Simulating Liquid SoundWill Moss

Hengchin Yeh

Part I: Fluid Simulation forSound Rendering

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 wavesFor these, wee work by Jason or Nikunj

Liquid Simulation

Sound GenerationMore detail in the second half

Sound is generated by bubblesOur fluid simulator must be able to handle

bubbles

Fluid Simulation TechniquesGrid Based (Eulerian)

Accurate to within the grid resolutionSlow

Particle Based (Lagrangian)FasterCan look a little strange

OthersShallow water equationsCoupled shallow water and particle based

Grid Based MethodsSplit the inviscid, incompressible Navier-Stokes

equations into the three partsAdvectionForcePressure

Correct within a factor of O(Δt)

Grid Based MethodsConsiders a constant grid and observes what

moves into an out of a cellStagger the grid points to avoid problems

Measure the pressure at the center of a grid cell Measure the velocity at the faces between the grid

points

u

x

Grid Based Methods

ui 12, j ui 12, jvi, j 12

pi, j pi1, j

pi, j 1

pi, j1

pi 1, j

vi, j 12

Grid Based MethodsNaturally 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 FedkiwFluid-fluid interactionsBoiling

Particle Based MethodsParticles are created by an emitter and exist

for a certain length of timeStore mass, position, velocity, external forces

and their lifetimeNo particle interactionsBased on smoothed particle hydrodynamics

[CITE]

Particle InteractionsNo particle interactions

Fast, system is decoupledCan only simulate splashing and

sprayingParticle Interactions

Theoretically n2 interactions Define a cutoff distance outside of

which particles do not interactAllows for puddles, pools, etc.

Particle InteractionsInteractions of liquids look

something like

Mathematically we model this with:

Smoothed Particle HydrodynamicsNavier-Stokes equations operate on continuous fields, but

we have particlesAssume each particle induces a smooth local fieldThe global fluid field is simply the sum of all the local fields

Equations of MotionSimple particle equations:

Reformulate Navier-Stokes equations in terms of forcesEach particle feels a force due to pressure, viscosi

ty and any external forces

BubblesBubbles are not inherently handled (like in Eule

rian 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 HydrodynamicsDemos

Simple SPH DemoAdding air particles

Boiling Pouring

Shallow Water EquationsReduce the problem to 2D

At each x and y in the grid, store the height of the fluidDrastically reduces the complexity of the Navier-Stokes equ

ations Runs in real time

Shallow Water EquationsOne value for each grid cell means no bubbles o

r breaking wavesExtension 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

HeuristicsAssume bubbles and foam form at regions of th

e surface where measureable quantities exceed a thresholdCould use curvature, divergence, Jacobian, etc.Generate bubble profiles for those regions heurist

ically based on the physical propertiesOther heuristics possible

Texture SynthesisUsed at UNC for generating realistic textures

for dynamic fluidsVideo

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 fluid-fluid 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., Feature-Guided 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

Cavitation InceptionTensile Strength Cavitation Nuclei

InsideVacuumGasVapor

Spherical Bubble

pi=pg+pv

ps

pL

R

p0

Hydrostatic pressure

Free Oscillation

=0

ps + pL > pi

pi

=0

Rmax

Rmin

R0

R0

pi

ContractingStart from wall speed =0 ps + pL > piInternal pressure builds up as ai

r is compressedadiabatically (PV = const. ) isothermally (PV=nRT)

Expandingwall speed =0ps + pL < piInternal pressure decreases

Rayleigh-Plesset EquationR-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-linearLinearization for R = R0+r

Solution without damping

Minnaert Resonance Frequence

DampingDamped Solution

Shifted resonance freq.

Damping factor

DampingRadiation

Viscosity

Thermal

Shifted Resonant Frequency

Large Bubble Assumption R > 0.1 mm, safely use Minnaert Fr

eq. 20hz ~ 20000hz 0.15m ~ 0.15mm

Pressure RadiationRelate R to pressure

Assume a Newtonian fluid of constant density

sound speed cwall speed amplitude U0

Result

is the acoustic pressure radiated by the source at unit distance from that source

Experiments

Nonspherical Bubble OscillationsSpherical Harmonics

Related to Oscillation modes

BurstBefore burst

ThinningInstabilityInterference

magnifiedMove around

very fast.

Burst when wall is still much thicker than 10 nm, the barrier

More IssueObstructionChange in Speed of SoundCouplingPopping excitation.

References [1] J. Ding et al., “Acoustical observation of bubble oscillations induced by bubble p

opping,” 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 lev

el 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 Mecha

nics Digital Archive, vol. 203, 2006, pp. 419-424. [6] R. Manasseh et al., “Passive acoustic bubble sizing in sparged systems,” Experi

ments in Fluids, vol. 30, Jun. 2001, pp. 672-682. [7] K. Lunde and R.J. Perkins, “Shape Oscillations of Rising Bubbles,” Applied Scie

ntific Research, vol. 58, Mar. 1997, pp. 387-408. [8]“Sound emission on bubble coalescence: imaging, acoustic and numerical exper

im”; 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.