A Two-Continua Approach to Eulerian Simulation of...
Transcript of A Two-Continua Approach to Eulerian Simulation of...
A Two-Continua Approach to EulerianSimulation of Water Spray
Copyright of figures and other materials in the paper belongs original authors.
Presented by MyungJin Choi
2015-01-26
Computer Graphics @ Korea University
Michael B. Nielsen et al.TOG 2013
MyungJin Choi | 2015-01-26| # 2Computer Graphics @ Korea University
• Current spray simulation pipelines typically employ a combination of Lagrangian (particle) and Eulerian (volumetric) methods
• the Eulerian methods being used for parts of the spray where individual droplets are not apparent
• Existing Eulerian methods are based on gas solvers that will exhibit hydrostatic equilibrium in certain scenarios
• We propose to simulate spray in the Eulerian domain
two-way coupling
air and water phases co-existing at each point in space
1. Abstract
MyungJin Choi | 2015-01-26| # 3Computer Graphics @ Korea University
• hydrostatic equilibrium
1. Abstract
MyungJin Choi | 2015-01-26| # 4Computer Graphics @ Korea University
• Water spray occurs in nature
waterfalls, water jets, air interacting with waves and splashes
• Simulating water spray turns out to be complex
the large number of droplets, the variation in droplet size
both the air and water phases must be accounted
• Several state of the art spray pipelines in the visual effects industry employ a combination of techniques
Lagrangian particles are used for parts of the spray
Eulerian grid methods are used for regions
2. Introduction
MyungJin Choi | 2015-01-26| # 5Computer Graphics @ Korea University
• Eulerian spray is currently simulated using gas solvers
• propose a two-way coupled Eulerian spray simulation method based on a two-continua representation
water droplets and air are represented as two distinct continua that co-exist at each point in space
each field has its own volume fraction and velocity
• the two-continua approach are not new
they have been developed and well studied in applied physics for more than a decade
2 Introduction
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• A two-continua Poisson equation that fits into the operator splitting methodology
• A diffusion equation modeling droplet mass flux and an unconditionally stable discretization thereof
• A semi-implicit discretization of the drag force which exhibits improved stability over explicit discretization
• An Eulerian algorithm for simulating spray as a two-continua for computer graphics
3. Contribution
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• Lagranian Representation
There are two pioneered techniques
• Modeling Waves and Surf
[Peachey / SIGGRAPH 1986]
• A Simple Model of Ocean Waves
[Fournier and Reeves / SIGGRAPH 1986]
4. Related Work
Peachey [1986] Fournier and Reeves [1986]
MyungJin Choi | 2015-01-26| # 8Computer Graphics @ Korea University
• Eulerian Representation
Simulation of cumuliform clouds based on computational fluid dynamics
• [Miyazaki et al. / EUROGRAPHICS 2002]
Physics Motivated Modeling of Volcanic Clouds as a Two Fluids Model
• [Mizuno et al. / IEEE Computer Society 2003]
4. Related Work
Miyazaki et al. [2002]Mizuno et al. [2003]
MyungJin Choi | 2015-01-26| # 9Computer Graphics @ Korea University
• Eulerian Representation
Simulation of atmospheric binary mixtures based on two-fluid model
• [Liu et al. / Graphical Models 2008]
4. Related Work
Liu et al. [2008]
MyungJin Choi | 2015-01-26| # 10Computer Graphics @ Korea University
• Eulerian Representation
Multiple interacting liquids [Lasasso et al. / SIGGRAPH 2006]
• multiphase simulation
• It is not feasible simulation method for water spray due to the large scale of scenes
4. Related Work
MyungJin Choi | 2015-01-26| # 11Computer Graphics @ Korea University
• Lagrangian and Eulerian flow
Realistic Animation of Fluid with Splash and Foam
• [Takahashi, T. et al. / EUROGRAPHICS 2003]
Stable but non-dissipative water
• [Song et al. / TOG 2005]
4. Related Work
Takahashi, T. et al. [2003] Song et al. [2005]
MyungJin Choi | 2015-01-26| # 12Computer Graphics @ Korea University
• Lagrangian and Eulerian flow
Practical animation of turbulent splashing water
• [Kim. et al. / SCA 2006]
Simulation of two-phase flow with sub-scale droplet and bubble effects
• [Mihalef. et al. / EUROGRAPHICS 2009]
4. Related Work
Kim et al. [2006] Mihalef et al. [2009]
MyungJin Choi | 2015-01-26| # 13Computer Graphics @ Korea University
• Lagrangian and Eulerian flow
Two-way coupled SPH and particle level set fluid simulation
• [Lasasso et al. / TVCG 2008]
4. Related Work
Lasasso et al. [2008]
MyungJin Choi | 2015-01-26| # 14Computer Graphics @ Korea University
• Instead of tracking individual droplets
air and water volume fractions 𝛼𝑤 , 𝛼a ∈ 0; 1
• The Velocities of air and water droplets are coupled
an interaction force and a pressure projection
• The fundamental equations to an operator splitting approach similar to the equation of Stable Fluid [Stam 1999]
5. Two-continua Method
A new Splitting approach
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• The dynamics of a two-continua mixture of water droplets and air is governed
Fluid Dynamics and Transport of Droplets and Sprays
• [Sirignano / Book]
• It ignore thermodynamical effects
5.1 Conservation of Mass and Momentum
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• Mass conservation
Assumptions
• 𝛼a ≡ 1 − 𝛼𝑤• 𝜌a and 𝜌𝑤 are constant
5.1 Conservation of Mass and Momentum
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• The pressure projection step becomes
• Inserting Eqs. (7) and (8) into Eq. (6)
The variable-coefficient Poisson Equation
• Solved in combination with Eqs. (3), (4) and (5)
5.1 Conservation of Mass and Momentum
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• Redefine P
Fluid Simulation for Computer Graphics [Bridson / Book 2008]
This means that
• 𝛼a𝜌a𝐠 cancels out from Eq. (4)
• 𝛼𝑤𝜌𝑤𝐠 is replaced by 𝛼𝑤 𝜌𝑤 − 𝜌a 𝐠 in Eq. (5)
5.1 Conservation of Mass and Momentum
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• The stress is proportional to the velocity gradient for a Newtonian fluid
Fundamentals of Multiphase Flow [Brennen / Book 2009]
• The magnitude of the mass flux:
in the direction of and proportional to the negative gradient of 𝛼𝑤
• The flux density thus becomes:
5.2 Droplet Diffusion
𝑑 𝒖𝑟𝑒𝑙
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• Inserting this flux density into the general formulation for a conservation law,
• The diffusion equation
5.2 Droplet Diffusion
MyungJin Choi | 2015-01-26| # 21Computer Graphics @ Korea University
• The drag force is parallel to and depends non-linearly on the relative velocity between water droplets and air
this makes an explicit discretization of the drag force
• it is subject to a stricter time step restriction
• Propose a semi-implicit discretization
does not impose further time step restrictions
5.3 Air and Water Droplet Interaction
MyungJin Choi | 2015-01-26| # 22Computer Graphics @ Korea University
• Drag Force
Simulation of two-phase flow with sub-scale droplet and bubble effects [Mihalef et al. / EUROGRAPHICS 2009]
5.3 Air and Water Droplet Interaction
MyungJin Choi | 2015-01-26| # 23Computer Graphics @ Korea University
• Redefine Eq. (12)
divide by the volume of a spherical particle 4
3𝜋𝑟3
multiply by the volume fraction 𝛼𝑤
• Two coupled ODEs at each grid point:
5.3 Air and Water Droplet Interaction
MyungJin Choi | 2015-01-26| # 24Computer Graphics @ Korea University
• Time Step Restrictions
WENO scheme
• Water Volume Fraction Advection (Discretization of the Mass Conservation Equation (Eq. (3)))
Fifth order accurate WENO
Third order accurate Runge Kutta
• Diffusion Discretization (Eq. (11))
Discretize implicitly in time using a first order accurate backward finite difference
Use second order accurate central difference approximations to the gradient and divergence operator
Discretize 𝛻𝛼𝑤 at voxel faces, half-way between voxel centers
6. Implementation
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• Interpolation to Voxel Faces
Use a second order accurate linear interpolation
• Mask Computation
Air and water masks are non-zero
• Velocity Advection and Forces (Discretization of the Momentum Conservation Equations (Eqs. (4) and (5)))
Apply the WENO+RK numerical scheme to each component of the water droplet velocity field
• Pressure Projection (Discretization of the Poisson Equation (Eq. (9)))
using second order accurate central finite differences of the gradient and divergence operators
6. Implementation
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• Velocity Extrapolation
Using a fast sweeping method
• Fast sweeping algorithms for a class of hamilton-Jacobi equations
• [Tsai et al. / journal on Numerical Analysis 2003]
• A Note on the Properties of the Discrete Water Droplet Volume Fractions
Maximum-principle-satisfying and positivity-preserving high order schemes for conservation laws: Survey and new developments
• [Zhang et al. / The Royal Society]
Allow 𝛼𝑤 ∉ [0; 1] as practice has shown that 𝛼𝑤 remains bounded
And that clamping 𝛼𝑤 to [0; 1]
6. Implementation
MyungJin Choi | 2015-01-26| # 27Computer Graphics @ Korea University
• Results are produced in the Windows operating system installed on a Mac Pro with 7GB of memory and two Intel Xeon quad-core 2.80GHz CPUs
• The images are produced by rendering 𝛼𝑤 with a ray marcher
• Use the open source vortex particle implementation
Synthetic turbulence using artificial boundary layers
• [Pfaff et al. / SIGGRAPH Asia 2009]
7. Result ans Discussion
MyungJin Choi | 2015-01-26| # 28Computer Graphics @ Korea University
7. Result ans Discussion
MyungJin Choi | 2015-01-26| # 29Computer Graphics @ Korea University
• Propose an Eulerian two-continua approach to spray simulation0
• Proposed algorithm fits into the operator splitting framework with increased stability behavior
• The development of turbulence models for spray
• A closer integration of and coupling between Eulerian and Lagrangian spray simulation methods
• The investigation of more accurate air-droplet interaction models
• The application of our two-continua approach to other phenomena such as air-dust and bubble flows to mention a few
Conclusion and Future Work