  We use approximate Bayesian computation (ABC) to estimate unknown parameter
values, as well as their uncertainties, in Reynolds-averaged Navier-Stokes
(RANS) simulations of turbulent flows. The ABC method approximates posterior
distributions of model parameters, but does not require the direct computation,
or estimation, of a likelihood function. Compared to full Bayesian analyses,
ABC thus provides a faster and more flexible parameter estimation for complex
models and a wide range of reference data. In this paper, we describe the ABC
approach, including the use of a calibration step, adaptive proposal, and
Markov chain Monte Carlo (MCMC) technique to accelerate the parameter
estimation, resulting in an improved ABC approach, denoted ABC-IMCMC. As a test
of the classic ABC rejection algorithm, we estimate parameters in a
nonequilibrium RANS model using reference data from direct numerical
simulations of periodically sheared homogeneous turbulence. We then demonstrate
the use of ABC-IMCMC to estimate parameters in the Menter
shear-stress-transport (SST) model using experimental reference data for an
axisymmetric transonic bump. We show that the accuracy of the SST model for
this test case can be improved using ABC-IMCMC, indicating that ABC-IMCMC is a
promising method for the calibration of RANS models using a wide range of
reference data.
