Three-Dimensional Smoothed Particle Hydrodynamics Simulation for Liquid Droplet with Surface Tension

We provide a basic method of Smoothed Particle Hydrodynamics (SPH) to simulate liquid droplet with surface tension in three dimensions. Liquid droplet is a simple case for surface tension modeling. Surface tension works only on fluid surface. In SPH method, we simply apply the surface tension on the boundary particles of liquid. The particle on the 3D boundary was detected dynamically using Free-Surface Detection algorithm. The normal vector and curvature of the boundary surface were calculated simultaneously with 3D boundary surface reconstruction using Moving Least-Squares (MLS) method. Before the reconstruction, the coordinate system was transformed into a local coordinate system. Afterwards, the surface tension force which depends on curvature of the surface, was calculated and applied on the boundary particles of the droplet. We present the simulation result of droplet motion with gravity force. By using the basic method of SPH for fluid modeling, and a combination of 3D Free-Surface Detection algorithm with MLS method, we can simulate droplet phenomena with expected result.Comment: 9 pages, 4 figure, Selected Paper from the International Symposium on Computational Science 201

다상유동 및 고체 파편물의 수력학적 거동 해석을 위한 GPU 기반의 SPH-DEM 연계해석 코드 개발

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 에너지시스템공학부, 2020. 8. 김응수.In the late phase severe accident of LWR, the massive corium releases out of the reactor pressure vessel (RPV) and falls to the coolant if the In-Vessel Retention (IVR) strategy fails. The melt jet can be fragmented into debris particles based on the assumption that the ex-vessel pool is sufficiently deep. It is known that there are various three-phase flow issues associated with the fragmented debris particles under the influence of phase change of cavity coolant. In such cases, the vapor phase forms a sharp and dynamic interface with the liquid phase while the transient relocation behavior of debris particles is the main concern. Thus, coupling Lagrangian-based multi-phase CFD techniques and Discrete Element Method (DEM) can be an effective approach in terms of numerical modeling of such behaviors. In this respect, an integrated numerical code for incompressible 3-phase flow has been developed in this study by two-way phase coupling of multi-phase Smoothed Particle Hydrodynamics (SPH) and DEM model. Smoothed Particle Hydrodynamics (SPH) is one of the best-known meshless CFD methods in which the fluid system is represented as the finite number of Lagrangian particles. The SPH code developed in this study proposes a new density estimation model and improves the surface tension model for accurate simulation of incompressible two-phase flow behavior. The demonstration of its applicability has been performed through several V&V simulations including multi-phase dam-break and sloshing simulations. Discrete Element Method (DEM) is a direct simulation method for a rigid body that can analyze the translation, rotation, and collision behavior of solid particles in detail. The soft-sphere collision model with Hertz-Mindlin contact force equations is adopted for developed DEM code in this study. To precisely estimate the wall boundary interactions of bed-formed debris particles, a versatile wall boundary model is newly proposed in this study that also covers the sliding and rolling behavior of solid particles. The inter-particle collision behavior and sliding & rolling motion of particles are well proven in several V&V cases. The numerical code system for incompressible 3-phase flow is newly developed by two-way phase coupling of the above two models (SPH-DEM). The unresolved coupling approach between two methods was adopted for the analysis of the overall behavior of particulate solid debris. The fundamental validation of the phase coupled model was performed for both single-particle behavior and particulate granular flow such as dam-breaking motion of particle-fluid. The SPH-DEM coupled code in this study has been parallelized based on Graphical Process Unit (GPU) in order to overcome the inherent efficiency problem of the Lagrangian-based numerical method. Parallel mapping and reduction are applied for solving discretized summation equations of each SPH particle, solving contact force equations for each DEM particle, and also for solving coupling equations between SPH and DEM particles. The efficiency of code parallelization was evaluated through the scalability analysis based on the benchmark calculation. Finally, the simulation of the vapor-driven leveling behavior of spherical solids was performed as a case study to demonstrate the applicability of the developed code. The time-variant surface shape of solid particles was compared with the benchmark experiments both qualitatively and quantitatively. The effect of gas flow rate on the tendency of leveling behavior also has been analyzed. The developed numerical system in this study is expected to be a good alternative for the simulation of such phenomena that were difficult to handle with traditional numerical methods since the numerical schemes used in the code have a high potential for simulation of complicated physics with highly deformable geometry. In addition, this validated code system can contribute to hydrodynamic modeling studies for severe accident technology by performing numerical experiments on conditions that hard to be conducted on a laboratory scale.가압경수로 중대사고 후기 과정에서 핵연료 용융물 노내 억류 전략의 실패로 용융 핵연료가 용기 바깥쪽 침수 공동으로 재배치되면, 공동의 깊이나 제트의 속도 등의 조건에 따라 핵연료 파편화가 발생할 수 있다. 이러한 핵연료 파편물에서 발생하는 붕괴열에 의해 냉각재의 상변화가 이루어지면, 이상유동 자연대류와 고체 파편물이 공존하는 3상유동 시스템이 된다. 핵연료 용융물의 제트 파쇄, 하강 및 퇴적, 평탄화 등 일련의 과정에서 다양한 형태의 파편물 3상 냉각 거동이 발생할 수 있으며, 핵연료 파편물의 붕괴열이 충분히 제거되지 못하면 파편물의 응집 및 재용융이 발생할 수 있다. 중대사고 완화의 관점에서 이러한 3상 냉각 거동에 대한 상세한 이해가 필요하지만, 이상유동 자연대류, 이상유동 열전달, 고화, 용융, 비등, 유동-입자 상호작용 등을 포함하는 거동 자체의 복잡성으로 인해 관련 현상들의 예측 및 평가는 큰 불확실성을 내포하고 있다. 전통적인 핵연료 파편물을 포함한 3상 거동 해석은 다른 중대사고 해석과 마찬가지로 경험적 방법론에 의존한 보수적인 접근 방법이나 고체 입자와 이상 유체를 모두 연속체로 가정하는 다유체 모델을 기반으로 이루어졌다. 최근에는 이러한 방법론들이 가지는 본질적인 한계를 극복하고자 고체 입자들 사이의 충돌이나 회전을 별개로 다루는 이산요소법(Discrete Element Method, DEM)과 격자 기반의 오일러리안 전산유체해석(CFD) 기법을 연계하는 형태로 서로 간의 상호작용을 해석하는 연구들이 많이 이루어지고 있다. 한편, 최근에는 하드웨어 및 소프트웨어의 성능이 비약적으로 좋아지면서 격자(Grid)에 기반하지 않고 질점 하나하나의 움직임을 따라가면서 유동에 대한 지배방정식을 해석하는 라그랑지안 유체 해석기법의 응용이 늘어나고 있다. 라그랑지안 해석 기법에서는 다상유동 해석 시에 액체와 기체 두 상을 완전히 별개의 영역으로 해상하여 지배방정식을 풀기 때문에 계면 마찰력이나 항력, 양력 등에 대한 별도의 상관식 없이 제 1 원리 기반으로 유동을 해석할 수 있어, 이상유동에 대한 보다 근본적인 해석이 가능하다. 핵연료 파편물의 생성 및 3상 냉각 거동과 관련된 현상들은 대부분 기체 상의 생성이나 이상유동 자연대류에 영향을 받는 현상들로 액체 기체 사이의 인터페이스가 복잡하고 역동적인 경향이 있기 때문에, 액체-기체 다상유동에 효과적인 라그랑지안 기반의 유체해석 기법과 강체 이산요소법(DEM)을 연계하면 효과적인 3상 유동 해석 체계를 구축할 수 있다. 하지만, 핵연료 파편물을 포함한 3상 냉각 거동과 관련하여 라그랑지안 입자 기반 유체해석 기법을 활용한 연구는 세계적으로도 아직 수행된 바가 없다. 이러한 필요성에 따라, 본 연구에서는 대표적인 입자 기반의 유체해석 방법론 중 하나인 완화입자유체동역학(Smoothed Particle Hydrodynamics, SPH) 기법과 강체의 충돌, 병진, 회전 운동을 직접적으로 다루는 이산요소법(DEM)의 연계를 통해 고체 입자를 포함한 3상 유동 해석을 위한 라그랑지안 해석 체계를 구축하였다. 고체 입자와 유체 사이의 연계는 mm 이하의 스케일을 가지는 핵연료 파편물의 형상적 특성을 고려하여 두 상 사이의 겹침을 허용하여 운동량 교환을 모델링하는 비해상(unresolved) 방식으로 이루어졌다. 또한, SPH 유체 모델, DEM 강체 모델, SPH-DEM 연계 모델 각각에 대한 검증을 다양한 스케일에서 다양한 실험 연구들과의 비교를 통해 수행하였다. 한편, 완화입자유체동역학(SPH) 기법과 이산요소법(DEM) 모델은 라그랑지안 해석 기법의 특성상 컴퓨터 성능의 비약적인 발전에도 불구하고 오일러리안 해석 기법에 비해 계산 효율 및 시간에 대한 상대적인 제약이 존재한다. 특히 액체-기체의 이상유동 해석을 다룰 경우 기체 상의 밀도가 작기 때문에 라그랑지안 유체해석 기법에서는 더 작은 시간 간격이 요구된다. 이에 본 연구에서는 그래픽 처리 장치 (Graphics Processing Unit, GPU)를 활용하여 SPH 해석, DEM 해석, SPH-DEM 연계 해석이 모두 각 라그랑지안 질점에 대해 동시에 수행될 수 있도록 GPU 기반의 연계코드 병렬화 및 가속화를 수행하였다. 마지막으로 개발한 라그랑지안 기반의 3상유동 해석 체계의 유용성 입증을 위해 원자로 중대사고 후기 과정에서 발생할 수 있는 핵연료 파편물 층(debris bed)의 평탄화(self-leveling) 거동에 대한 검증 해석을 수행하였다. 시간에 따른 파편물 층 표면의 형상 변화를 타 기관에서 수행된 기체주입 실험과 비교하는 형태로 검증이 이루어졌다. 분석 결과, 본 연구에서 개발한 SPH-DEM 연계해석 코드가 고체 입자 상을 포함한 수력학적 3상 거동을 정량적으로, 정성적으로 잘 해석하는 것을 확인하였다. 본 연구에서 개발한 라그랑지안 기반의 SPH-DEM 3상유동 해석 체계는 원자로 중대사고의 해석적 관점에서 기존의 수치해석 기법들이 다루기 어려웠던 현상들에 대한 대안 또는 상호 보완의 역할을 할 수 있다. 또한, 본 연구에서 개발한 코드는 제 1원리 기반의 물리 법칙을 기반으로 유동 및 강체의 거동을 해석하기 때문에 실험으로 구현하기 어려운 조건이나 스케일에 대한 수치적 재현이 가능하고, 이를 바탕으로 기존의 스케일링 법칙을 검증하거나 실험 결과가 없는 영역에서 수치 데이터를 생성하여 기존의 상관식을 개선하는데 활용할 수 있다. 이러한 점에서 본 연구는 원자로 중대사고의 해석이나 안전성 평가와 관련하여 불확실성을 저감하는데 기여한다.Chapter 1 Introduction 1 1.1 Background and Motivation 1 1.2 Previous Studies 3 1.2.1 Numerical Studies on Particulate Debris Bed 3 1.2.2 SPH-DEM Phase Coupling 4 1.3 Objectives and Scope 5 Chapter 2 Fluid Phase: Smoothed Particle Hydrodynamics 8 2.1 Smoothed Particle Hydrodynamics (SPH) 9 2.1.1 SPH Particle Approximation 9 2.1.2 SPH Particle Approximation of Derivatives 10 2.1.3 Kernel Function 11 2.1.4 Accuracy of SPH Approximation 12 2.1.5 Governing Equations for Incompressible Flow 14 2.2 Multi-phase SPH Governing Equations 16 2.2.1 Treatment of Multi-Phase Flow 16 2.2.2 Normalized Density Model 18 2.2.3 Continuum Surface Force (CSF) Model 19 2.3 Implementation of SPH Model 21 2.3.1 Algorithm of SPH Code 21 2.3.2 Nearest Neighboring Particles Searching (NNPS) 22 2.3.3 Time Integration 23 2.4 V&V Simulations 24 2.4.1 Rayleigh-Taylor Instability 25 2.4.2 Bubble Terminal Velocity 25 2.4.3 Dam-Break Simulation 25 2.4.4 Centralized Sloshing Simulation 26 Chapter 3 Solid Phase: Discrete Element Method 45 3.1 Discrete Element Method (DEM) 46 3.2 DEM Contact Force 47 3.2.1 Soft-sphere Contact Model 47 3.2.2 Contact Force Model 48 3.2.3 Hertz-Mindlin Contact Force Model 49 3.3 Wall Boundary Conditions 52 3.3.1 Versatile Wall Boundary Model 52 3.3.2 Particle Collision with the Wall 54 3.3.3 Sliding and Rolling on the Wall Boundary 56 3.4 DEM Implementation Algorithm 57 3.4.1 Contact Detection 58 3.4.2 Estimation of Relative Velocity 59 3.4.3 Calculation of Contact Force 60 3.4.4 Wall Boundary Conditions and Time Integration 60 3.5 V&V and Simulations 61 3.5.1 Conservation of Momentum and Angular Momentum 62 3.5.2 Conservation of Energy in Elastic Collision 63 3.5.3 V&V Simulations for Wall Boundary Model 63 3.5.4 Granular Collapse of Spherical Particles 64 Chapter 4 Two-way Phase Coupling of SPH and DEM 76 4.1 Unresolved Coupling of SPH and DEM 76 4.2 Governing Equations 78 4.2.1 SPH Particles: Locally Averaged N-S Equations 78 4.2.2 DEM Particles: Coupling Forces Acting on Solid Particles 80 4.2.3 SPH Particles: Reaction Force from Momentum Exchange 82 4.3 Algorithm of SPH-DEM Coupled Model 83 4.4 V&V Simulations for SPH-DEM Coupled Model 84 4.4.1 Single DEM Particle Behavior 85 4.4.2 Pressure Drop through Packed Bed 87 4.4.3 Granular Flow in Liquid: 3D Dam-Break 89 Chapter 5 GPU Parallelization of Coupled SPH-DEM Code 103 5.1 Parallelization of Governing Equations 104 5.1.1 GPU-based Parallelization 104 5.1.2 Parallelization of SPH-DEM Governing Equations 104 5.2 Parallelization of NNPS and Contact Detection 105 5.3 Results of GPU Parallelization 107 5.3.1 Speedup in Computation Time 107 5.3.2 Parallelization Factors 107 Chapter 6 Code Application to Vapor-Driven Leveling Behavior of Spherical Debris 113 6.1 Self-Leveling Behavior of Debris Bed 114 6.1.1 Self-Leveling Issue in LWR 114 6.1.2 Self-Leveling Behavior in Terms of Debris Coolability 114 6.2 Benchmark Experiment 116 6.3 SPH-DEM Simulation Setup 118 6.3.1 Properties and Simulation Conditions 118 6.3.2 Sequence of SPH-DEM Leveling Simulation 120 6.3.3 Determination of Inclined Angle 121 6.4 Validation Results and Discussions 121 6.4.1 SPH-DEM Simulation Results 121 6.4.2 Validation Result 122 6.4.3 Effect of Gas Flow Rate 122 Chapter 7 Summary 129 7.1 Summary 129 7.2 Recommendations 131 References 134 국문 초록 142Docto

Smoothed Particle Hydrodynamics for Navier-Stokes Fluid Flow Application

The aim of this publication is to introduce the particle based computational fluid dynamics (CFD) method smoothed particle hydrodynamics (SPH) and introduce an applicable and valid SPH implementation for practical cases. For this purpose, current research approaches are combined regarding performance and numerical stability.  The principles of the method, the mathematical basics and the discretization of the Navier-Stokes equations are clarified. Furthermore, the implementation of method-specific boundary conditions, wall, inlet and outlet, as well as several correction procedures and a surface tension setup into the present code framework are described. The advantages and validity of the method are shown based on different cases. The free surface fluid behavior of a dam break is compared to experimental data of the time dependent water level of selected positions. A Karman vortex street is validated by its Strouhal number for different Reynolds numbers. The frequency of an oscillating drop is analysed and compared to the analytical solution.  The SPH is utilized for pipe flows influenced by a backward facing step and shows an expected qualitative flow field

Pairwise Force SPH Model for Real-Time Multi-Interaction Applications

In this paper, we present a novel pairwise-force smoothed particle hydrodynamics (PF-SPH) model to allow modeling of various interactions at interfaces in real time. Realistic capture of interactions at interfaces is a challenging problem for SPH-based simulations, especially for scenarios involving multiple interactions at different interfaces. Our PF-SPH model can readily handle multiple kinds of interactions simultaneously in a single simulation; its basis is to use a larger support radius than that used in standard SPH. We adopt a novel anisotropic filtering term to further improve the performance of interaction forces. The proposed model is stable; furthermore, it avoids the particle clustering problem which commonly occurs at the free surface. We show how our model can be used to capture various interactions. We also consider the close connection between droplets and bubbles, and show how to animate bubbles rising in liquid as well as bubbles in air. Our method is versatile, physically plausible and easy-to-implement. Examples are provided to demonstrate the capabilities and effectiveness of our approach

Enhanced SPH modeling of free-surface flows with large deformations

The subject of the present thesis is the development of a numerical solver to study the violent interaction of marine flows with rigid structures. Among the many numerical models available, the Smoothed Particle Hydrodynamics (SPH) has been chosen as it proved appropriate in dealing with violent free-surface flows. Due to its Lagrangian and meshless character it can naturally handle breaking waves and fragmentation that generally are not easily treated by standard methods. On the other hand, some consolidated features of mesh-based methods, such as the solid boundary treatment, still remain unsolved issues in the SPH context. In the present work a great part of the research activity has been devoted to tackle some of the bottlenecks of the method. Firstly, an enhanced SPH model, called delta-SPH, has been proposed. In this model, a proper numerical diffusive term has been added in the continuity equation in order to remove the spurious numerical noise in the pressure field which typically affects the weakly-compressible SPH models. Then, particular attention has been paid to the development of suitable techniques for the enforcement of the boundary conditions. As for the free-surface, a specific algorithm has been designed to detect free-surface particles and to define a related level-set function with two main targets: to allow the imposition of peculiar conditions on the free-surface and to analyse and visualize more easily the simulation outcome (especially in 3D cases). Concerning the solid boundary treatment, much effort has been spent to devise new techniques for handling generic body geometries with an adequate accuracy in both 2D and 3D problems. Two different techniques have been described: in the first one the standard ghost fluid method has been extended in order to treat complex solid geometries. Both free-slip and no-slip boundary conditions have been implemented, the latter being a quite complex matter in the SPH context. The proposed boundary treatment proved to be robust and accurate in evaluating local and global loads, though it is not easy to extend to generic 3D surfaces. The second technique has been adopted for these cases. Such a technique has been developed in the context of Riemann-SPH methods and in the present work is reformulated in the context of the standard SPH scheme. The method proved to be robust in treating complex 3D solid surfaces though less accurate than the former. Finally, an algorithm to correctly initialize the SPH simulation in the case of generic geometries has been described. It forces a resettlement of the fluid particles to achieve a regular and uniform spacing even in complex configurations. This pre-processing procedure avoids the generation of spurious currents due to local defects in the particle distribution at the beginning of the simulation. The delta-SPH model has been validated against several problems concerning fluid-structure interactions. Firstly, the capability of the solver in dealing with water impacts has been tested by simulating a jet impinging on a flat plate and a dam-break flow against a vertical wall. In this cases, the accuracy in the prediction of local loads and of the pressure field have been the main focus. Then, the viscous flow around a cylinder, in both steady and unsteady conditions, has been simulated comparing the results with reference solutions. Finally, the generation and propagation of 2D gravity waves has been simulated. Several regimes of propagation have been tested and the results compared against a potential flow solver. The developed numerical solver has been applied to several cases of free-surface flows striking rigid structures and to the problem of the generation and evolution of ship generated waves. In the former case, the robustness of the solver has been challenged by simulating 2D and 3D water impacts against complex solid surfaces. The numerical outcome have been compared with analytical solutions, experimental data and other numerical results and the limits of the model have been discussed. As for the ship generated waves, the problem has been firstly studied within the 2D+t approximation, focusing on the occurrence and features of the breaking bow waves. Then, a dedicated 3D SPH parallel solver has been developed to tackle the simulation of the entire ship in constant forward motion. This simulation is quite demanding in terms of complexities of the boundary geometry and computational resources required. The wave pattern obtained has been compared against experimental data and results from other numerical methods, showing in both the cases a fair and promising agreement

Toward the development of a virtual spray test-rig using the Smoothed Particle Hydrodynamics method

In this work we present the numerical simulation of air-assisted liquid atomization at high pressure us- ing the Smoothed Particle Hydrodynamics (SPH) method. Different post-processing tools are applied to facilitate the comparison with experimental observations. This allows to quantitatively validate the nu- merical method against the experiment, in terms of (i) frequency of the Kelvin–Helmholtz instability that develops on the jet surface, and (ii) statistical distribution of the jet intact length. The qualitative com- parison also shows a good prediction of the jet global instability and of the fragmented liquid lumps, with regards to length and time scales. In addition, the post-processing tools also give access to the local parameters of the generated spray in the vicinity of the nozzle, which are not easily accessible in a real experiments. Using these tools, 1D profiles and 2D maps of the liquid phase properties such as the vol- ume fraction, the droplet concentration, the Sauter Mean Diameter (SMD) and the droplet sphericity are presented. Because of the Lagrangian nature of the SPH method, it is also possible to monitor the whole atomization cascade as a causal tree, from the primary instabilities to the spray characteristics. This tree contains various information such as the fragmentation spectrum and the breakup activity, which are of great interest for researchers and engineers. Hence, the capability of the Smoothed Particle Hydrody- namics (SPH) method for simulating air-assisted atomization at high ambient pressure is demonstrated as well as its applicability to realistic configurations. This is a first step towards the development of a complete virtual spray test-rig

Development of GPU-based SPH Framework for Hydrodynamic Interactions With Non-spherical Solid Debris

일본의 후쿠시마 사고 이후 원자로 중대 사고에 대한 연구의 필요성과 대처 능력 확보에 대한 중요성이 점점 증가하고 있다. 사고 시 발생할 수 있는 노심 용융물 거동에 대한 평가는 용융물-콘크리트 상호작용(MCCI, Molten Core Concrete Interaction)과 증기 폭발로부터의 원자로 노심 냉각성 및 건전성에 따른 재임계 측면에서 매우 중요하다. 특히 OPR 1000의 경우, 사전 충수 조건(Wet cavity condition)을 기본적인 원자로 외벽 냉각 대응 전략으로 채택함으로써 핵연료-냉각재 상호작용(FCI, Fuel Coolant Interaction) 반응이 필연적으로 발생하는 것으로 알려져 있다. [Jin, 2014] FCI 현상은 임의 형태의 핵연료 고체 파편물과 냉각재의 상호작용뿐만 아니라, 냉각재 비등 현상 등도 포함하는 다유체, 다상 현상으로 그 현상이 매우 복잡하다. 이 과정에서 원자로 건물 하부에 고체 파편물이 퇴적되어 잔해 층이 형성되고, 그 냉각성에 따라 사고의 다음 진행 상황에 영향을 줄 수 있다. 이러한 비구형 고체 파편물 거동에 대한 이해를 위해 강체 개념을 적용한 고체 해석 체계는 좋은 접근법이 될 수 있다. 따라서 본 연구에서는 유체와 고체 간 수력학적 상호작용 해석을 위해 입자유체동역학(SPH, Smoothed Particle Hydrodynamics) 기법과 강체역학(RBD, Rigid Body Dynamics) 기법을 연계하여 라그랑지안 해석 체계를 구축하였다. 완화입자유체동역학 기법은 해석 유체를 유한개의 입자로 표현함으로써 유동을 해석하는 라그랑지안 해석 기법 중 하나이다. 개별 입자들의 움직임으로 유동을 해석하므로 비선형의 대류항에 대한 계산이 필요 없으며, 입자가 추가되거나 사라지지 않는 한 해석 계의 전체 질량은 자동으로 보존된다. 이러한 라그랑지안 기법의 특성으로 SPH 방법은 자유 표면 유동, 다유체 유동, 다상 유동, 형태 변화가 큰 유동 등에 대해 해석 장점을 갖는다. 본 연구에서는 SPH 기법을 적용한 in-house SOPHIA 코드를 사용하여 비압축 다상 유동 해석을 수행하였으며, 벤치마크 데이터와의 비교에서 좋은 검증 해석 결과를 보였다. 강체역학은 외력에 의해 형태가 변하지 않는 강체의 개념을 이용하여 고체의 병진 운동과 회전 운동을 해석하는 연구 분야이다. 본 연구에서는 이산요소법(DEM, Discrete Element Method) 분야에서 오랜 시간 동안 널리 사용되고 검증되었던 Hertz-Mindlin 충돌 모델을 적용하여 강체 간 충돌 해석을 구현하였다. 강체는 유한개의 입자들로 표현할 수 있으며, 강체 간 충돌은 각 강체를 구성하고 있는 입자쌍의 작은 중첩을 기반으로 계산된다. 본 연구에서는 입자기반의 강체역학 해석 코드를 이용하여 단일 강체 및 다중 강체 충돌에 대해 검증 해석을 수행하였으며, 해석해 및 벤치마크 데이터 결과와 잘 일치함을 확인하였다. 원자력 분야에서 발생할 수 있는 비구형 고체와 유체간 상호작용 해석을 위해 앞서 설명한 SPH 기법과 강체역학 연계 해석 코드를 개발하였다. 본 연구에서 적용한 완전 해상 방식(Fully resolved approach)은 유체-고체의 상이 분리되어 있고, 제 1 원리를 만족하므로 고체의 형상에 따른 상관식과 표면 적분이 필요하지 않다는 장점이 있다. 또한 고체 경계면에서의 정확한 압력 계산을 위해 유체 입자 정보를 기반으로 노이만 압력 경계 조건을 적용하였다. 본 연구에서는 이러한 해상 방식의 유체-강체 연계 해석 코드를 이용하여 비구형 고체와 유체의 수력학적 상호작용에 대한 검증 해석을 수행하였으며, 선행된 실험과의 비교에서 좋은 결과를 보였다. 유동 해석을 위해 본 연구에 적용한 SPH 방법에서는 수식들이 매우 선형적이고 외연적(Explicit)으로 계산을 수행하기 때문에 각 입자에 대한 계산이 독립적으로 수행되어도 문제가 없다. 따라서 SPH 방법은 계산 병렬화에 최적화된 방법으로 잘 알려져 있으며, 대규모 고해상도 해석을 위해 이는 필수적이다. 또한 입자 기반의 강체 계산을 위해서는 효율적인 계산 알고리즘이 필요하다. 따라서 본 연구에서는 대규모 계산과 높은 연산 효율성을 위해 그래픽처리장치(GPU, Graphic Processing Unit)를 이용하여 계산 병렬화를 수행하였으며, 이를 이용한 다중 고체와 유체의 상호작용 해석에서 좋은 계산 성능을 확인하였다. 본 연구에서 수행한 비구형 고체와 유체의 수력학적 상호작용을 위한 GPU 기반의 SPH 해석 코드 개발 연구를 통해 원자로 중대사고 시 발생할 수 있는 냉각재와 핵연료 고체 파편물의 수력학적 상호작용 뿐만 아니라, 고체 파편물 간 역학적 상호작용에 대해 효율적인 해석 체계를 개발하였다. 이를 통해 습식 공동(wet cavity)에서 발생하는 핵연료 고체 파편물의 퇴적 작용, 쓰나미 사고로 인한 해안 구조물의 수력학적 상호작용, 그리고 침수 사고 시 원자로 건물 내 부유물의 거동 등 원자력 분야에서 발생할 수 있는 다양한 고체-유체의 수력학적 상호작용에 대한 해석적 연구에 적용하고 기여할 수 있을 것으로 기대한다.Since the Fukushima accident, the necessity for researches on severe accidents and the importance of securing the ability to cope with the accidents have been increasing. The evaluation of the molten core behavior that may occur during the accident is very important in terms of re-criticality according to the coolability and integrity of the reactor core from the MCCI (Molten Core Concrete Interaction) and steam explosion. In the case of OPR 1000, especially, FCI (Fuel Coolant Interaction) is known to occur unconditionally because the wet cavity condition has been adopted as a basic strategy for ex-vessel cooling. [Jin, 2014] FCI is a highly complicated phenomenon, which includes multi-fluid, multi-phase interaction between the arbitrary shape of solid debris and coolant as well as coolant boiling. In this process, the debris bed is formed at the bottom of the containment, and its coolability influences the next phase of the accident. For the understanding on the solid debris behavior, a solid system with a rigid body can be a good approach for the non-spherical solid debris analysis. Therefore, in this study, Smoothed Particle Hydrodynamics (SPH) method and Rigid Body Dynamics (RBD) are coupled in a fully Lagrangian manner for the hydrodynamic interactions between fluid and solid. Smoothed Particle Hydrodynamics (SPH) is one of the Lagrangian-based analysis methods which represents the fluid flow as a finite number of particles. Since the flow is analyzed by the motion of individual particles, there is no need to calculate the nonlinear convective term, and the total mass of the system is automatically conserved as long as particles are not added or removed. Through these Lagrangian nature, it is well known that the SPH method is effective for the free surface flow, multi-fluid and multi-phase flow, and highly deformable flow. In this study, the incompressible multi-phase flow analysis has been performed using the in-house SPH code, SOPHIA code, and V&V simulation results showed good agreement with the benchmark data. Rigid Body Dynamics (RBD) is a research field that analyses the translation and rotation of a solid body by using the concept that a rigid body doesn’t change its shape by external forces. In this study, the collision calculation between rigid bodies is implemented by applying the Hertz-Mindlin contact force model commonly used and verified for a long time in the Discrete Element Method (DEM) field. A rigid body can be expressed as a group of finite particles, and the contact forces between solid bodies are calculated based on the small overlap of the particle pairs. Using the particle-based RBD analysis code implemented in this study, V&V simulations on single- and multi- rigid body collisions have been performed and showed good agreement with the analytical solution and the benchmark data. To analyze the hydrodynamic interactions between non-spherical solids and fluids that can occur in the nuclear field, the integrated code has been developed by coupling RBD with SPH code. Since a fully resolved approach adopted in this study as a phase coupling method satisfies the 1st principle and the fluid-solid phase is entirely separated from each other, there is no need for the surface integral and empirical correlations depending on the solid geometry. In addition, the Neumann pressure boundary condition is implemented for accurate pressure estimation at the solid interface using the fluid particle properties. By applying the resolved SPH-RBD coupled code, V&V simulations were carried out on the hydrodynamic interactions of non-spherical solid-fluid and showed good agreement with the experimental data. In the SPH method, since the numerical expression are highly linear and the calculations are performed explicitly, there is no problem even if the calculations for each particle are performed independently. Therefore, the SPH is well known as an optimized method for parallelization, and it is essential for large scale high-resolution simulations. In addition, an efficient computational algorithm is required for particle-based rigid body calculation. In this study, therefore, the parallelization was performed using a Graphical Processing Unit (GPU) for large-scale calculations and high computational efficiency, and it showed a good performance in analyzing the interactions of a large number of solids and fluids particles. Through the researches on the development of a GPU-based SPH framework for the hydrodynamic interaction of non-spherical solids and fluids in this study, an efficient analysis system has been developed for not only the hydrodynamic interaction of solid corium debris with coolant but also the mechanical interaction between solid debris which can occur at the severe accidents in the nuclear reactor. By using this, it is expected that the integrated code will contribute to analytical researches on various accident scenarios that may occur in the nuclear field such as solid fuel debris sedimentation in the wet cavity, hydrodynamic interactions with coastal structures caused by the Tsunami, and the behavior of floating objects in the reactor building at the flooding accident, etc.Chapter 1 Introduction 1 1.1 Background and Motivation 1 1.2 Previous Studies 3 1.2.1 Numerical Studies on FCI Premixing Jet Breakup 3 1.2.2 Numerical Studies on Fluid-Solid Coupling with RBD 4 1.3 Objectives and Scope 5 Chapter 2 Smoothed Particle Hydrodynamics (SPH) 9 2.1 SPH Overview 9 2.1.1 Basic Concept of SPH 9 2.1.2 SPH Particle Approximation 10 2.1.3 SPH Kernel Function 12 2.1.4 SPH Governing Equations 13 2.2 SPH Multi-phase Models 16 2.2.1 Normalized Density Approach 16 2.2.2 Treatments for Multi-phase Flow 17 2.2.3 Surface Tension Force Model 18 2.3 SPH Code Implementation 20 2.3.1 Nearest Neighbor Particle Search (NNPS) 20 2.3.2 Algorithm of SPH Code 21 2.3.3 Time Integration 21 2.3.4 GPU Parallelization 22 Chapter 3 Rigid Body Dynamics (RBD) 30 3.1 RBD Overview 30 3.2 Collision Models of Rigid Body 31 3.2.1 Monaghan Boundary Force (MBF) Model 31 3.2.2 Ideal Plastic Collision Model 33 3.2.3 Impulse-based Boundary Force (IBF) Model 35 3.2.4 Penalty-based Contact Model 37 3.2.5 Determination of Collision Model 40 3.3 Algorithm of RBD 41 3.3.1 Calculation of Rigid Body Information 41 3.3.2 Contact Detection 42 3.3.3 Contact Normal Calculation 42 3.3.4 Contact Force Calculation 45 3.3.5 Summation of Rigid Body Particles 46 3.3.6 Time Integration 47 3.4 GPU Parallelization 48 3.4.1 Algorithm 1: Atomic Operation 49 3.4.2 Algorithm 2: Sorting 50 3.5 Code V&V Simulations 51 3.5.1 Conservation of Momentum & Angular Momentum 51 3.5.2 Conservation of Kinetic Energy in Elastic Collision 52 3.5.3 Bouncing Block 53 3.5.4 Sliding Block on a Slope 55 3.5.5 Collapse of Stacked Multi-body 57 Chapter 4 Two-way Coupling of SPH-RBD 75 4.1 Resolved Approach 75 4.2 Governing Equations 75 4.2.1 Solid Phase 75 4.2.2 Fluid Phase 78 4.3 Algorithm of SPH-RBD Code 78 4.4 Code V&V Simulations 81 4.4.1 Karman Vortex Problem 81 4.4.2 Water Entry 84 4.4.3 Sinking & Rotating Body 85 4.4.4 Floating & Falling Body 85 4.4.5 Collapse of Stacked Multi-body with Fluid 87 4.4.6 Code Application to Non-spherical Debris Sedimentation 89 Chapter 5 Conclusion 110 5.1 Summary 110 5.2 Recommendations 112 Nomenclature 114 Bibliography 117 국문 초록 127박

An improved SPH scheme for cosmological simulations

We present an implementation of smoothed particle hydrodynamics (SPH) with improved accuracy for simulations of galaxies and the large-scale structure. In particular, we combine, implement, modify and test a vast majority of SPH improvement techniques in the latest instalment of the GADGET code. We use the Wendland kernel functions, a particle wake-up time-step limiting mechanism and a time-dependent scheme for artificial viscosity, which includes a high-order gradient computation and shear flow limiter. Additionally, we include a novel prescription for time-dependent artificial conduction, which corrects for gravitationally induced pressure gradients and largely improves the SPH performance in capturing the development of gas-dynamical instabilities. We extensively test our new implementation in a wide range of hydrodynamical standard tests including weak and strong shocks as well as shear flows, turbulent spectra, gas mixing, hydrostatic equilibria and self-gravitating gas clouds. We jointly employ all modifications; however, when necessary we study the performance of individual code modules. We approximate hydrodynamical states more accurately and with significantly less noise than standard SPH. Furthermore, the new implementation promotes the mixing of entropy between different fluid phases, also within cosmological simulations. Finally, we study the performance of the hydrodynamical solver in the context of radiative galaxy formation and non-radiative galaxy cluster formation. We find galactic disks to be colder, thinner and more extended and our results on galaxy clusters show entropy cores instead of steadily declining entropy profiles. In summary, we demonstrate that our improved SPH implementation overcomes most of the undesirable limitations of standard SPH, thus becoming the core of an efficient code for large cosmological simulations.Comment: 21 figures, 2 tables, accepted to MNRA