Bursting on a vortex tube with initial axial core-size perturbations

We simulate and analyze the evolution of a rectilinear vortex tube with initial axial core-size perturbations at circulation-based Reynolds number of 5000. The initial variations in the core size are associated with axial gradients in the azimuthal velocity, which generates azimuthal vorticity. This azimuthal vorticity propagates as twist waves in the axial direction. Varying the initial core-size ratio $A$ shows that the propagation speed of the twist waves varies linearly with $A$ and approaches linear stability results of the long-wave limit of Kelvin waves on rectilinear vortex tubes as $A\to 1$. The simulations show that when two twist waves of opposite handedness meet the core expands radially, forming a pair of local ringlike structures with opposite-signed azimuthal vorticity through a process termed vortex bursting. An analysis of the vorticity dynamics during bursting reveals that initially the flow behaves qualitatively like a head-on collision of two isolated vortex rings, with the azimuthal vorticity dynamics driving radial growth. During bursting, however, the localized radial expansion of the core is also accompanied by an increase in the radial vorticity component, which ultimately arrests the bursting and reverses the sign of the azimuthal vorticity. Through long-time simulations of the periodic tube, we demonstrate that after the primary bursting event the twist waves reverse their direction and interact again, leading to further bursting events. The evolution of the perturbed tubes is then accompanied by sustained elevated enstrophy levels and thus accelerated energy decay as compared to undisturbed Lamb-Oseen vortices of identical initial circulation and energy. Overall, this work provides the first detailed qualitative and quantitative insights into the mechanisms and evolution of vortex bursting on rectilinear vortex tubes. To further assess the relevance and prevalence of bursting in practical settings, subsequent investigations in the stability and sensitivity of our results to varying Reynolds number, nonrectilinear vortex center lines, and external strain fields are needed.

Figure 1

Sketch of the initial vortex tube within the extent of the computational domain.

Figure 2

Visualization of vortex lines colored by azimuthal vorticity ${\omega }_{\theta }$ at different times for the case $A=3.0$. In all cases, the vortex lines are seeded from a disk at the left end of the domain ($z/{\sigma }_0=-10$) and their geometry is computed from the instantaneous vorticity vector field.

Figure 3

Twist density $\tau =\frac{{\omega }_{\theta }}{r{\omega }_z}$ contours in the $(r,z)$ plane for the case $A=3.0$, showing only the region $z<0$. Red and blue colors correspond to positive and negative values of $\tau$, respectively. We set the twist field manually to zero when $|\mathit{\omega }/({\mathrm{\Gamma }}_0/{\sigma }_0^2\left)\right|<0.0026$, to prevent spurious contours arising from the numerical division of two small numbers.

Figure 4

(a) Radial profile of twist density at different times at $z/{\sigma }_0=-2.5$. (b) The distribution of twist density $\tau$ at center line $r=0$ at different times for the case $A=3.0$. The dark gray dashed lines denote the positions of the maximum value of $\tau$, defined as the wave fronts. (c) The wave speed calculated by tracking the position of the peak value of $\tau$ along the center line, as a function of the initial core-size ratio $A$.

Figure 5

Bursting process for the case $A=3.0$, visualized within a subdomain centered around the bursting region. (a)–(c) Vortex lines seeded from selected positions and colored by azimuthal vorticity at different stages of bursting. Red and blue colors indicate positive and negative values, respectively. (d)–(f) A reduced version of figures (a)– (c), where the bulk of the vortex lines are visible in the background but only a select set of vortex lines, seeded from a rake at $z=0$, are fully rendered. The seeds for the latter lines are shown by green dots.

Figure 6

Visualization of the vorticity field in the $(r,z)$ plane for $A=3.0$ at four different times during bursting. Each subplot shows on the left a filled contour plot of the azimuthal vorticity field in a closeup of the $(r,z)$ plane around $z=0$, overlaid with black contours of constant $r{u}_{\theta }$ surfaces (vortex surfaces), and on the right the radial profiles at $z=0$ of nondimensional axial vorticity ${\omega }_z/({\mathrm{\Gamma }}_0/{\sigma }_0^2)$ (blue line), radial velocity $u_r/({\mathrm{\Gamma }}_0/{\sigma }_0)$ (orange line), and axial velocity gradient $\frac{\partial u_z}{\partial z}/({\mathrm{\Gamma }}_0/{\sigma }_0^2)$ (green line). In each subplot the vertical radial axes are aligned and the dashed blue line indicates the radial location of the peak value of ${\omega }_z$ at $z=0$. In the line plot, the horizontal axis shows the values of the nondimensional quantities on the center plane.

Figure 7

Contour plots of the different terms in the evolution equation for ${\omega }_{\theta }$ for the case $A=3.0$, overlaid with black contours of constant $r{u}_{\theta }$ surfaces (vortex surfaces). From left to right, we show ${\omega }_{\theta }$, its total time derivative $D{\omega }_{\theta }/Dt$, and the two components constituting the total time derivative, namely $-2\frac{u_{\theta }{\omega }_r}{r}$ and $\frac{u_r{\omega }_{\theta }}{r}$, all in nondimensional form. From top to bottom, we show the fields at three different times during bursting. The color maps for the last three columns are the same.

Figure 8

Velocity field in the $(r,z)$ plane ($r>0$) illustrated using line-integral convolution [44], overlaid with colors showing ${\omega }_{\theta }$ at three different times during the late stage of bursting for $A=3.0$. Red and blue indicate positive and negative values of ${\omega }_{\theta }$, respectively. The black arrows are annotated by hand to indicate the direction of the velocity field.

Figure 9

Visualization of the azimuthal vorticity field in the $(r,z)$ plane, overlaid with black contours of constant $r{u}_{\theta }$ surfaces (vortex surfaces). The top row shows three different times for $A=4.333$ [(a)–(c)], whereas the bottom two rows show six different times for $A=5.4$ [(d)–(i)].

Figure 10

(a) Time evolution of the first radial moment of ${\omega }_z$ at $z=0$ ($I$) rescaled by its initial value for different core-size ratios $A$. The value of $A$ increases from 1.286 with lightest blue to 5.4 with darkest blue and the legend is shown in (b). The dashed line denotes the value for $A=1$, i.e., the unperturbed case. (b) The rescaled evolution of $I$ for different values of $A$. (c) The values of $I$ at $t_{\text{start}}$, $t_{\text{start}}$, and their difference rescaled by ${\sigma }_{\text{min}}$ plotted against initial core-size ratio $A$. (d) The times associated with the start of bursting, the peak of bursting, and the bursting duration plotted against initial core-size ratio $A$.

Figure 11

(a)–(c) 3D volume rendering of the azimuthal vorticity field at selected times during bursting for $A=3.0$ (a), $A=4.333$ (b), and $A=5.4$ (c). The right subfigure of (c) shows the azimuthal vorticity field in the $(r,z)$ plane for $A=5.4$, with the green dashed circles indicating the side structures where instability develops. The bottom row (d) shows a similar visualization of the 3D azimuthal vorticity field for the case $A=4.333$ with nonaxisymmetric noise added to the initial condition.

Figure 12

Plots of radially averaged enstrophy density $\overline{\mathit{\omega }·\mathit{\omega }}=\frac{\int \mathit{\omega }·\mathit{\omega }rdr}{\int rdr}$ as a function of $z/{\sigma }_0$ (horizontal axis) and nondimensional time (vertical axis) for $A=3.0$ (a) and $A=4.333$ (b). The inset figures show 3D azimuthal vorticity field at $t^{*}=160$ for $A=3.0$ and for $A=4.333$.

Figure 13

(a) Time evolution of global enstrophy ($\varepsilon =\int \mathit{\omega }·\mathit{\omega }dV$) for different core-size ratios $A$. The gray dots denote the value of enstrophy at $t_{\text{start}}^{*}$ and the gray triangles denote the value of enstrophy at $t_{max}^{*}$, as defined in Sec. 4c2. (b) The evolution of global energy $E=1/2\int \mathit{\psi }·\mathit{\omega }dV$ for different core-size ratios $A$, where $\mathit{\psi }$ is the stream function. The dashed line represents the evolution of energy for an unperturbed Lamb-Oseen vortex tube whose initial core size is chosen to match the initial total energy for each value of $A$. (c) The time it takes for energy to decay to 95% of its initial value as a function of $A$ for our numerically simulated vortices (solid blue line) compared with Lamb-Oseen vortices with the same initial energy of our simulated vortices at each $A$ (dashed orange line).