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et al. ] have all taken this route. However, our chief concern now is in
what happens to vorticity in our regular fluid solver, based on velocity and
pressure.
In fact we already know that we run the risk of numerical dissipation
in our Eulerian advection schemes: we saw that for first-order linear inter-
polation, the error behaves like additional viscosity, and so it should be no
surprise that the vorticity of the velocity field similarly gets dissipated. So
far we've dealt with this by increasing the sharpness of the interpolation—
though even this doesn't fully avoid dissipation. We will get to a method
that can virtually eliminate dissipation in the advection stage in Chapter
10, but this is not the only source of vorticity dissipation.
The other big source lies in the time-splitting algorithm itself. We
mentioned before that our algorithm for separately advecting and then
projecting velocity is only first-order accurate in time; it turns out this
can be a fairly problematic error when attempting to capture small-scale
vortices. As a motivating example, imagine starting with just a 2D rigid
rotation of constant-density fluid around the origin, say
u
0
=(
−
y, x
)
.
Ignoring boundary conditions and body forces, the exact solution of the
Navier-Stokes equations, given this initial velocity field, is for
u
to stay
constant—the rotation should continue at exactly the same speed. How-
ever, if we advance it with our time-splitting algorithm, things go wrong.
Even with
perfect
error-free advection, for a time step of Δ
t
=
1
2
π
which
corresponds in this velocity field to a counterclockwise rotation of 90
◦
,we
get this intermediate velocity field:
u
A
=(
x, y
)
.
It no longer has any vorticity (easy to check) and moreover is divergent:
our advection step transferred all the energy from rotation to expansion.
It's not hard to verify that the pressure solution is
x
2
+
y
2
2
πρ
p
=
.
After updating the intermediate velocity field with this pressure, we end
up with
u
n
+1
=0
.
