Realisability condition for the velocity structure function in the scaling range of turbulent flows

Realisability conditions based on the Cauchy-Schwarz inequality are developed for the moments of the longitudinal velocity structure function, (δu(r))ⁿ (δu(r) = u(x,t) − u(x,+r,t)), for homogeneous isotropic turbulence in the inertial range, when they are assumed to follow a power-law form (∼ r^ζn). While these conditions cannot be used to assess the validity of any phenomenology behind the development of scaling laws in the form (δu)ⁿ ∼ r^ζn, they provide a simple and objective way to assess the realisability of the exponent ζ_n. In particular, they can also be used to assess the realisability of ζ_n obtained empirically from either experimental or numerical data. Application of these realisability conditions to existing data shows that the estimated exponents ζ_n as well as multifractal model expressions of _ζn are not realisable. Interestingly, the β-model and the Kolmogorov prediction (i.e. ζ_n = n/3) are both realisable, but only the latter is Reynolds number independent. This is expected since it is valid when the Reynolds number is infinitely large.