^{1}and Heinz Pitsch

^{1,2}

### Abstract

The small-scale interactions between turbulence, chemistry, and soot have a profound effect on the soot formation, growth, and destruction processes in turbulent reacting flows. In this work, the small-scale subfilter interactions between turbulence and soot are modeled using a presumed subfilter PDF for the statistical moments of the soot number density function. Due to a very large (infinite) Schmidt number, soot is confined to very thin structures. In addition, soot is formed initially from Polycyclic Aromatic Hydrocarbons, which exhibit a strong sensitivity to the local scalar dissipation rate in the flow field. These interactions of soot with the gas-phase chemistry, molecular transport, and turbulence result in very high spatial and temporal intermittency. Therefore, the soot subfilter PDF is presumed to be a pair of delta distributions with a sooting mode and a non-sooting mode. In addition to the mean values of the scalars, one additional parameter is needed to specify this distribution. The presumed soot subfilter PDF approach is evaluated *a priori* using a recent two-dimensional direct numerical simulation (DNS) database of soot evolution in a nonpremixed flame. The analysis shows that predictions of the soot intermittency as well as the soot source terms are improved with this presumed soot subfilter PDF approach compared to simply using the mean values of the soot scalars. Several choices for specifying the additional parameter of the PDF are also evaluated with the database. The results show that the parameter is best specified using the second moment of the soot number density. In addition, the transport equation for the second moment of the number density is briefly discussed. A model for the lone unclosed term is proposed and evaluated with the DNS data.

The authors gratefully acknowledge funding from the National Aeronautics and Space Administration (NASA) and the Strategic Environmental Research and Development Program (SERDP). M.E.M. gratefully acknowledges funding from the National Defense Science and Engineering Graduate (NDSEG) fellowship program.

I. INTRODUCTION

II. MODELING FRAMEWORK

A. Soot modeling

B. Presumed PDF approach

III. SOOT SUBFILTER PDF

IV. A PRIORI ANALYSIS

A. DNS database

B. Total intermittency

C. Coagulation source term

D. Evaluation of subfilter intermittency

V. SECOND MOMENT TRANSPORT EQUATION

VI. CONCLUSIONS

### Key Topics

- Intermittency
- 63.0
- Photon density
- 34.0
- Turbulence simulations
- 29.0
- Turbulent flows
- 18.0
- Chemical thermodynamics
- 17.0

## Figures

Filtered total intermittency at *t* = 15 ms for the various evaluations of the sub-filter intermittency for a filter size of Δ/*h* = 64. The lines correspond to a perfect model, that is, the filtered DNS itself. The sample mean and normalized sample error are indicated on each of the plots; the exact mean from the filtered DNS is 0.7510. For the single delta distribution, the maximum DNS intermittency for an LES intermittency of zero is about 0.98; the minimum DNS intermittency for an LES intermittency of one is about 0.65.

Filtered total intermittency at *t* = 15 ms for the various evaluations of the sub-filter intermittency for a filter size of Δ/*h* = 64. The lines correspond to a perfect model, that is, the filtered DNS itself. The sample mean and normalized sample error are indicated on each of the plots; the exact mean from the filtered DNS is 0.7510. For the single delta distribution, the maximum DNS intermittency for an LES intermittency of zero is about 0.98; the minimum DNS intermittency for an LES intermittency of one is about 0.65.

CDF of the filtered total intermittency for a filter width Δ/*h* = 64 at the time *t* = 15 ms. The solid line is the DNS data; the dashed line is the single delta distribution; the dotted line is the double delta distribution with ω evaluated from the total number density *M* _{0,0}; and the dashed-dotted line is the double delta distribution with ω evaluated from the total soot volume *M* _{1,0}.

CDF of the filtered total intermittency for a filter width Δ/*h* = 64 at the time *t* = 15 ms. The solid line is the DNS data; the dashed line is the single delta distribution; the dotted line is the double delta distribution with ω evaluated from the total number density *M* _{0,0}; and the dashed-dotted line is the double delta distribution with ω evaluated from the total soot volume *M* _{1,0}.

Mean filtered total intermittency and normalized error of the filtered total intermittency from the subfilter model as a function of the filter width at *t* = 15 ms. The solid line is the single delta distribution, and the dashed and dotted lines are the double delta distribution with the subfilter intermittency evaluated from *M* _{0,0} and *M* _{1,0}, respectively. The dashed-dotted line corresponds to the mean filtered total intermittency as evaluated from the DNS.

Mean filtered total intermittency and normalized error of the filtered total intermittency from the subfilter model as a function of the filter width at *t* = 15 ms. The solid line is the single delta distribution, and the dashed and dotted lines are the double delta distribution with the subfilter intermittency evaluated from *M* _{0,0} and *M* _{1,0}, respectively. The dashed-dotted line corresponds to the mean filtered total intermittency as evaluated from the DNS.

Filtered source terms at *t* = 15 ms for the various models for the sub-filter intermittency for a filter size of Δ/*h* = 64. The lines correspond to a perfect model. The sample geometric standard deviation is indicated on each of the plots.

Filtered source terms at *t* = 15 ms for the various models for the sub-filter intermittency for a filter size of Δ/*h* = 64. The lines correspond to a perfect model. The sample geometric standard deviation is indicated on each of the plots.

CDF of the filtered number density coagulation source term for a filter width of Δ/*h* = 64 at the time *t* = 15 ms. The lines are the same as in Fig. 2. The solid and dotted lines are nearly indistinguishable.

CDF of the filtered number density coagulation source term for a filter width of Δ/*h* = 64 at the time *t* = 15 ms. The lines are the same as in Fig. 2. The solid and dotted lines are nearly indistinguishable.

Geometric standard deviation of the filtered source term from the subfilter model as a function of filter width at *t* = 15 ms. The different lines are the same as in Fig. 3. The single delta distribution peaks at a standard deviation of about 30 at a filter width Δ/*h* = 256.

Geometric standard deviation of the filtered source term from the subfilter model as a function of filter width at *t* = 15 ms. The different lines are the same as in Fig. 3. The single delta distribution peaks at a standard deviation of about 30 at a filter width Δ/*h* = 256.

Exact correlation between the second moment of the number density and the divergence of the velocity and the proposed model (Eq. (19)) from the filtered DNS data at *t* = 15 ms. The lines correspond to a perfect model. The normalized sample error is indicated in each of the plots.

Exact correlation between the second moment of the number density and the divergence of the velocity and the proposed model (Eq. (19)) from the filtered DNS data at *t* = 15 ms. The lines correspond to a perfect model. The normalized sample error is indicated in each of the plots.

Model results for the filtered product of the number density and its coagulation source term. In the scatter plot, the line is a perfect model, and the geometric standard deviation is indicated. In the CDF, the solid line is the DNS, and the dashed line is the LES model. The two lines are nearly indistinguishable, especially at larger values of the product. Both plots correspond to the time *t* = 15 ms and a filter width Δ/*h* = 64.

Model results for the filtered product of the number density and its coagulation source term. In the scatter plot, the line is a perfect model, and the geometric standard deviation is indicated. In the CDF, the solid line is the DNS, and the dashed line is the LES model. The two lines are nearly indistinguishable, especially at larger values of the product. Both plots correspond to the time *t* = 15 ms and a filter width Δ/*h* = 64.

## Tables

Relevant parameters of the two-dimensional DNS of Bisetti *et al.* (Ref. 8).

Relevant parameters of the two-dimensional DNS of Bisetti *et al.* (Ref. 8).

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