Hierarchical model form uncertainty quantification for turbulent combustion modeling
Introduction
Uncertainties in turbulent combustion simulations arise from three distinct sources: uncertainties in operating parameters such as boundary conditions, uncertainties in model parameters such as chemical kinetic rate coefficients, and structural uncertainties associated with the form of various component models. Mueller et al. [1] conducted the first formal uncertainty quantification (UQ) estimation in turbulent combustion simulations, where they considered the propagation of uncertainties in chemical kinetic rates, specifically the parametric uncertainty in the pre-exponential factors in the Arrhenius rate expressions, through LES calculations of the Sandia D flame. In that work, the turbulent combustion model was used to reduce the dimensionality of the uncertainty from hundreds of kinetic rate coefficients to a lower-dimensional space that could be propagated efficiently through LES. In a later study, Khalil et al. [2] considered the propagation of uncertainties in a few turbulence model parameters through LES calculations of a turbulent nonpremixed bluff body flame through the construction of response surfaces. In more recent work, Ji et al. [3] propagated the uncertainties associated with kinetic rate coefficients through LES calculations by constructing low-dimensional response surfaces utilizing an active subspaces approach, as opposed to reducing the dimensionality of the uncertainty using the physical model as done by Mueller et al. [1]. All of these studies focused on parametric uncertainties.
While the above works are certainly novel, they do not address the more difficult challenge in uncertainty quantification: a “well-to-wheel” analysis of various models and their uncertainties and how these ultimately affect simulation outputs. This requires determination of the errors in models associated with not only their parameters but also their structural form. The first work approaching this challenge in the turbulent combustion community to date is that of Mueller and Raman [4]. In that work, an upstream error estimate in a flamelet model was derived from discrepancies with experimental measurements and propagated downstream to assess the impacts on soot volume fraction predictions. While the first of its kind, the approach relied on comparisons of forward model predictions with experimental measurements and would presumably be specific to that particularly geometry and set of operating parameters. A more general approach to quantifying model form uncertainty was later developed by Mueller and Raman [5] based on the concept of peer models. In this approach, model variability is used as a measure for model error by comparing two models with divergent assumptions, which may or may not be explicitly known. In that work, the approach was used to show that the model assumption in the subfilter dissipation rate model was more important than the model coefficient. The issue with this method is that it does not assess the uncertainty associated with individual model assumptions but rather the uncertainty in an aggregated set of model assumptions. More general model error or model form uncertainty estimation techniques are required that are derived directly from physical arguments and assess the uncertainty associated with individual assumptions.
Stated differently, the core issue of model form uncertainty quantification is the translation of model assumptions into mathematical statements of uncertainty. The first step then is the identification of the model assumptions. In cases where the model assumptions are not explicitly known, then a different approach to the one developed in this work is required (e.g., such as the peer models approach [5] discussed above). In the current approach, the model assumptions are explicitly known and can be individually isolated. Generally, the assumptions lead to a hierarchy of models of varying fidelity, and the hierarchy can be leveraged to estimate model uncertainties. One could utilize a data-based approach with such a hierarchy, in which a higher-fidelity model could be directly evaluated and the results of the model used to calibrate an uncertainty estimate for a lower-fidelity model [6]. However, the higher-fidelity models are often computationally inaccessible, necessitating the need for the lower-fidelity models in the first place. In this work, a physics-based approach is instead developed in which only the physical principles of the higher-fidelity model are used to derive error estimates for the lower-fidelity model. Specifically, the higher-fidelity model is used to identify a “trigger” parameter that indicates when the lower-fidelity model error will be large and derives an estimate for the model error based on this trigger parameter.
In this paper, this hierarchical approach to model form uncertainty quantification is applied to turbulent nonpremixed combustion modeling. In the next section, the generic hierarchical approach is demonstrated. In the following section, the specific application to turbulent nonpremixed combustion modeling is discussed in detail. Finally, using this “hierarchical” models approach, quantification of turbulent combustion model form errors is undertaken for LES of a canonical turbulent nonpremixed jet flame.
Section snippets
“Hierarchical” models approach to model form uncertainty quantification
Consider a hierarchy of models in order of increasing fidelity: . In order for these models to form a hierarchy, must be some well-defined limit of and must be some well-defined limit of . Here, well-defined means that an explicit assumption is made that is known at each level of the hierarchy. Without loss of generality, assume that the highest-fidelity model is parameterized by three quantities x1, x2, and x3 and the lower-fidelity models are each the corresponding limit
Application to turbulent combustion modeling
Turbulent nonpremixed combustion models follow a natural hierarchy as described above. From bottom-up, the equilibrium chemistry model invokes only thermodynamics. The steady flamelet model [7] considers finite-rate chemistry and transport but constrained to some steady-state balance on a reduced-order manifold defined in terms of the mixture fraction Z. Conditional Moment Closure (CMC) [8] allows for the inclusion of flow history effects but is still constrained to a reduced-order manifold
Test case: DLR flame
This hierarchical approach to model form uncertainty quantification is assessed with LES in a turbulent nonpremixed simple jet flame of CH4/H2/N2 [11]. The simulations were run using NGA, which is a structured, low Mach number solver [12], [13], using a non-uniform mesh with 256 × 144 × 64 grid points in the axial, radial, and circumferential directions, respectively. For the steady flamelet model, FlameMaster [14] was used to generate solutions of the steady flamelet equations using GRI-3.0
Comparison with experiment
In order to validate the LES calculations, Fig. 2 compares the mixture fraction, temperature, and mass fraction of carbon monoxide for the equilibrium chemistry and steady flamelet models with experimental data at . The downstream mixing rate is slightly overpredicted by the equilibrium chemistry model (note the overprediction of mixture fraction for r/D > 2) through the accumulation of small errors in the density upstream, which becomes increasingly severe downstream, but the steady
Conclusions
In this paper, a physics-based approach for estimating model form uncertainty has been developed by taking advantage of natural model hierarchies. In these hierarchies, a single, isolated assumption leads from one level of the hierarchy to the next, and insights from the higher-fidelity model are used to develop both a “trigger” parameter that indicates when the model error becomes large and an explicit functional dependence of the model error on the “trigger” parameter. This approach was
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors gratefully acknowledge funding from the Princeton University School of Engineering and Applied Sciences Project X Fund. The simulations presented in this article were performed on computational resources managed supported by the Princeton Institute for Computational Science and Engineering (PICSciE) and the Princeton University Office of Information Technology’s Research Computing department.
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