Statistics of incompressible hydrodynamic turbulence: an alternative approach

Using a recent alternative form of the Kolmogorov-Monin exact relation for fully developed hydrodynamics (HD) turbulence, the incompressible energy cascade rate $\varepsilon$ is computed. Under this current theoretical framework, for three-dimensional (3D) freely decaying homogeneous turbulence, the statistical properties of the fluid velocity (${\bf u}$), vorticity ($\boldsymbol \omega= \boldsymbol\nabla \times {\bf u}$) and Lamb vector ($\boldsymbol{\cal L}= \boldsymbol \omega \times {\bf u})$ are numerically studied. For different spatial resolutions, the numerical results show that $\varepsilon$ can be obtained directly as the simple products of two-point increments of ${\bf u}$ and $\boldsymbol{\cal L}$, without the assumption of isotropy. Finally, the results for the largest spatial resolutions show a clear agreement with the cascade rates computed from the classical 4/3 law for isotropic homogeneous HD turbulence.