论文标题
在三维湍流中速度循环的概率分布
Probability Distribution of Velocity Circulation in Three Dimensional Turbulence
论文作者
论文摘要
我们阐述了上一篇论文\ cite {m20a}中提出的湍流统计场理论。我们澄清并简化了该理论和研究数字配置（Instanton）的数学特性的基本能量泵浦方程，这些方程确定了PDF的尾部，以最高雷诺数的最高reynolds数量，围绕大环$ c $ c $ c $ c $ c $ c $ c $。对于平面环循环问题，发现了欧拉方程中的Clebsch Instanton的明确分析解决方案。该涡度的解决方案的术语与正常方向的三角洲函数成正比，与环路界定的最小表面。研究了$δ$在整个Navier-Stokes方程中涡度中的平滑功能，并发现了平滑奇异性的指数曲线。 流通的PDF现在是一个无限的指数项的无限总和。 $ a_c $。领先的术语与调整后的$ r^2 = 0.9999 $拟合了DNS中在六个以上的数量级中发现的PDF尾巴。循环矩的比率$ m_8/m_6 $适合调整后的$ r^2 = 0.9996 $的面积依赖性。 因此，我们的理论以高度或信心解释了DNS。 对于平坦的环路，我们得出二维积分方程，以依赖于循环的比例$γ_0[c] $作为循环形状的函数（矩形环的纵横比
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
