3.1.1. Vorticity distribution

To find the position of the vortex core centre, we integrated the squared vorticity along the $y$ and $z$ directions within the area $S$ given by $x=0$ , $y\in [-\unicode{x03C0},\unicode{x03C0}]$ , $z\in [0,\unicode{x03C0}]$ :

(3.1) \begin{equation} q_{y}=\int _0^{\unicode{x03C0} } |\boldsymbol{\omega }|^{2}\,\text{d}z,\quad q_{z}=\int _{-\unicode{x03C0} }^{\unicode{x03C0} } |\boldsymbol{\omega }|^{2}\,\text{d}y, \end{equation}

where $|\boldsymbol{\omega }|^{2}=\omega ^{2}_{x}+\omega ^{2}_{y}+\omega ^{2}_{z}$ . Then we find the $y$ and $z$ coordinates where $q_y$ and $q_z$ reach their extrema, respectively. A subsequent three-point Gaussian smoothing locates the unique axial and radial positions corresponding to the maximum integrated square vorticity. This point is identified as the centre of the vortex core. Based on this position, the (azimuthal) vorticity distributions along a $z$ -directional line passing the vortex core centre at different moments are plotted in figure 7. It can be seen that compared to the main peak vorticity, secondary peaks are smoothed in a very short time. (For convenience, we refer to the peak where the maximum vorticity is located as the main peak, and the rest as the secondary peak.) Especially for cases $n=2,3,5$ , the layered structures with alternating positive and negative vorticity reorganise into a smoother distribution very soon. The larger $n$ , which means more layers, the higher the initial decay rate of the main peak vorticity. It is natural to make a relation with viscosity, as it has a tendency to uniformise the vorticity field. According to the form of the viscous term $\nu\, \boldsymbol{\nabla} ^2 \boldsymbol{u}$ in the Navier–Stokes equation, the viscous effect is related to the first-order spatial derivative of the vorticity distribution.

Figure 7.

Evolution of the distribution of the azimuthal vorticity along a $z$ -directional line passing the vortex core centre: (a) $n=1$ , (b) $n=2$ , (c) $n=3$ , (d) $n=5$ .

3.1.2. Instantaneous streamline pattern and velocity distribution

Streamlines and velocity contours are plotted in figure 8 to show the flow field evolution. From the streamline patterns, the far-field part of layered zero-flux vortex rings is similar to vortex rings with Gaussian distribution, but the near-field patterns are completely different. At the initial moment ( $t=0$ ), it can be seen that all cases have a unified structure, namely the closed streamline area at and outside the velocity concentration zone, which is the key to achieving zero flux. The zero-flux characteristic requires that on any meridian plane, the circulation is zero, so these two closed streamline areas have opposite directions. The innermost vortex ring carrying main peak vorticity has properties similar to an ordinary vortex ring with non-zero circulation, affecting the near-field fluids. When flow fields begin to evolve ( $t=2.4,4.8$ ), due to the self-induction of the innermost vortex rings, the closed streamline areas with considerably greater velocities tend to grow along the direction of vortex ring motion. The trend of growth slows down as the number of layers increases, possibly due to more and more weak influence of the innermost vortex rings, as shown by the dashed areas in figure 8.

Figure 8.

Streamline patterns on the meridian plane $x=0$ of cases $n=1,2,3,5$ at $t=0.0, 2.4, 4.8$ . Backgrounds are velocity contours. The area surrounded by the dashed black line is the maximum closed streamline area containing considerably greater velocities.

Through careful observation of figure 8, it can be reasonable to assume that velocity has only a component along the $\phi$ direction near the vorticity core centre on each meridian plane. Therefore, we may obtain an approximation of the initial velocity distribution through the vorticity distribution. Since $\omega _\theta$ is the only non-zero component of the vorticity field, we have $\omega _\theta \approx \partial (r u_\phi)/r\,\partial r$ on the region with strongest velocities on any meridian plane. Substituting (2.14) into the approximate expression of vorticity, we can obtain an equation concerning velocity,

(3.2) \begin{equation} \frac {\sin(2\unicode{x03C0} n r/r_{0})}{r}\approx \frac {1}{r}\frac {\partial (r u_\phi)}{r\,\partial r}, \end{equation}

which yields

(3.3) \begin{equation} u_\phi \approx \frac {r_{0}}{2\unicode{x03C0} n r}\left (1-\cos\frac {2\unicode{x03C0} n r}{r_{0}}\right). \end{equation}

From this equation, it can be seen that on any meridian plane the velocity distribution is also layered, which is consistent with the velocity contours in figure 8.

A two-dimensional infinite plane can be mapped into a Riemannian sphere through stereo projection, with the north pole of the sphere corresponding to infinity. The Poincaré index theorem states that the index sum of fixed points for any continuous vector field on a sphere should be $2$ . It is easy to verify that for a circular vortex filament, the cross-section streamline has two centres on the sphere after being mapped, so the index sum is $2$ . For the cross-section streamline of a vortex ring with a Gaussian distribution under periodic boundary conditions, the sum of the near-field fixed point index is also $2$ . However, because of the influence brought about by periodic boundary conditions, there are two additional saddles in each periodic area (four half-saddles on the boundary at the initial moment), so the index sum within the periodic area is $0$ . For streamline patterns of a layered zero-flux vortex ring, the topological characteristics are clearly different from ordinary vortex rings. We select the instantaneous streamline pattern of case $n=2$ at $t=4.8$ for illustration, as shown in figure 9. Without considering the impact of boundary conditions, it can be seen that the index sum of zero-flux vortex rings also satisfies the index theorem, but is more complex compared to ordinary vortex rings. Due to the coexistence of anti-sign circulation areas, there exist two saddles and four centres. Unlike the separation of wall-attached flows, there is no source or sink in incompressible vortical flows, so the separation line starts from a saddle point and terminates at another saddle point rather than a node. Although the magnitude in the outer closed streamline area is very small, as can be seen from the velocity contours, the far-field fluids still contribute to flow properties. This area exists mainly due to the layered secondary vorticity structures, as an overall feature that offsets the influence from the main peak vorticity.

Figure 9.

Instantaneous streamline pattern of case $n=2$ at $t=4.8$ on plane $x=0$ . Background is the velocity contour. The black dots represent saddles generated by periodic boundary conditions, the red dots represent original saddles of the flow field, and the yellow dots represent centres. The thick black solid line in the middle represents the separation line, and the area enclosed by the dashed line corresponds to the innermost vortex ring. Here, $\pm 1$ represents the index corresponding to the fixed point.

3.2.1. The evolution of layered vorticity structures within the vortex core

We take the meridian plane $x=0,\ z\lt 0$ for analysis. On this plane, the vorticity vector has a non-zero component only along the $x$ direction, and the main peak vorticity is positive. We consider the region $|\boldsymbol{\omega }|\gt 0.01\,|\boldsymbol{\omega }|_{{max}}$ as the effective volume of the vortex ring, where $|\boldsymbol{\omega }|_{{max}}$ is the maximum vorticity magnitude of the initial vorticity field.

Figure 10.

Vorticity contours on the meridian plane for $n=1$ at typical moments. The solid line represents the boundary of the positive vorticity area at $\omega _x=0.01\,|\boldsymbol{\omega }|_{{max}}$ , and the dashed line represents the boundary of the negative vorticity area at $\omega _x=-0.01\,|\boldsymbol{\omega }|_{{max}}$ . Times are (a) $t=0$ , (b) $t=0.8$ , (c) $t=1.6$ , (d) $t=2.4$ , (e) $t=4.0$ , (f) $t=4.8$ .

Figure 10 shows the evolution of vorticity contours when $n=1$ . At the beginning stage of evolution, the main peak vorticity undergoes a rapid decay, and the innermost vortex ring that holds it has a self-induced velocity, causing the vortex core centre to translate (figure 10 $a{-}c$ ). A shell structure is formed in the late stage of evolution, in which the innermost vortex ring is wrapped by an external negative vorticity area, as can be seen from figure 10( $d{-}\!f$ ). The external negative vorticity area is similar to a hollow vortex ring, while the internal hollow area contains an ordinary vortex ring with positive vorticity, and there is also a weak-vorticity gap between two areas, indicating that the vorticity field is very smooth. Although the motion of the internal vortex ring is weak in the axial direction due to the limitation of the external vorticity region, it evolves like an ordinary vortex ring. From figure 10( $a{-}\!f$ ), it can be seen that the whole vortex ring has a significant trend to become thicker. This is because vorticity diffusion leads to the homogenisation of the vorticity field, causing the vortex core to gradually thicken. When the boundary of the vortex core is considered as $|\boldsymbol{\omega }|=0.1\,|\boldsymbol{\omega }|_{{max}}$ , a cracking process of the shell structure can be observed, as shown in figure 11. During this process, the outer-side vorticity structure carrying opposite vorticity gradually disappears under the action of vorticity diffusion and cancellation, exposing the inner-side vortex ring.

Figure 11.

Vorticity isosurfaces $|\boldsymbol{\omega }|=0.1\,|\boldsymbol{\omega }|_{{max}}$ and vorticity contours on the meridian plane for $n=1$ at typical moments. There are two surfaces from (a) to (e). The solid line represents the boundary of the positive vorticity area at $\omega _x=0.1\,|\boldsymbol{\omega }|_{{max}}$ , and the dashed line represents the boundary of the negative vorticity area at $\omega _x=-0.15\,|\boldsymbol{\omega }|_{{max}}$ . Times are (a) $t=0.8$ , (b) $t=0.88$ , (c) $t=0.96$ , (d) $t=1.04$ , (e) $t=1.12$ , (f) $t=2.04$ .

Figure 12 shows the evolution of vorticity contours when $n=2$ . At the initial stage, the vorticity field maintains its original structure, but the peak vorticity rapidly decays (figure 12 a,b). When the vortex core centre moves, the symmetry of vorticity distribution on the meridian plane is broken. Then the presence of a larger vorticity gradient in the direction of vortex centre movement leads to stronger cancellation of positive and negative vorticity. Therefore, during the vortex centre movement, vorticity of layered structures is continuously transported to the forward direction of the vortex core to supplement the vorticity consumed. This seems to be an ‘unshell’ process of the vortex ring, as can be seen in figure 12( $c$ ). This process is driven by the starting of the innermost vortex ring carrying main peak vorticity, going layer by layer from the inside out. In addition, vorticity fusion occurs between multi-layer structures because of the insufficient vorticity cancellation (figure 12 c,d). In the late stage, vorticity cancellation causes the negative vorticity area to break apart at the place with a larger vorticity gradient. After the outermost shell disappears, only the positive vorticity area is left as a vorticity concentration area (figure 12 e,f).

Figure 12.

Vorticity contours on the meridian plane for $n=2$ at typical moments. The solid line represents the boundary of the positive vorticity area at $\omega _x=0.01\,|\boldsymbol{\omega }|_{{max}}$ , and the dashed line represents the boundary of the negative vorticity area at $\omega _x=-0.01\,|\boldsymbol{\omega }|_{{max}}$ . Times are (a) $t=0$ , (b) $t=0.40$ , (c) $t=0.84$ , (d) $t=0.96$ , (e) $t=4.40$ , (f) $t=4.80$ .

Figure 13.

Vorticity contours on the meridian plane for $n=3$ at typical moments. The solid line represents the boundary of the positive vorticity area at $\omega _x=0.01\,|\boldsymbol{\omega }|_{{max}}$ , and the dashed line represents the boundary of the negative vorticity area at $\omega _x=-0.01\,|\boldsymbol{\omega }|_{{max}}$ . Times are (a) $t=0$ , (b) $t=0.28$ , (c) $t=0.36$ , (d) $t=0.40$ , (e) $t=0.80$ , (f) $t=3.20$ .

Figure 13 shows the evolution of vorticity contours when $n=3$ . Similar to case $n=2$ , during the initial stage, peak vorticity rapidly decays with its structure maintained (figure 13 a,b). Afterwards, when the movement of the vortex core centre begins, the negative vorticity area outside the innermost vortex ring is ‘unshelled’ due to vorticity cancellation (figure 13 b,c). Then vorticity fusion occurs between the innermost and outer-side positive vorticity areas (figure 13 c,d). At the same time, it can be seen that when there is interaction between the main peak and adjacent vorticity areas, the further positive vorticity area is also cancelled (figure 13 $d$ ), ultimately forming a shell structure with the outer-side negative vorticity area surrounding the inner-side positive vorticity area (figure 13 $e$ ). The intermediate layered structures are all diffused and cancelled by the interaction during this process. After that, there is a weak vorticity gap between the outside hollow vortex ring carrying negative vorticity and the inside vortex ring carrying positive vorticity. Subsequently, the vortex ring carrying main peak vorticity continues to thicken under the action of viscosity until vorticity cancellation occurs with the outside negative vorticity area (figure 13 e,f).

Figure 14 shows the evolution of vorticity contours when $n=5$ . As the number of layers increases, the influence of main peak vorticity weakens, and the interaction among layered vorticity structures strengthens. In the initial stage, there is also a rapid decay process (figure 14 a,b), but this stage is very short compared to the cases mentioned above. The multi-layered structures have extremely large vorticity gradients, resulting in strong interactions among layered structures. Sufficient vorticity cancellation occurs in a short period of time, leaving only the structure with the outer-side negative vorticity area surrounding the inner-side positive vorticity area. This process is driven by the effect that viscosity will gradually flatten the entire vorticity field through diffusion and cancellation, which is especially significant in areas with large vorticity gradients. It can be seen that there is a large gap of weak vorticity area between the negative and positive vorticity areas after sufficient vorticity cancellation (figure 14 $c$ ). Subsequently, the vortex ring carrying main peak vorticity slowly evolves, while the outer-side hollow vortex ring gradually diffuses (figure 14 $d{-}\!f$ ).

Figure 14.

Vorticity contours on the meridian plane for $n=5$ at typical moments. The solid line represents the boundary of the positive vorticity area at $\omega _x=0.01\,|\boldsymbol{\omega }|_{{max}}$ , and the dashed line represents the boundary of the negative vorticity area at $\omega _x=-0.01\,|\boldsymbol{\omega }|_{{max}}$ . Times are (a) $t=0$ , (b) $t=0.08$ , (c) $t=0.40$ , (d) $t=1.56$ , (e) $t=1.68$ , (f) $t=4.80$ .

Overall, the evolution of layered zero-flux vortex rings has a certain regularity and can be divided mainly into four stages: peak-vorticity-weakening stage, internal-interaction stage, shell-structure stage and free-travelling stage. In the first stage, the main process is the weakening of peak vorticity under viscous effect. The second stage is the interaction among layered vorticity structures, which mainly includes two mechanisms: vorticity cancellation and fusion. When the interaction is fully completed, the vortex ring with initial layered vorticity field becomes a shell structure and enters the third stage, which is a relatively smooth process. The whole shell structure evolves until the outer-side area carrying opposite vorticity is cancelled by the inner-side vorticity area. Ultimately, the vortex ring degenerates into an ordinary vortex ring, and the subsequent evolution is the fourth stage. It is worth noting that the entire evolution process is basically confined to a limited area, and the vortex ring motion is very inconspicuous.

3.2.2. The expansion and contraction of the vortex ring volume

In order to identify the different stages of flow field evolution more clearly, we investigated the evolution of vortex ring volume that is defined in § 3.2.1 and represents the volume of concentrated vorticity. The results are shown in figure 15. The four stages mentioned in the previous subsubsection can be clearly distinguished from this figure. For case $n=1$ , due to the absence of an internal-interaction stage, the volume of the vortex ring first increases in the peak-vorticity-weakening stage, then decreases due to the cancellation of vorticity between the innermost vortex ring and the outer-side vorticity area. For cases $n=2,3,5$ , the curves exhibit a similar trend. The first stage of growth is brought by the peak-vorticity-weakening stage, then due to sufficient interaction among layered structures, vorticity cancellation leads to a drop in volume, forming a shell structure. Subsequently, the evolution of the shell structure leads to an increase in volume due to vorticity diffusion. When it reaches a certain level, the process of vortex cancellation begins again, causing the volume to decrease, and ultimately leaving only the vortex ring carrying main peak vorticity to evolve independently.

Figure 15.

The evolution of the vortex ring volume. The purple segment $\textrm {I}$ represents the peak-vorticity-weakening stage, the orange segment $\textrm {II}$ represents the internal-interaction stage, the green segment $\textrm {III}$ represents the shell-structure stage, and the blue segment $\textrm {IV}$ represents the free-travelling stage.

3.2.3. The motion of the vortex core centre

Figure 16 shows trajectories of the vortex core centre and the evolution of velocity magnitude and its components yielded from trajectories. The velocity magnitude and its components are normalised by initial velocity magnitude. The seven marked points from left to right on the curve of figure 16( $a$ ) correspond to the position of the vortex core centre at times $t=0,0.8,1.6,2.4,3.2,4.0,4.8$ , respectively. Overall, vortex rings with different $n$ all have an axial self-induced velocity, indicating that although the circulation of layered zero-flux vortex rings vanishes, they cannot remain stationary due to the small vortical impulse. Intuitively, this velocity clearly comes from the innermost vortex ring carrying main peak vorticity. With more vorticity layers inside the vortex core, the circulation of the innermost vortex ring is then smaller. As a result, the self-induced velocity exhibited by the case with larger $n$ is smaller, as can be seen in figure 16( $a$ ). For ordinary vortex rings, under the action of viscosity, they will gradually become thicker, their radius will increase, and then the self-induced velocity will decrease. But the radii of layered zero-flux vortex rings first increase and then decrease. This trend approaches completely degenerating into a decreasing trend when $n$ is larger. For example, in case $n=5$ , it is almost impossible to see an increase in radius. From figure 16( $b$ ), one can see that the velocity of the vortex core centre is mainly dominated by the axial self-induced velocity, but the impact of radial velocity cannot be neglected, which causes a visible movement of the vortex core centre in the radial direction. In the late stage of evolution, vortex core centres seem to have a steady motion, as can be seen in both figures 16( $a$ ) and 16( $b$ ).

Figure 16.

The motion of the vortex core centre, where the $*$ denotes the normalised form based on initial $|\boldsymbol{u}|$ : (a) trajectories, (b) evolution of velocity magnitude and its components.

3.3.1. Total enstrophy

Figure 17.

Evolution of (a) total enstrophy and (b) total kinetic energy. Here, $\alpha$ and $\beta$ are fitting power exponents of $W^{*}$ and $K^{*}$ in the late stage of evolution, respectively.

Figure 17( $a$ ) shows the evolution of total enstrophy. Typically, an ordinary vortex ring evolves under slow vorticity diffusion until vorticity cancellation occurs at the axis. What is different from an ordinary vortex ring is that in the evolution of a zero-flux layered vortex ring, strong internal interaction happens at the initial stage, which causes strong vorticity cancellation and diffusion, especially for larger $n$ . Therefore, the decay of total enstrophy of zero-flux vortex rings is more rapid than for ordinary vortex rings. Particularly at the internal-interaction stage, the evolution of total enstrophy shows a sharp decay compared to other stages, which can be observed as the higher decay rate of total enstrophy during $10^{-2}\lt t\lt 10^{0}$ . When the interaction between layered vorticity structures is fully completed, the decay rate of total enstrophy recovers to the former level ( $t\gt 10^{0}$ ), as can be seen in figure 17( $a$ ). In the late stage, the evolution of total enstrophy decays with an approximate power law

(3.4) \begin{align} W^{*}\sim t^{-1.8\pm 0.1}. \end{align}

For homogeneous and incompressible flows without external force, the vorticity dynamics equation becomes

(3.5) \begin{equation} \frac {d \boldsymbol{\omega }}{\text{d}t}=\boldsymbol{\omega }\boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}+\nu\, \boldsymbol{\nabla} ^{2}\boldsymbol{\omega }. \end{equation}

The dot product of the vorticity vector with the above equation yields

(3.6) \begin{equation} \frac {\text{d}}{\text{d}t}\left (\frac {|\boldsymbol{\omega }|^{2}}{2}\right)=\boldsymbol{\omega }\boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\omega }+\nu \boldsymbol{\omega }\boldsymbol{\cdot} \boldsymbol{\nabla} ^{2}\boldsymbol{\omega }. \end{equation}

Substituting $\boldsymbol{\nabla} \boldsymbol{u}=\mathbb{D}+\mathbb{A}$ and $\boldsymbol{\nabla} ^{2}\boldsymbol{\omega }=-\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \boldsymbol{\omega })$ into (3.6), where $\mathbb{D}$ is the symmetric strain rate tensor, and $\mathbb{A}$ is the anti-symmetric rotation tensor, we have

(3.7) \begin{equation} \frac {\text{d}}{\text{d}t}\left (\frac {|\boldsymbol{\omega }|^{2}}{2}\right) = \boldsymbol{\omega }\boldsymbol{\cdot} \mathbb{D}\boldsymbol{\cdot} \boldsymbol{\omega }-\nu \boldsymbol{\omega }\boldsymbol{\cdot} \boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \boldsymbol{\omega }. \end{equation}

Integrating the above equation over the periodic box, we can obtain

(3.8) \begin{equation} \frac {\text{d}W}{\text{d}t}=\frac {\text{d}}{\text{d}t}\int \frac {|\boldsymbol{\omega }|^{2}}{2}\,\text{d}V =\int \left (\lambda _1\omega _x^2+\lambda _2\omega _y^2+\lambda _3\omega _z^2\right)\,\text{d}V-\nu \int |\boldsymbol{\nabla} \times \boldsymbol{\omega }|^{2}\,\text{d}V, \end{equation}

where $\lambda _1,\lambda _2,\lambda_3$ are the three eigenvalues of $\mathbb{D}$ . From the observation of previous results, it can be inferred that the stretching and compression of the vorticity line are not strong, so we may suppose that the three primary strain rates are relatively small (i.e. $\lambda _1\to 0,$ $\lambda _2\to 0,\ \lambda _3\to 0$ ). Therefore, the time rate of change of total enstrophy can be finally approximated as

(3.9) \begin{equation} \frac {\text{d}W}{\text{d}t}\approx -\nu \int |\boldsymbol{\nabla} \times \boldsymbol{\omega }|^{2}\,\text{d}V. \end{equation}

The time rate of change of total enstrophy with different $n$ and the corresponding curve of $ -\nu \int |\boldsymbol{\nabla} \times \boldsymbol{\omega }|^{2}\,\text{d}V$ are shown in figure 18( $a$ ). The results indicate that this approximation is quite good.

Figure 18.

Evolution of (a) $\text{d}W/\text{d}t$ and $-\nu \int |\boldsymbol{\nabla} \times \boldsymbol{\omega }|^{2}\,\text{d}V$ , (b) $\text{d}K/\text{d}t$ and $-2\nu W$ , for cases $n=1,2,3,5$ .

3.3.2. Total kinetic energy

Figure 17( $b$ ) shows the evolution of total kinetic energy, defined as

(3.10) \begin{equation} K=\int \frac {|\boldsymbol{u}|^{2}}{2}\,\text{d}V. \end{equation}

Similar to total enstrophy, the decay of total kinetic energy is significantly stronger than that of ordinary vortex rings. In the late stage of evolution, it also exhibits an approximate power-law decay:

(3.11) \begin{equation} K^{*}\sim t^{-0.8\pm 0.1}. \end{equation}

The power-law decay of total kinetic energy comes from the property of total enstrophy. Integrating the momentum equation (2.1) after dot product with the velocity over the periodic box, we have

(3.12) \begin{equation} \frac {\text{d}K}{\text{d}t}=\int \frac {\text{d}}{\text{d}t}\left (\frac {|\boldsymbol{u}|^{2}}{2}\right)\text{d}V=-\nu \int \boldsymbol{u}\boldsymbol{\cdot} (\boldsymbol{\nabla} \times (\boldsymbol{\nabla} \times \boldsymbol{u}))\,\text{d}V=-2\nu W. \end{equation}

Since $\beta \approx \alpha +1$ (referring to figure 17), we again confirm the power-law decay of $W^{*}$ and $K^{*}$ based on (3.12). Also, the time rate of change of total kinetic energy and $-2\nu W$ for different $n$ are plotted in figure 18( $b$ ). It can be seen that the two match very well. The underlying mechanism for the power-law evolution of the total kinetic energy and the enstrophy is interesting but not clear yet.

3.3.3. Total circulation on the meridian plane

For circulation, which is defined as

(3.13) \begin{equation} \text{$\Gamma $}=\displaystyle \oint _C\boldsymbol{u}\boldsymbol{\cdot} \text{d}\boldsymbol{x}, \end{equation}

the evolution is obviously formulated as

(3.14) \begin{equation} \frac {\text{d} \Gamma }{\text{d}t} =\frac {\text{d}}{\text{d}t}\oint _C \boldsymbol{u}\boldsymbol{\cdot} \text{d}\boldsymbol{x}=\oint _C \boldsymbol{a}\boldsymbol{\cdot} \text{d}\boldsymbol{x}=\oint _C (-\boldsymbol{\nabla} p-\nu\, \boldsymbol{\nabla} \times \boldsymbol{\omega })\boldsymbol{\cdot} \text{d}\boldsymbol{x}=-\nu \oint _C (\boldsymbol{\nabla} \times \boldsymbol{\omega })\boldsymbol{\cdot} \text{d}\boldsymbol{x}, \end{equation}

where $\boldsymbol{a}=\text{d}\boldsymbol{u}/\text{d}t$ is the acceleration vector, and the material loop ${C}$ can be chosen arbitrarily. For our purpose, we consider a loop on the meridian plane. For such a material loop $\mathcal{C}$ , one can obtain (Wu et al. Reference Wu, Ma and Zhou2015)

(3.15) \begin{equation} \frac {\text{d} \Gamma }{\text{d}t} =-\nu \oint _{\mathcal{C}} \text{d}\boldsymbol{x}\boldsymbol{\cdot} (\boldsymbol{\nabla} \times \boldsymbol{\omega })=-\nu \oint _{\mathcal{C}} (\text{d}\boldsymbol{x}\times \boldsymbol{\nabla})\boldsymbol{\cdot} \boldsymbol{\omega }=\nu \oint _{\mathcal{C}}\frac {\partial \omega }{\partial n}\,\text{d}s, \end{equation}

where $\text{d}s=|\text{d}\boldsymbol{x}|$ , and $n$ follows the direction of the main normal of the material loop. Apparently, the whole circulation enclosing both the upper and lower meridian planes remains zero because of the periodical boundary conditions. Moreover, when the vorticity core is far from both the central symmetry axis and the boundary of the box, the total circulation along the borderline of any meridian plane also remains unchanged, because $\partial \omega /\partial n \approx 0$ in the borderline since $\omega \approx 0$ outside the vortex core. For the zero-flux vortex ring at present, this total circulation itself remains zero because it is zero at the beginning. To observe the vorticity evolution better, we can define an ‘absolute vorticity flux’ in the whole meridian plane $\mathcal{S}$ as

(3.16) \begin{equation} \gamma =\frac {\text{1}}{\text{2}}\int _{\mathcal{S}}|\omega |\,\text{d}S, \end{equation}

and its ‘normalised’ version

(3.17) \begin{equation} \gamma ^{*}= \gamma / \gamma _{\text{0}}, \end{equation}

where $ \gamma _{\text{0}}= \gamma |_{t=\text{0}}$ , and the factor $1/2$ is added to remind us that there is the same absolute flux for positive and negative vorticity in the cross-section of the vortex core. It can be seen from (3.9) that the main role of viscosity is to cause the diffusion and cancellation of vorticity, i.e. to cause a change of vorticity distribution and $\gamma$ . While it is shown in figure 19 that the total circulation of layered zero-flux vortex rings does maintain zero during the temporal horizon of the numerical simulations as we just predicted, it can be seen that in the early stage of evolution, $\gamma ^{*}$ rapidly decreases, indicating strong vorticity cancellation during this process. Afterwards, $\gamma ^{*}$ enters a stable period, during which viscous diffusion dominants. In the late stage, $\gamma ^{*}$ shows slow downward trends, indicating that the cancellation of positive and negative vorticity is still going slowly. The strong cancellation of vorticity in the early stage is due to the intense interaction between layered vorticity structures. The time at the inflection point of $\gamma ^{*}$ , whose corresponding time can be found in figure 15, exactly indicates the formation of a shell structure. Here, $|\boldsymbol{\omega }|_{{max}}^{*}$ is related to vorticity stretching and compression, and its change is influenced by both vorticity diffusion and cancellation. In figure 19, it can be seen that in the early stage, the decay rates of $|\boldsymbol{\omega }|_{{max}}^{*}$ are comparable to those of $\gamma ^{*}$ , and the whole decay process of $|\boldsymbol{\omega }|_{{max}}^{*}$ is quite smooth, reflecting that vorticity stretching and compression during the evolution process are not strong.

Figure 19.

Evolution of total circulation $\Gamma$ , total absolute vorticity flux $\gamma ^{*}$ , and peak vorticity $|\boldsymbol{\omega }|_{{max}}^{*}$ for cases $n=1,2,3,5$ .