Towards a dynamical theory of multifractals in turbulence

Abstract

Making use of the exact equations for structure functions, supplemented by the equations for dissipa tive anomaly as well as an estimate for the Lagrangian acceleration of fluid particles, we obtain a main result of the multifractal theory of turbulence. The central element of the theory is a dissipation cut-off that depends on the order of the structure function. An expression obtained for the exponents $s_{n}$ in the scaling relations $\bar{(\frac{\partial u}{\partial x})^{n}}/\bar{(\frac{\partial u}{\partial x})^{2}}^{\frac{n}{2}} \propto Re^{s_{n}}$, between the velocity gradients $\frac{\partial u}{\partial x}$ and the Reynolds number $Re$, agrees well with experimental data.