三叶形涡旋结对初始涡量剖面的依赖性
Dependence of trefoil vortex knots upon the initial vorticity profile
AI总结：
本文通过六组Navier-Stokes三叶形涡旋结数值模拟，揭示了初始涡量剖面对螺旋度、涡量拟能演化和有限时间耗散行为的决定性影响，并阐明了不同剖面下重联过程中负螺旋度涡量片的形成机制。
摘要：
在$(2π)^3$域中的六组Navier-Stokes三叶形涡旋结展示了初始剖面形状如何影响涡量拟能$Z$、螺旋度${\cal H}$和耗散尺度的演化。即使所有算例都具有相同的三重对称轨迹、相同的初始环量和相同的黏性$ν$范围，仍会出现显著差异。将螺旋度密度$h=u\cdotω$映射到涡量等值面块上，可以显示重联过程中$h\lesssim0$薄片形成的位置。对于Gaussian/Lamb-Oseen剖面，螺旋度${\cal H}$显著增长，仅在细辫形成并随后在重联过程中衰减时出现短暂的涡量拟能增长。其余剖面为代数型。对于未截断的代数型算例，$h<0$涡量片与$\sqrtνZ(t)$在共同时间$t_x$处的$ν$无关收敛同时形成。对于具有最宽翼部的剖面，涡量拟能在重联期间加速增长，导致近似$ν$无关的收敛有限时间耗散率$ε=νZ$。通过将预算方程中的各项映射到中心线上，阐明了发散行为的来源。Lamb-Oseen剖面有六个中心线汇聚位置，伴随局部负螺旋度耗散$ε_h<0$和较小但为正的$h$。随后，这些局部$ε_h<0$斑块的总和导致全局${\cal H}$的正增长并抑制涡量拟能的产生。对于代数型剖面：只有三个中心线汇聚位置，每个位置具有较不局域的$ε_h<0$跨度以及一些$h<0$。这些跨度可能是$\sqrtνZ(t)$阶段开始时在三叶形下半部分形成的$h<0$涡量片的种子，可以解释涡量拟能的加速增长以及有限时间能量耗散$ΔE_ε$的证据。尽管存在初始对称性。
英文摘要：
Six sets of Navier-Stokes trefoil vortex knots in $(2π)^3$ domains show how the shape of the initialprofile influences the evolution of the enstrophy $Z$, helicity ${\cal H}$ and dissipation-scale. Significant differences develop even when all have the same three-fold symmetric trajectory, the same initial circulation and the same range of the viscosities $ν$. Maps of the helicity density $h=u\cdotω$ onto vorticity isosurfaces patches show where $h\lesssim0$ sheets form during reconnection. For the Gaussian/Lamb-Oseen profile helicity ${\cal H}$ grows significantly, with only a brief spurt of enstrophy growth as thin braids form then decay during reconnection. The remaining profiles are algebraic. For the untruncated algebraic cases,$h<0$ vortex sheets form in tandem with $ν$-independent convergence of $\sqrtνZ(t)$ at a common $t_x$. For those with the broadest wings, enstrophy growth accelerates during reconnection, leading to approximately $ν$-independent convergent finite-time dissipation rates $ε=νZ$. By mapping terms from the budget equations onto centerlines, the origins of the divergent behavior are illustrated. Lamb-Oseen has six locations of centerline convergence form with local negative helicity dissipation, $ε_h<0$, and small, but positive $h$. Later, the sum of these localized patches of $ε_h<0$ leads to a positive increase in the global ${\cal H}$ and suppression of enstrophy production. For the algebraic profiles: There are only three locations of centerline convergence, each with spans of less localized $ε_h<0$ and some $h<0$. Spans that could be the seeds for the $h<0$ vortex sheets that form in the lower half of the trefoil as the $\sqrtνZ(t)$ phase begins and can explain accelerated growth of the enstrophy and evidence for finite-time energy dissipation $ΔE_ε$. Despite the initial symmetries.
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