Statistical Properties of Turbulence in the Presence of a Smart Small-Scale Control

By means of high-resolution numerical simulations, we compare the statistical properties of homogeneous and isotropic turbulence to those of the Navier-Stokes equation where small-scale vortex filaments are strongly depleted, thanks to a nonlinear extra viscosity acting preferentially on high vorticity regions. We show that the presence of such smart small-scale drag can strongly reduce intermittency and non-Gaussian fluctuations. Our results pave the way towards a deeper understanding on the fundamental role of degrees of freedom in turbulence as well as on the impact of (pseudo)coherent structures on the statistical small-scale properties. Our work can be seen as a first attempt to develop smart-Lagrangian forcing or drag mechanisms to control turbulence.

Figure 1

(Top row) Left: vorticity amplitude in a 2D plane, from a simulation without control term ($\beta =0$). Right: regions where the vorticity amplitude is above 20% of its maximum over the flow volume. (Bottom row) Left: visualization of enstrophy plane from a simulation with active control term. Both visualizations are obtained using the same range in the color axis. Right: regions where the vorticity amplitude is above the control forcing threshold ${\omega }_p=0.2{\omega }_{\max }$. The control forcing amplitude is $\beta =5$. Both simulations are performed with $N=1024^3$ collocation points.

Figure 2

Energy spectra averaged on time for different simulations with $N=256$ at changing the control threshold, ${\omega }_p/{\omega }_{\max }=p$, with a fixed amplitude, $\beta =5$. Notice that $\beta =0$ corresponds to the uncontrolled full NSE. Inset: energy fluxes contributions, nonlinear term (${\mathrm{\Pi }}_{nl}$), control term (${\mathrm{\Pi }}_{f_c}$), and viscous term (${\mathrm{\Pi }}_{\nu }$) normalized to the mean energy input, ${ϵ}_f$. Here $p=0.2$ and $\beta =5$.

Figure 3

Log-log plot of the flatness, $F(r)$, versus $r$ from 30 different time snapshots at resolution $N=1024^3$. Results without control terms, $\beta =0$ (NSE). Data with active control ${\omega }_p/{\omega }_{\max }=0.2$, $\beta =50$. The dashed line at $F(r)=3$ represents the expected value for a Gaussian distribution. We also show the flatness from a posteriori pruning. Errors are evaluated as the standard deviation from 30 configurations. Inset: comparison of the PDFs of enstrophy, $x=\mathrm{\Omega }$, (solid lines) and shear intensity, $x=\mathcal{S}$ (dashed lines) measured from simulations of standard NSE (black lines) and from the system controlled with forcing threshold, ${\omega }_p/{\omega }_{\max }=0.2$ and amplitude $\beta =5$ (red lines).

Figure 4

Drag coefficient for the viscous dissipation $d_{\nu }$ (green line), the control forcing $d_c$ (black line), and their sum $d_{\mathrm{tot}}$ (cyan line). Results are shown as a function of the threshold, ${\omega }_p/{\omega }_{\max }=p$, and the volume fraction, $V$, where the control forcing is acting (upper scale). Here the resolution is $N=256^3$ and the control forcing amplitude is $\beta =5$. Errors are evaluated as the standard deviation of the temporal fluctuations.