  A data-driven closure modeling based on proper orthogonal decomposition (POD)
temporal modes is used to obtain stable and accurate reduced order models
(ROMs) of unsteady compressible flows. Model reduction is obtained via Galerkin
and Petrov-Galerkin projection of the non-conservative compressible
Navier-Stokes equations. The latter approach is implemented using the
least-squares Petrov-Galerkin (LSPG) technique and the present methodology
allows pre-computation of both Galerkin and LSPG coefficients. Closure is
performed by adding linear and non-linear coefficients to the original ROMs and
minimizing the error with respect to the POD temporal modes. In order to
further reduce the computational cost of the ROMs, an accelerated greedy
missing point estimation (MPE) hyper-reduction method is employed. A canonical
compressible cylinder flow is first analyzed and serves as a benchmark. The
second problem studied consists of the turbulent flow over a plunging airfoil
undergoing deep dynamic stall. For the first case, linear and non-linear
closure coefficients are both low in intrusiveness, capable of providing
results in excellent agreement with the full order model. Regularization of
calibrated models is also straightforward for this case. On the other hand, the
dynamic stall flow is significantly more challenging, specially when only
linear coefficients are used. Results show that non-linear calibration
coefficients outperform their linear counterparts when a POD basis with fewer
modes is used in the reconstruction. However, determining a correct level of
regularization is more complicated with non-linear coefficients. Hyper-reduced
models show good results when combined with non-linear calibration and an
appropriate sized POD basis.
