A Lagrangian-based flame index for the transported probability density function method

Zhen Lu , , , , Hu Zhou , Zhuyin Ren , Yue Yng , , e , Hong G. Im

a State Key Laboratory for Turbulent and Complex Systems, College of Engineering, Peking University, Beijing 100871, China

b BIC-ESAT, Peking University, Beijing 100871, China

c Clean Combustion Research Center, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

d Institute of Aero Engine, Tsinghua University, Beijing 10 0 084, China

e HEDPS-CAPT, Peking University, Beijing 100871, China

Keywords:Flame index Transported probability density function Partially premixed combustion

ABSTRACT We propose a new flame index for the transported probability density function (PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as the metrics to identify the combustion mode, with the Burke-Schumann solution as a reference. A priori validation of the flame index is conducted with a series of constructed turbulent partially premixed reactors. It indicates that the proposed flame index is able to identify the combustion mode based on the subgrid mixing information. The flame index is then applied the large eddy simulation/PDF datasets of turbulent partially premixed jet flames. Results show that the flame index separate different combustion modes and extinction correctly. The proposed flame index provides a promising tool to analyze and model the partially premixed flames adaptively.

Partial premixing plays a significant role in practical combustion devices [1] . Due to the inhomogeneity in the composition of reactant mixtures, different combustion modes, such as premixed,nonpremixed, and extinction/auto-ignition in between, may coexist in turbulent reacting flows. Therefore, distinct turbulent combustion models for premixed and nonpremixed modes are needed to model partially premixed flames [2–5] .

The flame index (or flame regime indicator) aims to identify the local and instantaneous combustion mode [2–8] . Upon identifying the combustion mode, an appropriate turbulent combustion model can be applied accordingly for partially premixed flames. We can further apply transport budget [9,10] and chemical mode analysis tools [11–13] to investigate extinction/re-ignition,flame propagation/auto-ignition, and turbulence-flame interactions.Takeno and coworkers [14] proposed to identify the combustion mode based on the gradients of fuel and oxidizer concentrations.Although the Takeno index was proposed for the direct numerical simulation (DNS), it has since been extended to the large eddy simulation (LES) with modeled subgrid gradients [2] . For simulations with detailed chemistry, mixture fraction and progress variable were proposed as replacements of fuel and oxidizer [15] . With the flamelet-based approaches, Knudsen and Pitsch [4] derived an indicator by comparing the contributions of different presumed manifolds in the transport equation. Alternatively, Wu and Ihme[5] determined the combustion mode via the drift from the presumed manifolds. These flame indices rely on resolved gradient to identify the combustion mode. However, the combustion mode is determined by the subgrid mixing process by definition.

In the present work, we propose a new Lagrangian-based flame index for the transported probability density function (PDF)method [16,17] . The transported PDF method models the molecular transport via the pairwise mixing of Lagrangian particles [16–19] . Accordingly, we propose to determine the combustion mode by the mixing status of the Lagrangian particles, using the Burke-Schumann solution as a reference. We tested the flame index with constructed partially premixed reactors and LES/PDF datasets of partially premixed jet flames. In addition to identifying the combustion mode, the new flame index provides a consistent way to model the scalar mixing rate adaptively for partially premixed flames.

whereφOdenotes the composition vector in the oxidizer stream,vcand vZarenc-dimensional vectors corresponding to the coefficients incandZdefinitions. In the present work, the progress variable is defined based on the product species mass fractions as[20]c=YCO2+YCO +YH2O +YH2, and the mixture fraction is calculated using Bilger’s formula [21] . ThecandZvariables are used as the primary reactive coordinate system, and their subgrid mixing fluxes provide the key information regarding the flame index.

As a simple example to show the trajectory of a Lagrangian particle in the(c,Z)space, we consider a one-dimensional counterflow flame with partially premixed boundary conditions of?lat the lean oxidizer side and?rat the rich fuel side. The counterflow flame solutions are obtained by Cantera [22] employing the GRI-3.0 mechanism [23] . The boundary temperature is set at 300 K for both sides and the strain ratea= 200 s?1.

Fig. 1. Progress variable against mixture fraction in counterflow flames with different equivalence ratio of lean and rich sides, at strain rate a = 200 s ?1 . The Burke-Schumann solution is plotted with the dotted line.

Figure 1 shows such trajectories of the Lagrangian particles in methane flames with different parametric pairs of(?l,?r). A limiting condition of pure nonpremixed flame(?l,?r)=(0,∞), at the Burke-Schumann limit, is shown as the dotted line. As partial premixing is added to the oxidizer boundary,e.g.,for?l= 0 to 0.6,the emergence of a premixed flame is noted by a distinctly steeper gradient in the curve in the lowerZrange. A similar statement can be made for the fuel side partial premixing,e.g.,for?r= ∞ to 1.3 shown in Fig. 1 , although the premixed mode tends to penetrate much deeper into theZstwith the fuel side premixing. Note that Fig. 1 only shows the solutions with?l= 0 or?r= ∞ , while there could be flames have both sides partially premixed.

In a turbulent reacting flow, each Lagrangian particle travels through the flow field, varyingcandZ. As evidenced in Fig. 1 ,direction of the Lagrangian particle movement in thec?Zspace indicates its combustion mode. A comparison of mixing fluxes oncandZdirections can be normalized to [ ?1,1] as

whereZst= 0.055 is the mixture fraction at stoichiometry,ZF=0.075 is the flame surface location predicted by a modified flame sheet solution that considers the intermediate species of CO and H 2 [24] . Note that the values ofβdepends on the definitions ofcandZ.

For particles in premixed flames, | dc/ dZ| = ∞ . This leads toγ=1 by Eq. (4) . As for particles in nonpremixed flames at the Burke-Schumann limit, | dc/ dZ| =βmakesγ= 0 . Trajectories of particles in partially premixed flames haveβ< | dc/ dZ| < ∞ . Therefore, particles in partially premixed flames have 0 <γ< 1 . Furthermore, nonpremixed flames bearing different strain effects have 0 < | dc/ dZ| <β, yielding ?1 <γ< 0 . And | dc/ dZ| = 0 for the cold mixing or extinction case, makingγ= ?1 . Note that analysis of turbulence-flame interactions and local auto-ignition needs the assistance of other tools, such as the transport budget and chemical mode analysis [9–13,25–27] .

In turbulent combustion modeling with the Lagrangian particles, a computational cell is represented as an ensemble of particles with different composition status. With the particle statusγ,a Lagrangian-based flame index (FI) is then determined as

Fig. 2. FI obtained with the presumed beta PDF of mixture fraction, for ( a ) ? r = ∞ and ( b ) ? r = 2 . 4 .

For the flame index in Eq. (6) ,γis integrated over the composition space. No distribution of the progress variable or mixture faction is presumed. The combustion mode of the flame is identified following the criteria forγ. In summary,

With the combustion mode information, we can model ?ωφin Eq. (2) adaptively for LES/PDF simulations.

In the followings, we present validations for the newly proposed flame index with two configurations. Constructed partially premixed reactors with presumed low-dimensional manifold and PDF are employed to validate the performance of the flame index at first. Then we apply the flame index to LES/PDF datasets of realistic partially premixed jet flames.

Here we apply the flame index to the LES/PDF datasets of turbulent partially premixed flames. The target flames DME-D and DME-F are piloted partially premixed flames at different Reynolds numbers [31] . The flames have a central jet of DME and air mixture in a volumetric ratio of 1:4, with a diameter ofD= 7.45 mm.The jet nozzle is surrounded by pilot flames with compositions and temperature corresponds to a premixed DME-air flame at equivalence ratio of 0.6. The LES/PDF simulations on the DME flames[32] are conducted in a computational domain of 60D×30D×2 π in the axialx, radialr, and azimuthalθdirections, discretized with a non-uniform structured grids with grid sizes of 128 ×72 ×32 . The low-Mach number, variable density Navier-Stokes equations for mass and momentum are solved in the LES. The hybrid mesh/particle PDF code [33] solves the position and composition evolution of the Lagrangian particles.np= 25 nominal particles are used in each cell. The transported specific volume approach [34] is employed to have two-way coupling between the LES and PDF solvers. In the present study, results of LES/PDF simulations using the IEM model are taken as examples. More details on the experiment setup and LES/PDF simulations can be found in Refs. [31,32] .

Fig. 3. Flame index of constructed cells with different fuel side partial premixing from ? = ∞ to 2.4. Solid lines are FI calculated with mixing models, dash dot lines are FI obtained from manifold.

Fig. 4. Snapshots of the contour of temperature (upper half) and flame index (lower half) on the x ?r plane cut from LES/PDF of ( a ) DME-D and ( b ) DME-F flames with the IEM model. Dash lines are the contours of the stoichiometric mixture fraction Z = Z st .

Figure 4 shows instantaneous contours of temperature and flame index for the DME-D and DME-F flames. We see that the proposed flame index can clearly separate the partially premixed mode on the fuel size, nonpremixed mode on the oxidizer side,and cold mixing/extinction for both DME-D and DME-F flames.With the high Reynolds number condition, the DME-F flame exhibits local extinction. As shown in the zoom-in subplots in Fig. 4 ,the temperature contours indicate extinction pockets, and the flame index detects the extinction asFI< 0 in the corresponding region. At the same time, the DME-D flame show continuos temperature and flame index profiles. Applications on the realistic turbulent partially premixed flames show that the proposed flame index is able to identify the combustion mode successfully. This can help to improve the modeling adaptively.

We propose a new flame index compatible with the transported PDF method to identify the combustion mode in a computational cell for turbulent combustion simulations. The flame index is based on the ratio between mixing of progress variable and mixture fraction in subgrid, which is modeled explicitly by the mixing models in PDF simulations. Constructed subgrid particles with the flamelet concept are employed to test the flame index for different partially premixed flames and pure nonpremixed flames. The test results indicated that for a precisely described particle distribution,different mixing models yield correct trend to report the combustion mode. We then apply the flame index to analyze LES/PDF simulations of turbulent partially premixed flames. It shows that the proposed flame index identifies different combustion modes of premixed, nonpremixed, partially premixed, and extinction successfully.

The proposed flame index not only serves as a tool to analyze the characteristics of partial premixed flames. It also allows us to model the turbulent partially premixed combustion adaptively with the Lagrangian particle approach. Development and implementation of refined mixing models and validation with experimental/DNS data requires additional studies in the future.

Declaration of Competing Interests

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

This work was sponsored by King Abdullah University of Science and Technology (KAUST) and the National Natural Science Foundation of China (Grant No. 91841302). The simulations utilized the computing resources at KAUST Supercomputing Laboratory.