论文标题
涡流湍流作为可解决的弦理论
Vortex Sheet Turbulence as Solvable String Theory
论文作者
论文摘要
我们研究Navier-Stokes的稳定涡流板解决方案，该溶液在固定能流量下消失的粘度极限。 我们将其称为动荡的极限。 这些稳定的流对应于Euler Hamiltonian的最小值，这是局部速度的切线不连续性的功能，该功能参数为$δ\ vec v_t = \ vec \ vec \nablaγ$。该观察结果意味着稳定流量代表了涡流板动力学的Gibbs分布的低温极限。正常位移$Δr_\ perp $作为哈密顿坐标，$γ$作为共轭动量。 无限数量的Euler保护法会导致该系统的退化真空，这解释了湍流统计的复杂性，并提供了相关的自由度（随机表面）。在湍流极限中，Navier-Stokes方程稳定解的最简单示例是球形涡流板，我们研究了。 这个稳定的解决方案家族提供了一个典型的示例，说明了我们最近的工作中提倡的欧拉instanton，这应该是造成\ ns {}方程在动荡极限中的耗散的原因。我们进一步得出结论，人们可以通过在封闭表面内的保守体积，能量泵送速率和能量耗散的情况下添加Lagrange乘数来从涡流板的Gibbs统计数据中获取湍流统计数据。我们的gibbs分布中的有效温度为零作为$ \ mbox {re}^{ - \ frac {1} {3}}} $，reynolds编号在湍流限制中。 \ textbf {此限制中的gibbs统计信息在二维（所谓的$ c = 1 $ critical matrix模型）中降低到可解决的字符串理论}。这为涡流湍流中的非扰动计算开辟了道路，其中一些我们在此处报告。
We study steady vortex sheet solutions of the Navier-Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as $Δ\vec v_t =\vec \nabla Γ$. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics. The normal displacement $δr_\perp$ of the vortex sheet as a Hamiltonian coordinate and $Γ$ as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier-Stokes equation in the turbulent limit is a spherical vortex sheet, which we investigate. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the \NS{} equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as $\mbox{Re}^{-\frac{1}{3}}$ with Reynolds number in the turbulent limit. \textbf{The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called $c=1$ critical matrix model)}. This opens the way for non-perturbative calculations in the Vortex Sheet Turbulence, some of which we report here.
