  A quadratic approximation manifold is presented for performing nonlinear,
projection-based, model order reduction (PMOR). It constitutes a departure from
the traditional affine subspace approximation that is aimed at mitigating the
Kolmogorov barrier for nonlinear PMOR, particularly for convection-dominated
transport problems. It builds on the data-driven approach underlying the
traditional construction of projection-based reduced-order models (PROMs); is
application-independent; is linearization-free; and therefore is robust for
highly nonlinear problems. Most importantly, this approximation leads to
quadratic PROMs that deliver the same accuracy as their traditional
counterparts using however a much smaller dimension -- typically, $n_2 \sim
\sqrt n_1$, where $n_2$ and $n_1$ denote the dimensions of the quadratic and
traditional PROMs, respectively. The computational advantages of the proposed
high-order approach to nonlinear PMOR over the traditional approach are
highlighted for the detached-eddy simulation-based prediction of the Ahmed body
turbulent wake flow, which is a popular CFD benchmark problem in the automotive
industry. For a fixed accuracy level, these advantages include: a reduction of
the total offline computational cost by a factor greater than five; a reduction
of its online wall clock time by a factor greater than 32; and a reduction of
the wall clock time of the underlying high-dimensional model by a factor
greater than two orders of magnitude.
