Robust prediction of dense gas flows under uncertain thermodynamic models

Abstract

A Bayesian approach is developed to quantify uncertainties associated with the thermodynamic models used for the simulation of dense gas flows, i.e. flows of gases characterized by complex molecules of moderate to high molecular weight, in thermodynamic conditions of the general order of magnitude of the liquid/vapor critical point. The thermodynamic behaviour of dense gases can be modelled through equations of state with various mathematical structures, all involving a set of material-dependent coefficients. For several organic fluids of industrial interest abundant and high-quality thermodynamic data required to specify such coefficients are hardly available, leading to undetermined levels of uncertainty of the equation output. Additionally, the best choice for the kind of equation of state (mathematical form) to be used is not always easy to determine and it is often based on expert opinion. In other terms, equations of state introduce both parametric and model-form uncertainties, which need to be quantified to make reliable predictions of the flow field. In this paper we propose a statistical inference methodology for estimating both kinds of uncertainties simultaneously. Our approach consists of a calibration step and a prediction step. The former allows to infer on the parameters to be input to the equation of state, based on the observation of aerodynamic quantities like pressure measurements at some locations in the dense gas flow. The subsequent prediction step allows to predict unobserved flow configurations based on the inferred posterior distributions of the coefficients. Model-form uncertainties are incorporated in the prediction step by using a Bayesian model averaging (BMA) approach. This consists in constructing an average of the predictions of various competing models weighted by the posterior model probabilities. Bayesian averaging also provides a useful tool for making robust predictions from a set of alternative calibration scenarios (Bayesian model-scenario averaging or BMSA). The proposed methodology is assessed for a class of dense gas flows, namely transonic flows around an isolated airfoil, at various free-stream thermodynamic conditions in the dense-gas region.

Estimates of turbulence modeling uncertainties in NACA65 cascade flow predictions by Bayesian model-scenario averaging

“…The approach used in this paper has been described in our previous work (de Zordo-Banliat, 2020; Merle and Cinnella, 2019). For the sake of self-containedness, we recall hereafter the principles of Bayesian model calibration and model mixures, with specific focus on BMSA.…”

“…According to Bayes’ rule: where p(Mi|Sk) is a user-defined prior and p(Dktrue¯|Mi,Sk) is the evidence for model M i and scenario S k , computed via Monte–Carlo integration. Here we follow Edeling et al (2014a) and Merle and Cinnella (2019) and choose p(Mi|Sk)=1/I.…”

“…To reduce the computational cost of BMSA predictions, it is interesting to consider a scenario-weighting criterion that allows to exclude a priori scenarios that are assigned low probability (below a user-defined threshold). Hereafter we consider the scenario-weighting criterion proposed in Merle and Cinnella (2019), which takes into account the hyperparameter σ η , a measure of model inadequacy, as mentioned in Section 2.1: …”