We investigate exact results of isotropic turbulence in three-dimensions when the pressure gradient is negligible. We derive exact two-point correlation functions of density in three-dimensions and show that the density-density correlator behaves as $ |{ x_1 - x_2}|^{ -\alpha_3}$, where $\alpha_3 = 2 + \frac{ \sqrt{ 33}}{ 6}$. It is shown that, in three-dimensions, the energy spectrum $E(k)$ in the inertial range scales with exponent $ 2 - \frac { \sqrt{ 33}}{ 12} \simeq 1.5212$. We also discuss the time scale for which our exact results are valid for strong 3D--turbulence in the presence of the pressure. We confirm our predictions by using the recent results of numerical calculations and experiment.
Categories: hep-th chao-dyn cond-mat nlin.CD