8.1. Theorems about vorticity

8.1.1. Kelvin’s theorem

Theorem 8.1 (Kelvin’s theorem)

In a barotropic or constant density flow, with negligible viscosity effects, the circulation on a material line \(\ell(t)\) is constant in time,

\[0 = \frac{d \Gamma_{\ell(t)}}{dt} = \dfrac{d}{dt} \oint_{\ell(t)} \mathbf{u} \cdot \hat{\mathbf{t}} \ .\]

Proof 1, using time derivative over time dependent domains

Using the expression of the time derivative over a work line integral, over a closed line with no discontinuity in the functions so that \(\mathbf{f}(\mathbf{r}_B) \cdot \mathbf{v}_B - \mathbf{f}(\mathbf{r}_A) \cdot \mathbf{v}_A = 0\),

\[\begin{split}\begin{aligned} \dfrac{d}{dt} \oint_{\ell(t)} \mathbf{u} \cdot \hat{\mathbf{t}} & = \oint_{\ell(t)} \partial_t \mathbf{u} \cdot \hat{\mathbf{t}} + \oint_{\ell(t)} \nabla \times \mathbf{u} \cdot \mathbf{u}_b \times \hat{\mathbf{t}} = && (1) \\ & = \oint_{\ell(t)} \hat{\mathbf{t}} \cdot \left\{ \partial_t \mathbf{u} + \boldsymbol\omega \times \mathbf{u} \right\} = && (2) \\ & = \oint_{\ell(t)} \hat{\mathbf{t}} \cdot \left\{ \partial_t \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} - \nabla \frac{|\mathbf{u}|^2}{2} \right\} = && (3) \\ & = - \oint_{\ell(t)} \hat{\mathbf{t}} \cdot \nabla \left\{ \nabla \Pi + \nabla \chi + \frac{|\mathbf{u}|^2}{2} \right\} + \oint_{\ell(t)} \nu \hat{\mathbf{t}} \cdot \nabla^2 \mathbf{u} = && (4) \\ & = \oint_{\ell(t)} \nu \hat{\mathbf{t}} \cdot \nabla^2 \mathbf{u} = && (5) \\ & = 0 \qquad \text{if $\nu = 0$} \ , \end{aligned}\end{split}\]

with \((1)\) \(\mathbf{u}_b = \mathbf{u}\) for material lines, and \(\boldsymbol\omega = \nabla \times \mathbf{u}\), and using the properties of the mixed vector product \(\mathbf{a} \cdot \mathbf{b} \times \mathbf{c} = \mathbf{b} \cdot \mathbf{c} \times \mathbf{a}\), \((2)\) \(\mathbf{u} \cdot \nabla \mathbf{u} = \boldsymbol\omega \times \mathbf{u} + \nabla \frac{|\mathbf{u}|^2}{2}\), \((3)\) from the momentum equation \(D_t \mathbf{u} = \mathbf{g} + \nu \nabla^2 \mathbf{u} - \nabla \Pi\), and \(\mathbf{g} = - \nabla \chi\), and \((4)\) \(\oint_{\ell(t)} \hat{\mathbf{t}} \cdot \nabla \Phi = \oint_{\ell(t)} d \Phi = 0\).

Proof 2, using material coordinates

8.1.2. Helmholtz’s theorems

A vortex line is defined as a line everywhere tangent to the vorticity vector field \(\boldsymbol\omega(\mathbf{r})\). A vortex tube is a 3-dimensional region of space with a lateral surface that is delimited by a set of vortex lines, and thus whose normal vector is orthogonal to the vorticity field, \(\hat{\mathbf{n}} \cdot \boldsymbol\omega = 0\) on \(S_{lat}\). It immediately follows first Helmholtz’s theorem.

Theorem 8.2 (First Helmholtz’s theorem)

The flux of the vorticity field is constant across all the sections of a vortex tube, at a given time.

Proof

As the vorticity is the curl of the velocity field, \(\boldsymbol = \nabla \times \mathbf{u}\), it’s divergence is identically zero, \(\nabla \cdot \boldsymbol\omega = 0\). Integrating over a volume delimited by the lateral surface of a vortex tube and 2 sections, and applying divergence theorem and the definition of the vortex tube,

\[\begin{split}\begin{aligned} 0 & = \int_V \nabla \cdot \boldsymbol\omega = \\ & = \oint_{\partial V} \hat{\mathbf{n}} \cdot \boldsymbol\omega = \\ & = \int_{S_1 } \hat{\mathbf{n}}_1 \cdot \boldsymbol\omega + \int_{S_2 } \hat{\mathbf{n}}_2 \cdot \boldsymbol\omega + \int_{S_{lat}} \underbrace{\hat{\mathbf{n}} \cdot \boldsymbol\omega}_{=0} = \\ \end{aligned}\end{split}\]

Reversing the unit normal on surface \(S_1\) in order to get an «oriented» vortex tube, with \(\hat{\mathbf{n}}|_{S_1} = - \hat{\mathbf{n}}_1\) pointing inward, and \(\hat{\mathbf{n}}|_{S_2} = \hat{\mathbf{n}}_2\), the theorem is proved,

\[\int_{S_1} \hat{\mathbf{n}} \cdot \boldsymbol\omega = \int_{S_2} \hat{\mathbf{n}} \cdot \boldsymbol\omega \ .\]

Theorem 8.3 (Second Helmholtz’s theorem)

Vortex lines are material lines.

Proof.

Let’s take two points on a vortex line,

\[\begin{split}\begin{aligned} & \mathbf{r}_1(t) \\ & \mathbf{r}_2(t) = \mathbf{r}_1(t) + \Delta \ell(t) \hat{\mathbf{t}}(t) + o(\Delta \ell(t)) \ , \end{aligned}\end{split}\]

with \(\boldsymbol\omega(\mathbf{r}_1(t), t) = \omega(\mathbf{r}_1(t), t) \hat{\mathbf{t}}(t)\). At time \(t + \Delta t\), the position of the two material points reads

\[\begin{split}\begin{aligned} \mathbf{r}_1(t + \Delta t) & = \mathbf{r}_1(t) + \mathbf{u}(\mathbf{r}_1(t), t) \Delta t + o (\Delta t) \\ \mathbf{r}_2(t + \Delta t) & = \mathbf{r}_2(t) + \mathbf{u}(\mathbf{r}_2(t), t) \Delta t \\ & = \mathbf{r}_2(t) + \left[ \mathbf{u}(\mathbf{r}_1(t), t) + \Delta \mathbf{r}_{12}(t) \cdot \nabla \mathbf{u}(\mathbf{r}_1(t), t) + o(|\Delta \mathbf{r}_{12}(t)|) \right] \Delta t + o(\Delta t) \ , \end{aligned}\end{split}\]

so that

\[\Delta \mathbf{r}_{12}(t + \Delta t) = \Delta \mathbf{r}_{12}(t) + \Delta \mathbf{r}_{12}(t) \cdot \mathbf{u}(\mathbf{r}_1(t),t) \Delta t + o(|\Delta \mathbf{r}_{12}(t)| \Delta t + o(\Delta t) \ .\]

Now, using the vorticity equation for inviscid flows

\[D_t \boldsymbol\omega = ( \boldsymbol\omega \cdot \nabla ) \mathbf{u} \ ,\]

the vorticity in \(\mathbf{r}_1(t+\Delta t)\) at time \(t+\Delta t\) is

\[\begin{split}\begin{aligned} \boldsymbol\omega(\mathbf{r}_1(t+\Delta t), t+\Delta t) & = \boldsymbol\omega(\mathbf{r}_1(t), t) + D_t \boldsymbol\omega (\mathbf{r}_1(t), t) \, \Delta t + o(\Delta t) = \\ & = \boldsymbol\omega(\mathbf{r}_1(t), t) + \boldsymbol\omega (\mathbf{r}_1(t), t) \cdot \nabla \mathbf{u}(\mathbf{r}_1(t), t) \, \Delta t + o(\Delta t) \ . \end{aligned}\end{split}\]

Comparing the evolution of the material segment \(\Delta \mathbf{r}_{12}\) and the vorticity vector in \(\mathbf{r}_1\), it’s immediate to realize that they have the same evolution, namely for \(\Delta t \rightarrow 0\)

\[d_t \mathbf{v} = \mathbf{v} \cdot \nabla \mathbf{u}(\mathbf{r}_1(t),t) \ .\]

As a consequence of Helmholtz’s theorems, vortex tubes are either closed or start and end at solid surfaces (production/destruction of vorticity, due to viscous term).

todo discuss using vorticity equation

A vortex tube with a section collapsing to zero, can be modeled as a zero section1, infinite vorticity vortex filament \(\ell_\Gamma\), so that

\[\begin{split}\lim_{\begin{aligned} S & \rightarrow 0 \\ |\boldsymbol\omega| & \rightarrow +\infty \end{aligned}} \int_{S} \hat{\mathbf{n}} \cdot \boldsymbol\omega = \Gamma \ .\end{split}\]

Vorticity in a irrotational 3-dimensional domain, \(\mathbf{r} \in V\), containing a vortex filament, \(\mathbf{r}_\Gamma \in \ell_\Gamma\), can be modelled using an impulsive two-dimensional2 Dirac’s delta function

\[\boldsymbol\omega(\mathbf{r},t) = \boldsymbol\Gamma(\mathbf{r}_\Gamma,t) \delta_{2}(\mathbf{r} - \mathbf{r}_\Gamma) \ ,\]

with \(\mathbf{r}_\Gamma \in \ell_\Gamma\), and \(\boldsymbol\Gamma = \Gamma \hat{\mathbf{t}}\), with \(\hat{\mathbf{t}}\) the unit vector tangent to the axis of the filament.

Circulation on a line winding once around a vortex filament with intensity \(\Gamma\) reads \(\oint_{\ell} \mathbf{u} \cdot \hat{\mathbf{t}} = \Gamma\).

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todo

A vortex surface …

Across a vortex surface the tangential velocity has a jump

\[\Delta \mathbf{u} = \dots\]

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1

A filament can be cut with a section orthogonal to its axis or with a generic angle w.r.t. it. Orthogonal cut gives the minimum area of the section \(S_0\), a generic cut with unit normal \(\hat{\mathbf{n}}\) gives \(S\) so that \(S_0 = \hat{\mathbf{n}} \cdot \hat{\mathbf{t}} S\).

2

The 2-dimensional Dirac’s delta \(\delta_2(\mathbf{r}-\mathbf{r}_\Gamma)\) has physical dimensions \(\left[ \delta_2 \right] = L^{-2}\), and it gives the value of the function in \(\mathbf{r}_\Gamma\) from the integral on a 2-dimensional surface \(S\), \(\int_{\mathbf{r} \in S} f(\mathbf{r}) \delta_2(\mathbf{r}-\mathbf{r}_\Gamma) = f(\mathbf{r}_\Gamma)\), if \(\mathbf{r}_\Gamma \in S\).