2.1. Closed vortex tubes

The helicity

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

is the integral of the helicity density $h = \boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {\omega }$, the dot product of the fluid velocity $\boldsymbol {u}$ and the vorticity $\boldsymbol {\omega }=\boldsymbol {\nabla } \times \boldsymbol {u}$, over a domain bounded by a vortex surface (Moffatt Reference Moffatt1969). We first consider a closed vortex tube whose centreline is a smooth closed curve. Moffatt & Ricca (Reference Moffatt and Ricca1992) derives the relation

(2.2)\begin{equation} H=n_{c} \varGamma^2=\varGamma^2(W_r(\mathcal{C})+T_w(\mathcal{R})) \end{equation}

between $H$ and the Călugăreanu–White invariant $n_c$, where $\varGamma$ denotes the strength of the vorticity flux along the centreline or vortex axis $\mathcal {C}$, and $n_c$ is a basic invariant of mathematical ribbons. By placing a ribbon $\mathcal {R}$ of edges $\mathcal {C}$ and $\mathcal {C}^{*}$ in the tube (see figure 1), $n_c$ can be further decomposed into two geometric quantities, i.e. the writhing number $W_r$ and the twist number $T_w$. Here, $W_r=W_r(\mathcal {C})$ is related only to the geometry of $\mathcal {C}$, representing the contribution of bending or knotting of $\mathcal {C}$ to $H$, while $T_w=T_w(\mathcal {R})$ is the geometrical quantity of the virtual ribbon placed in the tube (see Fuller Reference Fuller1971), representing the contribution of twisting of the vorticity field to $H$.

Figure 1. Schematic of a twisted vortex tube with finite twisting helicity. The dashed curve $\mathcal {C}$ represents the centreline of the vortex tube, and the solid curve line $\mathcal {C}^{*}$ denotes a twisted vortex line on the vortex tube, $\boldsymbol {N}_s$ denotes the radial unit vector from $\mathcal {C}$ to $\mathcal {C}^{*}$ and $\mathcal {R}$ denotes a ribbon with edges $\mathcal {C}$ and $\mathcal {C}^{*}$.

The writhing number is calculated by

(2.3)\begin{equation} W_r=\frac{1}{4 {\rm \pi}} \oint_{\mathcal{C}} \oint_{\mathcal{C}} \frac{\left(\boldsymbol{x}-\boldsymbol{x}^{*}\right) \boldsymbol{\cdot} \mathrm{d} \boldsymbol{x} \times \mathrm{d} \boldsymbol{x}^{*}}{\left|\boldsymbol{x}-\boldsymbol{x}^{*}\right|^{3}}, \end{equation}

where $\boldsymbol {x}$ and $\boldsymbol {x}^{*}$ denote two points on $\mathcal {C}$. The twisting number is calculated by

(2.4)\begin{equation} T_w=\frac{1}{2 {\rm \pi}} \oint_{\mathcal{C}}\left(\boldsymbol{N}_s \times \boldsymbol{N}_{s}^{\prime}\right) \boldsymbol{\cdot} \boldsymbol{T}\,\mathrm{d} s, \end{equation}

where $\boldsymbol {N}_s$ denotes a radial unit vector from $\mathcal {C}$ to $\mathcal {C}^{*}$ on $\mathcal {R}$, $\boldsymbol {N}_s^{\prime }=\mathrm {d} \boldsymbol {N}_s / \mathrm {d} s$, $\boldsymbol {T}$ denotes the unit tangent vector of $\mathcal {C}$ and $s$ is the arc-length parameter.

The twisting number $T_w = T_t + T_i$ can be further geometrically decomposed into the total torsion

(2.5)\begin{equation} T_t(\mathcal{C})=\frac{1}{2 {\rm \pi}} \oint_{\mathcal{C}} \tau(s) \,\mathrm{d} s, \end{equation}

which is only related to the centreline, and the intrinsic twist (Moffatt & Ricca Reference Moffatt and Ricca1992)

(2.6)\begin{equation} T_i(\mathcal{R})=\frac{[\theta]_\mathcal{R}}{2 {\rm \pi}}=\frac{1}{2 {\rm \pi}} \oint_{\mathcal{C}} \xi(s) \,\mathrm{d} s, \end{equation}

which is determined by the azimuthal variation $[\theta ]_\mathcal {R}$ of a vortex line. Here, $\tau$ is the torsion of the curve $\mathcal {C}$, and $\xi$ is the intrinsic rate of the azimuth change along $\mathcal {C}$. When the centreline passes through an inflection point where the curvature vanishes, $T_t(\mathcal {C})$ and $T_i(\mathcal {R})$ jump discontinuously through $\pm 1$ and $\mp 1$, respectively (Moffatt & Ricca Reference Moffatt and Ricca1992). Since the generation and elimination of the inflection point are common in the continuous deformation of $\mathcal {C}$ during the evolution of vortex tubes, it appears to be more suitable to use $T_w$ rather than $T_t$ to describe the internal twist of vortex lines within the vortex tube.

When multiple closed vortex tubes are linked to each other in the flow field, the topological decomposition of $H$ further introduces the Gauss linking number

(2.7)\begin{equation} L_{k,ij}= L_{k}(\mathcal{C}_{i},\mathcal{C}_{j}) = \frac{1}{4 {\rm \pi}} \oint_{\mathcal{C}_{i}} \oint_{\mathcal{C}_{j}} \frac{\left(\boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right) \boldsymbol{\cdot} \mathrm{d}\kern0.05em \boldsymbol{x}_{i} \times \mathrm{d} \boldsymbol{x}_{j}}{\left|\boldsymbol{x}_{i}-\boldsymbol{x}_{j}\right|^{3}} \end{equation}

between vortex tubes $i$ and $j$, where $\boldsymbol {x}_i$ and $\boldsymbol {x}_j$ denote points on $\mathcal {C}_i$ and $\mathcal {C}_j$, respectively. Thus, the full helicity decomposition (e.g. Berger & Field Reference Berger and Field1984; Scheeler et al. Reference Scheeler, Kleckner, Proment, Kindlmann and Irvine2014) reads as

(2.8)\begin{equation} H=\sum_{i \neq j} \varGamma_{i} \varGamma_{j} L_{k,ij}+\sum_{i} \varGamma_{i}^{2}(W_{r,i}+T_{w,i}), \end{equation}

where $\varGamma _{i}$, $W_{r,i}$ and $T_{w,i}$ are the circulation, writhing number and twisting number of vortex tube $i$, respectively.

2.2. Coherent knotted fields

The above discussion is based on the ideal closed vortex tube constituting vortex lines lying on nested torus vortex surfaces. When the vortex tubes reconnect or merge, the helicity decomposition (2.8) may break down due to the significant topological change.

Take the viscous vortex reconnection of a trefoil knotted vortex tube for example (e.g. Kleckner & Irvine Reference Kleckner and Irvine2013; Yao et al. Reference Yao, Yang and Hussain2021; Zhao et al. Reference Zhao, Yu, Chapelier and Scalo2021). In topology, the genus $g_v$ of the closed orientable surface that wraps the vortex changes from $g_v = 1$ for the initial trefoil knot to $g_v = 4$ for the interconnected structure during reconnection. After the reconnection is completed, the major vortical structure becomes two loops with $g_v=1$. For a vortical structure with $g_v>1$, the vorticity field must have one or more null points on the closed vortex surface based on the Poincaré–Hopf theorem (Milnor & Weaver Reference Milnor and Weaver1997). As illustrated in figure 2(a,b), vortex lines originate or annihilate at the null point. Furthermore, the reconnection triggers scale cascade to produce small-scale structures such as bridges and threads. The convoluted structures above do not satisfy an important assumption in the decomposition of (2.8) – all vortex lines lie on a family of surfaces nested around the central closed curve (Chui & Moffatt Reference Chui and Moffatt1995).

Figure 2. Schematics of vortex lines (black solid lines) integrated from the vorticity field with (a) no null point or (b) one null point on vortex surfaces. (c) Schematic for the generation of the generalized twist number during the bridging vortex reconnection, which is illustrated by a virtual surgery.

Thus, the exact helicity decomposition (2.8) is not applicable during the vortex reconnection, but a generalized helicity decomposition can still be useful to characterize geometric features of a coherent knotted field in which the vortex cores can be identified. After reconnection or merging of vortex tubes with the same circulation $\varGamma$, the vorticity is still concentrated in a large-scale tube-like structure. This coherent vortex structure can still be regarded as closed tubes, and each loop has a closed and smooth centreline and nearly constant circulation. The contribution of $W_{r}$ determined by the centreline to $H$ can be evaluated by (2.3). The remainder of the normalized total helicity can be defined as a generalized twist number (Yao et al. Reference Yao, Yang and Hussain2021)

(2.9)\begin{equation} \tilde{T}_{w}=\frac{H}{\varGamma^2}-\sum_{i} W_{r,i}-\sum_{i\neq j}L_{k,ij}. \end{equation}

We demonstrate that $\tilde {T}_{w}$ has a certain connection with $T_w$. As sketched on the left of figure 2(c), we consider a vortical structure with $g_v=2$ formed in reconnection. Between the two large-scale vortex rings at the top and bottom, some vortex lines are wound from one ring to the other through small-scale vortex tubes. We perform a virtual surgery on the vorticity field by cutting off the junction part and then adding two pairs of opposite vortex tubes. As sketched on the right of figure 2(c), the original structure is equivalent to two isolated vortex rings and an attached structure connecting the two rings and consisting of small-scale vortex tubes (e.g. threads). Therefore, the generalized twisting number can be re-expressed by

(2.10)\begin{equation} \tilde{T}_{w}=\sum_{i} T_{w,i}+T_w', \end{equation}

where $\sum _{i} T_{w,i}$ is the sum of twisting contributions of the virtual large-scale vortex rings, and $T_w'$ represents the twisting contribution from all small structures attached to the main vortex tube. This generalized twisting number characterizes the degree of entanglement of the vorticity field. By replacing $T_w$ by $\tilde {T}_w$, the helicity decomposition (2.8) becomes

(2.11)\begin{equation} H = \varGamma^2\left[\sum_{i\neq j}L_{k,ij}+\sum_{i} (W_{r,i}+T_{w,i}) +T_w'\right]. \end{equation}

3.1. Torus knots and links

In the present study the centreline $\mathcal {C}$ of vortex tubes is a torus unknot, knot or link $\mathcal {T}_{p,q}$. It is a discrete rotationally symmetric closed curve embedded on a two-dimensional torus $\varPi$ in $\mathbb {R}^{3}$. Namely, it has $q$-fold symmetry about the rotational symmetry axis of $\varPi$. The curve $\mathcal {C}$ characterized by the integer pair $(p,q)$ winds $q$ times around a circle in the interior of $\varPi$, and $p$ times around its axis of rotational symmetry. The ratio $w = |q/p|$ is the winding number, a measure of the knot and link topology (Oberti & Ricca Reference Oberti and Ricca2016). Each $\mathcal {T}_{p,q}$ is specified by a pair of integers $p$ and $q$. If $p$ and $q$ are co-prime, $\mathcal {T}_{p,q}$ is an isolated torus knot. If $|p| \leqslant 1$ or $|q| \leqslant 1$, $\mathcal {T}_{p,q}$ is an unknot which is ambient isotopic to a standard circle in topology. If $p$ and $q$ are not relatively prime, $\mathcal {T}_{p,q}$ is a torus link with $n$ components $\mathcal {T}_{p/n,q/n}$, where $n$ is the greatest common divisor of $p$ and $q$.

The parametric equation $\boldsymbol {c}(\zeta )=(c_{x}(\zeta ), c_{y}(\zeta ), c_{z}(\zeta ))$ of torus knots or unknots is

(3.1)\begin{equation} \left. \begin{gathered} c_{x}(\zeta) =R_{t}\left(1+\lambda \cos (q \zeta)\right) \cos (p \zeta) ,\\ c_{y}(\zeta) =R_{t}\left(1+\lambda \cos (q \zeta)\right) \sin (p \zeta), \\ c_{z}(\zeta) ={-}R_{t}\lambda \sin (q \zeta), \end{gathered} \quad \zeta \in[0,2 {\rm \pi}), \right\} \end{equation}

where $\lambda = r_t/R_t \in [0, 1)$ denotes the torus aspect ratio, and $R_{t}$ and $r_{t}$ are the major and minor radii of $\varPi$, respectively. Figure 3 illustrates three $\mathcal {T}_{1,6}$ embedded on $\varPi$ with different $\lambda$. The winding amplitude of $\mathcal {C}$ grows with $\lambda$ and vanishes for a standard circle with $\lambda = 0$.

Figure 3. Diagram of torus unknot $\mathcal {T}_{1,6}$ with (a) $\lambda =0.1$, (b) $\lambda =0.3$ and (c) $\lambda =0.5$. Black curves denote different torus unknots $\mathcal {C}$, and blue surfaces denote the torus $\varPi$ with the same $R_{t}$ and different $\lambda$.

Each component $\mathcal {T}_{p/n,q/n}$ of a torus link is a torus knot or unknot. Its parametric equation is

(3.2)\begin{equation} \left. \begin{gathered} c_{x}(\zeta) =R_{t}\left(1+\lambda \cos (q \zeta/n)\right) \cos (p \zeta/n) ,\\ c_{y}(\zeta) =R_{t}\left(1+\lambda \cos (q \zeta/n)\right) \sin (p \zeta/n), \\ c_{z}(\zeta) ={-}R_{t}\lambda \sin (q \zeta/n). \end{gathered} \quad \zeta \in[0,2 {\rm \pi}), \right\} \end{equation}

The rest of the link components are determined by rotating the curve (3.2) clockwise around the axis of rotational symmetry by $2{\rm \pi} / q$. Figure 4 depicts the torus unknot $\mathcal {T}_{1,3}$, torus knot $\mathcal {T}_{2,3}$ and torus link $\mathcal {T}_{2,4}$ embedded on the same $\varPi$ of $\lambda =0.5$. In this study we consider positive integer pairs $(p,q)$, and set constant $R_t=\sqrt {2}/2$. This choice of $R_t$ is to avoid the vortex touching the boundary at late times for large $R_t$ and to have self-intersection at early times for small $R_t$. Note that $\mathcal {T}_{p,-q}$ is the mirror image of $\mathcal {T}_{p,q}$. Thus, the topology and geometry of $\mathcal {T}_{p,q}$ are characterized by three parameters $p$, $q$ and $\lambda$.

Figure 4. Diagram of torus (a) unknot $\mathcal {T}_{1,3}$, (b) knot $\mathcal {T}_{2,3}$ and (c) link $\mathcal {T}_{2,4}$. Black curves denote different torus unknots $\mathcal {C}$, and blue surfaces denote the torus $\varPi$ with the same $R_{t}$ and different $\lambda$.

3.2. Initial configuration of vortex tubes

Based on the parametric equation (3.1) or (3.2) of the vortex centerline and the curved cylindrical coordinate system ($s, \rho, \theta$), we specify the vorticity of the vortex tube as (see Xiong & Yang Reference Xiong and Yang2020)

(3.3)\begin{equation} \boldsymbol{\omega}(s, \rho, \theta)=\varGamma f(\rho)\left[\boldsymbol{e}_{s}+\frac{\rho\eta}{1-\kappa \rho \cos \theta} \boldsymbol{e}_{\theta}\right], \end{equation}

where the circulation $\varGamma$ of the initial vortex tube is set to be $\varGamma _0 = 1$, the vorticity distribution

(3.4)\begin{equation} f(\rho)=\frac{1}{2 {\rm \pi}\sigma^{2}} \exp \left(-\frac{\rho^{2}}{2 \sigma^{2}}\right) \end{equation}

is a Gaussian function with the standard deviation $\sigma =\sigma _0=1 /(16 \sqrt {2 {\rm \pi}})\approx 0.025$, where $\boldsymbol {e}_{s}$ and $\boldsymbol {e}_{\theta }$ are the axial and azimuthal unit vectors, respectively, $\kappa$ is the curvature of $\mathcal {C}$ and $\eta (s)=\tau (s)+\xi (s)=(\boldsymbol {N}_s \times \boldsymbol {N}_{s}^{\prime }) \boldsymbol {\cdot } \boldsymbol {T}$ denotes the local twist rate of the vortex tube.

Note that $\eta$, the integrand in the definition of $T_w$ in (2.4), is constant for a uniformly twisted vortex tube, so that the twisting number can be tuned by varying

(3.5)\begin{equation} \eta=\frac{2 {\rm \pi}T_{w}}{l_c}, \end{equation}

where $l_c$ denotes the length of $\mathcal {C}$. In this study the initial vortex tubes have $T_w = 0$ by setting $\eta = 0$ with $\xi (s) = -\tau (s)$, which ensures that all vortex lines are initially parallel to $\mathcal {C}$ everywhere.

The vortex tube thickness can be characterized by $\sigma$ in (3.4), the Lamb–Oseen core size $a$ or the vortex core radius $r_c$, with the relation $r_c^2=2a^2=4\sigma ^2$ (Zhao & Scalo Reference Zhao and Scalo2021). In the present study the ratio between $r_c$ and the vortex average radius

(3.6)\begin{equation} \bar{R}(t) =\frac{1}{l_{c}} \oint_{ \mathcal{C} } R(s) \,\mathrm{d} s \end{equation}

is set to be $r_c/\bar {R}= 0.071$, where $R(s)$ is the distance between a point on $\mathcal {C}$ and the $q$-fold rotational symmetry axis of $\varPi$. The corresponding vortex tube is relatively thin, compared with those in recent numerical simulations in table 1.

Table 1. Comparison of vortex tube thicknesses and simulation box sizes in the present and recent numerical simulations.

In the numerical implementation, (3.3) is specified in a periodic box with dimensions $(2{\rm \pi} )^3$ in terms of Cartesian coordinates via the numerical algorithm transforming $\boldsymbol {x}$ into $(s, \rho, \theta )$ of Xiong & Yang (Reference Xiong and Yang2020). This numerical construction has been demonstrated to be very robust to generate a smooth vorticity field even for the thick vortex tube with self-intersection (Xiong & Yang Reference Xiong and Yang2019a). The velocity is calculated from the vorticity via the BS law in Fourier space

(3.7)\begin{equation} \boldsymbol{u}=\mathcal{F}^{{-}1}\left(\frac{i \boldsymbol{k} \times \hat{\boldsymbol{\omega}}}{|\boldsymbol{k}|^{2}}\right), \end{equation}

where $\mathcal {F}^{-1}$ denotes the inverse Fourier transform, $\boldsymbol {k}$ denotes the wavenumber in Fourier space, and $\hat {\boldsymbol {\omega }}=\mathcal {F}(\boldsymbol {\omega })$ denotes the Fourier coefficient of $\boldsymbol {\omega }$ with the Fourier transform $\mathcal {F}$.

3.3. Direct numerical simulation

We take the velocity fields of vortex torus knots/links as initial conditions and calculate each evolution using DNS. The velocity field $\boldsymbol {u}(\boldsymbol x, t)$ of an incompressible viscous flow with constant unit density is governed by the NS equations

(3.8)\begin{equation} \left. \begin{gathered} \frac{\partial \boldsymbol{u}}{\partial t}+(\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u}={-}\boldsymbol{\nabla} p+\nu \nabla^{2} \boldsymbol{u},\\ \displaystyle \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}=0, \end{gathered} \right\} \end{equation}

where $t$ denotes the time, $p$ the pressure and $\nu$ the kinematic viscosity.

The DNS for (3.8) is performed in the periodic box of side $L=2 {\rm \pi}$ with $N^3$ uniform grid points using a standard pseudo-spectral method. Here, the DNS box size should be much larger than the vortex average radius, i.e. $L\gg \bar {R}$, to avoid numerical artifacts due to periodic boundary conditions. Zhao et al. (Reference Zhao, Yu, Chapelier and Scalo2021) argued and validated that $L/\bar {R}=6.67$ is large enough. As listed in table 1, $L/\bar {R} = 8.89$ is used in the present study, which satisfies the criterion suggested by Zhao et al. (Reference Zhao, Yu, Chapelier and Scalo2021) and is relatively large in the current studies. Note that Zhao et al. (Reference Zhao, Yu, Chapelier and Scalo2021) and Zhao & Scalo (Reference Zhao and Scalo2021) utilized a compressible flow solver with the block-structured adaptive mesh refinement and the high-order compact finite difference scheme to achieve smaller $r_c/\bar {R}$ and larger $L/\bar {R}$ in the simulation of vortex knots than those in the present and other previous studies. The numerical solver removes aliasing errors using the two-third truncation method with the maximum wavenumber $k_{max } \approx N / 3$. The Fourier coefficient of $\boldsymbol u$ is advanced in time by a second-order Adams–Bashforth scheme. The time step is chosen to ensure the Courant–Friedrichs–Lewy number is less than 0.5 for numerical stability and accuracy. This numerical solver has been used and validated in various applications (e.g. Yang, Pullin & Bermejo-Moreno Reference Yang, Pullin and Bermejo-Moreno2010; Zheng, You & Yang Reference Zheng, You and Yang2017; Xiong & Yang Reference Xiong and Yang2019a). A non-dimensionalized time $t^*=t/(R_t^2/\varGamma _0)$ is used in post-processing.

We carry out a series of DNS at the vortex Reynolds number $ {\textit {Re}} \equiv \varGamma _0 / \nu =2000$ with various $p$, $q$, $\lambda$ and $N$, which are listed in table 2. To ensure that the grid resolution can fully resolve the flow evolution, $N$ is carefully chosen to be from $512$ to $2048$ for different cases. A grid convergence test is detailed in Appendix A, including a typical reconnection case of the trefoil knot and a transition case with highly intensive vortex interactions. Considering that the flow evolutions in over a half of cases have bridging reconnections or evolve into a turbulent-like state, we also list the lowest spatial resolution $k_{max}\eta _K$ for the DNS cases in the entire evolution in table 2 for reference, where $\eta _K=(\nu ^{3} / \varepsilon )^{1 / 4}$ denotes the Kolmogorov length scale, and $k_{max}\eta _K$ is always greater than $2$ in the temporal evolution, satisfying the criterion $k_{max } \eta _K \geqslant 1.5$ (Pope Reference Pope2000) for resolving the smallest scales in turbulence.

Table 2. Direct numerical simulation parameters: integers $p$ and $q$ for characterizing a torus knot/link, the torus aspect ratio $\lambda$, the number of grid points $N$ and the spatial resolution $k_{max}\eta _K$.

4.1. Effects of the torus aspect ratio

The evolution of a slender closed vortex tube with a specified thickness is generally determined by its centreline geometry, which is fully characterized by the integer pair $(p,q)$ and torus aspect ratio $\lambda$ in (3.1) or (3.2). Given the large parametric space of $p$, $q$ and $\lambda$, we first consider the vortex torus unknot $\mathcal {T}_{1,q}$ with the winding number $w=q$. It is a coiled vortex loop with different degrees of winding structure for different $q$ and $\lambda$.

We take vortex $\mathcal {T}_{1,3}$ as an example to illustrate the effect of $\lambda$ on the vortex centreline geometry and evolution. Figure 5 shows the initial configuration and the temporal evolution of $\mathcal {T}_{1,3}$ with $\lambda =0.1$, 0.3, 0.5 and 0.7 using the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ colour-coded by the helicity density $h$, where $\omega _0$ denotes the maximum initial vorticity magnitude.

Figure 5. Evolution of the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ for vortex $\mathcal {T}_{1,3}$ with (a) $\lambda =0.1$, (b) $\lambda =0.3$, (c) $\lambda =0.5$ and (d) $\lambda =0.7$ at $t^*=0$, 2.5, 5, 7.5 and 10 (from left to right). Surfaces are colour-coded by the helicity density. For vortex $\mathcal {T}_{1,3}$ with $\lambda =0.7$ at $t^*=10$, the central loop has $|\boldsymbol {\omega }|_{{max}}=34.35$ and $\varGamma =1.00$, and the secondary loop has $|\boldsymbol {\omega }|_{{max}} = 45.70$ and $\varGamma =1.06$.

Note that the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ contains $96\,\%$ of the circulation at the initial time. Its isocontour threshold is selected after tuning to identify the major vortical structures and distinguish signature stages in the temporal evolution of vortex loops, because in general, very thin or weak threads are not captured for too large thresholds (e.g. $|\boldsymbol {\omega }|>0.1\omega _0$), and the fine structures are merged into a bulky one for too small thresholds (e.g. $|\boldsymbol {\omega }|<0.01\omega _0$). We find that most geometric and topological features of vortical structures can be identified within a small range of isocontour values $0.01\omega _0 \leq |\boldsymbol {\omega }| \leq 0.1\omega _0$ at $t^*\leq 10$.

The vortex shape remains almost unchanged for small $\lambda =0.1$, and the vortex deformation becomes notable with increasing $\lambda$. For $\lambda = 0.3$ and $0.5$, three petal-like branches form and then relax via self-induction to a state close to the original shape. In addition to the overall movement along the $z$-axis as for ordinary vortex rings, these coiled vortex loops rotate counterclockwise around the axis of discrete rotational symmetry from the top view, and the rotational angular velocity grows with $\lambda$. For large $\lambda = 0.7$, the three petals are stretched with time, and then their tips are pinched off via the viscous vortex reconnection. Moreover, the evolution of coiled vortex loops can be affected by the vortex tube thickness and the Reynolds number – discussed in Appendices B and C.

Figure 6 shows the perspective and the close-up top views of vortex $\mathcal {T}_{1,3}$ for $\lambda =0.7$ before and after the reconnection at $t^*=6.5$ and $7.5$, respectively. Before reconnection, a pair of vortex tubes near the ‘neck’ of each petal approach each other into an antiparallel state. Then, the antiparallel parts are reconnected under the bridging mechanism (Melander & Hussain Reference Melander and Hussain1988; Kida & Takaoka Reference Kida and Takaoka1991), forming a structure with complex topology and genus $g_v > 1$. After the reconnection, a secondary vortex ring pinches off from each petal, and the central main part returns to a vortex loop of $\mathcal {T}_{1,3}$ with a smaller $\lambda$. Moreover, some small-scale threads connecting the central vortex loop and the three small secondary rings are stretched and gradually dissipated.

Figure 6. Close-up view of the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ for vortex $\mathcal {T}_{1,3}$ with $\lambda =0.7$ at (a) $t^*=6.5$ and (b) $t^*=7.5$. The surfaces are colour-coded by the helicity density. The black arrow indicates the direction of the vorticity of the large-scale vortex tube.

By comparing the evolution of vortices $\mathcal {T}_{1,3}$ with $\lambda = 0.5$ and $0.7$, a minimum critical torus aspect ratio $\lambda _c$ seems to exist for triggering the vortex reconnection. In the approximation of the vortex filament, the motion of a vortex tube is determined by the BS law and the geometry of the vortex core line. We use the method based on the normalized helicity density (Levy, Degani & Seginer Reference Levy, Degani and Seginer1990) to extract the vortex core line $\mathcal {C}(t)$ from DNS data. This method was also applied to study the dynamics of a trefoil vortex knot (Yao et al. Reference Yao, Yang and Hussain2021).

Figure 7 depicts different temporal evolutions of the location and velocity of $\mathcal {C}(t)$ for $\lambda =0.5$ and $0.7$, where the arrow length is proportional to the local speed. For vortex $\mathcal {T}_{1,3}$ with $\lambda =0.5$, the induction velocity on the core line stretches the petal-like branch at early times. The approaching speed of branch sides gradually decreases due to the reduced curvature in LIA-induced stretching, and then local vortex tubes repel each other as a relaxation process at late times. For $\lambda =0.7$, by contrast, the local vortex tubes on two sides of the branch keep approaching each other to trigger the antiparallel reconnection.

Figure 7. Local velocity on the vortex core line for vortex $\mathcal { T}_{1,3}$ with (a) $\lambda =0.5$ and (b) $\lambda =0.7$ at $t^*= 2, 4, 6$ and 7. The arrow length is proportional to the local speed.

We quantify the global geometry of $\mathcal {C}(t)$ by the average radius $\bar {R}(t)$, average torus aspect ratio

(4.1)\begin{equation} \bar{\lambda}(t) =\frac{R_{max}-R_{min}}{2\bar{R}} \end{equation}

and average $z$-coordinate

(4.2)\begin{equation} \bar{z}(t) =\frac{1}{l_{c}} \oint_{ \mathcal{C} } z(s) \,\mathrm{d} s. \end{equation}

As illustrated in figure 8, $R_{max}$ and $R_{min}$ are the maximum and minimum values of $R(s)$, respectively. Figure 9 plots the time evolution of $\bar {\lambda }(t)$, $\bar {z}(t)$ and $\bar {R}(t)$ of vortex $\mathcal {T}_{1,3}$ with $\lambda =0.1$, 0.3, 0.5 and 0.7, where the error bars denote the range between maximum and minimum values of $R(t)$ or $z(t)$ at a particular time; the $\lambda =0.7$ case only considers the central secondary ring after reconnection for $t^*>7$.

Figure 8. Illustration of $R_{max}$, $R_{min}$ and $\bar {R}$ of vortex $\mathcal {T}_{1,3}$ with $\lambda =0.5$ at $t^*=7.5$.

Figure 9. Evolution of (a) $\bar {R}(t)$, (b) $\bar {\lambda }(t)$ and (c) $\bar {z}(t)$ of vortex $\mathcal {T}_{1,3}$ with $\lambda =0.1$, 0.3, 0.5 and 0.7, where the $\lambda =0.7$ case only considers the central ring after reconnection for $t>7$. Error bars denote the range between minimum and maximum values.

The evolution of $\bar {R}(t)$ in figure 9(a) shows that the average radius for $\lambda = 0.1$ remains unchanged with the preservation of the geometric shape. For $\lambda = 0.3$ and 0.5, $\bar {R}$ first increases and then decreases, corresponding to the stretching and relaxation processes in the $x$–$y$ plane in figure 7, respectively. For $\lambda = 0.7$, the branches are highly stretched, and $\bar {R}$ grows faster than that in lower $\lambda$ cases under self-induced motion. Then, the vortex reconnection occurs, leaving a central secondary loop with a smaller $\bar {R}$, so the growth $\bar {R}$ is suddenly terminated.

The evolution of $\bar {\lambda }$ in figure 9(b) characterizes the winding amplitude of vortex loops (also see figure 3). For vortex loops with small initial $\lambda$ ($= 0.1,0.3$ and $0.5$), the variation of $\bar {\lambda }$ is generally small and quasi-periodic, so their evolution is stable. Here, the (Lyapunov) ‘stability’ of a vortex structure is defined in Ricca et al. (Reference Ricca, Samuels and Barenghi1999). If the vortex conserves topology, geometric signature (e.g. $p$, $q$ and $\bar {\lambda }$) and vortex coherency after travelling a considerable distance (much larger than their typical size) for some finite time, it is ‘stable’. If the vortex breaks up or unfolds in a short time (compared with the typical time scale), it is ‘unstable’. For $\lambda = 0.7$, $\bar {\lambda }(t)$ experiences large fluctuations in the early stage, and then it drops after the reconnection, because the central part of $\mathcal {T}_{1,3}$ becomes a vortex loop with a smaller $\lambda \approx 0.2$ (also see figure 5d).

The dominant motion of vortex loops is moving along the $z$-axis. The evolution of $\bar {z}$ in figure 9(c) shows the position and speed of $\mathcal {T}_{1,3}$ moving forward. The vortex loops without reconnection for $\lambda = 0.1$, 0.3 and 0.5 move at a constant speed, and the forward speed decreases with the increase of $\lambda$ (Barenghi et al. Reference Barenghi, Hänninen and Tsubota2006; Oberti & Ricca Reference Oberti and Ricca2019). For $\lambda = 0.7$, the vortex forward motion accelerates after the reconnection around $t^* = 7$, because the central loop is converted into the secondary one with a lower $\bar {\lambda }$, which has a faster forward speed. The range of $\bar {z}$ decreases before the reconnection, implying that the vortex tubes are approaching each other.

The results above suggest that the unstable vortex torus unknot with a larger initial torus aspect ratio tends to be transformed into a more stable one with a lower torus aspect ratio through the vortex reconnection.

In addition, the maximum vorticity magnitude $|\boldsymbol {\omega }|_{{max}}$ of the central parent loop or secondary offspring loops for $\mathcal {T}_{1,3}$ with $\lambda =0.7$ at $t^* = 10$ are calculated in a subdomain surrounding each of them. As listed in the caption of figure 5, $|\boldsymbol {\omega }|_{{max}}$ of the central and secondary loops are different. The non-uniform $|\boldsymbol {\omega }|$ along the vortex axis during and after reconnection is caused by deformation of the local vortex core and twisting of vortex tubes. The circulations of the central and secondary loops are calculated in the $x$–$z$ plane at $y={\rm \pi}$ and the $x$–$y$ plane at $z = \bar {z}(t^*=10)$, respectively. The circulation is almost conserved during the reconnection, which is consistent with the finding in Zhao & Scalo (Reference Zhao and Scalo2021) and supports the helicity decomposition in (2.9).

4.2. Flow statistics and helicity analysis

The geometric and topological changes of vortex loops alter the main flow statistics. Due to the viscous dissipation, the total energy $E_{tot}=\sum _{k}|\hat {\boldsymbol {u}}|^{2} / 2$ of all cases decreases monotonically (not shown). In general, the mean dissipation rate $\varepsilon =\nu \sum _{k}(|\boldsymbol {k}|\,|\hat {\boldsymbol {u}}|)^{2}$ decays and the total helicity $H$ is almost conserved for $\lambda = 0.1$, 0.3 and 0.5 in figure 10. By contrast, we observe a bump of $\varepsilon$ and a transient growth of $H$ around $t^* = 7$ due to the vortex reconnection in the case of $\lambda = 0.7$, similar to the observations during reconnection of other vortex knots in (e.g. Xiong & Yang Reference Xiong and Yang2019a). The notable variations of $\varepsilon$ and $H$ are expected to be more significant for larger $Re$ (see Yao et al. Reference Yao, Yang and Hussain2021; Zhao et al. Reference Zhao, Yu, Chapelier and Scalo2021).

Figure 10. Evolution of (a) the mean dissipation rate and (b) total helicity for vortex $\mathcal {T}_{1,3}$ with $\lambda =0.1$, 0.3, 0.5 and 0.7.

In order to elucidate the large helicity fluctuation during the bridging reconnection sketched in figure 2(c), we apply the positive–negative helicity decomposition $H^{\pm }= \int _{\mathcal {V}} h^{\pm } \,\mathrm {d} \mathcal {V}$ with

(4.3)\begin{equation} h^{+}= \left\{\begin{array}{@{}ll} h, & \text{if } h \geqslant 0, \\ 0, & \text{otherwise}, \end{array}\right. \end{equation}

and $h^{-}=h-h^{+}$, and the topological decomposition (2.11) to the case of $\lambda =0.7$.

In figure 11(a) both $|H^+|$ and $|H^-|$ surge at $t^*=6\sim 8$, and the growth of $|H^+|$ is slightly larger than $|H^-|$ due to the asymmetric configuration during reconnection. The uneven growths cause the transient growth of $H$, which is also reported in Yao et al. (Reference Yao, Yang and Hussain2021). After the reconnection, we simplify (2.11) by summing $W_r=\sum _{i} W_{r,i}$ and neglecting the total Gauss linking number as $L_k=0$, so the generalized twisting number becomes

(4.4)\begin{equation} \tilde{T}_{w}=H/\varGamma^2-W_r. \end{equation}

Figure 11. (a) Positive–negative and (b) topological decompositions of the total helicity of vortex $\mathcal {T}_{1,3}$ with $\lambda =0.7$.

Figure 11(b) shows $W_{r}$ and $\tilde {T}_{w}$ are almost conserved before reconnection. The conservation of $W_r$ for the evolution of vortex tubes without self-intersection is discussed in Appendix D. Then, $W_{r}$ is converted into $\tilde {T}_{w}$ during the reconnection, consistent with the finding for trefoil vortex knots (Yao et al. Reference Yao, Yang and Hussain2021; Zhao et al. Reference Zhao, Yu, Chapelier and Scalo2021).

4.3. Stability diagram of coiled vortex loops

This subsection presents a systematic study on the effect of the torus aspect ratio on the stability (Ricca et al. Reference Ricca, Samuels and Barenghi1999) of coiled vortex loops $\mathcal {T}_{1,q}$, extending the investigation in § 4.1. Note that $\mathcal {T}_{1,q}$ has been found as an intermediate or final topological state in the evolution of complex vortex knots (Scheeler et al. Reference Scheeler, Kleckner, Proment, Kindlmann and Irvine2014; Kleckner et al. Reference Kleckner, Kauffman and Irvine2016; Xiong & Yang Reference Xiong and Yang2019a; Liu, Ricca & Li Reference Liu, Ricca and Li2020).

For the coiled vortex loops $\mathcal {T}_{1,q}$, $q$ and $\lambda$ constitute the full geometric parameter space. We carried out DNS for 24 vortex torus unknots $\mathcal {T}_{1,q}$ with $q = 1\sim 6$ and $\lambda =$ 0.1, 0.3, 0.5 and 0.7. These coiled loops at $t^*=0$ and $10$ are visualized in figures 12(a) and 12(b), respectively. In the vortex evolution, we find that the coiled loops remain stable for smaller $q$ and $\lambda$, but break up into a central loop and $q$ small secondary loops for large $q$ and $\lambda$.

Figure 12. Coiled vortex loops $\mathcal {T}_{1,q}$ with $q = 1 \sim 6$ and $\lambda =$ 0.1, 0.3, 0.5 and 0.7 at (a) $t^*=0$ and (b) $t^*=10$. Each vortex is visualized by the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ colour-coded by the helicity density.

The deformation velocity of the vortex core line depends on the curvature according to the LIA (Hama Reference Hama1962; Kida Reference Kida1981). With the increase of $q$ and $\lambda$, the vortex loop becomes more coiled and experiences more severe deformation in the evolution. The vortex reconnection occurs as the initial $\lambda$ of vortex $\mathcal {T}_{1,q}$ exceeds a critical value $\lambda _c(q)$. The critical $\lambda _c(q)$ decreases with the increase of $q$; e.g. we observe $\lambda _c>0.5$ for $q\le 3$ and $0.3 < \lambda < 0.5$ for $4 \le q \le 6$ from reconnection events of $\mathcal {T}_{1,q}$ in figure 12(b).

Next, we focus on the cases of $\lambda =0.3$ and 0.5 and $q=4$, 5 and 6 close to the borderline cases with or without reconnection. For $\lambda =0.3$, a cycle of stretching and relaxation processes maintains the relative stability of vortices $\mathcal {T}_{1,4}$ and $\mathcal {T}_{1,6}$ in figure 13(a,b). For $\lambda =0.5$, the persistent stretching causes the reconnection of a pair of local vortex tubes with antiparallel vorticity directions for $\mathcal {T}_{1,4}$ and $\mathcal {T}_{1,6}$ in figure 13(c,d). The initial loop splits into $q$ small secondary loops moving outwards and leaving a coiled loop in the centre with smaller $\lambda$. The vortex reconnection for $\lambda =0.5$ can be further quantified by the transient growth of $\varepsilon$ and $H$ observed in figure 14. The reconnection occurs earlier and the dissipation peak is higher for the loops with larger $q$, because their initial coils are closer to each other and more reconnections occur at the same time. This trend is projected that $\lambda _c$ and the vortex reconnection time decrease with the further increase of $q$.

Figure 13. Evolution of the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ for vortices (a,c) $\mathcal {T}_{1,4}$ and (b,d) $\mathcal {T}_{1,6}$ with (a,b) $\lambda =0.3$ and (c,d) $\lambda =0.5$. The surfaces are colour-coded by the helicity density. For vortex $\mathcal {T}_{1,4}$ with $\lambda =0.5$ at $t^*=7.5$, the central loop has $|\boldsymbol {\omega }|_{{max}}=68.92$ and $\varGamma =1.02$, and the secondary loop has $|\boldsymbol {\omega }|_{{max}} = 81.71$ and $\varGamma =1.09$. For vortex $\mathcal {T}_{1,6}$ with $\lambda =0.5$ at $t^*=7.5$, the central loop has $|\boldsymbol {\omega }|_{{max}}=52.86$ and $\varGamma =0.995$, and the secondary loop has $|\boldsymbol {\omega }|_{{max}} = 53.78$ and $\varGamma =1.00$.

Figure 14. Evolution of (a) the mean dissipation rate and (b) total helicity for vortices $\mathcal {T}_{1,4}$, $\mathcal {T}_{1,5}$ and $\mathcal {T}_{1,6}$ with $\lambda =0.5$ and 0.7.

Figure 15 compares initial vortex loops and central loops left after reconnection for $\mathcal {T}_{1,4}$ and $\mathcal {T}_{1,6}$ with $\lambda =0.5$ and $\lambda =0.7$. We find that the vortices with larger $\lambda (t^*=0)$ in figure 15(c,d) than those in figure 15(a,b) are converted to more stable central secondary loops with smaller $\lambda (t^*=0)$. This relation is quantified by the average torus aspect ratio in (4.1) of the central secondary loop for all the reconnection cases at time $t^*=10$ in table 3.

Figure 15. Geometric transformation of vortices (a,b) $\mathcal {T}_{1,4}$ and (c,d) $\mathcal {T}_{1,6}$ with (a,c) $\lambda =0.5$ and (b,d) $\lambda =0.7$ from the initial vortex loop to the central secondary loop via vortex reconnection. Each vortex is visualized by the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ colour-coded by the helicity density.

Table 3. Average torus aspect ratio for central vortex loops after the first vortex reconnection.

Since the thickness of vortex tubes grows with time due to viscous diffusion, a vortex loop must have local reconnection or self-intersection eventually. Before a finite time, e.g. $t^*=10$ in figure 12(b), we observe a quasi-cycle of stretching and relaxation for small $\lambda$ and $q$ and clear reconnection events for large $\lambda$ and $q$. Based on this observation, we plot a stability diagram in figure 16 for the 24 vortex torus unknots with initial configurations shown in figure 12(a). The stable and unstable cases are marked by different symbols, and they are divided by a critical curve as an approximation of $\lambda _c$ in terms of $q$.

Figure 16. Stability diagram of coiled vortex loops, corresponding to the visualization in figure 12. The dashed line denotes the critical curve to demarcate stable and unstable regimes, and the arrows denote the geometric transformation illustrated in figure 15 via the vortex reconnection.

The transition route from the coiled vortex loop to the secondary loop via reconnection is marked by the arrows in figure 16, where the drop of $\lambda$ is according to the quantitative results in table 3. Again, we show the trend that a more unstable vortex loop can be reconnected into a more stable secondary loop, which appears to maintain the simple vortex topology and reduce the helicity variation of $\mathcal {T}_{1,q}$.

4.4. Transition to turbulence for large winding numbers

For large winding numbers $q$, the coiled loops in the unstable regime in figure 16 can have a transition to turbulence (or a turbulent-like state) after vortex reconnection. For example, the evolution of vortex $\mathcal {T}_{1,12}$ with $\lambda =0.5$ and $q=12$ in figure 17 differs from the bridging reconnections with $q=4$ and 6 in figure 13(c,d). At the initial time, the coils are very close to each other, particularly on the inner side. The adjacent vortex tubes on the inner side are nearly parallel, and they undergo strong vortex interactions at $t^*=2.5$ in the early stages. The tubular structure breaks up into a hierarchy of small-scale vortical structures at $t^*=5$, triggering sudden transition to turbulence from the inner to outer sides. At $t^*=10$, the thread-like vortex tubes are intertwined with each other, similar to the tangle of vortex tubes identified in turbulence (Xiong & Yang Reference Xiong and Yang2019b).

Figure 17. Evolution of the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ for vortex $\mathcal {T}_{1,12}$ with $\lambda =0.5$. Surfaces are colour-coded by the helicity density.

The time evolution of $H$, $H^+$ and $H^-$ of this vortex $\mathcal {T}_{1,12}$ in figure 18(a) shows that the total helicity generally decays in the early stage, and then undergoes a notable jump during the transition at $t^* = 2.5 \sim 5$. Both $|H^+|$ and $|H^-|$ surge before the transition due to the emergence of oppositely polarized helicity structures (Yao et al. Reference Yao, Yang and Hussain2021), and then decay after the transition due to viscous dissipation as in decaying turbulence. The time evolution of the energy spectrum in figure 18(b) shows the migration of $E(k)$ from low wavenumbers $k\le 50$ to high wavenumbers $k\ge 100$ during the transition, and a part of $E(k)$ is close to the $-5/3$ scaling at late times. The evolution of $E(k)$ is consistent with the flow visualization in figure 17, where large-scale tubular structures gradually break up into small-scale ones.

Figure 18. Evolution of (a) the total helicity with the positive–negative decomposition and (b) the energy spectrum of vortex $\mathcal {T}_{1,12}$ with $\lambda =0.5$.

This case demonstrates that even a single vortex loop with the trivial topology can undergo transition to turbulence and indicates that the geometry is more important than the topology for vortex evolution in viscous flows. Additionally in the stability diagram in figure 16, there is a sub-region for the transition in the unstable regime at large $q$, which is different from the simple reconnection sub-region at small $q$.

5.1. Vortex merging for small torus aspect ratios

The vortex $\mathcal {T}_{p,q}$ with $p \geqslant 2$ has the initial configuration of a knot or link, and it can undergo more complex topological transitions and helicity conversion than $\mathcal {T}_{1,q}$. In this section we focus on $\mathcal {T}_{2,q}$ with various torus aspect ratios.

We start from vortex $\mathcal {T}_{2,q}$ with small $\lambda =0.1$. The initial configuration of $\mathcal {T}_{2,q}$ with $q=1 \sim 6$ and $\lambda =0.1$ is visualized in figure 19, where $\mathcal {T}_{2,1}$ is topologically trivial, $\mathcal {T}_{2,3}$ and $\mathcal {T}_{2,5}$ are trefoil and cinquefoil knots, respectively, and $\mathcal {T}_{2,2}$, $\mathcal {T}_{2,4}$ and $\mathcal {T}_{2,6}$ are links containing two components. Table 4 lists the initial total writhing number $W_r=\sum _{i} W_{r,i}$ and linking number $L_k=\sum _{i\neq j}L_{k,ij}$ calculated by (2.3) and (2.7), respectively. From the helicity density distribution in figure 19 and the helicity components in table 4, the total helicity $H = W_r + L_k$ increases with $q$.

Table 4. Initial writhing and linking numbers of vortex $\mathcal {T}_{2,q}$ in figure 19.

Figure 19. Initial configurations of vortex $\mathcal {T}_{2,q}$ with $q=1 \sim 6$ and $\lambda =0.1$. Each vortex is visualized by the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ colour-coded by the helicity density.

For small $\lambda$, the cross-section of $\mathcal {T}_{p,q}$ has a pair of vortices which are close to each other and have nearly the same vortex-axis direction. This type of vortex pairs can merge together in viscous flows when the ratio of the Lamb–Oseen core size $a$ to the distance $b$ of vortex cores satisfies $a / b > 0.22$ (Le Dizes & Verga Reference Le Dizes and Verga2002). The initial configuration of $\mathcal {T}_{2,q}$ with $\lambda =0.1$ has a pair of adjacent vortex tubes, satisfying the merging criterion as

(5.1)\begin{equation} \frac{a}{b}=\frac{\sqrt{2}\sigma_0}{2\lambda R_t}=\frac{1}{16\sqrt{2{\rm \pi}}\lambda} = 0.249 >0.22, \end{equation}

where the estimation of $a$ is consistent with the method in Le Dizes & Verga (Reference Le Dizes and Verga2002), so the adjacent vortex tubes merge in the early stage of evolution. This criterion also suggests that the vortex merging for $\mathcal {T}_{2,q}$ occurs for $\lambda < 0.113$.

Taking $\mathcal {T}_{2,3}$ for example, the time evolution of the trefoil vortex in figure 20 shows the merging of adjacent vortex tubes at very early times. The knotted tube rapidly merges into a vortex ring. Figure 21 plots the contour of $|\boldsymbol {\omega }|$ on the $y$–$z$ slice at $x={\rm \pi}$. A pair of isolated and co-directional vortices at $t^*=0$ merge at $t^*=0.5$. This merging produces spiral structures on the periphery of the merged vortex core with the emergence of large opposite $h$, similar to the reconnections in figure 13 and in Yao et al. (Reference Yao, Yang and Hussain2021). The vortex merging is almost complete and the spiral arms are dissipated at $t^*=2.5$, leaving a simple vortex ring with doubled circulation $\varGamma _m=2\varGamma _0$. Then, the vortex lines continue to be coiled on the vortex ring after the merging in figure 20.

Figure 20. Evolution of the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ for vortex $\mathcal {T}_{2,3}$ with $\lambda =0.1$ at $t^*=0$, 0.1, 0.5, 2.5 and 20 (from a–e). Surfaces are colour-coded by the helicity density. Some typical vortex lines are plotted at $t^*=2.5$ and 20.

Figure 21. Evolution of $|\boldsymbol {\omega }|$ on the $y$–$z$ slice (marked by the dashed line in figure 20) for vortex $\mathcal {T}_{2,3}$ with $\lambda =0.1$ at (a) $t^*=0$, (b) $t^*=0.1$, (c) $t^*=0.5$ and (d) $t^*=2.5$.

The rapid topological transition causes very fast helicity conversion. Figure 22 plots the helicity topological decomposition of vortex $\mathcal {T}_{2,3}$ with $\lambda =0.1$ at very early times $t^* \leq 0.5$ and during the entire evolution. At $t^*\approx 0.35$, the circulation of the vortex doubles, and the falling $\varGamma ^2 W_r$ and surging $\varGamma ^2\tilde {T}_{w}$ imply that the writhe helicity is fully converted into the generalized twist helicity. In the final stage of the merging, the interaction of the spiral vortices causes $H$ to increase via the growth of $\tilde {T}_{w}$. The total helicity of the initial vortex $\mathcal {T}_{2,q}$ is $H = \varGamma ^2 (W_r+L_k)$ with zero twist, while it becomes $H = \varGamma ^2 \tilde {T}_{w}$ dominated by the twist after the vortex merging. This complete helicity conversion from $W_r$ to $T_w$ is much faster than that due to the stretching and compression of coiled vortex loops (see Scheeler et al. Reference Scheeler, van Rees, Kedia, Kleckner and Irvine2017).

Figure 22. Evolution of the helicity and its topological decomposition of vortex $\mathcal {T}_{2,3}$ with $\lambda =0.1$ (a) in the early stage $t^ * \leq 0.5$ and (b) during the entire evolution.

At late times, the twisted vortex lines gradually become parallel to the vortex core line under the viscous effect (see figure 20), which manifests in the decay of $T_w$ (Scheeler et al. Reference Scheeler, van Rees, Kedia, Kleckner and Irvine2017; Xiong & Yang Reference Xiong and Yang2020). Figure 23(a) plots the time evolution of $H$, $H^+$ and $H^-$ of vortex $\mathcal {T}_{2,3}$ with $\lambda =0.1$. All the helicity magnitudes surge during the incipient vortex merging, then quickly peak and slowly decay. The decay of $|H^-|$ is faster than $|H^+|$ so vortex lines gradually form a right-handed twisting structure. After the merging is complete around $t^*_m=2.5$, $H^+$ and $H^-$ relax to $H$ and zero, respectively; $H$ for $\mathcal {T}_{2,q}$ with different $q$ decays with time in a self-similar way, i.e. the curves of $H(t^*)/H(t^*_m)$ for various vortex knots/links against $t^*$ collapse (not shown). In figure 23(b) this decay law is well estimated by the model (E10) developed in Appendix E.

Figure 23. (a) Evolution of $H$, $H^+$ and $H^-$ of vortex $\mathcal {T}_{2,3}$ with $\lambda =0.1$. (b) Comparison of the DNS results (symbols) and model estimations (solid lines) from (E10) for vortex $\mathcal {T}_{2,q}$ with $q=1\sim 6$ and $\lambda =0.1$. The results of each case are represented in the same colour.

5.2. Topological degeneration for large torus aspect ratios

For relatively large torus aspect ratios, previous studies reported that the torus (Xiong & Yang Reference Xiong and Yang2019a; Yao et al. Reference Yao, Yang and Hussain2021) or non-torus (Kida & Takaoka Reference Kida and Takaoka1987; Kleckner & Irvine Reference Kleckner and Irvine2013; Zhao et al. Reference Zhao, Yu, Chapelier and Scalo2021) trefoil vortex knot with $\lambda = 0.5$ is untied into coiled vortex loops during the bridging reconnection. This unknotting process implies a possible route for the topological degeneration of vortex knots/links, i.e. the decrease of the topological complexity (Kleckner et al. Reference Kleckner, Kauffman and Irvine2016; Liu et al. Reference Liu, Ricca and Li2020) or values of $p$ and $q$ of the vortex core line in the evolution of $\mathcal {T}_{p,q}$.

We illustrate the topological degeneration of vortex $\mathcal {T}_{2,q}$ with $q=1\sim 6$ and $\lambda =0.5$ through vortex reconnection in figure 24. The vortex knots or links are untied into two coiled vortex loops $\mathcal {T}_{1,q}$ via the primary reconnection. Then, the upper vortex loop has a smaller $\lambda$ than that of the lower one, so it is more stable in the further evolution as implied by the stability diagram in figure 16.

Figure 24. Evolution of the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ for vortex $\mathcal {T}_{2,q}$ with $q=1\sim 6$ and $\lambda =0.5$ at $t^*=0$, 7, 13 and 20. Surfaces are colour-coded by the helicity density.

The lower loop $\mathcal {T}_{1,q}$ with large $q$, as discussed in § 4, can reconnect again and split into $q$ small vortex loops. In figure 24 the lower loop of $\mathcal {T}_{2,4}$ after unknotting has the secondary reconnection at $t^*=20$. At the same time, the secondary reconnection of $\mathcal {T}_{2,5}$ and $\mathcal {T}_{2,6}$ have been completed, and the lower vortex loops break up into small-scale, turbulent-like structures which are then gradually dissipated. We remark that the vortex breakup is also related to the vortex tube radius at very large $Re$, as suggested in Appendix B. The lower loops tend to be more stable for thinner vortex tubes.

The vortex reconnection event coincides with the occurrence of the large helicity fluctuation and the peak of the dissipation rate (see Xiong & Yang Reference Xiong and Yang2019a). Figure 25 shows the time evolution of $H$ and $\varepsilon$ of vortex $\mathcal {T}_{2,q}$ with $q=1 \sim 6$ and $\lambda =0.5$. For $\mathcal {T}_{2,1}$, $\mathcal {T}_{2,2}$ and $\mathcal {T}_{2,3}$ with single reconnection during the evolution, $H$ is almost conserved except for the fluctuation at the reconnection time. By contrast, $H$ of $\mathcal {T}_{2,5}$ and $\mathcal {T}_{2,6}$ decays shortly after the first reconnection due to vortex breakup, and the decay rate is mitigated when $H$ is low at late times. This indicates that the helicity stored in the lower loops is reduced in the vortex breakup, while the helicity for the stable upper loop is conserved. The number and time of peaks of $\varepsilon$ indicate that the evolution of $\mathcal {T}_{2,5}$ or $\mathcal {T}_{2,6}$ has two significant reconnections, and the first reconnection time of $\mathcal {T}_{2,q}$ decreases with the increase of $q$.

Figure 25. Evolution of (a) the total helicity and (b) dissipation rate of vortex $\mathcal {T}_{2,q}$ with $q=1 \sim 6$ and $\lambda =0.5$.

5.3. Transition to turbulence for moderate torus aspect ratios

Distinct from the vortex merging for small $\lambda$ and the topological degeneration for large $\lambda$, vortex $\mathcal {T}_{2,q}$ with moderate $\lambda$ can directly break up into turbulence via the first vortex reconnection. Figure 26 plots the time evolution of vortex $\mathcal {T}_{2,3}$ with $\lambda =0.3$. In the early stage, the vortex tubes approach each other and begin to reconnect, similar to the cases for large $\lambda$. Then, the vortex knot tends to separate into two loops with similar shapes at $t^* =7.5$.

Figure 26. Evolution of the isosurface of $|\boldsymbol {\omega }|=0.04\omega _0$ for vortex $\mathcal {T}_{2,3}$ with $\lambda =0.3$ at $t^*=0$, 2.5, 5, 7.5, 10, 12.5, 15, 17.5 and 20. Surfaces are colour-coded by the helicity density. Black solid lines at $t^*=7.5$ and 10 sketch the topology of vortex core lines.

The subsequent evolution and interaction of the two coaxial vortex loops depend on their $z$-direction forward velocity. Recall that the forward velocity of an isolated vortex ring with the constant circulation and thickness is inversely proportional to $R_t$. In addition, the forward velocity also decreases with the average torus aspect ratio $\bar {\lambda }$ for the same average radius $\bar {R}$ in figure 9(c). For vortex $\mathcal {T}_{2,q}$ with large $\lambda$ (see figure 24a–c), $\bar {\lambda }$ and $\bar {R}$ of the upper loop are much smaller than those of the lower loop. This geometric difference causes a large difference in the forward velocity between the upper and lower loops, i.e. the separation of the two loops after reconnection.

In contrast, the two vortex loops with similar $\bar {\lambda }$ and $\bar {R}$ after the first reconnection of $\mathcal {T}_{p,q}$ at $t^*=5$ have comparable forward velocities, so they cannot be fully separated after reconnection. During the incomplete separation, they can even reversely evolve into the former trefoil structure at $t^*=10$ in figure 26. The large-scale oscillating motion causes the strong interaction of coaxial upper and lower loops to trigger vortex breakup and transition to turbulence. In figure 27 the energy spectrum is broadened at high wavenumber region $k\ge 100$ in the time evolution, consistent with the generation of small-scale vortical structures in figure 26, and decays at small and moderate $k$. At $t^*=20$, the scaling of $E(k)$ is close to $-5/3$.

Figure 27. Energy spectra of vortex $\mathcal {T}_{2,3}$ with $\lambda =0.3$ at $t^*=0$, 10 and 20.

Figure 28(a) plots the time evolution of positive, negative and total helicities. In general, $H$ decays, and $|H^+|$ and $|H^-|$ grow with time. For comparison, $|H^-|$ decays to zero in the vortex merging case in figure 23(a). The statistically symmetric growths of $|H^+|$ and $|H^-|$ are consistent with the emergence of small-scale vortical structures, as statistically even generations of $h^+$ and $h^-$ in isotropic turbulence.

Figure 28. Evolution of the helicity with the (a) positive–negative and (b) topological decompositions for vortex $\mathcal {T}_{2,3}$ with $\lambda =0.3$.

Figure 28(b) shows the evolution of helicity components in (4.4). Before the reconnection, $W_r$ drops and $\tilde {T}_{w}$ rises as in the coiled loop in figure 11 and the trefoil vortex knot in figures 10 and 16 in Yao et al. (Reference Yao, Yang and Hussain2021). Subsequently, the incomplete reconnection and the transition to turbulence pose a great challenge to identify the vortex core line, so that the topological decomposition of helicity in (4.4) becomes invalid after $t^*=5$. It can be expected that twisting and winding of vortex lines are further intensified in the transition, which needs a new method to characterize in future work.

5.4. Route map of topological transitions

Vortex knots and links can serve as simplified model vortices for complex flows. We demonstrate that the evolution of even the simple vortex $\mathcal {T}_{2,q}$ with different torus aspect ratios has three independent topological transition routes: merging, reconnection and transition to turbulence. The evolutionary route map of $\mathcal {T}_{2,q}$ is sketched in figure 29.

1. (i) For small $\lambda$, vortex merging occurs at very early times. This rapid process transforms a knot or link, as $\mathcal {T}_{2,q}$ with $\lambda \rightarrow 0$, into a vortex ring, and converts all the initial writhe and link helicities into the generalized twist helicity.

2. (ii) For large $\lambda$, the vortex knot or link is untied into upper and lower coiled loops $\mathcal {T}_{1,q}$ during the primary vortex reconnection, along with a partial conversion of the writhe and link helicities to generalized twist helicity. Then, the upper one with small $\lambda$ tends to be stable, while the lower one with large $\lambda \in [\lambda _c(q),1)$ in the unstable regime in figure 16 has the secondary reconnection to split into more stable vortex loops with $\lambda \in (0,\lambda _c(q),1)$. In addition, the lower loop with a very large $q$ can have transition to a turbulent-like state.

3. (iii) For moderate $\lambda$, the vortex reconnection is incomplete. The two interconnected vortex loops with similar geometries have strong interaction to trigger the transition to turbulence and make the topological decomposition of $H$ invalid.

Figure 29. Evolutionary route map of vortex $\mathcal {T}_{2,q}$.