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 $\left(\mathbf{u}\right)$, vorticity $\left(\mathbf{\omega }=\mathbf{\nabla }\times \mathbf{u}\right)$, and Lamb vector $(\mathcal{L}=\mathbf{\omega }\times \mathbf{u})$ 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 $\mathbf{u}$ 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 four-thirds law for isotropic homogeneous HD turbulence.

Figure 1

Snapshot $\left(512\times 512\right)$ of the velocity (a), vorticity (b), and Lamb vector (c) modulus for run V on linear and logarithm scale, respectively.

Figure 2

For run IV: PDFs of the velocity (a), vorticity (b), and Lamb vector (c) components.

Figure 3

For run IV, PDFs of the transverse velocity (a) and Lamb (b) increments with $\ell =\eta ,\ell =5\eta ,\ell =10\eta$, and $\ell =20\eta$, where $\eta$ is the Kolmogorov dissipation scale.

Figure 4

Energy spectra for all runs in Table 1 as a function of wave number $k$.

Figure 5

For run IV: mean structure function $\langle F_{\ell }\rangle$ (black solid) and structure functions $F_{\ell }$ along different increment directions $\mathbf{\ell }$ (gray dotted) using Eq. (5). Inset plot: energy cascade rate along different directions (gray dotted) and mean cascade rate (black solid).

Figure 6

Energy cascade rates using Eq. (10) (left panel) and using Eq. (5) (right panel) as a function of $\ell$, for all runs.