Abstract: Smoothed Particle Hydrodynamics (SPH) is a particle-based Lagrangian numerical method, particularly suitable for simulating free surface flows of fluids with different constitutive equations. In this study, an SPH model was developed to solve the flow problem of granular materials in a dense regime, adopting the constitutive equations proposed by Pouliquen et al. and Jop et al. By introducing the weak compressibility of granular materials, the difficulty of resolving flows near static conditions was overcome: this approach requires solving a third-order algebraic equation to calculate the density of the granular material.For any viscoplastic fluid with a variable yield stress, a new viscosity regularization method was developed in this study to achieve accurate reproduction of flow stoppage and static conditions.The model was validated by solving uniform granular flow problems with known analytical solutions and by reproducingLubeet al.'s experimental data on granular column collapse. Introduction: Numerical simulation of free surface flows is generally a complex task—although grid-based methods such as the volume of fluid method have been successfully proposed in the past, meshless particle methods appear to offer a more effective approach to handling free surface problems. Among various particle methods, Smoothed Particle Hydrodynamics (SPH) has proven to be widely applicable. The method was originally proposed by Gingold and Monaghan and by Lucy for astrophysical applications, and was later applied to free surface flow simulation by Monaghan, Ferrari et al., and Violeau and Issa, among others. The SPH method has significant advantages in implementing different constitutive equations: Rodriguez-Paz and Bonet, Laigle et al., Pasculli et al., and Minatti and Pasculli employed Bingham-type fluids to implement SPH simulation of debris flows, whileBuiet al. used elastoplastic relationships to simulate soil failure flows. A comprehensive review of the SPH method and application examples can be found in the works of Monaghan, Liu and Liu, and Violeau. This study aims to leverage the above-mentioned characteristics of the SPH method to develop a numerical model for simulating complex free surface dense granular flows.Granular flows have a wide range of applications in industry, from pharmaceuticals to the semiconductor industry (according to Richard et al., granular materials are the second most handled material after water), and are also of great concern in civil engineering: in recent years, increasing attention has been paid to the disaster risks caused by solid materials carried by rivers during extreme events. At present, the scientific community has not yet established a universal constitutive equation for granular materials applicable to all flow types. However, various constitutive equations have been proposed in existing research, each of which can accurately reproduce a specific class of granular flow phenomena. The current study focuses on the constitutive equations proposed by Pouliquen et al. and Jop et al., which are applicable to a wide range of granular flow types dominated by friction in the dense regime. The SPH model proposed in this paper adopts a novel pressure calculation method, which is crucial for solving flow problems at low deformation strain rates. The innovation of this method lies in introducing the weak compressibility of granular materials and specifically optimizing it for the constitutive equation system studied.In addition, the SPH model features a novel viscosity regularization method, applicable to any viscoplastic fluid with a variable yield stress. Chapter 2 will present the governing equations describing the physical processes of the system and the constitutive equations adopted; Chapter 3 provides the general framework of the SPH method; Chapter 4 details the proposed SPH model; Chapter 5 validates the model by reproducing simple surface flow test cases with known analytical steady-state solutions and by reproducing the measured data from Lube et al.'s granular material collapse experiments; finally, Chapter 6 provides concluding remarks on the model and its performance. Figures:
Figure 1: Curve of relative error versus i arising from neglecting the 2.61d/H term in Equation 18. All curves are plotted under the conditions µ2/µs ≈ 1.5 and i0 = 2.65.
Figure 2: Schematic of the virtual particle method proposed by Morris et al.
Figure 3: Viscosity–strain rate relationship curves for the exact Bingham model and the Papanastasiou model according to Equations 45 and 46. In the figure, ˙γ0 = τy/η0, mp = b/˙γ0.
Figure 4: Relative error curves for the regularized viscosity using the Papanastasiou method in the constitutive equation of this study. The three curves differ in the value of the yield stress τy, which is set to be b orders of magnitude larger than the unit scale stress τ0 (τy = bτ0). The parameter mp is chosen such that atshear rate˙γ = 0, the viscosity value is one order of magnitude larger than the residual viscosity η0.
Figure 5: Schematic of uniform granular flow down an inclined plane.
Figure 6: Steady-state velocity profiles of simulated granular flow down an inclined plane. The solid lines represent analytical solutions. The left figure uses no artificial viscosity (α = 0), while the right figure uses artificial viscosity with α = 0.05. Steady state was reached after approximately 6.50 s of simulation time in both cases.
Figure 7: Steady-state pressure profiles of simulated granular flow down an inclined plane. The dashed line represents the analytical solution for incompressible granular material, and the solid line represents the analytical solution for compressible granular material using the equation of state. No artificial viscosity was used in this simulation (α = 0).
Figure 8: Steady-state particle distribution of granular flow down an inclined plane. Particles are color-coded by vx value, and the black dashed line indicates the fluid flow region. Particles below the line are boundary particles, and the side particles are standard particles replicated to implement periodic boundary conditions.
Figure 9: Schematic of the granular column collapse experimental setup and parameter definitions.
Figure 10: Front position–time relationship for granular column collapse simulations. ∆r represents the advance distance of the front relative to the initial position before column collapse.
Figure 11: Particle distribution of granular column collapse simulation under the condition a = 0.90. Particles are color-coded by velocity magnitude, and the reference simulation time is labeled in each figure. The black frame is used to track the initial shape of the deposit.
Figure 12: Particle distribution of granular column collapse simulation under the condition a = 2.75. Particles are color-coded by velocity magnitude, and the reference simulation time is labeled in each figure. The black frame is used to track the initial shape of the deposit.
Figure 13: Comparison between the scaling law of numerical simulation of sand column collapse and the experimental scaling law of Lube et al.
Conclusions: This paper proposes an SPH model for solving granular flow problems in a dense regime.The model adopts the constitutive equations proposed by Pouliquen et al. and Jop et al., which were specifically established for the dense flow regime. The study employs the Weakly Compressible SPH method (WCSPH), which can effectively reflect the characteristics of the constitutive equations—the equations predict that the bulk density of granular material will gradually decrease as flow intensity increases. To solve the pressure estimation problem that arises under near-static conditions, it is necessary to introduce the weak compressibility of granular materials. This measure introduces additional unknowns into the problem, and the algorithm needs to solve a third-order algebraic equation for each particle at each time step to calculate the density of the granular material. This paper proposes a novel viscosity regularization method for simulating near-static and flow stoppage conditions. The method is applicable to any viscoplastic model with a variable yield stress and calculates the regularized viscosity by solving a first-order ordinary differential equation.Practice has proven that this method can effectively reproduce near-static and flow stoppage states with reasonable accuracy. The SPH model was validated by comparing simulation results with analytical solutions for uniform flow on an inclined plane and by reproducing Lube et al.'s granular column collapse experimental data.In the dense flow regime, the model's calculation results agree well with known analytical solutions and observational data, demonstrating that the model can effectively handle various characteristics of the constitutive equations, such as spatially varying viscosity, pressure-dependent yield stress, and singular behavior under static conditions. Regarding extensions of the model, future upgrades can be made by introducing constitutive equations applicable to the collisional flow regime—in which particle spacing is larger than in the dense regime and flow intensity is generally higher.For conditions with larger aspect ratio a values, the model can still be validated using Lube et al.'s experimental data. References:
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