Making Fluids Incompressible - Fluid Simulation for Computer Graphics - page 45

Andnowin3Dforfluidgridcell( i, j, k ):

u n +1 u n +1

i− 1 / 2 ,j,k

v n +1 v n +1

i,j− 1 / 2 ,k

w n +1 w n +1

i,j,k− 1 / 2 i +1 / 2 ,j,k − i,j +1 / 2 ,k − i,j,k +1 / 2 −

+ + =0 ,

(4.10)

Δ x Δ x Δ x

u i +1 / 2 ,j,k −

1
Δ x

Δ t 1
ρ

p i +1 ,j,k − p i,j,k

Δ x

u i− 1 / 2 ,j,k −

Δ t 1
ρ

p i,j,k − p i− 1 ,j,k

Δ x

−

+ v i,j +1 / 2 ,k −

Δ t 1
ρ

p i,j +1 ,k − p i,j,k

Δ x

v i,j− 1 / 2 ,k −

Δ t 1
ρ

p i,j,k − p i,j− 1 ,k

Δ x

−

+ w i,j,k +1 / 2 −

p i,j,k +1 − p i,j,k

Δ t 1
ρ

Δ x

w i,j,k− 1 / 2 − =0 ,

Δ t 1
ρ

p i,j,k − p i,j,k− 1

Δ x

−

⎛ ⎞

⎠

6 p i,j,k − p i +1 ,j,k − p i,j +1 ,k − p i,j,k +1

⎝

− p i− 1 ,j,k − p i,j− 1 ,k − p i,j,k− 1

Δ t
ρ

=

Δ x 2

⎛
⎝

⎞
⎠

u i +1 / 2 ,j,k − u i− 1 / 2 ,j,k

Δ x

+ v i,j +1 / 2 ,k −

v i,j− 1 / 2 ,k
Δ x
+ w i,j,k +1 / 2 −

− . (4.11)

w i,j,k− 1 / 2
Δ x

Observe that Equations (4.9) and (4.11) are numerical approximations to
the Poisson problem

u .
If a fluid grid cell is at the boundary, recall that the new velocities on
the boundary faces involve pressures outside the fluid that we have to define
through boundary conditions: we need to use that here. For example, if
grid cell ( i, j + 1) is an air cell, then we replace p i,j +1 in Equation (4 . 9)
with zero. If grid cell ( i +1 ,j ) is a solid cell, then we replace p i +1 ,j with the
value we compute from the boundary condition there, as in formula (4.6).

− Δ t/ρ ∇·∇ p = −∇ ·