Sensitivity of vortex pairing and mixing to initial perturbations in stratified shear flows

The effects of different initial perturbations on the evolution of stratified shear flows that are subject to Kelvin-Helmholtz instability and vortex pairing have been investigated through direct numerical simulation. The effects of purely random perturbations of the background flow are sensitive to the phase of the subharmonic component of the perturbation that has a wavelength double that of the Kelvin-Helmholtz instability. If the phase relationship between the Kelvin-Helmholtz mode and its subharmonic mode is optimal, or close to it, then vortex pairing occurs. Vortex pairing is delayed when there is a phase difference, and this delay increases with increasing phase difference. In three-dimensional simulations vortex pairing is suppressed if the phase difference is sufficiently large, reducing the amount of mixing and mixing efficiency. For a given phase difference close enough to the optimal phase, the response of the flow to eigenfunctions perturbations is very similar to the response to random perturbations. The phase difference has a more significant effect on vortex pairing compared to the initial perturbation amplitude ratio between the KH and the subharmonic modes. In addition to traditional diagnostics, we show quantitatively that a nonmodal Fourier component in a random perturbation quickly evolves to be modal and describe the process of vortex pairing using Lagrangian trajectories.

Figure 1

Conceptual drawing of vortex pairing demonstrating the effect of phase of the subharmonic mode. (a) ${\theta }_{\text{sub}}=0,$ (b) ${\theta }_{\text{sub}}=-\frac{\pi }{2}$. The blue solid line is KH mode and the red dash-dotted line is the subharmonic mode. Blue circles denote the location of KH vortex centers and red squares are vortex centers associated with the subharmonic mode. The two arrows show the directions of the mean flow.

Figure 2

Nondimensional spanwise vorticity $(u_z-w_x)h_0/\mathrm{\Delta }U$ of two-dimensional simulations $R02D$ (phase of the subharmonic mode is ${\theta }_{\text{sub}}^M\approx 0$) and $R\frac{\pi }{2}2D$ (phase of the subharmonic mode is ${\displaystyle {\theta }_{\text{sub}}^M\approx -\frac{\pi }{2}}$) and their corresponding three-dimensional simulations $R03D$ and $R\frac{\pi }{2}3D$. The snapshots of three-dimensional simulations are plotted at $y=\frac{L_y}{2}$. Pairing is delayed in two-dimensional simulation $R\frac{\pi }{2}2D$ but completely eliminated in the three-dimensional simulation $R\frac{\pi }{2}3D$. Black stars are fluid particles located at vortex centres at $t=30$.

Figure 3

Evolution of $r$ for random perturbation simulations $R02D, R\frac{\pi }{4}2D$, and $R\frac{\pi }{2}2D$.

Figure 4

(a) Evolution of the phase of the subharmonic component. The phase before $t_M$ is shown as thin lines. The phase shift in simulation $E\frac{\pi }{2}2D$ begins after $t=150$ and is not shown in this figure. (b) $x$ coordinates of two fluid particles located at the two vortex centers at $t=30$ for random perturbation simulations and at $L_x/4$ and $3L_x/4$ for eigenfunction perturbation simulations. Pairing occurs at $t=249$ for simulation $E\frac{\pi }{2}2D$ and is not shown in this figure. (c) Growth rate of the subharmonic mode. The vertical dashed line indicates the saturation time of KH instabilities in simulation $R\frac{\pi }{4}2D$. The stars labeled as TG indicate the growth rate calculated using the TG equation with the time-dependent mean flow.

Figure 5

The trajectories of two fluid particles between $t=45$ and $t=135$. The stars indicate $t_{\text{kh}}$ and the triangles indicate $t_{\text{sub}}$.

Figure 6

Saturation time of the subharmonic mode, $t_{\text{sub}}$, as a function of the phase ${\theta }_{\text{sub}}^M$ for two-dimensional simulations perturbed by eigenfunctions and random perturbations. Note $t_{\text{sub}}$ is approximately equal to the time of pairing (see Table 2).

Figure 7

Kinetic energy of the subharmonic mode $K_{\text{sub}}$ and three-dimensional kinetic energy $K_{3d}$ in two-dimensional and three-dimensional simulations: (a) ${\theta }_{\text{sub}}^M\approx 0$, (b) ${\theta }_{\text{sub}}^M\approx -\frac{\pi }{2}$.

Figure 8

(a) The increase in total potential energy caused by fluid's motion $\mathrm{\Delta }P-D$, (b) the increase in mixing $M$.

Figure 9

The dependence of (a) the final amount of mixing $M$ and (b) the cumulative mixing efficiency $E_c$ on the phase difference between the primary KH and the subharmonic component, ${\theta }_{\text{sub}}^M$. In panel (b) the solid line shows $E_c^{t_{3d}-t_f}$, the cumulative mixing efficiency computed from $t_{3d}$ to $t_f$, and the dashed line delineates the overall cumulative mixing efficiency, $E_c^{t_0-t_f}$, the cumulative mixing efficiency computed from $t=0$ to $t_f$.

Figure 10

Effects of the initial perturbation amplitude on the emergence of pairing for the simulation $E02D$. The kinetic energy of the KH and the subharmonic mode for the initial ratios of (a) ${(K_{\text{sub}}/K_{\text{kh}})}_{t_0}=1.0$, (b) ${(K_{\text{sub}}/K_{\text{kh}})}_{t_0}=0.05$, and (c) ${(K_{\text{sub}}/K_{\text{kh}})}_{t_0}=30$. The dashed lines show the linear growth predictions from the Taylor Goldstein equations. Panel (d) shows the $x$ coordinate of two fluid particles initially located at $L_x/4$ and $3L_x/4$.