Development and testing of a model for turbulence-radiation interaction effects on the radiative emission

Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

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Development and testing of a model for turbulence-radiation interaction effects on the radiative emission G.C. Fraga a,∗, P.J. Coelho b, A.P. Petry a, F.H.R. França a a b

Mechanical Engineering Department, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil Mechanical Engineering Department, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

a r t i c l e

i n f o

Article history: Received 17 October 2019 Revised 20 January 2020 Accepted 22 January 2020 Available online 28 January 2020 Keywords: Turbulence-radiation interaction Radiative emission Pool fires Reynolds-Averaged Navier-Stokes simulations Taylor-series expansion

a b s t r a c t A new model to account for turbulence-radiation interaction on the radiation emission in the framework of Reynolds-Averaged Navier-Stokes simulations is proposed and tested. The model’s development is based on Reynolds decompositions of the temperature T and the Planck-mean absorption coefficient κ in the time-averaged radiative emission and on a multi-variable Taylor-series expansion of κ as a function of temperature and species concentration. The relative importance of the terms that arise from these processes are assessed using statistics extracted from high-resolution, fully-coupled large eddy simulations (LES) of large-scale pool fires. All high-order terms are related to resolved quantities or to quantities for which models are already available by curve fitting the LES-generated data. The model presents a total error in directly estimating the radiative emission of about 25%, offering a considerable improvement over existing approximations, which, at best, have total errors above 40%. An expression introduced for the mean absorption coefficient also shows a good accuracy, with an associated total error of 16%. When applied to the solution of the time-averaged radiative transfer equation, the new model again outperforms other approximations, especially in the flame region. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Radiative transfer is well-recognized to be important for combustion applications, due to the high temperatures involved and to the presence of species such as carbon dioxide, water vapor and soot, that absorb and emit radiative energy [1]. When dealing with turbulent reacting flows, additional care must be taken in the radiative transfer calculations, for neglecting turbulent fluctuations of the scalars can introduce considerable errors in the resulting radiation field [2,3]. What causes this error is the socalled Turbulence-Radiation Interaction (TRI), originated from the highly non-linear coupling between fluctuations of the temperature and the chemical composition of the medium and fluctuations of the radiation intensity. In principle, these errors can occur for any modeling approach that does not consider all turbulent eddies for solving the flow and thermal fields; however, previous studies have shown that TRI is much more important for ReynoldsAveraged Navier-Stokes (RANS) simulations, where all fluctuations

∗

Corresponding author. E-mail address: [email protected] (G.C. Fraga).

https://doi.org/10.1016/j.jqsrt.2020.106852 0022-4073/© 2020 Elsevier Ltd. All rights reserved.

are neglected, than for large eddy simulations (LES), which partially consider them [4–6]. Some models are available for TRI in the RANS framework. For the radiative absorption term, the optically thin fluctuation approximation (OTFA) [7,8] has been adopted for various types of problems, with a number of papers attesting its validity (for example, [9–14]). This approximation relies on the assumption that, if the mean free path for radiation is much larger than the turbulence length scale, then the local spectral radiation intensity Iη is weakly correlated to the spectral absorption coefficient κ η , resulting in κ η Iη  ≈ κ η Iη  (the angle brackets indicate an averaging operation). Other procedures to model the radiative absorption have also been developed [11,15,16], but usually involve complicated and computationally intensive formulations, so their application in practical problems is still limited. The OTFA greatly simplifies the modeling of TRI, reducing it to the treatment of κ η  and of the mean radiative emission, κ η Ibη , where Ibη is the spectral blackbody radiation intensity or Planck function. Both of these terms depend solely on one-point statistics—as opposed to the mean absorption term, which requires detailed knowledge of the instantaneous fields of temperature and species concentrations to be closed—and as such can be exactly re-

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G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

solved if the probability density function (PDF) of the fluctuating quantities are known; comprehensive reviews on PDFs and their application to model TRI can be found in Refs. [17,18]. However, PDF approaches for TRI modeling are generally adopted in conjunction with solution methods and turbulence-chemistry interaction models that already make use of probability density functions, and coupling them with infinitely-fast chemistry models (e.g., the eddy break-up [19,20] and eddy dissipation [21] models) can be cumbersome. Therefore, simple equations have been proposed to express the mean emission and the mean absorption coefficient as a function of quantities that are directly resolved in RANS calculations (namely, the mean temperature and mean species concentrations) [22–24]. To provide a measure of the local intensity of turbulent fluctuations, these expressions also invariably depend on the temperature variance, which in turn can be determined from a transport equation [25,26]. Based on an earlier theoretical work on TRI in a gray and homogeneous gas layer [27], Snegirev [22,23] developed a correction factor for the spectrally-integrated mean emission by decomposing the temperature and the Planck-mean absorption coefficient into mean and fluctuating components and neglecting all higher-order terms; to model the remaining unresolved terms, a Taylor-series expansion of the absorption coefficient around mean temperatures was performed, once again disregarding correlations of order higher than two. The resulting expression was originally employed in Ref. [23] for modeling TRI in buoyant diffusion flames of propane and air, but has since then been used, with and without modifications, for various other types of reacting flows. For example, Krishnamoorthy and collaborators [28–30] applied Snegirev’s model in uncoupled radiative transfer calculations to study different spectral models and angular discretization schemes for configurations including pool fires and airfuel and oxy-fuel diffusion jet flames, and Centeno et al. [31–33], who first adapted the model for the weighted-sum-of-gray-gases (WSGG) formulation, carried out a series of TRI analyses on natural gas furnaces. Other scenarios in which Snegirev’s model was adopted include whirling fires [34], non-reacting flows [35], forest and vegetation fires [36,37], and the supersonic flow surrounding a reentry vehicle [38]. A model for the time-averaged Planck-mean absorption coefficient (i.e., the spectrally-integrated form of κ η  defined as to ensure a consistent value for the total emission) can be derived following a similar approach as Refs. [22,23], and has been presented and tested by Coelho [24], with only moderate success. Another approach for this term, now for LES calculations, was proposed by Chatterjee et al. [39], who additionally first implemented Snegirev’s correction factor for modeling subgrid-scale (SGS) TRI. Further discussions on applications of Snegirev’s model for SGSTRI, which are not the focus of the present study, can be found in Refs. [40–42]. In this framework, this paper reports the development of a new model for the time-averaged radiative emission and Planckmean absorption coefficient on the lines of Snegirev’s model. However, while the derivation of that model involved assumptions that were not supported by empirical data, the approximations proposed in this study are based on statistical information extracted from high-resolution, fully-coupled LES computations of largescale, non-sooting pool fires. A similar effort has recently been undertaken by Krishnamoorthy and Rahman [43], using statistical data obtained from laser measurements on hydrogen-enriched oxy-methane flames, but it was restricted to modeling the blackbody radiation intensity auto-correlation and was based on a small number of data points. The model proposed here is able to approximate all correlations that compose the mean emission term, and it will be constructed from statistics collected at approximately 35 0 0 0 points in the fire and hot gas plume regions.

2. Turbulence-radiation interaction fundamentals and modeling The radiative transfer equation (RTE) expresses the variation of the spectral radiation intensity along a given path s in the medium. Neglecting scattering, as is usual in studies concerning TRI [2], the RTE is given as [44,45]

dIη = −κη Iη + κη Ibη . ds

(1)

Integrated over the entire wavenumber spectrum, this equation becomes

dI = −κG I + κ Ib , ds

(2)

∞

∞

in which I = 0 Iη dη and Ib = 0 Ibη dη are the total and blackbody ∞ radiation intensities, respectively, κG = 0 κη Iη dη/I is the incident∞ mean absorption coefficient, and κ = 0 κη Ibη dη/Ib is the Planckmean absorption coefficient. Time-averaging Eq. (2) yields

d I  = −κG I + κ Ib  , ds

(3)

where, as previously noted, the angle brackets indicate a timeaveraging operation. Because the local radiation intensity is dependent on the temperature and species concentration along the path, there is no general modeling procedure available for the first term on the right-hand side of this equation(the mean radiative absorption), and the most common treatment for it is through the OTFA [7,8], which yields κ G I ≈ κ G I—or, for the spectral absorption, κη Iη  = κη Iη . For the mean radiative emission (the second term on the right-hand side of Eq. (3)), an approximation analogous to the OTFA is not adequate, since the absorption coefficient and the blackbody intensity tend to be well-correlated for turbulent flames [24,46,47]. Furthermore, the auto-correlations of κ and of Ib both have important contributions to the overall TRI, with the latter usually being the strongest among all correlations [2,3,48], so they also need to be taken into account. Recognizing that Ib = σ T 4 /π , where σ is the Stefan-Boltzmann constant, the modeling of κ Ib  reduces to finding a suitable expression for κ T4 . Following a procedure similar to that of Cox [27] in the theoretical analysis of a homogeneous, isothermal hot gaseous layer, Snegirev [22,23] decomposed the temperature and the absorption coefficient into mean and fluctuating components, T = T  + T  and κ = κ + κ  (where the prime denotes a fluctuation from the mean), to obtain

κ T 4  = (κ + κ  )(T  + T  )4   T  2  T  3  T  4  4 = κT  1 + 6 +4 + 2 T  T 3 T 4          σT 2

σT 3

σT 4

 κ  T  4  , κT  T 2 κT 3 κT 4     κ        

+4

κ  T   σκ T

+6

κ  T  2  σκ T 2

+4

κ  T  3  σκ T 3

+

(4)

σκ T 4

in which the σT i and σκ T i terms contain all unresolved quantities— i.e., quantities that could not be directly obtained from the mean species, flow and thermal fields. The first four terms inside the parenthesis in the previous equation originate from the blackbody radiation intensity autocorrelation, Ib ∝T4 , and the last four arise from the crosscorrelation between fluctuations of the absorption coefficient and the blackbody intensity, κ  Ib  . In this framework, to facilitate future discussions, the σ i terms in Eq. (4) can be combined into the following ST 4 and Sκ T groups:

G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

S T 4 = 1 + σT 2 + σT 3 + σT 4 ,

(5)

S κ T = σκ T + σκ T 2 + σκ T 3 + σκ T 4 ,

(6)

so that Sκ T 4 = κ T 4 /(κT  ) = ST 4 + Sκ T . Additionally, it can be

difficult. Nevertheless, following those approximations, Eq. (10) becomes

  ∂T 

∂κ κ  T   = T  2  

shown that ST 4 = and Sκ T = κ  T 4 /(κT  ). There is not much information, experimental or otherwise, currently available on the relative magnitude of either the σ i or Si terms. Based on measurements [49] of confined flames, Cox [50] reports that σT 4 σT 2 , and, by making a correspondence with experimental findings for cross-correlations of the velocity components in turbulent boundary layers [51], that correlations of oddorder (e.g., T 3  and κ  T 2 ) are expected to be very small. These odd-order correlations can be demonstrated to be identically zero if a Gaussian PDF is assumed for the fluctuating quantities, although it has been shown that this assumption is not adequate for all regions of a turbulent flame [52]. Furthermore, in an analysis using experimental data from a series of CH4 -H2 -O2 diffusion flames, Krishnamoorthy and Rahman [43] found some contradictory results to those of Ref. [50], with the σT 2 term being at most on par with (σT 3 + σT 4 ). In [23], only the correlations of lowest order, T 2  and κ  T , were taken into account, leading to Snegirev’s model in its unclosed form: 4

  T  2  κ  T   κ T 4  = κT 4 1 + 6 + 4 . κT  T 2

(7)

An additional transport equation [25,26] has often been used to determine the temperature variance (for example, see [31– 33,38,53,54]), but evaluating its accuracy is outside the scope of the present paper—a brief discussion on this subject can be found in Ref. [43]. To obtain a model for κ  T , the Planck-mean absorption coefficient is expressed as a multi-variable Taylor series of the temperature T and the species concentrations X,



∞ n 1 κ (T , X ) = n! n=0



 

∂ n κ  n n−m m (T  ) (X  ) n m ∂ T −m ∂ X m 

,

T ,X 

m=0

(8)



n where m is the mth binomial coefficient of a nth-degree polynomial and the zeroth derivative of κ is taken as κ itself. Timeaveraging this equation results in



∞ n 1 κ = n! n=0

m=0

  n m

(T  )n−m (X  )m 

 ∂ n κ  . ∂ T n−m ∂ X m T ,X 

(9)

Then, by subtracting Eq. (9) from Eq. (8), an expression for κ  is obtained, which can be multiplied by T and time-averaged to ultimately yield



∞ n 1 κ  T   = n! n=1

m=0

  n (T  )1+n−m (X  )m  m

 ∂ n κ  . ∂ T n−m ∂ X m T ,X  (10)

To the best of the authors’ knowledge, all previous papers that applied Snegirev’s model without outright neglecting κ  T  (i.e., setting σκ T = 0) considered only the first term of the infinite series in Eq. (10) and ignored the dependence of the absorption coefficient on fluctuations of the medium composition. However, apart from the aforementioned comments regarding odd-order correlations, no theoretical or empirical justification has been offered for these simplifications, which are motivated mainly by practical reasons, since deriving a model for the T  − X  correlations can be

.

(11)

T ,X 

4

T 4 /T 4

3

Finally, the combination of Eqs. (7) and (11) leads to the closedform of Snegirev’s model for the mean radiative emission [23],

 κ T  = κT  4

4

  T  2  T  2  ∂κ  1 + 6CTRI,1 + 4CTRI,2 . κT  ∂ T T ,X  T 2 (12)

where CTRI,1 and CTRI,2 are constants introduced to compensate for the terms neglected and for the approximations involved in the derivation of this equation. By curve fitting the dependence between T4 /T4 and T 2 /T2 to data reported by Burns [48], Snegirev initially proposed CTRI,1 = 1.25, but later readjusted it to 2.50 following tests with buoyant propane flames [23]. This value has also been used by a number of researchers for modeling TRI in different types of reacting flows—for instance, in air-fuel and oxy-fuel furnaces [31–33,54], jet diffusion flames [55] and whirling fires [34]. Other values adopted for CTRI,1 include 0.92 [28], 1.00 [35] and 2.00 [43], with the latter two references being the only papers to provide justification for the choice of CTRI,1 . The second constant in Eq. (12) was defined as 1.00 in [23], based on the argument that the blackbody intensity auto-correlation dominates over the absorption coefficient-temperature correlation; likely also because of that, numerous papers have simply set CTRI,2 = 0 [28–30,38], which has the additional benefit of avoiding having to evaluate dκ /dT. An often overlooked issue that arises from Eq. (12) (or, for that matter, from Eq. (4) itself) is how to determine the time-averaged value of the Planck-mean absorption coefficient, κ, which is yet another unresolved quantity. In principle, Eq. (9) gives an exact expression for this quantity, but applying it in practice is unfeasible for the same reasons discussed after Eq. (10). A simplified form of that equation, obtained following similar assumptions to those that led to Eq. (11), was presented by Coelho [24],

κ = κ (T , X  ) +

 T  2  ∂ 2 κ  , 2 ∂T2  T ,X 

(13)

where κ (T, X) is the Planck-mean absorption coefficient evaluated at the mean temperature and mean species concentration. In [24], this equation was tested in a theoretical manner for diffusion flames of methane, and was found to provide a good approximation only for low levels of turbulence intensity. Unfortunately, however, studies that apply Snegirev’s model are usually not clear on whether κ is evaluated following Eq. (13) or through some other means; those that do specify this (e.g., [33,35,54]) generally adopt the simpler approach of completely neglecting turbulent fluctuations for this term and setting κ = κ (T , X  ). 3. Description of the test cases The statistical data used for the analyses to be conducted in Section 4 are obtained from numerical simulations of an ethanol and a methanol pool fire burning in an open atmosphere. In both scenarios, the pool was circular, measuring 0.5 m in diameter and 70mm in depth, and filled with pure liquid fuel. The mean mass burning rate and mean total heat release rate by combustion (per unit of pool surface area) were of 9.76g/(m2 s) and 261kW/m2 for the ethanol pool fire, respectively, and 10.5g/(m2 s) and 217kW/m2 for the methanol one, as numerically estimated in a previous paper [56]. A detailed description of the simulation of these cases, alongside a validation of their results, can be found in Ref. [56]. Here,

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only a brief summary is presented, followed by some comments regarding the sampling methodology. The calculations consisted of large eddy simulations carried out in a rectilinear, non-uniform spatial grid with about 15.7 million cells, divided into a finer region near the flame, with characteristic cell size of 0.005m, and a coarser region in the remainder of the domain, with cell size of 0.01m. A mesh quality analysis showed that the grid is capable of directly resolving at least 90% of the turbulent scales and to reproduce the inertial range of the velocity spectrum, as high-fidelity LES is expected to do [57]. The computational domain was a box of dimensions 1.5m × 1.5m × 3.3m, with all of its sides open to an atmospheric environment and the fuel pool positioned at the center of its bottom surface. Combustion was modeled as a single-step, infinitely-fast, complete reaction of fuel and air, which, despite being a relatively simple modeling approach, is widely used for pool fire simulations and by the fire community in general (see, for example, [43,48,58]). For the radiative heat transfer calculations, the participating species (CO2 and H2 O) were treated as non-gray with the WSGG model; once again, more accurate models are available, but it has been shown that the WSGG model with coefficients derived from recent spectroscopic databases yields results in good agreement with line-by-line reference data [59–61]. At every time step of the simulations, instantaneous (filtered) temperature and CO2 and H2 O molar fractions were recorded at approximately 35 0 0 0 points uniformly distributed in the flame region and in the hot gas plume. Further tests showed that increasing the number of sampling points has a negligible effect on the results. All statistical quantities of interest for the present study (i.e., time-averages, moments and correlations) are then computed over the last 10s of the transient data, which was found to be a large enough time-interval to adequately capture these statistics. A recent correlation [62], developed based on the HITEMP 2010 database [63], is employed to describe the dependence of κ on T and X; this correlation is more adequate for estimating the Planck-mean absorption coefficient than the WSGG model used in the simulations and is capable of very accurate predictions of this quantity for a wide range of temperatures and molar fractions [64]. In order to simplify the analysis, points with a negligible contribution to the radiative transfer are not considered; the criterion adopted for this purpose is a local mean temperature smaller than 500K, which approximately corresponds to neglecting all points where the local mean radiative emission is less than 1% the maximum mean emission in the domain. It should be noted that using LES-generated data for the analyses carried out in this paper implicitly neglects the contribution of subgrid-scale fluctuations to the radiative transfer. Although the importance of SGS-TRI is still a debatable subject, with some authors [65,66] arguing that it may have some relevance, previous studies [4–6,40] indicate that SGS fluctuations can safely be neglected for a reasonably fine spatial discretization. Furthermore, a preliminary investigation on the same pool fires studied here estimated that SGS-TRI effects were marginal for the level of grid resolution adopted for the present simulations [42]. 4. Results and discussion 4.1. Relative importance of unresolved terms in the mean radiative emission Fig. 1 plots the relation between the σT i terms in Eq. (4) and ST 4 (Eq. (5)), and between the σκ T i terms and Sκ T (Eq. (6)). Each dot in the figure corresponds to one data point pertaining to either the ethanol or the methanol pool fire simulations (which yielded very similar statistical data), and its value was determined following the procedure described in Section 3. Furthermore, to avoid

overvaluing points for which the radiative emission is relatively small, ST 4 and Sκ T have been normalized by the ratio between the local mean radiative emission and the maximum mean emission in the computational domain, i.e., SˆT 4 = ST 4 × κ T 4 /κ T 4 max and Sˆκ T = Sκ T × κ T 4 /κ T 4 max . Although the data points in Fig. 1 are fairly spread (especially for the σT i − SˆT 4 charts), there is a clear trend of the σ i terms to increase as the importance of TRI increases, as expected. The relations between σT i and SˆT 4 are broadly similar, with σT 2 , σT 3 and σT 4 having the same magnitude; while this is also true for the σκ T i − Sˆκ T charts, the values of σκ T i seem to be larger than those of σT i . To verify this, Fig. 2 plots the Sκ T /ST 4 ratio against the normalized mean radiative emission and against T 2 1/2 /T, which gives a measure of the intensity of turbulent fluctuations of temperature. From this figure, three distinct regimes can be identified: points with relatively low levels of turbulence (which include the region of largest emission), where Sκ T is negative and its absolute value is smaller than ST 4 ; points with intermediate T 2 1/2 /T, in which Sκ T is positive and it can be either greater or smaller than ST 4 ; and points where turbulent fluctuations (and, consequently, TRI effects) are high, that show Sκ T dominating over ST 4 . This contradicts in part the justification provided by Snegirev [23] for setting CTRI,2 = 1.0, in which it was argued that the blackbody intensity auto-correlation dominates the emission TRI effects. While this is true for the regions of largest radiative emission, at locations where TRI is the strongest the contribution of the blackbody intensity auto-correlation to the mean radiative emission is at most on par with, but mainly smaller than, the other correlations (i.e., Sκ T ). More information regarding the behavior of the emission correlations for these pool fires can be found in Ref. [67]. 4.2. Modeling the higher-order σ i terms The observation in Fig. 1 that the individual σT i and σκ T i terms have similar magnitude indicates that neglecting high-order terms is not an adequate approximation. To model the terms related to the blackbody intensity auto-correlation, previous papers [23,35,43] have tried to establish relations between the lowerorder σT 2 term and ST 4 through linear curve fitting based on statistical data obtained experimentally, theoretically or numerically; this is what gives rise to the different values of CTRI,1 in Eq. (12), as discussed in Section 2. To the best of the authors’ knowledge, no similar attempt has been made so far for the terms in the Sκ T group. In this study, each individual higher-order term pertaining to the ST 4 and Sκ T group is related to the corresponding lowestorder term. For this purpose, plots of the higher-order T i /Ti and κ  T i /κTi terms against T 2 /T2 and κ  T /κT, respectively, are shown in Fig. 3. Because the relations between these quantities are well-behaved, linear and quadratic curves are fitted to the LES data, as also depicted in the figure. These expressions are generated with the constraint y = 0 for x = 0 to ensure physical consistency, and their parametric constants are given in Table 1. Recalling the definitions of the σ i terms in Eq. (4), the relations established in Table 1 can then be used to derive approximations ap ap for ST 4 and Sκ T , denoted by S 4 and Sκ T , respectively:



ST 4 ≈ STap4 =

T

1 + 2.08σT 2

(Linear approximation),

1 + 1.28σT 2 + 0.27(σT 2 )

2

(Quadratic approximation),

(14)

 Sκ T ≈ SκapT =

3.88σκ T

(Linear approximation),

0.62σκ T + 1.33(σκ T )

2

(Quadratic approximation),

(15)

where the linear and quadratic approximations correspond to the linear and quadratic curves depicted in Fig. 3, respectively. Anal-

G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

5

Fig. 1. Relation between the σ i and the Si terms. Table 1 Parameters of the linear and quadratic expressions obtained by curve fitting the LES data in Fig. 3. Quadratic, c1 /c2 (y = c1 x + c2 x2 )

Linear, c1 (y = c1 x) y T  3  T 3 T  4  T 4

x= 1.06 2.26

 T2 T 2

y κ  T  2  κT 2 κ  T  3  κT 3 κ  T  4  κT 4

κ  T  

x = κT 

1.38 2.37

Fig. 2. Relative importance of the ST 4 (temperature auto-correlation) and Sκ T (absorption coefficient-temperature correlation) groups.

ap S 4 T

ap + Sκ T .

ap

κT4

T 

T  3  T 3 T  4  T 4

0.60

ogously, an approximation for Sκ T 4 is obtained as Sκ T 4 ≈ S

2 x = T 2

y

=

Note that the linear approximation proposed for ST 4 in Eq. (14) would imply CTRI,1 = 2.08 if the formulation of Snegirev (Eq. (12)) were used. This value is significantly larger than the one obtained from numerically-generated data for a non-reacting flow in Ref. [35] (CTRI,1 = 1.00), as well as the one initially proposed by Snegirev [23] (1.25) based on the data of Burns [48]; however, it matches well the value of CTRI,1 = 2.00 determined by Krishnamoorthy and Rahman [43] by fitting measurements for hydrogen-enriched oxy-methane flames. A more detailed discussion on the effect of using the different existing values of CTRI,1 for modeling the blackbody intensity auto-correlation for the present pool fires can be found in Ref. [67]. To assess the quality of the approximations in Eqs. (14) and (15), Fig. 4 plots the relative error associated to each of them, ap as well as to S 4 , against the intensity of temperature fluctuaκT tions; also shown in this figure is the error of neglecting all higherorder terms, i.e., of Eq. (7). In general, the quadratic approxima-

0.32/1.52 0.43/3.74

y κ  T  2  κT 2 κ  T  3  κT 3 κ  T  4  κT 4

κT x = κ T 

−0.09/1.13 0.02/2.23 −1.03/5.55

tion provides the best estimation of ST 4 , Sκ T and Sκ T 4 , and neglecting higher-order terms, the worst. The latter consistently underestimates ST 4 (since σT 4 is always positive and σT 3 is negative only for a small number of data points, cf. Fig. 3), as well as Sκ T 4 for moderate-to-high levels of turbulence. Conversely, the approximations given by Eqs. (14) and (15) may either under or overpredict these quantities. Moreover, as turbulence gets stronger, neglecting higher-order terms yields increasingly large errors for ST 4 and Sκ T 4 , while the linear and, especially, the quadratic approximations are less sensitive to the turbulence level. For Sκ T , the errors of the three approximations are large for 0.4 < T 2 1/2 /T < 0.6; this range, however, mostly corresponds to small values of Sκ T , so the elevated errors at these points should not be very important for the overall performance of the model under development. The total error in the prediction of ST 4 , Sκ T and Sκ T 4 , defined as  ap  T E = |Si − Si |/ |Si | for any of the Si groups, is also reported in Fig. 4 (this is preferable to a domain-average of the relative error as a measure of the overall accuracy of each approximation because the latter is highly influenced by large errors that occur when Si is very small). For all Si groups, the quadratic approximation has smaller total errors than the linear one, never surpassing 15%. Both approximations are far superior to considering only the lowest-order terms, which has a total error of almost 60% in Sκ T 4 . 4.3. Modeling the κ  − T  correlation The κ  − T  correlation can be exactly obtained from the infinite series in Eq. (10). However, this is not convenient, not only due to the number of terms that need to be evaluated, but also because most of those terms contain unresolved quantities in the RANS framework (namely, the higher-order central moments of temperam n ture and the various T  − X  correlations). Therefore, to devise a   model for κ T , Eq. (10) is first made dimensionless by dividing it by κT. Let’s then define λκ  T  ,n as the nth term of the infinite

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G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

Fig. 3. Relations between the higher-order T i /Ti and κ  T i /κTi terms and the lowest-order T 2 /T2 and κ  T /κT terms.

Fig. 4. Relative error associated to different approximations for ST 4 , Sκ T and Sκ T 4 .

series, such that ∞ κ  T   = λ   , κT  n=1 κ T ,n

with



n 1 λκ  T  ,n = n!

m=0

  n m

(16)

 ∂ n κ  (T  )1+n−m (X  )m  . κT  ∂ T n−m ∂ X m T ,X  (17)

Due to the nature of the expression for the Planck-mean absorption coefficient used in this study, which consists of a fifth-degree

polynomial of temperature with a linear relation with respect to X, n ranges from 1 to 6 [62]. The λκ  T  ,n terms are plotted in Fig. 5 against κ  T /(κT). These terms are all of the same magnitude—which indicates that neglecting all higher-order terms, as it was done to reach Eq. (11), is not a good approach—and they can be positive or negative de1+n−m pending on the signs of (T  ) (X  )m  and of the nth derivative of κ with respect to T. Furthermore, no single λκ  T  ,n can alone provide an accurate estimation of κ  T /(κT). On the other hand, the data points in the λκ  T  ,1 − κ  T  /(κT  ) chart are cluttered within a narrow region and give an almost direct proportionality between these terms, especially for intermediate-to-large

G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

7

Fig. 5. Magnitude of each non-zero term in the summation of Eq. (10) in dimensionless form.

Fig. 6. Comparison of the two terms that compose λκ  T  ,1 , (a) and (b), and relation between T 2 /T2 and X T /(XT) (c) alongside the linear curve fitted to the data (shown in red).

values of κ  T /(κT). Hence, Eq. (16) can be replaced by the following approximate linear relation

κ  T   ≈ 0.34λκ  T  ,1 , κT 

(18)

which is obtained by fitting the λκ  T  ,1 − κ  T  /(κT  ) data, again subjected to the constraint that the curve must pass through the origin. The total error (defined similarly to TE reported in Fig. 4) in the calculation of κ  T /(κT) using Eq. (18) is about 15%. Once established a direct relation to λκ  T  ,1 , the final obstacle in modeling κ  T /(κT) is to express the former as a function of resolved (or already modeled) quantities only. According to Eq. (17), λκ  T  ,1 is given as

  T  2  ∂κ  T  X   ∂κ  λκ  T  ,1 = + . κT  ∂ T T ,X  κT  ∂ X T ,X        ξIκ  T  

(19)

ξIIκ  T  

In the framework of RANS modeling, the first term on the righthand side of this equation, ξIκ  T   , is already resolved, since it

is possible to estimate the temperature variance from mean flow quantities [25,26], and ∂ T /∂ κ|T ,X  can be evaluated from the spectral model (in this discussion, κ is also assumed to be known; the modeling of this term is discussed in Section 4.4). On the other hand, the last term of Eq. (19), ξIIκ  T   , still requires modeling due to the presence of the T  − X  correlation.

Before discussing the treatment of ξIIκ  T   , it is worth compar-

ing the magnitude of the two terms that compose λκ  T  ,1 (which, in turn, is approximately proportional to κ  T /(κT)). Fig. 6a plots the ratio between ξIIκ  T   and ξIκ  T   as a function of λκ  T  ,1 , from which it is clear that the former dominates over the latter for all but the smallest values of λκ  T  ,1 . Note that ξIκ  T   represents the contribution of temperature fluctuations alone to Eq. (19)—and,

consequently, to κ  T —while ξIIκ  T   gives a measure of the importance of fluctuations in medium composition to that term (or, more specifically, the importance of fluctuations of T and X combined, through the presence of the T  − X  correlation). It is evident that considering only temperature fluctuations, as it was assumed in Refs. [22,23], is not enough for an accurate estimation of κ  T . Fig. 6b shows that there is no convenient way to directly relate ξIκ  T   and ξIIκ  T   due to the high scattering of the data; for the

same reason, it is not possible to propose an adequate curve fitting to express λκ  T  ,1 or κ  T  as a function of ξIκ  T   only (these

charts are not shown here). Conversely, the relation between the temperature variance and the T  − X  correlation, normalized by the corresponding time-average values, can be described by a simple linear curve, as plotted in red in Fig. 6c. The slope of this curve (generated using the same constraints as the previous fittings) is 2.08. Combining all the approximations discussed in this section, an expression for κ  T  in terms of κ, T 2  and resolved quantities is finally reached,

  ∂κ X  ∂κ κ  T   ≈ κ  T  ap = T  2  cκ T,1 |T ,X  + cκ T,2 |T ,X  , ∂T T  ∂ X (20) with cκ T,1 = 0.34 and cκ T,2 = 0.72. The relative error of this expression is plotted in Fig. 7, alongside the error of the approximation for κ  T  obtained following the approach proposed in Ref. [23], Eq. (11). The error associated to Eq. (20) is significantly lower than that of Eq. (11), which tends to overestimate the κ  − T  correlation for low levels of turbulence and heavily underestimate it for strong turbulence intensities. As it is also shown in Fig. 7, the to  ap tal error T E = |κ  T   − κ  T  |/ |κ  T  | of Eq. (11) is almost 300%, while, for the approximation presently developed, it is only about 25%.

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G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

developments already presented. Setting n = 2 in Eq. (21) gives

λκ,2 =

1 T  2 κ

2



   ∂ 2 κ  T  X   ∂κ  + κ ∂ X ∂ T T ,X  ∂ T 2 T ,X       ξIκ

ξIIκ

 2 1 X   ∂ 2 κ  + 2 κ ∂X2  T ,X    

(23)

II ξIκ

where the three terms on the right-hand side have been labeled analogously to Eq. (19). This equation can be simplified by recII = 0, since κ has a linear dependence on the ognizing that ξIκ

medium composition [62]. Furthermore, T X  can be expressed in terms of T 2  by using the fitting established in Fig. 6c. From these approximations, a model for κ as a function of the mean temperature, mean species concentration and temperature variance can be derived:



κ ≈ κ

ap

  X  ∂κ  + cκ ,2 , T  ∂ X ∂ T T ,X 

Fig. 7. Error in the approximations of κ  T  and κ proposed in this study (Eqs. (20) and (24)) and in previous papers (Eqs. (11) and (13)).

4.4. Modeling the mean absorption coefficient The process to develop a model for κ is similar to the one presented for κ  T  in Section 4.3. Here, we start by dividing Eq. (9) by κ (T, X), so that a set of λκ,n terms can be defined  such that κ/κ (T , X  ) = 1 + ∞ n=1 λκ,n . Hence,



n 1 λκ,n = n!

m=0

  n m

 ∂ n κ  (T  )n−m (X  )m  , κ ∂ T n−m ∂ X m T ,X 

(21)

where, for the correlation for the Planck-mean absorption coefficient presently adopted, 2 ≤ n ≤ 6. For simplicity, in Eq. (21) as well as in the remainder of this section the letter κ outside the average sign represents the absorption coefficient evaluated with mean quantities. The relation of each λκ,n term to κ/κ is shown in Fig. 8. Note first that TRI mostly contributes to decrease the absorption coefficient, κ < κ , which is the opposite effect it has on both the blackbody intensity auto-correlation and the mean emission term (see the signs of ST 4 and Sκ T in Fig. 1). This is in agreement with previous studies on other types of reacting flows, e.g., [24,47,66]. A more detailed discussion on how each individual correlation is affected by turbulent fluctuations for the pool fires studied here can be found in Ref. [67]. Fig. 8 shows that, although all λκ,n terms have the same magnitude (so neglecting the higher-order terms, as in Ref. [24], is not adequate), λκ,2 has a seemingly almost linear relation to κ/κ . In fact, the approximation

κ ≈ 1.0 + 0.68λκ,2 , κ

(22)

which provides the best linear fitting of the λκ,2 − κ/κ data, has a total error of less than 7%. Then, an expression for λκ,2 in terms of resolved quantities can be obtained from the

 

∂ κ 2 = κ ( X , T  ) + T   cκ ,1 ∂ T 2 T ,X  2

(24)

where cκ ,1 = 0.34 and cκ ,2 = 1.42. Fig. 7 plots the errors of this expression and of Eq. (13), the latter being the approximation for κ presented in Ref. [24]. The model developed here provides a much better approximation than that equation for all turbulence intensities. In particular, for strong levels of turbulence, Eq. (24) mostly underpredicts the mean absorption coefficient (i.e., overestimates the TRI effects on κ , since turbulent fluctuations contribute to decreasing this quantity, as previously noted), while Eq. (13) heavily   ap overpredicts it. The total errors (T E = |κ − κ|/ |κ|) of these approximations are 16% and 80%, respectively—for reference, by fully-neglecting fluctuations and assuming κ = κ (T , X  ), the associated TE is almost 120%. 4.5. Summary of the complete model Combining the expressions and fittings proposed in Sections 4.2–4.4, a complete model for approximating the TRI effects on the radiative emission is achieved. For convenience, let’s first introduce a correction factor β to the emission term ap

κ T 4  ≈ κ T 4  = βκ (X , T  )T 4 .

(25)

The correction factor is given as

β = βκ (1 + fT σT∗2 + fκ T σκ∗T ) ,

(26)

where β κ is a correction associated to the modeling of κ, derived from Eq. (24),

βκ = 1 +

  T  2  ∂ 2κ X  ∂κ cκ ,1 | + c T | ; κ , 2  T  ,  X   T  ,  X  κ ( X , T  ) T  ∂ X ∂T2 (27)

fT and fκ T are terms related to the blackbody intensity autocorrelation and the absorption coefficient fluctuation-blackbody intensity correlation, respectively,

fT = aT σT∗2 + bT ,

(28)

fκ T = aκ T σκ∗T + bκ T ;

(29)

G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

9

Fig. 8. Magnitude of each non-zero term in the summation of Eq. (9) in dimensionless form.

Fig. 9. Error in κ T4  as approximated by the new model proposed in this paper, by Snegirev’s model (Eq. (12), using different approaches for evaluating κ), and by fully-neglecting turbulent fluctuations.

Table 2 Constants of Eqs. (27)–(31). aT bT

0.273 1.28

aκ T bκ T

1.33 0.624

cκ ,1 cκ ,2

0.341 1.42

cκ T,1 cκ T,2

0.344 0.715

and σT∗2 and σκ∗T correspond to approximations (or are identical to, in the case of the former) for σT 2 and σ κ T ,

σT∗2 = σT 2 = 6 σ

T  2  , T 2

(30)





T   ∂κ  4 = cκ T,1 βκ κ (T , X  ) T  ∂T  2

∗ κT

 X  ∂κ  + cκ T,2 T  ∂ X 

T ,X 



,

(31)

T ,X 

where this last expression is obtained by applying Eq. (20) to the definition of σ κ T . In these equations, the terms denoted by the letters a, b and c are constants; although their values have already been given in previous sections, they are summarized in Table 2. Finally, note also that the model presented here has been derived by employing the quadratic approximations for ST 4 and Sκ T given in Eqs. (14) and (15), since they perform better than the linear ones, as discussed in Section 4.2. The relative and total errors of the new model are reported in Fig. 9. Although there are some points where the relative error exceeds 100%, the total error of the proposed model is about 26%. For comparison, Fig. 9 also shows the errors obtained in the case 4 of fully-neglecting TRI (i.e., taking κ T 4  = κ (T , X  )T  ) and of using Snegirev’s model, Eq. (12), with CTRI,1 = 2.5 and CTRI,2 = 1.0. For the latter, three different approaches for calculating κ are considered: neglecting TRI, κ = κ (T , X  ); employing the ap-

proximation given in Eq. (13); or fully-considering TRI for this term (which is recognizably unfeasible for RANS calculations, unless a PDF-based model is adopted). If TRI is altogether neglected, the mean emission is significantly underpredicted and the total error is almost 60%. Adopting Snegirev’s model as it is given in Eq. (12) using the exact value of κ reduces error, but TE is still about 50%. Interestingly, applying Eq. (12) with κ = κ (T , X  ) improves the results, with TE ≈ 40%; this occurs because neglecting turbulent fluctuations when evaluating κ tends to overestimate this quantity (see the discussion on the values of κ/κ (T, X) in Section 4.4), which in turn increases the radiative emission predicted by the model and yields values closer to the actual κ T4 . Eq. (13) overestimates κ even more than κ (T, X), in general (because ∂ 2 κ /∂ T2 |T, X > 0, cf. Eq. (13)); this results in the mean radiative emission computed by combining Eqs. (12) and (13) to be considerably larger than κ T4  for most points and causes the total error of this approximation to be the largest among all approaches shown in Fig. 9. 4.6. Application of the new model The performance of the model developed in this study is now assessed for the solution of the mean radiation field. The test cases correspond to the same pool fires from which the statistical data used for the analyses in the previous sections were extracted. The radiative transfer is solved in a decoupled manner, i.e., employing converged fields of mean temperature and mean species concentrations that were determined from the transient, fully-coupled LES calculations described in Section 3 and in Ref. [56]; the temperature variance is also obtained from the transient calculations. The finite volume method [68] is adopted for the numerical solution of the RTE, employing the same spatial grid of the LES computations and a total of twenty-four control angles for the directional discretization. Further refinements did not significantly alter the results, as reported in [56].

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Fig. 10. Mean emission, absorption and net radiative heat loss along three radial positions for the ethanol pool fire, as computed by fully considering resolved-scale TRI and by three approximations.

Although, in the same way as Snegirev’s model, the TRI model proposed in this paper was developed assuming the participating medium to be gray, it can still be applied for non-gray media if a global gas model (such as the WSGG or the spectral line-based WSGG models) is used for the spectral integration of the RTE. Then, a different correction factor can be derived for each individual gas j of the model by replacing all derivatives of κ with respect to T and X in Eqs. (25)–(31) by derivatives of (κ j wj ), where κ j and wj are the absorption coefficient and the blackbody weighting factor of gas j, respectively. A formulation of Snegirev’s model along these lines was first presented by Centeno et al. [33] in the framework of the WSGG model and has since been employed by other authors (e.g., [35,53,54]). It should be noted that, in principle, different model constants could be derived for each gas j by applying a methodology similar to the one described in Sections 4.2–4.4, but starting from the emission of each individual gas rather than from κ Ib ). While this would likely improve the performance of the TRI model, it would also make the constants dependent on the choice of spectral model; instead, here, the values given in Table 2 are used for all gases, which follows the same approach adopted in the aforementioned references for adapting Snegirev’s model to the WSGG formulation. Therefore, the medium is considered as a non-gray CO2 -H2 O mixture for the simulations in this section, with the spectral integration of the RTE carried out by the WSGG model and employing the correlations of Centeno et al. [69] and Dorigon et al. [59] for the calculations of the ethanol and the methanol pool fires, respectively. It is worth mentioning that an alternative method for coupling the TRI model to the WSGG formulation is to neglect spectral variations when evaluating β , having the same correction factor for all j (in this case, κ and its derivatives can be determined from the correlation for the Planck-mean absorption coefficient in Ref. [62], for example). For the cases considered in this paper, though, this approach led to much higher errors that the ones obtained with a j-dependent correction factor.

For the ethanol pool fire, Fig. 10 shows radial plots of the mean radiative emission, absorption and net loss, computed by different TRI approximations along three positions z above the pool surface. The emission and absorption reported in this figure are determined by the directional integration of the terms on the right-hand side   of Eq. (2) (i.e., Semi = 4π κ Ib d and Sabs = 4π κG Id , respectively, where denotes the solid angle), while the net radiative heat loss is Snet = Semi − Sabs . Three approximations are considered: the new model developed in this paper, given by Eqs. (25)–(31); Snegirev’s model, Eq. (12), with CTRI,1 = 2.50, CTRI,2 = 1.00 and κ = κ (T , X  ); and no model for TRI at all, where the RTE is simply solved using mean temperature and mean species concentration fields (“no-TRI”). The reference to which these approximations are compared is provided by a solution that takes all resolved-scale fluctuations into account (“full-TRI”), determined by time-averaging the instantaneous Semi , Sabs and Snet obtained in the transient LES calculation (note, however, that all these solution employ the OTFA for the treatment of the mean absorption). At the outer edge of the flame, which is the region with the strongest turbulence fluctuations, all TRI models underpredict the mean radiative emission and net loss to some degree. This is explained by the fact that, for high turbulence levels, the three approximations tend to mostly yield smaller κ T4  than the reference, as shown in Fig. 9. In the flame region, where Snet  is the largest, the new model provides in general a more accurate estimate of all radiative quantities than both Snegirev’s and noTRI models. The two latter approaches exhibit opposite trends in this region, with Snegirev’s model significantly overestimating Snet , Semi  and Sabs , while the no-TRI solution underestimates these quantities by just as much—this is also in accordance with the signs for the respective errors of κ T4  in Fig. 9 for low-tointermediate turbulence intensities. Farther away from the flame (z = 0.6m), the new model and Snegirev’s model perform similarly, both providing reasonable approximations for the radiative emission and net loss while significantly underestimating the ra-

G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

11

Fig. 11. Mean radiation loss fully-considering resolved-scale turbulent fluctuations (far-left) and difference between TRI models and this solution. Table 3 Domain-integrated mean emission, absorption and net radiative heat loss (only considering cells with T ≥ 500K) obtained by the approaches considered in this study. The relative difference to the full-TRI solution (in %) is shown within parenthesis. Ethanol pool fire

Full-TRI New model Snegirev’s model No-TRI

Methanol pool fire

∫Snet dV [kW]

∫Semi dV [kW]

∫Sabs dV [kW]

∫Snet dV [kW]

∫Semi dV [kW]

∫Sabs dV [kW]

8.48 9.42 (11.1) 9.99 (17.9) 2.40 (71.8)

20.6 21.4 (3.99) 23.5 (13.9) 6.07 (70.5)

12.1 12.0 (0.980) 13.5 (11.2) 3.67 (69.7)

7.00 7.76 (10.9) 8.47 (21.0) 2.22 (68.2)

15.3 16.6 (8.43) 18.5 (20.6) 5.20 (66.1)

8.33 8.86 (6.35) 10.0 (20.2) 2.98 (64.3)

diative absorption (this is why the models underpredict Semi  and overpredict Snet ); nevertheless, even for Sabs , they are significantly more accurate than the no-TRI approach. The results for the methanol pool fire follow a similar trend and are not shown here. For a quantification of the local error of the TRI models, a normalized difference δ to the full-TRI solution is plotted along a vertical plane that passes through the flame axis in Fig. 11, again for the ethanol fire. This difference is defined model full full as δ = |Snet  − Snet  |/ max(|Snet  | ), where the superscripts “full” and “model” indicate the results of the “full-TRI” solution and of any one of the TRI-models, respectively. For regions full in which the radiation loss is significant (contours of Snet  are also shown in Fig. 11), the new model offers a considerable improvement over Snegirev’s model, with δ less than 15% in all but some small locations near the pool surface. Interestingly, the local normalized differences associated to Snegirev’s model are even higher than those of the no-TRI approach both near the fuel pool and in a thin envelope surrounding the zone of largest Snet . However, appreciable errors occur in a much larger area of the domain if turbulent fluctuations are fully neglected (compare, for instance, the size of the regions with δ > 10% for the no-TRI and Snegirev’s model). Because of that, the domain-average value of δ (computed only considering points where T ≥ 500K) is 7.3% for the no-TRI model and 5.5% for Snegirev’s model; for the new model, the average δ is reduced by almost a half and is equal to 3.3%, once again showing that this model performs better than the other ones. The respective domainaverage normalized differences for the methanol pool fire are 8.3%, 6.2% and 4.3%.

Table 3 provides a comparison of the domain-integrated mean emission, absorption and net radiative heat loss obtained by the three TRI modeling approaches to the results of the full-TRI solution. Once again, the new model shows an improvement over both the no-TRI approach and Snegirev’s model, with errors in the domain-integrated radiative heat loss (which is directly proportional to the radiant fraction of the flame) of about 10%. Conversely, Snegirev’s model, although providing a considerable improvement in accuracy over the approach that neglects turbulent fluctuations, leads to errors of around 20%. The results and accuracy of both the new model and the one by Snegirev are sensitive to the values chosen for the constants that compose these approximations. Here, the values reported in Table 2 are used for the new model, which were determined based on the same set of flames that are considered in this section for the assessment of the TRI modeling approaches. Because those are likely the most favorable scenarios for the new model, the small errors observed in Figs 10 and 11 and in Table 3 should not be interpreted as to globally reflect its performance, and tests involving other reacting flow configurations are required. On the other hand, for Snegirev’s model the values of CTRI,1 and CTRI,2 recommended by Ref. [23] are adopted, determined from simulations of turbulent buoyant propane-air flames. Other values for these constants (especially for CTRI,1 ) have been proposed, as previously mentioned, but tests with them did not show an improvement in accuracy significant enough to make the model competitive with the approximation developed in this paper for the flames under study—for instance, using CTRI,1 = 2.08 (which was estimated in Section 4.2 to

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G.C. Fraga, P.J. Coelho and A.P. Petry et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106852

be its optimal value for the cases under study), Snegirev’s model still yields δ ≈ 15% in the flame region. Before concluding, an issue that was not considered in the present study and could give rise to additional errors in the calculation of the mean radiation field should be highlighted. In RANS simulations of turbulent reacting flows, the high-temperature reaction zones, which are the main contributors to the radiative emission, can not be properly resolved by an infinitely-fast chemistry model. Mathematically, though, this can be accommodated through the modeling of the temperature variance, that, as noted before, is usually done by introducing a transport equation for this quantity [25,26]. However, the conventional approach when deriving a transport equation for T 2  is to neglect the correlation between temperature fluctuations and the rate of reaction (see, for example, [23]), and this is the term responsible for accounting for the unresolved high-temperature reaction zones. This issue has previously been discussed in the framework of subgrid-scale TRI by Snegirev et al. [41], where it was referred to as turbulence-radiationreaction interaction (TRRI), but it has not been further studied in details so far. The calculations carried out for the new model and for Snegirev’s model in this section utilized fields of temperature variance extracted from the transient LES calculations, so the question of the modeling of T 2  was not investigated, and will be the subject of future studies.

5. Conclusions This paper proposed a new model to account for turbulenceradiation interaction effects on the mean radiative emission in the framework of RANS simulations. Its development was based on Reynolds-decompositions of the temperature and the Planck-mean absorption coefficient in the time-averaged radiative emission and on an expansion of κ in a multi-variable Taylor series, with each non-resolved term arising from these processes related to resolved quantities through fittings to transient data extracted from highresolution, fully-coupled large eddy simulations (LES) of large-scale pool fires. The ensuing model presented a total error in directly estimating κ T4  of about 25% for the studied configurations, offering a significant improvement in relation to existing approximations, which, at best, had total errors above 40%. An expression for the mean absorption coefficient was also introduced and the total error associated to it was of 16%, also performing better than an existing model, whose error reached 80%. When applied to the solution of the time-averaged form of the RTE, the mean emission, absorption and net radiative heat loss calculated with the new model were far closer to the results of a reference solution (in which resolvedscale turbulent fluctuations are fully considered) than other approximations, especially in the flame region. Finally, it should be noted that, although the model outlined in this paper was developed and tested for a small set of non-sooting pool fires, it can be extended to other types of reacting flows. In fact, this has already been done with Snegirev’s model, which was originally proposed for buoyant propane-air flames, but has since been applied (even without changes to its constants) to a number of other configurations, as noted in Section 2. In future studies, the present model will be tested in a decoupled fashion (similarly to the analyses carried out in Section 4.6) for pool fires of different sizes and different fuels—the LES calculations necessary for that are underway—and in fully-coupled RANS simulations.

Declaration of Competing Interest None.

CRediT authorship contribution statement G.C. Fraga: Conceptualization, Methodology, Software, Formal analysis, Investigation, Visualization, Writing - original draft. P.J. Coelho: Resources, Writing - review & editing, Supervision. A.P. Petry: Supervision. F.H.R. França: Resources, Writing - review & editing, Supervision, Funding acquisition, Project administration. Acknowledgements Author GCF thanks CNPq for a doctorate scholarship. Author FHRF thanks CNPq for research grant 302686/2017-7. This work was also supported by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2019. References [1] Modest MF, Haworth DC. Radiative Heat Transfer in Turbulent Combustion Systems: Theory and Applications. Springer; 2016. [2] Coelho PJ. Numerical simulation of the interaction between turbulence and radiation in reactive flows. Prog Energy Combust Sci 2007;33(4):311–83. doi:10. 1016/j.pecs.20 06.11.0 02. [3] Coelho PJ. Turbulence-radiation interaction: from theory to application in numerical simulations. J Heat Transf 2012;134(3):031001. doi:10.1115/ ihtc14-23339. [4] Coelho P. Approximate solutions of the filtered radiative transfer equation in large eddy simulations of turbulent reactive flows. Combust Flame 2009;156(5):1099–110. doi:10.1016/j.combustflame.2008.10.006. [5] Roger M, Coelho PJ, da Silva CB. The influence of the non-resolved scales of thermal radiation in large eddy simulation of turbulent flows: a fundamental study. Int J Heat Mass Transf 2010;53(13–14):2897–907. doi:10.1016/ j.ijheatmasstransfer.2010.02.002. [6] Miranda FC. Large eddy simulation of turbulent reacting flows with radiative heat transfer, Ph.D. thesis. Technischen Universität Darmstadt; 2018. [7] Kabashnikov V, Kmit G. Influence of turbulent fluctuations on thermal radiation. J Appl Spectrosc 1979;31(2):963–7. doi:10.10 07/bf0 0614832. [8] Kabashnikov VP, Myasnikova GI. Thermal radiation in turbulent flows – temperature and concentration fluctuations. Soviet Res 1985;17(6):116–25. [9] Coelho P. Evaluation of a model for turbulence/radiation interaction in flames using a differential solution method of the radiative transfer equation. In: 12th international heat transfer conference; 2002. p. 705–10. [10] Coelho PJ. Detailed numerical simulation of radiative transfer in a nonluminous turbulent jet diffusion flame. Combust Flame 2004;136(4):481–92. doi:10.1016/j.combustflame.20 03.12.0 03. [11] Wang A, Modest MF, Haworth DC, Wang L. Monte carlo simulation of radiative heat transfer and turbulence interactions in methane/air jet flames. J Quant Spectrosc RadiatTransf 2008;109(2):269–79. doi:10.1016/j.jqsrt.2007.08.030. [12] Gupta A, Modest MF, Haworth DC. Large-eddy simulation of turbulenceradiation interactions in a turbulent planar channel flow. J Heat Transf 2009;131(6):061704. doi:10.1115/1.3085875. [13] Consalvi JL, Demarco R, Fuentes A, Melis S, Vantelon JP. On the modeling of radiative heat transfer in laboratory-scale pool fires. Fire Saf J 2013;60:73–81. doi:10.1016/j.firesaf.2012.10.010. [14] Consalvi J, Nmira F. Absorption turbulence-radiation interactions in sooting turbulent jet flames. J Quant Spectrosc RadiatTransf 2017;201:1–9. doi:10.1016/ j.jqsrt.2017.06.024. [15] Tessé L, Dupoirieux F, Taine J. Monte Carlo modeling of radiative transfer in a turbulent sooty flame. Int J Heat Mass Transf 2004;47(3):555–72. doi:10.1016/ j.ijheatmasstransfer.20 03.06.0 03. [16] Coelho PJ. A general closure model for the time-averaged radiative transfer equation in turbulent flows. In: 2010 14th international heat transfer conference, Volume 5. ASME; 2010. p. 847–56. doi:10.1115/IHTC14-22461. [17] Haworth D. Progress in probability density function methods for turbulent reacting flows. Prog Energy Combust Sci 2010;36(2):168–259. doi:10.1016/j.pecs. 20 09.09.0 03. [18] Haworth DC, Pope SB. Transported probability density function methods for reynolds-averaged and large-eddy simulations. In: Turbulent combustion modeling. Springer Netherlands; 2011. p. 119–42. doi:10.1007/978- 94- 007- 0412- 1_ 6. [19] Spalding D. Mixing and chemical reaction in steady confined turbulent flames. Symp (Int) Combust 1971;13(1):649–57. doi:10.1016/s0 082-0784(71)80 067-x. [20] Spalding DB. Development of the eddy-break-up model of turbulent combustion. Symp (Int) Combust 1977;16(1):1657–63. doi:10.1016/s0082-0784(77) 80444-x. [21] Magnussen B, Hjertager B. On mathematical modeling of turbulent combustion with special emphasis on soot formation and combustion. Symp (Int) Combust 1977;16(1):719–29. doi:10.1016/s0082- 0784(77)80366- 4. [22] Snegirev AY. Statistical modeling of thermalradiation transfer in naturalconvection turbulent diffusion flames. 1. Model construction. J Eng Phys Thermophys 2003;76(2):287–98. doi:10.1023/A:1023644932518.

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