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Directly discretizing this gives
−
ψ
i
+3
/
2
,j
+1
/
2
+2
ψ
i
+1
/
2
,j
+1
/
2
−
ψ
i−
1
/
2
,j
+1
/
2
Δ
x
2
+
−
ψ
i
+1
/
2
,j
+3
/
2
+2
ψ
i
+1
/
2
,j
+1
/
2
−
ψ
i
+1
/
2
,j−
1
/
2
=
ω
i
+1
/
2
,j
+1
/
2
,
Δ
x
2
which in fact is our usual discretization of the Poisson problem—the same
matrix as the pressure solve! (It's easy to verify that in two dimensions,
∇×∇
ψ
.)
For stationary solid wall boundaries, where we want
u
ψ
=
−∇ · ∇
·
n
= 0, we will
t
=0,where
t
is the normal rotated 90
degrees—i.e., a tangent vector. This means that the tangential derivative
of
ψ
along the boundary should be zero, or in other words,
ψ
should be
equal to a constant value along the solid wall. The constant value can
be related to how fast the fluid is rotating around the solid (this can be
determined from the initial conditions; in the absence of viscosity it should
remain unchanged), but for simplicity we'll just take it to be zero. This
can feed into the discretization by simple setting
ψ
= 0 in solids, or more
accurately using the ghost fluid method discussed in Chapter 6.
One of the remarkable strengths of this method is that it suffers no
numerical dissipation of vorticity whatsoever. It can also be fairly eas-
ily extended to include forces such as buoyancy, changing the equation to
Dω/Dt
=
equivalently require that
∇
ψ
·
f
. That said, this method doesn't have the same flexibilities
in dealing with moving boundaries, free surfaces, variable densities, con-
trolled divergence, coupled solids, and more. More serious drawbacks come
in three dimensions: in addition to the vortex stretching term complicat-
ing the equations (a particle's vorticity no longer remains constant), the
potential we need to reconstruct the velocity field must be vector-valued.
The vector potential equation
∇×
ψ
=
ω
is significantly larger and
harder to solve than the simple pressure Poisson problem, and its boundary
conditions are significantly more complex.
In answer to this, Selle et al. [Selle et al. 05] created a method where a
regular velocity-pressure three-dimensional fluid simulation is augmented
with just a few vortex particles, or “spin particles” to put a distinct label
on them, sprinkled in turbulent regions. The spin particles are advected
with the flow as before, and to partially account for the vortex-stretching
term
ω
∇×∇
∇×
·∇
u
, have their vorticity vectors essentially rotated (by adding
Δ
tω
p
·∇
u
and then rescaling to preserve the magnitude). They in turn
add to the velocity field with an acceleration term which encourages the
flow to spin with that vorticity. This can be understood as a per-particle
