In this study, two series of pressurized turbulent jet sooting flames at 1, 3, and 5 bar with either fixed jet velocity or fixed Reynolds number are simulated to study the pressure effects on soot formation and evolution. Through a radiation flamelet progress variable approach with a conditional soot subfilter probability density function (PDF) model to consider the turbulence–chemistry–soot interactions, quantitatively good agreements are achieved for soot volume fraction (SVF) predictions compared with the experimental data, regardless different turbulent intensities and residence times. SVF source terms are then discussed to show the pressure effects on nucleation, condensation, surface growth, and oxidation at different axial positions in these flames. It is found that surface growth and oxidation increase by about three orders of magnitude from 1 to 5 bar, while nucleation and condensation only increase within one order of magnitude. The stronger SVF scaling on pressure than measured data is found to be attributed to the inaccurate surface growth and oxidation scaling on pressure. Further analysis indicates that (i) the uncertainty of C_{2}H_{2} prediction at elevated pressures is likely a major reason for the too strong surface growth scaling; and (ii) taking account of pressure effects in the conditional subfilter PDF modeling for turbulence–soot–chemistry interactions is likely a key to improve oxidation prediction. The results in this study open up the possibilities for improving future turbulent sooting flame modeling by improving C_{2}H_{2} chemistry and turbulence–chemistry–soot modeling at elevated pressures.

## I. INTRODUCTION

Soot, which could be produced in energy conversion and propulsion systems by combustion, is a major pollutant in air as the form of particulate matter.^{1} In addition, soot is also a major agent responsible for global warming.^{2} As a result, understandings on its formation and evolution in reacting flows are of significant practical interest. A variety of studies (either numerically or experimentally) have been conducted on the formation, growth, and oxidation processes of soot, most of which are based on laminar sooting flames due to the simple and well-controlled flow conditions. To be noted, most of these laminar sooting flame studies are focused on atmospheric pressure conditions.^{3}

To achieve higher efficiency, combustion is nowadays pushed to extremely high-pressure limits, such as supercritical combustion in sCO_{2} combustors and high-pressure combustion in pressure gain engines (e.g., rotating detonation engines). As such, it is fundamentally needed to understand the pressure effects on soot formation and evolution in combustion.^{4} A variety of pressurized sooting flame experiments have been conducted and clustered in the International Sooting Flame (ISF) workshop, together with different computational work toward these target flames (TF).^{5–7} In addition, Gulder and his co-workers measured sooting characteristics at high laminar flames for various fuels.^{8,9} At high-pressure conditions, one major concern is the dramatically increased soot emissions, as experimentally observed in many pressurized laminar^{10,11} and turbulent sooting flames.^{12,13} To explain the reason for high soot formation at high pressures, there exist a few computational works by using detailed chemistry^{14} and particle models.^{15–18}

For example, the computational works focused on the target pressurized laminar flames in ISF, such as the C_{2}H_{4}-fueled pressurized flames in the ISF Workshop 4 (ISF-4), including the target flame (TF) 2 (laminar pressurized diffusion flames with pressure range from 1 to 16 atm),^{6,12,19} TF3 (laminar premixed pressurized flames with pressure range from 0.1 to 3.0 MPa),^{20} and TF4 (laminar premixed pressurized flames with pressure range from 0.1 to 0.5 MPa).^{21} These laminar pressurized sooting flame studies indicate that together with narrower flame cross-sectional areas due to the enhanced buoyancy effect at elevated pressures, the flame centerline velocity is unchanged at elevated pressures. In this sense, the residence time is not likely to be the main reason for soot formation increase. Guo *et al.*^{7} indicate that the increased soot production at elevated pressures is due to the increasing contribution from large-sized polycyclic aromatic hydrocarbon (PAH) concentration. Recently, Zhou and Yang^{22} developed a diagnostic tool called soot-based global pathway analysis (SGPA) to track the controlling chemical reaction pathways from fuel to soot and found that elevated pressure significant affects the reaction pathways for C_{2}H_{2} and PAH production. As seen, the pressure effects on soot formation and evolution have been studied in several laminar flames both experimentally and numerically.

However, without considering the effect of turbulence, it is invalid to apply the above analysis to explain the sooting tendency in realistic high-pressure turbulent combustion devices. For this reason, studies on sooting characteristics in pressurized turbulent flames are needed. There exist a few experimental and computational works investigating the pressure effects on turbulent non-premixed flames without soot.^{23,24} However, there lack studies on the soot formation and evolution in pressurized turbulent flames due to two reasons. First of all, it is well recognized that developing predictive models for turbulent sooting flames is very challenging due to the complicated interactions between turbulence, detailed chemistry, and soot.^{25} Investigations on turbulent sooting flames rely on the direct numerical simulations (DNS) on sooting flames^{26–29} and the proposed advanced turbulence–chemistry–soot interaction modeling.^{30–32} For example, the DNS by Attili *et al.*^{27,28} indicates the high intermittence of soot spatial distribution, motivating the development of two delta functions to represent the non-sooting and sooting modes within the large-eddy simulation (LES) filter width.^{30} Furthermore, the DNS results by Attili *et al.*^{27} show fast oxidation in fuel-lean regions, which then motivates the development of the conditional soot subfilter probability density function (PDF) model.^{31} Second, measurements of key parameters such as temperature and major species in turbulent flames are significant for the comparison and validation of computational models. Although there are a variety of measurement of temperature and major species in atmospheric turbulent sooting flames, the measurement of these parameters is not easy for pressurized flames, due to the high soot volume fraction (SVF) at elevated pressures and thus the interfered combustion zone. Early work by Brookes and Moss^{33} measured SVF in a methane diffusion flame at 1 and 3 atm. A numerical simulation investigation toward this measurement by Roditcheva and Bai^{34} used an empirical soot model to study the pressure effects on turbulent methane jet sooting flames and found that the major pressure effect on soot yield is due to the enhanced soot surface growth. The successful measurement in a swirl gas turbine model combustor (DLR) with elevated pressures is another one of the few available soot data for pressurized turbulent sooting flames.^{13} The computational work in Chong *et al.*^{35} studied the pressure and dilution-jet effects on soot formation in this swirl gas turbine combustor. In this numerical study, SVF prediction has more than one order of magnitude error when comparing with experimental data at high pressures. It is reported in this study that the discrepancy is due to the chemical/physical models uncertainty, which, however, was not discussed in detail. To obtain more fundamental understanding of the pressure effects on soot formation and evolution in pressurized turbulent sooting flames, a simpler configuration without too complex geometry (e.g., gas turbine combustor) and too complex flow (e.g., swirling flow) for fundamental studies is needed.

Recently, Boyette *et al.*^{36,37} established a high-pressure turbulent sooting jet flame configuration (named as “KEN flames”) by diluting fuel C_{2}H_{4} with N_{2} and measured the soot volume fraction at three different pressures (1, 3, and 5 bar), with fixed jet velocity and fixed Reynolds number, respectively. This setup allows a non-sooting region for temperature and major species measurements. Tian *et al.*^{38} simulated KEN flames with constant Reynolds number, via a transported probability density function model and a sectional method for soot modeling. Good agreements have been achieved in their simulations by updating and tuning the nucleation reaction from C_{2}H_{2} to pyrene. In this study, we simulated the pressurized KEN flames with the pressure ranges of 1–5 bar with both fixed jet velocity and fixed Reynolds number conditions. A new conditional soot subfilter probability density function (PDF) model proposed by Yang *et al.*^{31} is used in this study to account for the turbulence–soot–chemistry interactions. At different pressures, prediction accuracy within a factor of up to six (and much smaller for some pressures) is achieved, better than turbulent soot prediction results in the literature,^{35,39,40} where a typical more than one order of magnitude difference was shown in these studies. Pressure effects on soot formation and evolution are then analyzed and discussed.

## II. METHODOLOGY AND SETUP

In this section, the experimental methodology (Sec. II A) and computational (Sec. II B) methodology to study the pressure effects on sooting characteristics in turbulent flames are introduced. In addition, the experimental setup (Sec. II A) and computational setup (Sec. II C) specifications for KEN flames are also shown in this section.

### A. Experimental Methodology

The experimental setup of KEN flames used in the current study is detailed in Boyette *et al.*^{37} It is briefly repeated here for completeness. The experiments were performed in a 8-m-tall vertical vessel. The inner diameter is 410 mm. Six ultraviolet fused silica windows sit on the test section. KEN flames are designed as turbulent piloted jet flames, where the fuel jet is composed of ethylene diluted with 65% (by volume) nitrogen. The co-flow air is issued at 294 K from a 250 mm co-flow nozzle. The fuel central inner inlet diameter is *D* = 3.4 mm. KEN flames are stabilized by a pilot flame. The piloted flame mixture is premixed C_{2}H_{4}/air with an equivalence ratio of 0.9. The surrounding co-flow is air, and the co-flow diameter is 250 mm. An Nd:YAG laser was used for igniting the pilot flames.

The jet velocity (*U _{j}*), air co-flow velocity (

*U*), and Reynolds number (

_{c}*Re*, defined by bulk jet velocity and central inner inlet diameter) of all the measured flames in this study are listed in Table I. The base flame (case 1) is an atmospheric pressure flame with a bulk jet Reynolds number

*Re*= 10 000, where the pilot provides 6% of the jet heat release. Two series of flames are measured to study the effects of pressure. The first series keeps the Reynolds number fixed, while the pressure increased to 3 bar (case 2) and 5 bar (case 3). To prevent flashback of the pilot flame in this series of flames at 3 and 5 bar, a pilot flame providing 12% and 18% of the heat release of the jet is used, respectively. In the second series, the jet velocity is unchanged when increasing pressure, but the Reynolds number is changed with the increased pressure to 30 000 (case 4) and 50 000 (case 5) due to the gas density change.

Case . | Pressure (bar) . | Re . | U (m/s)
. _{j} | U (m/s)
. _{c} |
---|---|---|---|---|

1 | 1 | 10 000 | 36.6 | 0.6 |

2 | 3 | 10 000 | 12.2 | 0.2 |

3 | 5 | 10 000 | 7.32 | 0.12 |

4 | 3 | 30 000 | 36.6 | 0.6 |

5 | 5 | 50 000 | 36.6 | 0.6 |

Case . | Pressure (bar) . | Re . | U (m/s)
. _{j} | U (m/s)
. _{c} |
---|---|---|---|---|

1 | 1 | 10 000 | 36.6 | 0.6 |

2 | 3 | 10 000 | 12.2 | 0.2 |

3 | 5 | 10 000 | 7.32 | 0.12 |

4 | 3 | 30 000 | 36.6 | 0.6 |

5 | 5 | 50 000 | 36.6 | 0.6 |

### B. Computational methodology

#### 1. Governing equations

Large-eddy simulation applies filtering operations to the evolution equations of mass, momentum, energy, and chemical scalars. For instance, the density-weighted (i.e., Favre) filtered evolution equation of a scalar *Q* is given by

and the three terms on the right-hand side (RHS) represent the subfilter contributions to the convective transport, molecular diffusive transport, and chemical source terms, respectively.

In this study, in addition to the filtered mass, momentum, and energy equations, the filtered evolution equations of mixture fraction *Z*, progress variable *C*, heat loss parameter *H*, lumped PAH species mass fraction $YP$, as detailed in Yang *et al.*,^{31} are solved. Additional equations including mixture fraction variation equation and soot moment equations for turbulent combustion and soot modeling will be introduced in Secs. II B 2 and II B 3, respectively,

where the overline indicates the unweighted (i.e., Reynolds) filtering operation, and the tilde implies a density-weighted (i.e., Favre) filtering. For each equation, the first term on the RHS is known as the subfilter scalar flux and contains the effects of unresolved turbulent transport; the second term is the filtered molecular diffusion fluxes. In this study, a Lagrangian dynamic sub-grid scale (SGS) model is used to close these terms.^{41} The last terms in these equations are the filtered source terms, including source terms due to sooting effects for mixture fraction *Z*, progress variable *C*, heat loss parameter *H*, and lumped PAH species mass fraction $YP$, which are discussed in the following section. $q\u0307\xafRAD$ is the heat loss source term due to radiation, which will be detailed in Sec. II B 2.

#### 2. Combustion modeling

In this study, the radiation flamelet progress variable (RFPV)^{42,43} approach is used to decouple the detailed flame structure from the computation of flow field. The non-premixed flame structure is computed *a priori* in the mixture fraction space, forming a series of steady non-premixed flamelet solutions with different radiation heat losses and mixture fraction scalar dissipation rates. Radiation is modeled with an optically thin gray gas approach, where the absorption coefficients for CO_{2}, H_{2}O, CH_{4}, and CO are obtained from Barlow *et al.*^{44} The KM2 chemical reaction model^{45} is used in this study when computing the non-premixed flamelet solutions. The source term of mixture fraction $m\u0307\xafZ$, progress variable $m\u0307\xafC$, heat loss function $m\u0307\xafH$, and radiation $q\u0307\xafRAD$ are all obtained from the RFPV database. The filtered PAH source term $m\u0307\xafP$ in Eq. (5) is decomposed into three terms with rescaling to consider the different PAH mass fraction values from the transport equation and the RFPV database, and more details can be found in Refs. 42 and 43.

#### 3. Soot modeling

A bivariate parameterization (volume and surface area) of the soot particle number density function (NDF) is employed to model detailed soot formation and evolution. To avoid computationally expensive solution of the entire soot NDF and yet capture the global statistical quantities of soot, the method of moments (MOM) is used. In MOM, the joint moments of the NDF are defined as

where summation over *i* implies summation over the entire two-dimensional state space and *N _{i}* is then the number density of particles of volume

*V*and surface area

_{i}*S*. The subscripts

_{i}*x*and

*y*denote the order of the moment in volume and surface area, respectively. In this study, four moments (i.e.,

*N*

_{0}, $M0,0,\u2009M1,0$, and $M0,1$) are transported

#### 4. Soot–turbulence–chemistry interactions

To model the subfilter soot–turbulence–chemistry interactions, the filtered source term $Q\u0307\xaf$ is closed through convolution with a density-weighted joint subfilter PDF $P\u0303(\xi i,Mj)$, which is a function of the vector of thermochemical variables *ξ _{i}* and the vector of soot variables

*M*:

_{j}The joint subfilter PDF $P\u0303(\xi i,Mj)$ is modeled by a presumed PDF approach in this work. Based on Bayes' theorem, the joint PDF $P\u0303(\xi i,Mj)$ is first decomposed into a thermochemical PDF $P\u0303(\xi i)$ and a conditional PDF of the soot scalars $P(Mj|\xi i)$ conditioned on the thermochemical scalars *ξ _{i}*:

where the conditional PDF $P(Mj|\xi i)$ characterizes the subfilter soot–turbulence–chemistry interaction. $P\u0303(\xi i)$ is modeled as a beta distribution for mixture fraction *Z* and further assuming no conditional variation for progress variable *C* and heat loss parameter *H* with respect to mixture fraction *Z.*^{31} In order to parameterize the beta distribution, the subfilter mixture fraction variance $Z\u20332\u0303$ is transported. The details and modeling for $Z\u20332\u0303$ can be found in Ihme and Pitsch.^{42}

In Mueller and Pitsch,^{43} the time scales of *ξ _{i}* and

*M*are argued to be disparate enough, and thus, the conditional PDF could be simplified to a marginal PDF: $P(Mj|\xi i)=P(Mj)$. The marginal PDF $P(Mj)$ is then calculated by two delta functions to represent the non-sooting and sooting modes within the large-eddy simulation (LES) filter width, considering the sooting intermittency in turbulent flames:

_{j}where *ω* is the subfilter intermittency, which could be obtained from a second-order moment of a soot scalar ($\omega =1\u2212Mj\xaf2/Mj2\xaf$), and $Mj*$ is selected such that the mean values of the soot scalars are obtained upon integration with this PDF. However, as shown in Yang *et al.*,^{31} soot evolution processes, notably soot oxidation and surface growth, are extremely fast and could be the same timescale as the thermochemical processes (e.g., heat release). In this sense, the simplification from the conditional PDF to a marginal PDF leads to large errors. Hence, a conditional soot subfilter PDF proposed by Yang *et al.*^{31} is used here

where $Mj**$ is selected such that $Mj*(Z)$ recovers $M\xafj$ upon convolution against the joint subfilter PDF. *H* is a Heaviside step function and *Z _{soot}* is chosen such that the model only activates the sooting mode at fuel-rich mixture fraction values when oxidation is slower than surface growth. More details are provided in Yang

*et al.*

^{31}

### C. Computational setup

The five KEN flame cases in Table I are simulated with NGA (Next Generation ARTS: Advanced Reactive Turbulent Simulator),^{47} a finite difference code for turbulent reacting flows. The solution database of non-premixed flamelet equation with radiative losses are computed with FlameMaster.^{48} The database is then parameterized by $Z\u0303,\u2009Z\u20332\u0303,\u2009C\u0303$, and $H\u0303$. A cylindrical axisymmetric computational domain with 300*D* in the axial direction and 74*D* in the radial direction is discretized into 192 × 96 × 32 in the axial, radial, and azimuthal directions, respectively. The grid points are concentrated near the mixing and combustion regions by stretching the grid (stretching rate 1.03) in the downstream direction axially and keeping uniform spaced grid in the circumferential direction, with 0.1 mm as the minimum grid size. In our previous studies^{31,32} with very similar flame configurations, mesh independence test was conducted to show that the current mesh resolution is able to resolve the flame structure and soot distribution, so we do not repeat it in this study because the convergence of time-averaged soot statistics requires very long computational time. In addition, these two previous studies indicate that the conditional soot subfilter PDF model used in this study is less sensitive to mesh resolution than the marginal soot subfilter PDF model. In this sense, the current mesh resolution should be sufficiently fine for the current turbulent sooting flames. To account for the turbulent inlet velocity profile, a separate pipe flow simulation with periodic boundary condition in the flow direction is first conducted with the mean axial velocity values as indicated in Table I and then enforced as the boundary conditions for the turbulent flame simulations. The simulations were run more than ten flow-through times to reach statistically stationary state, and then, another more than ten flow-through times were run to collect statistics until the soot statistics converge. One flow-through time takes about three days by hybrid MPI/OpenMP parallel computing with eight MPI nodes and 12 OpenMP threads per node on an AMD EPYC 7702 cluster. In total, about 40 days are needed for 20 flow-through times to reach statistical stationary state.

## III. RESULTS AND DISCUSSION

In this section, the predicted soot volume fraction (SVF) is compared with measured values to show the good accuracy of turbulent combustion and soot modeling in this work (Sec. III A). The SVF source terms, including nucleation, condensation, surface growth, and oxidation, are then analyzed (Sec. III B). Further scaling analysis is conducted on the pressure scaling of surface growth and oxidation terms to hint future improvements on turbulent sooting flame modeling at elevated pressures (Sec. III C).

### A. Soot volume fraction (SVF)

In this section, we quantitatively compared our predicted SVF with the experimental data to show the good accuracy of our LES predictions. Figure 1 presents the comparison between measured and predicted time-averaged mean and root mean square (RMS) of SVF fluctuation at the flame centerline, for the fixed Reynolds number flame series. For all three pressures, the predicted time-averaged SVF captures the measured data with the discrepancy within a factor of up to four. It is noted that without tuning any hyperparameters in the models, this level of prediction accuracy is considered to be superior, especially compared with the recent turbulent sooting numerical predictions where a typical more than one order of magnitude difference is shown.^{35,39,40} More specifically, our model provides rather accurate prediction on SVF for 3 bar, although it underpredicts four times at 1 bar, while over-predicts four times at 5 bar. It is then interesting to note that while the predicted mean SVF at 1 or 5 bar shows discrepancy, the RMS of SVF fluctuation for all three pressures matches very well with the measured data, implying the well-captured turbulence effects in our numerical modeling. The peak RMS positions, as shown both numerically and experimentally, locate downstream compared with the peak mean SVF, which indicates higher instantaneous SVF fluctuation beyond the peak mean SVF location, following the higher turbulence fluctuations downstream.

When fixing jet velocities (36.6 m/s) with increasing pressure, Reynolds numbers increase. In this series of flames, as shown in Fig. 2, the mean SVF of 3 bar is still well captured by the predicted data. At 5 bar with a Reynolds number of 50 000, the simulation over-predicts mean SVF again (by a factor of three). Consistent with the first series of flames, the predicted RMS of SVF fluctuation again matches well for both pressures, regardless the discrepancy of mean SVF at 1 and 5 bar. This indicates high fidelity of our model to capture soot fluctuation due to turbulence at different Reynolds numbers.

The predicted radial distributions of SVF in the two series of turbulent jet flames are also shown, compared with the measured data at six different axial positions, covering the soot formation region ($x/D=65$), concentrated SVF region ($x/D=81$), and soot depletion region ($x/D=92$). As seen in Fig. 3 (1 bar with *Re* = 10 000), at upstream and concentrated SVF positions (i.e., from $x/D=65$ to $x/D=81$), the soot radial distribution could be well captured by the simulation, although almost six times under-prediction is observed. At 3 bar with *Re* = 10 000 (Fig. 4), the predicted soot region is generally narrower than the experimental data, while the order of magnitude of SVF is captured very well. At 5 bar with *Re* = 10 000 (Fig. 5), the simulations over-predict soot volume fraction by five times. In this series of flames, a maximum six-time difference is observed in our simulations. Compared with the typically more than ten-time difference in other pressurized turbulent sooting flames,^{35,39,40} which is claimed by Chong *et al.*^{35} to be resulted from the uncertainty from the physical/chemical models, the model in this study thus shows strong predictability. The discrepancy is further analyzed in detail in Secs. III B and III C. In addition, the discrepancy at different pressure is consistent with the observation for the centerline results (i.e., Fig. 1).

In the other series of flame when fixing the jet velocity (36.6 m/s), the prediction discrepancy (see Figs. 6 and 7) is consistent with the first series that LES predicts accurately at 3 bar, while over-predicts three times at 5 bar. When the soot starts to nucleate at $x/D=65$ and $x/D=69$, simulations under-predict SVF, while near the peak SVF positions ($x/D=77$ and $x/D=81$), the radial SVF distribution is well predicted. At more downstream positions with $x/D=89$ and $x/D=92$, although the profile is captured by simulations, the SVF magnitude is over-predicted. As will be shown in the source term analysis in Sec. III B, this is due to the significantly increased soot surface growth source terms in the downstream positions. In this series of flames, the SVF distribution is well captured, and the major discrepancy is from the flame centerline prediction. For the Reynolds number of 50 000 case (Fig. 7), it is seen that generally, LES over-predicts SVF, in line with the first series of flames. Again, the radial distribution profile is well captured by LES. The observed difference in the predictions of SVF in this section is due to two sources: the effects of mixing/flow and the effects of soot source terms. Since both fixing inlet velocities and Reynolds number present similar discrepancy trend, residence time and turbulence intensities are thus not the reasons to induce discrepancies. In this sense, soot source terms will be analyzed in Sec. III B to explain the possible reasons.

In addition to evaluating the discrepancy between the predicted and measured SVF at different pressures, it is also important to figure out the reasons behind the higher sensitivity of the predicted SVF on pressure than the measured data for both flame series. From the previous results (e.g., Figs. 1 and 2), it is clearly seen that no matter fixing jet velocity or fixing Reynolds number, the increasing pressure significantly enhances soot production, especially from 1 to 3 bar. Further increasing from 3 to 5 bar could enhance soot production, while the increased magnitude becomes slight (i.e., the trend is saturating). Particularly for the simulation results, our model has a stronger dependence on pressure than the experimental data. More specifically, if we scale the maximum soot volume fraction at the centerline for the first series of flames by $fV=a1pn+b1$, the power *n* in experiment is 2.37, while the value of the power *n* in simulations is 3.95. This indicates that our numerical model is more sensitive to pressure than the experiment. The origin of this sensitivity is discussed in detail in Sec. III B by analyzing the SVF source terms.

### B. SVF source terms

To elaborate the possible reasons of our model prediction's stronger dependence on pressure, the soot volume fraction source terms, including nucleation, condensation, surface growth, and oxidation, are discussed at three different axial positions in this section. Note that the discussions in this section are based on the constant Reynolds number series of flames (*Re* = 10 000), due to the similar trend found (similar to the constant Reynolds flame series) in the constant jet velocity series of flames (*U _{j}* = 36.6 m/s). To split out the effects of flow and mixing, the soot volume fraction source terms are then plotted in the mixture fraction space. As seen in Fig. 8, when fixing the Reynolds number (

*Re*= 10 000), the increasing pressure significantly increases the rate of nucleation and condensation, which can be attributed to the higher PAH precursor concentrations at elevated pressures, similar to pressurized laminar sooting diffusion flames.

^{22}In addition, at $x/D=81$ where there are large SVF values for all three pressures, condensation is dominant over nucleation due to the large values of existing SVF values at this position. At more downstream ($x/D=92$) positions, nucleation is then stronger than condensation again due to the almost oxidized soot at this position. Because surface growth and oxidation source terms are dependent on surface area and hence SVF itself, the normalized surface growth and oxidation rates in the mixture fraction space are shown in Fig. 8. Note that the bimodal distribution of surface growth term at 3 bar and $x/D=69$ is due to the contributions from both the premixed pilot flame and main jet diffusion flame, as will discussed in Sec. III C. In addition, the normalized surface growth and oxidation rates are not enhanced by increasing pressure, suggesting that increased surface growth/oxidation is mainly due to the increased soot surface area.

Further presentation of the source term spatial contours in Figs. 9–12 at 1 and 5 bar shows that for fixed Reynolds number flames, nucleation and condensation, which are governed by the PAH specie concentration, locate more upstream. Surface growth and oxidation are more dependent on C_{2}H_{2} and soot surface area, and thus locate at more downstream positions at all the pressures. Compared with the close locations of soot formation and evolution zones for different source terms, increasing pressures lead to very different locations of nucleation and condensation from surface growth and oxidation (which is also true for the other series of flames with constant jet velocity, see Figs. 9–12). This is due to the fact that increasing pressure from 1 to 5 bar for both series of flames redistributes the peak nucleation and condensation position from centerline to flame wing (3 bar shows similar results as 5 bar and thus not detailed here). It is interesting to see that similar phenomena were found in laminar sooting diffusion flames, which are explained in the study of Zhou and Yang^{22} that it is due to the dominant indene (C_{9}H_{8}) relevant chemical reactions with increasing pressures.

In addition, increasing pressure from 1 to 5 bar with fixed Reynolds number (*Re* = 10 000) enhances nucleation and condensation only within the same order of magnitude. This phenomenon persists when increasing 1 to 5 bar with a fixed jet velocity (36.6 m/s) (Figs. 9 and 10). However, for surface growth, it is clearly seen that the SVF surface growth source terms increased by more than four orders of magnitude, together with oxidation source terms increasing about three orders of magnitudes, for both the fixed Reynolds number and fixed jet velocity when increasing pressure from 1 to 5 bar. In this sense, the varying residence time and turbulence intensity do not change the trend of SVF source term increasing with pressure. Hence, we can conclude that residence time and turbulence intensity are not the reasons to enhance soot production. In addition, due to the almost unchanged nucleation and condensation source terms with increased pressure, surface growth and/or oxidation, rather than nucleation and condensation, should be responsible for the significantly enhanced SVF with increased pressure. In this sense, the stronger pressure dependence of LES predictions than measurements should stem from the stronger pressure dependence of the surface growth model or the inaccurate pressure dependence of the oxidation model^{49} in this study.

### C. Scaling analysis for soot surface growth and oxidation

As pointed out by Kazakov *et al.*,^{50} pressure effects on soot formation are primarily two aspects. The first effect is to increase precursor concentration, while the second is on the increased soot surface areas due to the increased SVF. Soot surface areas are initially enhanced by the nucleation process. Then, together with the increased soot surface area, soot production is significantly enhanced by surface growth. In this study, nucleation process is only enhanced by one order of magnitude with pressure increasing from 1 to 5 bar for both series of flames (see Fig. 9). To understand the reasons of the stronger scaling on pressure in LES predictions than measurements, we then analyze surface growth and oxidation source terms in detail since Sec. III B indicates the strong surface growth and oxidation enhancement on increasing pressure. In the current HMOM framework, soot surface growth and oxidation source terms depend on only the gas-phase chemistry and a function of soot scalars

The rate coefficients for surface growth *k _{sg}* are calculated based on surface growth reactions,

^{51}which is highly relevant to the molar concentrations of C

_{2}H

_{2}, OH, H

_{2}O, H

_{2}, and H. For oxidation rate coefficient

*k*, it is computed based on soot oxidation surface reactions,

_{ox}^{51}which is a function of the molar concentrations of O

_{2}and OH. Function

*f*in HMOM is considered as multiplication of different order moments with the number of active sites per unit area on soot particles (1.7 $\xd71019\u2009m\u22122$ in this study).

^{46}Based on this equation, the filtered moment source terms due to surface growth in LES are calculated by convolution with the conditional subfilter PDF

The filtered rate coefficient $k\u0303sg/ox$ is basically the rate coefficient convoluted against the presumed thermochemical PDF, which is precomputed and tabulated. When solving transport equations for $M\xafj$ [i.e., Eq. (7)], the increased pressure significantly enhances moment source term thus promotes soot moments. Since $M**$ is positively correlated with $Mj\xaf$, the increased $ksg/ox$ by enhanced species concentration and the increased soot surface area finally enhance the soot surface growth as well as soot oxidation.

To directly show the change of $k\u0303sg$ due to pressure increase, we show $k\u0303sg$ in the $Z\u0303$–$C\u0303$ space to illustrate the pressure effect on $k\u0303sg$. As seen in Fig. 13, the peak $k\u0303sg$ is shown at $C\u0303>0.8$ for both pressures, indicating surface growth reactions require high temperatures. Note that the first peak in Fig. 13 indicates the premixed pilot flame (*Z* = 0.058), while the second peak at slightly richer region corresponds to the main jet diffusion flame. Increasing pressure from 1 to 5 bar pushes the concentrated $k\u0303sg$ from the first peak to the second peak in the $Z\u0303$–$C\u0303$ space, indicating the concentrated C_{2}H_{2} regions are near the premixed pilot flame positions in the 1 bar case.

In addition, $k\u0303sg$ increased by about two orders of magnitude from 1 to 5 bar. Since $k\u0303sg\u221dXC2H2Xsoot*$ (*X* stands for molar concentration per volume), accurate prediction on C_{2}H_{2} and soot^{*} (soot^{*} indicates soot radical sites) is then the key to correctly predict surface growth at different pressures for turbulent sooting flames. Note that soot^{*} is calculated by assuming quasi-steady-state soot^{*} in the surface growth reactions,^{51} which is a function of concentrations of OH, H_{2}O, H, C_{2}H_{2}, and H_{2}. Because the development of chemical kinetics of small radicals and major molecules (e.g., OH, H, H_{2}O, and H_{2}) is very mature (e.g., the well-studied H_{2} chemical kinetic mechanism^{52}) and the uncertainty involved with these small radicals/molecules are rather small, the scaling of C_{2}H_{2} prediction on pressure is the major uncertainty to explain the stronger SVF scaling on pressure from LES predictions than the scaling on pressure from experiment measurements. As pointed out by a high-pressure kinetic study of Shen *et al.*^{53} (i.e., the HP mechanism), updates on reactions such as C_{2}H_{2} + O and CH_{2}(S) + C_{2}H_{2} details could improve the predictions of C_{2}H_{2} at high pressures. As such, a future chemical mechanism such as the HP mechanism^{53} with PAH (e.g., the PAH chemistry in the KM2 mechanism^{45}) included should be developed and its application in turbulent pressurized sooting flame merits further studies.

Note that the stronger SVF scaling on pressure is primarily due to the combined effects of surface growth and oxidation. We apply the same analysis as the previous paragraph to oxidation, where $k\u0303ox\u221dXO2Xsoot*$ and $Xsoot*$ are functions of molar concentrations of O_{2} and OH based on the surface oxidation reactions.^{51} As discussed, the predictions on OH and O_{2} are accurate due to the mature chemical kinetics for these species. In this sense, by noting that the filtered oxidation rate coefficient is also highly relevant to the subfilter PDF [see Eq. (13)], we can conclude that the construction of the subfilter PDF might be further improved.

Note that as pointed out by Yang *et al.*,^{31} the application of this subfilter PDF with this Heaviside step function [$H(Z\u2212Zsoot)$] by properly assigning *Z _{soot}* has negligible effect on other source terms except oxidation. The subfilter PDF convolution with this Heaviside step function eliminates all the possible oxidation below

*Z*, assuming oxidation is fast enough such that no soot leaking can occur. As shown in Fig. 14, although the oxidation source term increase by about three orders of magnitude (see Fig. 12), the increase in $k\u0303ox$ is only within a factor of two, supplying the uncertainties involved in the conditional subfilter PDF model with the Heaviside step function. Assigning proper

_{soot}*Z*values to take pressure effects into account or proposing new function shape such as in Duvvuri

_{soot}*et al.*

^{49}to replace Heaviside step function merits future studies in pressurized turbulent sooting flames.

## IV. CONCLUSIONS

In this study, detailed large-eddy simulation (LES) of two series of turbulent sooting flames at elevated pressures of 1, 3, and 5 bar (KEN flames) with fixed jet velocities (*U _{j}* = 36.6 m/s) and fixed Reynolds number (

*Re*= 10 000) is conducted to analyze the pressure impacts on soot formation and evolution. Our numerical framework with radiation flamelet progress variable (RFPV) model as the turbulent combustion model, hybrid method of moments (HMOM) for the soot modeling, and a conditional soot subfilter PDF model for the turbulence–chemistry–soot interactions predict the soot volume fraction (SVF) at different positions for all three pressures with good accuracy (maximum discrepancy within a factor of up to six, which can be much smaller for some pressures). Both series of the flame simulations show stronger SVF scaling on pressure than the experimental observations, indicating the major causes for the stronger scaling are not due to residence time or turbulence intensities. SVF source terms, including nucleation, condensation, surface growth, and oxidation terms, are then discussed and analyzed. It is observed that the increasing pressure could increase the nucleation and condensation rates and also redistribute the peak nucleation and condensation from centerline to flame wing. Condensation is dominant over nucleation at the position where SVF peaks. Scaling analysis then shows that the major reason leading to stronger SVF scaling on pressures in simulations is due to the uncertainties involved in surface growth rate (and more precisely, C

_{2}H

_{2}prediction in chemistry) and oxidation rate (Heaviside step function in the subfilter conditional PDF model), regardless different residence time and turbulence intensities. Future work with high-pressure mechanisms with updated C

_{2}H

_{2}chemistry together with improved subfilter conditional PDF model for turbulence–chemistry–soot interactions are possible ways to further increase soot prediction fidelity for high-pressure turbulent jet flame simulations. In this study, detailed simulations of two series of turbulent sooting flames at elevated pressures of 1, 3, and 5 bar (KEN flames) with fixed jet velocities (

*U*= 36.6 m/s) and fixed Reynolds number (

_{j}*Re*= 10 000) are conducted to analyze the pressure impacts on soot formation and evolution. Our numerical framework with radiation flamelet progress variable (RFPV) model as the turbulent combustion model, hybrid method of moments (HMOM) for the soot modeling, and a conditional soot subfilter PDF model for the turbulence–chemistry–soot interactions predict the soot volume fraction (SVF) at different positions for all three pressures with good accuracy (maximum discrepancy within a factor of up to six, which can be much smaller for some pressures). Both series of the flame simulations show stronger SVF scaling on pressure than the experimental observations, indicating the major causes for the stronger scaling are not due to residence time or turbulence intensities. SVF source terms, including nucleation, condensation, surface growth, and oxidation terms, are then discussed and analyzed. It is observed that the increasing pressure could increase the nucleation and condensation rates and also redistribute the peak nucleation and condensation from centerline to flame wing. Condensation is dominant over nucleation at the position where SVF peaks. Scaling analysis then shows than the major reason leading to stronger SVF scaling on pressures in simulations are due to the uncertainties involved in surface growth rate (and more precisely, C

_{2}H

_{2}prediction in chemistry) and oxidation rate (Heaviside step function in the subfilter conditional PDF model), regardless different residence time and turbulence intensities. Future work with high-pressure mechanisms with updated C

_{2}H

_{2}chemistry together with improved subfilter conditional PDF model for turbulence–chemistry–soot interactions are possible ways to further increase soot prediction fidelity for high-pressure turbulent jet flames simulations.

## ACKNOWLEDGMENTS

S. Yang gratefully acknowledges the faculty start-up funding from the University of Minnesota and the grant support from NSF CBET 2038173. The authors acknowledge Mr. Anders Vaage for part of the RFPV database generation. Part of the simulation was conducted on the computational resources from the Minnesota Supercomputing Institute (MSI).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Dezhi Zhou:** Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **Shufan Zou:** Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). **Wesley R Boyette:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – review & editing (equal). **Thibault F Guiberti:** Data curation (equal); Formal analysis (equal); Investigation (equal). **William L. Roberts:** Conceptualization (supporting); Methodology (supporting); Project administration (equal); Supervision (equal). **Suo Yang:** Conceptualization (lead); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (lead); Resources (supporting); Software (supporting); Supervision (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.