1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
|
---
title: "Vorticity"
firstLetter: "V"
publishDate: 2021-04-03
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
date: 2021-04-03T09:24:42+02:00
draft: false
markup: pandoc
---
# Vorticity
In fluid mechanics, the **vorticity** $\va{\omega}$
is a measure of the local circulation in a fluid.
It is defined as the curl of the flow velocity field $\va{v}$:
$$\begin{aligned}
\boxed{
\va{\omega}
\equiv \nabla \cross \va{v}
}
\end{aligned}$$
Just as curves tangent to $\va{v}$ are called *streamlines*,
curves tangent to $\va{\omega}$ are **vortex lines**,
which are to be interpreted as the "axes" that $\va{v}$ is circulating around.
The vorticity is a local quantity,
and the corresponding global quantity is the **circulation** $\Gamma$,
which is defined as the projection of $\va{v}$ onto a close curve $C$.
Then, by Stokes' theorem:
$$\begin{aligned}
\boxed{
\Gamma(C, t)
\equiv \oint_C \va{v} \cdot \dd{\va{l}}
= \int_S \va{\omega} \cdot \dd{\va{S}}
}
\end{aligned}$$
## Ideal fluids
For an inviscid, incompressible fluid,
consider the *Bernoulli field* $H$, which is defined as:
$$\begin{aligned}
H
\equiv \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho}
\end{aligned}$$
Where $\Phi$ is the gravitational potential,
$p$ is the pressure, and $\rho$ is the (constant) density.
We then take the gradient of this scalar field:
$$\begin{aligned}
\nabla H
&= \frac{1}{2} \nabla \va{v}^2 + \nabla \Phi + \frac{\nabla p}{\rho}
\\
&= \va{v} \cdot (\nabla \va{v}) - \Big( \!-\! \nabla \Phi - \frac{\nabla p}{\rho} \Big)
\end{aligned}$$
Since $-\nabla \Phi = \va{g}$,
the rightmost term is the right-hand side of
the [Euler equation](/know/concept/euler-equations/).
We substitute the other side of said equation, yielding:
$$\begin{aligned}
\nabla H
&= \va{v} \cdot (\nabla \va{v}) - \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
= \va{v} \cdot (\nabla \va{v}) - \pdv{\va{v}}{t} - (\va{v} \cdot \nabla) \va{v}
\end{aligned}$$
We isolate this equation for $\pdv*{\va{v}}{t}$,
and apply a vector identity to reduce it to the following:
$$\begin{aligned}
\pdv{\va{v}}{t}
= \va{v} \cdot (\nabla \va{v}) - (\va{v} \cdot \nabla) \va{v} - \nabla H
= \va{v} \cross (\nabla \cross \va{v}) - \nabla H
\end{aligned}$$
Here, the definition of the vorticity $\va{\omega}$ is clear to see,
leading us to an equation of motion for $\va{v}$:
$$\begin{aligned}
\boxed{
\pdv{\va{v}}{t}
= \va{v} \cross \va{\omega} - \nabla H
}
\end{aligned}$$
More about this later.
Now, we take the curl of both sides of this equation, giving us:
$$\begin{aligned}
\nabla \cross \pdv{\va{v}}{t}
= \nabla \cross (\va{v} \cross \va{\omega}) - \nabla \cross (\nabla H)
\end{aligned}$$
On the left, we swap $\nabla$ with $\pdv*{t}$,
and on the right, the curl of a gradient is always zero.
We are thus left with the equation of motion of the vorticity $\va{\omega}$:
$$\begin{aligned}
\boxed{
\pdv{\va{\omega}}{t}
= \nabla \cross (\va{v} \cross \va{\omega})
}
\end{aligned}$$
Let us now return to the equation of motion for $\va{v}$.
For *steady* flows where $\pdv*{\va{v}}{t} = 0$, in which case
[Bernoulli's theorem](/know/concept/bernoullis-theorem/) applies,
it reduces to:
$$\begin{aligned}
\nabla H
= \va{v} \cross \va{\omega}
\end{aligned}$$
If a fluid has $\va{\omega} = 0$ in some regions, it is known as **irrotational**.
From this equation, we see that, in that case, $\nabla H = 0$,
meaning that $H$ is a constant in those regions,
a fact sometimes referred to as **Bernoulli's stronger theorem**.
Furthermore, irrotationality $\va{\omega} = 0$
implies that $\va{v}$ is the gradient of a potential $\Psi$:
$$\begin{aligned}
\va{v}
= \nabla \Psi
\end{aligned}$$
This fact allows us to rewrite the Euler equations in a particularly simple way.
Firstly, the condition of incompressibility becomes the well-known Laplace equation:
$$\begin{aligned}
0
= \nabla \cdot \va{v}
= \nabla^2 \Psi
\end{aligned}$$
And second, the main equation of motion for $\va{v}$ states
that the quantity $H + \pdv*{\Psi}{t}$ is spatially constant
in the irrotational region:
$$\begin{aligned}
\pdv{\va{v}}{t}
= \nabla \pdv{\Psi}{t}
= - \nabla H
\quad \implies \quad
\nabla \Big( H + \pdv{\Psi}{t} \Big)
= 0
\end{aligned}$$
## References
1. B. Lautrup,
*Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
CRC Press.
|
