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


• hydrostatic equilibrium

1. Abstract


• 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


• 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


• 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


• 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]


• 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]



Simulation of atmospheric binary mixtures based on two-fluid model

• [Liu et al. / Graphical Models 2008]

4. Related Work

Liu et al. [2008]



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


• 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]



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]



Two-way coupled SPH and particle level set fluid simulation

• [Lasasso et al. / TVCG 2008]

4. Related Work

Lasasso et al. [2008]


• 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


• 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


• Mass conservation

Assumptions

• 𝛼a ≡ 1 − 𝛼𝑤• 𝜌a and 𝜌𝑤 are constant



• 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)



• 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)



• 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


• Inserting this flux density into the general formulation for a conservation law,

• The diffusion equation

5.2 Droplet Diffusion


• 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


• Drag Force

Simulation of two-phase flow with sub-scale droplet and bubble effects [Mihalef et al. / EUROGRAPHICS 2009]



• 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:



• 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


• 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


• 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


• 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


7. Result ans Discussion


• 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

a two-continua approach to eulerian simulation of...

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