We compare the predictions of stochastic closure theory (SCT) (Birnir in J Nonlinear Sci 23:657–688, 2013a. https://doi.org/10.1007/s00332-012-9164-z) with experimental measurements of homogeneous turbulence made in the variable density turbulence tunnel (Bodenschatz et al. in Rev Sci Instrum 85(9):093908, 2014) at the Max Planck Institute for Dynamics and Self-Organization in Göttingen. While the general form of SCT contains infinitely many free parameters, the data permit us to reduce the number to seven, only three of which are active over the entire range of Taylor–Reynolds numbers. Of these three, one parameter characterizes the variance of the mean-field noise in SCT and another characterizes the rate in the large deviations of the mean. The third parameter is the decay exponent of the Fourier variables in the Fourier expansion of the noise, which characterizes the smoothness of the turbulent velocity. SCT compares favorably with velocity structure functions measured in the experiment. We considered even-order structure functions ranging in order from two to eight as well as the third-order structure functions at five Taylor–Reynolds numbers (\(R_\lambda \)) between 110 and 1450. The comparisons highlight several advantages of the SCT, which include explicit predictions for the structure functions at any scale and for any Reynolds number. We observed that finite-\(R_\lambda \) corrections, for instance, are important even at the highest Reynolds numbers produced in the experiments. SCT gives us the correct basis function to express all the moments of the velocity differences in turbulence in Fourier space. These turn out to be powers of the sine function indexed by the wavenumbers. Here, the power of the sine function is the same as the order of the moment of the velocity differences (structure functions). The SCT produces the coefficients of the series and so determines the statistical quantities that characterize the small scales in turbulence. It also characterizes the random force acting on the fluid in the stochastic Navier–Stokes equation, as described in this paper.

中文翻译：

---

均匀湍流中结构函数的雷诺数依赖性

我们将随机封闭理论（SCT）的预测（Birnir in J Nonlinear Sci 23：657–688，2013a。https://doi.org/10.1007/s00332-012-9164-z）与均质湍流的实验测量结果进行了比较在哥廷根马克斯·普朗克动力学与自组织研究所的变密度湍流隧道中（Bodenschatz等人，Rev Sci Instrum 85（9）：093908，2014）。尽管SCT的一般形式包含无限多个自由参数，但数据使我们可以将数目减少为七个，其中只有三个在整个泰勒-雷诺数范围内有效。在这三个参数中，一个参数表示SCT中平均场噪声的方差，另一个参数表示均值的大偏差率。第三个参数是噪声的傅立叶展开中傅立叶变量的衰减指数，这代表了湍流速度的平稳性。SCT与实验中测得的速度结构函数相比具有优势。我们考虑了偶数阶结构函数，阶数范围从2到8，以及三阶阶结构函数在5个Taylor-Reynolds数（\（R_ \ lambda \））介于110和1450之间。这些比较突出显示了SCT的几个优点，其中包括针对任何尺度和任何雷诺数的结构函数的显式预测。我们观察到有限- \（R_ \ lambda \）例如，即使在实验中产生最高的雷诺数时，校正也很重要。SCT为我们提供了正确的基函数，以表达傅立叶空间中湍流速度差的所有矩。这些证明是由波数索引的正弦函数的幂。在此，正弦函数的乘方与速度差（结构函数）的阶次相同。SCT产生级数的系数，因此确定表征小尺度湍流的统计量。如本文所述，它还描述了随机Navier-Stokes方程中作用在流体上的随机力。