leading to significant discrepancies between model predictions and in situ field conditions. Furthermore, existing scour hole prediction models rely heavily on idealized assumptions (e.g., uniform sediment distribution, steady flow), limiting their generalizability. The Rouse number model proposed by Sutherland et al. [11] has demonstrated remarkable effectiveness in characterizing vertical jet scouring. However, the model encounters notable limitations when applied to scenarios involving non-uniform sediments or dynamic flow conditions. Similarly, empirical formulas from European and American case studies (e.g., linear relationships between jet velocity and scour depth) exhibit significant errors in complex terrains.

To address the aforementioned limitations, this study begins by simulating the underwater jet scouring process using the CFD–DEM coupling methodology. The reliability of the developed numerical model in replicating the dynamic evolution of scour holes generated by submerged jets impinging on sand beds is systematically validated using a comparative analysis with established physical experimental data. Subsequently, key physical parameters are defined, and theoretical hypotheses regarding their interdependencies are proposed. Drawing upon the sediment incipient motion theory, empirical jet diffusion models, and the principle of energy conservation, a formula for predicting scour hole depth is systematically derived. Similarly, by integrating critical incipient motion conditions, jet velocity attenuation profiles, and the mass conservation law, a corresponding formula for estimating scour hole width is formulated. To verify the proposed hypotheses and establish empirical correlations for the key physical parameters, numerical simulations are conducted under various scenarios involving different nozzle diameters and jet velocities. Finally, the model-predicted values are juxtaposed against the simulation results to comprehensively evaluate the model’s predictive accuracy.

2. Numerical Model

2.1. Governing Equations for Fluid

Computational fluid dynamics (CFD) solves fluid flow problems using numerical methods. Its core lies in the spatiotemporal discretization of continuous physical quantities (velocity, pressure, temperature, etc.) into variables at finite discrete points. By discretizing the governing equations of fluid motion, algebraic equation systems are established and ultimately solved to obtain approximate solutions [16]. CFD analysis is based on the three fundamental conservation laws of fluid mechanics. The mass conservation law states that the total mass of a fluid system remains constant over any given time period. For fluid in a control volume, mass conservation can be mathematically expressed as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·(\rho v)=0$$

(3)

where $\rho$ represents the fluid density, t denotes time, v is the fluid velocity vector, and $\nabla ·\left(\rho v\right)$ describes the net mass flow per unit volume.

The momentum conservation law states that the total momentum of an isolated system of objects remains constant. For a control volume occupied by a fluid or other continuous medium, the law can be mathematically expressed as follows:

$${\displaystyle \frac{\partial (\rho v)}{\partial t}}+\nabla ·(\rho vv)=-\nabla p+\mu {\nabla }^{2}v+f$$

(4)

where $p$ is pressure, $\mu$ is the dynamic viscosity, and $f$ represents external forces (e.g., gravity, inertial forces).

In practical applications, turbulent flow is one of the primary characteristics in water jet problems. The multi-scale vortex structures inherent to turbulence demand more intricate descriptions of flow dynamics. Consequently, turbulence models (e.g., RANS, LES, or RNG $k-\epsilon$ models) are typically incorporated into the governing equations, enabling more accurate simulations of complex flow regimes.

2.2. Turbulence Model

Turbulence numerical simulation primarily employs two approaches: direct numerical simulation (DNS) and Reynolds-averaged Navier–Stokes (RANS) methods. The former directly solves the Navier–Stokes equations to resolve all turbulence scales but is restricted to fundamental research due to prohibitive computational costs. The latter, by applying time-averaging, drastically reduces computational demands and has become the cornerstone of engineering turbulence simulations [17]. The RNG k-ε model, by incorporating additional turbulence generation terms, exhibits enhanced capability in addressing high-Reynolds-number flows and rapidly evolving flow fields. This model is particularly well-suited for simulating complex flow regimes at high Reynolds numbers, including particle–gas coupling, gas–solid interactions, and high-speed flow scenarios [18].

Therefore, this study adopts the RNG k-ε model for fluid-particle coupled turbulence simulations. Turbulent kinetic energy equation (k equation) describes the changes in turbulent energy, including the influences of turbulent generation, dissipation, and transport. The equation of turbulent kinetic energy in the RNG k-ε model is as follows:

$${\displaystyle \frac{\partial (\rho k)}{\partial t}}+\nabla ·(\rho vk)=\nabla ·\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right)\nabla k\right]+P_k-\epsilon +{\varphi }_k$$

(5)

where $\rho$ is the fluid density, $k$ is the turbulent kinetic energy, $\mu$ is molecular dynamic viscosity, ${\mu }_t$ is turbulent viscosity, ${\sigma }_k$ is turbulent kinetic energy diffusion coefficient (typically taken as 1.0), $P_k$ is turbulence generation term (generation of turbulent energy), $\epsilon$ is turbulence dissipation rate, ${\varphi }_k$ is source term or additional term.

Turbulence dissipation rate equation. $\epsilon$ describes the process of turbulent energy dissipation. In the RNG k-ε model, a correction term related to the generation of turbulent kinetic energy has been added:

$${\displaystyle \frac{\partial (\rho \epsilon )}{\partial \tau }}+\nabla \times (\rho \nu \epsilon )=\nabla \left[\left(\mu +{\displaystyle \frac{{\mu }_{\tau }}{{\sigma }_{\epsilon }}}\right)\nabla \epsilon \right]+C_{1\epsilon }{\displaystyle \frac{\epsilon }{\kappa }}{\pi }_{\kappa }-C_{2\epsilon }{\displaystyle \frac{{\epsilon }^2}{\kappa }}+{\varphi }_{\epsilon }$$

(6)

where ${\sigma }_{\epsilon }$ is turbulent dissipation rate diffusion coefficient (typically taken as 1.3), $C_{1\epsilon }$ and $C_{2\epsilon }$ are model constants, ${\varphi }_{\epsilon }$ is source term or additional term.

The turbulent viscosity is calculated based on the turbulent kinetic energy and dissipation rate, and it describes the influence of turbulence in the fluid on the flow. The expression of the turbulent viscosity is as follows:

$${\mu }_t=\rho C_{\mu }{\displaystyle \frac{k^2}{\epsilon }}$$

(7)

where $C_{\mu }$ is model constant (typically taken as 0.9).

The source term ${\varphi }_k$ serving as a source term describes the evolution of non-equilibrium turbulence. ${\varphi }_{\epsilon }$ is a supplementary term that refines ε-prediction via coupling with turbulent kinetic energy k-generation, leveraging constants $C_{1\epsilon }$ and $C_{2\epsilon }$ to adjust energy balance in dynamically evolving flow regimes. The expression is as follows:

$${\varphi }_k=\rho C_{\mu }{\displaystyle \frac{{\eta }^3\left(1-\eta /{\eta }_0\right)}{1+\beta {\eta }^4}}·{\displaystyle \frac{{\epsilon }^2}{k}}$$

(8)

$${\varphi }_{\epsilon }=C_{\epsilon 1}{\displaystyle \frac{\epsilon }{k}}{\varphi }_k-C_{\epsilon 2}{\displaystyle \frac{{\eta }^3\left(1-\eta /{\eta }_0\right)}{1+\beta {\eta }^4}}·{\displaystyle \frac{{\epsilon }^3}{k^2}}$$

(9)

where turbulent strain rate parameter $\eta =sk/\epsilon$, s is the strain rate magnitude. ${\eta }_0$ is the critical value parameter (taken as 4.38), and $\beta$ is the stability parameter (taken as 0.012).

2.3. Governing Equations for Particles

In the study of sediment flow problems [19], the discrete element method (DEM) is widely applied to characterize the behavior of granular flows. Sediment particles are treated as assemblies of discrete small particles that play a critical role in their mutual interactions and motion [20,21,22,23,24]. The motion of particles adheres to Newton’s second law of motion, with the governing equations comprising translational and rotational components. By integrating these equations with the principle of momentum conservation, they effectively describe the movement of particles within a fluid medium [25]. The specific governing equations are as follows:

$$m_i{\displaystyle \frac{d\overrightarrow{v_i}}{dt}}=\overline{F_{if}}+{\displaystyle \sum _{j=1}^{n_i}\overline{F_{ij}}}$$

(10)

$$I_i{\displaystyle \frac{d\overline{{\omega }_i}}{dt}}=\overline{M_{if}}+{\displaystyle \sum _{j=1}^{n_i}\overline{M_{ij}}}$$

(11)

where $m_i$ is the mass of particle $i$, $v_i$ is the translational velocity vector of particle $i$, $\overline{F_{ij}}$ is the contact force exerted on particle $i$ by particle $j$, $\overline{F_{if}}$ is the fluid-particle interaction force acting on particle $i$, $I_i$ is the moment of inertia of particle $i$, $\overline{{\omega }_i}$ is the angular velocity of particle $i$, $\overline{M_{ij}}$ is the torque exerted on particle $i$ by particle $j$, and $\overline{M_{if}}$ is the fluid-induced torque acting on particle $i$.

The Hertz–Mindlin contact model is selected to characterize particle contact behavior. This model demonstrates strong versatility, simultaneously accounting for elastic deformation and frictional effects of particles, which is particularly essential for simulating sediment granular flow processes. When the distance between the centers of two particles with radius $R_i$ and $R_j$ exceeds the sum of their radius $\left(R_i+R_j\right)$, no contact force is generated. Conversely, when the distance is less than the sum of their radii, contact forces and torques arise between the particles. The Hertz–Mindlin model calculates these contact forces and torques using the following formulas:

$$F_n=k_n{\delta }^{3/2}$$

(12)

$$k_n={\displaystyle \frac{4}{3}}{E}^{*}\sqrt{R^{*}}$$

(13)

$$E^{*}={\left({\displaystyle \frac{1-v_1^2}{E_1}}+{\displaystyle \frac{1-v_2^2}{E_2}}\right)}^{-1}$$

(14)

$$R^{*}={\left({\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_2}}\right)}^{-1}$$

(15)

where $F_n$ is the normal force, $k_n$ is the normal elastic coefficient, $\delta$ is the overlap distance between particles, $E^{*}$ is the equivalent elastic modulus (calculated from the elastic moduli $E_1$ and $E_2$, and Poisson’s ratios ${\nu }_1$ and ${\nu }_2$ of the two particle materials), and $R^{*}$ is the equivalent radius (derived from the curvature radii $R_1$ and $R_2$ of the contacting particles).

Tangential force can be mathematically expressed as follows:

$$F_t=k_t{\delta }^{1/2}$$

(16)

$$k_t={\displaystyle \frac{8G^{*}\sqrt{R^{*}}}{2-v_1-v_2}}$$

(17)

$$G^{*}={\left({\displaystyle \frac{2-v_1}{G_1}}+{\displaystyle \frac{2-v_2}{G_2}}\right)}^{-1}$$

(18)

where $F_t$ is the tangential force, $k_t$ is the normal elastic coefficient, $G^{*}$ is the equivalent shear modulus (calculated from the shear moduli $G_1$ and $G_2$ of the two particle materials), and $G_1$ and $G_2$ are the shear moduli of the two particle materials. The normal moment can be expressed as noted in Equation (19):

$$M_n=r_n\times F_n$$

(19)

where $M_n$ is the normal moment (rotational component perpendicular to the contact surface), and $r_n$ is the normal contact radius (characteristic radius of the contact area under normal loading).

The tangential moment can be mathematically expressed as follows:

$$M_t=r_t\times F_t$$

(20)

where $M_t$ is the tangential moment, and $r_t$ is the normal contact radius.

The model proposed by Zhu et al. [26] is employed to calculate the rolling resistance. This model accounts for the nonlinear effects of interparticle contacts, enabling more accurate simulation of interactions between particles and walls. The model is expressed as follows:

$$T_r={\mu }_r{F}_{n}R$$

(21)

In the equation, $T_r$ represents the rolling resistance torque of the particle, ${\mu }_r$ is the rolling resistance coefficient, $F_n$ denotes the normal contact force between particles, and R refers to the effective radius of the particle.

2.4. Solid–Liquid Interphase Force Models

In solid–liquid two-phase flow simulation, the fluid–particle interaction force serves as the critical link bridging CFD and DEM. To characterize the two-phase coupling mechanism, this study employs the following interphase force models: the drag force (F_drag) is described using the Schiller–Naumann model, which is applicable for particle Reynolds numbers Re < 1000, with its expression given as follows:

$$F_{\mathrm{drag}}={\displaystyle \frac{1}{2}}{C}_D{A}_p{\rho }_f\left|v_f-v_p\right|(v_f-v_p)$$

(22)

where C_D is the drag coefficient, and the calculation formula is as follows:

$$C_D=\left\{\begin{array}{ll}{\displaystyle \frac{24}{R{e}_p}}\left(1+0.15R{e}_p^{0.687}\right),\hfill & R{e}_p<1000\hfill \\ 0.44,\hfill & R{e}_p\ge 1000\hfill \end{array}\right.$$

(23)

where A_p is particle projection area, ${\rho }_f$ is fluid density, and $v_f$ and $v_p$ represent the velocities of the fluid and particles, respectively.

The added mass force (F_added) accounts for the inertial force exerted on particles during fluid acceleration, with its expression given by:

$$F_{\mathrm{added}}={\displaystyle \frac{\pi }{6}}{\rho }_f{d}_p^3{\displaystyle \frac{D(v_f-v_p)}{Dt}}$$

(24)

where d_p is the particle diameter, D/Dt is the bulk derivative.

When the fluid velocity gradient is substantial (e.g., during the initial stage of jet impact), the influence of this force on particle motion is significant [27]. The Saffman lift force (F_saffman) describes the lateral lift force exerted on particles due to velocity gradients, applies to laminar shear flow, and its expression is as follows:

$$F_{\mathrm{saffman}}=1.61{\rho }_f^{0.5}{\mu }_f^{0.25}{d}_p^{1.5}{\left|\nabla v_f\right|}^{0.5}(v_f-v_p)$$

(25)

where ${\mu }_f$ is the hydrodynamic viscosity, $\nabla v_f$ is the fluid velocity gradient.

Near the jet shear layer, accurate prediction of particle lateral migration dynamics critically depends on the interphase force model; this study specifically accounts for the force arising from hydrodynamic hysteresis in the unsteady flow, which captures the history-dependent effects of relative acceleration, expressed as:

$$F_{\mathrm{basset}}=3\pi {\mu }_f{d}_p\sqrt{{\displaystyle \frac{{\rho }_f}{\pi }}}{\displaystyle {\int }_0^t\frac{d(v_f-v_p)/d\tau }{\sqrt{t-\tau }}}d\tau$$

(26)

where $\tau$ is the integration variable, representing a time instant between 0 and t (s).

3. Model Validation

3.1. Model Establishment

This study employs the software Cradle CFD (Cradle CFD|scFLOW) for research purposes. To validate the reliability of numerical simulation results, the underwater scouring process is reproduced by referencing the submerged jet scouring experiment conducted by Chen et al. [28], as illustrated in Figure 1. The numerical model is designed with the following parameters: length L = 1.3 m, a height H = 0.9 m, and a sand bed thickness C = 0.4 m. The width of the jet nozzle in the middle is D = 25 mm, the standoff distance of the submerged jet is h = 0.15 m, and the jet flow velocity is v = 5 m/s. The sand particles have a diameter of 1.8 mm and a density of 2650 kg/m³. This study focuses on two-dimensional scour in the flow direction and vertical direction.

3.2. Boundary Conditions

The computational domain is defined in a three-dimensional Cartesian coordinate system. The X-direction is assigned as a wall boundary, the Y-direction as a symmetry plane boundary, and the Z-direction has the top face configured as a velocity inlet (vertically downward at 5 m/s) and the bottom face as a wall boundary. The RNG k-ε turbulence model and a pressure-based solver are adopted. Pressure–velocity coupling is achieved using a SIMPLEC–PISO hybrid algorithm, which balances correction efficiency with iterative stability. A hybrid-order discretization scheme is employed. The convective terms are discretized using a second-order upwind scheme to ensure numerical stability, whereas the diffusive terms are treated with a central difference scheme to preserve accuracy. This configuration effectively balances the convergence behavior and computational accuracy requirements for simulating complex flow fields.

3.3. Validation Results

The scour duration is set to 10 s, focusing on the scour morphology, depth, and width. The reliability of the model is validated by comparing numerical results with experimental data. Numerical simulation results at three time points (t = 0.04 s, 20 s, and 600 s) during the jet scouring process are selected and compared with the submerged jet scouring experiments conducted by Chen et al. [28]. At t = 0, the jet initiates impingement on the sand bed surface, inducing migration of bed particles via bed-load movement (Figure 2). Over the subsequent 20 s, the scour hole undergoes rapid expansion in both vertical and radial dimensions, with copious sediment particles entrained from the cavity and deposited as lateral sand ridges. Equilibrium is attained at approximately t = 600 s, wherein no further suspended particles are ejected from the developed scour hole, and the scour depth stabilizes at a near-constant value. As shown in Figure 3, the numerical simulation results of this study are in good agreement with the physical model results in reproducing the morphology of scour holes, with an average error of 4.3% compared to the physical model.

Numerical simulation tests are carried out to verify the mesh independence study and time-step sensitivity, using different mesh size (2 mm, 4 mm, 6 mm, 8 mm, 10 mm) and time steps $\Delta t$ (10⁻⁴ s, 10⁻⁵ s, 10⁻⁶ s). It is found that when the time step is set to 10⁻⁶ s (Figure 4), the relative errors for scour depth are below 5%, satisfying the CFL stability condition. Furthermore, when the minimum mesh size is 10 mm, the error between the simulated and experimental values is 21.85%. Although this case yields the shortest computation time and the highest efficiency, the error exceeds 20%. When the minimum mesh size is 2 mm, the error between the simulated and experimental values is 3.31%, which is the smallest (Figure 4). Therefore, the mesh size is selected as 2 mm for subsequent numerical simulations. This analysis confirms that the numerical simulation results are affected by grid resolution and temporal resolution, and by conducting sensitivity analysis, the reliability of the model has been enhanced.

4. Scour Hole Morphology Prediction Model

4.1. Key Physical Parameters and Model Assumptions

Numerical simulation of vertical scouring of sand bed by water jets, combined with relevant studies [28,29], demonstrates that higher values of the average shear rate ($\overline{\tau }$), maximum shear rate (${\tau }_{\max }$), maximum shear velocity ($v_{*\max }$), and average shear velocity ($v_{*\mathrm{avg}}$) are associated with superior scouring effects. Therefore, these four key physical parameters are adopted as indicators for evaluating dredging effectiveness. However, it remains unclear which parameter primarily controls the scour depth and which factor dominantly governs the scour width. The following work integrates relevant research to conduct in-depth theoretical analyses, thereby clarifying the specific relationships among key physical quantities, initial conditions, and scour hole morphology. To facilitate the derivation of formulas, two approximate assumptions are proposed:

(a)

The maximum shear velocity $v_{*\max }$ is correlated with the jet velocity v and average shear velocity $v_{*avg}$, i.e., $v_{*\max }=f(v,v_{*avg})$.

(b)

The shear stress $\overline{\tau }$ depends on ${\tau }_{\max }$ and ${\tau }_{\mathrm{c}}$. When ${\tau }_{\max }>{\tau }_c$, it is assumed that $\overline{\tau }=f({\tau }_{\max }-{\tau }_c)$.

4.2. Derivation of Scour Hole Depth Prediction Formula

The maximum shear stress and critical shear stress are first derived based on sediment incipient motion theory [30]. Sediment incipient motion refers to the critical process by which bed surface sediment particles transition from a stationary state to a mobile state under hydrodynamic forces. When the shear stress generated by the flow surpasses the resistance of sediment particles, these particles start to slide, roll, or become suspended. The shear stress associated with this critical transition is referred to as the critical shear stress (${\tau }_c$), whereas the maximum shear stress that the flow can generate under given hydraulic conditions is defined as the maximum shear stress [31]. The prerequisite conditions for sediment incipient motion are noted as follows:

$${\tau }_{\max }=\rho ·v_{*\max }^2>{\tau }_c$$

(27)

where $\rho$ is the density of water, taken as 1000 kg/m³, and $v_{*\max }$ is the maximum shear velocity (m/s).

The critical shear stress for sediment can be obtained from the Shields curve as follows:

$${\tau }_c={\theta }_s({\rho }_s-\rho )g{d}_s$$

(28)

where $d_s$ is the sediment particle diameter (m), and ${\theta }_s$ is the Shields parameter corresponding to the sediment particle characteristics. For coarser sediment particles, ${\theta }_s$ typically ranges between 0.05 and 0.06.

According to relevant studies [32] on hydrodynamics and sediment transport mechanics, a significant nonlinear empirical relationship exists between the scour hole diffusion diameter and the jet travel distance when a jet acts on riverbed sediments. Based on regression analyses of laboratory flume experiments [27] and field observation data, this empirical relationship can be expressed as follows:

$$d=2.4·d_0·{\left({\displaystyle \frac{x}{d_0}}\right)}^{0.5}$$

(29)

where $d$ and $d_0$ represent the jet diffusion diameter and nozzle diameter, respectively.

The maximum shear velocity at the riverbed induced by a jet flow is a key parameter characterizing the flow’s ability to dislodge sediment [33]. Using empirical Equation (29), the maximum shear velocity acting on the bed surface at a jet distance x can be calculated as follows:

$$v_{*\max }={\displaystyle \frac{6.2·d_0·v_0}{x}}$$

(30)

where v₀ denotes the velocity at the nozzle outlet, namely the non-attenuated velocity.

Despite simplifying the actual scour profile to an inverted conical shape and overlooking irregularities caused by sediment inhomogeneity, stratification, or non-uniform flow fields, the geometric assumption that jet-induced scour holes are inverted cones remains widely adopted in numerous experimental studies of physical models and derivations of theoretical equation [12,27]. Using the previously derived scour hole depth h and diffusion diameter d, an approximate formula for calculating the scour hole volume V can be established as follows:

$$V={\displaystyle \frac{\pi d^{2}h}{4}}$$

(31)

Finally, according to the principle of energy conservation during jet scouring, the residual kinetic energy of the jet after traveling a distance x is partially converted into energy consumed by detaching sediment particles, overcoming the bed’s critical shear strength, and transporting dislodged sediment [34]. By integrating the previously derived diffusion diameter D and maximum shear velocity, a predictive formula for the scour hole depth h can be established as follows:

$$E_k={\displaystyle \frac{1}{2}}m{v}_{*avg}^2={\displaystyle \frac{1}{2}}\rho Qt{v}_{*avg}^2={\displaystyle \frac{1}{2}}\rho {\displaystyle \frac{\pi d^2}{4}}{v}_{*avg}^{3}t$$

(32)

where t is the scour hole formation time (s), and $v_{*avg}$ is the mean shear velocity at the sediment surface upon jet impact (m/s). Equation (32) simplifies the energy distribution process by assuming that jet energy is solely utilized for sediment detachment and transport, while overlooking turbulence and seepage that are common in natural bed layers with high porosity or high permeability. Thus, this assumption is applicable to consolidated deposited sediments where seepage effects are negligible.

During sediment incipient motion, the work performed by the sediment in the scour hole to overcome gravity is expressed as follows:

$$W_g={\gamma }_{s}V={\displaystyle \frac{{\gamma }_s\pi d^{2}h}{4}}$$

(33)

where the unit weight of sediment is ${\gamma }_s={\rho }_{s}g$.

Assuming the average shear stress on the scour hole wall is $\overline{\tau }$, the work performed to overcome frictional forces is expressed as follows:

$$W_f=\overline{\tau }\times S$$

(34)

where $S$ denotes the lateral surface area of the scour hole ($S=\pi dh$).

Based on the energy conservation principle, further derivation yields the following predictive equation for scour hole depth h:

$$h={\displaystyle \frac{1}{2}}·{\displaystyle \frac{\rho \pi d{v}_{*avg}^{3}t}{{\gamma }_s\pi d+4\pi \overline{\tau }}}$$

(35)

where h is the height of the corresponding cylinder. The average shear force is $\overline{\tau }=f({\tau }_{\max }-{\tau }_c)$, ${\tau }_c={\theta }_s({\rho }_s-\rho )g{d}_s$, and ${\tau }_{\max }=\rho ·v_{*\max }^2$.

4.3. Derivation of the Scour Hole Width Prediction Formula

As the depth of the scour hole increases, the diffusion angle of the high-pressure water jet also expands, leading to a larger effective diameter of sediment affected by the jet dredging [35]. Consequently, the diffusion process is subjected to greater hydrodynamic resistance from the surrounding water. At the point where $\rho ·v_{*\max }^2\left(x\right)\le {\tau }_c$, the sediment cannot reach incipient motion. For sediment of a known grain size, the critical shear stress can be determined from the Shields curve [36,37]. Therefore, the average jet velocity at the terminal point of the jet must satisfy the following:

$$\rho ·v_{*avg}^2(x)={\theta }_s({\rho }_s-\rho )g{d}_s$$

(36)

At a height ($x=x_0+h$) from the jet outlet, where the jet radius becomes d(x), the diameter of the scour hole can be determined based on the theoretical derivation of the Gaussian distribution and the principle of mass conservation as follows:

$$d^2(x)={\displaystyle \frac{d^2{v}_{*avg}}{v_{*avg}(x)}}$$

(37)

The derivation of the expressions for scour hole depth and width is highly complex; however, this derivation incorporates a broader range of practical factors, resulting in broader applicability. Additionally, because of the two assumptions and intricate coefficient relationships involved in the formula derivation process, the parameters in the resulting formulas require further approximation using numerical methods.

4.4. Assumption Verification

To ensure the accuracy of the formulas, numerical simulations are employed below to validate some of the previously mentioned assumptions and derive the corresponding empirical coefficients. The first assumption is $v_{*\max }=f(v,v_{*avg})$. Because adjusting the nozzle diameter can alter the jet velocity, submerged jet models with various nozzle diameters are simulated to establish the relationship between jet velocity and maximum/average shear velocities [38]. With the scouring time fixed at 8 s and other factors held constant, nozzle diameters are set to 30 mm, 40 mm, 50 mm, 75 mm, and 100 mm. The corresponding jet velocities, maximum shear velocities, and average shear velocities are presented in

Table 2. Comparison of velocity parameters under different operating conditions.

Nozzle Diameter	D30 mm	D40 mm	D50 mm	D75 mm	D100 mm
Nozzle jet velocity v (m/s)	23.58	13.26	8.492	3.774	2.123
Maximum shear velocity $v_{*\max }$ (m/s)	0.20	0.16	0.13	0.09	0.06
Average shear velocity $v_{*avg}$ (m/s)	0.091	0.071	0.058	0.0409	0.030

.

As shown in Table 2, under constant flow rate conditions, as the nozzle diameter increases, the initial nozzle jet velocity, maximum shear velocity, and average shear velocity all decrease progressively. Furthermore, a certain positive correlation exists among the jet velocity, maximum shear velocity, and average shear velocity. Two empirical formulas are derived using a multi-factor fitting method, which is expressed in the following form:

$$f_1(v,v_{*avg})=2.906\times v_{*avg}-0.00176\times v-0.023$$

(38)

$$f_2(v,v_{*avg})=0.00573\times \ln (v_{*vag})+0.0138$$

(39)

where v represents the initial jet velocity, $v_{*avg}$ represents the average shear velocity.

Data analysis reveals that both formulas can effectively predict the maximum shear velocity, as shown in Figure 5, validating the accuracy of the empirical formulas. The correlation coefficients ($R_1^2=0.999$, $R_2^2=0.993$) and significance levels (p < 0.01) for Equations (38) and (39) show high reliability. Given the higher accuracy of Equation (38), it will be adopted for calculations in the subsequent validation of the prediction model.

The second assumption is $\overline{\tau }=f({\tau }_{\max },{\tau }_c)$. Additionally, when the flow velocity at the sediment surface is less than the incipient motion velocity of the sediment, sediment transport ceases [39,40]. This implies that variations in jet velocity can induce changes in both the maximum and average shear stresses. Using the operating parameters from Table 2, the cross-sectional scour hole profiles under different conditions are compared in Figure 5. From the comparative scour hole cross-sectional shapes shown in Figure 6, it is evident that as the jet velocity progressively increases, the scour depth continues to deepen, and the scour width expands correspondingly. This phenomenon demonstrates that both the maximum and average shear stresses increase progressively as the jet velocity increases.

The simulation is configured with a sand grain size of 1.8 mm and a density of 2650 kg/m³. The critical shear stress of the sediment is determined as 1.94 Pa using the Shields curve. Parameters corresponding to different operating conditions are listed in

Table 3. Comparison of shear stress parameters under different operating conditions.

Physical Parameters	V1	V2	V3	V4	V5
Jet velocity (m/s)	23.58	13.26	8.492	3.774	2.123
Critical shear stress ${\tau }_c$ (Pa)	1.94	1.94	1.94	1.94	1.94
Maximum shear stress ${\tau }_{\max }$ (Pa)	41.18	25.27	15.85	7.59	3.22
Average shear stress $\overline{\tau }$ (Pa)	10.936	6.501	4.077	2.083	1.051

.

As shown in Figure 7, the average shear stress of the submerged jet scouring sediment exhibits a linear relationship with the difference between the maximum shear stress and the critical shear stress. Data analysis reveals a correlation coefficient $R^2$ of 0.98, indicating high reliability of the fitted formula. Using a multi-factor fitting method, the relationship is determined as follows:

$$\overline{\tau }=f({\tau }_{\max },{\tau }_c)=0.2607({\tau }_{\max }-{\tau }_c)+0.5811$$

(40)

where $\overline{\tau }$, ${\tau }_{\max }$, and ${\tau }_c$ represent the average shear stress, maximum shear stress, and critical shear stress, respectively.

4.5. Prediction Model Validation

To validate the accuracy of the scour hole prediction model, the submerged jet distance (i.e., the vertical distance from the nozzle exit to the sediment surface) is further adjusted, and the results of numerical simulations are compared and analyzed against formula-derived predictions.

Under constant particle parameters and jet velocity, simulations are conducted for six jet distances: h = 0.2 m, 0.25 m, 0.3 m, 0.35 m, 0.4 m, and 0.45 m. The scouring duration remained fixed at 8 s. A comparative analysis of the cross-sectional profiles of the scour holes under these conditions is presented in Figure 8.

As shown in Figure 8, the submerged jet distance significantly affects the scouring of the sand bed. When the target distance decreases, the velocity gradient within the jet core region intensifies, and the energy concentration effect strengthens. This rapidly elevates the bed shear stress beyond the critical threshold for sediment incipient motion, forming a high-efficiency erosion zone. In this scenario, the high-energy jet directly impacts surface sediment, inducing particle suspension and transport, thereby expanding both scour depth and lateral extent. Conversely, increasing the target distance exacerbates energy dissipation along the jet trajectory. Turbulent mixing and viscous resistance significantly attenuate the effective impact force, leading to a more diffuse bed shear stress distribution and reduced scouring efficiency [41]. Therefore, a smaller jet target distance amplifies the hydrodynamic forces acting on the bed sediment. This enhances near-bed water flow mobilization, allowing the sediment to reach its critical incipient velocity and triggering particle motion.

Figure 9 shows the cross-sectional profiles of scour holes formed under different jet target distances after 8 s of scouring. Observing the scour hole shapes under varying jet target distances reveals that their morphology undergoes certain changes depending on the target distance. Although the scour holes generally maintain an inverted conical shape, the trends in scour depth and scour width are inversely related. Specifically, the maximum scour depth induced by the nozzle jet decreases as the jet target distance increases, whereas the maximum scour width increases with greater target distances.

As observed in Figure 10, the scour hole depth gradually decreases as the submerged jet distance increases, whereas the scour width exhibits a corresponding increase. At a jet target distance of 0.2 m, the predicted maximum scour depth is 0.148 m (simulated value 0.152 m), and the predicted maximum scour width is 0.279 m (simulated value 0.242 m). When the jet target distance increases to 0.45 mm, the predicted maximum scour depth decreases to 0.108 m (simulated value 0.115 m), whereas the predicted maximum scour width increases to 0.365 m (simulated value 0.41 m). When the jet target distance increases by a factor of 2.25, the scour depth decreases by approximately 24%, whereas the scour width increases by 1.3 times. This demonstrates that the scour width is more sensitive to changes in the jet target distance compared to the scour depth. The predicted scour depths are consistently slightly smaller than the simulated values, with a maximum deviation of 7%, indicating close agreement. In contrast, the predicted scour widths show larger fluctuations, with a maximum relative error of 13%. In comparison, the prediction model exhibits higher prediction accuracy for scour hole depth than for scour width. The sources of error may stem from the inverted cone assumption failing to capture asymmetry, jet diffusion inadequately reflecting turbulence, and the width being affected by ambient flows. Furthermore, according to PIANC’s standards [42] for tolerance of errors in dredging projects, industry practice typically sets a tolerance of ±15% for errors in key dredging dimensions, and this is critical for avoiding over-dredging or under-dredging. Therefore, the 13% error in width and 7% error in depth of the prediction model in this study are consistent with industry standards for dredging operations.

5. Conclusions

This study focuses on the prediction of submerged-jet-induced scour hole morphology, with key findings and research advancements summarized below.

A submerged jet scouring numerical model is constructed based on the CFD–DEM coupling method. Validation against physical experimental data demonstrated an average relative error of 4.3% between simulated and experimental scour depths, confirming the model’s reliability in capturing fluid–solid coupling mechanisms, boundary condition settings, and turbulence parameter calibration. This model serves as an effective tool for visualizing jet scouring processes.

Through theoretical derivation and simplifying assumptions, key physical parameters (average shear rate, maximum shear rate, maximum shear velocity, and average shear velocity) obtained from numerical simulations are correlated with jet characteristics (nozzle diameter, jet velocity, and jet distance), and prediction models for scour depth and width are established. Numerical simulations are employed to validate the assumed conditions and optimize the formula coefficients, thereby ensuring scientific rigor and reliability. Comparative analysis with results indicates that the maximum discrepancy between the model predictions and CFD-simulated values for scouring hole depth is within 7%. Although the relative error in width prediction reaches 13%, it still complies with engineering accuracy standards.

Future work should further validate the prediction model by integrating physical model tests, verify the model using more practical engineering cases, and strengthen the theoretical foundation of underwater jet scouring.