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Vorticity

Vorticity

A central concept of fluid dynamics is vorticity, defined in three dimensions as

(12)\[{\vec \omega} = {\vec \nabla} \times {\vec v}\]

If you are comforable with the curl being a (pseudo)scalar in two dimenions, the definition is basically the same. Alternate definitions in 2d are:

\[ \omega_{2d} = \epsilon^{ij}\partial_i v_j \]

where \(\epsilon^{ij}\) is the totally antisymmetric tensor with \(\epsilon^{12} = 1\), we are using Einstein summation notation for repeated indices, and we are using the notation

\[ \partial_i \equiv \frac{\partial}{\partial x^i} \]

[For afficionados of differential forms, if we let \(v\) with lower indices be a 1-form, \(\omega = dv\) is a two-form. In 3 dimensions a two-form is dual to a vector; in 2 dimensions it is dual to a scalar. This is the path to defining vorticity in higher dimensions if one wanted to do that.

Vorticity is crucial for a number of reasons; physically it records how the shape of a fluid element distorts in a nontrivial manner.

Motivation and interpretation

A nice discussion is in the text, I would like to offer a more general one (see for example [Chorin et al., 1990].) The essential point is that a constant velocity field \({\vec v}\) is not by itself all that interesting (or particularly physical – the velocity usually goes to zero at the boundaries). What is interesting is when the velocity changes in space and time. Let us focus on characerizing the former.

Consider a velocity field \(v_i({\vec x})\). Let

\[ {\vec x} = {\vec x}_0 + \delta {\vec x} \]

A local sense of the variation of velocity can be found by doing a Taylor expansion. To first order in \(\delta {\vec x}\),

\[ v_i(x_0^j + \delta x^j) = v^i(x_0) + \delta x^j \partial_j v_i(x_0) + ... \]

Again, we are using Einstein summation notation here. Now we can write

(13)\[\begin{align} \partial_j v_i & = & \frac{1}{2}\left(\partial_j v_i + \partial_i v_j\right) + \frac{1}{2} \left(\partial_j v_i - \partial_i v_j\right) \\ & = & S_{ij} + \omega_{ij} \end{align}\]

\(S_{ij}\) is called the strain tensor, and \(\omega_{ij}\) is the vorticity tensor. In the second case, for \(d = 2\) dimensions, \(\omega = \epsilon^{ij} \omega_{ij}\) is the vorticity as defined in the text. In three dimensions,

\[ \omega^i \equiv \epsilon^{ijk} \partial_j v_k = ({\vec \nabla}\times{\vec v})^i \]

is the usual definiion of the vorticity. Here \(\epsilon^{ijk}\) is the totally antisymmetric tensor in \(d = 3\), such that \(\epsilon^{123} = 1\).

Returning to an interpretation of \(S\) and \(\omega\),

(14)\[\begin{align} \delta v_i(x_0) & = & v_i({\vec x}_0 + \delta {\vec x}) - v_i({\vec x}_0)\\ & = & S_{ij} \delta x^j + \omega_{ij} \delta x^j \end{align}\]

If we adopt the Lagrangian picture, with \(\delta x(t)\) corresponding to the physical location of a fluid parcel near \({\vec x}_0\), then \(\delta v_i = \frac{d}{dt} \delta x^i\), and we have

\[ \frac{d}{dt} \delta x^i = S_{ij} \delta x^i + \omega_{ij} \delta x^j \]

Now consider a blob of fluid surrounding \({\vec x}_0\) defined by a collection of such vectors. As a symmetric matrix, \(S_{ij}\) will have three real eigenvalues \(s^{\alpha}\) \(\alpha \in {1,2,3}\) and associated eigenvectors or principle axes \(h^{\alpha}_i\), such that \(S_{ij} h^{\alpha}_j = s^{\alpha} h^{\alpha}_i\). If \(\delta x\) points in one of these directions, it will stretch or shrink in time depending onthe sign of \(s^{\alpha}\). In other words, \(S_{ij}\) will cause he blob to stretch or shrink along the principle axes.

On the oher hand, if \(\epsilon^{ijk} \omega_{jk} \equiv \omega^i\), then we can use the identity

\[ \epsilon^{ijk} \epsilon_{ilm} = \delta^j_l \delta^k_m - \delta^j_m \delta^k_l \]

to show that

\[ \omega_{ij} = 2 \epsilon_{ijk} \omega^k \]

and so

\[ \omega_{ij} \delta x^j = 2 \epsilon_{ijk} \delta x^j \omega^k = 2 (\delta {\vec x} \times {\vec \omega})^i \]

If \(S_{ij}\) were vanishing, then

\[ \frac{d}{dt} \delta {\vec x} = 2 \delta{\vec x} \times {\vec \omega} \]

describes a rotation of \(\delta {\vec x}\) about the axis defined by \(\omega\), with angular velocity \(2|\omega|\).

Irrotational flows

When \({\vec \omega} = {\vec \nabla} \times {\vec v} = 0\) throughout the flow, we say the flow is irrotational. This flow will have special properties. In particular, in any simply connected region, we can write \({\vec v} = {\vec \nabla} \phi\). We call \(\phi\) the potential, and he associated flow potential flow.

Two-dimensional Incompressible flow

Any vector which is divergenceless can b written as a curl (in a simply connected region). In three dimensions this is no obvious simplification – you trade a vector for a vector - but in two dimensions, this means that you can write

\[ {\vec v} = - \partial_y \psi {\hat x} + \partial_x \psi {\hat y} \]

for some scalar streamfunction \(\psi\). In this case, \(\omega = \nabla^2 \psi\). This is a useful object in two dimensions and variants are important for large-scale atmospheric and oceanic flows: the streamfunction \(\psi\) can be written as the integral of the velocity and so becomes a measure of tansport across a line.

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