What is vorticity?

For example, consider the special case: For 2D flow,

\begin{displaymath}w = 0;\ \frac{\partial }{\partial z} = 0;\ \omega _{y}=\omega...
...\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}
\end{displaymath}

1.
Translation: $u =$ constant, $v =$ constant

\begin{figure}
\begin{center}
\epsfig{file=lfig74a.eps,height=3.0in,clip=}
\end{center}
\end{figure}

\begin{displaymath}\frac{\partial v}{\partial x} = 0,\frac{\partial u}{\partial y} = 0
\Rightarrow \omega _z = 0 \to \mbox{no vorticity}
\end{displaymath}

2.
Pure Strain: (no change in volume)

\begin{figure}
\begin{center}
\epsfig{file=lfig75a.eps,height=3.0in,clip=}
\end{center}
\end{figure}

\begin{displaymath}\frac{\partial u}{\partial x} = - \frac{\partial v}{\partial ...
...\frac{\partial v}{\partial x} = 0
\Rightarrow \omega _z = 0
\end{displaymath}

3.
Angular deformation

\begin{figure}
\begin{center}
\epsfig{file=lfig76a.eps,height=3.0in,clip=}
\end{center}
\end{figure}

\begin{displaymath}\vec{\omega} = 0 \mbox{\ only if\ } \frac{\partial u}{\partia...
...arrow \delta x = \delta y
(\mbox{for\ } \Delta x = \Delta y)
\end{displaymath}

4.
Pure Rotation

\begin{figure}
\begin{center}
\epsfig{file=lfig77a.eps,height=3.0in,clip=}
\end{center}
\end{figure}
Pure rotation with angular velocity $\Omega $

\begin{displaymath}\frac{\partial v}{\partial x} = \Omega ;\ \ \frac{\partial u}{\partial y} = -
\Omega ;\ \ \omega _z = 2\Omega
\end{displaymath}

i.e. vorticity $ \propto $ 2(angular velocity).
For irrotational Flow:

\begin{displaymath}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha...
...ghtarrow \Gamma \equiv 0 \mbox{\ for any closed contour \ } C
\end{displaymath}

Suppose that at $t=t_{o}$, the flow is irrotational, i.e. $\Gamma \equiv 0$ for all closed contours (material or not) $C$. Then for ideal fluid under conservative body forces, Kelvin's theorem states that $\Gamma \equiv 0$ for all closed material contours $C$ for all time $t$. i.e., once irrotational, always irrotational (Special case of Kelvin's theorem).

Karl P Burr
2003-07-07