Statistics of incompressible hydrodynamic turbulence: an alternative approach

Abstract

Using a recent alternative form of the Kolmogorov-Monin exact relation for fully developed hydrodynamics (HD) turbulence, the incompressible energy cascade rate $\epsilon$ is computed. Under this current theoretical framework, for three-dimensional (3D) freely decaying homogeneous turbulence, the statistical properties of the fluid velocity ($\mathbf{u}$), vorticity ($\mathbf{\omega }=\nabla \times \mathbf{u}$) and Lamb vector ($\mathcal{L}=\mathbf{\omega }\times \mathbf{u})$ are numerically studied. For different spatial resolutions, the numerical results show that $\epsilon$ can be obtained directly as the simple products of two-point increments of $\mathbf{u}$ and $\mathcal{L}$, 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.

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