Stretching Behavior of Knotted and Unknotted Flow Fields

There are many examples of flows with concentrated vortex lines, including tornadoes [1], smoke rings [2], flows inside the heart [3], and turbulence [4, 5]. Indeed, any incompressible flow can be regarded as a collection of vortex lines by the Helmoltz Theorem, and so understanding their behavior provides a complete description of such flows. As a result, there has been considerable theoretical, numerical, and experimental research focused on understanding the behavior of concentrated vortex filaments (e.g. [6, 7, 8, 9, 10, 11, 12, 13, 14]).

While a simple circular vortex ring is known to be quite stable [15, 16] – even if distorted [17, 16] – knotted and tangled vortices have been observed to be highly unstable in simulations [18, 19, 20, 21, 22] and experiments [23, 24]. In particular, tangled vortex lines are observed to rapidly stretch, leading to vortex reconnections which ultimately untie and/or dissipate the flow [23]. This behavior is quite reminiscent of the features of turbulent flows [25, 4, 26, 27], and has both enstrophy production (vortex stretching) and transport of energy to small scales, where it is dissipated (vortex reconnections).

The connection between vortex stretching and reconnections can be understood in terms of energy conservation: in order to stretch vortex lines without increasing energy, it is necessary to create regions of closely spaced counter-rotating vortices [18, 28]. As the vortices continue to stretch, these counter-rotating vortices must get closer together, ultimately resulting in reconnections which continue until non-stretching vortex state is reached [23, 18]. Moreover, vortex stretching (i.e. enstrophy production) is a key feature of turbulence; indeed it can be regarded as the key feature which separates 2D from 3D turbulence [29]. As a result, vortex stretching is intimately connected to the stability of flows, motivating research in to how and why collections of vortex lines self-stretch.

Previous work has modeled this stretching using either vortex filament models [30, 31] or direct numerical simulations of Newtonian or super-fluids using the Navier-Stokes [8] or Gross-Pitaevskii equations [32]. In either case, the non-linearity of fluid flows makes these models difficult to accurately simulate, and complicates the interpretation of the results.

Here we take a different approach: ‘freezing’ vortex generated flow fields, and investigating how material lines stretch when advected in this flow. Given that vortex lines themselves are transported by the flow [33], it follows that a flow field which stretches material lines will also stretch vortices. In this manuscript, we investigate if this stretching can be understood in terms of the properties of the flow field at a single instant in time, rather than as a consequence of the non-linear evolution of the flow. Using this approach, we find that simple unlinked and unknotted vortices have a flow field which produces linear stretching, while the flow field of knotted vortices produces regions of exponential stretching. Moreover, this stretching can be attributed to the generation of ‘bights’ in the material lines (Fig. 1), and that exponential stretching is only possible if these bights are continuously produced. This result offers a potential connection to existing results in two-dimensional topological mixing [34], and appears to explain why knotted vortices are themselves unstable. Moreover, the concept of bights provides a simple mechanism for understanding how lines in vortex dominated flow fields should stretch over time, with potential applications to a variety of fluid flow problems.

Before discussing the stretching produced by the flow fields of complex vortex geometries, it is useful to consider the case of a straight line vortex. The flow field of an infinite vortex along with the $z$-axis is given by:

$\displaystyle\bm{u}$

$\displaystyle=\frac{\Gamma}{2\pi\rho}\hat{\bm{\phi}},$

(1)

where $\Gamma$ is the circulation, $\rho$ and $\phi$ are cylindrical coordinates defined in the usual way, and $\hat{\bm{\phi}}$ is a unit vector in the azimuthal direction. Here, and in the rest of this manuscript, we will use dimensionless spatial and time coordinates, and a dimensionless circulation of $\Gamma=2\pi$ (i.e. for an infinite straight line we obtain: $\bm{u}=\hat{\bm{\phi}}/\rho$).

The evolution of a small circular material line in this flow field is shown in Fig. 1. How can we understand the stretching of this line? The time derivative of the total length of the material line, $L$, is given by:

	$\displaystyle\dot{L}$	$\displaystyle=\oint\underbrace{\bm{\nabla}_{\hat{\bm{T}}}\bm{u}}_{\frac{\partial\bm{u}[\bm{r}(s)]}{\partial s}}\cdot\hat{\bm{T}}\ ds$		(2)
		$\displaystyle=-\oint\bm{u}\cdot\kappa\hat{\bm{N}}\ ds,$		(3)

where $\bm{r}(s)$ is the material line displacement as a function of arc length coordinate, $s$. $\hat{\bm{T}}$, $\hat{\bm{N}}$, and $\kappa$ are the Frenet-Serret tangent vector, normal vector, and curvature, respectively, which obey the relationships $\hat{\bm{T}}=\frac{\partial\bm{r}}{\partial s}$ and $\kappa\hat{\bm{N}}=\frac{\partial\hat{\bm{T}}}{\partial s}$. Eqn. 3 is obtained using integration by parts, and would have an additional term for an open material line. (Note that integrand of eqn. 3 does not give the local stretching rate; see Appendix A.1.)

Any section of material line advected in the flow of an infinite line vortex will tend to align or anti-align with $\bm{u}$ as $t\rightarrow 0$ (e.g. Fig. 1; see also supplemental materials for a proof). When this happens, $\bm{u}\cdot\hat{\bm{N}}\rightarrow 0$, and from eqn. 3 we would expect that these regions to produce no stretching.

A closed material line, however, must change direction with respect to $\bm{u}$ at two or more points to form a closed path. Over time, these regions will form compact 180^∘ bends, which we will refer to as ‘bights’, by analogy with a term using in knot tying. Formally, we will define a bight as the point at which $\bm{u}\cdot\hat{\bm{T}}$ changes sign along a material line.

Near the bight, $\bm{u}\cdot\hat{\bm{N}}\neq 0$, and so the bights are responsible for stretching or shortening the material line, depending on whether they are a ‘leading’ bight ($\hat{\bm{N}}\cdot\hat{\bm{u}}<0$) or a ‘trailing’ bight ($\hat{\bm{N}}\cdot\hat{\bm{u}}>0$). (Note that for open material lines, the ends of the material line also function similar to bights; see Supplemental Materials for details.)

If we assume that all the stretching can be attributed to bights – each of which is assumed to be a compact 180^∘ bend – we would obtain a simple expression for the stretching rate:

$\displaystyle\dot{L}(t\rightarrow\infty)$

$\displaystyle\approx-2\sum_{\rm bights}\hat{\bm{N}}\cdot\bm{u},$

(4)

where $\bm{u}$ and $\hat{\bm{N}}$ are computed at the bights and an overall factor of 2 is obtained by integrating $\bm{u}\cdot\kappa\hat{\bm{N}}$ around the 180^∘ bend. As an example, consider a material line which is a circle displaced from a single vortex line (Fig. 1). If this circle has center displacement, $\rho_{0}$, and radius, $a$, it will form a leading bight near the location closest to the vortex, at $\rho=\rho_{0}-a$, and a trailing bight near the spot furthest from the vortex, at $\rho=\rho_{0}+a$. Under the assumption that each bight is a 180^∘ bend in the material line, we would predict a stretching rate of:

$\displaystyle\dot{L}(t\rightarrow\infty)=\frac{\Gamma}{\pi}\left[\frac{1}{\rho_{0}-a}-\frac{1}{\rho_{0}+a}\right].$

(5)

As can be seen in Fig. 1c, the stretching rate converges on this result after only a fraction of a turn around the vortex.

This behavior is not unique to circular material lines: as shown in the supplementary materials, any material line should tend to align with the flow field of a straight line vortex over time. As a result, it will form some number of compact bights, and in the long time limit these will determine the stretching rate.

It is not clear that such a formulation should apply to more complicated geometries, for which we can not assume that material lines will always align with $\bm{u}$. In this section we consider the stretching of material lines in the flow fields of more complex vortex shapes, including rings, distorted rings, and knots.

An alternative method to probe for exponential stretching is to compute the finite time Lyapunov exponent (FTLE) of the flow field [38]. The FTLE characterizes the exponential stretching rates of parcels of fluid in the direction of maximum strain.

We compute the FTLE using the Cauchy strain tensor; for an initial point, $\bm{x}(t=0)$, this is given by:

$\displaystyle C_{ij}(T)$

$\displaystyle=\sum_{k}\frac{\partial x_{k}(T)}{\partial x_{i}(0)}\frac{\partial x_{k}(T)}{\partial x_{j}(0)},$

(8)

where $\bm{x}(T)$ is the point $\bm{x}(0)$ transported forward in time an interval $T$, and in practice the derivatives are computed by attaching infinitesimal unit vectors to each point (see Appendix B.4).

The FTLE is then given by:

$$\Lambda_{T}(\bm{x})=\frac{1}{T}\log\sqrt{\lambda_{max}(\bm{x})},$$

(9)

where $\lambda_{max}(\bm{x})$ is the largest eigenvalue of the deformation tensor, $C_{ij}$, computed for an initial position $\bm{x}$.

FTLE maps for knotted vortices of varying aspect ratio are shown in Fig. 4. In each case, the regions of highest exponential growth appear in the regions between the vortices. This is despite the fact that the regions of highest instantaneous shear are immediately adjacent to the vortices; these regions do not produce new bights, and so produce only linear stretching in the long time limit. (If the FTLE is computed for larger $T$, the linearly stretching regions near the core will have a smaller values, but limits of numerical precision prevent us from accurately computing the exponent in these regions; the simulations are stopped when the longest stretched unit vectors are stretched by a factor of $\sim 10^{10}$.)

Most notably, the FTLE plots reveal ridges of high stretching – unstable manifolds – which connect sections of the vortex (e.g. in the center of Fig. 4(e-f). These appear to be the regions responsible for producing new bights; we note that if the material line does not cross one of these ridges we do not observe exponential stretching. For example, for $a=0.3$, $r_{p}=0.75$, we observe only linear growth (Fig. 3). This material line lies just outside the FTLE ridge; for a slightly smaller value of $r_{p}=0.65$ which does cross the ridge we do observe exponential growth after an initial linear period. Additionally, we note that the quantitative rate of exponential stretching observed for both toroidal and poloidal material lines has roughly the same time constant as the peak value on the ridges of high stretching observed in the FTLE plots.

In this manuscript we have compared the stretching of arbitrary material lines by vortical flow fields of varying topology and geometry. In particular we have shown that the flow fields of knotted vortices produce exponential stretching of material lines in certain regions, which is not seen in the stretching of flow fields of unknotted vortices. This change in the qualitative character of these flow fields is confirmed by computing the FTLE of the flow fields. The FTLE results demonstrate that it is the regions between the vortices which are responsible for exponential stretching. Intuitively, this can be explained by the production of ‘bights’ in these regions. Conversely when a region of flow is dominated by a single vortex the rapid strain prevents the formation of new bights, ultimately producing less stretching over the long term even though the local strain rate may be much higher.

These results suggest previously unknown connections between previous results in the physics of mixing and vortex dominated fluid flows. In 2D flows, it is well known that the topology of a time-dependent flow fields is connected to long term exponential stretching [34, 39]. In this case, exponential stretching is produced by a ‘stretching and folding’ action reminiscent of a taffy puller. Our work suggests that topology in space alone can play a similar role in 3D flows; in our case the ‘folding’ of the material lines is indicated by the production of bights.

These results may also offer an explanation for the apparent instability of knotted vortices. It has previously been observed – in both experiments and simulations – that linked and knotted vortices are highly unstable to self-stretching [18, 19, 20, 23, 24, 21, 22]. Although our results do not model the self-stretching of vortex lines (indeed, we treat the vortices as fixed), they do demonstrate that vortex topology has a dramatic effect on the stretching behavior of the a flow field. As vortex lines themselves are transported by flows, it follows that we would also expect a qualitative difference in their evolution.

Although in principle one could model the stretching produced by vortices whose shape evolves in time, as previously noted knotted vortices will rapidly approach reconnection events. Thus, a dynamic model would be limited in the total amount of time which the vortices remain knotted, making it difficult or impossible to separate exponential from non-exponential stretching. Interestingly, our results suggest that a more complex model is not required: the advection of material lines by a ‘frozen’ flow field may be sufficient to indicate their long term stability. This also suggests that the non-linearities present in the full Navier-Stokes equation are not required to explain the difference in stability between knotted and unkontted vortices.

These results have potential relevance for a wide variety of vortex-dominated flows, such as tornadoes, flows around aerodynamic surfaces, and turbulence. For example, we note that enstrophy production – i.e. vortex stretching – is one of the hallmarks of 3D turbulence, and is one of the features which distinguishes 2D from 3D flows [29]. Although further work is required to show that these results hold for a wider variety of flow configurations – e.g. multiple vortices of different circulation, those with finite core size, etc. – it does suggest the particular shape the ‘tangled’ vortices in a turbulent flow may offer clues as to when and why vortex stretching occurs.

Finally, we note that our work does not rule out the possibility that that some complex unkotted configuration of vortex lines may produce exponential stretching, or that all knotted vortex fields do. In future work it may be possible to prove this using newly developed 3D analogues of the techniques used in 2D topological mixing [40]. Nonetheless, the current results demonstrate a clear connection between vortex topology and stretching behavior mediated by bight production.