1. Computer Animation Rick Parent Computer Animation Algorithms and Techniques Computational Fluid Dynamics

2. Computer Animation Rick Parent Fluids conservation of mass momentum energy viscous v. non-viscous compressible v. incompressible types turbulent laminar flow steady state flow characteristics gravity density pressure heat velocity properties

3. Computer Animation Rick Parent Models grid-based (Eulerian) particle-based (Lagrangian) Hybrid dddd dddd dddd dddd

4. Computer Animation Rick Parent Notation Gradient Divergence

5. Computer Animation Rick Parent Divergence Theorem The integral of the flow field’s divergence over the volume of the cell is the same as the intgral of the flow field over the cell’s boundary

6. Computer Animation Rick Parent Conservation of Mass dx

7. Computer Animation Rick Parent Conservation of Mass If incompressible:

8. Computer Animation Rick Parent Conservation of Momentum dx dy in x

9. Computer Animation Rick Parent Conservation of Momentum

10. Computer Animation Rick Parent Euler Equations - 2D Velocity: u,v Coordinates: x,y Density: Pressure: p

11. Computer Animation Rick Parent Euler Equations - incompressible form

12. Computer Animation Rick Parent Stable Fluids Velocity gridDensity grid Move densities around using velocity grid and dissipate densities Move velocities around using velocity grid and dissipate velocities

13. Computer Animation Rick Parent Stable Fluids Source Force AdvectionDissipation