  Aerodynamic optimization is ubiquitous in the design of most engineering
systems interacting with fluids. A common approach is to optimize a performance
function defined by a choice of an aerodynamic model, e.g., turbulence RANS
model, and at nominal operating conditions. Practical experience indicates that
such a deterministic approach may result in considerably sub-optimal designs
when the adopted aerodynamic model does not lead to accurate flow predictions
or when the actual operating conditions differ from those considered in the
design. One approach to address this shortcoming is to consider an average or
robust design, wherein the statistical moments of the performance function,
given the uncertainty in the operating conditions and the aerodynamic model, is
optimized. However, when the number of uncertain inputs is large or the
performance function exhibits significant variability, an accurate evaluation
of these moments may require a large number of forward and/or adjoint solves,
at each iteration of a gradient-based scheme. This, in turn, renders the design
computationally expensive, if not infeasible. To tackle this difficulty, we
consider a variant of the stochastic gradient descent method where, in each
optimization iteration, a stochastic approximation of the objective,
constraints, and their gradients are generated. This is done via a small number
of forward/adjoint solves corresponding to random selections of the uncertain
parameters and aerodynamic model. The methodology is applied to the robust
optimization of the standard NACA-0012 subject to parametric and turbulence
model uncertainty. With a cost that is a small factor larger than that of the
deterministic approach, the stochastic gradient approach significantly improves
the performance (mean and variance) of the aerodynamic design for a wide range
of operating conditions and turbulence models.
