Small scale reconstruction in three-dimensional Kolmogorov flows using four-dimensional variational data assimilation

We apply the four dimensional variational method to reconstruct the small scales of three-dimensional turbulent velocity fields with a moderate Reynolds number, given a time sequence of measurement data on a coarse set of grid points. The results show that, reconstruction is successful when the resolution of the measurement data, given in terms of the wavenumber, is at the order of the threshold value $k_c = 0.2η_K^{-1}$ where $η_K$ is the Kolmogorov length scale of the flow. When the data are available over a period of one large eddy turn-over time scale, the filtered enstrophy and other small scale quantities are reconstructed with a $30\%$ or smaller normalized point-wise error, and a $90\%$ point-wise correlation. The spectral correlation between the reconstructed and target fields is higher than $80\%$ for all wavenumbers. Minimum volume enclosing ellipsoids (MVEEs) and MVEE trees are introduced to quantitatively compare the geometry of non-local structures. Results show that, for the majority samples, errors in the locations and the sizes of the reconstructed structures are within $15\%$, and those in the orientations is within $15^\circ$. This investigation demonstrates that satisfactory reconstruction of the scales two or more octaves smaller is possible if data at large scales are available for at least one large eddy turn-over time. In comparison, a direct substitution scheme results in three times bigger point-wise discrepancy in enstrophy. The spectral difference between the reconstructed and target velocity fields is more than ten times higher than what is obtained with the four dimensional variational method.