  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.
