  A deep learning (DL) closure model for large-eddy simulation (LES) is
developed and evaluated for incompressible flows around a rectangular cylinder
at moderate Reynolds numbers. Near-wall flow simulation remains a central
challenge in aerodynamic modeling: RANS predictions of separated flows are
often inaccurate, while LES can require prohibitively small near-wall mesh
sizes. The DL-LES model is trained using adjoint PDE optimization methods to
match, as closely as possible, direct numerical simulation (DNS) data. It is
then evaluated out-of-sample (i.e., for new aspect ratios and Reynolds numbers
not included in the training data) and compared against a standard LES model
(the dynamic Smagorinsky model). The DL-LES model outperforms dynamic
Smagorinsky and is able to achieve accurate LES predictions on a relatively
coarse mesh (downsampled from the DNS grid by a factor of four in each
Cartesian direction). We study the accuracy of the DL-LES model for predicting
the drag coefficient, mean flow, and Reynolds stress. A crucial challenge is
that the LES quantities of interest are the steady-state flow statistics; for
example, the time-averaged mean velocity $\bar{u}(x) = \displaystyle \lim_{t
\rightarrow \infty} \frac{1}{t} \int_0^t u(s,x) ds$. Calculating the
steady-state flow statistics therefore requires simulating the DL-LES equations
over a large number of flow times through the domain; it is a non-trivial
question whether an unsteady partial differential equation model whose
functional form is defined by a deep neural network can remain stable and
accurate on $t \in [0, \infty)$. Our results demonstrate that the DL-LES model
is accurate and stable over large physical time spans, enabling the estimation
of the steady-state statistics for the velocity, fluctuations, and drag
coefficient of turbulent flows around bluff bodies relevant to aerodynamic
applications.
