Modeling flame extinction and reignition in large eddy simulations with fast chemistry

Abstract

This work seeks to support the validation of large eddy simulation models used to simulate fire suppression. The emphasis in the present study is on the prediction of flame extinction and the prevention of spurious reignition using a fast chemistry, mixing-controlled combustion model applicable to realistic fire scenarios of engineering interest. The configuration provides a buoyant, turbulent methane diffusion flame within a controlled co-flowing oxidizer. The oxidizer allows for the supply of a mixture of air and nitrogen, including conditions for which oxygen-dilution in the oxidizer leads to flame extinction. Measurements to support model validation include local profiles of thermocouple temperature and oxygen mole fraction, global combustion efficiency, and the limiting oxygen index. The present study evaluates the performance of critical-flame-temperature-based extinction and reignition models using the Fire Dynamics Simulator, an open-source fire dynamics solver. Alternate model cases are explored, each offering a unique treatment of extinction and reignition. Comparisons between simulated results and experimental measurements are used to evaluate the capability of these models to accurately describe flame extinction. Of the considered cases, those that include provisions to prevent spurious reignition show excellent agreement with measured data, whereas a baseline case lacking explicit reignition treatment fails to predict extinction.

1. Introduction

The numerical simulation of fire phenomena using classical computational fluid dynamics (CFD) methods has advanced significantly over recent years. This progress has led to an increasing demand for the use of CFD tools to analyze and predict the performance of fire protection systems. For fire suppression applications in particular, this demand remains largely unmet as modern CFD solvers have not yet been shown to adequately model fire suppression physics in configurations of practical interest, namely in the buoyant turbulent diffusion flames characteristic of most fire safety applications. This limitation is the result of the complex physical processes that govern turbulent fire suppression, comprising both localized extinction events and the potential reignition of unburned fuel following such extinction events. Neither of these phenomena are easily modeled. Further contributing to the issue is the general unavailability of detailed experimental data suitable for model validation, also inhibiting model development.

In recognition of these constraints, numerous experimental and computational studies have investigated the extinction behaviors of diffusion flames in various configurations. These works have successfully identified the primary physical mechanisms for extinction (thermal, aerodynamic, and kinetic quenching) [1–9], while others have made progress toward developing simple formulations to model flame extinction in cases applicable to realistic fire scenarios [10–13]. Additional studies have highlighted the primary features of flame reignition events, which may follow localized extinction in large-scale turbulent flames [14–19].

As noted in previous works, the primary difficulty associated with modeling flame extinction and reignition in fire applications is that both phenomena are controlled by small-scale quantities, including the flame temperature and the fuel-oxidizer mixing rate at the flame sheet, that cannot be resolved numerically in configurations of practical interest. Compounding this issue is the question of how to discern in the model the difference between primary ignition at the fuel source (which is often desirable and necessary to achieve a stabilized flaming condition in the model) and reignition occurring downstream of localized extinction events (which is undesirable when such reignition is spurious or non-physical).

The issue of spurious reignition, its significance in determining simulated extinction performance, and available means for its prevention are the primary focus of the present work, which comprises a comprehensive evaluation of simulated flame suppression performance via comparisons of numerical results against recent experimental measurements [20, 21].

The selected configuration features the suppression of a buoyant turbulent methane diffusion flame via nitrogen dilution of the oxidizer, including conditions ranging from complete combustion through total extinguishment. This simplified, but well-characterized configuration incorporates the essential features of a suppressed accidental fire (buoyancy-driven flow, turbulence, intense radiative emissions), while isolating the flame extinction physics of interest. In particular, simulations in the selected configuration highlight conditions for which localized flame extinction events may be followed by reignition events.

The present study utilizes the Fire Dynamics Simulator (FDS) [22], an open-source fire dynamics solver that is widely used throughout the fire safety consulting and design industries. The selected numerical framework incorporates a large eddy simulation (LES) approach utilizing the classical eddy-dissipation concept (EDC) treatment for mixing-controlled combustion. These modeling choices represent the most appropriate options for modeling fire phenomena in practical engineering applications: direct numerical simulations (DNS) cannot be applied to large-scale configurations of engineering interest, and the combustion of realistic fuel sources cannot be represented using detailed finite-rate chemical kinetics models. Modeling treatments for flame extinction and reignition use the concept of a critical flame temperature [23], which should be viewed as an empirically-determined fuel-specific quantity approximating the flame temperature at the limits of flammability (considering heat losses).

2. Modeling Approach

2.1. Solver

The Fire Dynamics Simulator (FDS) provides a variable density, low Mach-number Navier-Stokes solver tuned for buoyancy-driven flows with heat release. FDS features a block-structured, Cartesian staggered grid and a parallel computing capability using MPI and OpenMP protocols. The numerics are generally second-order accurate with explicit time integration. Combustion is first-order accurate and time-split from transport, consistent with the classical eddy-dissipation concept (EDC) model [24, 25]. As applied in the present work, combustion is treated with a single-step, global reaction involving lumped species for the fuel, oxidizer, and products [26].

A large eddy simulation (LES) framework is employed for subgrid-scale turbulence modeling, where the LES residual stress and transport terms are closed via gradient diffusion coupled with a modified version of the Deardorff eddy-viscosity model [27], using an algebraic model for the subgrid kinetic energy based on scale similarity [28], and constant values for the turbulent Schmidt and Prandtl numbers.

FDS models thermal radiation via the radiative transfer equation using a discrete-ordinates, finite-volume method with gray absorption properties [29]. In the present study, the description of thermal radiation is further simplified by modeling flame emission using the concept of a global radiative loss fraction (χ_rad); where χ_rad is prescribed in the simulations a priori from experimental measurements.

For a comprehensive discussion detailing the mathematical models employed in the FDS solver, the reader is referred to the FDS technical reference guide [30–32]. Particularly relevant to the present work, the local volumetric heat release rate due to combustion, Q̇‴, is given by

$$\dot{Q}‴=-\sum _k{\dot{m}}_k^{‴}\mathrm{\Delta }{h}_{f,k}^{◦},$$ (1)

where ${\dot{m}}_k^{‴}$ is the volumetric mass reaction rate and $\mathrm{\Delta }{h}_{f,k}^{◦}$ the mass-specific standard enthalpy of formation for each species, k, present in the prescribed combustion reaction. The volumetric mass reaction rate of fuel, ${\dot{m}}_{\mathit{\text{fuel}}}^{‴}$, is evaluated as

$${\dot{m}}_{\mathit{\text{fuel}}}^{‴}=-\overline{\rho }\frac{\text{min }({\stackrel{\sim }{Y}}_{\mathit{\text{fuel}}},{\stackrel{\sim }{Y}}_{\mathit{\text{ox}}}/s)}{{\tau }_{\mathit{\text{mix}}}},$$ (2)

where ρ̅ is the local mass density, Ỹ_fuel and Ỹ_ox are the local mass fractions of fuel and oxidizer, and s is the stoichiometric oxidizer-to-fuel mass ratio. Other ${\dot{m}}_k^{‴}$ are then determined from ${\dot{m}}_{\mathit{\text{fuel}}}^{‴}$ based on prescribed stoichiometric reaction coefficients.

In Eq. 2, τ_mix is a modeled characteristic mixing time-scale with alternate definitions based on the resolution of the flow-field [33]. For all numerical resolutions considered in the present study, τ_mix is governed by turbulent advection and defined as

$${\tau }_{\mathit{\text{mix}}}=\frac{C_u\mathrm{\Delta }}{\sqrt{(2/3)k_{\mathit{\text{sgs}}}}},$$ (3)

where C_u = 0.4 is a model coefficient calibrated to match experimental flame height measurements [32], Δ = (Δx Δy Δz)^1/3 is the LES filter width defined by the local grid resolution (Δx, Δy, Δz), and k_sgs is the modeled subgrid turbulent kinetic energy derived from the turbulence sub-model.

Using the Deardorff model [27], the turbulent viscosity, μ_t, is given by

$${\mu }_t=\overline{\rho }{C}_{\upsilon }\mathrm{\Delta }\sqrt{k_{\mathit{\text{sgs}}}},$$ (4)

where C_υ = 0.1 is a model coefficient set to match a value reported in the literature [34]. Turbulent transport coefficients for mass and energy are then derived from μ_t via constant Schmidt and Prandtl numbers, Sc_t = Pr_t = 0.5.

The subgrid turbulent kinetic energy is evaluated as

$$k_{\mathit{\text{sgs}}}=\frac{1}{2}({[\stackrel{\sim }{u}-\hat{u}]}^2+{[\stackrel{\sim }{\upsilon }-\hat{\upsilon }]}^2+{[\stackrel{\sim }{w}-\hat{w}]}^2),$$ (5)

where [ũ, υ̃, w̃] are the cell-mean values of the x, y, and z-components of the local velocity field, and [û, υ̂, ŵ] are weighted averages of their respective cell-mean counterparts over the adjacent computational cells (representing a test-filtered velocity field over length scale 2Δ) [28, 30].

2.2. Configuration

The configuration for the present study follows the experimental facility developed by White et al., illustrated in Fig. 1 [20, 21]. This configuration provides the suppression of a buoyant, turbulent, methane diffusion flame via nitrogen dilution of the oxidizer. Flames are stabilized above a 5-cm-wide by 50-cm-long slot burner surrounded by a 50-cm-wide by 75-cm-long co-flowing oxidizer. Surrounding the fuel port is a thin 5-cm-wide strip of ceramic fiberboard, which encourages horizontally directed entrainment at the flame base and promotes transition to fully turbulent conditions within the flame. Methane flows at a fixed rate of 1.0 g/s to yield a flame with an unsuppressed total heat release rate of 50 kW.

Figure 1.

(a) Diagram of experimental facility. (b) Plan-view of fuel/oxidizer port surface.

The co-flowing oxidizer is supplied at a fixed rate of 85 g/s, including a variable flow of nitrogen gas mixed into the oxidizer to provide controlled flame suppression. Suppression potential is monitored via the oxygen mole-fraction in the oxidizer stream $(X_{O_2}^{\mathit{\text{ox}}})$, measured using a sampling probe connected to a Servomex 540E paramagnetic oxygen analyzer (uncertainty ±1250 ppm; response time ±5 s). The flow rate of nitrogen supplied to the oxidizer varies between 0–40 g/s yielding changes in $X_{O_2}^{\mathit{\text{ox}}}$ from the ambient $X_{O_2}^{\mathit{\text{ox}}}=0.21$ to as low as $X_{O_2}^{\mathit{\text{ox}}}=0.11$, a value sufficient to achieve global flame extinguishment.

Available measurement data to support model validation include local thermocouple temperatures, measured via exposed-junction 1.0 mm bead-diameter K-type thermocouple probes (uncertainty ±2 K; response time ±3 s); local oxygen mole fractions, measured using a sampling probe connected to the aforementioned oxygen analyzer; global radiative loss fraction, measured via heat flux transducer combined with infrared flame imaging and an analytical multipoint radiation source model [20]; global combustion efficiency, measured via carbon dioxide generation calorimetry [21]; and the value of $X_{O_2}^{\mathit{\text{ox}}}$ at extinction, termed the limiting oxygen index (LOI).

The numerical configuration is depicted in Fig. 2, which visualizes the simulated flame at ambient condition $(X_{O_2}^{\mathit{\text{ox}}}=0.21)$. The computational domain is a 0.8-m-wide by 1.0-m-long by 1.0-m-tall box (x × y × z), which is large enough to prevent excessive boundary-condition effects on the simulated flame. The baseline numerical grid features a uniform resolution in each coordinate direction with Δx = Δy = Δz = 5.0 mm. The total number of computational cells is 6.4 million.

Figure 2.

Illustration of the numerical configuration; a 50 kW methane diffusion flame is visualized where the local volumetric heat release rate exceeds 200 kW/m³.

All edges of the computational domain feature an open, passive-flow boundary condition, with a background species of ambient air. The boundary conditions at the fuel and oxidizer ports utilize a prescribed mass-flux boundary condition (accounting for both convective and diffusive components), with values specified to match the experiment (fuel mass flow, ṁ_fuel = 1.0 g/s, and oxidizer mass flow, ṁ_ox = 85 g/s). The fuel and oxidizer, as well as any ambient entrainment, enter the domain with a constant temperature of T_∞ = 293 K. All solid surfaces within the domain, comprising the structural rim of the oxidizer port and the ceramic fiberboard around the fuel port, are described as inert, isothermal objects with constant temperature, T_∞ = 293 K, and with zero-flow boundary condition. At initial condition, the interior of the domain includes a uniform composition of air at T_∞ = 293 K with ambient relative humidity of 40%.

The value of $X_{O_2}^{\mathit{\text{ox}}}$ at the oxidizer-port boundary is varied by controlling the relative mass fluxes of air and nitrogen leaving the oxidizer-port, which are balanced to maintain a constant total oxidizer flow rate, ṁ_ox = 85 g/s. At the start of each simulation (t = 0 s), $X_{O_2}^{\mathit{\text{ox}}}=0.21$, remaining constant at this value for the first 5 s of simulation time, a duration sufficient to achieve steady conditions in the flame and oxidizer flow fields.

After t = 5 s, the value of $X_{O_2}^{\mathit{\text{ox}}}$ is steadily decreased at a rate of ${\mathit{\text{dX}}}_{O_2}^{\mathit{\text{ox}}}/\mathit{\text{dt}}=0.002{\mathrm{s}}^{-1}$. Simulations are conducted for a total duration of 60 s so that $X_{O_2}^{\mathit{\text{ox}}}=0.10$ at the end of the simulation, a value sufficient to achieve global flame extinction. Simultaneous with the reduction in $X_{O_2}^{\mathit{\text{ox}}}$, the value of χ_rad also steadily decreases over the course of the simulation, with a prescribed trend matching experimental data [20].

In addition to transient simulations, which feature continuously variable $X_{O_2}^{\mathit{\text{ox}}}$, alternative steady-state simulations are conducted to provide converged turbulence statistics at discrete $X_{O_2}^{\mathit{\text{ox}}}$ conditions. For such simulations, $X_{O_2}^{\mathit{\text{ox}}}=0.21$ again for the first 5 s of the simulation to establish a stabilized ambient flaming condition. After t = 5 s, $X_{O_2}^{\mathit{\text{ox}}}$ is decreased rapidly over a 3 s duration to a final specified value, which is then held constant for the remainder of the simulation.

Simulations are conducted using 40 processors on the University of Maryland Deepthought2 high-performance computing cluster. Typical simulations require roughly 2500 CPU hours using Intel Ivy Bridge E5-2680v2 2.80 GHz processors.

2.3. Resolution

To evaluate grid sensitivity in the present configuration, a series of simulations have been conducted featuring uniform grid resolution corresponding to Δx = 20.0 mm, 10.0 mm, 5.0 mm, and 2.5 mm. These simulations are conducted with constant $X_{O_2}^{\mathit{\text{ox}}}=0.21$ (no suppression) for a duration of 30 s. Simulated diagnostics include vertical centerline (z-direction), and lateral cross-flame profiles (x-direction) of various locally computed quantities including gas-phase temperature, z-component of flow velocity, and the resolved and subgrid turbulent kinetic energies. Time-mean and root-mean-square (rms) statistics for each quantity are gathered over the final 25 s of each simulation, during a period when the flow is statistically stationary. Run-time characteristics for these simulations are summarized in Table 1, comparing the total number of cells, processors, and computational cost applicable to each resolution case.

Table 1.

Run-time characteristics for simulations with varying grid resolution.

Δx (mm)	⧣ Cells (−)	⧣ Cores (−)	CPU Time (hr)
20.0	356,000	2	22
10.0	1,056,000	5	111
5.0	6,400,000	40	1,085
2.5	51,200,000	320	30,655

Figure 3 presents comparisons of vertical centerline and cross-flame profiles of the time-mean and rms gas temperature, T, among each resolution case. As shown, adequate grid convergence is observed for Δx ≤ 5.0 mm in the mean temperature field, though T_rms is under-predicted on the order of 50 K compared to the finest resolved case at Δx = 2.5 mm. These results indicate that there remain unresolved turbulent fluctuations in the temperature field for Δx ≥ 2.5 mm, though these fluctuations negligibly impact the mean field. The simulated peak mean value of T ≈ 1200 K closely resembles a value of 1191 K predicted by an empirical correlation developed for line-fire geometries from corresponding experimental measurements in a similar configuration [35].

Figure 3.

Simulated gas temperature (T) at selected grid resolutions; (a) mean z-profile, (b) rms z-profile, (c) mean x-profile at z = 0.25 m, (d) rms x-profile at z = 0.25 m.

Figure 4 presents similar resolution comparisons of the time-mean and rms z-velocity component magnitude, w. As shown, adequate grid convergence is observed for Δx ≤ 5.0 mm in both the mean and rms values. As with the results for temperature, the simulated far-field value of w ≈ 2.8 m/s agrees well with a value of 2.88 m/s predicted by empirical correlation [35].

Figure 4.

Simulated z-velocity magnitude (w) at selected grid resolutions; (a) mean z-profile, (b) rms z-profile, (c) mean x-profile at z = 0.25 m, (d) rms x-profile at z = 0.25 m.

The mean turbulent kinetic energy resolved by the numerical grid, k_t, may be approximated as

$$k_t=\frac{1}{2}(u_{\mathit{\text{rms}}}^2+{\upsilon }_{\mathit{\text{rms}}}^2+w_{\mathit{\text{rms}}}^2),$$ (6)

where u_rms, υ_rms, and w_rms, are respectively the root-mean-square statistics for the x, y, and z components of the flow velocity. The unresolved subgrid turbulent kinetic energy, k_sgs, may then be retrieved from the turbulence model (see Eq. 5).

From these diagnostics, and following the conventions given in previous work [36], a characteristic resolution criterion, M, may be defined as

$$M=\frac{k_t}{k_t+k_{\mathit{\text{sgs}}}},$$ (7)

where this ratio indicates the fraction of the modeled turbulent kinetic energy that is resolved by the numerical grid, taking a value of unity when k_sgs is negligible and decreasing toward zero as k_sgs begins to dominate k_t. Note that k_sgs is a modeled quantity and should not be expected to represent the true turbulent fluctuations in the unresolved flow. As a result, M = 1 does not imply that 100% of the turbulence in the flow is resolved by the grid. This criterion merely indicates the degree of influence that the turbulence model has on the flow field. Following established recommendations [34, 36], M ≥ 0.8 satisfies reasonably well-resolved LES. Vertical centerline and cross-flame profiles of M are presented in Fig. 5.

Figure 5.

Simulated turbulence resolution criterion (M) at selected grid resolutions; (a) mean z-profile, (b) mean x-profile at z = 0.25 m.

As shown in Fig. 5, for Δx ≤ 5.0 mm, M ≥ 0.8 for all z along the flame centerline and for all x along a representative cross-flame profile, though there is a small region very near to the fuel port for which M < 0.8. This downward peak near z = 1 cm is attributed to the laminar-turbulent transition that occurs at the flame base.

The limiting length scale of interest in the present configuration is the flame width (W_f, approximated as the burner width, W_b = 5 cm). For the baseline resolution of Δx = 5.0 mm, W_b/Δx = 10, indicating that this length scale is resolved over ten computational cells. In consideration of the previous results, this resolution may be considered marginal in the near-field region of the flame base (where W_f ≈ W_b), but adequate in the intermediate and far-field regions within and above the bulk of the flame (where W_f increases with elevation so that W_f > 10.0 mm and W_f/Δx > 20 above mid-flame height, z > 25 cm). While it is questionable whether the Δx = 5.0 mm resolution accurately captures the transition from laminar to turbulent flow near the flame base, this resolution is deemed adequate and most cost-effective for the aims of the present study.

2.4. Flame Extinction & Reignition

The fundamental limitation of any mixing-controlled combustion scheme (as presented in Eqs. 1–5), is that combustion is guaranteed to occur anywhere fuel and oxidizer are mixing, regardless of local conditions for temperature or fuel/oxidizer dilution. This represents the characteristic ‘mixed-is-burned’ treatment adopted by the EDC combustion framework and has the advantage that detailed heat transfer and chemical kinetics treatments are not necessary to model primary ignition at the fuel source. As a result however, additional modeling provisions must be made in order to account for flame extinction and prevent spurious reignition, each respectively addressed in the following sections.

2.4.1. Extinction

The baseline flame extinction algorithm available in FDS uses the concept of a critical flame temperature [10], which accounts for the thermal quenching mechanism, but neglects the aerodynamic and kinetic quenching mechanisms. For extinction via nitrogen dilution of the oxidizer, thermal quenching acts as the dominant mode of extinction; therefore a critical flame temperature based extinction model is sufficient for the present study.

The criterion for flame extinction in FDS incorporates an enthalpy balance where in order for combustion to occur, the potential heat release due to combustion must be sufficient to increase the mean temperature of the reactant mixture above a critical value, T_ext, defined as the critical flame temperature [10, 23]. Combustion is allowed to proceed if the inequality given by

$${\hat{Y}}_{\mathit{\text{fuel}}}(h_{\mathit{\text{fuel}}}(\stackrel{\sim }{T})+\mathrm{\Delta }{h}_{\mathit{\text{comb}}})+{\hat{Y}}_{\mathit{\text{ox}}}{h}_{\mathit{\text{ox}}}(\stackrel{\sim }{T})+{\hat{Y}}_{\mathit{\text{dil}}}{h}_{\mathit{\text{dil}}}(\stackrel{\sim }{T})>{\hat{Y}}_{\mathit{\text{fuel}}}{h}_{\mathit{\text{fuel}}}(T_{\mathit{\text{ext}}})+{\hat{Y}}_{\mathit{\text{ox}}}{h}_{\mathit{\text{ox}}}(T_{\mathit{\text{ext}}})+{\hat{Y}}_{\mathit{\text{dil}}}{h}_{\mathit{\text{dil}}}(T_{\mathit{\text{ext}}}),$$ (8)

is satisfied, where [Ŷ_fuel, Ŷ_ox, Ŷ_dil] and [h_fuel, h_ox, h_dil] are the mass fractions and mass-specific sensible enthalpies of the fuel, oxidizer, and diluent (defined as any non-reactant species, including any inert agents or products of combustion) present in the reactant mixture, T̃ is the initial cell temperature, and Δh_comb is the mass-specific enthalpy of combustion of the fuel. Note that Eq. 8 considers only the initial composition of the reactant mixture at the beginning of the combustion time step (after transport, but prior to any reaction). Extinction occurs if the local concentrations of fuel and oxidizer and/or the local cell temperature are too low to support combustion (thermal quenching).

The reactant mixture referred in Eq. 8 represents the portion of the computational cell that is subject to potential combustion. The composition of the reactant mixture is defined relative to the composition of the host computational cell via the following,

$${\hat{Y}}_{\mathit{\text{fuel}}}=\text{min }({\stackrel{\sim }{Y}}_{\mathit{\text{fuel}}},{\stackrel{\sim }{Y}}_{\mathit{\text{ox}}}/s),$$ (9)

$${\hat{Y}}_{\mathit{\text{ox}}}=s{\hat{Y}}_{\mathit{\text{fuel}}},$$ (10)

$${\hat{Y}}_{\mathit{\text{dil}}}=({\hat{Y}}_{\mathit{\text{ox}}}/{\stackrel{\sim }{Y}}_{\mathit{\text{ox}}})({\stackrel{\sim }{Y}}_{\mathit{\text{fuel}}}-{\hat{Y}}_{\mathit{\text{fuel}}}+{\stackrel{\sim }{Y}}_{\mathit{\text{dil}}}),$$ (11)

where [Ỹ_fuel, Ỹ_ox, Ỹ_dil] are the local mass fractions of the fuel, oxidizer, and diluent in the computational cell.

By Eqs. 9–11, the reactant mixture comprises a stoichiometric combination of fuel and oxidizer, with additional diluent included such that the ratio of diluent to oxidizer in the reactant mixture matches that in the host computational cell. These definitions (Eqs. 9–11) have the effect that excess fuel is treated as a diluent but excess oxidizer is not. As a result of this convention, combustion is likely to occur in a cell having a small amount of fuel in excess oxidizer, but is likely to be suppressed in a cell having a small amount of oxidizer in excess fuel. This treatment accommodates reasonable combustion and extinction performance in relatively coarse computational cells, as are frequently used in simulations of practical-scale fire configurations.

The critical flame temperature used in the present study, T_ext = 1600 K, is representative of typical hydrocarbon fuels [23]. If the criterion specified in Eq. 8 fails, combustion is suppressed and the reaction source terms for species generation/consumption and heat release are set to zero for that computational cell, at that time step ( $\dot{Q}‴={\dot{m}}_k^{‴}=0$ in Eqs. 1 and 2).

2.4.2. Reignition

By inspection of the combustion scheme presented in Eqs. 1–5, it is apparent that spurious reignition may be an issue anywhere fuel and oxidizer mix downstream of a localized extinction event. Including consideration for the previously defined extinction criterion (Eq. 8), reignition occurs as long as fuel and oxidizer are locally present in sufficient quantities to satisfy Eq. 8, which is possible even for low initial cell temperatures near ambient condition. An example of this fact is the initial ignition of primary combustion at the start of a simulation, which desirably does not require any specification for an ignition heat source. Still, any non-physical, low-temperature reignition of previously extinguished fuel can significantly degrade the suppression performance of the extinction model. For applications in which suppression is the primary modeling interest, such spurious reignition must be prevented.

Restricting available solutions, any ignition or extinction criterion that is implemented to prevent spurious reignition cannot be generally applied in the model without also suppressing primary ignition at the fuel source. This issue may be circumvented by modeling primary ignition, though this would require an explicit ignition source coupled with detailed reaction kinetics and a highly-resolved treatment for heat transfer at the fuel source. Such an option is generally unavailable in fire applications which often include fuel sources that cannot be described with detailed reaction kinetics and complex configurations for which resolved heat transfer at the fuel source would require prohibitive computational cost. Despite these challenges, recent works have made notable progress in modeling and distinguishing ignition, extinction, and reignition processes in LES applications [11–13].

Simulations in the present configuration highlight the noted issues surrounding spurious reignition, including conditions for which flame extinction leads directly to non-physical reignition if explicit treatment preventing such reignition is not provided in the model. As a preliminary modeling attempt, and following the convention of the critical flame temperature employed by the extinction criterion, a simple temperature based reignition criterion may be written as

$$\stackrel{\sim }{T}>T_{\mathit{\text{ign}}},$$ (12)

where T̃ is the initial temperature of the computational cell (before the combustion time step) and T_ign is a critical temperature for reignition. As with the extinction criterion, if the inequality in Eq. 12 is unsatisfied, combustion is suppressed and the reaction source terms for species generation/consumption and heat release are set to zero for that cell, at that time step ( $\dot{Q}‴={\dot{m}}_k^{‴}=0$ in Eqs. 1 and 2).

As previously noted, Eq. 12 cannot be generally applied in the solver without requiring an explicit treatment for primary ignition at the fuel source. To avoid this complication, implementations with selective application of Eq. 12 that still permit spontaneous primary ignition are explored in the following. Three simple modeling cases, respectively denoted M1, M2, and M3, and each utilizing Eqs. 8 and 12 to treat extinction and reignition in various capacities, are summarized as follows.

In each of the following cases, the baseline single-step global reaction scheme employed for primary combustion follows the EDC treatment and is defined as

$$\text{Fuel}+s\text{ Air}\to (1+s)\text{ Products},$$ (R1)

where reaction R1 is allowed to proceed only if the inequalities in both Eqs. 8 and 12 are satisfied, subject to the respective temperature criteria T_ext and T_ign.

The first case, M1, represents the standard EDC behavior for which T_ext = 1600 K and T_ign = 0 K everywhere in the computational domain. Here, Eq. 12 is universally satisfied such that M1 includes no provision to prevent spurious reignition. As previously stated, spurious reignition may occur if, downstream of a region where Eq. 8 is unsatisfied (and therefore reaction R1 is suppressed), the resulting unburned fuel propagates to a region where Eq. 8 is satisfied such that reaction R1 then spontaneously proceeds (potentially at non-physically low ignition temperatures).

The second case, M2, provides a novel development consisting of a simple extension of M1, but including a small prescribed ignition zone near the fuel source within which T_ign = 0 K. Elsewhere in the domain, T_ign = 900 K, a value approximately equal to an experimentally measured auto-ignition temperature for methane fuel [37]. Everywhere in the domain, T_ext = 1600 K, consistent with M1. The ignition zone is defined within a region that extends 5 mm below, 5 mm around, and 2 cm above the surface of the fuel port. A diagram illustrating the bounds of the ignition zone is provided in Fig. 6.

Figure 6.

Diagram of the ignition zone applied in modeling case M2.

The spatially-variable application of T_ign employed by M2 effectively eliminates the issue of spurious reignition. Reignition is allowed to occur, but only if the local cell temperature is high enough to support such a process (via Eq. 12).

The confined region of the ignition zone permits the convenience and simplicity of spontaneous primary ignition, but does so without subjecting the bulk of the computational domain to this convention. Despite these advantages, the specification of this zone is somewhat arbitrary and guidelines for its size and location do not yet exist. These are subjects of interest in the present work.

The third case, M3, follows conventions developed in recent affiliated works [11–13], separating ignition and reignition using a tiered reaction mechanism. Including the primary reaction, R1, two additional reactions are introduced as

$$\text{Fuel}+s\text{ Air}\to {\text{Fuel}}^{\ast }+s\text{ Air},$$ (R2)

$${\text{Fuel}}^{\ast }+s\text{ Air}\to (1+s)\text{ Products},$$ (R3)

where Fuel* is a secondary fuel species representing unburned fuel that has had the opportunity to react, but has been suppressed by Eq. 8. Here, reaction R2 acts as a mutually-exclusive alternate reaction pathway to reaction R1, transforming the primary Fuel into Fuel* if, and only if, R1 is suppressed. Reaction R3 then provides for the reignition of Fuel*, where the reignition criterion in Eq. 12 may be independently applied to R3 without affecting primary ignition in R1 (different T_ign may be separately applied to each reaction).

In M3, T_ext and T_ign apply uniformly throughout the domain, with reaction-specific values defined as follows. For reaction R1, T_ext = 1600 K and T_ign = 0 K, equivalent to the treatment in M1. For reaction R2, T_ext = 0 K and T_ign = 0 K, signifying that R2 is always unrestricted whenever R1 is suppressed. For reaction R3, T_ext = 1600 K and T_ign = 900 K, matching the treatment in M2 (outside the ignition zone). A summary of the critical temperature parameters, T_ext and T_ign, as applied in cases M1, M2, and M3 is displayed in Table 2.

Table 2.

Critical temperature parameters for model cases M1, M2, and M3.

Case	Reaction	Text (K)	Tign (K)
M1	R1	1600	0
M2	R1	1600	900
	R1	1600	0
M3	R2	0	0
	R3	1600	900

The tiered reaction mechanism adopted in case M3 is advantageous in that it does not require an arbitrarily specified ignition zone. To its disadvantage however, M3 requires the additional complexity of reactions R2 and R3 and the computational expense of transporting an additional reacting species in Fuel*.

It is also important to recognize the implications of the stoichiometric requirement for air in the fuel transformation reaction, R2. This convention is necessary for the system of reactions R1–R3 to reduce back to the classical EDC treatment (R1) in cases of zero or total extinction; however, in situations where reaction R1 is extinguished and there is insufficient air to support reaction R2, fuel transformation will fail, permitting the persistence of the primary fuel in conditions of extinction, and providing the potential for spurious reignition to later occur (as visualized in the results that follow). While technically permissible, this issue is limited to cases of extreme oxidizer dilution and in the majority of practical applications, M3 effectively eliminates spurious reignition.

3. Results and Discussion

3.1. Model Validation

A preliminary validation of each modeling case (M1, M2, and M3) against measured thermocouple temperatures (T_tc) and local oxygen mole fractions (X_O2) is presented in Fig. 7. Measured and simulated data are reported as lateral cross-flame profiles (x-direction) at selected elevations corresponding to z = 12.5 cm (~ L_f/4) and z = 25.0 cm (~ L_f/2) above the fuel port, for a partially-diluted oxidizer condition of $X_{O_2}^{\mathit{\text{ox}}}=0.18$. This validation condition serves to evaluate the influence that each extinction and reignition treatment may have on the flame structure for a partially suppressed condition where complete combustion is still expected.

Figure 7.

Simulated and measured mean x-profiles of local thermocouple temperature (T_tc) and oxygen mole fraction (X_O2) at selected elevations above the fuel port, and for $X_{O_2}^{\mathit{\text{ox}}}=0.18$; (a) T_tc at z = 12.5 cm, (b) T_tc at z = 25.0 cm, (c) X_O2 at z = 12.5 cm, (d) X_O2 at z = 25.0 cm.

As shown in Fig. 7, simulated T_tc and X_O2 for all three cases agree reasonably well with the measured values. In particular, peak T_tc at both z = 12.5 cm and z = 25.0 cm show good agreement, though the simulated profiles are slightly wider than measured at both elevations. Agreement for X_O2 at z = 12.5 cm is also good, though the simulated trends at z = 25.0 cm are slightly sharper than measured. These results validate that each case yields mean temperature and mixing fields that resemble those measured in the experiment, confirming that the simulated flame structure is not unreasonably skewed by the extinction and reignition treatments.

A preliminary verification test to evaluate the capacity for spurious reignition in each case is depicted in Fig. 8, which visualizes each simulated flame at $X_{O_2}^{\mathit{\text{ox}}}=0$ (pure nitrogen co-flow). For this condition, total flame extinction is definitively expected with no possibility of reignition, corresponding to a simulated combustion efficiency of η_comb = 0. Here, the simulated combustion efficiency is defined as

$${\eta }_{\mathit{\text{comb}}}=\frac{1}{\tau {\dot{m}}_{\mathit{\text{fuel}}}\mathrm{\Delta }{h}_{\mathit{\text{comb}}}}{\int }_{\tau }{\int }_V\dot{Q}‴\mathit{\text{dV dt}},$$ (13)

where Q̇‴ is the local volumetric heat release rate in each computational cell, ṁ_fuel is the mass flow rate of primary fuel from the fuel port, V is the volume of the computational domain, and τ is a time-averaging window.

Figure 8.

Spurious reignition in each modeling case for $X_{O_2}^{\mathit{\text{ox}}}=0$; (a) M1 suffers significant spurious reignition (η_comb = 0.69), (b) M2 thoroughly eliminates spurious reignition (η_comb = 0), (c) M3 suffers slight, but nearly negligible spurious reignition (η_comb = 0.01).

For case M1 in Fig. 8(a), as expected, there is significant spurious reignition and η_comb = 0.69. As shown, extinction is correctly predicted within the region of influence of the diluted oxidizer; however, as soon as the plume of unburned fuel mixes with sufficient air from the surrounding ambient (roughly 1 m above the fuel port) the inequality in Eq. (8) is satisfied and combustion proceeds spontaneously.

For case M2 in Fig. 8(b), there is no spurious reignition and η_comb = 0. For M3 however, in Fig. 8(c), there is a nearly negligible, but still finite occurrence of spurious reignition (η_comb = 0.01). This is due to the tiered reaction scheme implemented in M3, where the fuel transformation reaction (R2) includes a stoichiometric requirement for air. As there is no air present in the oxidizer at $X_{O_2}^{\mathit{\text{ox}}}=0$, reaction R2 fails within the diluted oxidizer, allowing the extinguished primary fuel to persist within the domain. Upon initial mixing with air in the ambient, most but not all of this primary fuel is eventually transformed to secondary fuel (via the extinction model). Spurious reignition in case M3 is thus significantly reduced compared to M1, though it is not as thoroughly eliminated as in M2.

Figure 9 presents simulated η_comb for each case plotted versus $X_{O_2}^{\mathit{\text{ox}}}$ and compared against experimental data [21], which depict a plateau of constant η_comb ≈ 1 for $X_{O_2}^{\mathit{\text{ox}}}>0.14$. Between $X_{O_2}^{\mathit{\text{ox}}}=0.14$ and $X_{O_2}^{\mathit{\text{ox}}}=0.125$, a sharp drop in η_comb occurs, tapering briefly before global extinction (η_comb = 0), occurring at $X_{O_2}^{\mathit{\text{ox}}}\approx 0.12$. Compared to the measured data, all simulated cases match the initial plateau in η_comb for $X_{O_2}^{\mathit{\text{ox}}}>0.14$. For $X_{O_2}^{\mathit{\text{ox}}}<0.14$, M1 fails to follow the experimental trend and significantly over-predicts η_comb, primarily due to the occurrence of spurious reignition, as illustrated in Fig. 8(a). Notably, M1 does not predict deviation from η_comb ≈ 1 until $X_{O_2}^{\mathit{\text{ox}}}<0.12$, where the experimental data (and M2 and M3) show global extinction. Also noteworthy, M1 fails to ever predict extinguishment.

Figure 9.

Simulated and measured combustion efficiency (η_comb) plotted versus $X_{O_2}^{\mathit{\text{ox}}}$; M1, which permits spurious reignition, fails to match the measured trend, while M2 and M3, which include provisions to prevent spurious reignition, match the measured data well.

As shown in Fig. 9, cases M2 and M3 follow the experimental η_comb trend very well, falling within measurement uncertainty (±0.04), and matching the extinction limit at LOI ≈ 0.12. Despite their different approaches at addressing spurious reignition (M2 via spatially defined ignition zone, and M3 via tiered reaction mechanism), the two cases also agree with one another remarkably well. In both cases, this agreement is achieved using relatively simple temperature-based extinction and reignition treatments, with established critical temperature values (T_ext = 1600 K and T_ign = 900 K). The results in Fig. 9 demonstrate the significance of adequately preventing spurious reignition in simulations with localized flame suppression.

With its tiered reaction mechanism, case M3 permits additional detailed diagnostics to compare the relative contributions of heat release from either primary combustion (reaction R1) or reignition (reaction R3). These results are presented in Fig. 10. As shown, combustion is initially dominated by R1 at ambient $X_{O_2}^{\mathit{\text{ox}}}$, though there is considerable reignition heat release from R3 (~30%), indicating that even at ambient conditions, M3 predicts localized extinction (though later reignition satisfies total η_comb ≈ 1). As $X_{O_2}^{\mathit{\text{ox}}}$ is reduced, η_{comb, R1} steadily and monotonically decreases while η_{comb, R3} gradually increases. For $X_{O_2}^{\mathit{\text{ox}}}>0.14$, these trends offset, resulting in constant total simulated η_comb ≈ 1. Near $X_{O_2}^{\mathit{\text{ox}}}\approx 0.135$, η_{comb, R3} attains a peak value of roughly 80%, dropping o abruptly with further declining $X_{O_2}^{\mathit{\text{ox}}}$. Along with the continuous decline in η_{comb, R1}, total η_comb also decreases, tapering once η_{comb, R3} drops to zero, occurring just before global extinction.

Figure 10.

Simulated and measured combustion efficiency (η_comb) among tiered reactions in modeling case M3; reaction R1 represents primary combustion, while reaction R3 represents reignition.

In agreement with the results of recent affiliated work [13], these trends clearly demonstrate that reignition treatment can dominate simulated suppression performance, where η_{comb, R3} is maximized and significantly greater than η_{comb, R1} throughout the region of rapidly declining η_comb in advance of the extinction limit. The simulated result that η_{comb, R3} > 0 at ambient $X_{O_2}^{\mathit{\text{ox}}}$ additionally suggests that local extinction and reignition events may be prevalent in turbulent fires, even for unsuppressed conditions. While these insights are interesting, it must be noted that no experimental measurements presently exist which could corroborate any distribution of total heat release amongst primary combustion and reignition. The present results are then useful to evaluate model performance, but should not be construed to represent actual flame behaviors until supporting measurements are made available.

3.2. Model Sensitivity

The sensitivity of modeling cases M1, M2, and M3 to variations in input parameters is examined in Fig. 11, which primarily compares simulated η_comb trends for each case among T_ext between 1500–1700 K, and T_ign between 800–1000 K. Specific to case M2, an additional sensitivity study considering the influence of the size of the prescribed ignition zone has also been performed.

Figure 11.

Sensitivity of simulated combustion efficiency (η_comb) to varying model parameters; (a) M1 T_ext sensitivity, (b) M2 ignition-zone size sensitivity, (c) M2 T_ext sensitivity, (d) M2 T_ign sensitivity, (e) M3 T_ext sensitivity, (f) M3 T_ign sensitivity.

Shown in Figs. 11(a), (c), and (e), respectively for cases M1, M2, and M3, a shift of T_ext by 100 K in either direction produces a corresponding shift in LOI by roughly 0.015. The same also results in a shift in the value of $X_{O_2}^{\mathit{\text{ox}}}$ at the inflection point of η_comb decline $(X_{O_2,\mathit{\text{crit}}}^{\mathit{\text{ox}}})$ by roughly 0.015, such that the entire η_comb trend is synchronously shifted to higher or lower $X_{O_2}^{\mathit{\text{ox}}}$ with variation in T_ext. Similarly shown in Figs. 11(d) and (f), respectively for cases M2 and M3, a shift of T_ign by 100 K in either direction results in a negligible change in LOI, but a shift in $X_{O_2,\mathit{\text{crit}}}^{\mathit{\text{ox}}}$ of roughly 0.01 (with M2 slightly more sensitive to variation in T_ign than M3). Shown in Fig. 11(b) (case M2), variations in ignition-zone elevation between 1.0–4.0 cm lead to a negligible change in LOI, with changes in $X_{O_2,\mathit{\text{crit}}}^{\mathit{\text{ox}}}$ of less than 0.005. Simulated suppression performance in each case is then most sensitive to T_ext, less sensitive to T_ign, and for M2, even less sensitive to ignition zone size.

Note that while simulated suppression performance is relatively insensitive to increasing size of the ignition zone, as shown in Fig. 11(b), there does exist a minimum effective size for the ignition zone below which spontaneous primary ignition is not achieved. For the present configuration, this ineffectiveness limit corresponds to an ignition zone one computational-cell high, extending 5 mm above the fuel port. This minimum size is likely to depend on the configuration of the fuel source, the selected value for T_ign, and the grid resolution (see Fig. 12 below), though no numerical or theoretical basis for its recommendation has yet been determined. A maximum size for the ignition zone however, may be discerned relative to the mean flame volume at ambient condition, where it would be desirable to limit the size of the ignition zone to be less than some small fraction of the mean flame volume.

Figure 12.

Sensitivity of simulated combustion efficiency (η_comb) to varying grid resolution; (a) case M1, (b) case M2, (c) case M3.

The sensitivity of each modeling case to variations in grid resolution is explored in Fig. 12, including comparisons between the baseline resolution, Δx = 5.0 mm, and coarsened resolutions, Δx = 10.0 mm and Δx = 20.0 mm (the refined resolution of Δx = 2.5 mm is deemed prohibitively expensive for the present analysis, with an estimated computational cost of greater than 100,000 processor-hours per simulation).

As depicted in Fig. 12(a), grid sensitivity in case M1 is relatively negligible, attributed to the dominance of spurious reignition over extinction within the domain. Comparatively, grid sensitivity in cases M2 and M3 is significant, respectively shown in Figs. 12(b) and (c). In both cases, grid coarsening leads to an over-prediction of suppression, attributed to the inherent grid sensitivity of the gas-phase cell temperature on which the reignition criterion is solely based and which tends to be under-predicted in coarse grids. Grid sensitivity is greatest in case M2, attributed to the bounds of the specified ignition zone, whose size does not scale with the underlying grid resolution so that for coarse grids, the relative size of the ignition zone becomes too small (contains too few cells) to support primary ignition.

3.3. Model Performance

Drawing from the present simulation results, a set of target objectives for modeling extinction and reignition in fire applications could include the following.

1. When desirable, fuel should be allowed to ignite spontaneously without the need to model complex ignition physics.

2. Combustion should be extinguished wherever extinction conditions are encountered, as determined by the chosen extinction model.

3. Following extinction events, fuel should not spuriously reignite once non extinction conditions are encountered elsewhere in the computational domain.

4. The model should ideally be free of arbitrary user input.

5. When desirable, the model should also be able to handle non-burning fuel sources, such as fuel leakages.

Here, objective (1) is necessary for reasons described in Sec. 2.4.2, where the modeling of primary ignition is generally unavailable in fire applications, which often include fuels that cannot be described with detailed reaction kinetics and complex configurations for which resolved heat transfer at the fuel source would require prohibitive computational cost. The current state-of-the-art solution to objective (1) in the classical EDC combustion treatment is to completely neglect ignition, where fuel burns spontaneously upon mixing with oxidizer. This convention then necessitates objectives (2) and (3), respectively for the provision of flame extinction and the prevention of spurious reignition, where the significance of objective (3) is readily demonstrated in the simulation results presented in Sec. 3.1.

Objective (4) recognizes the desire for a model that requires no arbitrary or configuration-dependent user input, while objective (5) accommodates potential specialized applications whose interests may include the modeling of fuel transport through a domain with spontaneous ignition occurring at a specified location other than the location of fuel injection. Note that objectives (4) and (5) are likely to conflict with one another, as user input is sure to be necessary to distinguish between a fuel source where spontaneous ignition is desired and a fuel leakage, where spontaneous ignition may be undesirable.

Of the modeling cases considered in the present study, all incorporate EDC mixing-controlled combustion, by which objective (1) is satisfied. Each case additionally includes a simple extinction criterion based on the concept of a critical flame temperature (see Sec. 2.4.1), satisfying objective (2). As discussed in Sec. 2.4.2 (and demonstrated in Sec. 3.1), this combination of EDC combustion treatment with simplistic extinction modeling unfortunately leads to spurious reignition and thus the failure of objective (3) in case M1.

While case M2 passes objective (3), it does so with the requirement that a prescribed ignition region be explicitly specified within the computational domain. While a prescribed ignition zone may be desirable in some applications (as recognized in objective (5), which only M2 passes), its requirement will be undesirable in many more cases where the appropriate spatial bounds of the ignition zone are not easily discernible. In general, the size and shape of the ignition zone must relate to the physical configuration of the fuel source and therefore no one-size-fits-all approach may be advised. Additional model sensitivities to the size of the ignition region and its performance dependency on grid resolution must also be considered (see Sec. 3.2). For these reasons, case M2 fails objective (4).

Case M3 addresses objective (3) using a tiered reaction mechanism (see Sec. 2.4.2), which avoids the constraint of arbitrary user input and therefore passes objective (4). Still, case M3 does require the added computational expense associated with the solution of an additional transport equation for the secondary fuel species. Additionally, though M3 adequately prevents spurious reignition in most situations, there are cases for which it fails to do so (see Fig. 8(c)). For this reason, M3 only qualifies objective (3).

The performance of each of the presently investigated extinction/reignition cases with respect to the stated modeling objectives is summarized in Table 3, where ✓, ✕, and ○ respectively indicate a passed, failed, or qualified objective.

Table 3.

Summary of objective performance for model cases M1, M2, and M3.

Objective	M1	M2	M3
(1)	✓	✓	✓
(2)	✓	✓	✓
(3)	✕	✓	○
(4)	✓	✕	✓
(5)	✕	✓	✕

From Table 3, it is readily apparent that there is no simple and generally applicable approach to achieve accurate modeling of extinction and reignition within the limitations of the EDC combustion framework. In the present configuration, for which objective (3) has been shown to be paramount to achieve reasonable simulated suppression performance, cases M2 and M3 (which pass or qualify this objective) are shown to yield accurate results when compared to measured global combustion efficiency data (see Fig. 9). While the reignition treatments adopted in these cases are simplistic, they produce significant improvements in model performance with manageable added complexity and computational cost.

The present results offer incentive to open new dialogues regarding the issue of spurious reignition in EDC combustion models, which until this investigation, have received limited or no attention in the broader research community, with all existing focus paid exclusively to extinction. The present results highlight that extinction modeling, while key to model performance, can be significantly affected and potentially dominated by the chosen reignition treatment.

4. Conclusions

The present work explores the capability of a large eddy simulation (LES)-based fire dynamics solver incorporating a mixing-controlled combustion framework to accurately describe flame response in a range of diluted oxidizer conditions, including a transition from complete combustion to total flame extinguishment. Three alternative cases for the implementation of critical flame temperature based extinction and reignition models are considered, including a baseline case without explicit reignition treatment (M1), a case including a novel spatially-variable reignition criterion (M2), and a case incorporating a tiered combustion mechanism with separate treatment of ignition, extinction, and reignition (M3). The extinction performance of these cases are compared against available experimental measurements, including local profiles of thermocouple temperature and oxygen mole fraction, as well as global combustion efficiency.

Comparisons of simulated results highlight the necessity for explicit treatment of reignition in mixing-controlled combustion models, where cases M2 and M3, which include provisions to prevent spurious reignition, yield extinction results that closely match the experimental data, while case M1, which does not restrict spurious reignition, fails to adequately predict flame extinction. While the reignition treatments adopted in cases M2 and M3 are relatively simplistic and ripe for further development, they are shown to produce significant improvements in modeled extinction performance in the present configuration with manageable parametric sensitivity, added complexity, and computational expense.

In comparing the extinction performance provided by each case, it is apparent that no simple and generally applicable approach is available to accurately model both extinction and reignition in a LES with mixing-controlled combustion. As a result, the requirement for configuration-dependent user input will be inevitable for certain applications using these types of models. In particular, additional in-flame experimental measurements are necessary to better differentiate and determine which of the available modeling treatments best represent both extinction and reignition phenomena in general applications.

Nomenclature

Symbols

Cu	model coefficient	(−)
Cυ	model coefficient	(−)
h	mass-specific sensible enthalpy	(J/kg)
k	mass-specific kinetic energy	(m2/s2)
Lf	flame height	(m)
LOI	limiting oxygen index	(mol/mol)
ṁ	mass flow rate	(kg/s)
ṁ‴	mass reaction rate	(kg/m3/s)
M	turbulence resolution criterion	(−)
Pr	Prandtl number	(−)
Q̇‴	volumetric heat release rate	(W/m3)
s	oxidizer to fuel mass ratio	(kg/kg)
Sc	Schmidt number	(−)
t	time	(s)
T	temperature	(K)
u	velocity, x-direction	(m/s)
υ	velocity, y-direction	(m/s)
V	volume	(m3)
w	velocity, z-direction	(m/s)
W	width	(m)
X	mole fraction	(mol/mol)
Y	mass fraction	(kg/kg)
Δ	LES filter width	(m)
Δhcomb	mass-specific enthalpy of combustion	(J/kg)
$\mathrm{\Delta }{h}_f^{◦}$	mass-specific standard enthalpy of formation	(J/kg)
[Δx, Δy, Δz]	numerical grid resolution	(−)
ηcomb	combustion efficiency	(−)
μ	dynamic viscosity	(kg/m/s)
ρ	mass density	(kg/m3)
τ	time-averaging window	(s)
τmix	turbulent mixing time scale	(s)
χrad	radiative loss fraction	(−)

Scripts

∞	ambient property
b	burner
crit	critical
dil	diluent
ext	extinction
f	flame
fuel	fuel
ign	ignition/reignition
k	indexing variable (species)
ox	oxidizer
rms	root mean square
sgs	sub-grid scale
t	turbulent
tc	thermocouple