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Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics. Carlos Eduardo Aguiar Instituto de Física - UFRJ. Outline Non-relativistic hydrodynamics. SPH equations. Applications. Relativistic hydrodynamics. Relativistic SPH. High energy nuclear physics. Fluid Dynamics. Hydrodynamic Equations.

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Smoothed Particle Hydrodynamics

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  1. Smoothed Particle Hydrodynamics Carlos Eduardo Aguiar Instituto de Física - UFRJ • Outline • Non-relativistic hydrodynamics. • SPH equations. • Applications. • Relativistic hydrodynamics. • Relativistic SPH. • High energy nuclear physics.

  2. Fluid Dynamics

  3. Hydrodynamic Equations Continuity equation Ideal fluid Euler’s equation

  4. Entropy Equation • no viscosity • no thermal conduction

  5. Energy Equation

  6. Ideal Gas

  7. Conservation Laws

  8. SPH • Developed to study gas dynamics in astrophysical systems. • Lagrangian method. • No grids. • Arbitrary geometries. • Equally applicable in 1, 2 and 3 space • dimensions. - L.Lucy, Astron.J. 82, 1013 (1977) - R.Gingold, J.Monaghan, MNRAS 181, 378 (1977) Reviews: - J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543 (1992) - L. Hernquist, N. Katz, Ap. J. Suppl. 70, 419 (1989)

  9. Smoothing h x 0 Error:

  10. Particles "Monte-Carlo" sampling

  11. Different ways of writing SP estimates (we omit the SP subscript from now on):

  12. Derivatives No need for finite differences and grids: D D i+1 i-1 i

  13. More than one way of calculating derivatives: Exact Galilean invariance

  14. Moving the Particles

  15. Euler's Equation Exact momentum conservation

  16. Entropy

  17. Energy

  18. Alternatively:

  19. SPH Equations

  20. Smoothing Kernels Gaussian: Spline:

  21. numerical calculation shock wave x Shock Waves

  22. Artificial Viscosity • Galilean invariant. • Vanishes for rotations. • Conserves linear and angular momentum.

  23. Local Resolution Length

  24. SPH Simulation of the Hubble Volume Mass density in a thin slice (100x100x20Mpc/h) at the present epoch. This is a view one would observe if the speed of light were infinite.

  25. SPH Simulation of Galaxy Formation Gas Dark Matter Density of gas and dark matter in a group of galaxies.

  26. SPH Simulation of Supernova Explosion 75 ms after bounce z (km) x (km) Herant et al., Ap.J. 435, 339 (1994)

  27. SPH Simulation of Supernova Explosion Herant et al., Ap.J. 435, 339 (1994)

  28. SPH Simulation of Stellar Collision Disruption of a main sequence star by a close encounter with a high velocity neutron star. Colors represent log (density).

  29. SPH Simulation of Colliding Asteroids An 8 m radius rock strikes the 1.6 km long asteroid Castalia at 5 km/s. Red is totally fractured rock, blue is intermediate fractured rock, and white particles represent the impactor.

  30. SPH Simulation of Projectile Impact on Sand Projectile: Aluminum cylinder 30x10 cm (2d). Initial velocity: 3 km/sec Particles Temperature

  31. Relativistic Hydrodynamics Energy-momentum conservation Baryon-number conservation

  32. Continuity equation: Entropy equation: s = entropy density (rest frame)

  33. Relativistic Euler equation: w = enthalpy per baryon Momentum equation:

  34. Energy equation:

  35. Relativistic SPH

  36. Momentum (per baryon number) Energy (per baryon number)

  37. ? Particle Velocity

  38. RSPH Equations

  39. Baryon-Free System entropy density: (Rest frame) (Lab frame)

  40. Baryon-Free RSPH

  41. Ultrarelativistic Pion Gas

  42. Ultrarelativistic Pions Rarefaction Wave

  43. SHASTA et al.

  44. Ultrarelativistic Pions Landau Solution

  45. Artificial Viscosity • E.Chow and J.Monaghan, Journal of • Computational Physics 134, 296 (1997) • See also: • S.Siegler and H.Riffert, astro-ph/9904070

  46. Shock Tube Ideal nucleon gas

  47. SHASTA et al.

  48. Shock Tube Ideal nucleon gas

  49. SHASTA et al.

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