Previous work using the PDF Method

The application of the PDF method to combustion has received considerable attention with most of the effort focused on non-premixed combustion investigations. Some of the more significant PDF studies and their relation to the progression of the PDF method are reviewed in this section.

The single step reaction of NO with O3 essentially occurs without significant heat release, this reaction was examined using the PDF method by Givi et al. [66] in two fundamental geometries: the plane jet and the axisymmetric jet. Although no experimental comparisons were able to be made, the study was an important precursor to PDF methods that included heat release. Nguyen and Pope [67] present PDF calculations of reacting hydrogen-air jet diffusion flames assuming a conserved scalar approach and infinitely fast chemistry, the mixture fraction PDF equation was solved using a stochastic Monte Carlo approach. Some of the obvious limitations in predicting finite-rate chemistry effects when combined with the conserved scalar approach in the PDF method are highlighted by Wouters et al. [68].

An intermediate between the conserved scalar approach with no chemistry computation and more detailed kinetic mechanisms is the ILDM chemistry approach. Norris and Pope [69] use three characteristic scalars to characterise the ILDM and use these three scalars in the joint PDF of composition, velocity and dissipation. By incorporating chemistry with only three characteristic scalars Norris and Pope [69] are able to predict blow-off to within 10% of the experimental blow-off jet velocity of a piloted non-premixed flame with a CO/H2/N2 fuel jet [70] as well as predict a degree of finite-rate chemistry effects near blow-off.

The simulation of the Delft piloted burner reported by Peeters et al. [71] has been examined by Nooren et al. [72] using an elliptic PDF code. Three mixing models were evaluated in this study: the IEM model [73], a revised version of the modified Curl (MC) model [74] and the mapping closure model [75, 76]. Due to the elliptic nature of the solution, constrained equilibrium [71] with low temperature corrections [77] and ILDM [78] chemistry were utilised to allow the flames to be computed at a realistic computation cost. The constrained equilibrium solution was shown to yield good predictions for the temperature PDF in mixture fraction space, whilst the ILDM solution went further in its predictive capabilities so that the OH PDF in mixture fraction space was well predicted including the super-equilibrium concentrations. It should be noted that the solution was reported to be sensitive to the mixing model; both the mapping closure and MC mixing

models produced satisfactory results whilst the IEM were reported to be not as good in certain regions of the flow. A constant value of C_φ =2.0 was used for all simulations by Nooren et al. [72].

In the past ten years recent advances in computing power has allowed PDF simulations using more complex methane chemistry ranging from skeletal mechanisms [79] to augmented reduced mechanisms with steady and quasi steady state species [80], to simulations with full GRI3.0 chemistry [81]. With the subsequent increase in chemical description corresponding increases in the predictive capability of kinetic sensitive parameters such as extinction and re-ignition, fuel rich side kinetics, minor species predictions such as OH and CO and pollutant formation such as NO have been reported [81, 82]. Simulations with more complex fuels than methane such as methanol using a augmented reduced mechanism based on a detailed kinetic mechanism [83] have also been carried out using the PDF method.

Typically PDF simulations have relied on geometrical symmetries allowing the problem to be solved on a 2D axisymmetric grid with a considerable saving in computational cost.

An extension of the PDF to 3D practically relevant gas turbine combustors has been reported by James et al. [84] with many complicating practical aspects incorporated into the calculation such as sprays, secondary dilution, high pressure (17.6 Bar), preheating of the primary air to 782K and direct modelling of the primary swirler. In order to make such large computations tractable a two-step mechanism is used for propane and jet-A fuels. Good agreement is reported for computed and experimental mean temperature profiles.

To model the unclosed micro-mixing term in the PDF equation there have been a number of models proposed such as: Curl’s mixing model [85, 86], the modified Curl (MC) mixing model [74, 87, 88], the interaction exchange with the mean (IEM) mixing model [73, 89], the mapping closure model [75], the ordered pairing model [90] and the Euclidean minimum spanning trees (EMST) mixing model [91]. The specifics of three of these mixing models (MC, IEM and EMST) will be reviewed in greater detail later in this

Chapter in Section 2.3.5. In general each of the above mixing models has certain advantages and shortcomings in different situations. In general for non-premixed combustion the EMST model has been shown by Cao et al. [92] to give the best results for the TNF piloted diffusion flame series compared to the MC and IEM mixing models.

The MC and IEM mixing models are shown to be able to produce extinction and re-ignition; however the scalar variance values are unacceptably under predicted. In general it is also shown by Cao et al. [92] that the MC mixing model outperforms the IEM mixing model. Further difference in the predictions of the EMST, IEM and MC mixing models applied to a partially stirred reactor (PaSR) have been examined by Ren and Pope [93], that confirm the EMST mixing model is more resilient to extinction than the IEM and MC mixing models when applied to the particular PaSR problem examined.

The relative performance of the mixing models in terms of the predicted fields for the mean scalar, scalar variance and finite-rate chemistry effects can be modified by changing the value of the value of C_φ which governs the ratio of the scalar to fluid mechanical time scales. In early PDF simulations [67, 94, 95, 96, 97] a value of C_φ =2.0 was used, this assumes that the ratio of scalar to fluid mechanical time scales is 2. The value of C_φ is proposed to be in the range 1.5-2.5 by Beguier et al. [98] for a range of shear flows, thereby an average value can be taken to be 2. More recent measurements than those of Beguier et al. [98] for the scalar time scale ratio carried out by Panchapakesan and Lumley [99] indicate that for the axisymmetric jet the scalar to velocity time scale ratio is within 15% of 1.5 over the entire radial profile up to

1 2 2.0 r r = .

By modifying the value of C_φ it is possible to increase or decrease the predicted scalar variance (such as ξ′′ ), finite-rate chemistry effects (such as extinction and re-ignition) and mean scalar predictions (such as mean flame length and ξ ). Generally by increasing C_φ more robust burning (less extinction) and decreased variance predictions are found.

One of the first parametric investigation of the effect of varying C_φ in a turbulent flame

with complex chemistry was conducted by Xu and Pope [79] who found that the optimal value of C_φ for the piloted flame series using the EMST mixing model was C_φ =1.5. Further work by Cao et al. [92] on the same piloted non-premixed flame geometry confirmed the results of Xu and Pope [79] for C_φ =1.5 using the EMST mixing model, but also optimal values of C_φ =3.8 and C_φ =3.3 were proposed for the MC and IEM mixing models respectively. In auto-igniting flows where the chemical source term dominates and mixing is as important, Masri et al. [100] and later Cao et al. [101] find there is very little sensitivity for C_φ in the range 1.5-2.5 for the EMST, MC and IEM mixing models, further simulations by Ren and Pope [65] examining the sensitivity to mixing model and C_φ at the particle level confirm this finding.

The optimal value of C_φ is not necessarily universal between investigations. Lindstedt et al. [80] report an optimal value of C_φ =2.3 using the MC mixing model and complex chemistry [102, 103] based on a reduced version of a 48 species 300 reaction mechanism.

Cao et al. [101] argue that with the GRI3.0 chemical mechanism, a flame is not able to be stabilised for C_φ ≤3.0 using the MC mixing model for flame E. Cao et al. [101] argue that the success of C_φ =2.3 by Lindstedt et al. [80] is probably not due to the implementation differences in the PDF equations or mixing models, but rather that the chemical mechanism used was significantly faster and more resilient to extinction than GRI 3.0. If the explanation provided by Cao et al. [101] were to hold then the methanol mechanism [104] used by Lindstedt and Louloudi [83] must also be too kinetically fast and resilient to extinction, as a value of C_φ =2.3 was also found to be near optimal for their study of the experimental piloted methanol flame reported by Lindstedt and Louloudi [83].

Further resolution of the controversy regarding the optimal or correct value of C_φ for non-premixed combustion may not be possible until identical simulations with a given PDF code is computed with the MC mixing model and both GRI3.0 and the Lindstedt

methane mechanism [102, 103] and compared with the results of Lindstedt et al. [80] and those of Cao et al. [101]. Such a study is already underway and being examined [105].

Using the joint PDF of velocity, turbulence frequency and composition Liu et al. [106]

has examined the effect of varying C_φ for the EMST mixing model and complex chemistry on the prediction for the Sydney bluff-body burner [107, 108, 109, 110, 111, 112, 113]. The results show that C_φ =1.0 gives optimal results for the mixture fraction variance but C_φ =2.0 gives better results for extinction and temperature variance fields, furthermore NO and OH is predicted well only for C_φ ≤1.5. This shows that the optimal choice of C_φ is often a compromise between many variables in a simulation as well as between different geometries (bluff-body and pilot diffusion flame).

Using a joint composition PDF approach coupled with a FV solver for the fluid and turbulence solutions, Merci et al. [114, 115, 116, 117, 118, 119, 120, 121] have examined the influence of different mixing models, values of C_φ and different turbulence models for the Delft piloted flame and the Sydney bluff body flame. These studies confirm those by Cao et al. [101] in that the EMST mixing model was found to be superior to the IEM and MC particularly for flames near extinction and increased values of C_φ or pilot heat release were required to obtain a stable flame for the IEM and MC mixing models. Also the comment is made by Merci et al. [120] concur with those of Fox [122] that the EMST is sometimes ‘too local’. This means that when predicting extinction, the EMST mixing model does not evolve the particle towards the inert mixing limit at the same rate that the experimental data indicates. The MC mixing model seems to do a better job than the EMST mixing model predicting the beginnings of extinction using a significantly increased value of C_φ to obtain good correlation with the experimental results. However, the MC model suffers from an unrealistically rapid transition to total flame blow-off due to the non-localness property of the MC mixing model in composition space. This unrealistically rapid transition to total blow-off is an issue the EMST mixing model does not suffer from.