Simulating Liquid Sound Will Moss Hengchin Yeh

Part I: Fluid Simulation for Sound 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 waves For these, wee work by Jason or Nikunj Liquid Simulation

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 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

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 n 2 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 Th ü rey, N., Sadlo, F., Schirm, S., M ü ller-Fischer, M., and Gross, M “ Real- time simulations of bubbles and foam within a shallow water framework ”. In Proceedings of the 2007 ACM Siggraph/Eurographics Symposium on Computer Animation M ü ller, M., Solenthaler, B., Keiser, R., and Gross, M “ Particle-based fluid- fluid interaction ”. In Proceedings of the 2005 ACM Siggraph/Eurographics Symposium on Computer Animation Bridson, R. and M ü ller-Fischer, M 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 Foster, N. and Fedkiw, R Practical animation of liquids. In Proceedings of the 28th Annual Conference on Computer Graphics and interactive Techniques SIGGRAPH '01

Part II: Bubble Sound

Cavitation Inception Tensile Strength Cavitation Nuclei Inside Vacuum Gas Vapor Spherical Bubble p i =p g +p v psps pLpL R p0p0 Hydrostatic pressure

Free Oscillation =0 p s + p L > p i pipi =0 R max R min R0R0 R0R0 pipi Contracting Start from wall speed =0 p s + p L > p i Internal pressure builds up as air is compressed adiabatically (PV = const. ) isothermally (PV=nRT) Expanding wall speed =0 p s + p L < p i Internal pressure decreases

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 = R 0 +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 U 0 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 [2] A. M S Plesset and A Prosperetti, “Bubble Dynamics and Cavitation,” Nov. 2003; [3] D. Lohse, “Bubble Puzzles,” Physics Today, vol. 56, 2003, pp [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 [5] T.B. Benjamin, “Note on shape oscillations of bubbles,” Journal of Fluid Mechanics Digital Archive, vol. 203, 2006, pp [6] R. Manasseh et al., “Passive acoustic bubble sizing in sparged systems,” Experiments in Fluids, vol. 30, Jun. 2001, pp [7] K. Lunde and R.J. Perkins, “Shape Oscillations of Rising Bubbles,” Applied Scientific Research, vol. 58, Mar. 1997, pp [8]“Sound emission on bubble coalescence: imaging, acoustic and numerical experim”; [9] T.G. Leighton, The acoustic bubble, London: Academic Press, [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