  Estimation of the initial state of turbulent channel flow from limited data
is investigated using an adjoint-variational approach. The data are generated
from a reference direct numerical simulation (DNS) which is sub-sampled at
different spatiotemporal resolutions. When the velocity data are at 1/4096 the
spatiotemporal resolution of DNS, the correlation coefficient between the true
and adjoint-variational estimated state exceeds 99 percent. The robustness of
the algorithm to observation noise is demonstrated. In addition, the impact of
the spatiotemporal density of the data on estimation quality is evaluated, and
a resolution threshold is established for a successful reconstruction. The
critical spanwise data resolution is proportional to the Taylor microscale,
which characterizes the domain of dependence of an observation location. Due to
mean advection, either the streamwise or temporal data resolution must satisfy
a criterion based on the streamwise Taylor microscale. A second configuration
is considered where the sub-sampled data are comprised of velocities in the
outer layer and wall shear stresses only. The near-wall flow statistics and
coherent structures, although not sampled, are accurately reconstructed, which
is possible because of the coupling between the outer flow and near-wall
motions. Finally, the most challenging configuration is addressed where only
the spatiotemporally resolved wall stresses are observed. The estimation
remains accurate within the viscous sublayer and deteriorates significantly
with distance from the wall. In wall units, this trend is nearly independent of
the Reynolds number considered, and is indicative of the fundamental difficulty
of reconstructing wall-detached motions from wall data.
