标题： 通过分数拉普拉斯算子的湍流建模 显示英文标题

标题： Turbulence Modeling via the Fractional Laplacian

摘要： 在此，我们推导出分数拉普拉斯算子作为一种表示湍流中平均摩擦力的方法：$ \rho \frac{D\bar{\bf u}}{Dt} = -\nabla p + \mu_\alpha \nabla^2\bar{\bf u} + \rho C_\alpha \iiint_{\!-\infty}^\infty \frac{ \bar{\bf u}{\scriptstyle(t,{\bf x}')} - \bar{\bf u}{\scriptstyle(t,{\bf x})} }{|{\bf x}'-{\bf x}|^{\alpha+3}} \,d{\bf x}' $，其中$\bar{\bf u}{\scriptstyle(t,{\bf x})}$是集合平均速度场，$\mu_\alpha$是一种增强的分子粘度，$C_\alpha$是一个湍流混合系数（单位为（长度）$^\alpha$/（时间））。 该推导基于玻尔兹曼动力学理论，该理论假设粒子速度的平衡概率分布为$f_\alpha^{eq}(t,{\bf x},{\bf u})$。 虽然历史上$f_\alpha^{eq}$被认为是麦克斯韦-玻尔兹曼分布，但我们表明该勒维$\alpha$稳定分布族的任何成员都是合适的替代方案。如果$\alpha=2$，则$f^{eq}_\alpha$是麦克斯韦-玻尔兹曼分布，大粒子速度非常不可能出现，并且恢复了纳维-斯托克斯方程（与$\mu_\alpha = \mu$和$C_\alpha = 0$相关）。 如果$0 < \alpha < 2$，那么$f^{eq}_\alpha$是一个 Lévy$\alpha$稳定分布，具有“重尾”特性，允许大的速度波动，如湍流中所示。 对于剪切湍流，选择$\alpha = 1$（$f_\alpha^{eq}$的柯西分布）会导致已知的对数速度剖面，即墙定律。 我们还给出了1D库埃特流和2D边界层流的例子，并在该动力学理论框架内讨论了湍流输运。 这项工作建立了一个新的湍流建模框架，可能有助于对湍流流动产生新的基本理解。 显示英文摘要

摘要： Herein, we derive the fractional Laplacian operator as a means to represent the mean friction force arising in a turbulent flow: $ \rho \frac{D\bar{\bf u}}{Dt} = -\nabla p + \mu_\alpha \nabla^2\bar{\bf u} + \rho C_\alpha \iiint_{\!-\infty}^\infty \frac{ \bar{\bf u}{\scriptstyle(t,{\bf x}')} - \bar{\bf u}{\scriptstyle(t,{\bf x})} }{|{\bf x}'-{\bf x}|^{\alpha+3}} \,d{\bf x}' $, where $\bar{\bf u}{\scriptstyle(t,{\bf x})}$ is the ensemble-averaged velocity field, $\mu_\alpha$ is an enhanced molecular viscosity, and $C_\alpha$ is a turbulent mixing coefficient (with units (length)$^\alpha$/(time)). The derivation is grounded in Boltzmann kinetic theory, which presumes an equilibrium probability distribution $f_\alpha^{eq}(t,{\bf x},{\bf u})$ of particle speeds. While historically $f_\alpha^{eq}$ has been assumed to be the Maxwell-Boltzmann distribution, we show that any member of the family of L\'evy $\alpha$-stable distributions is a suitable alternative. If $\alpha=2$, then $f^{eq}_\alpha$ is the Maxwell-Boltzmann distribution, with large particle speeds very unlikely, and the Navier-Stokes equations are recovered (with $\mu_\alpha = \mu$ and $C_\alpha = 0$). If $0 < \alpha < 2$, then $f^{eq}_\alpha$ is a L\'evy $\alpha$-stable distribution, with "heavy tails" that permit large velocity fluctuations, as in turbulence. For shear turbulent flows, the choice of $\alpha = 1$ (Cauchy distribution for $f_\alpha^{eq}$) leads to the logarithmic velocity profile known as the Law of the Wall. We also present examples of 1D Couette flow and 2D boundary layer flow, and we discuss turbulent transport within this kinetic theory framework. This work lays out a new framework for turbulence modeling that may lead to new fundamental understanding of turbulent flows.