Energy transfer in turbulence under rotation

It is known that rapidly rotating turbulent flows are characterized by the emergence of simultaneous upscale and downscale energy transfer. Indeed, both numerics and experiments show the formation of large-scale anisotropic vortices together with the development of small-scale dissipative structures. However the organization of interactions leading to this complex dynamics remains unclear. Two different mechanisms are known to be able to transfer energy upscale in a turbulent flow. The first is characterized by two-dimensional interactions among triads lying on the two-dimensional, three-component (2D3C)/slow manifold, namely on the Fourier plane perpendicular to the rotation axis. The second mechanism is three-dimensional and consists of interactions between triads with the same sign of helicity (homochiral). Here, we present a detailed numerical study of rotating flows using a suite of high-Reynolds-number direct numerical simulations (DNS) within different parameter regimes to analyze both upscale and downscale cascade ranges. We find that the upscale cascade at wave numbers close to the forcing scale is generated by increasingly dominant homochiral interactions which couple the three-dimensional bulk and the 2D3C plane. This coupling produces an accumulation of energy in the 2D3C plane, which then transfers energy to smaller wave numbers thanks to the two-dimensional mechanism. In the forward cascade range, we find that the energy transfer is dominated by heterochiral triads and is dominated primarily by interaction within the fast manifold where $k_z\ne 0$. We further analyze the energy transfer in different regions in the real-space domain. In particular, we distinguish high-strain from high-vorticity regions and we uncover that while the mean transfer is produced inside regions of strain, the rare but extreme events of energy transfer occur primarily inside the large-scale column vortices.

Figure 1

Energy spectrum averaged in time in the stationary state for both simulations A (inverse cascade) and B (split cascade). In the insets, the energy fluxes averaged on the same time range are presented. The gray areas indicate the forced wave numbers, while the dashed vertical lines represent the cutoff scale, $k_c$, used in the analysis of the sub-grid-scales (SGS) energy transfer in both simulations (see Sec. 6).

Figure 2

Snapshot of module squared, $u^2\left(\mathit{x}\right)={\left|\mathit{u}\left(\mathit{x}\right)\right|}^2$, of a total velocity field (left) and velocity decomposed on the slow (center) and fast (right) manifolds. Data correspond to simulation A.

Figure 3

Snapshot of module squared, $u^2\left(\mathit{x}\right)={\left|\mathit{u}\left(\mathit{x}\right)\right|}^2$, of a total velocity field (left) and velocity decomposed on the slow (center) and fast (right) manifolds. Data correspond to simulation B.

Figure 4

Snapshot the vorticity along the direction of the rotation axis $z,{\omega }_z$, for the two sets of simulations, A (inverse) and B (split). The large-scale columnar structures are parallel to the direction of the external angular velocity $\mathbf{\Omega }=\mathrm{\Omega }\widehat{z}$. In particular, the cyclonic structures, columns with positive (red) vorticity, rotate in the direction of $\mathbf{\Omega }$, while the negative (blue) columns observed in dataset A (Inverse) are the anticyclonic vortices rotating in the inverse direction of $\mathbf{\Omega }$. The two vorticity fields are filtered using a sharp cutoff in Fourier space at scales $k_c=12$ and $k_c=20$ respectively for the inverse and the split cascade (see Sec. 6).

Figure 5

Spectrum decomposition into fast and slow manifolds for set A (left panel) and set B (right panel). The gray areas represent the forced wave numbers, and the dashed vertical lines represent the cutoff scale, $k_c$, used in the analysis of the SGS energy transfer in both simulations (see Sec. 6). In the insets, there are the slow spectra further decomposed into their parallel $E_{2D}\left(k\right)$ and passive, $E_{\theta }\left(k\right)$, components.

Figure 6

Total energy flux $\mathrm{\Pi }\left(k\right)$ (solid line) and fluxes decomposed on the different slow-fast interactions, namely slow-slow interactions, ${\mathrm{\Pi }}_{{}_{S\leftrightarrows S}}\left(k\right)$ (empty squares), fast-fast interactions ${\mathrm{\Pi }}_{{}_{F\leftrightarrows F}}\left(k\right)$ (empty triangles), and coupling slow-fast interactions ${\mathrm{\Pi }}_{{}_{F\leftrightarrows S}}\left(k\right)$ (empty circles). (Left panel) Data from simulation A; (right panel) data from simulation B. The dashed vertical lines represent the cutoff scale, $k_c$, used in the analysis of the SGS energy transfer (see Sec. 6), which are respectively in the inverse and in the direct cascade regimes for simulation sets A and B.

Figure 7

Energy flux due to coupling slow-fast interactions, ${\mathrm{\Pi }}_{{}_{F\leftrightarrows S}}\left(k\right)$ (empty circles) decomposed on its two different contributions, the one coming from the evolution equations of the slow modes (upward triangles) and the one from the evolution equation of the fast modes (downward triangles). (Left panel) Data from simulation A; (right panel) data from simulation B. The dashed vertical lines represent the cutoff scale, $k_c$, used in the analysis of the SGS energy transfer (see Sec. 6).

Figure 8

Total energy flux $\mathrm{\Pi }\left(k\right)$ (solid line) and fluxes decomposed into the homochiral, ${\mathrm{\Pi }}^{HO}\left(k\right)$ (upward triangles) and heterochiral ${\mathrm{\Pi }}^{HE}\left(k\right)$ (downward triangles) bases. (Left panel) Data from simulation A; (right panel) data from simulation B. The dashed vertical lines represent the cutoff scale, $k_c$, used in the analysis of the SGS energy transfer (see Sec. 6).

Figure 9

(Left panel) PDFs of SGS energy transfer for the inverse cascade simulation, decomposed on the different slow-fast interactions: total interactions ${\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}$ (solid line), slow-slow interactions ${\overline{\mathrm{\Pi }}}_{{}_{S\leftrightarrows S}}^{\mathrm{\Delta }}$ (empty squares), fast-fast interactions ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows F}}^{\mathrm{\Delta }}$ (empty triangles), and coupling slow-fast interactions ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows S}}^{\mathrm{\Delta }}$ (empty circles). (Right panel) Standardized PDF to zero mean and $\sigma =1$ for the same data. All the SGS energy transfer are measured with the cutoff at $k_c=12$, in the inverse cascade regime of simulation A. The values for the skewness $\left(S\right)$ and the flatness $\left(F\right)$ for the PDF of the total interactions $\left({\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}\right)$ are $S=\langle x^3\rangle /{\langle x^2\rangle }^{3/2}\simeq -0.42$ and $F=\langle x^4\rangle /{\langle x^2\rangle }^2\simeq 14$.

Figure 10

Visualization of the total SGS energy transfer ${\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}$, across the filter scale corresponding to $k_c=12$, and its three terms coming from the slow-fast decomposition, namely slow ${\overline{\mathrm{\Pi }}}_{{}_{S\leftrightarrows S}}^{\mathrm{\Delta }}$, fast ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows F}}^{\mathrm{\Delta }}$, and coupling ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows S}}^{\mathrm{\Delta }}$. The data correspond to simulation A with the inverse cascade.

Figure 11

(Left panel) PDFs of SGS energy transfer for the split cascade simulation, decomposed on the slow-fast different interactions: total interactions ${\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}$ (solid line), slow-slow interactions ${\overline{\mathrm{\Pi }}}_{{}_{S\leftrightarrows S}}^{\mathrm{\Delta }}$ (empty squares), fast-fast interactions ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows F}}^{\mathrm{\Delta }}$ (empty triangles), and coupling slow-fast interactions ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows S}}^{\mathrm{\Delta }}$ (empty circles). (Right panel) Standardized PDF to zero mean and $\sigma =1$ for the same data. All the SGS energy transfer are measured with the cutoff at $k_c=20$, in the direct cascade regime of simulation B. The values for the skewness $\left(S\right)$ and the flatness $\left(F\right)$ for the PDF of the total interactions $\left({\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}\right)$ are $S=\langle x^3\rangle /{\langle x^2\rangle }^{3/2}\simeq 1.5$ and $F=\langle x^4\rangle /{\langle x^2\rangle }^2\simeq 68$.

Figure 12

Visualization of the total SGS energy transfer ${\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}$, across the filter scale corresponding to $k_c=20$, and its three contributions coming from the slow-fast decomposition, namely: slow ${\overline{\mathrm{\Pi }}}_{{}_{S\leftrightarrows S}}^{\mathrm{\Delta }}$, fast ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows F}}^{\mathrm{\Delta }}$, and coupling ${\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows S}}^{\mathrm{\Delta }}$. The data correspond to simulation B with a split energy cascade.

Figure 13

Mean value of SGS energy transfer, $\langle {\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}\rangle$, conditioned on the volume's regions where $Q\le \widehat{Q}$ (solid line). The same analysis is done considering only slow-slow interactions $\langle {\overline{\mathrm{\Pi }}}_{{}_{S\leftrightarrows S}}^{\mathrm{\Delta }}\rangle$ (empty squares), fast-fast interactions $\langle {\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows F}}^{\mathrm{\Delta }}\rangle$ (empty triangles), and coupling slow-fast interactions $\langle {\overline{\mathrm{\Pi }}}_{{}_{F\leftrightarrows S}}^{\mathrm{\Delta }}\rangle$ (empty circles). (Left panel) Data from dataset A with a cutoff at $k_c=12$, and (right panel) data from dataset B with a cutoff at $k_c=20$. In both cases, the field $Q$ is measured from a filtered velocity field with energy up to $k=7$.

Figure 14

PDFs of SGS energy transfer ${\overline{\mathrm{\Pi }}}^{\mathrm{\Delta }}$ (solid line) and the same PDF measured only on regions of space dominated by vorticity, $Q>0$, (downward triangles) and on regions dominated by strain, $Q<0$, (upward triangles). (Left panel) Data from dataset A with a cutoff at $k_c=12$, and (right panel) data from dataset B with a cutoff at $k_c=20$. In both cases, the field $Q$ is measured from a filtered velocity field with energy up to $k=7$.