A posteriori validation of sub-filter variance and unsteady NO-transport closures for predictive LES of NOx in gas-turbine combustors

The objective is to carry the atomization, sub-filter-variance and NO-transport developments of the 2008 review Large-eddy simulation of turbulent reacting flows (Pitsch, Desjardins, Balarac, Ihme) beyond their preliminary, a priori or single-condition demonstrations to predictive engine-relevant use. The review assembled the Stanford program toward predictive LES for aircraft-engine combustors — an accurate conservative level set/ghost-fluid method for turbulent atomization, a dynamic sub-filter mixture-fraction-variance model derived from a filter expansion and validated only a priori on homogeneous isotropic DNS, and an unsteady NO transport model with enthalpy-based radiation losses — and left its validation incomplete: mesh-refinement studies and parameter variations for the atomization DNS, a planned DNS database to develop and test LES sub-filter models, and use of the dynamic variance model in actual large-eddy simulations were all stated as future work, while the combustor NO prediction carried a documented baseline of roughly 38% overprediction of exit-plane averaged NO with radiation (a factor of about 4.2 without) in a Pratt & Whitney PW6000 combustor LES at take-off. The open problem is the review's remaining-work agenda for its own developments.

Background

Large-eddy simulation (LES) computes the large, energy-carrying turbulent motions of a flow directly and models the aggregate effect of the small unresolved motions, which are removed by a spatial filtering operation; everything the filter removes is called the sub-filter (or subgrid) content, and its statistics must be closed by models. In non-premixed (fuel and air enter separately) turbulent combustion, the local degree of fuel–air mixing is measured by the mixture fraction $Z$, a conserved scalar equal to zero in pure air and unity in pure fuel. In the flamelet/progress-variable (FPV) approach, the thermochemical state is reconstructed from a precomputed library of one-dimensional laminar counterflow diffusion flame solutions (flamelets) indexed by $Z$, a reaction progress variable marking how far chemistry has proceeded, and, when heat losses are included, enthalpy. Because the library is built for instantaneous local states while LES carries only filtered fields, the filtered chemical source must be averaged over sub-filter fluctuations of $Z$; this is usually done with a presumed filtered probability density function (a beta distribution) whose inputs are the filtered mixture fraction and its sub-filter variance $\widetilde{Z''^2}$. That variance is therefore a central closure quantity: it sets how much of the chemistry — including slow pollutant pathways — the flamelet library samples. A dynamic closure determines its own coefficients on the fly by comparing the model applied at the grid filter and at a coarser test filter (a Germano-type identity) instead of using fixed constants.

Nitric oxide ($NO$) is a regulated pollutant of gas-turbine combustors, produced mainly by the thermal pathway whose rate rises steeply (roughly exponentially) with temperature, and on chemical time scales that can exceed the residence time of a flame element, so that $NO$ is not in steady state with the local flame. Accurate $NO$ prediction therefore couples three demands: correct sub-filter mixing statistics (the variance and the scalar dissipation rate, the rate at which mixture-fraction fluctuations decay); unsteady rather than steady-state chemistry, typically through a transported $NO$ equation whose source comes from transient flamelet solutions; and radiative heat loss, since radiation cools the flame and the temperature error is amplified exponentially in the $NO$ rate. In liquid-fueled engines the fuel also atomizes: the liquid–gas interface can be tracked with a level-set function transported with conservation corrections (the accurate conservative level set, ACLS) combined with a ghost-fluid method (GFM) that treats the density jump across the interface sharply; farther downstream, droplets are represented by Lagrangian parcels whose evaporation feeds the gas-phase mixture fraction.

An a priori test evaluates a sub-filter model by filtering direct numerical simulation (DNS) data, whereas an a posteriori test runs the model inside an actual LES and compares the outcome against reference data; closures routinely pass the first and fail the second.

In a 2008 review, Pitsch, Desjardins, Balarac and Ihme surveyed a program toward predictive LES of turbulent reacting flows in aircraft engines, building on earlier work of the same group (the dynamic subgrid variance model and the FPV formulation of Pierce and Moin; the LES of realistic gas-turbine combustors of Moin and Apte). The program rested on three developments: the ACLS/GFM method for turbulent atomization (Desjardins, Moureau and Pitsch); a dynamic sub-filter variance model derived from a filter (Taylor-series) expansion and tested a priori with a box filter against homogeneous isotropic DNS ($256^3$ in the review's own tests; $512^3$ in the companion paper of Balarac, Pitsch and Raman); and an unsteady $NO$ transport model with enthalpy-based radiation losses, applied to Sandia flame D and a Pratt & Whitney combustor (Ihme and Pitsch). The review's remaining-work statements concern this program directly: mesh-refinement studies and parameter variations to complete the preliminary atomization DNS; generation of a planned DNS database (turbulent atomization of liquid jets, air-assisted atomization of co-annular liquid jets, and liquid jets in cross-flow) to develop and test LES sub-filter models; and future use of the method in large-eddy simulations once appropriate sub-filter models have been developed from the DNS results. Its documented engine baseline is the Pratt & Whitney PW6000 combustor LES at a single take-off condition, in which the exit-plane averaged $NO$ mole fraction is over-predicted by about 38% with radiation (ratio 1.3803 to the experimental measurement) and by a factor of about 4.2 without radiation (4.2254). Successor models developed since 2008 — a posteriori variance-closure tests in lifted flames, tabulated and improved NO models, engine-representative spray LES with NO and radiation, and mesh-resolution studies of primary atomization — have not achieved quantitatively predictive NOx LES for realistic combustors: tabulated NO models still carry about ±30% errors on laboratory flames, and the unresolved-variance modeling framework was still being rebuilt in 2024.

Problem Statement

The problem left open by the 2008 review of Pitsch, Desjardins, Balarac and Ihme is to carry the review's LES developments from their preliminary, a priori or single-condition demonstrations to predictive engine-relevant use. The review states its remaining work as separate future-work items concerning its own methods and models:

First, completion of the atomization DNS enabled by the ACLS/GFM method: mesh-refinement studies and parameter variations to complete the preliminary work, and generation of the planned DNS database — turbulent atomization of liquid jets, air-assisted atomization of co-annular liquid jets, and liquid jets in cross-flow — to provide data to develop and test sub-filter models for LES.

Second, use of the new dynamic sub-filter mixture-fraction-variance model — derived from a Taylor-series (filter) expansion and validated a priori only, with a box filter against homogeneous isotropic turbulence DNS ($256^3$ in the review's own tests; $512^3$ in the companion paper of Balarac, Pitsch and Raman) — in actual large-eddy simulations, including its integration with FPV closures and presumed probability density functions.

Third, advancement of quantitatively predictive $NO$ prediction in gas-turbine combustor LES with the unsteady flamelet/progress-variable $NO$ model including enthalpy-based radiation losses. This target concerns the model as stated, with the following documented closure approximations: equal species diffusivities in the transported-$NO$ equation (review Eq. (23)), and the scale-similarity closure approximating the correlation between steady-state $NO$ fluctuations and transient flamelet/progress-variable fluctuations (review Eq. (30)). The source-documented baseline is the Pratt & Whitney PW6000 combustor LES at a single take-off condition: with radiation, the exit-plane averaged $NO$ mole fraction is over-predicted by about 38% (ratio 1.3803 to the experimental measurement); without radiation, by a factor of about 4.2 (4.2254). The review reports no operating range beyond this single condition and sets no explicit completion criterion of its own for its remaining-work items.

The remaining work comprises three parts. The first: the atomization DNS completed — the planned database of turbulent atomization of liquid jets, air-assisted atomization of co-annular liquid jets, and liquid jets in cross-flow generated, with a mesh-refinement and parameter study establishing grid convergence. The second: the dynamic sub-filter variance model used a posteriori in actual large-eddy simulations, with its integration with FPV closures and presumed probability density functions. The third: the unsteady flamelet/progress-variable $NO$ model with enthalpy-based radiation losses made a posteriori predictive for the review's documented Pratt & Whitney PW6000 take-off case without recalibration of the model against that measurement, the target being that the radiation-inclusive exit-plane averaged $NO$ is reproduced to within about 19% of the experimental measurement (a factor of about 0.81 to 1.19 relative to that measurement; equivalently, the documented overprediction of about 38%, ratio 1.3803, is at least halved).

The open target is this remaining-work agenda for the review's own developments, pursued toward quantitatively predictive engine-relevant combustor LES; successor models developed since 2008 are relevant progress but the agenda is anchored to the review's specific methods and models, not to an arbitrary alternative closure class. The review's model remains the target of the advance; its documented approximations (review Eq. (23) and Eq. (30)) may be relaxed or replaced where the analysis requires, with the model's coefficients not fixed from the PW6000 measurement, so that the PW6000 exit-plane $NO$ result is a prediction rather than a fit.

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