  A multilevel Monte Carlo (MLMC) method for quantifying model-form
uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS)
simulations is presented. Two, high-dimensional, stochastic extensions of the
RANS equations are considered to demonstrate the applicability of the MLMC
method. The first approach is based on global perturbation of the baseline eddy
viscosity field using a lognormal random field. A more general second extension
is considered based on the work of [Xiao et al.(2017)], where the entire
Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For
two fundamental flows, we show that the MLMC method based on a hierarchy of
meshes is asymptotically faster than plain Monte Carlo. Additionally, we
demonstrate that for some flows an optimal multilevel estimator can be obtained
for which the cost scales with the same order as a single CFD solve on the
finest grid level.
