The Lagrangian and Eulerian transverse velocity structure functions of fully developed fluid turbulence are found based on the inviscid Navier-Stokes equation. The structure functions are shown to obey the scaling relations $S_n^L(\tau )\propto {\tau }^{{\xi }_n}$ and $S_n^E(l)\propto l^{{\zeta }_n}$ inside the inertial range. The scaling exponents ${\zeta }_n$ and ${\xi }_n$ are calculated analytically without using dimensional considerations. This result is a significant step toward understanding the nature of hydrodynamic turbulence. The obtained values are in agreement with recent numerical and experimental data.

• Received 22 September 2009

DOI:https://doi.org/10.1103/PhysRevLett.104.154501