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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
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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,”
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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.