Scale locality of the energy cascade using real space quantities

The classical energy cascade in turbulence as described by Richardson and Kolmogorov is predominantly a conjecture relying on the locality of interactions between scales of turbulence. This picture is generally accepted and assumes that energy and enstrophy transfers occur between neighboring scales of turbulence and that vortex stretching plays a major role in the dynamics of this energy cascade. Direct numerical simulation data for ${\text{Re}}_{\lambda }$ ranging from 37 to 1131 is used to gather evidence for the cascade by investigating the energy and enstrophy fluxes between scales and the interplay between vorticity at one scale and strain at an adjacent scale. This is achieved by using a bandpass filter to educe the turbulent structures at various length scales, allowing one to determine the fluxes between these scales and to interrogate the role of nonlocal (in physical space) vortex stretching. It is shown that the structures of a length scale $L$ mostly transfer their energy to structures of size $0.3L$ and that most of the enstrophy flux goes from structures of scale $L$ to $0.3L$. Furthermore, vortical structures of a length scale $L_{\omega }$ are stretched mostly by straining structures of size $3L_{\omega }$ to $5L_{\omega }$, and the stretching by eddies of sizes larger than $10L_{\omega }$ is negligible. The stretching is dominated by the most extensive principal strain rate of the straining structures. These observations are found to be independent of ${\text{Re}}_{\lambda }$ for the range investigated in this study. These results provide strong evidence for the classical view of an energy cascade transferring energy from large to small scales through a hierarchy of steps, each step consisting of the stretching of vortices by somewhat larger structures.

Figure 1

Isosurfaces of enstrophy (red) and straining (green) structures with a threshold value of $\mu +2\sigma$, where $\mu$ is the mean and $\sigma$ is the rms value. Panels (a) and (b) are for ${\text{Re}}_{\lambda }=140$ and panels (c) and (d) are for ${\text{Re}}_{\lambda }=1131$. In (a) and (c) $L_s=24\eta$ and $L_{\omega }=5\eta$, in (b) $L_s=75\eta$ and $L_{\omega }=24\eta$, and in (d) $L_s=750\eta$ and $L_{\omega }=150\eta$.

Figure 2

(a) Isolated single isosurface of enstrophy (red) and dissipation (green) structures with a threshold value of $\mu +2\sigma$ extracted from the case $\text{Re}=1131$. (b) The associated mid $x\text{−}y$ plane distribution.

Figure 3

Joint pdf of planarity $\mathcal{P}$ and filamentarity $\mathcal{F}$ of the strained (or enstrophy) structures seen in Fig. 1.

Figure 4

Normalized energy transfer function, ${\widehat{\mathrm{\Pi }}}_{V,b}^{L\to S}$, for various cases.

Figure 5

Normalized enstrophy flux, ${\widehat{F}}_b^{L\to S}$, for various cases.

Figure 6

Ratio (a) ${(S/L)}_{max}^E$ and (b) ${(S/L)}_{max}^{\mathrm{\Omega }}$ yielding the maximum energy or enstrophy transfer from eddies of a scale $L$ to a scale $S$.

Figure 7

pdf of alignment between the vorticity filtered at scale $L_{\omega }=5\eta$ and the principal directions of the strain filtered at scales $L_s$. Solid line $\alpha$ and dashed line $\beta$. The gray lines are for the case ${\text{Re}}_{\lambda }=1131$ computed using unfiltered fields.

Figure 8

Variation of probability of near perfect alignment of $\mathbf{\omega }$ with $\alpha$ versus $\mathcal{L}$ with (a) $L_{\omega }=5\eta$ and (b) $L_{\omega }=45\eta$.

Figure 9

Ratio ${\mathcal{L}}^{*}=L_s/L_{\omega }$ yielding the highest probability for perfect alignment of $\mathbf{\omega }$ with the $\alpha$ strain rate (lines). The shaded region shows the minimum and maximum values of ${(L/S)}_{max}^E$ yielding the highest energy transfer for the associated ${\text{Re}}_{\lambda }$.