Synchronization to Big Data: Nudging the Navier-Stokes Equations for Data Assimilation of Turbulent Flows

Nudging is an important data assimilation technique where partial field measurements are used to control the evolution of a dynamical system and/or to reconstruct the entire phase-space configuration of the supplied flow. Here, we apply it to the canonical problem of fluid dynamics: three-dimensional homogeneous and isotropic turbulence. By doing numerical experiments we perform a systematic assessment of how well the technique reconstructs large- and small-scale features of the flow with respect to the quantity and the quality or type of data supplied to it. The types of data used are (i) field values on a fixed number of spatial locations (Eulerian nudging), (ii) Fourier coefficients of the fields on a fixed range of wave numbers (Fourier nudging), or (iii) field values along a set of moving probes inside the flow (Lagrangian nudging). We present state-of-the-art quantitative measurements of the scale-by-scale transition to synchronization and a detailed discussion of the probability distribution function of the reconstruction error, by comparing the nudged field and the truth point by point. Furthermore, we show that for more complex flow configurations, like the case of anisotropic rotating turbulence, the presence of cyclonic and anticyclonic structures leads to unexpectedly better performances of the algorithm. We discuss potential further applications of nudging to a series of applied flow configurations, including the problem of field reconstruction in thermal Rayleigh-Bénard convection and in magnetohydrodynamics, and to the determination of optimal parametrization for small-scale turbulent modeling. Our study fixes the standard requirements for future applications of nudging to complex turbulent flows.

Popular Summary

When observing nature, we only have access to partial information. Truth is always filtered when measured. This is strikingly true when looking at fluid flows. From hurricanes to little whirls, motions of a wide variety of scales coexist in a flow, but in all of our pictures and measurements of the atmosphere and the oceans, all those tiny little whirls are lost between the pixels. In our work, we apply a technique called nudging to bring back to life all the motions and structures that were just too small to be caught by the cameras. We do this in the most high-dimensional and chaotic problem in fluid dynamics: homogeneous and isotropic turbulence.

Nudging involves adding a “relaxation term” to the equations of motion that penalizes the fields when they diverge from the given data. We perform synthetic experiments using direct numerical simulations where we have access to all the motions of the true field. We measure the large- and small-scale correlations in the reconstructed flow as a function of the quantity and type of data provided. We show under which circumstances we can recover motions that are not present in the supplied data.

The technique is thus able to handle high-dimensional turbulence problems and serves as a probe to identify key degrees of freedom in a flow. We discuss potential further applications of nudging to a series of applied flow configurations, including the problem of field reconstruction in thermal Rayleigh-Bénard convection and in magnetohydrodynamics as well as optimal parametrization for small-scale turbulent modeling.

Figure 1

Diagram outlining the nudging algorithm. In our numerical experiments the reference data come from a well-controlled direct numerical simulation, and the process of measuring is summarized by the filtering operation $\mathcal{I}$.

Figure 2

Top: Visualizations of (left) reference fields ${\mathit{u}}_{\mathrm{ref}}$, (middle) filtered or nudging fields $\mathcal{I}{\mathit{u}}_{\mathrm{ref}}$, and (right) nudged fields $\mathit{u}$ for Eulerian nudging. The parameters are $\alpha t_{\eta }=6.5$, $\tau /t_{\eta }=1.5$, $\varphi =0.05$. Bottom: Same as above but for Lagrangian nudging. The parameters are $\alpha t_{\eta }=6.5$, $\tau /t_{\eta }=1.5$, $\varphi =0.03$. All flows are visualized at the final time of the simulations; see Sec. 2b for more details.

Figure 3

(a) Evolution of the total energy for the reference field and for two nudged fields with different volume fraction, before and close to full synchronization, $\varphi =0.05$, 0.23. (b) Log-log plots of the energy spectra for the reference field (the truth) ${\mathit{u}}_{\mathrm{ref}}$), the nudging partial data $\mathcal{I}{\mathit{u}}_{\mathrm{ref}}$, the nudged field on the whole volume $\mathit{u}$, and the spectrum of the error field ${\mathit{u}}_{\mathrm{\Delta }}$ at the end of the simulation for the simulation with $\varphi =0.05$. (c) Same as (b) for the simulation with $\varphi =0.23$. Gray regions mark the two typical wave numbers $k_l$ and $k_r$ (see text). Note the transition to full synchronization for (c), where the spectrum of the error field ${\mathit{u}}_{\mathrm{\Delta }}$ is negligible at all scales. (d) Two-point correlation functions for the case with $\varphi =0.05$.

Figure 4

(a) Values of the velocity correlations ${\delta }_E$ (filled symbols) and the vorticity correlations ${\delta }_Z$ (empty symbols). (b) Values of the energy error $E_{\mathrm{\Delta }}/E_{\mathrm{ref}}$ (filled symbols) and the enstrophy errors $Z_{\mathrm{\Delta }}/Z_{\mathrm{ref}}$ (empty symbols). All values are plotted as a function of the volume fraction $\varphi$ and for different values of the nudging amplitude $\alpha$. Solid line in (a) represents the linear scaling expected for no effects of nudging (see text). The dashed red vertical lines mark the value ${\varphi }_c=0.2$ as the typical estimate for transition to maximum synchronization in these setups. Here and in all figures error bars are always plotted; when they are not visible it means that they are smaller than the symbol size. See Sec. 2c for the definitions of the different quantities and their errors.

Figure 5

Histograms of the point-by-point reconstructing error $|{\mathit{u}}_{\mathrm{\Delta }}|$ measured in the whole volume (top), only inside the nudging regions (bottom), and only outside (inset). The vertical dashed lines represent $⟨|{\mathit{u}}_{\mathrm{ref}}|⟩$.

Figure 6

Values of the velocity correlations ${\delta }_E$ (filled symbols) and the vorticity correlations ${\delta }_Z$ (empty symbols) as a function of the volume fraction $\varphi$ for different nudging protocols.

Figure 7

(a) Energy spectra for the nudged $\mathit{u}$, the reference ${\mathit{u}}_{\mathrm{ref}}$, and the error ${\mathit{u}}_{\mathrm{\Delta }}$ fields. The gray area indicates the nudged scales, $k<k_n$. (b) Point-by-point error between the phases of the nudged field and the reference one in the $k_z=0$ Fourier plane. (c) The same as for (b) but for the normalized amplitudes of the $z$ component. The red circle defines $k_n$. (d) Two-point correlation functions.

Figure 8

Values of the velocity correlations ${\delta }_E$ (filled symbols) and the vorticity correlations ${\delta }_Z$ (empty symbols) as a function of the maximum nudged wave number $k_n$ for different values of the nudging amplitude $\alpha$ and the interpolation time $\tau$. In (a) $\tau /t_{\eta }=0.77$ and in (b) $\alpha t_{\eta }=1.3$.

Figure 9

Values of the velocity correlations ${\delta }_E$ (filled symbols) and the vorticity correlations ${\delta }_Z$ (empty symbols) as a function of the maximum nudged wave number $k_n$ (spectral case) or the mean nudged wavelength $k_l$ (Eulerian case).

Figure 10

(a) Values of the velocity correlations ${\delta }_E$ (filled symbols) and the vorticity correlations ${\delta }_Z$ (empty symbols) as a function of the maximum nudged wave number $k_n$ for different Reynolds numbers. Red dashed line marks the $k_c\sim 0.2k_{\eta }$ value where transition to synchronization is obtained. (b) Energy spectra of the reference simulation RUN2 (round markers) and spectra of the difference field $u_{\mathrm{\Delta }}$ at different $k_n$ (triangular markers). The value of $k_n/k_{\eta }$ for each nudged simulation is annotated with arrows. Inset: Nonlinear and viscous contributions to the energy fluxes for the reference simulation RUN2.

Figure 11

Visualizations of the local energy field of (a) a reference simulation ${\mathit{u}}_{\mathrm{ref}}$, (b) a filtered field $\mathcal{I}{\mathit{u}}_{\mathrm{ref}}$, (c) and a nudged simulation $\mathit{u}$ of a rotating turbulent flow for $\varphi =0.15$, $\alpha t_{\eta }=5.0$, and $\tau /t_{\eta }=2.0$. The reference simulation has $E_{\mathrm{ref}}=1.3$, $\nu =0.002$, $N^3=256^3$, $L=2\pi$, $t_L=3.89$, $t_{\eta }=0.05$, $\mathrm{Re}=5000$, and $\mathrm{Ro}=0.06$. The forcing ${\mathit{f}}_{\mathrm{ref}}$ is a randomly generated, quenched in time, isotropic field with support on wave numbers with amplitudes $k\in [1,2]$ whose Fourier coefficients are given by ${\widehat{\mathit{f}}}_{\mathrm{ref}}(\mathit{k})=f_0{k}^{-7/2}{e}^{i{\theta }_{\mathit{k}}}$, where ${\theta }_{\mathit{k}}$ are random in $[0,2\pi )$ and $f_0=0.005$.

Figure 12

Values of the velocity correlations ${\delta }_E$ and the vorticity correlations ${\delta }_Z$ (empty symbols) as a function of the volume fraction $\varphi$ for homogeneous and isotropic turbulence (HIT) or under rotation. The HIT simulations have $\alpha t_{\eta }=6.5$ and $\tau /t_{\eta }=1.5$, and the rotating turbulence ones have $\alpha t_{\eta }=5$ and $\tau /t_{\eta }=2.0$.