Enhanced weakly-compressible MPS method for violent free-surface flows: Role of particle regularization techniques
Graphical abstract
Introduction
Continuum-based particle methods have been developed and applied for solving many scientific and engineering problems (see e.g., [1], [2], [3], [4]). The Lagrangian nature of these approaches has attracted many research projects involving complex fluid flow physics. In contrast to the traditional Eulerian methods identified with semi- or fixed grids, the moving particles carrying the flow and material properties provide an effective computational tool for capturing extreme flows and interface deformations. Nevertheless, the numerical stability and convergence of these particle methods are still open problems that require robust stabilization techniques (e.g., [5], [6]).
Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) (initially developed by Gingold and Monaghan [7] and Koshizuka and Oka [8], respectively) are two popular particle methods for fluid flow problems. Shakibaeinia and Jin [9] introduced the Weakly Compressible MPS (WC-MPS) method, with the equation of state being the function of the particle number density. This method has been the focus of many studies, particularly for simulating hydro-environmental problems (e.g., see [10], [11] for free-surface and multiphase fluid flows, and [12], [13] for sediment dynamic modeling). Jandaghian and Shakibaeinia [14] recently proposed a new version of the WC-MPS approach, denoted as the Enhanced WC-MPS (EWC-MPS). The EWC-MPS method, as a continuity-based model, employs a diffusive term and particle regularization techniques to reduce/eliminate numerical instability. In the present study, we further improve this method and investigate its stability and convergence for the long-term modeling of violent free-surface flows by introducing several new enhancements.
Numerical instabilities in particle methods mainly arise from errors due to the discretization methods and the Lagrangian motion, which lead to non-uniform particle distributions recognized as noisy pressure fields, particle clustering, and tensile instability. These drawbacks of the particle methods not only cause numerical instabilities but also affect the modeling accuracy, particularly in regions with high-pressure and/or tension forces and extreme flow deformations [15]. To decrease the high-frequency unphysical pressure fluctuations, particle methods implement higher-order operators (e.g., [16], [17], [18]) and artificial viscosity/diffusion terms (e.g., [19], [20]). Moreover, particle regularization techniques have been developed to assure uniform particle distributions and overcome the common pairing instabilities. Among these regularization techniques, the pair-wise Particle Collision (PC) (e.g., [11], [21], [22]), the Particle Shifting (PS) (e.g., [5], [23], [24], [25], [26]), and the transport-velocity formulations (e.g., [27], [28], [29]) approaches have shown to be promising in surmounting these instabilities by regulating the particles.
Unphysical high-frequency pressure noises, as the common numerical issue of the weakly compressible SPH and MPS methods, originate from the particle approximation and the kernel truncation errors [14], [20]. To reduce these pressure fluctuations, Monaghan and Gingold [19] inserted an artificial viscosity term into the momentum equation of the weakly compressible SPH method for shock simulations. The artificial viscosity term has shown to be an effective numerical technique for eliminating the pressure fluctuations and ensuring numerical stability, and is being implemented in well-developed SPH models (e.g., see [5], [30], [31], [32]). Molteni and Colagrossi [20] introduced a novel diffusive term defined as the Laplacian of the density field in the continuity equation to surmount the pressure oscillations in the weakly compressible SPH method. Antuono et al. [33] proposed the convergent form of this diffusive term using high-order Laplacian and gradient operators. In the context of the weakly compressible MPS, Jandaghian and Shakibaeinia [14] derived a new version of this diffusive term as a function of the particle number density for free-surface flows. Furthermore, high-order approximation operators have been employed to increase the accuracy of the numerical calculations and remove the pressure noises [17]. Oger et al. [16] improved the SPH method through a renormalization-based formulation for increasing the accuracy of the gradient approximations. Similarly, Khayyer and Gotoh [17] represented a first-order accuracy pressure gradient equation in the incompressible MPS framework. Duan et al. [18] implemented a second-order corrective matrix and showed that in their incompressible MPS model the stabilization errors are more dominant compared to the truncation errors. Jandaghian and Shakibaeinia [14] by comparing a conservative pressure gradient operator versus the non-conservative high-order pressure force illustrated that respecting the conservation properties of the system plays an important role in the numerical stability of the weakly compressible MPS method.
To increase the numerical stability, the particle methods adopt the PC technique developed based on the concept of the momentum transfer in the collision of physical solid or gas particles. This approach utilizes empirical coefficients to indirectly apply a repulsive force between a pair of colliding particles. Lee et al. [21] represented the PC technique as a particle regularization technique in the MPS formulations for simulating violent free-surface flows and impact loads. Shakibaeinia and Jin [12] developed the PC formulation for simulating sediment dynamic problems in the context of the multiphase weakly compressible MPS method. Further, Shakibaeinia and Jin [11] extended the application of the PC method to high-density ratio multiphase flows, and Xu and Jin [22] validated this technique as the particle stabilization approach of the weakly compressible MPS method for free-surface flows involving impact events.
Moreover, PS techniques aim at eliminating particle-clustering and tensile instability by moving particles to the area with less particle concentration. In the context of the incompressible SPH method, Xu et al. [23] originally implemented the PS method for solving internal flows. Lind et al. [24] represented the PS approach based on Fick's law of diffusion and corrected the PS vector for free-surface flows. Shadloo et al. [34] adopted the Lennard-Jones repulsive force to derive the PS vector and validated the improved SPH method for simulating rapid fluid flows over solid bodies. Zainali et al. [35] employed this PS formulation for multiphase incompressible SPH simulations. Khayyer et al. [25] proposed an optimized version of the PS scheme to enhance the performance of the incompressible SPH method for interfacial flows. Mokos et al. [36] developed the PS technique for multiphase violent fluid flows in the weakly compressible SPH framework. Duan et al. [26] implemented the PS technique within an incompressible multiphase MPS model considering the free-surface corrections of the PS vector. Sun et al. [37] appended the PS technique to the weakly compressible SPH method with special treatments of particles in the free-surface region. Further, Sun et al. [5] developed the consistent form of this PS method by including additional diffusive terms of the PS transport velocity within the governing equations. Jandaghian and Shakibaeinia [14] proposed the corrected-PS algorithm coupled with the PC technique (i.e., the hybrid method denoted as CPS+PC) for their enhanced weakly compressible MPS method, and the developed model was shown to be effective and essential for avoiding unphysical fluid fragmentations.
Violent free-surface flows are particularly challenging for the well-developed particle methods for capturing long-term mechanical behaviors (see e.g., [29], [38]). These complex flows are usually associated with breaking waves, impact events, and water sloshing, and are characterized by high Reynolds numbers, highly non-linear deformations, and fluid-fluid and fluid-solid impacts [30]. Hence, increasing stability and reducing the approximation errors of the particle methods while conserving the global momentum/energy of the system become critical issues for simulating these problems. For this purpose, in addition to employing higher-order formulations and diffusive terms, eliminating the particle-pairing instabilities requires rigorous particle regularization techniques [5]. The PS techniques (e.g., [36], [38]), transport-velocity algorithms (e.g., [29], [39]), and the PC method (e.g., [21], [22]) have been specifically developed and utilized to surmount the numerical instability of violent free-surface flows. Nevertheless, despite their success in improving the particle distributions, many existing particle-shifting methods and the associated free-surface detection algorithms introduce some new challenges in simulating such complex flows (e.g., [6], [14]). Particularly in weakly compressible particle methods, where the potential energy is dominant and several breaking events occur (e.g., water dam break and sloshing problems), the continuous shifting of the particles leads to an unphysical expansion of the fluid field [5], [32]. Further developments to overcome these numerical issues involve implementing more accurate particle classification algorithms and treatment of the free-surface particles (e.g., [6], [25], [32]). Moreover, including some additional advection terms in the governing equations may dissipate the excessive potential energy inserted via the particle-shifting technique [5]. On the other hand, the conservative PC technique has advantages over PS because of its simplicity and being free from many exceptions and boundary treatments; however, by using empirical coefficients, this technique has been shown to be less effective for highly violent flows (see e.g., [21], [22]).
The main objective of this paper is to propose a robust numerical tool based on the EWC-MPS method for simulating the long-term stability of violent free-surface flows while predicting the system's global energy. To accomplish that, we propose a conservative form of the three-dimensional EWC-MPS method by employing a higher-order gradient and Laplacian operators (Section 3.1). Moreover, we implement a modified diffusive term in the MPS framework (based on the formulation of [33]) to reduce the high-frequency pressure fluctuations and kernel truncation errors. Regarding the particle regularization techniques, we investigate the role and impact of PC and PS techniques on stability, accuracy, and conservation properties, then to address the issues with these techniques we propose and evaluate a new dynamic PC approach (Section 3.2.1) and a consistent form of the CPS technique (Section 3.2.2). By simulating and validating two-dimensional dam-breaks and three-dimensional water sloshing and obstacle impacts, we investigate and compare the performance of the proposed developments for the long-term stability of relevant problems (Section 4).
Section snippets
Governing equations
The mathematical model governs the fluid flows through the momentum and mass conservation laws. The Lagrangian form of the governing equations in the continuum mechanics is as follows [40]: In this system, the continuity equation updates the fluid density, ρ, by the divergence of the velocity, , stated as the mass volume expansion rate. The momentum equation updates the velocity vector, v, used to move the position of the material points, r. The gradient of
The discrete system with the conservation properties
The particle methods discretize the computational domain, Ω, into particles that represent the fluid phase, , and the solid walls, (i.e. ). In the EWC-MPS method using the symbolic notation, the approximation operator, , forms the differential equations (1) for a target particle i into: in which the model calculates the particle number density, , (as the substitute for the density, ) via the continuity equation supplied
Numerical results and discussions
In this study, we simulate challenging violent free-surface flow benchmark cases to evaluate the role of the proposed enhancements. The cases include the two-dimensional (2D) water dam-break and three-dimensional (3D) sloshing and obstacle impact problems. Simulation videos are avaiable in the supplementary materials.
Conclusion
We developed and validated the three-dimensional EWC-MPS method for simulating violent free-surface flows. In a conservative framework of governing equations, we introduced several enhancement techniques that are shown to be essential for dealing with fluid-fluid and fluid-solid impact events. To include the turbulence shear force, we adopted the SPS scheme with a higher-order gradient operator to estimate the magnitude of the strain rate tensor. Furthermore, we employed a convergent form of
CRediT authorship contribution statement
Mojtaba Jandaghian: Conceptualization, Methodology, Software, Formal analysis, Visualization, Validation, Writing – original draft, Writing – Review & Editing. Abdelkader Krimi: Conceptualization, Methodology, Writing – review & editing. Amir Reza Zarrati: Conceptualization, Methodology, Writing – review & editing. Ahmad Shakibaeinia: Supervision, Conceptualization, Methodology, Analysis, Writing – review & editing, Funding acquisition, Project administration.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) (Grant number: RGPIN-06308-2017) and Polytechnique Montréal, Canada. This study utilized the high-performance computing resources of Compute Canada and Calcul Quebec. The authors also acknowledge Nvidia for supporting this research with their GPU Grant Program.
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