The first-order, single-conditioned sub-grid Conditional Moment Closure (CMC) model for Large Eddy Simulation (LES) is applied to simulate a globally lean swirling methane flame in a gas turbine model combustor that has been studied experimentally. The time-averaged velocity, mixture fraction, temperature, major species and OH mass fractions, and the heat release rate are predicted well for most locations. A transient lift-off and flashback of the flame root due to localized extinction near the burner exit is observed that is qualitatively consistent with the experimental measurements. The time-averaged temperature is over-predicted very close to the fuel injection point, while it is accurately reproduced downstream. Comparisons of instantaneous conditionally-filtered temperature in mixture fraction space shows that the LES/CMC reproduces the large scatter of the experimental data points in temperature‒mixture fraction plane that span the full range unburnt to fully burnt, but to a smaller extent, suggesting a minor under-prediction of local extinction or the inaccuracy of the present first-order, coarse-grid CMC formulation to capture locally premixed flame propagation behaviour. Periodic variation of the heat release rate is observed with a frequency close to that of the measured Precessing Vortex Core (PVC). In general, the current LES/CMC model captures most features of this high-mixing-rate nominally non-premixed swirl flame in a gas turbine model combustor in agreement with experimental measurements.