Bayesian Parameter Estimation of a k-ε Model for Accurate Jet-in-Crossflow Simulations

Abstract

Reynolds-Averaged Navier-Stokes (RANS) models are not very accurate for high Reynolds number, compressible jet-in-crossflow interactions. The inaccuracy arises from the use of inappropriate model parameters and model-form error in the RANS model. In this work we pursue the hypothesis that RANS predictions could be significantly improved by using parameters inferred from experimental measurements of a supersonic jet interacting with a transonic crossflow. We formulate a Bayesian inverse problem to estimate 3 RANS parameters (C µ , C ǫ2 , C ǫ1) and use a Markov chain Monte Carlo (MCMC) method to develop a probability density function for them. The cost of MCMC is addressed by developing statistical surrogates for the RANS models. We find that only a subset of the (C µ , C ǫ2 , C ǫ1) space, R, supports realistic flow simulations. We use R as our prior belief when formulating the inverse problem. It is enforced with a classifier in our MCMC solution. We find that the calibrated parameters improve predictions of the entire flowfield substantially, compared to the nominal/literature values of (C µ , C ǫ2 , C ǫ1); further, this improvement is seen to hold for interactions at other Mach numbers and jet strengths for which we have experimental data to provide a comparison. We also quantify the residual error, which is an approximation of the model-form error; it is most easily measured in terms of turbulent stresses.

Turbulence Modeling in the Age of Data

“…Their studies included data from a number of wall-bounded flows. Lefantzi et al (2015) and Ray et al (2016) used a similar approach to infer the RANS model coefficients and also investigated the likelihood of competing closures while focusing on jet-in-cross-flow, which is a canonical flow in film cooling in turbo-machinery applications. More recently, Ray et al (2018) used experimental data and Bayesian inference to calibrate the parameters in a nonlinear eddy viscosity model.…”

Quantification of model uncertainty in RANS simulations: A review

“…In CFD applications, each evaluation involves a simulation that takes hours or even weeks to run depending on the complexity of the flow configuration. For example, RANS simulations of a jet in crossflow, which is a geometrically simple yet industrially relevant case, needed O(10 7 ) grid points and O(10 4 ) CPU hours to run on a high performance computing cluster [23,65]. Clearly, it is impractical to perform a full RANS simulation for each evaluation of likelihood in the MCMC sampling.…”

“…As in the uncertainty propagation discussed above, surrogate models are commonly used for likelihood evaluation in MCMC-based model uncertainty quantification to alleviate the high computational cost of RANS simulations [23,65,66]. Efficient sampling of high dimensional spaces with MCMC is a topic of active research, with many methods proposed in the past few years, e.g., by adaptively constructing local approximations during the sampling and by using the likelihood to inform the sampling [see, e.g., 81,82].…”

“…Consequently, such regions should be excluded when using MCMC to sample the posterior. Again, this can be done by building surrogate models from simulation data [23,65,83]. The fraction of excluded regions increases exponentially with the dimension of the sample space.…”

Robust Bayesian Calibration of a RANS Model for Jet-in-Crossflow Simulations

“…The experimental and computational setup used in this calibration study have been described fully elsewhere 17 and we provide a summary below. The data used here is obtained from a set of wind tunnel experiments conducted by Beresh et al 3,4 .…”