AI 中文总结
本文比较了欧拉系综理论对自由衰减湍流的预测与数值模拟及实验，验证了吸引子、能量谱标度及衰减规律，并指出理论在特定条件下与实验高度一致。
AI 中文摘要
我们将自由衰减不可压缩湍流的欧拉系综解的理论预测与模拟和实验进行了比较：能量谱的标度函数、二阶结构函数的指数以及衰减规律。预测具有固定的形状；仅对其归一化和尺度进行拟合。在 Rodhiya 和 Sreenivasan 的最终 $4096^3$ 直接数值模拟中，湍流吸引子由体长增长的一条抛物线定律所选择。沿着该吸引子，塌缩的谱仅依赖于瞬时雷诺数，对于 Saffman 和 Loitsyansky 初始谱而言是相同的；外推到无限雷诺数时，在涡量范围的体部（耗散范围以下），理论与模拟在 1% 以内一致。能量平衡精确成立。理论的能量积分在红外区域发散，因此能量和涡量取决于有限流动如何截断谱：对于尖锐截断 $k_0$，只有当截断进入谱的 $k^{-7/2}$ 尾部时，衰减规律（能量 $\propto t^{-5/4}$，涡量 $\propto t^{-9/4}$）才达到。这样的截断只是一个估计；可靠的检验是谱的体部。拟合模拟数据时，截断位于体部的下边缘，在那里谱仍携带初始数据，衰减指数依赖于初始红外谱。在马克斯·普朗克风洞中，在 $Re_\lambda$ 高达 5779 时，外推到无限雷诺数的纵向指数与大分离处的奇数系综（带一个尺度参数）匹配，而标准交叉形式在那里失效。Matsuzawa 等人的自由衰减湍流团块太小、太不均匀且太弱，无法显示吸引子。该理论在本系列的前两篇论文中推导。
英文摘要
We compare the predictions of the Euler-ensemble solution of freely decaying incompressible turbulence with simulations and experiments: the scaling function of the energy spectrum, the index of the second-order structure function and the decay laws. The predictions have a fixed shape; only their normalization and scale are fitted. In the final $4096^3$ direct numerical simulations of Rodhiya and Sreenivasan, the turbulent attractor is selected by a parabola law for the growth of the bulk length. Along it the collapsed spectrum depends only on the instantaneous Reynolds number, identically for Saffman and Loitsyansky initial spectra; extrapolated to infinite Reynolds number, it agrees with the theory to 1% in the bulk of the enstrophy range, below the dissipation range. The energy balance holds exactly. The energy integral of the theory diverges in the infrared, so energy and enstrophy depend on how a finite flow cuts off the spectrum: with a sharp cutoff $k_0$, the decay laws, energy $\propto t^{-5/4}$ and enstrophy $\propto t^{-9/4}$, are reached only when the cutoff enters the $k^{-7/2}$ tail of the spectrum. Such a cutoff is only an estimate; the reliable test is the bulk of the spectrum. Fitted to the simulations, the cutoff lies at the lower edge of the bulk, where the spectrum still carries the initial data, and the decay exponents depend on the initial infrared spectrum. In the Max Planck wind tunnel at $Re_λ$ up to 5779, the longitudinal index extrapolated to infinite Reynolds number matches the odd ensemble with one scale parameter at large separations, where standard crossover forms fail. The freely decaying turbulent blob of Matsuzawa et al. is too small, too inhomogeneous and too weak to display the attractor. The theory is derived in the first two papers of this series.
Comments19 pages, 6 figures, 1 table
