Turbulence中Navier-Stokes方程的解析和数值研究
Analytic and Numerical Study of Navier-Stokes Loop Equation in Turbulence
摘要：
本文发展了非平面回路方程的解析方法，并在以前的论文\cite{M19a}，\cite{M19b}，\cite{M19c}中导出了该方程。我们在极小曲面上找到了涡量分布$Ω(r)$的二次积分方程。然而，对最小曲面没有修正：对于任意非平面环，它仍然由平均外曲率等于零定义。对Navier-Stokes方程中含有粘性项的回路方程进行了分析。该项为$Ω(r\in C)$创建边界条件。领先的粘度校正项混合的时刻$\left< Γ^p \right>$与$\left< Γ^{p-1} \right>$类似于在\cite{S19}中观察到的双分形行为，并明确打破了时间反演对称性。我们还发展了具有任意弯曲回路的回路方程的数值方法，并给出了\Mathematica笔记本建立三角化极小曲面，然后数值求解这些方程。因此，我们得到的预测，未来的数值实验，将计算涡度分布沿环路。
英文摘要：
We developed analytic approach to the non-planar loop equation, which we derived in previous papers \cite{M19a},\cite{M19b},\cite{M19c}. We found quadratic integral equation for the vorticity distribution $Ω(r)$ we introduced on a minimal surface. There are no corrections to the minimal surface though: it is still defined by mean external curvature equal to zero, for arbitrary non-planar loop. We also analyzed the loop equations with viscosity term in Navier-Stokes equations. This term creates boundary condition for $Ω(r\in C)$. The leading viscosity correction term mixes the moments $\left< Γ^p \right>$ with $\left< Γ^{p-1} \right>$ resembling the bi-fractal behavior observed in \cite{S19} and explicitly breaking the time reversal symmetry. We also develop numerical approach to the loop equation with arbitrary curved loop and present \Mathematica notebook building triangulated minimal surface and then numerically solving these equations. As a result we obtain predictions for future numerical experiments which will compute vorticity distribution along the loop.
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