  Bayesian uncertainty quantification (UQ) is of interest to industry and
academia as it provides a framework for quantifying and reducing the
uncertainty in computational models by incorporating available data. For
systems with very high computational costs, for instance, the computational
fluid dynamics (CFD) problem, the conventional, exact Bayesian approach such as
Markov chain Monte Carlo is intractable. To this end, the ensemble-based
Bayesian methods have been used for CFD applications. However, their
applicability for UQ has not been fully analyzed and understood thus far. Here,
we evaluate the performance of three widely used iterative ensemble-based data
assimilation methods, namely ensemble Kalman filter, ensemble randomized
maximum likelihood method, and ensemble Kalman filter with multiple data
assimilation for UQ problems. We present the derivations of the three ensemble
methods from an optimization viewpoint. Further, a scalar case is used to
demonstrate the performance of the three different approaches with emphasis on
the effects of small ensemble sizes. Finally, we assess the three ensemble
methods for quantifying uncertainties in steady-state CFD problems involving
turbulent mean flows. Specifically, the Reynolds averaged Navier--Stokes (RANS)
equation is considered the forward model, and the uncertainties in the
propagated velocity are quantified and reduced by incorporating observation
data. The results show that the ensemble methods cannot accurately capture the
true posterior distribution, but they can provide a good estimation of the
uncertainties even when very limited ensemble sizes are used. Based on the
overall performance and efficiency from the comparison, the ensemble randomized
maximum likelihood method is identified as the best choice of approximate
Bayesian UQ approach~among the three ensemble methods evaluated here.
