3Dynamics
IB Fluid Dynamics
3.7 Vorticity equation
The Navier-Stokes equation tells us how the velocity changes with time. Can we
obtain a similar equation for the vorticity? Consider the Navier-Stokes equation
for a viscous fluid,
ρ
∂u
∂t
+ u · ∇u
= −∇p − ∇χ + µ∇
2
u
We use a vector identity
u · ∇u =
1
2
∇|u
2
| − u × ω,
and take the curl of the above equation to obtain
∂ω
∂t
− ∇ × (u × ω) = ν∇
2
ω,
exploiting the fact that the curl of a gradient vanishes. We now use the fact that
∇ × (u × ω) = (∇·ω)u + ω · ∇u − (∇ · u)ω −u ·∇ω.
The divergence of a curl vanishes, and so does
∇ · u
by incompressibility. So we
get
∂ω
∂t
+ u · ∇ω −ω · ∇u = ν∇
2
ω.
Now we use the definition of the material derivative, and rearrange terms to
obtain
Proposition (Vorticity equation).
Dω
Dt
= ω ·∇u + ν∇
2
ω.
Hence, the rate of change of vorticity of a fluid particle is caused by
ω ·∇u
(amplification by stretching or twisting) and
ν∇
2
ω
(dissipation of vorticity by
viscosity). The second term also allows for generation of vorticity at boundaries
by the no-slip condition. This will be explained shortly.
Consider an inviscid fluid, where ν = 0. So we are left with
Dω
Dt
= ω ·∇u.
So if we take the dot product with ω, we get
D
Dt
1
2
|ω|
2
= ω ·∇u ·ω
= ω(E + Ω)ω
= ω
i
(E
ij
+ Ω
ij
)ω
j
.
Since
ω
i
ω
j
is symmetric, while Ω
ij
is antisymmetric, the second term vanishes.
In the principal axes, E is diagonalizable. So we get
D
Dt
1
2
|ω|
2
= E
1
ω
2
1
+ E
2
ω
2
2
+ E
3
ω
2
3
.
wlog, we assume
E
1
>
0 (since the
E
i
’s sum to 0), and imagine
E
2
, E
3
<
0. So
the flow is stretched in the
e
1
direction and compressed radially. We consider
what happens to a vortex in the direction of the stretching,
ω
= (
ω
1
,
0
,
0). We
then get
D
Dt
1
2
ω
2
1
= E
1
ω
2
1
.
So the vorticity grows exponentially. This is vorticity amplification by stretching.
This is not really unexpected — as the fluid particles have to get closer to
the axis of rotation, they have to rotate faster, by the conservation of angular
momentum.
This is in general true — vorticity increases as the length of a material
line increases. To show this, we consider two neighbouring (Lagrangian) fluid
particles, x
1
(t), x
2
(t). We let δ`(t) = x
2
(t) − x
1
(t). Note that
Dx
2
(t)
Dt
= u
2
(x
2
),
Dx
1
(t)
Dt
= u
1
(x
1
).
Therefore
Dδ`(t)
Dt
= u(x
2
) − u(x
1
) = δ` · ∇u,
by taking a first-order Taylor expansion. This is exactly the same equation as
that for
ω
in an inviscid fluid. So vorticity increases as the length of a material
line increases.
Note that the vorticity is generated by viscous forces near boundaries.
ω
When we make the inviscid approximation, then we are losing this source of
vorticity, and we sometimes have to be careful.
We have so far assumed the density is constant. If the fluid has a non-uniform
density ρ(x), then it turns out
Dω
Dt
= ω ·∇u +
1
ρ
2
∇ρ × ∇p.
This is what happens in convection flow. The difference in density drives a
motion of the fluid. For example, if we have a horizontal density gradient and a
vertical pressure gradient (e.g. we have a heater at the end of a room with air
pressure varying according to height), then we get the following vorticity:
∇p
∇ρ
hot, light cold, dense
ω
×