Turbulence - Fluid Simulation for Computer Graphics - page 126

The here is a parameter that can be adjusted to control the effect of
vorticity confinement. The Δ x factor, as mentioned above, makes this
physically consistent: as we refine the grid and Δ x tends to zero, the
erroneous numerical dissipation of vorticity also tends to zero so our fix
should too.
Let's step through numerically implementing this. We begin by av-
eraging velocities from the MAC grid to the cell centers (as discussed in
Chapter 2) and then use central derivatives to approximate the vorticity: 1
ω i,j,k = w i,j +1 ,k −

w i,j− 1 ,k
2Δ x

v i,j,k +1 − v i,j,k− 1

2Δ x

− ,

u i,j,k +1 − u i,j,k− 1

2Δ x

w i +1 ,j,k − w i− 1 ,j,k

2Δ x

− ,

.

v i +1 ,j,k − u i,j +1 ,k −

v i− 1 ,j,k
2Δ x

u i,j− 1 ,k
2Δ x

−

The gradient of is similarly estimated with central differences at the

grid cell centers, for use in defining

ω

N :

∇ ω i,j,k =

.

ω | i +1 ,j,k − ω | i− 1 ,j,k

,

ω | i,j +1 ,k − ω | i,j− 1 ,k

,

ω | i,j,k +1 − ω | i,j,k− 1

2Δ x 2Δ x 2Δ x

When we normalize this to get N , we should of course guard against a
divide-by-zero by using, for example,

∇ ω i,j,k

N i,j,k =

+10 − 20 M , ∇ ω i,j,k

where M is a characteristic value of units m − 1 s − 1 for the simulation—
nothing to be too concerned about; M =1 / (Δ x Δ t ) is fine just to make
sure this scales properly. Finally, we take the cross-product to get f conf at
the grid cell centers; we can take the appropriate averages to apply this to
the different components of velocity on the MAC grid.
Ideally we would connect the confinement parameter with the ex-
pected numerical dissipation of vorticity. However, this has yet to be done,
but in the meantime serves as another tweakable parameter for the sim-
ulation. If set too high, the simulation can go quasi-unstable, reducing
the velocity field to essentially a random turbulent chaos; more moderate
values encourage fine-scale vortices and keep the flow more lively.

1 Note that the null-space problem we discussed earlier isn't particularly alarming
here: we just will lack the ability to “see” and boost the very smallest vortices. We still
get the benefit of boosting slightly larger ones.