Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence

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 zeta (n) depends on n and, except for zeta 3 ( = 1 ), needs to be determined. In this Letter, we show that applying Holder's inequality to the power-law form ( delta u ) n <overbar></mml:mover> similar to ( <mml:mfrac> r L</mml:mfrac> ) zeta n (with r / L ? 1; L is an integral length scale) leads to the following mathematical constraint: zeta 2 p = p zeta 2. When we further apply the Cauchy-Schwarz inequality, a particular case of Holder's inequality, to | ( delta u ) 3 <overbar></mml:mover> | with zeta 3 = 1, we obtain the following constraint: zeta 2 <= 2 / 3. Finally, when Holder's inequality is also applied to the power-law form ( | delta u | ) n <overbar></mml:mover> similar to ( <mml:mfrac> r L</mml:mfrac> ) zeta n (this form is often used in the extended self-similarity analysis) while assuming zeta 3 <mml:mo>= 1, it leads to zeta 2 <mml:mo>= 2 <mml:mo>/ 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 Holder's inequality. On the other hand, scaling exponents, except for zeta (3), predicted by current small-scale intermittency models do not comply with Holder'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 Holder's inequality.