A further investigation on the data assimilation-based small-scale reconstruction of turbulence

Existing works have shown that the small-scale errors of turbulence can be completely eliminated through data assimilation (DA), provided that all the large-scale Fourier modes below a critical wavenumber k(c) asymptotic to 0.2 eta(-1) are continuously enforced, where eta is the Kolmogorov length scale. Here, we further explore the DA-based small-scale reconstruction problem, for which the large-scale data are insufficient. Under such conditions, an unexpected artificial jump in the energy spectrum is observed. To alleviate this issue and improve the reconstruction accuracy, several approaches have been attempted, including ensemble averaged assimilation, temporally sparse data assimilation (TSDA), and filtering the penalty term in the assimilation. It is shown that ensemble averaging can tangibly reduce the reconstruction error, but the resulted energy spectrum is invariably lower than the true spectrum; TSDA can effectively remove the jump in the energy spectrum, but the reduction of the reconstruction error is limited. Filtering the penalty term can also rectify the energy spectrum, but it makes the reconstruction error larger. Based on these observations, we re-scale the ensemble averaged solution according to the rectified energy spectrum. Both the energy spectrum and the small-scale reconstruction accuracy have been improved by the re-scaled ensemble average method. Furthermore, we also test the current approach in the spatial nudging-based reconstruction of turbulence. Again, enhanced predictions are obtained for both the energy spectrum and the instantaneous turbulent field, invariably demonstrating the effectiveness and robustness of the proposed method.