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Eulerian Method for Water and Bubbles

1. Multi-phase flows

We handles the multi-phase navier-stokes solver which includes the buoyancy and surface tension. Our system is designed to use the merits of Navier-Stokes simulation schemes developed for the computer graphics animation.

Multi-phase Navier-Stokes equation

2. Bubbles

We introduced one more subject pertaining to liquid animation - the bubbles. Bubbles are pockets of air enclosed by liquid. Bubbles exist everyplace where liquid and air coexist. There are two flows to consider - i.e. those occurring inside and outside of the bubble bodies. Differences in specific gravity between the two fluids generate buoyancy forces. Surface tension forces are exerted at the interfaces between the two fluids. We developed a new fluid animation method in which liquid and gas interact with each other, using the example of bubbles rising in water.

 

3. Minimum-stress surface tension

In the multi-phase fluid configuration, we can know that the surface will be constructed like blue lines. First find the stress-zero surface of each cell like red line. Second, make the tendency toward the stress-zero surfaces. These tendencies are equivalent to the surface tension forces and defined on the simulation grids. Finally, we can add them to the velocity as body forces. Then, Navier-Stokes solver will handle this naturally.

Minimum-stress surface tension method
The velocity field induced
by the surface tension forces

4. Numerical simulation of buoyancy

We numerically simulated buoyancy effects instead of empolying some experimental equations.
Our method was easily implemented as a part of multi-phase simulation system.

Step A
Step B
Step C ((AVI/1.46MB))

[Step A]
Add gravity forces to the velocity field, since the buoyancy comes from the density difference of two materials.
[Step B]
Correct the velocity field by solving the hyperbolic partial differential equations.
[Step C]
Trace the interfaces following the velocity field.

5. Mass conservation

Since our method is partially based on the VOF (volume-of-fluid) scheme, we could explicitly correct the total mass of enclosed fluids after each interface capturing step.

   

6. Result

Finally we could animate the rising bubbles in liquid within the Navier-Stokes fluid simulation scheme. We empolyed some particles to provide liveliness for the animation scenes. Interfaces were rendered by the vertex-shader techniques.
7. Related Publications

[1] Animation of Bubbles in Liquid, J. M. Hong, C. H. Kim, In Proceedings of Eurographics 2003.
[2] Surface Tensions in Liquid Animation, J. M. Hong, C. H. Kim, Sketches & Applications in Siggraph 2003.

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