Role of helicity for large- and small-scale turbulent fluctuations

The effects of the helicity on the dynamics of turbulent flows are investigated. The aim is to disentangle the role of helicity in fixing the direction, the intensity, and the fluctuations of the energy transfer across the inertial range of scales. We introduce an external parameter $\alpha$ that controls the mismatch between the number of positive and negative helically polarized Fourier modes. We present direct numerical simulations of Navier-Stokes equations from the fully symmetrical case, $\alpha =0$, to the fully asymmetrical case, $\alpha =1$, when only helical modes of one sign survive. We found a singular dependency of the direction of the energy cascade on $\alpha$, measuring a positive forward flux as soon as only a few modes with different helical polarities are present. Small-scale fluctuations are also strongly sensitive to the degree of mode reduction, leading to a vanishing intermittency already for values of $\alpha \sim 0.1$. If the analysis is restricted to sets of modes with the same helicity sign, intermittency is vanishing for the modes belonging to the minority set, and it is close to that measured on the original Navier-Stokes equations for the other set.

Figure 1

Evolution of energy at varying $\alpha$. All simulations reach a stationary state, except for the case at $\alpha =0.999$, where a constant increase of energy is observed, signaling the existence of a stable inverse energy transfer.

Figure 2

(a) Log-log plot of $E^{+}\left(k\right)={\sum }_{\left|\mathit{k}\right|=k}{\left|u_{\mathit{k}}^{+}\right|}^2$ vs $k$ at changing $\alpha$. (b) Log-log plot of $E^{-}\left(k\right)={\sum }_{\left|\mathit{k}\right|=k}(1-{\gamma }_{\mathit{k}}){\left|u_{\mathit{k}}^{-}\right|}^2$ vs $k$ at changing $\alpha$. Inset: Rescaled $E^{-}\left(k\right)$ with factor $(1-\alpha )$. (c) Semilog plot of flux of energy. Inset: Flux of helicity, at changing $\alpha$.

Figure 3

Isovorticity surfaces for (a) $\alpha =0$, (b) $\alpha =0.5$. The color palette is proportional to the intensity of the helicity, red for high positive values $\left(\sim {10}^3\right)$ to blue for high negative values $\left(\sim -{10}^3\right)$.

Figure 4

Top: Excess kurtosis of transverse SF for $r=\eta$ at changing $\alpha$: $(\square )$ decimation of negative helical modes only; $(◯)$ decimation of either positive or negative helical modes with $50\%$ probability; $\left(▵\right)$: a posteriori decimation of negative helical modes from a velocity field of standard nondecimated NSEs. Bottom: Logarithmic local slopes for fourth-order moment. $\alpha =0$, i.e., the full NSE $(\square )$. $\alpha =0.5$: The whole field $(◯)$ or its ${\mathit{h}}_{\mathit{k}}^{+} \left(▵\right)$ and ${\mathit{h}}_{\mathit{k}}^{-} (\diamond )$ components. The horizontal line represents the standard intermittent correction as predicted from Ref. [38].