Matthew T. Reagan

Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection

Abstract A spectral formalism has been developed for the “non-intrusive” analysis of parametric uncertainty in reacting-flow systems. In comparison to conventional Monte Carlo analysis, this method quantifies the extent, dependence, and propagation of uncertainty through the model system and allows the correlation of uncertainties in specific parameters to the resulting uncertainty in detailed flame structure. For the homogeneous ignition chemistry of a hydrogen oxidation mechanism in supercritical water, spectral projection enhances existing Monte Carlo methods, adding detailed sensitivity information to uncertainty analysis and relating uncertainty propagation to reaction chemistry. For 1-D premixed flame calculations, the method quantifies the effect of each uncertain parameter on total uncertainty and flame structure, and localizes the effects of specific parameters within the flame itself. In both 0-D and 1-D examples, it is clear that known empirical uncertainties in model parameters may result in large uncertainties in the final output. This has important consequences for the development and evaluation of combustion models. This spectral formalism may be extended to multidimensional systems and can be used to develop more efficient “intrusive” reformulations of the governing equations to build uncertainty analysis directly into reacting flow simulations.

Spectral stochastic uncertainty quantification in chemical systems

Uncertainty quantification (UQ) in the computational modelling of physical systems is important for scientific investigation, engineering design, and model validation. We have implemented an ‘intrusive’ UQ technique in which (1) model parameters and field variables are modelled as stochastic quantities, and are represented using polynomial chaos (PC) expansions in terms of Hermite polynomial functions of Gaussian random variables, and (2) the deterministic model equations are reformulated using Galerkin projection into a set of equations for the time evolution of the field variable PC mode strengths. The mode strengths relate specific parametric uncertainties to their effects on model outputs. In this work, the intrusive reformulation is applied to homogeneous ignition using a detailed chemistry model through the development of a reformulated pseudospectral chemical source term. We present results analysing the growth of uncertainty during the ignition process. We also discuss numerical issues pertaining to the accurate representation of uncertainty with truncated PC expansions, and ensuing stability of the time integration of the chemical system.

Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions

A stochastic projection method (SPM) is developed for quantitative propagation of uncertainty in compressible zero-Mach-number flows. The formulation is based on a spectral representation of uncertainty using the polynomial chaos (PC) system, and on a Galerkin approach to determining the PC coefficients. Governing equations for the stochastic modes are solved using a mass-conservative projection method. The formulation incorporates a specially tailored stochastic inverse procedure for exactly satisfying the mass-conservation divergence constraints. A brief validation of the zero-Mach-number solver is first performed, based on simulations of natural convection in a closed cavity. The SPM is then applied to analyze the steady-state behavior of the heat transfer and of the velocity and temperature fields under stochastic non-Boussinesq conditions.

Analysis of parametric uncertainty propagation in detailed combustion chemistry

Publisher Summary This chapter illustrates the nonintrusive analysis of parametric uncertainty. Quantification of spectral modes adds a new level of understanding, identifying the significance of individual uncertain parameters and determining where within the evolution of the system each parameter has the most influence. In this study, a technique of uncertainty quantification through spectral projection methods is used to examine the propagation of parametric uncertainty through 0D combustion chemistry and 1D premixed flame, with a focus on higher-order information and the correlations among parameters. Compared to conventional Monte Carlo analysis, this method quantifies the extent, dependence, and propagation of uncertainty through the model system and allows correlation of uncertainties in specific parameters to the resulting total uncertainty in product concentrations and flame structure. Analysis of 0D and 1D hydrogen-oxygen combustion demonstrates that known empirical uncertainties in model parameters may result in large uncertainties in the final output, and nonlinearities in system response behavior make the order of the analysis an important consideration. Furthermore, this analysis is readily extendable to multiple dimensions and greater numbers of uncertain parameters, while preserving the integrity of the realization engine through the nonintrusive operation of the method.