We study the statistically steady states of the forced dissipative three-dimensional homogeneous isotropic turbulence at scales larger than the forcing scale in real separation space. The probability density functions (p.d.f.s) of longitudinal velocity difference at large separations are close to, but deviate from, Gaussian, measured by their non-zero odd parts. The analytical expressions of the third-order longitudinal structure functions derived from the Kármán–Howarth–Monin equation prove that the odd-part p.d.f.s of velocity differences at large separations are small but non-zero. Specifically, when the forcing effect in the displacement space decays exponentially as the displacement tends to infinity, the odd-order longitudinal structure functions have a power-law decay with an exponent of $-$ 2, implying a significant coupling between large and small scales. Under the assumption that forcing controls the large-scale dynamics, we propose a conjugate regime to Kolmogorov's inertial range, independent of the forcing scale, to capture the odd parts of p.d.f.s. Thus, dynamics of large scales departs from the absolute equilibrium, and we can partially recover small-scale information without explicitly resolving small-scale dynamics. The departure from the statistical equilibrium is quantified and found to be viscosity-independent. Even though this departure is small, it is significant and should be considered when studying the large scales of the forced three-dimensional homogeneous isotropic turbulence.

中文翻译：

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强迫三维均匀各向同性湍流中大尺度统计平衡的偏离

我们研究了比真实分离空间中的强迫尺度更大的强迫耗散三维均匀各向同性湍流的统计稳态。大间距处纵向速度差的概率密度函数 (pdf) 接近但偏离高斯分布，通过其非零奇数部分测量。由Kármán-Howarth-Monin方程导出的三阶纵向结构函数的解析表达式证明，大间距下速度差的奇数部分pdf很小但不为零。具体来说，当位移空间中的强迫效应随着位移趋于无穷大而呈指数衰减时，奇数阶纵向结构函数具有幂律衰减，其指数为 $-$ 2，意味着大尺度和小尺度之间存在显着的耦合。在强迫控制大尺度动力学的假设下，我们提出了一种与柯尔莫哥洛夫惯性范围共轭的体系，与强迫尺度无关，以捕获 pdf 的奇数部分。因此，大尺度动力学偏离绝对平衡，我们可以部分恢复小尺度信息，而无需明确解决小尺度动态。对统计平衡的偏离进行了量化，发现其与粘度无关。尽管这种偏差很小，但它很重要，在研究大尺度的强迫三维均匀各向同性湍流时应该考虑到这一点。