In this paper we are concerned with obtaining estimates for the error in Reynolds-Averaged Navier-Stokes (RANS) simulations based on the Launder-Sharma k − ε turbulence closure model, for a limited class of flows. In particular we search for estimates grounded in uncertainties in the space of model closure coefficients, for wall-bounded flows at a variety of favourable and adverse pressure gradients. In order to estimate the spread of closure coefficients which reproduces these flows accurately, we perform 13 separate Bayesian calibrations-each at a different pressure gradient-using measured boundary-layer velocity profiles, and a statistical model containing a multiplicative model inadequacy term in the solution space. The results are 13 joint posterior distributions over coefficients and hyper-parameters. To summarize this information we compute Highest Posterior-Density (HPD) intervals, and subsequently represent the total solution uncertainty with a probability-box (p-box). This p-box represents both parameter variability across flows, and epistemic uncertainty within each calibration. A prediction of a new boundary-layer flow is made with uncertainty bars generated from this uncertainty information, and the resulting error estimate is shown to be consistent with measurement data.

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A novel deterministic symbolic regression method SpaRTA is introduced to infer algebraic stress models for the closure of RANS equations directly from high-fidelity LES or DNS data. The models are written as tensor polynomials and are built from a library of candidate functions. The machine-learning method is based on elastic net regularisation which promotes sparsity of the inferred models. By being data-driven the method relaxes assumptions commonly made in the process of model development. Model-discovery and cross-validation is performed for three cases of separating flows, i.e. periodic hills (Re=10595), converging-diverging channel (Re=12600) and curved backward-facing step (Re=13700). The predictions of the discovered models are significantly improved over the k-ω SST also for a true prediction of the flow over periodic hills at Re=37000. This study shows a systematic assessment of SpaRTA for rapid machine-learning of robust corrections for standard RANS turbulence models.

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 an 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.

A flexible non-intrusive approach to parametric uncertainty quantification problems is developed, aimed at problems with many uncertain parameters, and for applications with a high cost of functional evaluations. It employs a Kriging response surface in the parameter space, augmented with gradients obtained from the adjoint of the deterministic equations. The Kriging correlation parameter optimization problem is solved using the Subplex algorithm, which is robust for noisy functionals, and whose effort typically increases only linearly with problem dimension. Integration over the resulting response surface to obtain statistical moments is performed using sparse grid techniques, which are designed to scale well with dimensionality. The efficiency and accuracy of the proposed method is compared with probabilistic collocation, direct application of sparse grid methods, and Monte-Carlo initially for model problems, and finally for a 2d compressible Navier-Stokes problem with a random geometry parameterized by 4 variables.