Scaling of turbulent velocity structure functions: plausibility constraints

Abstract

The $n$ th-order velocity structure function $S_n$ in homogeneous isotropic turbulence is usually represented by $S_n \sim r^{\zeta _n}$ , where the spatial separation $r$ lies within the inertial range. The first prediction for $\zeta _n$ (i.e. $\zeta _3=n/3$ ) was proposed by Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941) using a dimensional argument. Subsequently, starting with Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85), models for the intermittency of the turbulent energy dissipation have predicted values of $\zeta _n$ that, except for $n=3$ , differ from $n/3$ . In order to assess differences between predictions of $\zeta _n$ , we use the Hölder inequality to derive exact relations, denoted plausibility constraints. We first derive the constraint $(p_3-p_1)\zeta _{2p_2} = (p_3 -p_2)\zeta _{2p_1} +(p_2-p_1)\zeta _{2p_3}$ between the exponents $\zeta _{2p}$ , where $p_1 \leq p_2 \leq p_3$ are any three positive numbers. It is further shown that this relation leads to $\zeta _{2p} = p \zeta _2$ . It is also shown that the relation $\zeta _n=n/3$ , which complies with $\zeta _{2p} = p \zeta _2$ , can be derived from constraints imposed on $\zeta _n$ using the Cauchy–Schwarz inequality, a special case of the Hölder inequality. These results show that while the intermittency of $\epsilon$ , which is not ignored in the present analysis, is not incompatible with the plausible relation $\zeta _n=n/3$ , the prediction $\zeta _n=n/3 +\alpha _n$ is not plausible, unless $\alpha _n =0$ .