Anomalous Scaling in Kolmogorov--1941 Turbulence

We show that the Kolmogorov-1941 picture of fully developed hydrodynamic turbulence (with the scaling of the structure functions $S_n(R) \propto R^{n/3}$) necessarily leads to an anomalous scaling for correlation functions which include the rate of energy dissipation $\varepsilon (t,{\bf r})$, these correlation functions being described by an independent index. The mechanism for anomalous scaling, suggested on the basis of the Navier-Stokes equation, is the multi-step interaction of eddies from the inertial interval with eddies at the viscous scale via a set of eddies of intermediate scales.