- Title
- Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence
- Creator
- Djenidi, L.; Antonia, R. A.; Tang, S. L.
- Relation
- Physics of Fluids Vol. 33, Issue 3, no. 031703
- Publisher Link
- http://dx.doi.org/10.1063/5.0039643
- Publisher
- A I P Publishing
- Resource Type
- journal article
- Date
- 2021
- Description
- 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, (𝛿𝑢), can be described by a simple power-law 𝑟𝜁𝑛, where the scaling exponent ζn depends on n and, except for 𝜁₃(=1), needs to be determined. In this Letter, we show that applying Hölder's inequality to the power-law form (𝛿𝑢)³⎯⎯⎯⎯⎯⎯⎯⎯ ∼(𝑟/𝐿) 𝜁𝑛 (with 𝑟/𝐿≪1; L is an integral length scale) leads to the following mathematical constraint: 𝜁2𝑝=𝑝𝜁₂. When we further apply the Cauchy–Schwarz inequality, a particular case of Hölder's inequality, to |(𝛿𝑢)³⎯⎯⎯⎯⎯⎯⎯⎯| with 𝜁₃=1, we obtain the following constraint: 𝜁₂≤2/3. Finally, when Hölder's inequality is also applied to the power-law form (|𝛿𝑢|)³⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ∼(𝑟/𝐿)𝜁𝑛 (this form is often used in the extended self-similarity analysis) while assuming 𝜁₃=1, it leads to 𝜁₂=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.
- Subject
- fluid turbulence; finite Reynolds number; theoreticl and modeling approaches; mathematical constraints
- Identifier
- http://hdl.handle.net/1959.13/1433620
- Identifier
- uon:39305
- Identifier
- ISSN:1070-6631
- Language
- eng
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