  Advanced measurement techniques and high performance computing have made
large data sets available for a wide range of turbulent flows that arise in
engineering applications. Drawing on this abundance of data, dynamical models
can be constructed to reproduce structural and statistical features of
turbulent flows, opening the way to the design of effective model-based flow
control strategies. This review describes a framework for completing
second-order statistics of turbulent flows by models that are based on the
Navier-Stokes equations linearized around the turbulent mean velocity. Systems
theory and convex optimization are combined to address the inherent uncertainty
in the dynamics and the statistics of the flow by seeking a suitable
parsimonious correction to the prior linearized model. Specifically, dynamical
couplings between states of the linearized model dictate structural constraints
on the statistics of flow fluctuations. Thence, colored-in-time stochastic
forcing that drives the linearized model is sought to account for and reconcile
dynamics with available data (i.e., partially known second order statistics).
The number of dynamical degrees of freedom that are directly affected by
stochastic excitation is minimized as a measure of model parsimony. The
spectral content of the resulting colored-in-time stochastic contribution can
alternatively be seen to arise from a low-rank structural perturbation of the
linearized dynamical generator, pointing to suitable dynamical corrections that
may account for the absence of the nonlinear interactions in the linearized
model.
