三维Turbulence中速度环量的概率分布
Probability Distribution of Velocity Circulation in Three Dimensional Turbulence
摘要：
本文阐述了\cite{M20a}中提出的湍流统计场论。我们澄清并简化了该理论的基本能量泵送方程，并研究了奇异场配置（瞬子）的数学性质，这些性质决定了各向同性湍流中大回路$C$周围速度环流在最高雷诺数下PDF的尾部。本文给出了一个欧拉方程中的Clebsch瞬子的显式解析解。这个涡量解有一个与三角函数成比例的项，该三角函数在环所限定的最小曲面的法线方向上。本文研究了全Navier-Stokes方程中涡量的$δ$函数的光滑化，发现光滑奇异性的指数分布，环流的PDF现在是指数递减项$\EXP{- n |w|}\sqrt{\frac{n}{|w|}}$和$ w = \fracΓ{Γ_0[C]}$的无穷和，以及$ Γ_0[C] \sim \sqrt{A_C} $和最小面积$A_C$的无穷和。前导项与调整后的$R^2 = 0.9999$相吻合，即DNS中发现的PDF尾部超过六个数量级。在$100 $到$500$ Kolmogorov尺度的方环惯性范围内，环流力矩比$M_8/M_6$与调整后的$R^2=0.9996$和DNS的面积关系符合。因此，我们的理论解释DNS与高度或信心。对于一个平坦的环路，我们导出了作为环路形状（矩形环路的纵横比）的函数的环流尺度$Γ_0[C] $的依赖关系的二维积分方程
英文摘要：
We elaborate the statistical field theory of Turbulence suggested in the previous paper \cite{M20a}. We clarify and simplify the basic Energy pumping equation of that theory and study mathematical properties of singular field configuration (instanton) which determine the tails of PDF for the velocity circulation around large loop $C$ in isotropic turbulence at highest Reynolds numbers. Explicit analytic solution is found for the Clebsch instanton in an Euler equation for a planar loop circulation problem. This solution for vorticity is has a term proportional to a delta function in normal direction to the minimal surface bounded by the loop. The smoothing of $δ$ functions in the vorticity in the full Navier-Stokes equations is investigated and exponential profile of smoothed singularity is found. The PDF for circulation is now an infinite sum of decreasing exponential terms $\EXP{- n |w|}\sqrt{\frac{n}{|w|}}$, with $ w = \fracΓ{Γ_0[C]}$, and $ Γ_0[C] \sim \sqrt{A_C} $ with minimal area $A_C$. The leading term fits with adjusted $R^2 = 0.9999$ the PDF tail found in DNS over more than six orders of magnitude. The area dependence of the ratio of the circulation moments $M_8/M_6$ fits with adjusted $R^2=0.9996$ the DNS in inertial range of square loop sizes from $100 $ to $500$ Kolmogorov scales. Thus, our theory explains DNS with high degree or confidence. For a flat loop we derive two-dimensional integral equation for the dependence of the scale $Γ_0[C] $ of circulation as a function of the shape of the loop (aspect ratio for rectangular loop
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