Predictive RANS simulations via Bayesian Model-Scenario Averaging

The turbulence closure model is the dominant source of error in most Reynolds-Averaged Navier–Stokes simulations, yet no reliable estimators for this error component currently exist. Here we develop a stochastic, a posteriori error estimate, calibrated to specific classes of flow. It is based on variability in model closure coefficients across multiple flow scenarios, for multiple closure models. The variability is estimated using Bayesian calibration against experimental data for each scenario, and Bayesian Model-Scenario Averaging (BMSA) is used to collate the resulting posteriors, to obtain a stochastic estimate of a Quantity of Interest (QoI) in an unmeasured (prediction) scenario. The scenario probabilities in BMSA are chosen using a sensor which automatically weights those scenarios in the calibration set which are similar to the prediction scenario. The methodology is applied to the class of turbulent boundary-layers subject to various pressure gradients. For all considered prediction scenarios the standard-deviation of the stochastic estimate is consistent with the measurement ground truth. Furthermore, the mean of the estimate is more consistently accurate than the individual model predictions.ANR UF

Estimation of Model Error Using Bayesian Model-Scenario Averaging with Maximum a Posterori-Estimates

International audienceThe lack of an universal modelling approach for turbulence in Reynolds-Averaged Navier–Stokes simulations creates the need for quantifying the modelling error without additional validation data. Bayesian Model-Scenario Averaging (BMSA), which exploits the variability on model closure coefficients across several flow scenarios and multiple models, gives a stochastic, a posteriori estimate of a quantity of interest. The full BMSA requires the propagation of the posterior probability distribution of the closure coefficients through a CFD code, which makes the approach infeasible for industrial relevant flow cases. By using maximum a posteriori (MAP) estimates on the posterior distribution, we drastically reduce the computational costs. The approach is applied to turbulent flow in a pipe at Re= 44,000 over 2D periodic hills at Re=5600, and finally over a generic falcon jet test case (Industrial challenge IC-03 of the UMRIDA project)

Space-dependent turbulence model aggregation using machine learning

In this article, we propose a data-driven methodology for combining the solutions of a set of competing turbulence models. The individual model predictions are linearly combined for providing an ensemble solution accompanied by estimates of predictive uncertainty due to the turbulence model choice. First, for a set of training flow configurations we assign to component models high weights in the regions where they best perform, and vice versa, by introducing a measure of distance between high-fidelity data and individual model predictions. The model weights are then mapped into a space of features, representative of local flow physics, and regressed by a Random Forests (RF) algorithm. The RF regressor is finally employed to infer spatial distributions of the model weights for unseen configurations. Predictions of new cases are constructed as a convex linear combination of the underlying models solutions, while the between model variance provides information about regions of high model uncertainty. The method is demonstrated for a class of flows through the compressor cascade NACA65 V103 at Re~3e5. The results show that the aggregated solution outperforms the accuracy of individual models for the quantity used to inform the RF regressor, and performs well for other quantities well-correlated to the preceding one. The estimated uncertainty intervals are generally consistent with the target high-fidelity data. The present approach then represents a viable methodology for a more objective selection and combination of alternative turbulence models in configurations of interest for engineering practic

Stochastic turbulence modeling in RANS simulations via Multilevel Monte Carlo

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.Comment: 40 page

Space-dependent aggregation of data-driven turbulence models

A machine-learning approach for data-driven Reynolds-Averaged Navier--Stokes (RANS) predictions of turbulent flows including estimates of turbulence modelling uncertainties is developed by combining a Bayesian symbolic identification methodology for learning customised RANS model corrections for selected classes of flows and a space-dependent model-aggregation algorithm that combines the predictions of a set of competing machine-learned RANS models by means of weighting functions depending on a vector of local flow features. The customised model corrections are learned by using the SBL-SpaRTA algorithm, recently proposed by \citet{cherroud2022sparse}, which delivers sparse model correction terms in analytical form and whose parameters are described by probability distribution functions. This makes the learned models naturally interpretable and endowed with a measure of uncertainty. The learned models are subsequently aggregated by training Random Forests Regressors (RFR), which associates a model performance score with a set of local flow features. The scores can be interpreted as the probability that a candidate model performs better than its competitors, given the flow behavior at a given location. Predictions of new flows are then formulated as a locally weighted average of the solutions of a set of machine-learned models. An uncertainty measure is obtained by propagating through the models their posterior parameter distributions and by combining competing model predictions to estimate the inter-model variance. The proposed space-dependent model aggregation procedure (X-MA) is applied to flows of varying complexity, showing significant improvement with respect to the baseline model and good generalizability to unseen flows. The results make X-MA an attractive candidate for the development of a generalizable data-driven turbulence modelling framework with quantified uncertainty