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Simulating Liquid Sound

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.

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Simulating Liquid Sound

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  1. Simulating Liquid Sound Will Moss Hengchin Yeh

  2. Part I: Fluid Simulation forSound Rendering

  3. 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

  4. 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

  5. Sound Generation • More detail in the second half • Sound is generated by bubbles • Our fluid simulator must be able to handle bubbles

  6. 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

  7. Grid Based Methods • Split the inviscid, incompressible Navier-Stokes equations into the three parts • Advection • Force • Pressure • Correct within a factor of O(Δt)

  8. u x 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

  9. Grid Based Methods

  10. 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

  11. 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]

  12. 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.

  13. Particle Interactions • Interactions of liquids look something like • Mathematically we model this with:

  14. 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

  15. 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

  16. 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

  17. Smoothed Particle Hydrodynamics • Demos • Simple SPH Demo • Adding air particles • Boiling • Pouring

  18. 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

  19. 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

  20. 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

  21. 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

  22. Texture Synthesis • Used at UNC for generating realistic textures for dynamic fluids • Video

  23. 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

  24. Part II: Bubble Sound

  25. Spherical Bubble • Cavitation Inception • Tensile Strength • Cavitation Nuclei • Inside • Vacuum • Gas • Vapor pL pi=pg+pv R ps p0 Hydrostatic pressure

  26. Free Oscillation ps + pL > pi Rmax • 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 R0 =0 pi R0 Rmin =0 pi

  27. Rayleigh-Plesset Equation • R-P eq. • Work done by pressure difference = Kinetic Energy (Speed of wall) + Viscosity damping μ + (Acoustic radiation) + (Thermal damping)

  28. Linearization of R-P eq. • R-P eq. is non-linear • Linearization for R = R0+r • Solution without damping • Minnaert Resonance Frequence

  29. Damping • Damped Solution • Shifted resonance freq. • Damping factor

  30. Damping • Radiation • Viscosity • Thermal

  31. Shifted Resonant Frequency • Large Bubble Assumption • R > 0.1 mm, safely use Minnaert Freq. • 20hz ~ 20000hz  0.15m ~ 0.15mm

  32. 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

  33. Experiments

  34. Nonspherical Bubble Oscillations • Spherical Harmonics • Related to Oscillation modes

  35. Burst • Before burst • Thinning • Instability • Interference magnified • Move around very fast. • Burst when wall is still much thicker than 10 nm, the barrier

  36. More Issue • Obstruction • Change in Speed of Sound • Coupling • Popping excitation.

  37. 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.  

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