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A diffuse-interface smoothed particle hydrodynamics method for compressible multiphase flows
Abstract
This study incorporates the well-known diffuse-interface concept into the meshless smoothed particle hydrodynamics (SPH) method for compressible multiphase flow simulations. The diffuse-interface SPH method combines the merits of the diffuse-interface and SPH methods in treating material interface. The material interface location is explicitly tracked because of the Lagrangian nature of SPH. On the other hand, introducing the diffuse-interface method helps reduce the artificial numerical oscillations due to the presence of discontinuity near the material interface. The Lagrangian form of the five-equation model is solved, while the volume fraction is explicitly updated via SPH kernel approximation instead of solving the advection equation. A diffuse-interface zone with a certain width is artificially established at the material interface of different fluids. A smooth variation of physical properties across the interface zone is assumed, and the mixing rules are proposed, similar to the grid-based method, to reconstruct the physical properties. Moreover, the volume adaptive scheme (VAS) is adopted to maintain the uniform distribution of particles in strong compressible flows. A set of one-dimensional multi-fluid shock tube problems and typical two-dimensional numerical experiments in compressible multiphase flows is presented to investigate the performance of the proposed method. Numerical results suggest the capability of the new method to capture the material interfaces and the shock-interface interactions without spurious numerical oscillations.
Smoothed particle hydrodynamics (SPH) is a meshless, particle-based approach that has been increasingly applied for modelling of various fluid-flow phenomena. Concerning multiphase flow computations, an advantage of the Lagrangian SPH over Eulerian approaches is that the advection step is straightforward. Consequently, the interphasial surface can be explicitly determined from the positions of particles representing different phases; therefore, there is no need for the interface reconstruction step. In this review paper, we briefly recall the basics of the SPH approach, and in particular the physical modelling and numerical implementation issues. We also mention the weaknesses of the approach and some remedies to overcome them. Then, we demonstrate the applicability of SPH to selected interfacial flow cases, including the liquid column break-up, gas–liquid flow regimes in a channel capturing the transitions between them and the wetting phenomena. Concerning the two-fluid modelling, it is illustrated with sediment transport in the presence of surface waves. Various other applications are briefly recalled from the rich and growing literature on the subject, followed by a tentative list of challenges in multiphase SPH.
In this study, we establish a hybrid high-order smoothed particle hydrodynamics (SPH) framework (MLS-TENO-SPH) for compressible flows with discontinuities, which is able to achieve genuine high-order convergence in smooth regions and also capture discontinuities well in non-smooth regions. The framework can be either fully Lagrangian, Eulerian or realizing arbitary-Lagrangian-Eulerian (ALE) feature enforcing the isotropic particle distribution in specific cases. In the proposed framework, the computational domain is divided into smooth regions and non-smooth regions, and these two regions are determined by a strong scale separation strategy in the targeted essentially non-oscillatory (TENO) scheme. In smooth regions, the moving-least-square (MLS) approximation is used for evaluating high-order derivative operator, which is able to realize genuine high-order construction; in non-smooth regions, the new TENO scheme based on Vila's framework with several new improvements will be deployed to capture discontinuities and high-wavenumber flow scales with low numerical dissipation. The present MLS-TENO-SPH method is validated with a set of challenging cases based on the Eulerian, Lagrangian or ALE framework. Numerical results demonstrate that the MLS-TENO-SPH method features lower numerical dissipation and higher efficiency than the conventional method, and can restore genuine high-order accuracy in smooth regions. Overall, the proposed framework serves as a new exploration in high-order SPH methods, which are potential for compressible flow simulations with shockwaves.
Complex multiphase flow problems in ocean engineering have long been challenging topics. Problems such as large deformations at interfaces, multi-media interfaces, and multiple physical processes are difficult to simulate. Mesh-based algorithms could have limitations in dealing with multiphase interface capture and large interface deformations. On the contrary, the Smoothed Particle Hydrodynamics (SPH) method, as a Lagrangian meshless particle method, has some merit and flexibility in capturing multiphase interfaces and dealing with large boundary deformations. In recent years, with the improvement of SPH theory and numerical models, the SPH method has made significant advances and breakthroughs in terms of theoretical completeness and computational stability, which starts to be widely used in ocean engineering problems, including multiphase flows under atmospheric pressure, high-pressure multiphase flows, phase-change multiphase flows, granular multiphase flows and so on. In this paper, we review the progress of SPH theory and models in multiphase flow simulations, discussing the problems and challenges faced by the method, prospecting to future research works, and aiming to provide a reference for subsequent research.
Fluid-Structure Interaction (FSI) is a crucial problem in ocean engineering. The smoothed particle hydrodynamics (SPH) method has been employed recently for FSI problems in light of its Lagrangian nature and its advantage in handling multi-physics problems. The efficiency of SPH can be greatly improved with the Adaptive Particle Refinement (APR) method, which refines particles in the regions of interest while deploying coarse particles in the left areas. In this study, the APR method is further improved by developing several new algorithms. Firstly, a new particle refinement strategy with the refinement scale ratio of 4 is employed for multi-level resolutions, and this dramatically decreases the computational costs compared to the standard APR method. Secondly, the regularized transition sub-zone is deployed to render an isotropic particle distribution, which makes the solutions between the refinement zone and the non-refinement zone smoother and consequently results in a more accurate prediction. Thirdly, for complex FSI problems with free surface, a new free-surface detection method based on the Voronoi diagram is proposed, and the performance is validated in comparison to the conventional method. The improved APR method is then applied to a set of challenging FSI cases. Numerical simulations demonstrate that the results from the refinement with scale ratio 4 are consistent with other studies and experimental data, and also agree well with those employing the refinement scale ratio 2. A significant reduction in the computational time is observed for all the considered cases. Overall, the improved APR method with a large refinement scale ratio and the new free-surface detection strategy shows great potential in simulating complex FSI problems efficiently and accurately.
The smoothed particle hydrodynamics (SPH) research community has pursued simulating cavitating flows during the past decades, but so far there are no accurate and stable SPH-based cavitation models. This paper aims to present an attempt to predict cavitation phenomena within the SPH framework. To this end, an equation-of-state-based (EoSB) cavitation model is proposed in the SPH context to capture the inception and development of cavitating flows. In particular, the SPH technique named volume adaptive scheme (VAS) is employed to guarantee isotropic particle distribution when cavitating regions rapidly expand or shrink. Besides, with the purpose of preventing particle clumping and avoiding spurious flow voids induced by negative pressures, two SPH techniques called particle shifting technique (PST) and tensile instability control (TIC) are respectively adopted in the SPH model to further improve the numerical accuracy and stability. Finally, in order to make the present SPH model more applicable to problems with a high Reynolds number, a large eddy simulation (LES) model is also employed to take into account turbulence effects. It is evidently demonstrated that the present SPH model can provide a basically accurate prediction for several cavitation phenomena including cavitating areas and pressure distributions.
There are many compressible flows involving strong discontinuities in nature and engineering applications. Their behaviors are complicated by the existence of shock waves, rarefaction waves and contact discontinuities. To investigate these flows, a shock-capturing scheme based on smoothed particle hydrodynamics (SPH) is proposed. In this scheme, Roe’s approximate Riemann solver cooperated with a novel limiter is embedded into the SPH governing equations to capture shocks. This limiter is simple and effective to control numerical dissipations, and the use of this limiter eliminates the tunable artificial viscosity as required by the conventional SPH. Additionally, to restore the accuracy limited by heterogeneous particle distribution that is usually encountered in the simulation of strongly-compressible flows, the gradient operator in the continuity equation is corrected by a renormalization procedure. Last but not least, for initial particle distribution, unlike the equal mass particle distribution commonly used in compressible SPH simulations, the equal spacing particle distribution is adopted in this scheme, which makes it much easier to model multidimensional problems. The present scheme has been verified by a set of benchmark tests involving contact discontinuities, extreme shock waves and strong rarefaction waves, some of which are firstly simulated by the SPH method.
In this work, a particle regeneration technique is developed for Smoothed Particle Hydrodynamics (SPH). In traditional SPH, the particle disorder phenomenon will occur when dealing with the strongly compressible flow problem. To solve this, in the present work, uniformly distributed background particles filled in the computational domain are adopted. The particle regeneration technique is that the fluid particles replaced by the background particles when the fluid density changes to a specific limitation. The fluid variables of the background particles are approximated by the fluid variables of the initial particles in their support domain. For the multiphase flow, the multiphase interface is calculated by an interface reproducing algorithm, in which, we defined an indicator variable, and set the indicator value discontinuity between different materials. By setting the threshold value, the multiphase interface is reconstructed. Meanwhile, the momentum equation for the Riemann SPH is modified to eliminate the instability in the light phase. Several numerical examples are studied to verify the present algorithm.
This paper presents a brief review of grand challenges of Smoothed Particle Hydrodynamics (SPH) method. As a meshless method, SPH can simulate a large range of applications from astrophysics to free-surface flows, to complex mixing problems in industry and has had notable successes. As a young computational method, the SPH method still requires development to address important elements which prevent more widespread use. This effort has been led by members of the SPH rEsearch and engineeRing International Community (SPHERIC) who have identified SPH Grand Challenges. The SPHERIC SPH Grand Challenges (GCs) have been grouped into 5 categories: (GC1) convergence, consistency and stability, (GC2) boundary conditions, (GC3) adaptivity, (GC4) coupling to other models, and (GC5) applicability to industry. The SPH Grand Challenges have been formulated to focus the attention and activities of researchers, developers, and users around the world. The status of each SPH Grand Challenge is presented in this paper with a discussion on the areas for future development.
Geometrical Volume-of-Fluid (VoF) methods mainly support structured meshes, and only a small number of contributions in the scientific literature report results with unstructured meshes and three spatial dimensions. Unstructured meshes are traditionally used for handling geometrically complex solution domains that are prevalent when simulating problems of industrial relevance. However, three-dimensional geometrical operations are significantly more complex than their two-dimensional counterparts, which is confirmed by the ratio of publications with three-dimensional results on unstructured meshes to publications with two-dimensional results or support for structured meshes. Additionally, unstructured meshes present challenges in serial and parallel computational efficiency, accuracy, implementation complexity, and robustness. Ongoing research is still very active, focusing on different issues: interface positioning in general polyhedra, estimation of interface normal vectors, advection accuracy, and parallel and serial computational efficiency.
This survey tries to give a complete and critical overview of classical, as well as contemporary geometrical VOF methods with concise explanations of the underlying ideas and sub-algorithms, focusing primarily on unstructured meshes and three dimensional calculations. Reviewed methods are listed in historical order and compared in terms of accuracy and computational efficiency.
Numerical simulation of bubble dynamics and cavitation is challenging; even the seemingly simple problem of a collapsing spherical bubble is difficult to compute accurately with a general, three-dimensional, compressible, multicomponent flow solver. Difficulties arise due to both the physical model and the numerical method chosen for its solution. We consider the 5-equation model of Allaire et al. [1], the 5-equation model of Kapila et al. [2], and the 6-equation model of Saurel et al. [3] as candidate approaches for spherical bubble dynamics, and both MUSCL and WENO interface-capturing methods are implemented and compared. We demonstrate the inadequacy of the traditional 5-equation model of Allaire et al. [1] for spherical bubble collapse problems and explain the corresponding advantages of the augmented model of Kapila et al. [2] for representing this phenomenon. Quantitative comparisons between the augmented 5-equation and 6-equation models for three-dimensional bubble collapse problems demonstrate the versatility of pressure-disequilibrium models. Lastly, the performance of pressure disequilibrium model for representing a three-dimensional spherical bubble collapse for different bubble interior/exterior pressure ratios is evaluated for different numerical methods. Pathologies associated with each factor and their origins are identified and discussed.
We are interested in multiphase flows involving the liquid and vapor phases of one species and a third inert gaseous phase. We describe these flows by a hyperbolic single-velocity multiphase flow model composed of the phasic mass and total energy equations,
the volume fraction equations, and the mixture momentum equation. The model includes stiff mechanical and thermal relaxation source terms for all the phases, and chemical relaxation terms to describe mass transfer between the liquid and vapor phases of the species that may undergo transition. First, we present an analysis of the characteristic wave speeds associated to the hierarchy of
relaxed multiphase models corresponding to different levels of
activation of infinitely fast relaxation processes, showing that sub-characteristic conditions hold. We then propose a mixture-energy-consistent finite volume method for the numerical solution of the multiphase model system. The homogeneous portion of the equations is solved numerically via a second-order wave propagation scheme based on robust HLLC-type Riemann solvers. Stiff relaxation source terms are treated by efficient numerical procedures that exploit algebraic equilibrium conditions for the relaxed states. We present
numerical results for several three-phase flow problems,
including two-dimensional simulations of liquid-vapor-gas flows with interfaces and cavitation phenomena.
We develop a parallel fast neighbor search method and communication strategy for particle-based methods with adaptive smoothing-length on distributed-memory computing systems. With a multi-resolution based hierarchical data structure, the parallel neighbor search method is developed to detect and construct ghost buffer particles, i.e. neighboring particles on remote processor nodes. In order to migrate ghost buffer particles among processor nodes, an undirected graph is established to characterize the sparse data communication relation and is dynamically recomposed. By the introduction of an edge coloring algorithm from graph theory, the complex sparse data exchange can be accomplished within optimized frequency. For each communication substep, only efficient nonblocking point-to-point communication is involved.
We consider two demonstration scenarios: (i) fluid dynamics based on smoothed-particle hydrodynamics with adaptive smoothing-length, (ii) a recently proposed physics-motivated partitioning method [Fu et al., JCP 341 (2017): 447-473]. Several new concepts are introduced to recast the partitioning method into a parallel version. A set of numerical experiments is conducted to demonstrate the performance and potential of the proposed parallel algorithms. The proposed methods are simple to implement in large-scale parallel environment and can handle particle simulations with arbitrarily varying smoothing-lengths. The implemented SPH solver has good parallel performance, suggesting the potential for other scientific applications.
A novel projection-based particle method is presented for simulation of multiphase flows characterized by large density ratios and discontinuous density fields at the phase interface. The method considers a multi-fluid continuous system and comprises of a specific computational algorithm utilizing the recently developed Optimized Particle Shifting (OPS [1]) scheme to maintain the regularity of particles at the phase interface and free-surface. The method is founded on an improved version of Moving Particle Semi-implicit (MPS [2]) as a projection-based particle method. A set of previously developed improved schemes are also adopted and hence the proposed method is referred to as improved MPS + OPS. Validations are made both qualitatively and quantitatively in terms of accuracy, energy conservation properties as well as convergence properties by consideration of several benchmark tests.
A wide variety of interface capturing methods have been introduced for simulating two-phase flows throughout the years. However, there is a noticeable dearth of literature focusing on objective comparisons between these methods, especially when they are coupled to the momentum equation and applied in physically relevant regimes. In this article, we compare two techniques for simulating two-phase flows that possess attractive qualities, but belong to the two distinct classes of diffuse interface (DI) and volume of fluid (VOF) methods. Both of these methods allow for mass-conserving schemes that can naturally capture large interfacial topology changes omnipresent in realistic two phase flows. The DI solver used in this work is based on a conservative and bounded phase field method, developed recently. Similar to level set methods, this diffuse interface method takes advantage of the smoothness of the phase field in computing curvature and surface tension forces. Geometric VOF methods track the fractional tagged volume in a cell. The specific geometric VOF scheme used here is a discretely conservative and bounded implementation that uses geometric algorithms for unsplit advection and interface reconstruction, while employing height functions for normal and curvature calculation. We present a quantitative comparison of these methods on Cartesian meshes in terms of their accuracy, convergence rate, and computational cost using canonical two-dimensional (2D) two-phase test cases: a very dense drop moving through a quiescent gas, the Rayleigh-Taylor instability, an equilibrium static drop, an oscillating drop and the damped surface wave. We further compare these methods in their ability to resolve thin films by simulating the impact of a water drop on a deep water pool. Using results of these studies, we suggest qualitative guidelines for selection of schemes for two-phase flow calculations.
In this paper we develop a Lagrangian Inertial Centroidal Voronoi Particle (LICVP) method to extend the original CVP method \cite{fu2017physics} to dynamic load balancing in particle-based simulations. Two new concepts are proposed to address the additional problems encountered in repartitioning the system. First, a background velocity is introduced to transport Voronoi particle according to the local fluid field, which facilitates data reuse and lower data redistribution cost during rebalancing. Second, in order to handle problems with skew-aligned computational load and large void space, we develop an inertial-based partitioning strategy, where the inertial matrix is utilized to characterize the load distribution, and to confine the motion of Voronoi particles dynamically adapting to the physical simulation. Intensive numerical tests in fluid dynamics simulations reveal that the underlying LICVP method improves the incremental property remarkably without sacrifices on other objectives, i.e. the inter-processor communication is optimized simultaneously, and the repartitioning procedure is highly efficient.
This study aims to investigate the capability of smoothed particle hydrodynamics (SPH), a fully Lagrangian mesh-free method, to simulate the bulk blood flow dynamics in two realistic left ventricular (LV) models. Three dimensional geometries and motion of the LV, proximal left atrium and aortic root are extracted from cardiac magnetic resonance imaging and multi-slice computed tomography imaging data. SPH simulation results are analyzed and compared with those obtained using a traditional finite volume-based numerical method, and to in vivo phase contrast magnetic resonance imaging and echocardiography data, in terms of the large-scale blood flow phenomena usually clinically measured. A quantitative comparison of the velocity fields and global flow parameters between the in silico models and the in vivo data shows a reasonable agreement, given the inherent uncertainties and limitations in the modeling and imaging techniques. The results indicate the capability of SPH as a promising tool for predicting clinically relevant large-scale LV flow information.
A numerical inconsistency has emerged for multi-phase smoothed particle hydrodynamics simulations when using very high resolution, made possible by graphical processing units. In violent flows unphysical voids and phase separation occur ultimately leading to numerical instability. New Fickian-based particle shifting algorithms with a selectively activated free-surface correction are developed for air–water simulations to prevent the creation of unnatural voids and maintain numerical stability through nearly uniform distributions. Using the shifting algorithm without surface correction in the air phase is recommended, with marginal improvements if the shifting algorithm is not applied in water. However, maintaining shifting in water would avoid possible void occurrence. The improvement is demonstrated using a dry-bed dam break and a sloshing tank case. A 3D case involving the impact of the water flow on an obstacle is compared with experimental data. The multi-phase SPH scheme gives closer agreement with experiment than a single-phase simulation.
This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.
This paper assesses some recent trends in the novel numerical meshless method smoothed particle hydrodynamics, with particular focus on its potential use in modelling free-surface flows. Due to its Lagrangian nature, smoothed particle hydrodynamics (SPH) appears to be effective in solving diverse fluid-dynamic problems with highly nonlinear deformation such as wave breaking and impact, multi-phase mixing processes, jet impact, sloshing, flooding and tsunami inundation, and fluid–structure interactions. The paper considers the key areas of rapid progress and development, including the numerical formulations, SPH operators, remedies to problems within the classical formulations, novel methodologies to improve the stability and robustness of the method, boundary conditions, multi-fluid approaches, particle adaptivity, and hardware acceleration. The key ongoing challenges in SPH that must be addressed by academic research and industrial users are identified and discussed. Finally, a roadmap is proposed for the future developments.
Although classical WENO schemes have achieved great success and are widely accepted, they exhibit several shortcomings. They are too dissipative for direct simulations of turbulence and lack robustness when very-high-order versions are applied to complex flows. In this paper, we propose a family of high-order targeted ENO schemes which are applicable for compressible-fluid simulations involving a wide range of flow scales. In order to increase the numerical robustness as compared to very-high-order classical WENO schemes, the reconstruction dynamically assembles a set of low-order candidate stencils with incrementally increasing width. While discontinuities and small-scale fluctuations are efficiently separated, the numerical dissipation is significantly diminished by an ENO-like stencil selection, which either applies a candidate stencil with its original linear weight, or removes its contribution when it is crossed by a discontinuity. The background linear scheme is optimized under the constraint of preserving an approximate dispersion-dissipation relation. By means of quasi-linear analyses and practical numerical experiments, a set of case-independent parameters is determined. The general formulation of arbitrarily high-order schemes is presented in a straightforward way. A variety of benchmark-test problems, including broadband waves, strong shock and contact discontinuities are studied. Compared to well-established classical WENO schemes, the present schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. They are particularly suitable for DNS and LES of shock-turbulence interactions.
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.
High-fidelity numerical simulation of compressible multi-phase flows is of great challenge due to its competing requirements for resolving complex flow structures with low dissipation and capturing moving interfaces as well as other discontinuities sharply. Recently, a novel hybrid scheme, combining the standard targeted essentially non-oscillatory (TENO) scheme with Tangent of Hyperbola for INterface Capturing (THINC) scheme as two building blocks for smooth and non-smooth regions respectively and thus named as TENO-THINC, has been proposed and shows great potential for resolving complex single-phase fluids. In this work, a high-order finite-volume method, based on the TENO-THINC scheme for spatial reconstruction and the Harten-Lax-van Leer contact (HLLC) approximate Riemann solver for flux evaluation, is developed for simulating compressible multi-phase flows with a reduced five-equation formulation of the diffuse-interface model. The TENO-THINC scheme deploys the THINC reconstruction to resolve the physical discontinuities as well as the material interfaces within a few cells, and is desired to resolve the interface evolution more reliably in multi-phase flow simulations. Several algorithms have been implemented and elaborated for ensuring the numerical robustness of extreme simulations with high density and pressure ratios. Numerical results of the 1D and 2D challenging benchmark tests show that the TENO-THINC scheme is more robust than the standard TENO scheme and less dissipative than both the TENO and WENO-JS schemes. This property is essential for the long-term simulations of compressible multi-phase flows.
A method to simulate unsteady multi-fluid flows in which a sharp
interface or a front separates incompressible fluids of different density
and viscosity is described. The flow field is discretized by a conservative
finite difference approximation on a stationary grid, and the interface is
explicitly represented by a separate, unstructured grid that moves
through the stationary grid. Since the interface deforms continuously,
it is necessary to restructure its grid as the calculations proceed. In addition
to keeping the density and viscosity stratification sharp, the tracked
interface provides a natural way to include surface tension effects. Both
two- and three-dimensional, full numerical simulations of bubble
motion are presented.
Water entry of marine structures has long been an important problem in ocean engineering. Among the different techniques to predict fluid-structure interactions during water entry, smoothed particle hydrodynamics (SPH) method gradually becomes a promising method that is able to solve the impact pressure and the splashing fluid jets simultaneously. However, for three-dimensional (3D) problems, SPH method is computationally expensive due to the huge number of particles that are needed to resolve the local impact pressure accurately. Therefore, in this work an axisymmetric SPH model is applied to solve different water entry problems with axisymmetric structures including spheres and cones with different deadrise angles. Importantly, the Volume Adaptive Scheme (VAS) is added to guarantee the homogeneousness of particle volumes during the simulation. The axisymmetric SPH model with VAS scheme will be introduced in detail and the numerical results will be sufficiently validated with experimental data to demonstrate the high robustness and accuracy of the SPH model for solving 3D axisymmetric water entry problems in an efficient way.
In the current study, an improved numerical model is proposed in the compressible fields based on Smoothed Particle Hydrodynamics (SPH), which is comprised of MUSCL interpolation in multiphase flow, enhanced particle regeneration technique (PRT) and the particle shifting technique (PST) in compressible flows. The PRT is specially proposed to deal with compressible problems, in which the volume of particles have large variation during the whole simulation process. Different to the conventional PRT [1], an interface control (IC) method is proposed to deal with the mass conservation problem which may result in the unphysical movement of interface when modeling multifluids. The multiphase MUSCL interpolation aims at dealing with over-dissipation problem that exists in the Godunov-type SPH which may result in the wrong detection of wave front. To avoid large discontinuity between different fluids, two kinds of extrapolation schemes (constant extrapolation and isentropic extrapolation) are discussed and compared at the inspiration of the ghost fluid method (GFM). The proposed MUSCL-based compressible SPH model is validated and discussed in several challenging test cases, such as multiphase shock tube like problem, the shock wave impacting on multifluids interface problem and Rychtmyer-Meshkov instability etc., in which good agreements are obtained.
The discontinuity across interface of multi-phase flows with large density ratios usually poses great challenges for numerical simulations. The smoothed particle hydrodynamics (SPH) is a meshless method with inherent advantages in dealing with multi-phase flows without the necessity of tracking the moving interfaces. In this paper, we develop a new weakly-compressible SPH model for multi-phase flows with large density ratios while allowing large CFL numbers. In the present SPH model, the continuity equation is first modified by eliminating the influence from particles of different phases based on the simple fact that different phases will not contribute when calculating the density for immiscible multi-phase flows; thus, the modified continuity equation will only consider the influence from neighboring particles of the same phase. The pressure and density of the particles of other phases are then re-initialized by using the Shepard interpolation function. The present multi-phase SPH model has been tested by four numerical examples, including the two-phase hydrostatic water, standing waves, liquid sloshing, and dam breaking. It has been demonstrated that the present multi-phase SPH model can obtain satisfactory results stably, even at large CFL numbers, and this means that large time steps can be employed. Therefore, the present multi-phase SPH model can significantly save computational cost through using large time steps, especially for large-scale problems with a large number of particles.
Solving compressible flows containing both smooth and discontinuous flow structures still remains a big challenge for finite volume methods, especially on unstructured grids where one faces more difficulties in building high-order polynomial reconstruction and limiting projection to suppress numerical oscillations in comparison with the case of structured grids. As a result, most of the current finite volume schemes on unstructured grids are of second order and too dissipative to resolve fine structures of complex flows. In this paper, we report two novel hybrid schemes to resolve vortical and discontinuous solutions on unstructured grids by reducing numerical dissipation. Different from conventional shock capturing schemes that use polynomials and limiting projections for reconstruction, the proposed schemes employ two second-order schemes, i.e. a polynomial and a sigmoid function as candidate reconstruction functions to approximate smooth and discontinuous solutions respectively. As the polynomial function, the MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted, while being a sigmoid function, the multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used to mimic the discontinuous solution structure. With these candidates for reconstruction, a single-step boundary variation diminishing (BVD) algorithm, which aims to minimize numerical dissipation, is designed on unstructured grids to select the final reconstruction function. The resulting two variant schemes, MUSCL-THINC/QQ-BVD schemes with two and three candidates respectively, are algorithmically simple and show great superiority to other existing schemes in capturing discontinuous and vortical flow structures for single and multiphase compressible flows on unstructured grids. The performance of the proposed schemes has been extensively verified through benchmark tests of single and multi-phase compressible flows, where discontinuous and vortical flow structures, like shock waves, contact discontinuities and material interfaces, as well as vortices and shear instabilities of different scales, coexist simultaneously. The numerical results show that the proposed schemes that hybrids two second-order schemes are capable of capturing sharp discontinuous profiles without numerical oscillations and resolving vortical structures along shear layers and material interfaces with significantly improved solution quality superior to other schemes of even higher order reconstructions.
In the present work, the single-phase and weakly-compressible δ-SPH model is further extended to simulate multiphase and strongly-compressible flows. This is motivated by the fact that traditional SPH models can meet some difficulties when modeling strongly-compressible flows with large volume variations (e.g. expansion and collapse of cavitation bubbles). Due to the strong compressibility of the fluid, the energy equation should be considered in the governing equations. In that case, the pressure is solved based on both density and internal energy. To stabilize the pressure field, density and energy diffusive terms should be applied. Large variations of particle volumes in the compressible phase would result in large variations of particle spacing. Therefore, particle smoothing lengths are adjusted in time to maintain appropriate neighboring particles. To ensure good properties of accuracy and conservation when particles with different smoothing lengths interact, corrected SPH operators are utilized to discretize the governing equations. Moreover, in order to limit the particle volume variations and maintain a homogeneous volume distribution in the entire flow field, especially near the interface between different phases of different compressibility, a new volume adaptive scheme is proposed to control particle volumes. The volumes which are over-expanded or over-compressed will be split or merged with others, maintaining a small particle volume variation in the flow. Finally, the proposed SPH model is validated with several challenging benchmarks including expansion and collapse of underwater-explosion bubbles or cavitation bubbles. All the SPH results are compared with other numerical solutions with good agreements.
We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front also can be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand sides, by using techniques from hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general Hamilton-Jacobi-type problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.
This article presents an extension of coupled volume-of-fluid and level-set method (VOSET) for simulating free surfaces flows in arbitrary 2D polygon meshes. A series of techniques are introduced for geometric calculations in convex polygons. Newton iteration is adopted for the interface reconstruction in polygons, and incremental remapping approach is employed for the propagation of the volume fractions. The interface tracking test results suggested a second order accuracy in mixed and hexagonal grids. For the validation purpose, a Rayleigh-Taylor instability problem, a liquid column collapse problem and a single bubble rising problem were numerically studied, and the obtained results show excellent agreements with experimental data and benchmark solutions in literatures. Finally the proposed VOSET method was applied in simulating the working process of a flow-focusing microfluidic droplet generator.
A novel formulation for the diffusive term in the continuity equation is proposed to improve the stability of smoothed particle hydrodynamics weakly compressible scheme avoiding the introduction of empirical parameters. Densities at particle-particle interface have been computed by means of a first-order consistent total variational diminishing reconstruction and a one-dimensional Roe's approximate Riemann solver is applied to add the correct amount of diffusion. Results of numerical tests also demonstrate that the proposed method is able to guarantee consistency both inside the fluid and close to the free surface. Furthermore, a numerical analysis of several flux limiter functions has been conducted, finding that the choice of this function is a critical point to guarantee the accuracy of the method. It has been assessed, through the monitoring of the internal energy, that the van Albada limiter is more effective in dissipating spurious density fluctuations.
This paper is motivated by the cavitation phenomenon, which occurs when bubbles collapse near a hydraulic machine surface. The bubble compression close to the wall has been addressed as the fundamental mechanism producing cavitation damage, whose general behavior is characterized by the emission of pressure waves and the formation of a micro jet. In order to simulate the collapse of a gas bubble in water, it is proposed a multiphase and compressible model developed in SPH-ALE. This model does not diffuse the interface and guarantees the continuity of normal velocity and pressure at the interface between both fluids, allowing it to deal with interfaces of simple contact. The model solves the mass, momentum and energy conservation equations of Euler system using an equation of state for each phase without phase change. The compressible model was validated through monodimensional configurations, such as shock tube test cases for monophase and multiphase flows. Bubble collapse simulations in 2D are presented highlighting the principle features, i.e. pressure waves and micro jets. Also, it is analyzed the effect of the initial distance between the bubble and the wall (H0). Limitations and perspectives of SPH-ALE method on this particular subject are also discussed.
In this paper we present a new multi-resolution parallel framework, which is designed for large-scale SPH simulations of
fluid dynamics. An adaptive rebalancing criterion and monitoring system is developed to integrate the CVP partitioning method
as rebalancer to achieve dynamic load balancing of the system. A localized nested hierarchical data structure is developed in
cooperation with a tailored parallel fast-neighbor-search algorithm to handle problems with arbitrarily adaptive smoothing-length
and to construct ghost buffer particles in remote processors. The concept of “diffused graph” is proposed in this paper to improve
the performance of the graph-based communication strategy. By utilizing the hybrid parallel model, the framework is able to exploit
the full parallel potential of current state-of-the-art clusters based on Distributed Shared Memory (DSM) architectures. A range
of gas dynamics benchmarks are investigated to demonstrate the capability of the framework and its unique characteristics. The
performance is assessed in detail through intensive numerical experiments at various scales.
Simulation of compressible flows became a routine activity with the appearance of shock-/contact-capturing methods. These methods can determine all waves, particularly discontinuous ones. However, additional difficulties may appear in two-phase and multimaterial flows due to the abrupt variation of thermodynamic properties across the interfacial region, with discontinuous thermodynamical representations at the interfaces. To overcome this difficulty, researchers have developed augmented systems of governing equations to extend the capturing strategy. These extended systems, reviewed here, are termed diffuse-interface models, because they are designed to compute flow variables correctly in numerically diffused zones surrounding interfaces. In particular, they facilitate coupling the dynamics on both sides of the (diffuse) interfaces and tend to the proper pure fluid-governing equations far from the interfaces. This strategy has become efficient for contact interfaces separating fluids that are governed by different equations of state, in the presence or absence of capillary effects, and with phase change. More sophisticated materials than fluids (e.g., elastic-plastic materials) have been considered as well.
We propose efficient single-step formulations for reinitialization and extending algorithms,
which are critical components of level-set based interface-tracking methods.
The level-set field is reinitialized with a single-step (non iterative) "forward tracing" algorithm.
A minimum set of cells is defined that describes the interface, and reinitialization employs only data from these cells. Fluid states are extrapolated or extended across the interface by a single-step "backward tracing" algorithm. Both algorithms, which are motivated by analogy to ray-tracing, avoid multiple block-boundary data exchanges that are inevitable for iterative reinitialization and extending approaches within a parallel-computing environment. The single-step algorithms are combined with a multi-resolution conservative sharp-interface method and validated by a wide range of benchmark test cases. We demonstrate that the proposed reinitialization method achieves second-order accuracy in conserving the volume of each phase. The interface location is invariant to reapplication of the single-step reinitialization. Generally, we observe smaller absolute errors than for standard iterative reinitialization on the same grid. The computational efficiency is higher than for the standard and typical high-order iterative reinitialization methods. We observe a 2- to 6- times efficiency improvement over the standard method for serial execution. The proposed single-step extending algorithm, which is commonly employed for assigning data to ghost cells with ghost-fluid or conservative interface interaction methods, shows an about 10-times efficiency improvement over the standard method while maintaining same accuracy. Despite their simplicity, the proposed algorithms offer an efficient and robust alternative to iterative reinitialization and extending methods for level-set based multi-phase simulations.
We propose a coupled volume-of-fluid and level set (VOSET) method for interfacial flow simulations on unstructured triangular grids. In this method, the volume fraction advection is performed using a Lagrangian–Eulerian remapping algorithm, and the level set function is calculated by a simple iterative geometric operation. The present VOSET method can not only satisfy mass conservation but also predict the surface tension with good accuracy, thus combining the advantages and overcoming the disadvantages of volume-of-fluid (VOF) and level set (LS) methods. Finally, the present method is verified by well known Zalesak's slotted-disk revolution, single vortex flow, dam break and single bubble rising problems. The results illustrate that the present method can accurately simulate incompressible two-phase flows for unstructured triangular grids.
In this paper, we propose a novel domain decomposition method for large-scale simulations in continuum mechanics by merging the concepts of Centroidal Voronoi Tessellation (CVT) and Voronoi Particle dynamics (VP). The CVT is introduced to achieve a high-level compactness of the partitioning subdomains by the Lloyd algorithm which monotonically decreases the CVT energy. The number of computational elements between neighboring partitioning subdomains, which scales the communication effort for parallel simulations, is optimized implicitly as the generated partitioning subdomains are convex and simply connected with small aspect-ratios. Moreover, Voronoi Particle dynamics employing physical analogy with a tailored equation of state is developed, which relaxes the particle system towards the target partition with good load balance. Since the equilibrium is computed by an iterative approach, the partitioning subdomains exhibit locality and the incremental property. Numerical experiments reveal that the proposed Centroidal Voronoi Particle (CVP) based algorithm produces high-quality partitioning with high efficiency, independently of computational-element types. Thus it can be used for a wide range of applications in computational science and engineering.
The paper provides a comparative investigation on accuracy and conservation properties of two particle regularization schemes, namely, the Dynamic Stabilization (DS) [1] and generalized Particle Shifting (PS) [2] schemes in simulations of both internal and free-surface flows in ISPH (Incompressible SPH) context. The paper also presents an Optimized PS (OPS) scheme for accurate and consistent implementation of particle shifting for free-surface flows. In contrast to PS, the OPS does not contain any tuning parameters for free-surface, consistently resulting in perfect elimination of shifting normal to an interface and resolves the unphysical discontinuity beneath the interface, seen in PS results.
An efficient method for the simulation of compressible multimaterial flows with a general form of equation of state is presented for explosive detonation and airblast applications. Multimaterial flows are modeled with a volume-fraction type approach for immiscible fluids governed by the compressible Euler equations on three-dimensional unstructured grids. The five-equation quasi-conservative system is discretized in space using an edge-based finite volume approach with a second-order accurate HLLC approximate Riemann solver and temporal discretization with an explicit multistage Runge–Kutta method. The computational model is robust enough to handle flows with strong shocks, while being general enough to model materials with different equations of state and physical states. Numerical tests demonstrate the accuracy of the method for strong shock and interface interactions. A program burn method is implemented to describe the conversion of solid unreacted explosive to reacted gases in condensed phase detonations. The accuracy of the burn model is validated by comparison with published numerical results of flow profiles during detonation and for near-field airblast. Numerical simulations of hemispherical and plate-shaped explosive charge detonations are performed to investigate the influence of charge shape on airblast. The predicted pressure and impulse from simulation compare well with published experimental data.
Single fluid schemes that rely on an interface function for phase identification in multicomponent compressible flows are widely used to study hydrodynamic flow phenomena in several diverse applications. Simulations based on standard numerical implementation of these schemes suffer from an artificial increase in the width of the interface function owing to the numerical dissipation introduced by an upwind discretization of the governing equations. In addition, monotonicity requirements which ensure that the sharp interface function remains bounded at all times necessitate use of low-order accurate discretization strategies. This results in a significant reduction in accuracy along with a loss of intricate flow features. In this paper we develop a nonlinear transformation based interface capturing method which achieves superior accuracy without compromising the simplicity, computational efficiency and robustness of the original flow solver. A nonlinear map from the signed distance function to the sigmoid type interface function is used to effectively couple a standard single fluid shock and interface capturing scheme with a high-order accurate constrained level set reinitialization method in a way that allows for oscillation-free transport of the sharp material interface. Imposition of a maximum principle, which ensures that the multidimensional preconditioned interface capturing method does not produce new maxima or minima even in the extreme events of interface merger or breakup, allows for an explicit determination of the interface thickness in terms of the grid spacing. A narrow band method is formulated in order to localize computations pertinent to the preconditioned interface capturing method. Numerical tests in one dimension reveal a significant improvement in accuracy and convergence; in stark contrast to the conventional scheme, the proposed method retains its accuracy and convergence characteristics in a shifted reference frame. Results from the test cases in two dimensions show that the nonlinear transformation based interface capturing method outperforms both the conventional method and an interface capturing method without nonlinear transformation in resolving intricate flow features such as sheet jetting in the shock-induced cavity collapse. The ability of the proposed method in accounting for the gravitational and surface tension forces besides compressibility is demonstrated through a model fully three-dimensional problem concerning droplet splash and formation of a crownlike feature.
We develop a shock- and interface-capturing numerical method that is suitable for the simulation of multicomponent flows governed by the compressible Navier–Stokes equations. The numerical method is high-order accurate in smooth regions of the flow, discretely conserves the mass of each component, as well as the total momentum and energy, and is oscillation-free, i.e. it does not introduce spurious oscillations at the locations of shockwaves and/or material interfaces. The method is of Godunov-type and utilizes a fifth-order, finite-volume, weighted essentially non-oscillatory (WENO) scheme for the spatial reconstruction and a Harten–Lax–van Leer contact (HLLC) approximate Riemann solver to upwind the fluxes. A third-order total variation diminishing (TVD) Runge–Kutta (RK) algorithm is employed to march the solution in time. The derivation is generalized to three dimensions and nonuniform Cartesian grids. A two-point, fourth-order, Gaussian quadrature rule is utilized to build the spatial averages of the reconstructed variables inside the cells, as well as at cell boundaries. The algorithm is therefore fourth-order accurate in space and third-order accurate in time in smooth regions of the flow. We corroborate the properties of our numerical method by considering several challenging one-, two- and three-dimensional test cases, the most complex of which is the asymmetric collapse of an air bubble submerged in a cylindrical water cavity that is embedded in 10% gelatin.
This article describes a frame-invariant vector limiter for Flux-Corrected Transport (FCT) numerical methods. Our approach relies on an objective vector projection, and, because of its intrinsic structure, the proposed approach can be generalized with ease to higher-order tensor fields. The proposed concept is applied to nodal finite element formulations and the so-called algebraic FCT paradigm, but the ideas pursued here are very general and also apply to more general instantiations of flux-corrected transport. Specifically, we consider the arbitrary Lagrangian-Eulerian (ALE) equations of compressible inviscid flows. In addition to the geometric conservation law (GCL) and the local extreme diminishing (LED) property of the original scalar limiters, the proposed approach ensures frame invariance (objectivity) for vectors. Particularly, we use an ALE strategy based on a two-stage, Lagrangian plus mesh remap (data transfer based on conservative interpolation), in which remap and limiting are performed in a synchronized way. The proposed approach is however of general applicability, is not limited to a specific ALE implementation, and can easily be generalized to computations with standard (monolithic) ALE or Eulerian reference frames. The significance of the frame-invariant limiter for vectors is demonstrated in computations of compressible materials under extreme load conditions. Extensive testing in two and three dimensions demonstrates that the proposed limiter greatly enhances the robustness and reliability of the existing methods under typical computational scenarios.