Towards a priori models for differential diffusion in turbulent non-premixed flames

Introduction

Background

Most of the world's energy needs are met by the combustion of fossil fuels, which account for approximately 80% of global energy production [1]. In partial premix combustion, the fuel and oxidant are partially mixed before combustion takes place.

Challenges in turbulent combustion

Differential diffusion effects and soot formation
Computational cost of detailed species transport
Reduced-order modeling of turbulence-chemistry interaction 6

Only the use of the flame equations with unit Lewis number shows a good agreement (See Fig. 3.17 in [16]). For the latter, a unit Lewis number flame calculation was found to be representative of the flame structure.

Figure 1.1: Diagram illustrating turbulence-chemistry interactions (adapted from Bray [9].)

Summary of contributions

Average flame position (a) and fuel consumption-based flame speed (b) for the four simulations of the two-dimensional lean H2/air flame. A more quantitative comparison of the four test cases (i)–(iv) is made in Figure 9 , which shows different turbulence statistics for the fuel mass fraction and its (normalized) source term. Figures 9(a) and 9(b) show a comparison of the area-weighted conditional means of the fuel mass fraction and its normalized source term for the four cases (i)–(iv), respectively.

Theoretical Framework and Computational Methodology

Direct Numerical Simulations

Governing equations
Radiation model
Numerical approach

The scalars (Yi, T) are transported using the third-order exact boundedness scheme QUICK (BQUICK) [97], which guarantees the boundedness of the transported scalars. The flamelet model 18estimates the order of accuracy of the preconditioned iterative method for integra-.

The flamelet model

Generalized coordinate transformation
Transformed species transport equations
Convection in mixture fraction
One-dimensional curved flamelets
One-dimensional flat flamelets
Flamelet-based modeling of turbulent non-premixed com-

The test case with unit Lewis numbers leads to large differences in the structure of the flame. An a posteriori analysis of the grid resolution for chemistry will be performed in section 5.6.

Figure 2.1: Two-dimensional schematic of the coordinate transformation (adapted from Xuan & Blanquart [82]).

Assessment of the Constant Non-Unity Lewis Number As-

Introduction

In addressing these issues, the full spectrum of combustion configurations must be considered, including (but not limited to) the differences between premixed and diffusion flames, laminar and turbulent flames, and light and heavy fuels. While some theoretical analysis will be done using one-dimensional flames, the goal is not to suggest changes to one-dimensional steady-state codes (such as Chemkin, FlameMaster, and Cantera).

Test cases and chemical models

Selected flames
Lewis numbers
Chemical mechanisms

Two sets of constant Lewis numbers (test cases (ii) and (iii)) are extracted from one-dimensional FlameMaster simulations. In test case (iii), the Lewis numbers of the intermediates are taken at their highest mass fractions, while the highest temperature criterion is used for the reactant/product species.

Premixed hydrogen/air flame

One-dimensional preliminary analysis
Two-dimensional simulations
Three-dimensional simulations

Variation of the Lewis numbers by the lean laminar premixed H2/air flame (a), the fuel mass fraction (b), and the source term (c) against temperature for mixture-average transport (black), Unit Lewis numbers (red), and Lewis numbers corresponding to Tmax(blue) and Yi,max(green). Mean flame position (a) and fuel consumption based flame speed normalized by the respective laminar, unstretched flame speed (b), for the four simulations (i) – (iv) of the three-dimensional lean H2/air flame.

Figure 2. Variation of the Lewis numbers through the lean laminar premixed H 2 /air flame (a), fuel mass fraction (b) and its source term (c) against temperature for mixture-average transport (black), unity Lewis numbers (red), and Lewis numbers correspond

Three-dimensional simulations

Finally, Figures 11(c) and 11(d) show a comparison of the PDFs for the normalized fuel mass fraction and its normalized source term, respectively. Figures 3.14 and 3.14b show a comparison of the area-weighted conditional mean values of the fuel mass fraction and its normalized source term for the four cases (i)-(iv), respectively.

Laminar diffusion ethylene/air flame

Description of the configuration
Two-dimensional simulations

The test case with unity Lewis numbers leads to large differences in flame structure. b) Benzene Lewis number (red) and mass fraction (black). The flame heights for the experimental data and test cases (i)–(iv) are shown in Table 5. Figure 15(a) shows that the shape of the temperature profiles does not change significantly.

Figure 11. Turbulence statistics of the four three-dimensional heptane–air flames corresponding to test cases (i)–(iv)

Computational cost

Scaling of the time required to calculate the diffusion coefficients with the number of species. Most parts of the code (eg velocity solver, scalar transport) scale linearly with the number of grid points.

Figure 17. Scaling of the time required to compute the diffusion coefficients with the number of species

Summary

For the streamwise velocity component, an excellent agreement with the refined cases was found at the base of the jet (x/d/15). The slope of the points calculated from case RAD is less than that observed for the experiments.

Effective Lewis numbers in turbulent non-premixed flames

Introduction

Second, there is limited work to accommodate differential diffusion effects within simulations where turbulence-chemical interactions are unresolved, as in LES (see Chapter 1). In their experiments, species measurements show a marked decrease in differential diffusion effects with (i) increasing axial distance from the exiting nozzle and (ii) increasing jet Reynolds number.

Review of experimental data

Z is the difference between Peters' definition of the mixture fraction and ZTNF, calculated by (i) the measured species only (solid line) and (ii) the measured species with YCH4 calculated by Eq. To take into account the disturbances in the CH4signal, a modified form of Eq. 4.1) is considered, where the CH4 mass fraction is calculated as.

Figure 4.1: Comparison of mixture fraction definitions computed using the optimal flamelet (see Sec

Extracting effective Lewis numbers

Model for the effective Lewis numbers
Flamelet equations
Error maps
Optimal flamelet parameters

Extracting Effective Lewis Numbers 53 In Eq. 4.4), the “correction terms” include diffusion, correction diffusion and differential diffusion contributions. As can be seen, the shape of the imposed χTNF profile is close to the experimentally measured profiles.

Figure 4.3: Colormaps of L 2 error by Eq. (4.8) for flames B at x/d = 15 (left), C at x/d = 30 (center), and E at x/d = 45 (right)

Discussion

Experimental uncertainties
Spatial resolution
Chemical model
Choice of error norms
Choice of species for the error norm
Additional biases

In summary, while the experimental uncertainties are large (see Table 4.2), the influence of the effective Lewis numbers is still larger. To investigate the effect of the choice of error metric in the present analysis, the following rates L1andL∞ are considered in addition to Eq.

Figure 4.7: Comparison of H 2 mass fraction conditioned on Z TNF for flame C at x/d = 30, against the optimal solution (black line), and the filtered optimal solution (blue line).

Review of scaling based on the Reynolds number

Scaling based on the Reynolds number
Estimating turbulent quantities
Assessment of the scaling based on the Reynolds number

Assessment of scaling based on the Reynolds number 62. The optimal parameters obtained using the L1 and L2 standards are generally close to each other. As you can see, the rF values for the different flames remain close throughout the entire flame length, with small differences at the tip.

Figure 4.10: Radial flame location r F (left) for flames B through E (green, red, blue, and black lines, respectively), and jet half-width r 12 (right) for flames D and E (blue and black lines, respectively)

Scalings based on the Karlovitz number

Turbulent scales interacting with the flame
Flame length scale-based scaling
Flame time scale-based scaling
Assessment of the scalings based on the Karlovitz number . 71

The data file storage rate for the enhanced cases is five times that of the base case (see Section 5.4). Significant differences between the BL cases and the RAD case are observed on the rich side of the mixture.

Figure 4.16: Optimal χ st values normalized by (U jet − U pilot )/d (left) and U jet /d (right), as a function of downstream direction for flames B, C, D, and E.

Direct Numerical Simulation of Sandia flame B: grid resolu-

Introduction

Yooet al.[50] investigated the stabilization mechanism of a turbulent lifted ethylene jet flame (Reynolds number of 10,000) using more than 1.29 billion grid points. However, such a naive approach would lead to a large number of grid points without clear advantages.

Boundary conditions

The fuel pipe flow is repeated every max, and the composition is taken directly from the reported values [72]. Finally, no-slip and no-penetration conditions are used for the fuel pipe and pilot walls.

A priori design of the grid

Resolving the turbulence
Resolving the chemistry
Baseline grid

The determination of this factor requires an assessment of the axial growth of the smallest turbulence scales. Thus, the average grid size should be considered instead of ∆r for preliminary estimation of grid resolution for chemistry.

Figure 5.1: Top: burning branch of the S-shaped curve obtained solving flamelet equa- equa-tions [66] using the FlameMaster code [79] with the GRIMech3.0 chemical model [135].

Test cases and total run time

More specifically, for the BL case 243.0d/Water units of simulation time are considered, while for each of the refined cases only 45.5d/Water of simulation time are used, due to the much higher computational cost. Each data file obtained with the baseline network is 11.3 GB, while the data files from each of the refined cases is twice that amount.

Figure 5.3: Instantaneous (left) and averaged (right) temperature field for case BL. The averaging is carried out both in time and the azimuthal direction

Convergence of statistics

Randomness and time-correlation
Confidence intervals

For both quantities, the mean of the ratioσN/hXiN initially grows with the sample separation time, and then plateaus when the separation time is greater than the decorrelation time. As expected, for large sample separation times, the standard deviation becomes independent of the sample size and the separation distance.

Figure 5.5: Solid lines: two-time auto-correlation for the centerline U for x/d = 1 (black), x/d = 10 (blue) and x/d = 30 (red)

Grid independence

A posteriori analysis of the base of the jet
Unconditional statistics
Flame structure

The central rms values are equal to zero for the refined cases at x/d=45 and different for the BL case. To investigate whether the differences between the BL case and the refined cases shown in Fig. 5.16 are grid resolution effects, the conditional median and rms profiles for the same species are considered in Fig. 5.22.

Figure 5.8: Contour maps of ∆x/η (a), ∆r/η (b), r∆θ/η (c), and ∆/η (d), for the base of the jet (case BL)

Summary

Several reasons could explain the discrepancies between the values of χoptst from case RAD and those from measurements of flame B, including experimental uncertainties and biases introduced by the models used (eg the chemical model). Figure 7.13 shows a comparison of radial profiles of the mean flow velocity and the effective velocity for case RAD and the measurements for flames D and E. The location of rF is also shown.

Validation of the DNS with experimental data

Assessment of the optically-thin assumption for case RAD

RADCAL is the recommended radiation model for the target flames of the TNF workshop [72], and is very popular, due to the ease with which radiation heat transfer can be included in simulations. Thus, the correct radiation case is expected to be in "between" cases BL and RAD.

Unconditional statistics

Radial profiles
Centerline profiles

Conditional statistics

Scalar profiles
Probability density functions
Differential diffusion parameter
Scalar dissipation

To determine the convergence of the statistics, the mean and the wg values are calculated for the last 200d/Ujet (solid lines) and the last 100d/Ujet (dashed lines) units of simulation time. When comparing unconditional statistics from different simulations, one must ensure that the simulation time is sufficient to "average out" the relatively slow, large spatial structures.

Figure 5.14: Centerline mean and rms velocity (left) and mixture fraction (right). Solid black, BL; green, Res-x; red, Res-r; blue, Res-z

Summary

As a result of the differences for the eu0andl values from the RAD case compared to the fits for the higher Reynolds number flames, the slope of γopt was found to be lower than that of the C-E flames. Due to the missing species effect, the UL case was found to be non-zero at all four downstream locations.

Analysis of effective Lewis numbers from the DNS of San-

Extracting effective Lewis numbers

Methodology
Error maps
Optimal flamelet parameters

The values extracted from case RAD are lower than the values extracted from the measurements of flame B. The values extracted from case RAD are lower than those from the measurements of flame B.

Figure 7.1: Contour maps of the L 2 error, given by Eq. (4.8), for case RAD, at x/d = 7.5 (left), x/d = 15 (center), and x/d = 30 (right)

Scalings for γ

Turbulence and flame quantities
Scaling based on the Reynolds number
Scaling based on the Karlovitz number

In (b), the downstream location of the measured points for flame B is normalized by the stoichiometric flash point location from case RAD (xF≈59.3d). Again, the agreement between case RAD and the flames with higher Reynolds number improves once the downstream distance is normalized by the location of the flame tip.

Figure 7.8: Comparison of the optimal scalar dissipation rate, χ opt st with the values h χ| Z st i from case RAD, as a function of the downstream direction x/d.

Summary

The mean profiles were found to be in good agreement at the base of the jet (x/d/15), and small differences were observed moving downstream up to x/d≈45. The large differences observed beyond x/d≈45 were found to be due to the limited run time of the refined cases, compared to the BL case.

Conclusions and Outlook

Assessment of the constant Lewis number assumption

For the two cases of constant non-unity Lewis numbers, the Lewis numbers were precalculated using the solutions to the flame equations. Finally, the two constant non-unit Lewis number methods were found to perform well, with neither consistently better than the other.

Effective Lewis numbers in turbulent non-premixed flames

Direct Numerical Simulation of Sandia flame B

Comparisons of radial velocity profiles from the baseline and refined cases showed good agreement for the bottom of the jet. While the mean CH profiles are in excellent agreement for the first station, some shape and size differences were observed for the other stations.

Validation with experimental data

Both stations, the RAD case was found to be in very good agreement with experiments. Differential diffusion was found to have a greater influence on flame structure than radiation heat loss.

Analysis of effective Lewis numbers for the DNS of Sandia flame B . 153

Direct Numerical Simulations of Sandia flame B
Assessment of the flamelet assumptions
A posteriori assessment of models for γ

Limitations and directions for future work 156 The cross-dependence of optimal flame parameters is described a priori. First, table chemistry DNS should be performed on the same flames used to develop these priority models.

The TNF Workshop

Scalar and velocity measurements

As a result, a richer set of experimental data exists for these flames, while fewer measurements are available for flames A-C [72]. It has the same chemistry and burner geometry as the higher Reynolds number flames, and can be used to investigate the effect of modeling choices.

Chosen test cases

In this appendix, Sandia flame A is considered to investigate the effects of radiative heat loss, thermal diffusion, and an alternative choice for the chemical model.

Effect of radiation heat loss

Comparison of chemical models

Effect of thermal diffusion

Modeling of thermal diffusion
Comparisons

Symbols: try; black line, ABL; red line, ARAD+TD. where kTi is the thermal diffusion ratio of the ith kind [181]. Figure C.5 shows the Favre-averaged radial profiles, where it can be seen that thermal diffusion has only a small effect on the scalars.

Figure C.5: Comparison of Favre-average temperature (left) and species mass fractions (right)

Effect of radiation and thermal diffusion

The impact of detailed transport and thermal diffusion effects of multiple components on soot formation in ethylene/air flames.Proc. The effect of flame structure on soot formation and transport in turbulent, non-premixed flames using direct numerical simulation. Combustion.