Publications

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In this paper, we present a Bayesian framework for estimating joint densities for large eddy simulation (LES) sub-grid scale model parameters based on canonical forced isotropic turbulence direct numerical simulation (DNS) data. The framework accounts for noise in the independent variables, and we present alternative formulations for accounting for discrepancies between model and data. To generate probability densities for flow characteristics, posterior densities for sub-grid scale model parameters are propagated forward through LES of channel flow and compared with DNS data. Synthesis of the calibration and prediction results demonstrates that model parameters have an explicit filter width dependence and are highly correlated. Discrepancies between DNS and calibrated LES results point to additional model form inadequacies that need to be accounted for. Copyright © 2016 John Wiley & Sons, Ltd.

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The thermal decomposition of H2O2 is an important process in hydrocarbon combustion playing a particularly crucial role in providing a source of radicals at high pressure where it controls the 3rd explosion limit in the H2-O2 system, and also as a branching reaction in intermediatetemperature hydrocarbon oxidation. As such, understanding the uncertainty in the rate expression for this reaction is crucial for predictive combustion computations. Raw experimental measurement data, and its associated noise and uncertainty, is typically unreported in most investigations of elementary reaction rates, making the direct derivation of the joint uncertainty structure of the parameters in rate expressions difficult. To overcome this, we employ a statistical inference procedure, relying on maximum entropy and approximate Bayesian computation methods, and using a two-level nested Markov Chain Monte Carlo algorithm, to arrive at a posterior density on rate parameters for a selected case of laser absorption measurements in a shock tube study, subject to the constraints imposed by the reported experimental statistics. The procedure constructs a set of H2O2 concentration decay profiles consistent with these reported statistics. These consistent data sets are then used to determine the joint posterior density on the rate parameters through straightforward Bayesian inference. Broadly, the method also provides a framework for the replication and comparison of missing data from different experiments, based on reported statistics, for the generation of consensus rate expressions.

The reaction of OH with H2 is a crucial chain-propagating step in the H2-O2 system thus making the specification of its rate, and its uncertainty, important for predicting the high-temperature combustion of hydrocarbons. In order to obtain an uncertain representation of this reaction rate in the absence of actual experimental data, we perform an inference procedure employing maximum entropy and approximate Bayesian computation methods to discover hypothetical data from a target shock-tube experiment designed to measure the reverse reaction rate. This method attempts to invert the fitting procedure from noisy measurement data to parameters, with associated uncertainty specifications, to arrive at candidate noisy data sets consistent with these reported parameters and their uncertainties. The uncertainty structure of the Arrhenius parameters is obtained by fitting each hypothetical data set in a Bayesian framework and pooling the resulting joint parameter posterior densities to arrive at a consensus density. We highlight the advantages of working with a data-centric representation of the experimental uncertainty with regards to model choice and consistency, and the ability for combining experimental evidence from multiple sources. Finally, we demonstrate the utility of knowledge of the joint Arrhenius parameter density for performing predictive modeling of combustion systems of interest.

The thermal decomposition of H2O2 is an important process in hydrocarbon combustion playing a particularly crucial role in providing a source of radicals at high pressure where it controls the 3rd explosion limit in the H2-O2 system, and also as a branching reaction in intermediatetemperature hydrocarbon oxidation. As such, understanding the uncertainty in the rate expression for this reaction is crucial for predictive combustion computations. Raw experimental measurement data, and its associated noise and uncertainty, is typically unreported in most investigations of elementary reaction rates, making the direct derivation of the joint uncertainty structure of the parameters in rate expressions difficult. To overcome this, we employ a statistical inference procedure, relying on maximum entropy and approximate Bayesian computation methods, and using a two-level nested Markov Chain Monte Carlo algorithm, to arrive at a posterior density on rate parameters for a selected case of laser absorption measurements in a shock tube study, subject to the constraints imposed by the reported experimental statistics. The procedure constructs a set of H2O2 concentration decay profiles consistent with these reported statistics. These consistent data sets are then used to determine the joint posterior density on the rate parameters through straightforward Bayesian inference. Broadly, the method also provides a framework for the replication and comparison of missing data from different experiments, based on reported statistics, for the generation of consensus rate expressions.

The development of scramjet engines is an important research area for advancing hypersonic and orbital flights. Progress towards optimal engine designs requires both accurate flow simulations as well as uncertainty quantification (UQ). However, performing UQ for scramjet simulations is challenging due to the large number of uncertain parameters involved and the high computational cost of flow simulations. We address these difficulties by combining UQ algorithms and numerical methods to the large eddy simulation of the HIFiRE scramjet configuration. First, global sensitivity analysis is conducted to identify influential uncertain input parameters, helping reduce the stochastic dimension of the problem and discover sparse representations. Second, as models of different fidelity are available and inevitably used in the overall UQ assessment, a framework for quantifying and propagating the uncertainty due to model error is introduced. These methods are demonstrated on a non-reacting scramjet unit problem with parameter space up to 24 dimensions, using 2D and 3D geometries with static and dynamic treatments of the turbulence subgrid model.

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Here, we present the results of an application of Bayesian inference and maximum entropy methods for the estimation of the joint probability density for the Arrhenius rate para meters of the rate coefficient of the H₂/O₂-mechanism chain branching reaction H + O₂ → OH + O. Available published data is in the form of summary statistics in terms of nominal values and error bars of the rate coefficient of this reaction at a number of temperature values obtained from shock-tube experiments. Our approach relies on generating data, in this case OH concentration profiles, consistent with the given summary statistics, using Approximate Bayesian Computation methods and a Markov Chain Monte Carlo procedure. The approach permits the forward propagation of parametric uncertainty through the computational model in a manner that is consistent with the published statistics. A consensus joint posterior on the parameters is obtained by pooling the posterior parameter densities given each consistent data set. To expedite this process, we construct efficient surrogates for the OH concentration using a combination of Pad'e and polynomial approximants. These surrogate models adequately represent forward model observables and their dependence on input parameters and are computationally efficient to allow their use in the Bayesian inference procedure. We also utilize Gauss-Hermite quadrature with Gaussian proposal probability density functions for moment computation resulting in orders of magnitude speedup in data likelihood evaluation. Despite the strong non-linearity in the model, the consistent data sets all res ult in nearly Gaussian conditional parameter probability density functions. The technique also accounts for nuisance parameters in the form of Arrhenius parameters of other rate coefficients with prescribed uncertainty. The resulting pooled parameter probability density function is propagated through stoichiometric hydrogen-air auto-ignition computations to illustrate the need to account for correlation among the Arrhenius rate parameters of one reaction and across rate parameters of different reactions.

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One of the most widely-used procedures for dimensionality reduction of high dimensional data is Principal Component Analysis (PCA). More broadly, low-dimensional stochastic representation of random fields with finite variance is provided via the well known Karhunen-Loève expansion (KLE). The KLE is analogous to a Fourier series expansion for a random process, where the goal is to find an orthogonal transformation for the data such that the projection of the data onto this orthogonal subspace is optimal in the L2 sense, i.e., which minimizes the mean square error. In practice, this orthogonal transformation is determined by performing an SVD (Singular Value Decomposition) on the sample covariance matrix or on the data matrix itself. Sampling error is typically ignored when quantifying the principal components, or, equivalently, basis functions of the KLE. Furthermore, it is exacerbated when the sample size is much smaller than the dimension of the random field. In this paper, we introduce a Bayesian KLE procedure, allowing one to obtain a probabilistic model on the principal components, which can account for inaccuracies due to limited sample size. The probabilistic model is built via Bayesian inference, from which the posterior becomes the matrix Bingham density over the space of orthonormal matrices. We use a modified Gibbs sampling procedure to sample on this space and then build probabilistic Karhunen-Loève expansions over random subspaces to obtain a set of low-dimensional surrogates of the stochastic process. We illustrate this probabilistic procedure with a finite dimensional stochastic process inspired by Brownian motion.

Monte Carlo (MC) sampling is a common method used to randomly sample a range of scenarios. The associated error follows a predictable rate of convergence of $1/\sqrt{N}$, such that quadrupling the sample size halves the error. This method is often employed in performing global sensitivity analysis which computes sensitivity indices, measuring fractional contributions of uncertain model inputs to the total output variance. In this study, several models are used to observe the rate of decay in the MC error in the estimation of the conditional variance, the total variance in the output, and the global sensitivity indices. The purpose is to examine the rate of convergence of the error in existing specialized, albeit MC-based, sampling methods for estimation of the sensitivity indices. It was found that the conditional variances and sensitivity indices all follow the $1/\sqrt{N}$ convergence rate. Future work will test the convergence of observables from more complex models such as ignition time in combustion.

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The objective of this work is to investigate the efficacy of using calibration strategies from Uncertainty Quantification (UQ) to determine model coefficients for LES. As the target methods are for engineering LES, uncertainty from numerical aspects of the model must also be quantified. 15 The ultimate goal of this research thread is to generate a cost versus accuracy curve for LES such that the cost could be minimized given an accuracy prescribed by an engineering need. Realization of this goal would enable LES to serve as a predictive simulation tool within the engineering design process.

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