Role of internal structures within a vortex in helicity dynamics

Abstract

Helicity, an invariant under ideal-fluid (Euler) evolution, has a topological interpretation in terms of writhe and twist for a closed vortex tube, but accurately quantifying twist is challenging in viscous flows. With a novel helicity decomposition, we present a framework to construct the differential twist that establishes the theoretical relation between the total twisting number and the local twist rate of each vortex surface. This framework can characterize coiling vortex lines and internal structures within a vortex – important in laminar–turbulence transition, and in vortex instability, reconnection and breakdown. As a typical example, we explore the dynamics of vortex rings with differential twist via direct numerical simulation (DNS) of the Navier–Stokes equations. Two twist waves with opposite chiralities propagate towards each other along the ring and then collide whence the local twist rate rapidly surges. Local vortex surfaces are squeezed into a disk-like dipole structure containing coiled vortex lines, leading to vortex bursting. We derive a Burgers-equation-like model to quantify this process, which predicts a bursting time that agrees well with DNS.

1. Introduction

A complex flow field can be modelled as a collection of flux tubes, such as hydrodynamic flows (Moffatt & Tsinober 1992; Kleckner & Irvine 2013), superfluids (Koplik & Levine 1996; Kleckner, Kauffman & Irvine 2016) and plasmas (Cirtain et al. 2013). In particular, the vortex tube is a candidate elementary structure of turbulence (Hussain 1986; Moffatt, Kida & Ohkitani 1994; Pullin & Saffman 1998) (see figure 1a). Prototypical examples include rings in jets and wakes, and ‘typical eddies’ in turbulent boundary layers (Robinson 1991). Vortex line coiling within a vortex tube – a topological manifestation of the helicity (Moffatt 1969; Moffatt & Ricca 1992) – plays an essential role in flow evolution, such as laminar–turbulence transition (Fritts, Arendt & Andreassen 1998; Ruan et al. 2022), vortex instability (Mayer & Powell 1992; Pradeep & Hussain 2001) and vortex reconnection (Zhao et al. 2021; Yao & Hussain 2022) and breakdown (Leibovich 1978).

Figure 1. Schematic of closed vortex tubes with complex internal structures in transition and turbulence. (a) Conceptual model of the hairpin vortex in boundary layer transition and a collection of coiled and linked vortex rings in fully developed turbulence, where the flow visualization data were reported in Zheng, Yang & Chen (2016). (b) Closed vortex tube with differential twist on coaxial vortex surfaces and along the vortex centreline. The vortex surfaces are represented by VSF isosurfaces of different isocontour values, with embedded vortex lines (red solid). (c) A segment of the vortex tube in (b), where the vorticity is constructed based on the curved cylindrical coordinates $(s,\rho,\theta )$, and the vortex centreline $\mathcal {C}$ (blue dash-dotted) is described in the Frenet–Serret frame ($\boldsymbol {T}$, $\boldsymbol {N}$, $\boldsymbol {B}$).

Coiled vortex lines in a vortex tube can generate twist-wave packets, and their propagation and collision (Melander & Hussain 1994) can lead to bursting – causing an increase in the local enstrophy and energy dissipation. Vortex bursting has been found in aircraft trailing vortices (Tombach 1973), and addressed in theoretical (Arendt, Fritts & Andreassen 1997), experimental (Cuypers, Maurel & Petitjeans 2003) and numerical studies, but most are restricted to the configuration of vortex columns (Melander & Hussain 1994; Ji & van Rees 2022). By contrast, the vortex ring is more common in practical flows (Shariff & Leonard 1992) and has a well-defined topological interpretation of helicity in terms of the writhe $W_r$ and twist $T_w$ (Moffatt & Ricca 1992). Helicity and its decomposition provide a powerful diagnostic tool to understand the complex three-dimensional (3-D) flow dynamics.

Whether helicity conservation can be extended to real dissipative flows is of particular interest and has been extensively studied recently (Kleckner & Irvine 2013; Scheeler et al. 2017; Kerr 2018a; Meng, Shen & Yang 2023). For example, Kleckner & Irvine (2013) experimentally observed that the knotted vortex is quite unstable and transferred to unlinked, coiled vortex rings through viscous reconnection. Numerical studies (Yao, Yang & Hussain 2021; Zhao et al. 2021) reveal that the helicity is not conserved during this process: while the initial writhe helicity is destroyed, the local twist rapidly surges at the reconnection site and then travels along the two separated rings. Therefore, studying twist-wave propagation and bursting can shed light not only on the extreme events in turbulence and transition (Moffatt 2021; Buaria & Pumir 2022), but also on the helicity dynamics of flux tubes with complex internal structures.

Studying twist-wave propagation and bursting in a vortex ring, or more generally, in a closed vortex tube, which can be knotted and linked (Ricca, Samuels & Barenghi 1999; Kleckner & Irvine 2013; Kerr 2018b; Yao et al. 2021), is challenging. First, it is difficult to directly construct a twist wave with a precise amplitude and distribution in a closed vortex tube. Second, in real flows, vortex lines within vortex tubes can have differential twist, i.e. different local twist rates on coaxial vortex surfaces or along the vortex centreline (see figure 1). The differential twist of vortex tubes with finite thickness cannot be characterized by the existing helicity decomposition (Moffatt & Ricca 1992), nor could it be directly measured in previous experiments (Kleckner & Irvine 2013; Scheeler et al. 2017; Angriman et al. 2021) or numerical simulations (Yao et al. 2021; Shen et al. 2022; Yao et al. 2022). The existing ribbon model (Moffatt & Tsinober 1992; Chui & Moffatt 1995) for twisting is restricted to a vortex tube with uniform twist, and it cannot characterize the internal twisting structure of vortex tubes.

We develop a novel helicity decomposition – along with numerical construction and measurement methods – for the differential twist. Moreover, the vortex-surface field (VSF) (Yang & Pullin 2010, 2011) is used to track and measure the twist of vortex lines. These methods facilitate the first quantitative study of bursting vortex rings with differential twist.

2. Twisting helicity for differential twist

We introduce here a definition for the differential twist and explain its relation to the helicity. The total helicity

(2.1)\begin{equation} H=\int_\mathcal{V} h \,\mathrm{d} \mathcal{V}, \end{equation}

is the volume integral of the helicity density $h=\boldsymbol u \boldsymbol {\cdot } \boldsymbol \omega$ (Moreau 1961; Moffatt 1969), with the fluid velocity $\boldsymbol u$ and the vorticity $\boldsymbol \omega =\boldsymbol {\nabla } \times \boldsymbol u$. The helicity of a closed vortex tube can be topologically morphed into $H=\varGamma ^2 (W_r+T_w)$ (Moffatt & Ricca 1992), with $\varGamma$ the total circulation. Note that, while $W_r$ can be obtained from a measurement of the vortex tube centrelines alone, it is difficult to characterize and directly measure $T_w$ in practical flows. As sketched in figure 1(b), the nested coaxial vortex tubes without self-intersection are distinguished by different isosurfaces of a normalized VSF $\phi _v \in [0,1]$. The limiting surface with $\phi _v =1$ represents the vortex centreline $\mathcal {C}$. In figure 1(c), the vortex tube is represented in the curved cylindrical coordinate system $(s,\rho,\theta )$ (Xiong & Yang 2019, 2020). Here, $s\in [0,L_C)$ denotes the arclength along $\mathcal {C}$, $L_C$ the length of $\mathcal {C}$, $\rho$ the radial distance from $\mathcal {C}(s)$ and $\theta$ the azimuthal angle from $\boldsymbol N(s)$ in the plane $S_C$ spanned by $\boldsymbol N(s)$ and $\boldsymbol B(s)$, where the unit normal $\boldsymbol N(s)$, binormal $\boldsymbol B(s)$ and tangent $\boldsymbol T(s)$ constitute the Frenet–Serret frame on $\mathcal {C}$. For such vortex tubes with uniform $\boldsymbol \omega$ in $\theta$, $(s,\rho,\theta )$ is simplified to $(s,\phi _v)$.

We derive the contribution of coiled vortex lines on different coaxial vortex surfaces to the total helicity. The twisting helicity (Moffatt & Ricca 1992) of an isolated closed vortex tube can be expressed as

(2.2)\begin{equation} H_{T}=\varGamma^2 T_w, \end{equation}

where $\varGamma$ denotes the circulation and $T_w$ the total twist number of the vortex tube. For closed vortex tubes with uniform twist along $\phi _v$, the twisting number (Fuller 1971; Chui & Moffatt 1995)

(2.3)\begin{equation} T_{w}=\frac{1}{2 {\rm \pi}} \oint_{\mathcal{C}}(\boldsymbol{N}_s \times \boldsymbol{N}_s^{\prime}) \boldsymbol{\cdot} \boldsymbol{T} \,\mathrm{d} s, \end{equation}

is defined by a ribbon model. Here, the ribbon edges are the vortex centreline $\mathcal {C}$ and a vortex line $\mathcal {C}^{*}$. Moreover, $\boldsymbol {N}_s$ denotes a radial unit vector from $\mathcal {C}$ pointing to $\mathcal {C}^{*}$ in plane $S_C$ (see figure 1c), and $\boldsymbol {N}_s^{\prime }=\mathrm {d} \boldsymbol {N}_s / \mathrm {d} s$; $\boldsymbol {T}$ is the unit tangent vector of $\mathcal {C}$. This definition requires that every vortex line has the same value of $T_w$ calculated from (2.3), so it is restricted to characterizing a vortex tube with uniform twist or a differential twist along the vortex centreline.

In order to characterize the differential twist both along the vortex centreline and on different vortex surfaces, we establish a more complete definition of $T_w$ than (2.3). For generalized closed tubes with differential twist, we introduce the local twist rate

(2.4)\begin{equation} \eta\left(s,\phi_{v}\right)=\left(\boldsymbol{N}_{s} \times \boldsymbol{N}_{s}^{\prime}\right) \boldsymbol{\cdot} \boldsymbol{T}, \end{equation}

for a vortex line on a vortex surface of $\phi _v$ at different locations, and the circulation

(2.5)\begin{equation} \varGamma_\phi=\varGamma_\phi(\phi_v) \in [0,\varGamma], \end{equation}

through the tube enclosed by a vortex surface of $\phi _v$. If $\eta$ is circumferentially uniform on each vortex surface, i.e. $\eta$ is constant on the intersection of the isosurface of $\phi _v$ and the plane $S_C$ normal to $\mathcal {C}$, we define the twisting number

(2.6)\begin{equation} T_{\phi}\left(\phi_{v}\right)=\frac{1}{2{\rm \pi}}\oint_{\mathcal{C}} \eta\left(s,\phi_{v}\right)\,\mathrm{d} s, \end{equation}

for each vortex surface.

We first calculate the twisting helicity $\Delta H_{T}(\varPhi )$ for a single vortex surface of $\phi _v=\varPhi$ (see figure 2) with a given constant $\varPhi$. This surface with infinitesimal thickness has $\varGamma _{\phi }(\varPhi )$ and $T_\phi (\varPhi )$. As illustrated in figure 2

(2.7)\begin{equation} \Delta H_{T}(\varPhi) = \bar{H}_{T,1} - \bar{H}_{T,2}, \end{equation}

of a twisted vortex tube can be obtained by the difference of twisting helicities for two adjacent co-axial virtual vortex tubes 1 and 2 with

(2.8a,b)\begin{equation} \bar{\varGamma}_1 = \varGamma_{\phi}\left(\varPhi\right), \quad \bar{T}_{w,1} =T_{\phi}\left(\varPhi\right), \end{equation}

and

(2.9a,b)\begin{equation} \bar{\varGamma}_2 = \varGamma_{\phi}\left(\varPhi+\Delta\phi_v\right), \quad \bar{T}_{w,2} =T_{\phi}\left(\varPhi\right), \end{equation}

respectively, where the overline denotes the quantity in a virtual tube. All co-axial vortex surfaces inside the two virtual tubes have the same twist distribution as the vortex surface $\phi _v=\varPhi$, so the two virtual tubes have uniform twist along $\phi _v$ and their twisting helicities can be obtained by (2.2).

Figure 2. Schematic for calculating the twisting helicity of a vortex surface with infinitesimal thickness by two adjacent co-axial virtual tubes with uniform twist along $\phi _v$. The vortex surface of $\phi _v=\varPhi$ is peeled off from the vortex tube with differential twist along $s$ and $\phi _v$. All co-axial vortex surfaces inside the two virtual tubes have the same twist distribution as the vortex surface of $\phi _v=\varPhi$.

Substituting (2.2), (2.8a,b) and (2.9a,b) into (2.7) yields

(2.10)\begin{equation} \Delta H_{T}(\varPhi) = \varGamma_\phi(\varPhi)^2 T_\phi(\varPhi) - \varGamma_\phi(\varPhi+\Delta\phi_v)^2 T_\phi(\varPhi). \end{equation}

Applying the Taylor expansion of $\varGamma _\phi (\varPhi +\Delta \phi _v)$ to (2.10) yields

(2.11)\begin{equation} \Delta H_{T}(\varPhi) ={-}2\varGamma_\phi(\varPhi)\frac{\mathrm{d}\varGamma_\phi(\varPhi)}{\mathrm{d}\phi_v}T_\phi(\varPhi)\Delta\phi_v+O(\Delta\phi_v^2). \end{equation}

Then, we obtain

(2.12)\begin{equation} \frac{\mathrm{d} H_{T}(\varPhi)}{\mathrm{d} \phi_v}=\lim_{\Delta \phi_v\rightarrow 0} \frac{\Delta H_{T}(\varPhi)}{\Delta \phi_v}={-}2\varGamma_\phi(\varPhi)\frac{\mathrm{d}\varGamma_\phi(\varPhi)}{\mathrm{d}\phi_v}T_\phi(\varPhi). \end{equation}

Thus each vortex surface of $\phi _v$ in a vortex tube with differential twist along $s$ and $\phi _v$ has

(2.13)\begin{equation} \mathrm{d} H_{T}(\phi_v)={-}2\varGamma_\phi(\phi_v)\frac{\mathrm{d}\varGamma_\phi(\phi_v)}{\mathrm{d}\phi_v}T_\phi(\phi_v) \,\mathrm{d} \phi_v. \end{equation}

Finally, we obtain the total twisting helicity

(2.14)\begin{equation} H_{T}={-}\frac{1}{\rm \pi}\int_{0}^{1}\varGamma_\phi(\phi_v)\frac{\mathrm{d}\varGamma_\phi(\phi_v)}{\mathrm{d}\phi_v} \left( \oint_{\mathcal{C}} \eta(s,\phi_v) \,\mathrm{d} s \right) \,\mathrm{d}\phi_v, \end{equation}

of a vortex tube with the total circulation $\varGamma$ and differential twist along $s$ and $\phi _v$ by the integration $\int \,\mathrm {d} H_T$ with (2.6), which is a circulation-weighted average of twisting numbers over all co-axial vortex surfaces. Substituting (2.14) into (2.2) yields

(2.15)\begin{equation} T_w={-}\frac{1}{{\rm \pi}\varGamma^2}\int_{0}^{1}\varGamma_\phi(\phi_v)\frac{\mathrm{d}\varGamma_\phi(\phi_v)}{\mathrm{d}\phi_v} \left( \oint_{\mathcal{C}} \eta(s,\phi_v) \,\mathrm{d} s \right) \,\mathrm{d}\phi_v. \end{equation}

This equation is further verified with several numerical examples in Appendix A.

3. Construction of differential twist

We construct the vorticity field $\boldsymbol {\omega }$ for a closed vortex tube with differential twist. This construction method with its numerical algorithm is an extension of that in Xiong & Yang (2019, 2020) by incorporating variations of the core size and local twist rate in terms of $s$ and $\phi _v$.

First, the tube centreline $\mathcal {C}$ is described by a given parametric equation

(3.1)\begin{equation} \boldsymbol{x}=\boldsymbol{c}(s)+\rho \cos \theta \boldsymbol{N}(s)+\rho \sin \theta \boldsymbol{B}(s). \end{equation}

The Frenet–Serret formulas on $\mathcal {C}$ are

(3.2)\begin{equation} \left. \begin{aligned} \frac{\mathrm{d} \boldsymbol{T}}{\mathrm{d} s} & =\kappa \boldsymbol{N}, \\ \frac{\mathrm{d} \boldsymbol{N}}{\mathrm{d} s} & ={-}\kappa \boldsymbol{T}+\tau \boldsymbol{B}, \\ \frac{\mathrm{d} \boldsymbol{B}}{\mathrm{d} s} & ={-}\tau \boldsymbol{N}, \end{aligned} \right\} \end{equation}

where $\kappa$ is the curvature and $\tau$ is the torsion of $\mathcal {C}$.

Based on coordinates ($s, \rho, \theta$), we specify

(3.3)\begin{equation} \boldsymbol{\omega}(s, \rho, \theta)= \omega_s(s,\rho)\boldsymbol{e}_{s}+\omega_\rho(s,\rho,\theta)\boldsymbol{e}_{\rho}+\omega_\theta(s,\rho,\theta)\boldsymbol{e}_{\theta}, \end{equation}

of a vortex tube, where the local frame is spanned by unit vectors

(3.4)\begin{equation} \left. \begin{aligned} \boldsymbol{e}_{s} & =\boldsymbol{T}, \\ \boldsymbol{e}_{\rho} & =\cos \theta \boldsymbol{N}+\sin \theta \boldsymbol{B}, \\ \boldsymbol{e}_{\theta} & ={-}\sin \theta \boldsymbol{N}+\cos \theta \boldsymbol{B}. \end{aligned} \right\} \end{equation}

By setting the variable initial core size $\sigma (s)$ and local twist rate $\eta (s,\phi _v)$, vorticity components $\omega _s(s,\rho )$ and $\omega _\theta (s,\rho,\theta )$ are determined by introducing $\sigma (s)$ and $\eta (s,\phi _v)$ into the construction method in Xiong & Yang (2020) and Shen et al. (2022) and then $\omega _\rho (s,\rho,\theta )$ is solved from the divergence-free constraint. Thus the vorticity of closed vortex tubes with differential twist and variable thickness is specified as

(3.5) \begin{equation} \boldsymbol{\omega}(s, \rho, \theta)=\varGamma f(s,\rho)\left[\underbrace{\boldsymbol{e}_{s}}_{\textit{flux}}+\underbrace{\frac{\mathrm{d} \sigma(s)}{\mathrm{d} s}\frac{\rho\boldsymbol{e}_{\rho}}{\sigma(s)(1-\kappa(s) \rho \cos \theta)}}_{\textit{tube thickness}}+\underbrace{\frac{\rho \eta(s, \phi_v) \boldsymbol{e}_{\theta}}{1-\kappa(s) \rho \cos \theta}}_{\textit{twist}} \right], \end{equation}

with the Gaussian kernel function

(3.6) \begin{equation} f(s,\rho)=\left\{ \begin{array}{@{}lll} \dfrac{1}{2{\rm \pi}\sigma(s)^2}\exp\left[ \dfrac{-\rho^2}{2\sigma(s)^2}\right] , & s \in [0,L_C), & \rho \in [0,R_v),\\ 0, & s \in [0,L_C), & \rho \in [R_v,+\infty), \end{array} \right. \end{equation}

and the initial normalized VSF

(3.7)\begin{equation} \phi_{v}(s,\rho)= 2{\rm \pi}\sigma(s)^2f(s,\rho) \in [0,1], \end{equation}

where the three terms on the right-hand side of (3.5) represent the vorticity flux, tube thickness and twist terms of $\boldsymbol \omega$, respectively.

If $\kappa (s) = 0$ and $\eta (s,\phi _v) = 0$, (3.5) degenerates into the vorticity for a straight vortex tube with a variable core size (Ji & van Rees 2022). If $\sigma (s)$ and $\eta (s,\phi _v)$ are constants, (3.5) degenerates into a constant-thickness vortex tube with uniform twist (Xiong & Yang 2020; Shen et al. 2022).

As proved below, the vector field constructed by (3.5) is solenoidal, which can be used as a vorticity or magnetic field.

Theorem 1 The vector field $\boldsymbol {\omega }$ constructed by (3.5) is divergence free.

Proof. In the curved cylindrical coordinate system, by applying the inverse function theorem to the Jacobian matrix (Xiong & Yang 2020) between $(s, \rho, \theta )$ and $(x,y,z)$, we derive

(3.8)\begin{equation} \left. \begin{aligned} & \boldsymbol{\nabla} s=\frac{\boldsymbol{T}}{1-\kappa \rho \cos \theta}, \\ & \boldsymbol{\nabla} \rho=\cos \theta \boldsymbol{N}+\sin \theta \boldsymbol{B}, \\ & \boldsymbol{\nabla} \theta=\frac{-\tau}{1-\kappa \rho \cos \theta} \boldsymbol{T}+\frac{1}{\rho}\left(-\sin \theta \boldsymbol{N}+\cos \theta \boldsymbol{B}\right). \end{aligned} \right\} \end{equation}

Taking the divergence of (3.3) yields

(3.9)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\omega}= \boldsymbol{\nabla}\boldsymbol{\cdot} \left(\omega_s\boldsymbol{e}_{s}\right)+\boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_\rho\boldsymbol{e}_{\rho})+\boldsymbol{\nabla}\boldsymbol{\cdot} \left(\omega_\theta\boldsymbol{e}_{\theta}\right), \end{equation}

and using (3.4) yields

(3.10) \begin{align} \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_s\boldsymbol{e}_{s}) &=\left(\frac{\partial \omega_s}{\partial s}\boldsymbol{\nabla} s +\frac{\partial \omega_s}{\partial \rho}\boldsymbol{\nabla} \rho\right) \boldsymbol{T}+ \omega_s \frac{\text{d} \boldsymbol{T}}{\text{d} s}\boldsymbol{\cdot}\boldsymbol{\nabla} s,\end{align}

(3.11) \begin{align}\boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_\rho\boldsymbol{e}_{\rho}) &=\left(\frac{\partial \omega_\rho}{\partial s}\boldsymbol{\nabla} s +\frac{\partial \omega_\rho}{\partial \rho}\boldsymbol{\nabla} \rho+\frac{\partial \omega_\rho}{\partial \theta}\boldsymbol{\nabla} \theta\right) (\cos \theta \boldsymbol{N}+\sin \theta \boldsymbol{B})\nonumber\\ &\quad+ \omega_\rho \left(-\sin\theta\boldsymbol{\nabla}\theta\boldsymbol{\cdot}\boldsymbol{N}+\cos\theta\frac{\text{d} \boldsymbol{N}}{\text{d} s}\boldsymbol{\cdot}\boldsymbol{\nabla} s +\cos\theta\boldsymbol{\nabla}\theta\boldsymbol{\cdot}\boldsymbol{B}+\sin\theta\frac{\text{d} \boldsymbol{B}}{\text{d} s}\boldsymbol{\cdot}\boldsymbol{\nabla} s\right), \end{align}

(3.12)\begin{align} \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_\theta\boldsymbol{e}_{\theta}) &=\left(\frac{\partial \omega_\theta}{\partial s}\boldsymbol{\nabla} s +\frac{\partial \omega_\theta}{\partial \rho}\boldsymbol{\nabla} \rho+\frac{\partial \omega_\theta}{\partial \theta}\boldsymbol{\nabla} \theta\right) (-\sin \theta \boldsymbol{N}+\cos \theta \boldsymbol{B})\nonumber\\&\quad+ \omega_\theta \left(-\cos\theta\boldsymbol{\nabla}\theta\boldsymbol{\cdot}\boldsymbol{N}-\sin\theta\frac{\text{d} \boldsymbol{N}}{\text{d} s}\boldsymbol{\cdot}\boldsymbol{\nabla} s -\sin\theta\boldsymbol{\nabla}\theta\boldsymbol{\cdot}\boldsymbol{B}+\cos\theta\frac{\text{d} \boldsymbol{B}}{\text{d} s}\boldsymbol{\cdot}\boldsymbol{\nabla} s\right). \end{align}

Substituting (3.2) and (3.4) into (3.10), (3.11) and (3.12), and considering the orthogonality of the Frenet–Serret frame yields

(3.13)\begin{equation} \left. \begin{aligned} & \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_s\boldsymbol{e}_{s}) =\frac{1}{1-\kappa\rho\cos\theta}\frac{\partial \omega_s}{\partial s},\\ & \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_\rho\boldsymbol{e}_{\rho}) =\frac{\partial \omega_\rho}{\partial \rho}+\frac{1-2\kappa\rho\cos\theta}{\rho(1-\kappa\rho\cos\theta)}\omega_\rho,\\ & \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_\theta\boldsymbol{e}_{\theta}) =\frac{1}{\rho}\frac{\partial \omega_\theta}{\partial \theta}+\frac{\kappa\sin\theta}{1-\kappa\rho\cos\theta}\omega_\theta. \end{aligned} \right\} \end{equation}

For $\rho \geqslant R_v$, we have $\boldsymbol {\omega }=0$ from (3.6). For $\rho < R_v$, substituting (3.5) and (3.6) into (3.13) yields

(3.14)\begin{equation} \left. \begin{aligned} & \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_s\boldsymbol{e}_{s}) =\frac{\varGamma(\rho^2-2\sigma^2)}{2{\rm \pi}\sigma^5(1-\kappa\rho\cos\theta)}\frac{\mathrm{d} \sigma}{\mathrm{d} s}\exp\left( \frac{-\rho^2}{2\sigma^2}\right),\\ & \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_\rho\boldsymbol{e}_{\rho}) =\frac{\varGamma(2\sigma^2-\rho^2)}{2{\rm \pi}\sigma^5(1-\kappa\rho\cos\theta)}\frac{\mathrm{d} \sigma}{\mathrm{d} s}\exp\left( \frac{-\rho^2}{2\sigma^2}\right),\\ & \boldsymbol{\nabla}\boldsymbol{\cdot} (\omega_\theta\boldsymbol{e}_{\theta}) =0, \end{aligned} \right\} \end{equation}

after some algebra. Finally we obtain $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {\omega }=0$.

The numerical implementation is detailed in Appendix A. Typical examples constructed by (3.5) in figure 3 show coiled vortex lines with differential twist lying on various closed vortex tubes. Furthermore, we develop a numerical method to measure the local twisting rate on a vortex surface for given $\boldsymbol \omega$ and $\phi$. The algorithm is based on multiple vortex lines in terms of the discrete arclength on the VSF isosurface, which is detailed in Appendix B. Thus, we can quantify the evolution of coiling vortex lines on different vortex surfaces in a viscous evolution.

Figure 3. Closed vortex tubes with various internal structures. These closed vortex tubes with arbitrary topology, differential twist and variable thickness are constructed by (3.5): (a) trivial ring, (b) trefoil knot and (c) figure-eight knot. They are visualized by VSF isosurfaces with embedded vortex lines. The inner and outer tubes in (a) are two VSF isosurfaces with different colours; the surfaces in (b,c) are colour coded by $h$.

4. Results

4.1. Evolution of vortex ring with differential twist

We highlight the role of differential twist in helicity and vortex dynamics via direct numerical simulation (DNS) of bursting of vortex rings. Initial twisted vortex rings with a radius $R_0=1$ are constructed by (3.5), with initial $\varGamma =\varGamma _0=1$ and $\sigma =\sigma _0=1/(8\sqrt {2{\rm \pi} })$. The initial local twist rate $\eta (s,\phi _{v})=\eta _0=A \sin (s/R_0)$ varies along $\mathcal {C}$. We use the constructed vorticity fields in (3.5) as initial conditions, and calculate their evolutions using DNS. The 3-D incompressible Navier–Stokes equations are solved in the vorticity–velocity form (Wu, Ma & Zhou 2015) using the pseudo-spectral method in a periodic box of size $L=2 {\rm \pi}$ on $N^{3}$ uniform grid points. The numerical solver removes aliasing errors using the two-third truncation method with the maximum wavenumber $k_{max} \approx N / 3$. The time integration is treated by the explicit second-order Runge–Kutta scheme in physical space, with the adaptive time step ensuring the small enough Courant–Friedrichs–Lewy number for numerical stability and accuracy. The vortex Reynolds number is set to ${\textit {Re}}\equiv \varGamma /\nu =2000$. To ensure that the grid resolution can fully resolve the flow evolution, $N$ is carefully chosen to be $512$, $768$ and $1024$ for the initial twist amplitudes $A=10$, 20 and 30, respectively. For each case, we carried out the grid convergence test and confirmed that the DNS results converge for $N$ to ensure that the grid resolution fully resolves the flow evolution.

In addition, the VSF evolution is calculated using the two-time method (Yang & Pullin 2011) and its implementation is reported in Appendix C. The Lagrangian-like evolution of the twisted vortex ring with $A=20$ and $Re=2000$ is visualized by the isosurface of $\phi _v=0.5$ in figure 4. At $t=0$, two twist waves of vortex lines with opposite chiralities travel in opposite directions. Each wave packet is similar to a Kelvin wave with zero azimuthal wavenumber (Arendt et al. 1997; Fabre, Sipp & Jacquin 2006). Then, they collide and burst at the upper symmetric plane $SP_1$, forming a disk-like vortex dipole structure. Meanwhile, the axial gradient of the core size near the bursting site regenerates secondary twist waves, which propagate backward and cause secondary bursting at the lower symmetric plane $SP_2$ after $t = 2$. Note that, as shown in Ji & van Rees (2022) for a vortex column, such successive bursting can also be triggered by a vortex ring with initial core-size perturbation (see Appendix D).

Figure 4. Lagrangian-like evolution of vortex surfaces and lines. The visualization shows the evolution of the VSF isosurface (colour coded by $h$) of $\phi _v = 0.5$ for $A=20$ and $Re=2000$. Some attached vortex lines are integrated from points on the isosurface. Note that $t=t/(R_0^2/\varGamma _0)$ is non-dimensionalized here. The close-up view shows vortex lines (colour coded by $h$) on the VSF isosurface (translucent) of $\phi _v=0.7$ at $t=0.75$ in vortex bursting.

In figure 5(a), the enstrophy $\varOmega (t) = \int _\mathcal {V} |\boldsymbol {\omega }|^{2}/2 \,\mathrm {d} \mathcal {V}$ in the viscous evolution decays and shows a bump during bursting. For comparison, viscous diffusion of a vortex column shows exponential decay of $\varOmega$. Due to the initial symmetry, $H$ remains zero, and the positive and negative parts $H^\pm =\int _{\mathcal {V}} h^{\pm } \,\mathrm {d} \mathcal {V}$ of $H$ characterize the amplitudes of the counter-rotating waves. Before bursting, the core dynamics induced meridional flow (Melander & Hussain 1994) uncoils vortex lines and thickens the local vortex tube to form an axial core-size gradient. The vortex tube with the axial core-size variation then re-coils the vortex lines. Thus, $|H^\pm |$ first decays and then rebounds in figure 5(b). The viscous decay of $H^\pm$ (solid line) is faster than that of the uniform helicity model (dash-dotted line), because $\eta$, which is proportional to the viscous decay rate of twist (Yao et al. 2021; Shen et al. 2022), is more locally concentrated and larger than in the latter.

Figure 5. Flow statistics. (a,b) Evolution of (a) $\varOmega$ and (b) $H$ (black), $H^+$ (red) and $H^-$ (blue) for various $A$. The dash-dotted lines denote modelling results $\varOmega (t)=\varOmega _{0} \exp [-\int _{0}^{t}2 \nu (\sigma _0^{2}+2\nu t)^{-1} \,\mathrm {d} t]$ and $H(t)=H_0 \exp [-\int _{0}^{t}2 \nu (\sigma _0^{2}+2\nu t)^{-1} \,\mathrm {d} t]$ for a uniformly twisted vortex column with $A=20$. The peak heights of $\varOmega$ and $|H^\pm |$ grow with $A$, while the height of the secondary peak of $|H^\pm |$ is the highest for $A=20$. (c–e) Local twist rates on different VSF isosurfaces along the vortex centreline at (c) $t=0.25$, (d) 0.5 and (e) 1.5 for $A=20$. Arrows in (c,e) denote the propagating direction of twist-wave packets. The inset in (d) shows entire profiles of $\eta$.

4.2. Vortex bursting

During the evolution, the twist propagates along $\mathcal {C}$ and varies with $\phi _v$. Figure 5(c) plots the distribution of $\eta$ along $s$ on different vortex surfaces at $t=0.25$ for $A=20$ and $Re=2000$. At early times, two peaks of $\eta$ approach each other and evolve towards a discontinuity, similar to shock formation. The propagation speed of twist waves grows with $\phi _v$; i.e. the waves travel faster on an inner vortex surface than on an outer surface.

We develop an inviscid model for the propagation of twist vortex waves, which can predict when the vortex bursting occurs. The twist waves are modelled as travelling waves along the vortex centreline, so that their propagation speed equals the axial velocity of the local fluid. Thus, we have

(4.1)\begin{equation} \eta(s,\phi_v,t)=F(s-u_s t,\phi_v), \end{equation}

with a function $F$. The axial velocity (Yao et al. 2021) of a uniformly twisted vortex tube with constant $\eta$ is obtained by the Biot–Savart law as

(4.2)\begin{equation} u_s(\rho)=\frac{\varGamma \eta}{2 {\rm \pi}} \exp \left(\frac{-\rho^{2}}{2 \sigma^{2}}\right). \end{equation}

Substituting the initial VSF profile $\phi _{v}(\rho )$ into (4.2) and replacing the constant $\eta$ by a varying one yield the axial velocity

(4.3)\begin{equation} u_s(s,\phi_v)=\frac{\varGamma\phi_v\eta(s,\phi_v,t)}{2{\rm \pi}}, \end{equation}

of a vortex surface with differential twist. Based on (4.1) and (4.3), we obtain a Burgers-like equation

(4.4) \begin{equation} \left. \begin{array}{ll@{}} \dfrac{\partial \eta}{\partial t}+\dfrac{\varGamma \phi_{v} \eta}{2 {\rm \pi}} \dfrac{\partial \eta}{\partial s}=0, & s \in[0,L_C),\quad t>0,\\ \eta\left(s,\phi_{v}, 0\right)=\eta_{0}\left(s,\phi_{v}\right), & \end{array} \right\} \end{equation}

where $L_C$ denotes the length of $\mathcal {C}$ and $\eta _{0}(s,\phi _{v})$ the given initial $\eta$. Note that this model is an inviscid approximation, and the twist wave can have dispersion in viscous flows.

From the solution to (4.4)

(4.5)\begin{equation} \eta(s,\phi_v,t)=\eta_0\left(s-\frac{\varGamma\phi_v\eta}{2{\rm \pi}}t,\phi_v\right) ,\quad t< t_b, \end{equation}

we obtain that $\partial \eta _{0}/\partial s$ becomes infinite at the blow-up time

(4.6)\begin{equation} t_{b}(\phi_v)=\min \left.\left[\left(-\frac{\varGamma \phi_{v}}{2 {\rm \pi}} \frac{\partial \eta_{0}}{\partial s}\right)^{{-}1}\right]\right|_{{\partial \eta_{0}}/{\partial s}<0}. \end{equation}

With $\eta _0(s,\phi _{v})=A \sin (s/R_0)$, (4.6) gives an estimation of the vortex bursting time $t_b(\phi _v) = 2{\rm \pi} R_0/(\varGamma _0A\phi _v)$ for an isosurface of $\phi _v$. It decreases with $\phi _v$, so the bursting develops gradually from the vortex centreline to its outer surfaces, and the earliest blow-up time is $t_{b}(\phi _v=1)={\rm \pi} /10 \approx 0.314$ for $A=20$. The comparison of the DNS and modelling results for the evolution of $\eta$ (see figure 6) shows that (4.5) provides a satisfactory estimate of the local twist rate.

Figure 6. Comparison of DNS (symbols) and modelling (solid lines) results of $\eta$ on VSF isosurfaces of $\phi _v =0.1$ (red), $0.5$ (green) and $0.9$ (blue) at $t=0.25$. The model predictions calculated from (4.5) capture different propagation speeds on different vortex surfaces, in close agreement with the DNS results.

In figure 5(d), the coiling of vortex lines gradually accumulates on both sides of $SP_1$ at $s/L_C=0.5$ after $t = {\rm \pi}/10$. The surge of $\eta$ characterizes incipient vortex bursting. Consistent with the model of $t_b$, bursting first occurs near the vortex centreline (with large $\phi _v$). In particular, the spikes of $\eta (s,\phi _v=0.9)$ have a maximum value around 160 at $t=0.5$ (see the inset) and are more than 10 times the averaged initial amplitude $2A/{\rm \pi}$. As illustrated in the close-up view in figure 4, vortex surfaces are flattened on $SP_1$ and rolled up at their edge, forming a disk-like structure with highly spiral vortex lines. The local flow topology at the bursting site is similar to the statistically preferential state of the bi-axial strain in turbulence (Meneveau 2011).

The formation and decay of the disk structure significantly alter the radial tube size near $SP_1$, triggering the generation of secondary counter-twist waves (Ji & van Rees 2022). In figure 5(e), new twist waves are first generated at large $\phi _v$ near $\mathcal {C}$. Subsequently, the chirality of twist waves on outer vortex surfaces is reversed from inner to outer layers. The secondary twist waves gradually intensify and cause the secondary bursting on $SP_2$.

4.3. Effect of initial twist amplitude

During bursting, larger $A$ (or higher $Re$) can cause a more complex vortex dynamics. Increasing $A$ from 20 to 30 in figure 7(a), the VSF visualization reveals that vortex reconnection occurs within larger disk structures. As sketched in figure 7(b), the spiral vortex lines are pressed onto the disk, and the reconnection of each line at two locations (marked by dashed circles) pinches off a vortex loop from the rolling-up edge of the disk. Strongly coiled vortex lines are uncoiled immediately after reconnection, causing a drop in local core size and twist rate of the main tube. The major secondary ring structure with vanishing flux consists of the pinched-off vortex loops. It is also called the vortex dipole tube (Hussain & Stout 2013) whose cross-section is a pair of concentrated vorticity regions with opposite signs. The successive reconnection of vortex lines is asymmetric in the $\theta$-direction due to the curved vortex centreline, which is distinctly different from the symmetric reconnection in bursting of a rectilinear vortex tube (Ji & van Rees 2022).

Figure 7. Vortex reconnection for the bursting vortex ring. (a) Evolution of the VSF isosurface $\phi _v = 0.3$ for $A=30$ with some attached vortex lines (colour coded by $h$) before and after reconnection (from $t=0.9$ to $t=1.4$). (b) Schematic of the vortex reconnection that occurs in the bursting disk. Red and blue lines represent right- and left-handed coiled vortex lines, respectively. Dashed circles mark reconnection locations of a vortex line. Translucent green sections illustrate the sudden loss of the vortex tube thickness after the reconnection, where the dipole tube formed after the reconnection consists of vortex dipoles.

In figure 5(b), the second peak of $|H^\pm |$ of $A=20$ around $t=3$ after bursting is the highest, even slightly higher than $A=30$, implying that secondary twist waves are weakened by vortex reconnection. The amplitude of the secondary twist wave is positively correlated with the axial gradient of the vortex core size (Ji & van Rees 2022). Although stronger twist-wave collision can produce a larger disk, the reconnection significantly reduces the core size (and its axial gradient) of the disk (see figure 7b). After reconnection, the reduction of the core-size gradient inhibits the regeneration of strong twist waves and pre-empts subsequent bursting.

The cancellation and regeneration of right- or left-handed twist waves on different vortex surfaces can be quantified based on $T_\phi ^+$ or $T_\phi ^-$, defined as $T_\phi ^\pm (\phi _v)=\oint _{\mathcal {C}} \eta ^\pm \,\mathrm {d} s$ with

(4.7)\begin{equation} \eta^{+}= \begin{cases}\eta, & \text{if } \eta \geqslant 0, \\ 0, & \text{otherwise},\end{cases} \end{equation}

and $\eta ^-=\eta -\eta ^+$. The evolution of $T_\phi ^+$ is shown in figure 8. After the first bursting, secondary twist waves regenerate successively from inner to outer vortex surfaces. The regeneration of $T_\phi ^+$ for $A=30$ with slope 2/3 is weaker than for $A=20$ with slope 4/3, confirming weakened secondary twist waves.

Figure 8. Effect of the vortex reconnection on the strength of twist waves. Evolution of $T_\phi ^+$ on different VSF isosurfaces for $A=20$ ((a), without reconnection) and $A=30$ ((b), with reconnection) at $Re=2000$, along with guide lines (dotted) after the vortex bursting with slopes of $4/3$ and $2/3$, respectively.

5. Discussion

We develop a helicity decomposition that allows computation of the differential twist within vortex tubes. The decomposition is used to study the propagation of twist waves within a vortex ring and the bursting due to their collision. In particular, we establish a theoretical relation (2.15) between $H$ and $\eta$ of vortex lines on different coaxial vortex surfaces, along with the numerical measurement of $\eta$ based on VSF; DNS cases of bursting vortex rings are set up with differential twists. Two twist waves with opposite helical chiralities collide on the ring cross-section, where the local twist rate surges by over 10 times the average initial amplitude. The propagation speed is faster on inner vortex surfaces than on outer ones. The dynamics of vortex bursting and the bursting time are modelled by a Burgers-like equation. During the bursting, local vortex surfaces are squeezed to form a disk-like dipole structure with strongly coiled vortex lines. With larger initial twisting rates, vortex reconnection pinches dipole vortex rings off from the rolling-up edge of the bursting disk and significantly reduces the core-size gradient to inhibit subsequent bursting.

The rapid coiling and stretching of vortex lines can destabilize their vortical structures and trigger transition. As a heuristic model problem, the propagation of twist waves and bursting of a vortex ring can be further used to study extreme events of the vorticity/helicity dynamics in transition and turbulence and to explore the possible formation of finite-time singularities in the Euler dynamics. Moreover, the construction and diagnostic methods of the differential twist provide a complete framework for understanding the topological fluid dynamics of various closed vortex/magnetic tubes with delicate internal structures.