PROBABILITY DENSITY AND SCALING EXPONENTS OF THE MOMENTS OF LONGITUDINAL VELOCITY DIFFERENCE IN STRONG TURBULENCE
PROBABILITY DENSITY AND SCALING EXPONENTS OF THE MOMENTS OF LONGITUDINAL VELOCITY DIFFERENCE IN STRONG TURBULENCE
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强湍流中纵向速度差矩的概率密度和尺度指数
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发表时间：
1997
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通讯作者：
V. Yakhot
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作者：
V. Yakhot
我们考虑了几种均匀和各向同性湍流的情况，它们的湍流产生机制不同。Navier-Stokes方程和Burgers方程中的平流项相似。提出了均匀各向同性三维湍流中的纵向结构函数{S}{n}(R)$受一维(1D)运动方程的支配，其中强烈的非局域压力贡献由伽利略不变破缺项来解释.所得到的方程不涉及从实验数据中获得的参数，给出了结构函数的标度指数和振幅，与实验数据非常吻合。导出的概率密度函数$P(保证数学{Delta}u，r)保证数学{
E}P(EnsureMath{-}Ensureath{Delta}u，r)，$BUT$P(EnsureMath{Delta}u，r)=P(EnsureMath{Delta}u，Ensureath{-}r)，$符合Navier-Stokes方程的对称性。随着位移r的减小，不能用标度不变形式表示的概率密度从大尺度上的高斯函数平滑地变化到接近指数的函数，从而显示出小尺度间歇的开始。结果表明，考虑到结构函数的次优贡献，${S}_{n}(r)ensuremath{propto}{r}^{{ensuremath{xi}}_{n}}$对于速度差矩的幅值的推导是至关重要的。
We consider a few cases of homogeneous and isotropic turbulence differing by the mechanisms of turbulence generation. The advective terms in the Navier-Stokes and Burgers equations are similar. It is proposed that the longitudinal structure functions ${S}_{n}(r)$ in homogeneous and isotropic three-dimensional turbulence are governed by a one-dimensional (1D) equation of motion, resembling the 1D Burgers equation, with the strongly nonlocal pressure contributions accounted for by Galilean invariance-breaking terms. The resulting equations, not involving parameters taken from experimental data, give both scaling exponents and amplitudes of the structure functions in an excellent agreement with experimental data. The derived probability density function $P(ensuremath{Delta}u,r)ensuremath{
e}P(ensuremath{-}ensuremath{Delta}u,r),$ but $P(ensuremath{Delta}u,r)=P(ensuremath{-}ensuremath{Delta}u,ensuremath{-}r),$ in accord with the symmetry properties of the Navier-Stokes equations. With decrease of the displacement $r,$ the probability density, which cannot be represented in a scale-invariant form, shows smooth variation from the Gaussian at the large scales to close-to-exponential function, thus demonstrating onset of small-scale intermittency. It is shown that accounting for the subdominant contributions to the structure functions ${S}_{n}(r)ensuremath{propto}{r}^{{ensuremath{xi}}_{n}}$ is crucial for a derivation of the amplitudes of the moments of the velocity difference.