2.2.1. Set-up for helical vortex tubes with varying core size

The centreline of the vortex tube is defined as a right-handed circular helix with radius $R$ and pitch $p$ (distance of one complete helical turn), winding around the $z$-axis. The centreline is parametrized using arc length $s$, so that its definition in Cartesian coordinates becomes

(2.3)\begin{equation} \boldsymbol{c}(s) = R \cos(\theta(s))\,\hat{\boldsymbol{e}}_x + R \sin(\theta(s))\, \hat{\boldsymbol{e}}_y + p\,\frac{\theta(s)}{2{\rm \pi}}\,\hat{\boldsymbol{e}}_z, \end{equation}

with

(2.4)\begin{equation} \theta(s) = \frac{s}{\sqrt{R^2+\left(\dfrac{p}{2{\rm \pi}}\right)^2}} \end{equation}

the angular coordinate varying along the helix.

We construct a helical vortex tube around the centreline using a Gaussian core vorticity profile, where the vortex lines are initially untwisted, and the core size is allowed to vary with $s$. To do so, we define the unit tangent vector at a point $\boldsymbol {c}(s)$ along the centreline as $\hat {\boldsymbol {e}}_s \equiv {{\rm d}\boldsymbol {c}}/{{\rm d}s}$. The point $\boldsymbol {c}(s)$ and vector $\hat {\boldsymbol {e}}_s$ uniquely define a plane $\mathcal {P}(s)$ normal to the centreline. Each location $\boldsymbol {x}$ in the computational domain lies in one or more of these planes. We identify the closest plane to $\boldsymbol {x}$ by the arc length parameter $\hat {s}{(\boldsymbol {x})}$, defined as $\hat {s}{(\boldsymbol {x})} = \min _{s}|\boldsymbol {x}-\boldsymbol {c}(s)|$, and solved for numerically in practice. Then for each point $\boldsymbol {x}$, we can form a local cylindrical coordinate system $(\hat {\boldsymbol {e}}_s, \hat {\boldsymbol {e}}_{\rho }, \hat {\boldsymbol {e}}_{\phi })$ where $\hat {\boldsymbol {e}}_{\rho } \equiv ({\boldsymbol {x}-\boldsymbol {c}(\hat {s}{(\boldsymbol {x})})})/{|\boldsymbol {x}-\boldsymbol {c}(\hat {s}{(\boldsymbol {x})})|}$ and $\hat {\boldsymbol {e}}_{\phi } \equiv \hat {\boldsymbol {e}}_{s} \times \hat {\boldsymbol {e}}_{\rho }$. The vorticity components are defined in this local orthonormal coordinate system as

(2.5)\begin{equation} \boldsymbol{\omega}(\boldsymbol{x}) = \frac{\varGamma_0}{{\rm \pi} \sigma(\hat{s}{(\boldsymbol{x})})^2}\exp\left( -\frac{\rho(\boldsymbol{x})^2}{\sigma(\hat{s}{(\boldsymbol{x})})^2}\right) \left[\frac{\rho(\boldsymbol{x})}{\sigma(\hat{s}{(\boldsymbol{x})})}\left.\frac{{\rm d}\sigma}{{\rm d}s}\right|_{s=\hat{s}{(\boldsymbol{x})}}\hat{\boldsymbol{e}}_{\rho} + \hat{\boldsymbol{e}}_s\right]\!, \end{equation}

where $\rho (\boldsymbol {x}) = |\boldsymbol {x} - \boldsymbol {c}(\hat {s}{(\boldsymbol {x})})|$, $\varGamma _0$ is the circulation of the vortex tube, and $\sigma (s)$ is a function controlling the size of the vortex core.

Here, we choose $\sigma (s)$ so that the core exhibits an isolated, sinusoidal bump of wavelength $\lambda$ where the core size increases smoothly from $\sigma _{min}$ to $\sigma _{max}$ and back. The remainder of the vortex tube, spanning a distance $p - \lambda$ along the $z$-axis, has constant core size $\sigma _{min}$. We define the non-dimensional initial core size variation as $A = \sigma _{max}/\sigma _{min}$, and the average core size parameter as $\sigma _0 = \frac {1}{2}(\sigma _{max} + \sigma _{min})$. A schematic of the helical vortex tube is shown in figure 1.

Figure 1.

Sketch of the projection of the initial helical vortex tube on the $r\unicode{x2013}z$ plane.

In this work, we consider helical vortex tubes with small radius-to-pitch ratios $0.02 \leq R/p \leq 0.0625$ and core size ratios $1 \leq A \leq 4.3$. For all cases, we fix $p = 20 \sigma _0$ and $\lambda = 10\sigma _0$. The circulation-based Reynolds number is fixed to be $Re_{\varGamma } = {\varGamma _0}/{\nu } = 5000$ for all cases considered. The rectangular computational domain has size $L_x \times L_y \times L_z$ and has periodic boundary conditions in the $z$ direction, and is unbounded in the $x$ and $y$ directions (Chatelain & Koumoutsakos Reference Chatelain and Koumoutsakos2010). There is one complete helical turn inside the domain, i.e. $p = L_z$, and the transverse extent of the domain $L_x = L_y$ varies with the radius of the helical tube. The parameters for all the cases are summarized in table 1. To ensure the accuracy of each simulation, we calculate the instantaneous errors in the effective viscosity for each simulation (van Rees et al. Reference van Rees, Leonard, Pullin and Koumoutsakos2011), which peaks at $1.7\,\%$ across all cases. Based on our previous experience, this bound is consistent with well-resolved direct numerical simulations (van Rees et al. Reference van Rees, Leonard, Pullin and Koumoutsakos2011). In all the results shown below, time is non-dimensionalized with the circulation $\varGamma _0$ and the average core size, i.e. $t^* = t ({\varGamma _0}/{\sigma _0^2})$.

Table 1.

List of straight and helical vortex tube cases and associated identifiers considered in this work. For the case denoted by ${{\dagger}}$, $\sigma _0/h = 57.6$; for the rest of the cases, $\sigma _0/h = 38.4$. For the cases denoted by $\star$, $L_x = L_y = 0.625 L_z$; for the other cases, $L_x = L_y =0.5L_z$.

2.2.2. Set-up for vortex rings with varying core size

To construct the initial condition for vortex rings with varying core size around their circumference, we start from the map $\boldsymbol {f} : \mathbb {R}^3 \to \mathbb {R}^3$ defined as

(2.6)\begin{equation} \boldsymbol{f}(\tilde{\boldsymbol{x}}) = (R+\tilde{x})\cos\left(\frac{\tilde{z}}{R}\right)\hat{\boldsymbol{e}}_x + (R+\tilde{x})\sin\left(\frac{\tilde{z}}{R}\right)\hat{\boldsymbol{e}}_y - \tilde{y}\,\hat{\boldsymbol{e}}_z. \end{equation}

This transformation maps a circular cylindrical surface of radius $|\tilde {x}|$, centred around the origin and aligned with $\hat {\boldsymbol {e}}_z$, to the surface of a torus with major radius $R$ and minor radius $|\tilde {x}|$, centred at the origin, and lying in the plane with normal $\hat {\boldsymbol {e}}_z$. In cylindrical coordinates, this becomes

(2.7)\begin{equation} \boldsymbol{f}(\tilde{\boldsymbol{x}}) = (R+\tilde{r} \cos(\tilde{\theta})) \cos\left(\frac{\tilde{z}}{R}\right)\hat{\boldsymbol{e}}_x + (R+\tilde{r} \cos(\tilde{\theta})) \sin\left(\frac{\tilde{z}}{R}\right)\hat{\boldsymbol{e}}_y - \tilde{r} \sin(\tilde{\theta})\,\hat{\boldsymbol{e}}_z, \end{equation}

where $\tilde {r} = \sqrt {\tilde {x}^2 + \tilde {y}^2}$ and $\tilde {\theta } = \arctan (\kern0.7pt \tilde {y}/\tilde {x})$. A sketch of the mapping is shown in figure 2.

Figure 2.

Sketch showing the mapping $\boldsymbol {f}$ from a straight tube to a circular ring.

We then define our vorticity field prior to mapping as

(2.8)\begin{equation} \tilde{\boldsymbol{\omega}}(\tilde{\boldsymbol{x}}) = \frac{\varGamma_0}{{\rm \pi}\, \sigma(\tilde{z})^2}\exp\left(-\frac{\tilde{r}^2}{\sigma(\tilde{z})^2}\right)\left(\frac{\tilde{r}\, \sigma'(\tilde{z})}{\sigma(\tilde{z})}\,\hat{\boldsymbol{e}}_{r} +\hat{\boldsymbol{e}}_{z}\right)\!, \end{equation}

where $\hat {\boldsymbol {e}}_{r}$ and $\hat {\boldsymbol {e}}_{z}$ are the radial and axial unit vectors in cylindrical coordinates, $\varGamma _0$ is the circulation of the vortex tube, and $\sigma (\tilde {z})$ is the core size function. This vorticity field is identical to the one used to investigate bursting on straight-line vortex tubes in Ji & van Rees (Reference Ji and van Rees2022), and coincides with the vorticity field of the helical tube when $R \to 0$.

Finally, the transformed vorticity field $\boldsymbol {\omega }(\boldsymbol {x})$ associated with the vortex ring initial condition is obtained as

(2.9)\begin{equation} \boldsymbol{\omega}(\boldsymbol{x}) = J^{-1}(\tilde{\boldsymbol{x}})\,\boldsymbol{F}(\tilde{\boldsymbol{x}})\,\tilde{\boldsymbol{\omega}}(\tilde{\boldsymbol{x}}), \end{equation}

where $\boldsymbol {F} = {{\rm d}\boldsymbol {f}}/{{\rm d}\kern0.7pt \tilde {\boldsymbol {x}}}$ is the Jacobian of the transformation $\boldsymbol {f}$, $J = \det ( \boldsymbol {F} )$ is the determinant of the Jacobian, and $\tilde {\boldsymbol {x}}$ in all terms on the right-hand side depends on $\boldsymbol {x}$ through the inverse map $\tilde {\boldsymbol {x}} = \boldsymbol {f}^{-1}(\boldsymbol {x})$. This transformation ensures $\boldsymbol {\nabla }_{\boldsymbol {x}} \boldsymbol {\cdot } \boldsymbol {\omega }(\boldsymbol {x}) = \boldsymbol {\nabla }_{\tilde {\boldsymbol {x}}} \boldsymbol {\cdot } \tilde {\boldsymbol {\omega }}(\tilde {\boldsymbol {x}})$, so that the transformed vorticity field is divergence-free. We can find closed-form expressions for all quantities related to this mapping, so an explicit expression for (2.9) is immediately available:

(2.10)$$\begin{gather} \omega_r = \frac{\varGamma}{{\rm \pi} r \left(\sigma(R\theta)\right)^3} \exp\left({-\frac{(r-R)^2+z^2}{\left(\sigma(R\theta)\right)^2}} \right)(r-R)R\,\sigma'(R\theta), \end{gather}$$

(2.11)$$\begin{gather}\omega_{\theta} =\frac{\varGamma}{{\rm \pi} \left(\sigma(R\theta)\right)^2} \exp\left({-\frac{(r-R)^2+z^2}{\left(\sigma(R\theta)\right)^2}}\right)\!, \end{gather}$$

(2.12)$$\begin{gather}\omega_z = \frac{\varGamma}{{\rm \pi} r \left(\sigma(R\theta)\right)^3} \exp\left({-\frac{(r-R)^2+z^2}{\left(\sigma(R\theta)\right)^2}}\right)Rz\,\sigma'(R\theta), \end{gather}$$

where $\sigma '(z) = {{\rm d} \sigma (z)}/{{\rm d}z}$. In the absence of core size perturbations, the core profile reduces to that of a standard Gaussian vortex ring.

The core size function $\sigma (\tilde {z})$ is chosen of a similar functional form as for the helical vortex tubes described above, with a perturbed section on the ring initially centred at $\theta =0$ with arc length $\lambda = {\rm \pi}R = 10 {\rm \pi}\sigma _0$, where the average core size is $\sigma _0 = \frac {1}{2}(\sigma _{min} + \sigma _{max}) = 0.1R$. The ratio between the maximum and the minimum core size is denoted as $A$, as before. We consider two different core size ratios, $A=3$ (denoted case $A3$) and $A=4.3$ (denoted case $A4$). For all simulations, the rectangular computational domain is a cube with side length $L = 3R$ and unbounded boundary conditions on all faces. The rings are centred at the $x$–$y$ plane with normal $\hat {\boldsymbol {e}}_z$, and initial vertical position at $\frac {1}{6}L_z$. The spatial resolution of case $A3$ is $512^3$ ($\sigma _0/h=19.2$), and for $A4$ we use $768^3$ ($\sigma _0/h=25.6$), so that the effective viscosity for each simulation remains below $5.0\,\%$. As for the helical tubes, time is non-dimensionalized with the circulation $\varGamma _0$ and the average core size, i.e. $t^* = t ({\varGamma _0}/{\sigma _0^2})$.

3.3.1. Effect of helical centreline for moderate core size ratio $A = 3$

Figure 5 shows the 3-D vorticity magnitude field in the bursting region for cases with core size ratio $A = 3$ and (from top to bottom) $R/p = 0$, $R/p = 0.02$ and $R/p=0.04$. For each case, four temporal snapshots of the flow are shown from left to right. For the case with a straight centreline, bursting appears as an annular structure with high vorticity magnitude, exhibits axisymmetric radial growth from $t^* =45$ to $t^* = 65$, and remains stable during growth and subsequent decay. On the helical centrelines with $R/p \ge 0.02$, the bursting structure initially (at $t^*=45$) appears similar to the straight vortex tube, though slightly tilted due to the time-varying orientation of the centreline normal plane. At later times, the bursting structure on the helical vortex tubes loses its axisymmetry and becomes highly distorted, as seen in figure 5.

Figure 5.

The 3-D volume rendering of the vorticity magnitude field at different times (increasing from left to right) during bursting for vortex tubes with $A = 3$ and different $R/p$ values (increasing from top to bottom).

To investigate the onset of the loss of axisymmetry, we probe the structure of the vorticity field during bursting for case $A3.R2$ in more detail. To facilitate the analysis, we first identify the centroid of the bursting structure. At a given snapshot of the simulation, this bursting centroid $\boldsymbol {x}_b$ is defined as the centroid of vorticity magnitude in a truncated rectangular domain $V_t$ containing the bursting region ($0.8L_z< z/\sigma _0 <1.2L_z$), so that $\boldsymbol {x}_b = \int _{V_t} |\boldsymbol {\omega }|\,\boldsymbol {x} \,{\rm d}V/\int _{V_t} |\boldsymbol {\omega }|\,{\rm d}V$. After identifying the bursting structure centroid, we extract the local Frenet frame at that location on the corresponding unperturbed helical tube (case $A1.R2$), for which the centreline remains a helix over time. We use that Frenet frame of the undisturbed vortex to estimate the local tangent, normal, and binormal vectors of the bursting case $A3.R2$, which we refer to below as $\boldsymbol {T}$, $\boldsymbol {N}$ and $\boldsymbol {B}$, respectively (see figure 6a).

Figure 6.

Evolution of the vorticity field during bursting for the case $A3.R2$. (a,d) Volume rendering of the vorticity magnitude field at $t^* = 65$, annotated with the local Frenet frame and a sketch of the planes (a) normal to the $\boldsymbol {N}$ vector, and (d) normal to the $\boldsymbol {B}$ vector. (b,e) A 3-D schematic explaining how the rotation of the centreline tilts the azimuthal vorticity that constitutes the bursting ring pair. The colours show the vorticity component normal to the highlighted planes (in grey), which match with (a,d). (c,f) Three different time instances of the cross-section of the vorticity field (c) normal to the $\boldsymbol {N}$ plane and (f) normal to the $\boldsymbol {T}$ plane, matching with (a,b) and (d,e), respectively.

During bursting, the twist waves continuously transport azimuthal vorticity into the bursting region, forming a structure akin to a vortex ring pair that expands radially over time (Ji & van Rees Reference Ji and van Rees2022). Simultaneously, due to the self-induced translation and rotation of the centreline, the Frenet frame rotates primarily around the binormal ($\boldsymbol {B}$) axis as time evolves. Abstractly, we can then imagine the bursting structure to be the superposition of opposite-signed, growing vortex ring pairs that emanate on either side of the bursting plane. Each vortex ring pair has an orientation that is determined by the centreline tangent vector at the time of the ring pair's formation. This process is sketched in figures 6(b) and 6(e), showing two ring pairs generated at subsequent time instances. Because the outer pair is generated at an earlier time than the inner pair, the outer pair has a larger diameter. Further, because the centreline tangent direction changes over time, the inner pair has a different orientation compared to the outer pair. Figure 6(b) also highlights the plane normal to the $\boldsymbol {N}$ direction, and the rings are coloured according to the azimuthal vorticity component in that plane. Figure 6(e) instead highlights the plane normal to the $\boldsymbol {B}$ direction, and the colouring here is changed accordingly. From these two sketches, we can observe that the rotation of the centreline affects the vorticity field differently between the $\boldsymbol {N}$-normal plane in figure 6(b) and the $\boldsymbol {T}$-normal plane in figure 6(e). In figure 6(b), the legs of the vortex rings intersecting the $\boldsymbol {N}$-normal plane are unaffected by the temporally changing orientation, so azimuthal vorticity is accumulated over time. In figure 6(e), however, we observe that as the ring pair changes orientation, opposite-signed azimuthal vorticity components lead to distortion and partial cancellation of the vorticity in the $\boldsymbol {B}$-normal plane.

The effects of these dynamics are shown in figures 6(c) and 6(f), which show actual slices of the simulated vorticity field normal to the $\boldsymbol {N}$ plane (top) and normal to the $\boldsymbol {B}$ plane (bottom), coloured by the azimuthal vorticity component (the component normal to each plane). For the $\boldsymbol {N}$-normal plane (top), the vortex ring pair (represented as dipoles in this slice) is shown to grow roughly symmetrically about the bursting structure orientation, with both rings in the pair remaining approximately equally strong in this slice. On the other hand, in the $\boldsymbol {T}$-normal plane (bottom), the positive normal vorticity (in red) dominates but the negative normal vorticity (in blue) is weaker, consistent with the cancellation hypothesis arising from figure 6(e). Because of this difference in strength, the positive azimuthal vorticity entrains the weaker, negative vorticity in this plane, as shown in figure 6(f). Returning to bursting as a 3-D structure, this difference between the two planes explains how the centreline dynamics introduces a strong non-axisymmetry into the bursting structure. This is the onset of the non-axisymmetric evolution of the bursting structure that governs $A=3$ (figure 5) and leads to instabilities for $A=4.3$ discussed in the next subsubsection.

3.3.2. Effect of helical centreline for higher core size ratio $A = 4.3$

Figure 7 shows the 3-D vorticity magnitude field in the bursting region for cases with core size ratio $A = 4.3$, where from top to bottom, $R/p = 0$, $R/p = 0.02$ and $R/p=0.04$, and for each case, four temporal snapshots of the flow are shown from left to right. For a vortex tube with straight centreline ($R/p=0$), an increase in the core size ratio from $A=3$ to $A=4.3$ leads to the successive generation of multiple concentric bursting ring pairs (Ji & van Rees Reference Ji and van Rees2022). These successive ring pairs do not significantly affect the bursting dynamics compared with $A=3$, as the bursting structure remains axisymmetric and stable. When the centreline becomes increasingly helical as $R/p$ increases, however, the bursting structure becomes distorted due to the centreline dynamics, as discussed in the previous subsubsection for $A=3$. Consequently, the successively generated bursting ring pairs are no longer concentric at $t^* = 45$ (figure 7); instead, they can be thought of as being aligned on different planes. Further, due to cancellation, the ring pairs can no longer be identified as separate vortical structures, as in the case of $R/p=0$. A figure equivalent to figure 6, but for case $A=4.3$, is shown in supplementary material § 4. This supports the claim that the bursting dynamics on helical vortices at $A=4.3$ is similar to that for $A=3$, though with stronger secondary structures due to the increased strength of the azimuthal vorticity field. At times beyond $t^*=45$, we observe breakup of the bursting structure into small-scale structures by $t^* = 65$ (figure 7), even at the smallest radius-to-pitch ratio $R/p=0.02$. This can be contrasted with the head-on collision of two opposite-signed vortex rings, where experiments and numerical simulations revealed that an iterative cascade instability leads to the breakdown of the vortex ring core into a turbulent cloud (McKeown et al. Reference McKeown, Ostilla-Mónico, Pumir, Brenner and Rubinstein2018). In our case, the breakdown of the bursting structure is driven by the swirling flow and deformation due to the centreline dynamics, as discussed in § 3.3.1. At even later times, the small-scale structures develop further as the bursting structure breaks down (see the flow visualizations at $t^*=100$ in figure 7); their effect on the long-time flow evolution will be discussed in the next subsection.

Figure 7.

The 3-D volume rendering of the vorticity magnitude field at different times (increasing from left to right) during bursting for vortex tubes with $A = 4.3$ and different $R/p$ values (increasing from top to bottom).

3.4.1. Enstrophy density integrated over $x\unicode{x2013}y$ planes

To get an overview of the flow evolution, we compute the time evolution of enstrophy density integrated over $x\unicode{x2013}y$ planes, $\varOmega (z,t) = \iint \boldsymbol {\omega } \boldsymbol {\cdot } \boldsymbol {\omega } \,{\rm d}\kern0.06em x\,{\rm d}y$, which is shown in figure 8 for two helical turns of case $A3.R4$. The twist wave propagation is associated with the streaks of light colour, since there is a low value of $\varOmega$ at the twist wave front. The bursting events are associated with regions of high values of $\varOmega$ and are outlined with black ellipses. There is a clear shift in bursting location between the first and third bursting events (denoted by the green dashed line), associated with the axial flow induced by the helical centreline discussed in § 3.2. Further, the enstrophy density highlights the core instabilities after the second bursting event as irregular streaks of high enstrophy that linger around the bursting region even after the twist wave reverses. The presence of these instabilities inhibits further strong bursting events: the enstrophy density highlights a weak tertiary bursting around $t^*=240$ followed by an uneventful flow evolution, characterized by a largely uniform enstrophy density distribution.

Figure 8.

(a) Colour plot of enstrophy density integrated over $x\unicode{x2013}y$ planes, $\varOmega (z,t) = \iint \boldsymbol {\omega } \boldsymbol {\cdot } \boldsymbol {\omega } \,{\rm d}\kern0.06em x\,{\rm d}y$, for the case $R/p = 0.04$, $A = 3$, for two helical turns. The horizontal axis represents the $z$ extent of the vortex tube, and the vertical axis represents time. (b) An appropriately scaled schematic of the helical tube projected onto the $r\unicode{x2013}z$ plane.

3.4.2. Global enstrophy and energy

As in the straight tube case, bursting events on helical vortex tubes are associated with a significant increase in the global enstrophy $\varepsilon = \int \boldsymbol {\omega } \boldsymbol {\cdot } \boldsymbol {\omega } \,{\rm d}V$ (figure 9a) and thus an accelerated energy dissipation (figure 9b) compared to unperturbed tubes. For the small $R/p$ values considered, the time evolution of global enstrophy and energy follows trends very similar to those for the corresponding straight tube case with the same core size ratio $A$.

Figure 9.

Time evolution of (a) global enstrophy $\varepsilon =\int \boldsymbol {\omega } \boldsymbol {\cdot } \boldsymbol {\omega } \,{\rm d}V$, (b) total energy $E = \frac {1}{2} \int \boldsymbol {\psi } \boldsymbol {\cdot } \boldsymbol {\omega } \,{\rm d}V$, (c) $\gamma (t^*) = E_{{small\text -scale}} / E$, for straight and helical vortex tubes with various values of initial core-size ratio $A$ and radius-to-pitch ratio $R/p$.

Bursting leads to the generation of small-scale vortical structures, especially when instabilities disrupt the core of the vortex tube. To quantify the amount of energy contained in the small scales, we consider the energy contained for wavenumber $k \leq 2{\rm \pi} /\sigma _0$, i.e. $E_{{small\text -scale}} = \int _{2 {\rm \pi}/\sigma _0}^{\infty }E(k)\,{\rm d}k$, as a fraction of the global energy $E$. We denote this quantity $\gamma (t^*) = E_{{small\text -scale}} / E$, and plot its evolution in figure 9(c) for the cases $A3.R0$, $A3.R4$, $A4.R0$ and $A4.R4$, as well as the reference cases $A1.R0$ and $A1.R4$ that do not undergo bursting. For the reference cases ($A=1$), the percentage of energy contained in these scales smaller than the mean core size is almost zero during the long-time evolution of the flow. In contrast, for the cases $A>1$, the fraction of small-scale energy increases significantly until $t^* \approx 170$, which is the time during which the dominant bursting events occur. For $t^*\gtrsim 170$, the small-scale energy fraction decays as the main bursting events are over, and viscous dissipation becomes dominant in the flow evolution. The time evolution of $\gamma$ for straight and helical vortex tubes with identical values of $A$ are comparable in their trends and peak values. Together with the plots in figures 9(a) and 9(b), this indicates that the global enstrophy and energy evolution, as well as the distribution of the total energy across the dominant length scales, is largely determined by $A$ rather than the specific shape of the centreline.

3.4.3. Global helicity

The global helicity $H = \int \boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {\omega } \,{\rm d}V$ is proportional to the degree of linking of the vortex lines (Moffat & Ricca Reference Moffat and Ricca1992). For a single thin vortex bundle, helicity is related to the geometry of the centreline of the vortex tube and the surrounding vortex lines through the expression $H = \varGamma _0^2 (Wr + Tw)$, where $\varGamma _0$ is the circulation of the vortex tube, $Wr$ is the writhe of the centreline of the tube and $Tw$ is a measure of the total twist of the vortex lines (Moffat & Ricca Reference Moffat and Ricca1992; Scheeler et al. Reference Scheeler, van Rees, Kedia, Kleckner and Irvine2017). In the context of this work, the vortex lines are initially untwisted in all cases. Consequently the initial global helicity can be expressed analytically as $H(t^*=0) = \varGamma _0^2\,Wr(t^*=0)$, where the writhe of the helical centreline is $Wr(t^*=0) = 1 - \cos (\phi _{h})$, with $\phi _h = \arctan (2{\rm \pi} R/p)$ (White & Bauer Reference White and Bauer1986). The initial helicity thus is zero for straight vortex tubes ($R/p=0$), and increases with $R/p>0$; notably, it does not depend on the core size perturbation parameter $A$.

The time evolution of the helicity for three different $A$ values, at both $R/p=0$ and $R/p=0.04$, is shown in figure 10(a). As expected, for straight vortex tubes ($R/p=0$), the global helicity remains identically zero due to symmetry for all values of $A$. For the unperturbed ($A=1$) helical vortex tube with $R/p=0.04$, the helicity remains constant, consistent with the notion that helicity is preserved for an untwisted vortex bundle with writhe (Scheeler et al. Reference Scheeler, van Rees, Kedia, Kleckner and Irvine2017). For helical vortex tubes with $R/p=0.04$ and $A>1$, the helicity initially remains constant across the different perturbation amplitudes, but then deviates from these values starting around the initial bursting event.

Figure 10.

(a) The time evolution of the global helicity $H= \int \boldsymbol {\omega } \boldsymbol {\cdot } \boldsymbol {u} \,{\rm d}V$. (b) The time evolution of the mean of the positive and negative global helicity $\frac {1}{2}(H^+ + H^-)$.

To analyse how bursting affects the helicity evolution, we define a simple sign-based composition of the helicity as $H(t) = H^+(t) - H^-(t)$, where $H^{\pm } = \int|h^{\pm }| \,{\rm d}V$, and

(3.1)\begin{equation} h^+= \begin{cases} h & \text{if } h \geq 0,\\ 0 & \text{otherwise}, \end{cases} \end{equation}

and $h^- = h^+-h$. This approach is then able to capture the positive and negative twist waves in the domain independently, and was also used in Shen et al. (Reference Shen, Yao, Hussain and Yang2022, Reference Shen, Yao, Hussain and Yang2023). Further, we can define the average between the components of different signs as $\bar {H} = \frac {1}{2}(H^++H^-)$, which is visualized in figure 10(b) for $R/p=0$ and $R/p=0.04$ at three different values of $A$. For unperturbed vortex tubes ($A=1$), there are no twist waves, and $\bar {H} = 0$ for $R/p=0$ (solid green line). For the helical tube $R/p=0.04$ with $A=1$ (dash-dotted green line), $\bar {H}$ is very small as it captures only the helicity components associated with the persisting writhe of the tube. For $A>1$, the values $\bar {H}$ are two orders of magnitude larger than $H$ and oscillate slowly, reaching peaks around each bursting instance (blue and orange dash-dotted lines in figure 10b). In fact, compared to the other global metrics used in this work, the evolution of $\bar {H}$ yields the clearest identification of the twist wave dynamics and bursting events across all parameters studied here. For each $A$, $\bar {H}$ is slightly higher for the helical tubes compared with the straight tube due to the persisting writhe of the centreline. Notably, the evolution of $\bar {H}$ differs significantly between $A=3$ and $A=4.3$, but does not vary strongly with $R/p$. As for the other global metrics discussed above, this implies that the centreline geometry does not have a strong effect on the global flow evolution of a vortex with a given initial core size ratio.

Returning to the global helicity $H = H^+ - H^-$ in figure 10(a), we observe that the variations of $H$ for $R/p=0.04$ after bursting, i.e. the differences between $H^+$ and $H^-$, are negligible compared to the magnitudes of $H^+$ and $H^-$ individually. Small physical or numerical perturbations in the flow evolution during bursting on helical vortex tubes are easily picked up in the total helicity evolution plotted in figure 10(a), yet are insignificant compared to the dominant and physically relevant variations in $H^+$ and $H^-$.