Todd A. Oliver

Bayesian Uncertainty Analysis with Applications to Turbulence Modeling

Abstract In this paper, we apply Bayesian uncertainty quantification techniques to the processes of calibrating complex mathematical models and predicting quantities of interest (QoIs) with such models. These techniques also enable the systematic comparison of competing model classes. The processes of calibration and comparison constitute the building blocks of a larger validation process, the goal of which is to accept or reject a given mathematical model for the prediction of a particular QoI for a particular scenario. In this work, we take the first step in this process by applying the methodology to the analysis of the Spalart–Allmaras turbulence model in the context of incompressible, boundary layer flows. Three competing model classes based on the Spalart–Allmaras model are formulated, calibrated against experimental data, and used to issue a prediction with quantified uncertainty. The model classes are compared in terms of their posterior probabilities and their prediction of QoIs. The model posterior probability represents the relative plausibility of a model class given the data. Thus, it incorporates the models ability to fit experimental observations. Alternatively, comparing models using the predicted QoI connects the process to the needs of decision makers that use the results of the model. We show that by using both the model plausibility and predicted QoI, one has the opportunity to reject some model classes after calibration, before subjecting the remaining classes to additional validation challenges.

Estimating uncertainties in statistics computed from direct numerical simulation

Rigorous assessment of uncertainty is crucial to the utility of direct numerical simulation (DNS) results. Uncertainties in the computed statistics arise from two sources: finite statistical sampling and the discretization of the Navier–Stokes equations. Due to the presence of non-trivial sampling error, standard techniques for estimating discretization error (such as Richardson extrapolation) fail or are unreliable. This work provides a systematic and unified approach for estimating these errors. First, a sampling error estimator that accounts for correlation in the input data is developed. Then, this sampling error estimate is used as part of a Bayesian extension of Richardson extrapolation in order to characterize the discretization error. These methods are tested using the Lorenz equations and are shown to perform well. These techniques are then used to investigate the sampling and discretization errors in the DNS of a wall-bounded turbulent flow at Reτ ≈ 180. Both small (Lx/δ × Lz/δ = 4π × 2π) and la...Rigorous assessment of uncertainty is crucial to the utility of DNS results. Uncertainties in the computed statistics arise from two sources: finite statistical sampling and the discretization of the Navier-Stokes equations. Due to the presence of non-trivial sampling error, standard techniques for estimating discretization error (such as Richardson extrapolation) fail or are unreliable. This work provides a systematic and unified approach for estimating these errors. First, a sampling error estimator that accounts for correlation in the input data is developed. Then, this sampling error estimate is used as part of a Bayesian extension of Richardson extrapolation in order to characterize the discretization error. These methods are tested using the Lorenz equations and are shown to perform well. These techniques are then used to investigate the sampling and discretization errors in the DNS of a wall-bounded turbulent flow. For both cases, it is found that while the sampling uncertainty is large enough to make the order of accuracy difficult to determine, the estimated discretization errors are quite small. This indicates that the commonly used heuristics provide ad- equate resolution for this class of problems. However, it is also found that, for some quantities, the discretization error is not small relative to sampling error, indicating that the conventional wisdom that sampling error dominates discretization error for this class of simulations needs to be reevaluated.

An Unsteady Adaptation Algorithm for Discontinuous Galerkin Discretizations of the RANS Equations

An adaptive method for high-order discretizations of the Reynolds-averaged NavierStokes (RANS) equations is examined. The RANS equations and Spalart-Allmaras (SA) turbulence model are discretized with a dual consistent, discontinuous Galerkin discretization. To avoid oscillations in the solution in under-resolved regions, particularly the edge of the boundary layer, artificial dissipation is added to the SA model equation. Two adaptive procedures are examined: a standard output-based adaptation algorithm that requires the steady state solution to estimate the error and a new, unsteady approach that allows the mesh to be adapted without requiring a steady state solution. Results show that the combination of a dual consistent discretization with artificial dissipation and adaptation has significant promise as a practical method for obtaining high-order RANS solutions.

Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms

The effects of dual consistency on discontinuous Galerkin discretizations of solution and solution gradient dependent source terms are examined. Two common discretizations are analyzed: the standard weighting technique for source terms and the mixed formulation. It is shown that if the source term depends on the first derivative of the solution, the standard weighting technique leads to a dual inconsistent scheme. A straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated. The mixed formulation, where the solution gradient in the source term is replaced by an additional variable that is solved for simultaneously with the state, leads to an asymptotically dual consistent discretization. Numerical results for a one-dimensional test problem confirm that the dual consistent and asymptotically dual consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional output.

Bayesian analysis of syngas chemistry models

Syngas chemistry modelling is an integral step toward the development of safe and efficient syngas combustors. Although substantial effort has been undertaken to improve the modelling of syngas combustion, models nevertheless fail in regimes important to gas turbine combustors, such as low temperature and high pressure. In order to investigate the capabilities of syngas models, a Bayesian framework for the quantification of uncertainties has been used. This framework, given a set of experimental data, allows for the calibration of model parameters, determination of uncertainty in those parameters, propagation of that uncertainty into simulations, as well as determination of model evidence from a set of candidate syngas models. Here, three syngas combustion models have been calibrated using laminar flame speed measurements from high pressure experiments. After calibration the resulting uncertainty in the parameters is propagated forward into the simulation of laminar flame speeds. The model evidence is then used to compare candidate models for the given set of experimental conditions and results. Additionally, the technique MUM-PCE, an interesting uncertainty minimisation method for kinetics models, has been compared to the Bayesian method for this application to the prediction of syngas laminar flame speeds. This comparison shows the importance of model form error and experimental error representations in the uncertainty quantification context, for these choices significantly affect uncertainty quantification results.

Impact of Turbulence Model Irregularity on High-Order Discretizations

A high-order, discontinuous Galerkin nite element discretization for the the Reynoldsaveraged Navier Stokes equations coupled with the Spalart-Allmaras turbulence model is presented and applied to subsonic airfoil test cases. In particular, grid renement studies are conducted to examine the accuracy of the scheme. The results of these renement studies demonstrate that optimal order of accuracy is not obtained. Adjoint based error estimates are used to investigate this loss of accuracy. The results demonstrate that the turbulent/non-turbulent interface at the boundary layer edge region, where the turbulence model solution is not smooth, contribute signicantly to the total error in the computed drag.

Accounting for uncertainty in the analysis of overlap layer mean velocity models

When assessing the veracity of mathematical models, it is important to consider the uncertainties in the data used for the assessment. In this paper, we study the impact of data uncertainties on the analysis of overlap layer models for the mean velocity in wall-bounded turbulent flows. Specifically, the tools of Bayesian statistics are used to calibrate and compare six competing models of the mean velocity profile, including multiple logarithmic and power law forms, using velocity profile measurements from a zero-pressure-gradient turbulent boundary layer and fully developed turbulent pipe flow. The calibration problem is formulated as a Bayesian update of the joint probability density function for the calibration parameters, which are treated as random variables to characterize incomplete knowledge about their values. This probabilistic formulation provides a natural treatment of uncertainty and gives insight into the quality of the fit, features that are not easily obtained in deterministic calibration ...

Manufactured Solutions for the Favre-Averaged Navier-Stokes Equations with Eddy-Viscosity Turbulence Models

The Method of Manufactured Solutions is applied to verify the implementation of eddy viscosity turbulence models for closure of the Favre-averaged Navier-Stokes equations. In the Method of Manufactured Solutions, the governing equations are modied by the addition of source terms such that the exact solution|i.e., the manufactured solution|is known a priori. Given the exact solution, order of accuracy studies are conducted to verify that the discrete solution converges to the exact solution at the expected rate. The goal of this work is to verify the implementation of turbulence models in the Fully-Implicit NavierStokes ow solver. The turbulence model of interest for this work is the Spalart-Allmaras one-equation model, and two manufactured solutions have been examined. The rst solution is based on trigonometric functions, as commonly used in manufactured solution literature. This solution is appropriate for use in unbounded ows, enabling verication of the implementation of the free shear ow form of the model. The second solution has been newly developed in this work and is intended for use in wall-bounded ows. While other manufactured solutions for the Spalart-Allmaras model for wall-bounded ow have appeared in the literature, these solutions are shown to have features that make them illsuited to verication. To avoid such features, the wall-bounded solution developed here is loosely based on the behavior of the model solution in the inner portion of a zero-pressure boundary layer. Results obtained using both solutions show that the Fully-Implicit NavierStokes ow solver is achieving the expected second-order accuracy.

Adjoint-based sensitivity analysis of flames

Simulation of chemically reacting flows using detailed chemistry introduces a large number of chemistry model parameters. While not all significantly affect the target outcomes of a simulation, the parameters that do are not always known a priori. In order to improve simulations for specified target outcomes, termed quantities of interest (QoIs), the sensitivity of these QoIs to the model parameters are needed. However, evaluating the sensitivities is computationally expensive, especially for complex fuels that may involve many parameters. For these simulations, the forward sensitivity method requires the solution of an additional number of governing equations proportional to the number of parameters. Here, an adjoint sensitivity approach is formulated where the computational cost scales as the number of QoIs and not the number of parameters. Specifically, adjoint equations are derived for laminar, incompressible, variable density reacting flow and applied to hydrogen flame simulations. From the solution of the corresponding adjoint equations, sensitivity of the QoIs to chemistry model parameters is calculated. The one-dimensional simulation results show that the adjoint sensitivity results closely match those of forward sensitivity methods, thus providing validation of the adjoint method. The two-dimensional simulation results indicate the most sensitive parameters for two QoIs, flame tip temperature and NOx emission. For these tests, the adjoint method reduces computational expense compared to forward sensitivity methods by a factor proportional to the number of QoIs over the number of parameters, here 2/172. Such savings can be more drastic for cases that involve complex fuels, such as combustion of jet fuel, requiring thousands of chemistry model parameters. Further, this sensitivity information can be used in development of experiments by pointing out which are the critical chemistry model parameters.

Representing Model Inadequacy: A Stochastic Operator Approach

Mathematical models of physical systems are subject to many uncertainties such as measurement errors and uncertain initial and boundary conditions. After accounting for these uncertainties, it is often revealed that discrepancies between the model output and the observations remain; if so, the model is said to be inadequate. In practice, the inadequate model may be the best that is available or tractable, and so despite its inadequacy the model may be used to make predictions of unobserved quantities. In this case, a representation of the inadequacy is necessary, so the impact of the observed discrepancy can be determined. We investigate this problem in the context of chemical kinetics and propose a new technique to account for model inadequacy that is both probabilistic and physically meaningful. A stochastic inadequacy operator