The dominant view in the theory of fluid turbulence assumes that, once the effect of the Reynolds number is negligible, moments of order n of the longitudinal velocity increment, ($\delta u$), can be described by a simple power-law $r^{{\zeta }_n}$, where the scaling exponent ζ_n depends on n and, except for ${\zeta }_3(=1)$, needs to be determined. In this Letter, we show that applying Hölder's inequality to the power-law form $\overline{{(\delta u)}^n}\sim {(\frac{r}{L})}^{{\zeta }_n}$ (with $r/L⪡1$; L is an integral length scale) leads to the following mathematical constraint: ${\zeta }_{2p}=p{\zeta }_2$. When we further apply the Cauchy–Schwarz inequality, a particular case of Hölder's inequality, to $\left|\overline{{(\delta u)}^3}\right|$ with ${\zeta }_3=1$, we obtain the following constraint: ${\zeta }_2\le 2/3$. Finally, when Hölder's inequality is also applied to the power-law form $\overline{{(|\delta u|)}^n}\sim {(\frac{r}{L})}^{{\zeta }_n}$ (this form is often used in the extended self-similarity analysis) while assuming ${\zeta }_3=1$, it leads to ${\zeta }_2=2/3$. The present results show that the scaling exponents predicted by the 1941 theory of Kolmogorov in the limit of infinitely large Reynolds number comply with Hölder's inequality. On the other hand, scaling exponents, except for ζ₃, predicted by current small-scale intermittency models do not comply with Hölder's inequality, most probably because they were estimated in finite Reynolds number turbulence. The results reported in this Letter should guide the development of new theoretical and modeling approaches so that they are consistent with the constraints imposed by Hölder's inequality.

中文翻译：

---

流体湍流惯性范围内标度指数的数学约束

流体湍流理论中的主要观点认为，一旦雷诺数的影响可忽略不计，纵向速度增量的n阶矩即为（$\delta ü$），可以用简单的幂定律来描述 ${\mathrm{[R}}^{{\zeta }_{ñ}}$，其中标度指数ζ _ñ取决于ñ，而除${\zeta }_3（=\mathrm{1个}）$，需要确定。在这封信中，我们证明了将霍尔德不等式应用于幂律形式$\overline{{（\delta ü）}^{ñ}}〜{（\frac{\mathrm{[R}}{\mathrm{大号}}）}^{{\zeta }_{ñ}}$ （和 $\mathrm{[R}/\mathrm{大号}⪡\mathrm{1个}$; L是整数长度标尺）导致以下数学约束：${\zeta }_{\mathrm{2个}p}=p{\zeta }_{\mathrm{2个}}$。当我们进一步将Cauchy-Schwarz不等式（Hölder不等式的一个特例）应用于$\left|\overline{{（\delta ü）}^3}\right|$ 和 ${\zeta }_3=\mathrm{1个}$，我们得到以下约束： ${\zeta }_{\mathrm{2个}}\le \mathrm{2个}/3$。最后，当霍尔德不等式也适用于幂律形式时$\overline{{（\left|\delta ü\right|）}^{ñ}}〜{（\frac{\mathrm{[R}}{\mathrm{大号}}）}^{{\zeta }_{ñ}}$ （此形式通常用于扩展的自相似性分析），同时假设 ${\zeta }_3=\mathrm{1个}$，它导致 ${\zeta }_{\mathrm{2个}}=\mathrm{2个}/3$。目前的结果表明，由1941年的Kolmogorov理论预测的缩放指数在无限大的雷诺数范围内符合Hölder不等式。在另一方面，标度指数，除了ζ ₃，由目前的小规模间歇模型所预测的不符合持有人的不平等，最有可能是因为他们在有限雷诺数湍流估计。本信中报告的结果应指导新的理论和建模方法的开发，以使其与霍尔德不等式所施加的约束相一致。