  Multi-fidelity optimization methods promise a high-fidelity optimum at a cost
only slightly greater than a low-fidelity optimization. This promise is seldom
achieved in practice, due to the requirement that low- and high-fidelity models
correlate well. In this article, we propose an efficient bi-fidelity shape
optimization method for turbulent fluid-flow applications with Large-Eddy
Simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) as the high- and
low-fidelity models within a hierarchical-Kriging surrogate modelling
framework. Since the LES-RANS correlation is often poor, we use the full LES
flow-field at a single point in the design space to derive a custom-tailored
RANS closure model that reproduces the LES at that point. This is achieved with
machine-learning techniques, specifically sparse regression to obtain high
corrections of the turbulence anisotropy tensor and the production of
turbulence kinetic energy as functions of the RANS mean-flow. The LES-RANS
correlation is dramatically improved throughout the design-space. We
demonstrate the effectiveness and efficiency of our method in a
proof-of-concept shape optimization of the well-known periodic-hill case.
Standard RANS models perform poorly in this case, whereas our method converges
to the LES-optimum with only two LES samples.
