Fluid Simulation

Fluid Simulation Koray Balcı Işık Barış Fidaner

Level set • Implicit signed distance function • Negative inside, positive outside • Isocontour at zero is the surface.

• Evolves in time by eqn:

Particles • Level set methods tend to lose volume • Particles are added to prevent this loss.

Only level sets

Level sets with particles

• Each particle adds to level set equation by: (r: radius, s: sign of particle (inside / outside)) • Then surface is smoothed by a function S.

3D grid of cells • Pressures at the center of cells • Velocities at the faces of cells • High resolution Euler grid • Low resolution Navier-Stokes subgrid

The Navier-Stokes Equations (u: velocity field) 1) Incompressibility. No fluid flows out of the cell, mass is conserved.

(Ď…: viscosity, Ď : density, p: pressure, g: gravity) 2) Velocity and pressure fields are related through conservation of momentum.

Solving Navier-Stokes Velocity field u is updated after each time step through 4 operations:

u(t)  add force  advect  diffuse  project  u(t+1) i) Initial velocity field at a time step

ii) Firstly, forces are added to the velocity field.

Solving Navier-Stokes iii) Semi-Lagrangian method for stability of fluid movement. Velocity is traced back in one time step. Thus, momentum is carried forward in time.

iv) Diffusion (viscousity) of the fluid is realized in this step.

Solving Navier-Stokes v) Finally, incompressibility is enforced.

By using Helmholtz-Hodge Decomposition, a vector field can be divided into a divergence free vector field and the gradient of a scalar field.

In this step, we find the divergence-free part of the velocity field.

Time step limitations • Timestep must be short enough that no significant change occurs.

• Width of a cell, divided by maximum velocity • CFL (Courant-Friedrichs-Levy) condition: