Temporally sparse data assimilation for the small-scale reconstruction of turbulence

TL;DR

This paper demonstrates that temporally sparse data assimilation (TSDA) can effectively reconstruct small-scale turbulence features with larger time steps than traditional methods, maintaining accuracy even with significant large-scale errors.

Contribution

The study introduces TSDA, a novel approach that relaxes the assimilation time step, enabling efficient small-scale turbulence reconstruction with less frequent data updates.

Findings

TSDA maintains accuracy with larger time steps than TCDA.

Error decay in TSDA is driven by large-scale error relaxation.

TSDA performs better with higher large-scale errors.

Abstract

Previous works have shown that the small-scale information of incompressible homogeneous isotropic turbulence (HIT) is fully recoverable as long as sufficient large-scale structures are continuously enforced through temporally continuous data assimilation (TCDA). In the current work, we show that the assimilation time step can be relaxed to values about 1 $\sim$ 2 orders larger than that for TCDA, using a temporally sparse data assimilation (TSDA) strategy, while the accuracy is still maintained or even slightly better in the presence of non-negligible large-scale errors. The one-step data assimilation (ODA) is examined to unravel the mechanism of TSDA. It is shown that the relaxation effect for errors above the assimilation wavenumber $k_a$ is responsible for the error decay in ODA. Meanwhile, The errors contained in the large scales can propagate into small scales and make the high-wavenumber ($k>k_a$) error noise decay slower with TCDA than TSDA. This mechanism is further confirmed by incorporating different levels of errors in the large scales of the reference flow field. The advantage of TSDA is found to grow with the magnitude of the incorporated errors. Thus, it is potentially more beneficial to adopt TSDA if the reference data contains non-negligible errors. Finally, an outstanding issue raised in previous works regarding the possibility of recovering the dynamics of sub-Kolmogorov scales using direct numerical simulation (DNS) data at Kolmogorov scale resolution is also discussed.