Triangular instability of a strained Batchelor vortex

Abstract

We investigate the triangular instability of a Batchelor vortex subjected to a stationary triangular strain field generated by three satellite vortices, in the presence of weak axial flow. The analysis combines theoretical predictions with numerical simulations. Theoretically, the instability arises from resonant coupling between two quasi-neutral Kelvin modes with azimuthal wavenumbers $m$ and $m+3$ with the background strain. Numerically, we solve the linearized Navier-Stokes equations around a quasi-steady base flow to identify the most unstable modes, and compare their growth rates and frequencies with theoretical predictions for a Reynolds number $Re = 10^4$ and a straining strength $ε= 0.008$. In the absence of axial flow, only the mode pair $(m_A, m_B) = (-1,2)$ (and its symmetric counterpart) is unstable. However, we show that additional combinations such as $(0,3)$, $(1,4)$, and $(2,5)$, which are otherwise strongly damped by the critical layer in the absence of axial flow, also become unstable once axial flow exceeds a certain threshold, as the critical layer damping is significantly reduced. Furthermore, we show that the most unstable mode in the no-axial-flow case, originating from the second branch of $m = -1$ and the first branch of $m = 2$, becomes less unstable as axial flow increases. It is eventually overtaken by a mode from the first branches of both wavenumbers, which then remains the dominant unstable mode across a wide range of axial flow strengths and Reynolds numbers. A comprehensive instability diagram as a function of the axial flow parameter is presented.

Figures (29)

• Figure 1: Flow configuration of the hub vortex and the three satellite vortices
• Figure 2: Comparison of the axial vorticity and axial velocity of the quasi-steady base flow obtained from DNS (solid black line) and theory (red dashed line) for ${Re} = 1000$ and $\epsilon = 0.008$. Panels (a) and (b) show the leading-order term and the triangular-straining correction of the axial vorticity, respectively. Panels (c) and (d) present the corresponding leading-order and correction terms for the axial velocity.
• Figure 3: Effects of axial flow on the dispersion curves (${\text{Re}}(\omega)$ vs. $k$) of Kelvin modes of the Batchelor vortex, computed at ${Re} = 10^4$ by integration along the real axis. Each point represents a mode, with its greyscale intensity corresponding to the damping rate $-{\text{Im}}(\omega)$: darker points indicate lower damping ($=0$) and lighter points higher damping ($=0.1$). Top panels show results for $W_0 = 0$ (Lamb–Oseen vortex), and bottom panels for $W_0 = -0.1$. Results are shown for (a) $m = -1$, (b) $m = 2$, (c) $m = 0$, and (d) $m = 3$.
• Figure 4: Effects of axial flow on the crossing points of the dispersion curves (${\text{Re}}(\omega)$ vs. $k$) for $m_A = -1$ and $m_B = 2$. Results are shown for (a) $W_0 = 0$, (b) $W_0 = -0.1$, and (c) $W_0 = -0.3$. Each point represents a mode, with its greyscale intensity corresponding to the damping rate $-{\text{Im}}(\omega)$: darker points indicate lower damping ($=0$) and lighter points higher damping ($=0.1$).
• Figure 5: Plots of the epicyclic frequencies $\omega^+$ and $\omega^-$ (solid lines) and the critical frequency $\omega_c$ (dashed line) as functions of the radial coordinate $r$. Results are shown for (a) $m = 2$, $kW_0 = 0$, with an inset showing a zoomed-in view of the indicated region; (b) $m = 2$, $kW_0 = -4$. The blue and red shaded areas represent the domains of regular neutral core modes and singular neutral core modes, respectively. In (b), the yellow shaded region corresponds to regular ring modes. Hatched regions denote the regions where modes are localized. The dotted region in the inset of (a) highlights where singular neutral modes encounter a critical point given by $\omega_c$.