Pressure corrected SPH for fluid animation
|
|
- Tamsin Griffith
- 6 years ago
- Views:
Transcription
1 Pressure corrected SPH for fluid animation Kai Bao, Hui Zhang, Lili Zheng and Enhua Wu Analyzed by Po-Ram Kim 2 March 2010
2 Abstract We present pressure scheme for the SPH for fluid animation In conventional SPH method, EOS are used For volume conservation, high speeds of sound are required à numerical instability In the paper, a new extra pressure correction scheme is proposed Po-Ram Kim 2 March 2010 # 2
3 Abstract Smoother pressure distribution and more efficient simulation are achieved The proposed method has been used to simulate free surface problems Surface tension and fluid fragmentation can be well handled Po-Ram Kim 2 March 2010 # 3
4 Introduction SPH is a method to capture characteristics and thus becomes more and more widely used To simulate free surface flow, one crucial problem is to ensure the incompressibility of the flow There are two main ways to achieve this The divergence of the velocity field à zero Poisson equation : time consuming solution Keeping the fluid density constant Poisson equation : time consuming solution Moving Particle Semi-implicit (MPS) method Po-Ram Kim 2 March 2010 # 4
5 Introduction In the conventional SPH method EOS are used to directly associate the pressure with particle density The time-consuming solution of Poisson equation is avoided : widely used But, it proved to be hard to guarantee the incompressibility of free surface flow Tait s equation The volume of fluid is generally well conserved But, a high sound speed has to be used àthe time step is too small little difference in density will lead to large variation in pressure Po-Ram Kim 2 March 2010 # 5
6 Introduction In the paper An iterative pressure correction scheme is proposed to be used along with the EOS The local pressure disturbance is made to propagate to the neighboring areas Smoother pressure distribution Incompressible fluid More accurate and efficient simulation Smaller sound speeds and larger time steps are made possible Enhancement in the surface tension model Po-Ram Kim 2 March 2010 # 6
7 Previous Work In Eulerain methods The incompressibility is easy to enforce While the mass conservation for small features is not well guaranteed Widely used in the physically based animation and many fluid phenomena Po-Ram Kim 2 March 2010 # 7
8 Previous Work In Largrangian particles(sph) First introduced for highly deformable bodies Grids are required during the computation Mass conservation is naturally guaranteed Surface tracking techniques are required Large deformation and violent fragmentation can be handled Interactive simulation of free surface flow was achieved using SPH by Muller et al Po-Ram Kim 2 March 2010 # 8
9 Previous Work Free-surface flow Highly viscous fluids Po-Ram Kim 2 March 2010 # 9
10 Previous Work Solid-fluid coupling An adaptive sampling technique Po-Ram Kim 2 March 2010 # 10
11 Methodology Basic SPH Formations Lagrangian form of the Navier Stokes equation Conservation of mass Conservation of momentum v : velocity vector, p : pressure, ρ : fluid density, g : gravitational acceleration vector, : kinetic viscosity Po-Ram Kim 2 March 2010 # 11
12 Methodology Basic SPH Formations To evaluate the value f at an arbitrary position x an interpolation is applied with the neighboring particles : Particle approximation j i f j : the value of f at the position of particle j, W : smoothing kernel function m : mass, ρ : density Po-Ram Kim 2 March 2010 # 12
13 Methodology Basic SPH Formations By applying the SPH particle approximation to the momentum equation(equation (2)) j i p j : the pressure of particle i, : direction gradient to particle i Po-Ram Kim 2 March 2010 # 13
14 Methodology Density Computation Two main approaches to determine the density of particles in the traditional SPH The density summation method Tracking the evolution of the density through the continuity equation Po-Ram Kim 2 March 2010 # 14
15 Methodology Density Computation The density summation method (ßwidely used) Advantage It conserves the mass exactly Disadvantage It suffers from particle deficiency near the boundary In the paper, the continuity density approach (equation(5)) is used Po-Ram Kim 2 March 2010 # 15
16 Methodology Equations of State Incompressible fluid liquids Compressible fluid Gases A theoretically incompressible flow is practically compressible Artificial compressibility is introduced Weakly compressible The pressure is determined with EOS This approach for free surface flow àthe volume of the flow is hard to be well conserved Po-Ram Kim 2 March 2010 # 16
17 Methodology Equations of State Equation of State k p = c 2, c : the sound of speed, ρ0 : reference density Tait s equation àthe variations of density remain small àthe volume of the fluid is generally well conserved Po-Ram Kim 2 March 2010 # 17
18 Methodology Equations of State Tait s equation Small deviation in density field will result in large fluctuation in pressure Noisy pressure distribution will be obtained Numerical instability Po-Ram Kim 2 March 2010 # 18
19 Methodology Equations of State Time step CFL Viscous condition External force condition force condition With the Tait s equation, A high speed of sound is required To keep density fluctuation low à Small time step has to be used To keep the density variation under the order of 1% Sound speed = 10 * (maximum possible velocity) In ref[8] Time step = 4.52*10-4 Po-Ram Kim 2 March 2010 # 19
20 Methodology Pressure Correction Equation For a truly incompressible flow dr ρ = constant à = 0 dt equation(1) à Ñ à divergence-free field v = 0 To obtain a divergence free field, the classical prejec tion method is used Ñ 2 r p = Ñ v dt * v* : intermediate velocity field without applying the pressure in momentum equation However, solving poisson equation proves to be very time consuming Po-Ram Kim 2 March 2010 # 20
21 Methodology Pressure Correction Equation To resolve The noisy pressure disturbance Instability arising from the EOS To avoid the expensive solution of global Poisson equation A flexible pressure correction equation is presented Po-Ram Kim 2 March 2010 # 21
22 Methodology Pressure Correction Equation By substituting the equation(6) into continuity equation(equation(1)), the following equation could be obtained dp dt 2 + r c Ñ v = 0 (9) Variational method di ( f) = di( f) dt dt p dp = k p dt dr 1 = dt k equation(1) 1 k p = dp dt k dp dt + p c ( r - r ) 2 p dr dt dp dt = 0 + rñ v rñ v = = 0 0 Po-Ram Kim 2 March 2010 # 22
23 Methodology Pressure Correction Equation Equation(9) can be written in SPH form as If the computation is convergent, RHS of equation(10) should be zero A pressure correction value could be obtained by Since the pressure correction scheme is iterative, a counting number n is introduced Po-Ram Kim 2 March 2010 # 23
24 Methodology Pressure Correction Equation The pressure at the new iteration is written as ω is the relaxation factor with a value under 1.0 With the pressure correction value, the velocity correction value can be obtained with the momentum equation v is the kinetic viscosity Po-Ram Kim 2 March 2010 # 24
25 Methodology Pressure Correction Equation The velocity correction can be obtained by The velocity is updated with Ω is the relaxation factor with a value under 1.0 Po-Ram Kim 2 March 2010 # 25
26 Methodology Pressure Correction Equation In each SPH time step Equations (11) and (15) are solved iteratively until convergent During the iteration Pressure disturbance will propagate to the neighboring particles Smoother pressure distribution will be obtained The pressure correction scheme actually provides a combination of the EOS method and the global pressure Poisson method With larger speed of sound, less pressure correction iterations will be required Po-Ram Kim 2 March 2010 # 26
27 Methodology Surface Tension Model Surface tension plays a fundamental role in many fluid phenomena Fluid breaking Droplet dynamics The surface tension results from the uneven molecular forces of attraction near the surface The surface tension will lead to a net force in the direction of surface normal Po-Ram Kim 2 March 2010 # 27
28 Methodology Surface Tension Model In SPH method, widely used form σ : Tension coefficient Smoother surface tension force Po-Ram Kim 2 March 2010 # 28
29 Results and Discussions All the simulations are performed within a single thread Intel Core2 Q6700 CPU 8GB RAM The reference densities in all the simulations All the 2D results are rendered with OpenGL All the 3D results with POVRay Po-Ram Kim 2 March 2010 # 29
30 Results and Discussions Divergence The particles are represented by dots The velocities of the particles are displayed with line segments starting from the positions of the particles Located in the rectangle of The initial spacing of the particles is fluid particles are used in the simulation A speed of sound of 40 is taken The time step is second Po-Ram Kim 2 March 2010 # 30
31 Results and Discussions Divergence Initial velocity : (0.5,0.0) Figure1(a-1) Figure1(a-2) Figure1(a-3) Figure1(a-4) Po-Ram Kim 2 March 2010 # 31
32 Results and Discussions Divergence Initial velocity : (0.5,0.5) Figure1(b-1) Figure1(b-2) Figure1(b-3) Figure1(b-4) Po-Ram Kim 2 March 2010 # 32
33 Results and Discussions Divergence We consider the computation has been convergent Usually, several times of the iterations are enough c = 5 and dt = second exactly the same correction results are obtained Po-Ram Kim 2 March 2010 # 33
34 Results and Discussions Pressure Distribution Dam-break flow Initial height of water body = 2m Initial width of water body = 1m Initial particle spacing = 0.02m Total 5000 fluid particles are used In figure 2 Figure 2a : the pressure correction scheme is NOT used Figure 2b : the pressure correction scheme is used Purple color : the highest pressure Red color : the lowest pressure Po-Ram Kim 2 March 2010 # 34
35 Results and Discussions Pressure Distribution c = 102gH 62.6m/second dt = c = 30 m/second dt = figure2 Po-Ram Kim 2 March 2010 # 35
36 Results and Discussions Pressure Distribution As shown from the Figure 2, Without the pressure correction, the pressure fields obtained are unphysically noisy The pressure noise is significantly reduced Smoother pressure distribution is achieved Po-Ram Kim 2 March 2010 # 36
37 Results and Discussions Surface Tension The initial side length of cube is m The initial spacing of the particles is m About 12K particles in total are used in the simulation. dt = sec It takes about 0.3 second for one time step of simulation As the energy is damped by viscous and numerical dissipation The particles are stable at a spherical shape It takes very long time to achieve Po-Ram Kim 2 March 2010 # 37
38 Results and Discussions Surface Tension Evolution of a drop initially in cube shape under effect of surface tension with zero gravity 0.0sec sec sec se c 0.114sec 2.0sec Figure 3 Po-Ram Kim 2 March 2010 # 38
39 Results and Discussions Dam-break on a Wet Bed Initial value Height = 0.45m Length = 0.32m Width = 0.4m Initial water depth on the bed region = Initial spacing of the particles = 0.006m Total # of particle = 267k Time step = sec Simulation time = 7.5 sec Po-Ram Kim 2 March 2010 # 39
40 Results and Discussions Dam-break on a Wet Bed The free surface shape is the main focus of this simulation At the initial time A mushroom shape in free surface Two breaking waves enclosing voids will be generated 0.0sec 0.19sec 0.34sec 0.72sec Figure 4 Po-Ram Kim 2 March 2010 # 40
41 Results and Discussions Dam-break Flow on Complicated Topography Digital elevation model (DEM) data is used To generate the terrain surface Then a distance field is generated to enforce the solid boundary condition Initial velocity of water body is 3m/sec The initial spacing of particles is 0.04m Total # of particle is 330k particles The time step is second When water interacts with the terrain surface Violent breakage and fragmentation occur Wave propagation and reflection are well produced Po-Ram Kim 2 March 2010 # 41
42 Results and Discussions Dam-break Flow on Complicated Topography 0.0sec 0.5sec 1.25sec 2.9sec 4.75sec 7.9sec Po-Ram Kim 2 March 2010 # 42
43 Results and Discussions CASA 2009 A simulation of dambreak with an obstacle in CASA 2009 shape is carried out When the water flows over the obstacle, violent breaking is produced When the flow settles down The shape of the terrain obstacle becomes visible The initial spacing of particles is 0.005m The time step is second Total # of particle is 310k particles Po-Ram Kim 2 March 2010 # 43
44 Results and Discussions CASA sec 0.34sec 1.14sec 4.64sec Figure 6 Po-Ram Kim 2 March 2010 # 44
45 Conclusions and Future Work In the paper, A pressure correction equation is proposed for free surface flow The pressure disturbance incurred by the EOS is reduced No solution of pressure Poisson equation is required More accurate and efficient simulation is achieved The improved SPH method has been used in free surface and surface tension problem simulation Po-Ram Kim 2 March 2010 # 45
46 Conclusions and Future Work Our ongoing work Investigation of numerical properties of the pressure correction scheme Its applications to more fluid phenomena, such as multi-phase flow Po-Ram Kim 2 March 2010 # 46
Modeling, Simulating and Rendering Fluids. Thanks to Ron Fediw et al, Jos Stam, Henrik Jensen, Ryan
Modeling, Simulating and Rendering Fluids Thanks to Ron Fediw et al, Jos Stam, Henrik Jensen, Ryan Applications Mostly Hollywood Shrek Antz Terminator 3 Many others Games Engineering Animating Fluids is
More informationMath background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
More informationFluid Animation. Christopher Batty November 17, 2011
Fluid Animation Christopher Batty November 17, 2011 What distinguishes fluids? What distinguishes fluids? No preferred shape Always flows when force is applied Deforms to fit its container Internal forces
More informationCSCI1950V Project 4 : Smoothed Particle Hydrodynamics
CSCI1950V Project 4 : Smoothed Particle Hydrodynamics Due Date : Midnight, Friday March 23 1 Background For this project you will implement a uid simulation using Smoothed Particle Hydrodynamics (SPH).
More informationAn Overview of Fluid Animation. Christopher Batty March 11, 2014
An Overview of Fluid Animation Christopher Batty March 11, 2014 What distinguishes fluids? What distinguishes fluids? No preferred shape. Always flows when force is applied. Deforms to fit its container.
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More informationPDE Solvers for Fluid Flow
PDE Solvers for Fluid Flow issues and algorithms for the Streaming Supercomputer Eran Guendelman February 5, 2002 Topics Equations for incompressible fluid flow 3 model PDEs: Hyperbolic, Elliptic, Parabolic
More informationComputational Astrophysics
Computational Astrophysics Lecture 1: Introduction to numerical methods Lecture 2:The SPH formulation Lecture 3: Construction of SPH smoothing functions Lecture 4: SPH for general dynamic flow Lecture
More informationSoft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies
Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed
More informationParticle-based Fluids
Particle-based Fluids Particle Fluids Spatial Discretization Fluid is discretized using particles 3 Particles = Molecules? Particle approaches: Molecular Dynamics: relates each particle to one molecule
More informationImprovement of Calculation Stability for Slow Fluid Flow Analysis Using Particle Method *
Materials Transactions, Vol. 58, No. 3 (2017) pp. 479 to 484 2017 Japan Foundry Engineering Society Improvement of Calculation Stability for Slow Fluid Flow Analysis Using Particle Method * Naoya Hirata
More informationSmoothed Particle Hydrodynamics (SPH) Huamin Wang
Smoothed Particle Hydrodynamics (SPH) Huamin Wang Fluid Representation Fluid is represented using a set of particles. Particle i has position x i, velocity v i, and mass m i. Animation Example Using 10
More informationPhysics-Based Animation
CSCI 5980/8980: Special Topics in Computer Science Physics-Based Animation 13 Fluid simulation with grids October 20, 2015 Today Presentation schedule Fluid simulation with grids Course feedback survey
More informationDan s Morris s Notes on Stable Fluids (Jos Stam, SIGGRAPH 1999)
Dan s Morris s Notes on Stable Fluids (Jos Stam, SIGGRAPH 1999) This is intended to be a detailed by fairly-low-math explanation of Stam s Stable Fluids, one of the key papers in a recent series of advances
More informationCHAPTER 4. Basics of Fluid Dynamics
CHAPTER 4 Basics of Fluid Dynamics What is a fluid? A fluid is a substance that can flow, has no fixed shape, and offers little resistance to an external stress In a fluid the constituent particles (atoms,
More informationA PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM
A PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM Hyun Geun LEE 1, Jeong-Whan CHOI 1 and Junseok KIM 1 1) Department of Mathematics, Korea University, Seoul
More informationMulti-physics CFD simulation of three-phase flow with MPS method
APCOM & ISCM 11-14 th December, 2013, Singapore Abstract Multi-physics CFD simulation of three-phase flow with MPS method *Ryouhei Takahashi¹, Makoto Yamamoto 2 and Hiroshi Kitada 1 1 CMS Corporation,
More informationThe Effect of Baffles on Fluid Sloshing inside the Moving Rectangular Tanks
The Effect of Baffles on Fluid Sloshing inside the Moving Rectangular Tanks Krit Threepopnartkul and Chakrit Suvanjumrat * Department of Mechanical Engineering, Faculty of Engineering, Mahidol University,
More informationSPH Molecules - a model of granular materials
SPH Molecules - a model of granular materials Tatiana Capone DITS, Univeristy of Roma (la Sapienza) Roma, Italy Jules Kajtar School of Mathematical Sciences Monash University Vic. 3800, Australia Joe Monaghan
More informationInvestigation of an implicit solver for the simulation of bubble oscillations using Basilisk
Investigation of an implicit solver for the simulation of bubble oscillations using Basilisk D. Fuster, and S. Popinet Sorbonne Universités, UPMC Univ Paris 6, CNRS, UMR 79 Institut Jean Le Rond d Alembert,
More informationConservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion
http://www.nd.edu/~gtryggva/cfd-course/ http://www.nd.edu/~gtryggva/cfd-course/ Computational Fluid Dynamics Lecture 4 January 30, 2017 The Equations Governing Fluid Motion Grétar Tryggvason Outline Derivation
More informationSimulation of Particulate Solids Processing Using Discrete Element Method Oleh Baran
Simulation of Particulate Solids Processing Using Discrete Element Method Oleh Baran Outline DEM overview DEM capabilities in STAR-CCM+ Particle types and injectors Contact physics Coupling to fluid flow
More informationFluid Dynamics. Part 2. Massimo Ricotti. University of Maryland. Fluid Dynamics p.1/17
Fluid Dynamics p.1/17 Fluid Dynamics Part 2 Massimo Ricotti ricotti@astro.umd.edu University of Maryland Fluid Dynamics p.2/17 Schemes Based on Flux-conservative Form By their very nature, the fluid equations
More informationDiscussion panel 1: Enforcing incompressibility in SPH
Discussion panel 1: Enforcing incompressibility in SPH Weakly compressible or not weakly compressible, this is the question. Introduction: Andrea Colagrossi Chairman : Antonio Souto Iglesias, Physical/Mathematical
More informationPhysically Based Simulations (on the GPU)
Physically Based Simulations (on the GPU) (some material from slides of Mark Harris) CS535 Fall 2014 Daniel G. Aliaga Department of Computer Science Purdue University Simulating the world Floating point
More informationThe Bernoulli theorem relating velocities and pressures along a streamline comes from the steady momentum equation for a constant density fluid,
Flow Science Report 0-14 ADDING FLOW LOSSES TO FAVOR METHODOLOGY C.W. Hirt Flow Science, Inc. June 014 Background From time to time it has been observed that large velocities may appear in the vicinity
More information( ) Notes. Fluid mechanics. Inviscid Euler model. Lagrangian viewpoint. " = " x,t,#, #
Notes Assignment 4 due today (when I check email tomorrow morning) Don t be afraid to make assumptions, approximate quantities, In particular, method for computing time step bound (look at max eigenvalue
More informationA unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation
A unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation S. Bordère a and J.-P. Caltagirone b a. CNRS, Univ. Bordeaux, ICMCB,
More informationThe Evolution of SPH. J. J. Monaghan Monash University Australia
The Evolution of SPH J. J. Monaghan Monash University Australia planetary disks magnetic fields cosmology radiation star formation lava phase change dam break multiphase waves bodies in water fracture
More informationFluid-soil multiphase flow simulation by an SPH-DEM coupled method
Fluid-soil multiphase flow simulation by an SPH-DEM coupled method *Kensuke Harasaki 1) and Mitsuteru Asai 2) 1), 2) Department of Civil and Structural Engineering, Kyushu University, 744 Motooka, Nishi-ku,
More informationSPH for the Modeling of non-newtonian Fluids with Thermal-Dependent Rheology
SPH for the Modeling of non-newtonian Fluids with Thermal-Dependent Rheology G. Bilotta 1,2 1 Dipartimento di Matematica e Informatica, Università di Catania, Italy 2 Istituto Nazionale Geofisica e Vulcanologia,
More informationThe Lattice Boltzmann Method for Laminar and Turbulent Channel Flows
The Lattice Boltzmann Method for Laminar and Turbulent Channel Flows Vanja Zecevic, Michael Kirkpatrick and Steven Armfield Department of Aerospace Mechanical & Mechatronic Engineering The University of
More informationTransport equation cavitation models in an unstructured flow solver. Kilian Claramunt, Charles Hirsch
Transport equation cavitation models in an unstructured flow solver Kilian Claramunt, Charles Hirsch SHF Conference on hydraulic machines and cavitation / air in water pipes June 5-6, 2013, Grenoble, France
More informationSimulating Interfacial Tension of a Falling. Drop in a Moving Mesh Framework
Simulating Interfacial Tension of a Falling Drop in a Moving Mesh Framework Anja R. Paschedag a,, Blair Perot b a TU Berlin, Institute of Chemical Engineering, 10623 Berlin, Germany b University of Massachusetts,
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationLevel Set and Phase Field Methods: Application to Moving Interfaces and Two-Phase Fluid Flows
Level Set and Phase Field Methods: Application to Moving Interfaces and Two-Phase Fluid Flows Abstract Maged Ismail Claremont Graduate University Level Set and Phase Field methods are well-known interface-capturing
More informationNumerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
More informationFEniCS Course. Lecture 6: Incompressible Navier Stokes. Contributors Anders Logg André Massing
FEniCS Course Lecture 6: Incompressible Navier Stokes Contributors Anders Logg André Massing 1 / 11 The incompressible Navier Stokes equations u + u u ν u + p = f in Ω (0, T ] u = 0 in Ω (0, T ] u = g
More information2 Equations of Motion
2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More informationLaminar Boundary Layers. Answers to problem sheet 1: Navier-Stokes equations
Laminar Boundary Layers Answers to problem sheet 1: Navier-Stokes equations The Navier Stokes equations for d, incompressible flow are + v ρ t + u + v v ρ t + u v + v v = 1 = p + µ u + u = p ρg + µ v +
More informationGas Dynamics: Basic Equations, Waves and Shocks
Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks Susanne Höfner Susanne.Hoefner@fysast.uu.se Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks
More informationA Study on Numerical Solution to the Incompressible Navier-Stokes Equation
A Study on Numerical Solution to the Incompressible Navier-Stokes Equation Zipeng Zhao May 2014 1 Introduction 1.1 Motivation One of the most important applications of finite differences lies in the field
More informationTarget Simulations. Roman Samulyak in collaboration with Y. Prykarpatskyy, T. Lu
Muon Collider/Neutrino Factory Collaboration Meeting May 26 28, CERN, Geneva U.S. Department of Energy Target Simulations Roman Samulyak in collaboration with Y. Prykarpatskyy, T. Lu Center for Data Intensive
More informationSimulation of T-junction using LBM and VOF ENERGY 224 Final Project Yifan Wang,
Simulation of T-junction using LBM and VOF ENERGY 224 Final Project Yifan Wang, yfwang09@stanford.edu 1. Problem setting In this project, we present a benchmark simulation for segmented flows, which contain
More information2. FLUID-FLOW EQUATIONS SPRING 2019
2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid
More informationThomas Pierro, Donald Slinn, Kraig Winters
Thomas Pierro, Donald Slinn, Kraig Winters Department of Ocean Engineering, Florida Atlantic University, Boca Raton, Florida Applied Physics Laboratory, University of Washington, Seattle, Washington Supported
More informationToy stars in one dimension
Mon. Not. R. Astron. Soc. 350, 1449 1456 (004) doi:10.1111/j.1365-966.004.07748.x Toy stars in one dimension J. J. Monaghan 1 and D. J. Price 1 School of Mathematical Sciences, Monash University, Clayton
More informationMULTIGRID CALCULATIONS FOB. CASCADES. Antony Jameson and Feng Liu Princeton University, Princeton, NJ 08544
MULTIGRID CALCULATIONS FOB. CASCADES Antony Jameson and Feng Liu Princeton University, Princeton, NJ 0544 1. Introduction Development of numerical methods for internal flows such as the flow in gas turbines
More informationDirect numerical simulation of compressible multi-phase ow with a pressure-based method
Seventh International Conference on Computational Fluid Dynamics (ICCFD7), Big Island, Hawaii, July 9-13, 2012 ICCFD7-2902 Direct numerical simulation of compressible multi-phase ow with a pressure-based
More informationGodunov methods in GANDALF
Godunov methods in GANDALF Stefan Heigl David Hubber Judith Ngoumou USM, LMU, München 28th October 2015 Why not just stick with SPH? SPH is perfectly adequate in many scenarios but can fail, or at least
More informationQuick Recapitulation of Fluid Mechanics
Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change
More informationPhysical Diffusion Cures the Carbuncle Phenomenon
Physical Diffusion Cures the Carbuncle Phenomenon J. M. Powers 1, J. Bruns 1, A. Jemcov 1 1 Department of Aerospace and Mechanical Engineering University of Notre Dame, USA Fifty-Third AIAA Aerospace Sciences
More informationRelativistic Hydrodynamics L3&4/SS14/ZAH
Conservation form: Remember: [ q] 0 conservative div Flux t f non-conservative 1. Euler equations: are the hydrodynamical equations describing the time-evolution of ideal fluids/plasmas, i.e., frictionless
More informationNumerical Simulations. Duncan Christie
Numerical Simulations Duncan Christie Motivation There isn t enough time to derive the necessary methods to do numerical simulations, but there is enough time to survey what methods and codes are available
More informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More informationNavier-Stokes Equation: Principle of Conservation of Momentum
Navier-tokes Equation: Principle of Conservation of Momentum R. hankar ubramanian Department of Chemical and Biomolecular Engineering Clarkson University Newton formulated the principle of conservation
More informationCalculating equation coefficients
Fluid flow Calculating equation coefficients Construction Conservation Equation Surface Conservation Equation Fluid Conservation Equation needs flow estimation needs radiation and convection estimation
More informationSmooth Particle Hydrodynamic (SPH) Presented by: Omid Ghasemi Fare Nina Zabihi XU Zhao Miao Zhang Sheng Zhi EGEE 520
Smooth Particle Hydrodynamic (SPH) Presented by: Omid Ghasemi Fare Nina Zabihi XU Zhao Miao Zhang Sheng Zhi EGEE 520 OUTLINE Ø Introduction and Historical Perspective: Ø General Principles: Ø Governing
More informationCHAPTER 2 INVISCID FLOW
CHAPTER 2 INVISCID FLOW Changes due to motion through a field; Newton s second law (f = ma) applied to a fluid: Euler s equation; Euler s equation integrated along a streamline: Bernoulli s equation; Bernoulli
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationChapter 10. Solids and Fluids
Chapter 10 Solids and Fluids Surface Tension Net force on molecule A is zero Pulled equally in all directions Net force on B is not zero No molecules above to act on it Pulled toward the center of the
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 43 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Treatment of Boundary Conditions These slides are partially based on the recommended textbook: Culbert
More informationPhysics Motivated Modeling of Volcanic Clouds as a Two Fluids Model
Physics Motivated Modeling of olcanic Clouds as a Two Fluids Model Ryoichi Mizuno Yoshinori Dobashi Bing-Yu Chen Tomoyuki Nishita The University of Tokyo {mizuno, robin, nis}@nis-lab.is.s.u-tokyo.ac.jp
More informationDaniel J. Jacob, Models of Atmospheric Transport and Chemistry, 2007.
1 0. CHEMICAL TRACER MODELS: AN INTRODUCTION Concentrations of chemicals in the atmosphere are affected by four general types of processes: transport, chemistry, emissions, and deposition. 3-D numerical
More informationNumerical Simulation of the Hagemann Entrainment Experiments
CCC Annual Report UIUC, August 14, 2013 Numerical Simulation of the Hagemann Entrainment Experiments Kenneth Swartz (BSME Student) Lance C. Hibbeler (Ph.D. Student) Department of Mechanical Science & Engineering
More informationINTER-COMPARISON AND VALIDATION OF RANS AND LES COMPUTATIONAL APPROACHES FOR ATMOSPHERIC DISPERSION AROUND A CUBIC OBSTACLE. Resources, Kozani, Greece
INTER-COMPARISON AND VALIDATION OF AND LES COMPUTATIONAL APPROACHES FOR ATMOSPHERIC DISPERSION AROUND A CUBIC OBSTACLE S. Andronopoulos 1, D.G.E. Grigoriadis 1, I. Mavroidis 2, R.F. Griffiths 3 and J.G.
More informationFluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 17 Laminar and Turbulent flows
Fluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture - 17 Laminar and Turbulent flows Welcome back to the video course on fluid mechanics. In
More informationFluid Dynamics. Massimo Ricotti. University of Maryland. Fluid Dynamics p.1/14
Fluid Dynamics p.1/14 Fluid Dynamics Massimo Ricotti ricotti@astro.umd.edu University of Maryland Fluid Dynamics p.2/14 The equations of fluid dynamics are coupled PDEs that form an IVP (hyperbolic). Use
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationLecturer, Department t of Mechanical Engineering, SVMIT, Bharuch
Fluid Mechanics By Ashish J. Modi Lecturer, Department t of Mechanical Engineering, i SVMIT, Bharuch Review of fundamentals Properties of Fluids Introduction Any characteristic of a system is called a
More informationCCC Annual Report. UIUC, August 19, Argon Bubble Behavior in EMBr Field. Kai Jin. Department of Mechanical Science & Engineering
CCC Annual Report UIUC, August 19, 2015 Argon Bubble Behavior in EMBr Field Kai Jin Department of Mechanical Science & Engineering University of Illinois at Urbana-Champaign Introduction Argon bubbles
More informationSummary PHY101 ( 2 ) T / Hanadi Al Harbi
الكمية Physical Quantity القانون Low التعريف Definition الوحدة SI Unit Linear Momentum P = mθ be equal to the mass of an object times its velocity. Kg. m/s vector quantity Stress F \ A the external force
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationKELVIN-HELMHOLTZ INSTABILITY BY SPH
II International Conference on Particle-based Methods - Fundamentals and Applications PARTICLES 2011 E. Oñate and D.R.J. Owen (Eds) KELVIN-HELMHOLTZ INSTABILITY BY SPH M. S. Shadloo, M. Yildiz Faculty
More informationSmoothed Particle Hydrodynamics (SPH) 4. May 2012
Smoothed Particle Hydrodynamics (SPH) 4. May 2012 Calculating density SPH density estimator Weighted summation over nearby particles: ρ(r) = N neigh b=1 m bw (r r b, h) W weight function with dimension
More informationd v 2 v = d v d t i n where "in" and "rot" denote the inertial (absolute) and rotating frames. Equation of motion F =
Governing equations of fluid dynamics under the influence of Earth rotation (Navier-Stokes Equations in rotating frame) Recap: From kinematic consideration, d v i n d t i n = d v rot d t r o t 2 v rot
More informationChapter 4: Fundamental Forces
Chapter 4: Fundamental Forces Newton s Second Law: F=ma In atmospheric science it is typical to consider the force per unit mass acting on the atmosphere: Force mass = a In order to understand atmospheric
More information.u= 0 ρ( u t +(u. )u)= ρ g p+.[µ( u+ t u)]
THETIS is a numerical simulation tool developed by University of Bordeaux. It is a versatile code to solve different problems: fluid flows, heat transfers, scalar transports or porous mediums. The potential
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation
More informationIntroduction to Physical Acoustics
Introduction to Physical Acoustics Class webpage CMSC 828D: Algorithms and systems for capture and playback of spatial audio. www.umiacs.umd.edu/~ramani/cmsc828d_audio Send me a test email message with
More information7 The Navier-Stokes Equations
18.354/12.27 Spring 214 7 The Navier-Stokes Equations In the previous section, we have seen how one can deduce the general structure of hydrodynamic equations from purely macroscopic considerations and
More informationHeat Transfer Benchmark Problems Verification of Finite Volume Particle (FVP) Method-based Code
PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, 16 17 MAY 2016 Heat Transfer Benchmark Problems Verification of Finite Volume
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis
More information[N175] Development of Combined CAA-CFD Algorithm for the Efficient Simulation of Aerodynamic Noise Generation and Propagation
The 32nd International Congress and Exposition on Noise Control Engineering Jeju International Convention Center, Seogwipo, Korea, August 25-28, 2003 [N175] Development of Combined CAA-CFD Algorithm for
More informationDetailed 3D modelling of mass transfer processes in two phase flows with dynamic interfaces
Detailed 3D modelling of mass transfer processes in two phase flows with dynamic interfaces D. Darmana, N.G. Deen, J.A.M. Kuipers Fundamentals of Chemical Reaction Engineering, Faculty of Science and Technology,
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationTreecodes for Cosmology Thomas Quinn University of Washington N-Body Shop
Treecodes for Cosmology Thomas Quinn University of Washington N-Body Shop Outline Motivation Multipole Expansions Tree Algorithms Periodic Boundaries Time integration Gravitational Softening SPH Parallel
More informationNeeds work : define boundary conditions and fluxes before, change slides Useful definitions and conservation equations
Needs work : define boundary conditions and fluxes before, change slides 1-2-3 Useful definitions and conservation equations Turbulent Kinetic energy The fluxes are crucial to define our boundary conditions,
More informationINTERNAL GRAVITY WAVES
INTERNAL GRAVITY WAVES B. R. Sutherland Departments of Physics and of Earth&Atmospheric Sciences University of Alberta Contents Preface List of Tables vii xi 1 Stratified Fluids and Waves 1 1.1 Introduction
More informationSome notes about PDEs. -Bill Green Nov. 2015
Some notes about PDEs -Bill Green Nov. 2015 Partial differential equations (PDEs) are all BVPs, with the same issues about specifying boundary conditions etc. Because they are multi-dimensional, they can
More informationFEM-Level Set Techniques for Multiphase Flow --- Some recent results
FEM-Level Set Techniques for Multiphase Flow --- Some recent results ENUMATH09, Uppsala Stefan Turek, Otto Mierka, Dmitri Kuzmin, Shuren Hysing Institut für Angewandte Mathematik, TU Dortmund http://www.mathematik.tu-dortmund.de/ls3
More informationNumerical simulation of wave breaking in turbulent two-phase Couette flow
Center for Turbulence Research Annual Research Briefs 2012 171 Numerical simulation of wave breaking in turbulent two-phase Couette flow By D. Kim, A. Mani AND P. Moin 1. Motivation and objectives When
More information(Refer Slide Time 1:25)
Mechanical Measurements and Metrology Prof. S. P. Venkateshan Department of Mechanical Engineering Indian Institute of Technology, Madras Module - 2 Lecture - 24 Transient Response of Pressure Transducers
More information2 GOVERNING EQUATIONS
2 GOVERNING EQUATIONS 9 2 GOVERNING EQUATIONS For completeness we will take a brief moment to review the governing equations for a turbulent uid. We will present them both in physical space coordinates
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationSemi-Lagrangian Formulations for Linear Advection Equations and Applications to Kinetic Equations
Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR.
More informationSimulation of unsteady muzzle flow of a small-caliber gun
Advances in Fluid Mechanics VI 165 Simulation of unsteady muzzle flow of a small-caliber gun Y. Dayan & D. Touati Department of Computational Mechanics & Ballistics, IMI, Ammunition Group, Israel Abstract
More informationInterpreting Differential Equations of Transport Phenomena
Interpreting Differential Equations of Transport Phenomena There are a number of techniques generally useful in interpreting and simplifying the mathematical description of physical problems. Here we introduce
More informationNumerical Studies of Droplet Deformation and Break-up
ILASS Americas 14th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2001 Numerical Studies of Droplet Deformation and Break-up B. T. Helenbrook Department of Mechanical and
More information