  We investigate the applicability of the data assimilation (DA) to large eddy
simulations (LESs) based on the lattice Boltzmann method (LBM). We carry out
the observing system simulation experiment of a two-dimensional (2D) forced
isotropic turbulence, and examine the DA accuracy of the nudging and the local
ensemble transform Kalman filter (LETKF) with spatially sparse and noisy
observation data of flow fields. The advantage of the LETKF is that it does not
require computing spatial interpolation and/or an inverse problem between the
macroscopic variables (the density and the pressure) and the velocity
distribution function of the LBM, while the nudging introduces additional
models for them. The numerical experiments with $256\times256$ grids and $10\%$
observation noise in the velocity showed that the root mean square error of the
velocity in the LETKF with $8\times 8$ observation points ($\sim 0.1\%$ of the
total grids) and 64 ensemble members becomes smaller than the observation
noise, while the nudging requires an order of magnitude larger number of
observation points to achieve the same accuracy. Another advantage of the LETKF
is that it well keeps the amplitude of the energy spectrum, while only the
phase error becomes larger with more sparse observation. We also see that a
lack of observation data in the LETKF produces a spurious energy injection in
high wavenumber regimes, leading to numerical instability. Such numerical
instability is known as the catastrophic filter divergence problem, which can
be suppressed by increasing the number of ensemble members. From these results,
it was shown that the LETKF enables robust and accurate DA for the 2D LBM with
sparse and noisy observation data.
