Weakly Compressible Smoothed Particle Hydrodynamics Research Articles

Comparative study on volume conservation among various SPH models for flows of different levels of violence

The weakly-compressible smoothed particle hydrodynamics (WCSPH) models offer distinct advantages in simulating free-surface flow. Within the rapidly evolving WCSPH field, Classical WCSPH model, based on the artificial viscous term, and its derived models, which additionally introduce numerical diffusive terms into the continuity equations - including the δ-SPH, δ-SPHC, δ-SPHC+, δ-LES-SPH, and δ-SPHF models have gained prominence recently. The numerical diffusive term is found to alter the particle distributions in the free-surface region, leading to a change in the mean water level that reflects the volume conservation, which might result in inaccuracies in some simulations. This study evaluates the volume conservations of the aforementioned WCSPH models for flows at quasi-hydrostatic states evolved from varying levels of flow violence under prolonged simulations, through a series of numerical tests involving hydrostatic, standing-wave, and dam-break cases. For the hydrostatic and standing-wave cases, the performance of different types of WCSPH models remains consistent overall. The Classical WCSPH and δ-LES-SPH models exhibit minimal changes in volume, with almost no decrease in the mean water level occurring. The δ-SPH, δ-SPHC, δ-SPHC + models demonstrate decreases in the mean water level converging to around 0.2 times the SPH particle diameter (Δx), indicating favorable volume conservation. Conversely, the δ-SPHF model consistently exhibits a decrease in the mean water level exceeding 1 Δx at t = 2000s in a linear fashion, resulting in a noticeable reduction in volume. Under the dam-break case, most of the mean water levels simulated by different models experience a small increase, which is close to the decrease observed under the hydrostatic and standing-wave cases with corresponding models. Only the δ-SPHF model continues to show a continuous decreasing tendency. In summary, considering volume conservation, the δ-LES-SPH model demonstrates the best performance (excluding the Classical WCSPH model as it cannot simulate violent flow), followed by the δ-SPHC, δ-SPHC+, δ-SPH and δ-SPHF models.

Comparative study of WCSPH, EISPH and explicit incompressible-compressible SPH (EICSPH) for multi-phase flow with high density difference

This study presents three Smoothed Particle Hydrodynamics (SPH) methods capable of handling high-density differences in violent incompressible multiphase flows. The conventional Weakly Compressible SPH (WCSPH) is reformulated into a quasi-Lagrangian framework based on Arbitrary Lagrangian–Eulerian (ALE) context. The Explicit Incompressible SPH (EISPH) method is extended to handle multiphase flows and reformulated in the ALE framework. The Explicit Incompressible-Compressible SPH (EICSPH) method, which can handle both compressible and incompressible phases, is developed by combining these two approaches in a fully explicit algorithm. The proposed methods are validated and compared through four benchmark problems including Rayleigh-Taylor instability, hydrostatic, liquid sloshing and dam break problems. Firstly, the stability and accuracy of interface modeling of the three methods were verified through Rayleigh-Taylor instability with small density differences. In the hydrostatic problem with a large density difference, the results from all three methods exhibit good agreement in the pressure distribution. Particularly, EICSPH demonstrates the faster convergence when compared to WCSPH, with EISPH showing fastest convergence overall. In the transient sloshing problem, all three methods quantitatively converged to the experimental results and exhibited good agreement, although slight pressure noise was observed in WCSPH initially. Finally, in the dam break problem, all three methods successfully simulated sharp interfaces without non-physical voids. The temporal variations of pressure and height were well predicted by all three methods, with EISPH showing better conformity to the water-air interface morphology obtained from other incompressible numerical methods compared to WCSPH and EICSPH. Additionally, the impact of air speed of sound was examined in EICSPH, where while the difference in pressure variation at the probe was minimal, differences were observed in energy dissipation and the progression of the free surface due to the air cushion effect.

Numerical simulation of water wave propagation using DualSPHysics

In this study, the study of water waves was carried out using one of major the mesh-free CFD i.e. Smoothed Particle Hydrodynamics (SPH). DualSPHysics ver 5.0 which is an open-source SPH solver based on Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) was used to reproduce three dimension of water waves in large and medium numerical wave tanks (NWT). This study analyses water wave profiles and the accuracy of DualSPHysics for regular waves with period 1.15 second and two NWT were used, i.e., 35 and 15 m. In addition, advanced post-processing using VisualSPHysics was used to mimic water waves in real conditions. Single water depth was used, i.e. 0.4 meters for large and medium NWT. The results showed SPH has limitations for reproducing large NWT. However medium NWT successfully reproduce water wave with advanced post-processing using VisualSPHysics.

Numerical investigations on water entry and/or exit problems using a multi-resolution Delta-plus-SPH model with TIC

As a classical Fluid-Structure Interaction (FSI) problem, water entry and/or exit of marine structures involve large deformations of free-surface and splashing, which have long been a great challenge for numerical modeling. Smoothed Particle Hydrodynamics (SPH) is a meshless Lagrangian particle method that has natural advantages in dealing with the non-linear variation of free-surface and moving interfaces. However, the characteristics of weakly-compressible SPH itself may lead to deficiencies such as pressure noise, tensile instability, heterogeneous particle distributions and computational inefficiency. Therefore, in this work, an acoustic damper term is formulated in the momentum equation to eliminate the amount of acoustic pressure waves induced by liquid impacts. Meanwhile, the Adaptive Particle Refinement (APR) is applied to obtain sufficiently fine particle resolution and reduce the computational cost. For the tensile instability issues induced by negative pressures, especially at the water exit stage, the Tensile Instability Control (TIC) is adopted into the SPH simulations to suppress its adverse impact. Furthermore, the Particle Shifting Technique (PST) is beneficial to the two schemes above, and it is also employed in this study. That is to say, PST can effectively solve the problem of particle disorder, which increases the accuracy and robustness of APR and reduces the momentum conservation error caused by TIC. Finally, we propose an improved Free-Surface Detection Method (FSDM) to allow the flow separation from the structure surface at the water entry stage that is impacted by PST. The fairly good agreements between the numerical results and the experimental data indicate the present SPH model can be treated as a reliable numerical tool for accurately solving such problems of structures penetrating fluid interfaces.

On the partition noise in chosen particle weighted methods and its consequences for weakly-compressible flow models

The Meshless Finite Volume (MFV) method, which is a member of the particle weighted methods (PWM) family, is analysed in the context of the so-called partition noise problem. This issue is inherent to PWM-type discretisation which is based on compact support kernel functions and particle sampling. The partition noise exhibits itself as spurious oscillations observed in the solution; they are caused by unphysical source terms in the discretised equations and approximate local volumes computation. Unfortunately, decreasing the discrete length scale which controls the spatial resolution reduces the partition noise only weakly (and never completely). When choosing numerical schemes, a special care is required since in naive implementations the partition noise may even lead to loss of stability. However, even when stability is assured, PWM does not fulfil the assumptions of the Lax equivalence theorem (i.e. the discretised equations are not consistent), therefore convergence in the classical meaning cannot be guaranteed.As an example of a problematic use of MFV, we investigate its application to the solution of fluid flow equations at low Mach numbers, aiming to mimic truly incompressible flow conditions. Taken here as isothermal Euler equations, they are the basis of the so-called weakly-compressible (WC) flow models. Such modelling does not involve any elliptic-type equation and, due to the algorithmic and computational reasons, seems to be well-suited for particle methods. The WC flow model imposes, however, a serious challenge for PWM since: (i) the method is not gauge invariant and the disparity between the speed of sound and the convective velocity scale causes a significant spurious source of momentum; (ii) the inaccuracy in the local volumes approximation leads to errors in the computed density which are comparable to physically relevant slight variations, inherent to the WC approach. We propose specific numerical and modelling measures: controlled relaxation of initial particle distribution, particle volumes computation using an optimal kernel function, a choice of gradient limiting techniques and a carefully chosen background pressure. Although these improvements significantly increase the accuracy (however at remarkably increased computational cost), the theoretical 2nd-order convergence is achieved only at lower spatial resolutions when the truncation errors dominate. The accuracy of mesh-based methods is achieved in PWM only for the pressure field; the errors in the velocity field are much larger and convergence is at most 1st-order. We point some contradictory requirements of the model equations and the numerical method. One of the conclusions is that it is doubtful to construct a convergent truly Lagrangean method of PWM type at reasonable computational cost. The present findings also apply to conservative variants of the weakly-compressible Smoothed Particle Hydrodynamics which, being first order consistent in the best case, suffers from the partition noise even more severely than the considered MFV approach which is theoretically second order consistent. The conclusions pose a question about a rationale for the use of WC-PWM approach in chosen applications, in particular free surface and interfacial flows.

Multiscale Smoothed Particle Hydrodynamics based on a domain-decomposition strategy

A multi-resolution algorithm for weakly-compressible Smoothed Particle Hydrodynamics is hereby proposed. The approach chosen is based on a domain decomposition to subdivide the computational domain into regions with different resolutions. Each sub-problem is closed by appropriate Dirichlet boundary conditions that are enforced via buffer regions, populated by particles whose physical quantities are obtained by means of an interpolation over adjacent sub-domains.The algorithm has been implemented into the DualSPHysics open-source code and it has been tested and validated through a series of different study cases. The capability of the numerical scheme to simulate multiscale fluid flow has been demonstrated by solving the flow past a cylinder for a Reynolds number of 9,500 and a ratio between the largest and smallest particle size equal to 28. Furthermore, the proposed SPH multi-resolution algorithm can also be used for flow around moving objects, such as an oscillating cylinder in cross-flow, and free-surface flow, such as the simulation of a triangular wedge impacting on the free surface of a quiescent liquid.

An efficient correction method in Riemann SPH for the simulation of general free surface flows

In this paper, a new efficient weighted correction method for weakly-compressible smoothed particle hydrodynamics is proposed and successfully implemented in the simulation of violent free-surface flow exhibiting breaking and impact events for the first time. It is well known that the original kernel gradient correction (KGC) encounters numerical instability resulting from matrix inversion. The present method remedies this issue by introducing a weighted average of the KGC matrix and the identity matrix, other than directly applying the KGC matrix, to achieve numerical stability meanwhile decrease numerical dissipation. To ensure momentum conservation, the correction is implemented in a particle-average pattern by rewriting the pressure term of the Riemann solution. Furthermore, the proposed weighted KGC scheme is incorporated into the dual-criteria time-stepping framework developed by Zhang et al. (2020) to achieve optimized computational efficiency. A set of numerical examples in both two- and three-dimensions are investigated to demonstrate that the present method can significantly reduce numerical dissipation meanwhile exhibit a smooth pressure field for general free-surface flows.

SPH Modeling of Tsunami Wave Overtopping over the Reef-Flat with a Vertical Revetment

Artificial revetments are commonly constructed on reef-flats to protect the rear infrastructure. Evaluating the performance of a revetment under extreme wave conditions requires accurate prediction of wave overtopping. To simulate tsunami wave overtopping on an artificial revetment above a coral reef-flat, a numerical model based on the weakly compressible smoothed particle hydrodynamics (WCSPH) method is developed. This numerical model includes an additional dissipative term in the continuity equation and a dynamic boundary condition enhancement to correct the pressure distortion from particles at the boundary. Reef topography is simplified as a steep reef-face and a horizontal reef-flat, and tsunami waves are described as solitary waves. This research explores the characteristics of solitary wave overtopping on the revetment above the reef-flat with varying slope angles and reef-flat lengths.

A Lagrangian free-stream boundary condition for weakly compressible smoothed particle hydrodynamics

This paper proposes the first Lagrangian free-stream boundary condition for weakly-compressible smoothed particle hydrodynamics (WCSPH), significantly decreasing the domain-size effect for simulating flows around static or moving geometries and increasing computational efficiency. The boundary condition is implemented based on several numerical techniques, as follows. First, a spatio-temporal identification and an ensuring technique precisely identify surface particles without misjudgment. Second, the far-field corrections of density and velocity precisely target surface particles. Third, the improved in- and outlet boundaries guarantee the smooth inflow profile and free-outflow particles without the issue of the truncated kernel. The test cases of flows over a flat plate and around a circular cylinder have validated the accuracy and computational efficiency of the present method. In order to demonstrate its applicability, the method has also been tested with more complex examples, such as a fluid-structure interaction (FSI) problem and a 3-D flow around a sphere.

Smoothed Particle Hydrodynamics-Based Study of 3D Confined Microflows

In this study, we investigate the performance of the smoothed particle hydrodynamics (SPH) method regarding the computation of confined flows in microchannels. Modeling and numerical simulation with SPH involve the representation of flowing matter as distinct mass points, leading to particle discretization of the Navier–Stokes equations. The computational methodology exhibits similarities with other well-established particle methods, such as molecular dynamics (MD), dissipative particle dynamics (DPD), and smooth dissipative particle dynamics (SDPD). SPH has been extensively tested in the simulation of free-surface flows. However, studies on the performance of the method in internal flow computations are limited. In this work, we study flows in microchannels of variable cross-sections with a weakly compressible SPH formulation. After preliminary studies of flows in straight constant cross-section ducts, we focus on channels with sudden expansion and/or contraction. Flow models based on periodic or various inlet/outlet boundary conditions and their implementations are discussed in the context of 2D and 3D simulations. Numerical experiments are conducted to evaluate the accuracy of the method in terms of flowrate, velocity profiles, and wall shear stress. The relation between f and Re for constant cross-section channels is computed with excellent accuracy. SPH captured the flow characteristics and achieved very good accuracy. Compressibility effects due to the weakly compressible smoothed particle hydrodynamics (WCSPH) formulation are negligible for the flows considered. Several typical difficulties and pitfalls in the application of the SPH method in closed conduits are highlighted as well as some of the immediate needs for the method’s improvement.

Hydrodynamic investigation on the submerged tunnel suspended from a fixed platform using SPH method

In this paper, the coupled dynamic motion characteristics of a tunnel element suspended from a platform have been studied at a 1:50 scaled model by using the weakly compressible Smoothed Particle Hydrodynamic (WCSPH) method. The corresponding numerical model was validated with the experimental published data of the motion characteristics of the tunnel suspended from a fixed platform, and the case of a floating structure with its mooring lines. Multiple linear wave conditions and different sinking depths of the submerged were involved in the numerical model for tunnel dynamic response and cable behavior investigation. Effects of the mooring system, the wave parameters, the tunnel sinking depth and its mooring configurations were studied. The numerical results were arranged to inform about the dynamic characteristics and mooring behavior for different case studies. The results indicate that the tunnel motions decrease with the increment of the tunnel sinking depth and decrease with the decreasing mooring angle. The natural periods of the tunnel-platform system play key roles in the wave condition impacts on the tunnel motions. It was found that the additional mooring system anchoring the tunnel to seabed played a specific role in reducing the tunnel motions in roll and sway, the motion amplitude of the tunnel in heave is mainly controlled by the suspension cables, particularly when subjected to lower sinking depth and longer period waves.

SPHydro: Promoting smoothed particle hydrodynamics method toward extensive applications in ocean engineering

This paper aims at presenting a general-purpose-oriented and fully parallelized meshless framework to simulate complex Fluid–Structure Interaction (FSI) problems in ocean engineering. In this framework, a Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) solver is combined with several advanced pre- and post-processing techniques. Based on the framework, we have been developing our in-house WCSPH-FSI package named SPHydro for solving hydrodynamic problems involving complex FSI processes in an accurate, efficient, and convenient manner. Three benchmarks are performed to qualitatively and quantitatively validate the accuracy and convergence of SPHydro. In addition, several practical applications are also provided to further highlight the generality and applicability of SPHydro in ocean engineering simulations. It is demonstrated that SPHydro holds satisfactory performance in solving complex FSI problems in ocean engineering and that the present framework can be further developed to tackle more complex FSI problems for general engineering applications due to its high flexibility and extensibility.

An implementation of the SPH method and its application to two-dimensional dam break cases

A basic implementation of the weakly compressible smoothed particle hydrodynamics (W-SPH) method is used to model the open surface flow phenomena of two different types of hydraulic problems. They were both two-dimensional approximations of known hydraulics problems: dam break over dry bed and dam break over wet bed with shallow and deep variants. For all models, the results were compared quantitatively with data from waterfront profiles from previously published experiments. A qualitative approach was used to assess the general profile shape comparing the numerical hydraulic profile results with published photographs of all setups shown. The numerical model converged to a stable solution with relatively low-resolution setups (2 000 to 20 000 particles) and a good correlation was found with the experimental and numerical references, even without the application of correction algorithms. Numerical oscillations in the density field did produce small artificial waves and vortices, but did not interfere with the behavior of the bulk of the fluid except for some delay in the speed of surface waves. The choice of developing an entirely original, self-contained source code, although very insightful, does mean that the neighboring particle search algorithm used considerably increments the associated computational cost. However, using this insight along with the basis of the comparison with experimental results, optimization priorities for the source code were found and some guidelines to choose fluid resolution and boundary density can be applied for future cases where the flow phenomena are similar to those studied. Also, the experience of writing the SPH source code that was used to solve the numerical models, shows the path for future developments on fluid flow modeling using this method.

Wave generation and wave–structure impact modelling with WCSPH

The reliable prediction of plunging wave induced load, which represents a challenging task in offshore engineering, is of key relevance for the safety of existing offshore structures as well for the reliable design of new ones. This work shows the early advances in the development of a FOSS derived model, obtained by independently introducing relevant modifications in the original model SPHERA v.9.0.0 (RSE SpA) based on the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This derived model is validated on single-phase laboratory flume tests from technical literature (i.e., dam break and regular waves) and on a laboratory flume test of plunging wave impacting an offshore structure. The derived model allows modelling two-phase flows with a large density ratio, as shown by an additional two-phase dam break simulation. Therefore it seems promising for application to two-phase plunging impact to improve the prediction of impact load on the structure.

How to train your solver: Verification of boundary conditions for smoothed particle hydrodynamics

The weakly compressible smoothed particle hydrodynamics (WCSPH) method has been employed to simulate various physical phenomena involving fluids and solids. Various methods have been proposed to implement the solid wall as well as inlet and outlet boundary conditions. However, error estimation and the formal rates of convergence for these methods have not been carefully discussed or examined. In this paper, we use the method of manufactured solutions to verify the convergence properties of a variety of commonly employed solid, inlet, and outlet boundary implementations. In order to perform this study, we propose various manufactured solutions for different domains. On the basis of the convergence offered by these methods, we systematically propose a convergent WCSPH scheme along with suitable methods for implementing the boundary conditions. We also demonstrate the accuracy of the proposed scheme by using it to solve the flow past a circular cylinder. Along with other recent developments in the use of adaptive resolution, this paves the way for accurate and efficient simulation of incompressible or weakly compressible fluid flows using the SPH method.

Numerical simulation of drop deformation under simple shear flow of Giesekus fluids by SPH

PurposeThe purpose of this study is to investigate the effects of the shear-thinning viscoelastic behavior of the surrounding matrix on droplet deformation by weakly compressible smoothed particle hydrodynamics (WC-SPH). Also, the effect of the presence of another droplet is examined.Design/methodology/approachA modified consistent weakly compressible SPH method is proposed. After code verification, a complete parameter study is performed for a drop under the simple shear flow of a Giesekus liquid. The investigated parameters are 0.048≤Ca ≤ 14.4, 0.1≤c ≤ 10, 0.04≤De ≤ 10, 0≤α ≤ 1 and 0.12≤Re ≤ 12.FindingsIt is demonstrated that the rheological behavior of the surrounding fluid could dramatically affect the droplet deformation. It is shown that the droplet deformation is increased by increasing Re and Ca. In contrast, the droplet deformation is decreased by increasing a, De and polymer content. Also, it is indicated the presence of another droplet could drastically affect the flow field, and the primary stress difference (N1) is resonated between two droplets.Originality/valueThe main originality of this paper is to introduce a new consistent WC-SPH algorithm. The proposed method is very versatile for tackling the shear-thinning viscoelastic multiphase problems. Furthermore, a complete parameter study is performed for a drop under the simple shear flow of Giesekus liquid. Another novelty of the current paper is studying the effect of the presence of a second droplet. To the best of the authors’ knowledge, this is performed for the first time.

A coupled SPH-PD model for fluid–structure interaction in an irregular channel flow considering the structural failure

Understanding fluid–structure interaction (FSI) is important because it dominates diverse natural phenomena and engineering problems. This paper presents an integrated particle model for FSI problems involving irregular channel flows and crack propagation in structures. The proposed model is implemented as follows: (1) we couple weakly compressible smoothed particle hydrodynamics (WCSPH) with bond-based peridynamics (BBPD) in a partitioned approach (this framework has a much simpler algorithm than the previously reported SPH-PD method); (2) we propose a novel periodic boundary conditions (PBCs) algorithm for modeling flows in non-regular channels; and (3) we incorporate crack propagation in structural responses under fluid dynamics, which was rarely considered in previous works. The new framework has been validated and illustrated to be effective and versatile in diverse FSI problems, including hydrostatic pressure-induced solid deformation, violent free-surface flows and channel flows interacting with flexible structures. Compared with conventional grid-based methods, this particle framework is more user-friendly, since no extra effort is required to update meshes, even when a discontinuity appears during the modeling process. The extendibility and potential of this framework is further demonstrated by the simulation of fluid-driven deformation and crack propagation in elastomers.

The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations

The choice of a time integration scheme is a crucial aspect of any transient fluid simulation, and Smoothed-Particle Hydrodynamics (SPH) is no exception. The influence of the time integration scheme on energy balance is here addressed. To do so, explicit expressions allowing to compute the deviations from the energy balance, induced by the time integration scheme, are provided. These expressions, computed a posteriori, are valid for different integration methods. Besides, a new formulation that improves energy conservation by enhancing stability, based on an implicit integration scheme, is proposed. Such formulation is tested with the simulation of a two-dimensional non-viscous impact of two jets, with no artificial dissipation terms. To the best of our knowledge, this is the first stable simulation of a non-dissipative system with a weakly-compressible SPH method. A viscous case, the Taylor–Green vortex, has also been simulated. Results show that an implicit time integration scheme also behaves better in a viscous context.

Techniques for second-order convergent weakly compressible smoothed particle hydrodynamics schemes without boundaries

Despite the many advances in the use of weakly compressible smoothed particle hydrodynamics (SPH) for the simulation of incompressible fluid flow, it is still challenging to obtain second-order convergence even for simple periodic domains. In this paper, we perform a systematic numerical study of convergence and accuracy of kernel-based approximation, discretization operators, and weakly compressible SPH (WCSPH) schemes. We explore the origins of the errors and issues preventing second-order convergence despite having a periodic domain. Based on the study, we propose several new variations of the basic WCSPH scheme that are all second-order accurate. Additionally, we investigate the linear and angular momentum conservation property of the WCSPH schemes. Our results show that one may construct accurate WCSPH schemes that demonstrate second-order convergence through a judicious choice of kernel, smoothing length, and discretization operators in the discretization of the governing equations.

Study of the weakly-compressible SPH method for improving pressure distribution of violent fluid-structure impact flows

Abstract When a traditional Weakly-Compressible Smoothed Particle Hydrodynamics (WCSPH) model is used to simulate free surface flow with a large Reynolds number, an unstable numerical calculation due to high random pressure oscillations will result, while an accurate pressure field is of vital significance for simulating violent fluid-structure interactions. Riemann-based SPH and Delta-SPH are widely used to solve this problem. In this paper, to enhance computational efficiency, the SPH method is implemented on a General Processing Unit (GPU) platform using Compute Unified Device Architecture (CUDA). Parallelized SPH programs including the standard SPH method, Riemann-based SPH and Delta-SPH are verified by a dam break model with large Reynolds number and violent deformation of free surfaces. The results show that all SPH methods can vividly reflect the whole process of splashing, rolling and backward jet flow; both the Riemann-based SPH and the Delta-SPH methods are effective in alleviating the problem of inhomogeneous pressure distribution in the simulation process; the Riemann-based SPH method has better stability even with relatively large particle spacing, and it has higher accuracy in simulating impact pressure. When the number of particles reaches 100,000, compared with a single-thread Central Processing Unit (CPU) implementation, the speedups obtained with NVIDIA Titan V with high computing cores and Quadro K2200 with low computing cores are thousands and hundreds, respectively.