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Recalling that the curl of a gradient is automatically zero (see Appendix A
for identities such as this) and assuming that
g
is constant or the gradient
of some potential, and substituting in vorticity, reduces this to
∂ω
∂t
+
∇×
(
u
·∇
u
)=
ν
∇·∇
ω.
The advection term can be simplified with some work (and exploiting the
divergence-free condition
∇·
u
= 0) to eventually get, in three dimensions:
∂ω
∂t
+
u
·∇
ω
=
−
ω
·∇
u
+
ν
∇·∇
ω.
(9.1)
This is known as the
vorticity equation
, which you can see has the material
derivative
Dω/Dt
on the left-hand side, a viscosity term on the right-hand
side, and a new term
ω
·∇
u
, which we can write in components as
⎞
⎛
⎝
ω
1
∂u
∂x
+
ω
2
∂v
∂x
+
ω
3
∂w
⎠
∂x
ω
1
∂u
∂v
+
ω
2
∂v
∂v
+
ω
3
∂w
ω
·∇
u
=
.
∂v
ω
1
∂u
∂z
+
ω
2
∂v
∂z
+
ω
3
∂w
∂z
This term is sometimes called the
vortex-stretching
term from a geometric
point of view which we won't get into in this topic. In two dimensions, the
vorticity equation actually simplifies further: the vortex-stretching term is
automatically zero. (This is easy to verify if you think of a 2D flow as
being a slice through a 3D flow with
u
and
v
constant along the
z
-direction
and
w
= 0.) Here it is in 2D, now written with the material derivative to
emphasize the simplicity:
Dω
Dt
=
ν
∇·∇
ω.
(9.2)
In fact, if we are talking about inviscid flow where viscosity is negligible (as
we have done throughout this topic except in Chapter 8 on highly viscous
flow), the 2D vorticity equation reduces simply to
Dω/Dt
=0. Thatis,
vorticity doesn't change, but is just advected with the flow.
It turns out you can build an attractive fluid solver based on vortic-
ity, particularly in 2D where the equation is even simpler, though there
are decidedly non-trivial complications for boundary conditions and recon-
structing the velocity field for advection from vorticity (more on this in
the next chapter). For example, Yaeger et al. [Yaeger et al. 86], Gamito
et al. [Gamito 95], Angelidis et al. [Angelidis and Neyret 05, Angelidis
et al. 06], Park and Kim [Park and Kim 05], and Elcott et al. [Elcott
