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Simulating Liquid Sound
Will Moss
Hengchin Yeh
Part I: Fluid Simulation for
Sound Rendering
Liquid Simulation
Solve the Navier-Stokes equations
where v is the flow velocity, ρ is the fluid density, p is the pressure, T is the
(deviatoric) stress tensor, and f represents body forces
Liquid Simulation
Generally, graphics people assume the fluid is incompressible
and inviscid (no viscosity)
Looks fine for water and other liquids.
Cannot handle shockwaves or acoustic waves
For these, wee work by Jason or Nikunj
Sound Generation
More detail in the second half
Sound is generated by bubbles
Our fluid simulator must be able to handle bubbles
Fluid Simulation Techniques
Grid Based (Eulerian)
Accurate to within the grid resolution
Slow
Particle Based (Lagrangian)
Faster
Can look a little strange
Others
Shallow water equations
Coupled shallow water and particle based
Grid Based Methods
Split the inviscid, incompressible Navier-Stokes
equations into the three parts
Advection
Force
Pressure
Correct within a factor of O(Δt)
Grid Based Methods
Considers a constant grid and observes what moves
into an out of a cell
Stagger the grid points to avoid problems
u
x
Measure the pressure at the center of a grid cell
Measure the velocity at the faces between the grid points
Grid Based Methods
pi, j 1
vi, j 1 2
pi 1, j
pi, j
ui 1 2 , j
vi, j 1 2
pi, j 1
ui 1 2 , j
pi 1, j
Grid Based Methods
Naturally handle bubbles
Just grid cells that are empty with liquid surrounding
them
Must take rendering into account
Used in boiling simulations (Kim, et al)
Demos
Early Foster and Fedkiw
Fluid-fluid interactions
Boiling
Particle Based Methods
Particles are created by an emitter and exist for a
certain length of time
Store mass, position, velocity, external forces and their
lifetime
No particle interactions
Based on smoothed particle hydrodynamics [CITE]
Particle Interactions
No particle interactions
Fast, system is decoupled
Can only simulate splashing and spraying
Particle Interactions
Theoretically n2 interactions
Define a cutoff distance outside of which
particles do not interact
Allows for puddles, pools, etc.
Particle Interactions
Interactions of liquids look something
like
Mathematically we model this with:
Smoothed Particle Hydrodynamics
Navier-Stokes equations operate on continuous fields, but we
have particles
Assume each particle induces a smooth local field
The global fluid field is simply the sum of all the local fields
Equations of Motion
Simple particle equations:
Reformulate Navier-Stokes equations in terms of
forces
Each particle feels a force due to pressure, viscosity and
any external forces
Bubbles
Bubbles are not inherently handled (like in Eulerian
approaches)
Add an air particle to the system
Create air particles at the surface, so they can be incorporated
into the fluid
Add a interaction force and a surface tension force to the
particles
Smoothed Particle Hydrodynamics
Demos
Simple SPH Demo
Adding air particles
Boiling
Pouring
Shallow Water Equations
Reduce the problem to 2D
At each x and y in the grid, store the height of the fluid
Drastically reduces the complexity of the Navier-Stokes equations
Runs in real time
Shallow Water Equations
One value for each grid cell means no bubbles or
breaking waves
Extension to the method by Thuerey, et. Al
Simulate the bubbles as particles interacting with the
fluid
Can also simulate foam on the surface with SPH
particles
Video
Small Bubbles?
What about small scale bubbles?
Increase the resolution
Computationally expensive
Use finer grid sizes near the surface
Complicated, still expensive
Use a heuristic near the surface
Inaccurate, but faster
We have seen before, sounds can be inaccurate and still
portray the necessary feeling
Heuristics
Assume bubbles and foam form at regions of the
surface where measureable quantities exceed a
threshold
Could use curvature, divergence, Jacobian, etc.
Generate bubble profiles for those regions heuristically
based on the physical properties
Other heuristics possible
Texture Synthesis
Used at UNC for generating realistic textures for
dynamic fluids
Video
References
Thürey, N., Sadlo, F., Schirm, S., Müller-Fischer, M., and Gross, M. 2007. “Real-time
simulations of bubbles and foam within a shallow water framework”. In Proceedings
of the 2007 ACM Siggraph/Eurographics Symposium on Computer Animation
Müller, M., Solenthaler, B., Keiser, R., and Gross, M. 2005. “Particle-based fluidfluid interaction”. In Proceedings of the 2005 ACM Siggraph/Eurographics
Symposium on Computer Animation
Bridson, R. and Müller-Fischer, M. 2007. Fluid simulation: SIGGRAPH 2007 course
notes
Narain, R., Kwatra, V., Lee, H.P., Kim, T., Carlson, M., and Lin, M.C., FeatureGuided Dynamic Texture Synthesis on Continuous Flows, Eurographics Symposium
on Rendering 2007.
Foster, N. and Fedkiw, R. 2001. Practical animation of liquids. In Proceedings of the
28th Annual Conference on Computer Graphics and interactive Techniques
SIGGRAPH '01
Part II: Bubble Sound
Spherical Bubble
Cavitation Inception
Tensile Strength
Cavitation Nuclei
Inside
Vacuum
Gas
Vapor
pL
pi=pg+pv
ps
R
p0
Hydrostatic pressure
Free Oscillation
ps + pL > pi
Contracting
Start from wall speed =0
ps + pL > pi
Internal pressure builds up as air is
compressed
adiabatically (PV = const. )
isothermally (PV=nRT)
Expanding
wall speed =0
ps + pL < pi
Internal pressure decreases
Rmax
R0
=0
pi
R0
Rmin
=0
pi
Rayleigh-Plesset Equation
R-P eq.
Work done by pressure difference =
Kinetic Energy (Speed of wall)
+ Viscosity damping μ
+ (Acoustic radiation)
+ (Thermal damping)
Linearization of R-P eq.
R-P eq. is non-linear
Linearization for R = R0+r
Solution without damping
Minnaert Resonance Frequence
Damping
Damped Solution
Shifted resonance freq.
Damping factor
Damping
Radiation
Viscosity
Thermal
Shifted Resonant Frequency
Large Bubble Assumption
R > 0.1 mm, safely use Minnaert Freq.
20hz ~ 20000hz 0.15m ~ 0.15mm
Pressure Radiation
Relate R to pressure
Assume a Newtonian fluid of constant density
sound speed c
wall speed amplitude U0
Result
is the acoustic pressure radiated by the source at
unit distance from that source
Experiments
Nonspherical
Bubble Oscillations
Spherical Harmonics
Related to Oscillation modes
Burst
Before burst
Thinning
Instability
Interference
magnified
Move around very
fast.
Burst when wall is
still much thicker
than 10 nm, the
barrier
More Issue
Obstruction
Change in Speed of Sound
Coupling
Popping excitation.
References
[1] J. Ding et al., “Acoustical observation of bubble oscillations induced by bubble popping,”
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), vol. 75, Apr. 2007, pp.
041601-7.
[2] A. M S Plesset and A Prosperetti, “Bubble Dynamics and Cavitation,” Nov. 2003;
http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.fl.09.010177.001045.
[3] D. Lohse, “Bubble Puzzles,” Physics Today, vol. 56, 2003, pp. 36-41.
[4] S. Nagrath et al., “Hydrodynamic simulation of air bubble implosion using a level set
approach,” Journal of Computational Physics, vol. 215, Jun. 2006, pp. 98-132.
[5] T.B. Benjamin, “Note on shape oscillations of bubbles,” Journal of Fluid Mechanics Digital
Archive, vol. 203, 2006, pp. 419-424.
[6] R. Manasseh et al., “Passive acoustic bubble sizing in sparged systems,” Experiments in
Fluids, vol. 30, Jun. 2001, pp. 672-682.
[7] K. Lunde and R.J. Perkins, “Shape Oscillations of Rising Bubbles,” Applied Scientific
Research, vol. 58, Mar. 1997, pp. 387-408.
[8]“Sound emission on bubble coalescence: imaging, acoustic and numerical experim”;
http://espace.library.uq.edu.au/view/UQ:120769.
[9] T.G. Leighton, The acoustic bubble, London: Academic Press, 1994.
[10] H.C. Pumphrey and P.A. Elmore, “The entrainment of bubbles by drop impacts,” Journal of
Fluid Mechanics Digital Archive, vol. 220, 2006, pp. 539-567.