Evaluation and optimization-based modification of a model for the mean radiative emission in a turbulent non-reactive flow

International Journal of Heat and Mass Transfer 114 (2017) 664–674

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Evaluation and optimization-based modification of a model for the mean radiative emission in a turbulent non-reactive flow G.C. Fraga, F.R. Centeno, A.P. Petry, F.H.R. França ⇑ Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Rua Sarmento Leite, 425, 90050-170 Porto Alegre, RS, Brazil

a r t i c l e

i n f o

Article history: Received 10 March 2017 Received in revised form 8 June 2017 Accepted 9 June 2017

Keywords: Turbulence-radiation interaction Radiative emission Large-eddy simulation Weighted-sum-of-gray-gases model Optimization

a b s t r a c t An analysis on turbulence-radiation interaction (TRI) effects and an evaluation and modification of a model for the mean radiative emission are presented, in the context of a non-reactive channel flow of a high temperature homogeneous participating gas. Large-eddy simulation is adopted to generate transient data that can be compared to independent calculations initialized with mean temperature and flow fields. Both the gray gas and the weighted-sum-of-gray-gases models are used to solve the radiative heat transfer, as a means of investigating how the consideration or not of the spectral variation of radiative properties influences TRI effects. Results show an overall small impact of TRI on both the mean radiative heat flux and the mean radiative heat source, but relatively greater effects are observed when the spectral dependence of the problem is not neglected. The model for the radiative emission does not have a good accuracy for the cases studied in this paper, probably because it was developed for higher temperature fluctuation intensities than the range predominant in the simulations. A modification on the values of two coefficients associated with the model, performed based on an optimization methodology, leads to a considerable reduction in the error in the predicted mean radiative heat flux compared to solutions fully neglecting turbulent fluctuations. Although the improvement in the estimation of the mean radiative heat source is not so substantial, in most of the domain this quantity is better predicted with the modified model than without it. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The numerical coupling of thermal radiation and turbulence is a challenging task even for simple problems, which involves the modeling of instantaneously variating quantities, integrations in time, space, along angular directions and over the whole spectrum of radiation. Moreover, there is a highly nonlinear dependence between fluctuations of radiation intensity and fluctuations of temperature and species concentration. This nonlinear coupling is referred to as turbulence-radiation interaction (TRI), and its importance has been demonstrated experimentally, theoretically and numerically for a number of applications [1]. In some situations, TRI effects can lead to substantial errors when turbulent fluctuations are neglected, i.e., the time-averaged radiation intensity and heat fluxes may differ significantly from those same quantities computed from mean temperature and species concentration profiles [2–4]. Therefore, turbulence-radiation interaction should be ⇑ Corresponding author. E-mail addresses: [email protected] (G.C. Fraga), frcenteno@mecanica. ufrgs.br (F.R. Centeno), [email protected] (A.P. Petry), frfranca@mecanica. ufrgs.br (F.H.R. França). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.038 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

accounted for, either by fully or partially solving the turbulent fluctuations, as in direct numerical simulation (DNS) or large-eddy simulation (LES), or by adopting a model capable of representing TRI effects based only on mean quantities, which may be obtained from simpler Reynolds-averaged Navier Stokes (RANS) calculations. As the conditions of the simulations become more complex, incorporating, for example, the coupled solution of the flow field and the radiative heat transfer, or including chemical reaction effects, computational costs can quickly become prohibitive and a series of compromises must be made in the solution; thus, the necessity of reliable, yet simplified, models increases. Some assumptions are widely accepted and employed for a number of different problems, as the negligence of scattering in the radiative heat transfer solution, and the optically thin fluctuation approximation (OTFA). Of these two, while neglecting scattering is a more general approximation, not limited to studies on TRI, the OTFA is directly related to modeling of terms introduced by turbulent fluctuations. This approximation, proposed by Kabashnikov and Kmit [5], supposes primarily that the correlation between the local spectral radiation intensity and the spectral absorption coefficient is null if the mean free path for radiation is much larger than the inte-

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model and the time-averaged radiative emission obtained from the LES computations. As far as the authors’ knowledge, this is the first investigation on the performance of the Snegirev approximation employing transient data generated by LES, and also the first study to propose a modification of this approximation based on an optimization methodology. Although other researchers have proposed and utilized modified expressions of the Snegirev approximation, these were based either on theoretical considerations, ad hoc corrections or experimental results [14–16]. 2. Problem statement The problem studied consists of a turbulent non-reactive flow of a homogeneous participating gas inside a square duct. Geometry and boundary conditions are similar to [18], and are shown in Fig. 1. The lateral dimensions of the duct are equal to 0.5 m corresponding to a hydraulic diameter Dh (defined by the ratio between four times the cross-sectional area and the wetted perimeter of the cross-section) equal to 0.5 m; the length of the duct is 5.25 m, or 10:5
Dh . At the front opening of the duct, gas at a spatially uniform and temporally constant temperature of 1200 K is injected with a Reynolds number (based on the hydraulic diameter and the mean bulk velocity) of 5100. Turbulence fluctuations are imposed at the inlet using the Synthetic Eddy Method (SEM) [19], considering a range of turbulence intensities, IT, between 0 and 20%. For the remaining parameters necessary for the implementation of the SEM boundary conditions, a constant number of eddies of 1000 and an eddy length scale of 0.1025 m (equal to the hydraulic diameter multiplied by the Von Kármán constant) are used, following sensibility analyses conducted in [19] and by the present authors. The incoming gas, as well as the gas initially inside the computational domain at the start of the simulations, is homogeneous and its composition is either of carbon dioxide or of water vapor – i.e., no dilution with other species is studied (partial pressure equals to 1 atm in all simulations). The other initial conditions are gas temperature of 600 K uniform throughout the domain, null velocity field and pressure equals to atmospheric pressure. On the opposite surface of the inlet, an open condition to an outside environment kept at atmospheric pressure and composed of the same species as the gas flowing inside the duct is imposed. For the radiative heat transfer calculations, this environment is assumed to behave as a blackbody with a temperature equal to the bulk mean temperature calculated at the cross-sectional plane of grid cells adjacent to the exit surface. The determination of this temperature requires an iterative procedure, updating the outside temperature as equal to the bulk mean temperature computed by the preceding simulation until convergence is reached. A no-slip condition is imposed at the duct walls. A constant and uniform value of 400 K is prescribed for their temperature; for the

open condition at the domain outlet (prescribed pressure and temperature) no-slip condition and prescribed temperature at the domain laterals prescribed velocity and temperature at the domain inlet

0.5

y

z

m

0.5 m

gral turbulence length scale; such assumption greatly simplifies the radiation absorption term and has proven to be valid in a variety of situations [2,6,7]. Although attempts to model other correlations between fluctuating radiative quantities are reported in the literature, none so far has been successful enough outside the cases for which they were designed to be broadly accepted. In [8], a simplified expression for the correlation between the blackbody radiation intensity and the emissivity was proposed (that paper considered only the radiation coming from a surface emitter), by decomposing the instantaneous temperature inside the time-averaged Planck function in mean and fluctuating components and neglecting correlations of odd order. The resulting expression was later adapted for a turbulent participating medium by Snegirev [9], who used experimental data [1] to make further simplifications and model the temperature selfcorrelation and the absorption-temperature correlation (both components of the mean radiative emission term) as functions of the mean temperature, the temperature variance – that can be solved in the RANS framework through an additional transport equation [10] – and the mean absorption coefficient. With this approximation, the author found good agreement with experimental results for buoyant turbulent flames, employing the Monte Carlo method together with both the gray gas and the weighted-sum-of-graygases (WSGG) spectral models to solve the radiative heat transfer part of the problem. After its proposal, a few other studies examined the Snegirev modeling approach for different configurations. In [11], the approximation was tested for turbulent diffusive flames for different turbulence levels through a theoretical analysis based on an assumed-shape probability density function (PDF) of the mixture fraction, also using the gray gas model, resulting in errors of up to 15% for stoichiometric mixtures at higher turbulence intensities when compared to a solution fully including the effects of turbulent fluctuations. Centeno et al. [12], adopted the model for the solution of a non-premixed methane-air flame and reported deviations from experimental data of less than 10% for mean species concentration and mean flame temperature. The same type of flame was investigated in [13], in which the approximation of Snegirev [9] was explicitly formulated in the framework of the WSGG model, obtaining again good agreement with experimental results. Modeling approaches exclusively for the temperature selfcorrelation, generally derived from the approximations of Cox [8] and Snegirev [9], were studied or adopted by various authors for different configurations, with mostly positive results [14–17]. Conversely, remaining correlations arising from turbulence-radiation interaction have not been subject to much study, since their contributions are usually less important to the phenomenon. For example, an approximation for the absorption coefficient selfcorrelation, based on a representation in Taylor series of the functional dependence of the absorption coefficient with temperature, was tested in [11]. The error relative to a solution fully considering turbulent fluctuations was approximately 10% for moderate turbulence levels, but rapidly increased as the turbulence fluctuation got higher. The present study applies large-eddy simulation (LES) in a low Mach number formulation to firstly assess the importance of TRI in a turbulent non-reacting channel flow of a high temperature homogeneous participating gas; both the weighted-sum-of-graygases model and the gray gas model are adopted as a means to investigate how considering or not the spectral variation of radiative properties influences turbulence-radiation interaction effects. Afterwards, the modeling approach of Snegirev [9] for the mean radiative emission, as formulated for the WSGG model by Centeno et al. [13], is evaluated and modified in the framework of the same problem. This modification is proposed following an optimization process based on minimizing the difference between the Snegirev

5m 5.2

x

Fig. 1. Geometry and boundary conditions of the problem.

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radiative transfer calculations, the walls are treated as black surfaces. 3. Mathematical formulation 3.1. Transport equations and turbulence modeling Mass, momentum and energy transport equations are solved for a three-dimensional, transient, non-isothermal, compressible flow of a single non-reacting species in a Cartesian coordinate system. The low Mach number approximation is introduced to facilitate numerical solution [20], allowing to separate the total pressure in a background component, which is solved by the equation of state (for which it is assumed ideal gas behavior), and a pressure perturbation, whose solution is given by an additional Poisson equation [21]. To capture the transient fluctuations of the quantities involved, large-eddy simulation is adopted, through the application of a lowpass box filter of width D ¼ ðdxdydzÞ1=3 (where dx; dy and dz are the dimensions of the grid cells in the directions of the coordinate axes) to the transport equations. A Favre or mass-weighted filter is used to separate cross-correlations between density and other quantities [10,22]. A constant turbulent Prandtl number of 0.5 is assigned, and, to close the LES transport equations, the dynamic Smagorinsky model [23,24] is employed to compute the turbulent viscosity. 3.2. Radiation modeling For the determination of the thermal radiation field, the radiative heat transfer equation (RTE) is solved. For a non-scattering medium, the RTE is given by [7,25]:

dIg ¼ jg Ibg  jg Ig ds

ð1Þ

Z

1





jg Gg  4pIbg dg

ð2Þ

0

where Gg is the spectral incident radiation, given by the integration of the spectral intensity over all solid angles X:

Z

Gg ¼

4p

I g dX

3.2.1. The gray gas (GG) model In the gray gas model, the absorption coefficient of the medium is assumed to be independent of the radiation wavenumber. Therefore, the spectral dependence of the RTE can be dropped, leading to:

dI ¼ jðIb  IÞ ds

ð4Þ

where I and Ib are the total radiative intensity and the total blackbody radiative intensity, respectively, and j is the absorption coefficient of the participating medium. Its value is computed using a correlation proposed in [27]:

j ¼ pa

5 X cGG;i T i

ð5Þ

i¼0

where T is the local temperature; pa is the partial pressure of the gas; and cGG;i are the coefficients of the correlation, whose values vary depending on the gaseous species and are given in Table 1. 3.2.2. The weighted-sum-of-gray-gases (WSGG) model The weighted-sum-of-gray-gases model is a relatively simple methodology capable of accounting for spectral variations in the solution of the radiative heat transfer problem, that has shown good agreement with line-by-line integration (benchmark solution) results for a number of different situations [13,28,29]. In this model, the spectrum of radiation is replaced by N j gray gases with uniform absorption coefficients and by transparent windows. The RTE for each gray gas j may then be written as [30]:

dIj ¼ jj aj Ib  jj Ij ds where Ij and

ð6Þ

jj are the radiative intensity and the absorption coefth

where g is the wavenumber; s is the coordinate along the path of radiation; jg is the spectral absorption coefficient of the medium; and Ig and Ibg are the spectral radiation intensity and the blackbody spectral radiation intensity, respectively. The coupling between the energy equation and the RTE is given by the volumetric radiative heat source term, Sr , defined as the negative of the radiative heat flux vector divergence:

Sr ¼  r  q ¼

ing for the spectral dependence of the radiative properties influences the magnitude of TRI effects in the problem.

ð3Þ

The global solution of the radiative transfer problem requires the spatial and spectral integrations of Eqs. (1) and (2). For the spatial integration, the finite volume method [26] is employed. This method divides the unit sphere surrounding a given point in a finite number of solid angles inside which the intensity is assumed constant relative to direction; the RTE is then individually solved for each angle and the continuous integral over the entire unit sphere is approximated as a summation over the results for all discrete angles. The spectral integration, or the integration over all radiation wavenumbers, is performed using both the gray gas (GG) and the weighted-sum-of-gray-gases (WSGG) models, described in the following sections. The differences between these modeling approaches allow the investigation of how neglecting or account-

ficient of the j gas, respectively, and aj is the temperature coefficient. This parameter corresponds to the fraction of blackbody radiation emitted at the local temperature of the medium in the th

wavenumber interval represented by the j by the polynomial relation:

aj ¼

4 X

gas, and is described

bj;k T k

ð7Þ

k¼0

The polynomial coefficients bj;k in this equation, as well as the pressure absorption coefficient jp;j of each gray gas, given by the ratio between jj and the partial pressure of the participating species (jp;j ¼ jj =pa , with a ¼ H2 O or CO2), are obtained from fitting global radiation data (typically, of total emittance). In the present study, the values of these coefficients were taken from [31] and are given in Tables 2 and 3. The correlations are valid for temperatures between 400 K and 2500 K and pressure path lengths between 0:001 atm m 6 pa S 6 10 atm m, where S is the path length. In the present test cases, the partial pressures of H2O and

Table 1 Polynomial coefficients for the determination of the absorption coefficient in the gray gas model. Valid for 400 K 6 T 6 2500 K and 0:1 atm 6 pa 6 1 atm [27].

cGG;0 (m1 atm1) cGG;1 (m1 atm1 K1) cGG;2 (m1 atm1 K2) cGG;3 (m1 atm1 K3) cGG;4 (m1 atm1 K4) cGG;5 (m1 atm1 K5)

CO2

H2O

6.4750
101 4.2895
101 6.6089
104 4.4190
107 1.3796
1010 1.6484
1015

7.5702
101 1.9716
101 2.1998
104 1.2429
109 3.5385
1011 3.9662
1015

667

G.C. Fraga et al. / International Journal of Heat and Mass Transfer 114 (2017) 664–674 Table 2 WSGG model coefficients for a participating gas composed entirely of carbon dioxide. Valid for 400 K 6 T 6 2500 K and 0:001 atm m 6 pa S 6 10 atm m [31].

jp;j (atm

1

1

m bj;0 bj;1 (K1) bj;2 (K2) bj;3 (K3) bj;4 (K4)

)

j¼1

j¼2

j¼3

j¼4

0.138 0.0999 64.41
105 86.94
108 41.27
1011 67.74
1015

1.895 0.00942 10.36
105 2.277
108 2.134
1011 6.497
1015

13.301 0.14511 30.73
105 37.65
108 18.41
1011 30.16
1015

340.811 0.02915 25.23
105 26.10
108 9.965
1011 13.26
1015

Table 3 WSGG model coefficients for a participating gas composed entirely of water vapor. Valid for 400 K 6 T 6 2500 K and 0:001 atm m 6 pa S 6 10 atm m [31].

jp;j (atm1 m1) bj;1 bj;2 bj;3 bj;4

bj;0 (K1) (K2) (K3) (K4)

j¼1

j¼2

j¼3

j¼4

0.171 0.06617 55.48
105 48.41
108 22.27
1011 40.17
1015

1.551 0.11045 0.576
105 24.00
108 17.01
1011 30.96
1015

5.562 0.04915 70.63
105 70.12
108 26.07
1011 34.94
1015

49.159 0.23675 18.91
105 0.907
108 4.082
1011 9.778
1015

CO2 are both 1 atm, so considering the path length as equal to the hydraulic diameter (Dh ¼ 0:5 m) or the duct length (10:5
Dh ¼ 5:25 m) the limits of the correlations are not violated. For the solution of the radiative heat transfer problem, Eq. (6) is solved N j þ 1 times (j ¼ 0 denotes the transparent windows) and the total radiation intensity is obtained as a summation over the intensities computed with each gray gas and transparent window. PN j To determine a0 , the constraint j¼0 aj ¼ 1, derived from the conservation of energy, is used. 3.3. Turbulence-radiation interaction (TRI) Turbulence-radiation interaction originates from the highly non-linear coupling between turbulence-induced fluctuations of radiation intensity and of temperature and species concentration [2]. Denoting a mean quantity with an overscore, the timeaveraging of the spectrally integrated form of the RTE results in:

dI ¼ jI þ jIb ds

ð8Þ

Decomposition of both terms on the right-hand side of Eq. (8) into mean and fluctuating components followed by further timeaveraging leads to a number of correlations involving the absorption coefficient, the temperature and the local radiation intensity that requires modeling. In the framework of the WSGG model and considering a medium with constant chemical composition, the correlation between absorption coefficient and local radiation intensity – i.e., the radiative absorption – may be replaced by a simple multiplication of both mean quantities (for each gray gas j), jj Ij . Furthermore, for the set of assumptions adopted in this work, the mean absorption coefficient is equal to its instantaneous constant value, jj ¼ jj . Thus, the treatment of the time-averaged RTE reduces to modeling the mean radiative emission. An approximation for this term proposed by Snegirev [9], based on an expression developed in [8], is investigated in the present study. Recognizing that jIb is proportional to jT 4 , this methodology consists of decomposing both temperature and absorption coefficient into mean and fluctuating components, time-averaging the resulting expression and expanding it in Taylor series. After neglecting terms of order higher than two, the mean radiative emission is approximated as [9]:

0

jIb  jIb ðTÞ@1 þ 6C TRI;1

1  2 T 0 @ j A þ 4C TRI;2 T2 jT @T T

T0

2

ð9Þ

where Ib ðTÞ is the blackbody radiation intensity evaluated at the local mean temperature of the medium and C TRI;1 and C TRI;2 are constants. This expression, which will be referred to here as the Snegirev model or the Snegirev approximation, reduces the mean radiative emission to a summation between its value, evaluated with mean quantities, and two other terms: the temperature selfcorrelation and the absorption coefficient-temperature correlation (second and third terms inside the parenthesis, respectively). Snegirev [9], initially estimated C TRI;1 as 1.25 by fitting the dependence between T 4 =T 4 and T 2 =T 2 from data reported in [1], but, after tests with turbulent buoyant propane flames, readjusted it to 2.5. Conversely, recognizing that the temperature selfcorrelation is expected to dominate the TRI effects over the absorption coefficient-temperature correlation, the value of C TRI;2 was set to 1. Other authors adopted different values for constants C TRI;1 and C TRI;2 in the implementation of the Snegirev model; however, little theoretical or experimental justification has been given for these modifications [14–16]. An adaptation of the Snegirev approximation in the framework of the WSGG model was proposed by Centeno et al. [13], by introducing a correction factor aj associated with the mean radiative th

emission for the j

gray gas:

jj aj Ib  aj jj aj ðTÞIb ðTÞ

ð10Þ

where the expression for aj is analogous to the term inside the parenthesis in Eq. (9):

aj ¼ 1 þ 6C TRI;1

 daj  þ 4C TRI;2  T2 aj ðTÞT dT T

T0

2

T0

2

ð11Þ

In the preceding equations, aj ðTÞ denotes the temperature coefficient of gas j evaluated at the local mean temperature, and, as already stated, jj ¼ jj for the cases simulated in this study. Since it is derived from the Snegirev approximation, the factor aj in Eq. th

(11) is denominated the Snegirev correction factor (for the j gas) in the present paper.

G.C. Fraga et al. / International Journal of Heat and Mass Transfer 114 (2017) 664–674

The set of equations described in the previous section are numerically solved using the open-source Fortran-based computer fluid dynamics code Fire Dynamics Simulator (FDS). The core algorithm of solution is an explicit predictor-corrector scheme, with second order accuracy both in time and in space; the computational domain is spatially discretized adopting staggered, structured, rectilinear meshes [32]. Since it is not possible to achieve grid independence in LES computations [33,34], the timeaveraged Measure of Turbulence Resolution, MTR, is used as an indicator of mesh quality [21]. This is an adaptation for the FDS code of the M criterion of Pope [35], that estimates the fraction of the flow’s kinetic energy that remains unresolved by the numerical grid. Using an uniform spatial mesh of 121 cells in the flow’s developing direction and 36 cells in the two transversal directions, MTR  0:1 is observed for most of the simulations (with a few exceptions that do not surpass the recommended value of 0.2 [21,35]), indicating that, on average, at least 90% of the flow kinetic energy is directly resolved by the numerical grid. Another verification of quality of the spatial grid is performed by comparing the mean radiative and total heat fluxes at the domain walls computed with increasingly refined meshes. For the same grid that resulted in the MTR values described earlier, differences of about 0.5% relative to a mesh with double the number of volumes are obtained. Since the analysis conducted in this paper is based on deviations between results of transient and one timestep simulations using the same mesh, and considering that 0.5% is well under what would be indicative of significant TRI effects, this magnitude of error is considered to be tolerable. Similar comparisons of mean wall radiative and total heat fluxes are performed for successively refined levels of temporal and angular discretizations. Using an uniform time-step dt of 1
103 s results in a maximum difference of 0.02% compared to a simulation with dt ¼ 5
104 s, while an angular discretization consisting of 100 solid angles leads to deviations below 0.1% relative to the discretization with double the number of angular divisions. Furthermore, the Courant-Friedrichs-Lewy criterion, CFL, in a modified formulation that accounts for constraints of mass density and fluid volume realizability [21] is also verified; for dt ¼ 1
103 s, its value is kept approximately at 0.1 for all simulations, well under the recommendation for numerical stability of CFL 6 1 [21].

5. Results and discussion The present study is divided in three parts, all centered around the solution of the problem described in Chapter 2. In the first one, the magnitude of TRI effects is analyzed for both the gray gas and the weighted-sum-of-gray-gases models. In the second part, the approach proposed by Snegirev [9] for modeling the mean radiative emission is implemented for the WSGG model, and the effectiveness of such approximation is evaluated. Finally, an optimization process is proposed for defining new values of constants C TRI;1 and C TRI;2 of the Snegirev approximation, and this modified model is subsequently tested. All evaluations presented next are based on comparisons between mean quantities related to the radiative transfer problem (namely, the wall heat fluxes and the volumetric radiative heat source) computed from solutions explicitly resolving turbulent fluctuations, and the mean results obtained while neglecting these fluctuations or while approximating their effect through the Snegirev model – with or without optimized C TRI;1 and C TRI;2 values. For this purpose, a relative difference w/ is defined, in which / can be any mean quantity:

 TRI  /  /Approx   w/ ¼   /Approx

ð12Þ

In this equation, /TRI is the value of / computed including the effects of turbulent fluctuations. These results are generated by instantaneously solving the problem using LES for a total time of 20s and averaging the quantities of interest over the last 15 s of simulation; analyses performed in longer lasting cases show that this time interval is sufficient to provide averages that are independent of the initial development of the flow and, thus, are representative of the transient problem. In the following sections, this approach is denominated the ‘‘with TRI” solution. It should be noted that, although this approach is referred to as the solution explicitly resolving turbulent fluctuations, TRI on the subgrid level is not accounted for. However, previous studies [36–39] have shown that, for reasonably good grid resolution and for problems with low fluctuation intensities (such as the cases analyzed in this paper), the effect of subgrid scale fluctuations may be neglected. On the other hand, /Approx in Eq. (12) is the mean wall heat flux or mean radiative heat source calculated completely disregarding TRI, /nTRI , or implementing the Snegirev approximation (either in its original or modified formulation), /Cor . Results fully neglecting turbulence-radiation interaction are obtained from an independent one time-step calculation initialized with mean temperature and flow fields determined from data of the transient ‘‘with TRI” solution; this methodology to produce mean fields where turbulent fluctuations are not considered has also been adopted by other authors [18,40,41], and is named here the ‘‘without TRI” solution. Solutions adopting the Snegirev approximation are generated in a similar manner, but modifying the radiative emission term to include the correction factor aj , as given by Eq. (10).

5.1. Assessment of turbulence-radiation interaction Fig. 2a and b shows the relative difference wqr between the mean wall radiative heat fluxes obtained by ‘‘with TRI” and ‘‘without TRI” solutions along the longitudinal direction of the domain for cases where the participating medium is composed of water vapor and of carbon dioxide, respectively. The turbulence intensity of the incoming flow is 10% and three spectral modeling approaches are studied: the WSGG model; the GG model with a temperature-dependent absorption coefficient as given by Eq. (5), denominated GGj(T); and a variation of this last model, named GGj, for which j has a constant value, determined by evaluating Eq. (5) at the temperature of the gas at the domain’s entrance (leading to j ¼ 3:524 m1 for H2O and j ¼ 16:855 m1 for CO2).

2.0 1.0

ψ qr (%)

4. Numerical methods and grid quality analysis

ψ qr (%)

668

0.5 0.0

1.5 1.0 0.5 0.0

0

2

4

0

2

x (m)

x (m)

(a)

(b)

WSGG

GGκ(T )

4

GGκ

Fig. 2. Relative difference between mean quantities obtained by the ‘‘with TRI” and ‘‘without TRI” solutions along the domain length, with IT ¼ 10%. From left to right: H2O and CO2.

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G.C. Fraga et al. / International Journal of Heat and Mass Transfer 114 (2017) 664–674

Table 4 Relative difference, in %, between solutions considering and neglecting TRI for mean radiative and convective heat fluxes at the wall (averaged value over all surfaces) and for the mean radiative heat source (averaged value over the entire domain). Model

a

IT (%)

CO2

H2O

wq r

wqc

wSr

wqr

wq c

wS r

WSGG

0 5 10 15 20

0.66 0.96 1.20 1.41 1.55

0.75 1.04 1.12 1.24 1.25

6.51 27.62 17.58 5.65 5.65

–a 0.67 0.80 0.89 0.97

–a 0.92 1.01 1.06 1.11

–a 6.29 6.54 15.06 25.19

GGj(T)

0 5 10 15 20

0.54 0.68 0.78 0.90 1.02

0.32 0.44 0.49 0.54 0.60

1.68 3.65 2.59 5.67 10.41

0.15 0.17 0.18 0.19 0.20

0.59 0.61 0.63 0.66 0.69

2.29 0.92 0.80 0.95 0.99

GGj

0 5 10 15 20

0.46 0.51 0.59 0.70 0.78

0.46 0.52 0.57 0.63 0.67

1.51 2.05 1.78 2.27 3.17

0.47 0.50 0.57 0.62 0.67

0.91 0.92 0.96 1.01 1.01

14.83 10.18 11.61 14.83 21.92

H2O, IT ¼ 0% case was not simulated due to convergence issues.

A more general view is provided by Table 4, which presents the domain-averaged values of wqr for all studied cases. The domain-averaged relative difference between the mean wall radiative heat flux considering and neglecting TRI effects does not exceed 2% for any spectral model, gas composition or turbulence intensity, while the maximum local wqr over all cases is 2.3% Besides being small values, comparable to the deviations observed in the mesh quality analysis, wqr is of the same magnitude as the corresponding difference observed for the mean wall convective heat flux, wqc , as shown in Table 4 (the overall maximum local wqc is equal to 1.9%). Since turbulence-radiation interaction is promoted by the nonlinearities in the coupling between equations that govern the problem, and considering the high nonlinearity of radiative heat transfer [7], it would be expected greater deviations between ‘‘with TRI” and ‘‘without TRI” solutions to be found for the mean radiative heat flux than for the mean convective heat flux. Therefore, there is evidence that the contribution of turbulence-radiation interaction is not significant for the configuration studied here, which is in agreement with findings of other works on non-reactive flows [18,42–44]. Greater differences between ‘‘with TRI” and ‘‘without TRI” results are observed for the mean radiative heat source. Table 4 shows that domain-averaged values of wSr may be up to 27%, above what is generally reported for non-reactive problems and comparable to deviations obtained by studies on reacting flows [2]. However, it should be noted that close-to-zero values of both Sr TRI and Sr nTRI in some cells of the computational domain tend to result in elevated local wSr despite the absolute difference between those quantities being negligible. Although this occurs in a very small number of points, specially near the walls, it has an effect of increasing the domain-averaged difference presented in Table 4. When only the core region of the flow is considered (i.e., data corresponding to cells inside the thermal boundary layer region is neglected), the domain-averaged wSr does not exceed 0.5% for any of the simulated cases. Another approach to the analysis is to define a normalized difference for the mean radiative heat source,    TRI cSr ¼ ðSr TRI  Sr nTRI Þ=Sr TRI avg , where Sr avg is the domain-average of the mean radiative heat source obtained from the ‘‘with TRI” solution; following this methodology, the maximum local value of cSr throughout all simulated cases is 41.4%.

Table 4 shows that, in general, TRI effects have a stronger impact for the WSGG model than for the GG model, indicating that accounting for or neglecting spectral variations of the radiative properties in the solution of the radiative heat transfer influences the turbulence-radiation interaction phenomenon. This may be explained by the different response of distinct individual spectral bands of the participating gas to fluctuations of temperature, depending on how the Planck function varies with this quantity at the center of the band [45]. As a consequence, turbulent fluctuations may strongly affect the mean spectral radiation intensity in some parts of the spectrum, while in other regions this influence might not be significant. Since it can represent the spectral dependence of the radiation intensity field, the weighted-sum-of-graygases model is thus capable of capturing this behavior in the integration of Ig over all wavelengths, whereas the gray gas model is not. In the studied cases, the fact that the WSGG model generally leads to higher values of wqr and wSr implicates that, after summing all positive and negative contributions of temperature fluctuations for each individual band, there is an overall increase in TRI effects compared to calculations completely neglecting these contributions. It should be noted, however, that the decrease in the importance of turbulence-radiation interaction caused by using the GG model is particular to the specific cases analyzed in this paper and must not be generalized; in fact, studies of turbulenceradiation interaction in other problems have reported the opposite result (see, for example, [4]). It is interesting to observe that, between the two variants of the gray model, the GGj(T) model presents greater wqr and wSr for simulations with carbon dioxide, while, with water vapor, the deviations are higher for the GGj model. An explanation for that lies on the shape of the absorption coefficient-blackbody radiation intensity correlation for these participating species. In the GGj model, this correlation is zero, since the absorption coefficient is assumed constant; however, as Fig. 3 illustrates, for H2O the correlation is negative over the entire range of gas temperatures of the problem studied here (i.e., an increase in temperature results in a decrease of j and in an increase of Ib ), resulting in a negative contribution to the mean RTE when solved neglecting turbulenceradiation interaction effects, which leads to a reduction of the difference between ‘‘with TRI” and ‘‘without TRI” solutions. Conversely, for CO2, the j-Ib correlation is positive for the temperature range predominant in the computational domain

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the WSGG model; i.e., .j ¼ jj aj Ib =jj aj ðTÞIb ðTÞ. An expression for .j is derived by decomposing the temperature in the numerator of its definition into mean and fluctuating components and retaining all higher order terms, which ultimately leads to:

κ (m−1 )

30

20

20

10

10

Ib (kW/m2 )

40

30

P4

.j ¼ P4 k¼0

k¼0 bj;k T

0 400

600

800

ð13Þ

kþ4

where bj;k are the polynomial coefficients of the WSGG model, given in Tables 2 and 3, and C k are functions of the time-averaged temper-

1000 1200

T (K) CO2

bj;k C k

m

ature and of its central moments T 0 :

H2 O

C k ¼ T ð4þkÞ þ

Fig. 3. Absorption coefficient of CO2 and H2O in the GGj(T) model, computed following Eq. (5), and blackbody radiation intensity, in the range of temperatures of the simulations.

4þk X

ð4 þ kÞ! m T ð4þkmÞ T 0 ð4 þ k  mÞ!m! m¼2

ð14Þ

(400–650 K); thus, there is a positive contribution to the mean RTE and an increase in wqr and wSr , compared to the GGj model.

Although computed individually for each gray gas, it is possible to determine global correction factors a and . from the values of aj and .j . This is more convenient for the comparisons that follow and may be achieved through the expression

5.2. Evaluation of the Snegirev approximation

. ¼ Pj¼1 4

In this section, the mean radiative emission approximation proposed by Snegirev [9], as formulated for the WSGG model by Centeno et al. [13], is examined. The GG model, because it is a less complete spectral model and for which the magnitude of TRI effects has shown to be relatively smaller, is not considered in this or in the subsequent analyses. To assess the accuracy of the Snegirev approximation, the correction factor aj given in Eq. (11) is initially compared to an analytical factor .j , defined as to assure the mean radiative emission computed neglecting TRI effects is exactly corrected to the value obtained considering turbulent fluctuations, for each gray gas in

for the global analytical correction factor; the equation for the global Snegirev correction factor, a, is analogous. The upper limits of the sums in Eqs. (13) and (15), as well as the number of coefficients C k , are defined by the number of gray gases used to represent the participating medium in the WSGG model. Using data obtained from the transient LES computations, it is possible to directly calculate all terms in Eqs. (11)–(15) for every grid cell, thus determining the fields of a and . in the entire computational domain. Afterwards, these correction factors may be compared through a relative difference wCor ¼ j1  .=aj. Following this procedure, Table 5 shows that domain-averages of wCor are of

P4

.j jj aj ðTÞ j¼1 jj aj ðTÞ

ð15Þ

Table 5 Performance of the Snegirev approximation. wCor (%)a

wqr (%)a

wSr (%)a

Wqr a

WSr a

% of domain with WSr < 1

CO2, IT ¼ 0% CO2, IT ¼ 5% CO2, IT ¼ 10% CO2, IT ¼ 15% CO2, IT ¼ 20% H2O, IT ¼ 5% H2O, IT ¼ 10% H2O, IT ¼ 15% H2O, IT ¼ 20%

1.28 1.58 1.88 2.13 2.34 1.21 1.39 1.52 1.63

0.91 1.32 1.64 1.93 2.12 1.18 1.42 1.59 1.75

10.64 6.32 5.03 5.16 5.44 13.03 16.18 29.29 55.00

1.45 1.48 1.48 1.48 1.48 1.93 1.94 1.94 1.95

2.18 2.34 2.30 2.41 6.25 2.78 3.10 3.89 4.14

7.87 7.62 8.60 9.68 10.16 3.56 4.52 6.23 6.85

CO2 ,

IT = 20 %

0.25

0.25

0.25

0.25

10 8 6 4

ψ Cor (%)

H2 O,

IT = 20 %

y (m)

CO2 ,

IT = 10 %

y (m)

H2 O,

IT = 10 %

0.25

y (m)

CO2 ,

IT = 0 %

y (m)

Averaged quantities over the entire computational domain.

y (m)

a

Case

2

1

2

3

4

5

0

x (m) Fig. 4. Fields of the relative difference between the Snegirev and the analytical global correction factors, a and ., along the mean longitudinal section of the domain for selected cases.

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approximation is 1.4–2 times greater than the corresponding error computed while completely neglecting TRI effects; for the mean radiative heat source, the results are even worse, with a domainaveraged WSr higher than 6 for one of the cases simulated. A similar trend is observed for the maximum local values of wqr and cSr , that are equal to 3.0% and 71.5% when implementing the Snegirev approximation, which is 32% and 73% higher than the maximum local deviations found without the correction, respectively. Furthermore, despite the fact that in some volumes WSr is less than one (i.e., there is a local improvement in the predicted mean radiative heat source by implementing the Snegirev approximation), this condition is observed in only a small percentage of the domain – at best, according to Table 5, in just about 10% of the domain, and, at worst, in 3.5%.

about 2% for every case simulated. It is also noted a strong relation between wCor and the intensity of temperature fluctuations, qffiffiffiffiffiffiffi 2

T 0 =T, which can be verified by comparing Figs. 4 and 5, where fields of both quantities are shown along the mean longitudinal section of the computational domain for selected cases. At higher intensities of temperature fluctuations, the difference between a and . is accentuated, indicating that the Snegirev approximation is less accurate when the temperature fluctuations are increased. This is a curious and seemly contradictory result because, since the Snegirev approximation was developed in the framework of reactive flows, which generally exhibit much greater values of qffiffiffiffiffiffiffi 2

T 0 =T than non-reactive problems, a better agreement between a and . was expected to be achieved as temperature fluctuations intensified. The performance of the Snegirev approximation when implemented to the solution of the radiative heat transfer problem is evaluated next. For this purpose, the fields of aj computed earlier are applied to model the radiative emission, following Eq. (10). Table 5 gives the domain-averaged values of the ratio between the difference, relative to the ‘‘with TRI” solution, of mean quantities obtained by the simulation implementing the Snegirev model and the corresponding difference of mean results computed while fully neglect  ing TRI effects, measured as W/ ¼ ð/Cor  /TRI Þ=ð/nTRI  /TRI Þ.

5.3. Optimization of the constants of the Snegirev approximation Considering that, for the problem studied, the original formulation of the Snegirev model does not provide an overall improvement of the predicted mean quantities compared to results fully neglecting turbulent fluctuations, new values for the constants C TRI;1 and C TRI;2 are proposed (the general form of Eq. (9) is kept unaltered). These values are obtained through an optimization methodology based on finding the set (C TRI;1 ; C TRI;2 ) that minimizes the difference between correction factors a and .. For this purpose, a quadratic deviation f between these factors, summed over all volumes inside the domain and over every case that implement the WSGG model, is defined as

According to this definition, values of W lower than unity indicate an improvement of the Snegirev approximation over the ‘‘without TRI” solution, whereas W > 1 signifies that the implementation of the model actually worsen the predicted mean quantities. Table 5 evidences that, although differences between a and . are small, the Snegirev approximation is not capable to reduce the overall discrepancies between ‘‘without TRI” and ‘‘with TRI” results. In fact, in average over all volumes, the absolute error of the mean wall radiative heat flux obtained with the Snegirev

CO2 ,

IT = 20 %

ð16Þ

where the dependence of f and a with C TRI;1 and C TRI;2 is explicitly stated.

0.25

0.25

0.25

0.10 0.08

 T  2 /T

H2 O,

IT = 20 %

y (m)

IT = 10 %

y (m)

CO2 ,

y (m)

IT = 10 %

0.25

y (m)

H2 O,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qX X ½aðC TRI;1 ; C TRI;2 Þ  .2 Cases Volumes

0.12 0.25

y (m)

CO2 ,

IT = 0 %

f ðC TRI;1 ; C TRI;2 Þ ¼

0.06 0.04 0.02 1

2

x (m)

3

4

5

0.00

Fig. 5. Fields of the temperature fluctuation intensity along the mean longitudinal section of the domain for selected cases.

Table 6 Performance of the Snegirev approximation with modified constants C TRI;1 and C TRI;2 (C TRI;1 ¼ 1; C TRI;2 ¼ 0:9).

a

Case

wCor (%)a

wqr (%)a

wSr (%)a

Wqr a

WSr a

% of domain with WSr < 1

CO2, IT ¼ 0% CO2, IT ¼ 5% CO2, IT ¼ 10% CO2, IT ¼ 15% CO2, IT ¼ 20% H2O, IT ¼ 5% H2O, IT ¼ 10% H2O, IT ¼ 15% H2O, IT ¼ 20%

0.09 0.12 0.15 0.18 0.20 0.03 0.03 0.03 0.04

0.01 0.02 0.02 0.03 0.03 0.04 0.05 0.06 0.06

0.71 1.17 0.30 0.31 0.45 0.28 0.33 0.61 3.36

0.03 0.03 0.03 0.03 0.03 0.07 0.07 0.07 0.07

0.29 0.31 0.35 0.38 1.51 0.09 0.10 0.13 0.13

97.88 97.68 97.18 97.00 96.56 99.81 99.74 99.56 99.41

Averaged quantity over the entire computational domain.

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From the earlier LES computations, the field of . is known throughout the domain for all cases, as well as the mean temperature and central moments required to calculate a for a given set of C TRI;1 and C TRI;2 . Therefore, it is possible to map values of the quadratic deviation for different pairs of constants. Eq. (16) is evaluated for C TRI;1 and C TRI;2 ranging from 0 to 10 (f rapidly increases as either constant approaches 0 or 10; hence, these limits are deemed adequate), with increments of 0.1 between each new calculation, from which it follows that the set that minimizes the quadratic deviation is C TRI;1 ¼ 1 and C TRI;2 ¼ 0:9. Table 6 indicates that, employing these new set of constants, the relative difference between the Snegirev and the analytical correction factors is reduced in almost two orders of magnitude compared to wCor computed with the original C TRI;1 and C TRI;2 . The performance of the Snegirev approximation with updated constants for the solution of the radiative transfer problem is also significantly improved. For every case considered, Wqr (calculated using the same definition presented earlier, but with updated values of C TRI;1 and C TRI;2 ) is kept below one throughout the domain. Table 6 shows that, when averaged over the computational domain, the error in the estimation of this quantity is at least 93% smaller than the error fully neglecting turbulent fluctuations. For the mean radiative heat source, in all but one of the simulated cases there is an overall good performance of the Snegirev model with the new set of constants, i.e., a domain-averaged WSr smaller than one; locally, for each case, more than 95% of the domain has the condition WSr < 1. Compared to the ‘‘with TRI” solution, the domain-averaged relative difference of the predicted mean wall heat flux is lower than 0.1% for all simulations, while for the mean

1.10

the relation of the temperature self-correlation, T 4 =T 4 , with the qffiffiffiffiffiffiffi 2

temperature fluctuation intensity, T 0 =T. For the investigated problem, both quantities may be calculated for each numerical volume from the transient LES computations; the relation between them is given (as gray points) in Fig. 6. This figure condenses data from all nine cases that adopt the WSGG model and shows that the temperature fluctuation intensities in this work are significantly lower than the average intensity of the data based on which the Snegirev approximation was developed [1,9]. In fact, the entire qffiffiffiffiffiffiffi 2

range of T 0 =T in the present study corresponds to only the initial spectrum of temperature fluctuation intensities reported by Burns qffiffiffiffiffiffiffi [1] (0 <

2

T 0 =T < 0:6). Therefore, although the curve-fitting 2

T 4 =T 4 ¼ 1 þ 7:5T 0 =T 2 originally proposed by Snegirev [9], that led to the definition of the value of C TRI;1 ¼ 7:5=6 ¼ 1:25, cf. Eq. (9), presented an overall good agreement with the results in [1], this fitting has a poor performance for the smaller range of temperature fluctuation intensities observed in the cases considered here, as illustrated by Fig. 6. When using the modified value of C TRI;1 ¼ 2:5, introduced as an ad hoc adjustment also by Snegirev [9], the concordance with the data is even worse. On the other 2

CTRI,1 = 1.0 CTRI,1 = 2.5 CTRI,1 = 1.25

4

T 4 /T

radiative heat source this difference is approximately 1% for all cases but one. As for the maximum differences, the local wqr and cSr have maximums (over the nine cases of Table 6) of 0.09% and 6.1%, respectively; this is a reduction of approximately 90% compared to the maximum local deviations obtained without any correction. The elevated discrepancy between the value of C TRI;1 proposed by Snegirev [9] and the one obtained from the optimization methodology may be explained by a difference in the shape of

1.05

1.00 0.00

0.05

 T  2 /T

0.10

Fig. 6. Relation between the temperature self-correlation and the temperature fluctuation intensity. Gray points represent data obtained from the simulations, 2 while the lines are curve-fits in the form T 4 =T 4 ¼ 1 þ 6C TRI;1 T 0 =T 2 , for different values of C TRI;1 .

hand, Fig. 6 shows that the curve-fit T 4 =T 4 ¼ 1 þ 6T 0 =T 2 , which assumes C TRI;1 ¼ 1, provides a good representation of the dependence between the temperature self-correlation and the temperature fluctuation intensity for the problem. Following the same methodology adopted to plot Fig. 6, Fig. 7 illustrates the relation of the last two terms of the Snegirev approximation, without the associated constants, with the ratio between the mean radiative emission and the radiative emission evaluated using the mean temperature field (which, by definition, is equal to .). As stated in Section 3.3, the second and third terms of Eq. (9) are responsible to model the temperature self-correlation and the absorption coefficient-temperature correlation, respectively. To represent the latter correlation in the framework of the WSGG model, the last term of Eq. (11) is normalized as to account for the contribution of all four gray gases, similarly to Eq. (15). The dis-

1.10

κIb /κIb (T )

κIb /κIb (T )

1.10

1.05

1.00

1.05

1.00 0.00

0.05

−1 −0.5

0.10

 T  2 /T

4

j=1

4

(a)

 2 daj dT

κj TT

j=1

CO2

H2O

0

0.5

  

κj aj (T )

×10−2

T

(b)

Fig. 7. Relation between second (a) and third (b) terms of the Snegirev approximation (without the associated constants) with the global analytical correction factor.

G.C. Fraga et al. / International Journal of Heat and Mass Transfer 114 (2017) 664–674

tribution shown in Fig. 7a is very similar to the relation between qffiffiffiffiffiffiffi 02

4

T =T and T =T 4 given in Fig. 6, indicating that the temperature self-correlation dominates over the absorption coefficienttemperature correlation as the term with the greatest contribution to the mean radiative emission. Furthermore, comparisons between Fig. 7a and b evidence that the second term of Eq. (11) is one order of magnitude higher than the third term of the equation. These findings are in agreement with the observations of [1,9] and of other studies [2]. Additionally, Fig. 7b evidences the difficulty of proposing a simple curve-fitting for the absorption coefficient-temperature correlation, as it was proposed for the temperature self-correlation, due to the relatively larger spread of the data. Fig. 7 also presents the different behavior of simulations considering carbon dioxide or water vapor as the participating medium, given as gray and blue points in the distribution, respectively. Turbulence-radiation interaction effects on the radiative emission are more significant for CO2 than for H2O, probably because both the temperature self-correlation and the absorption-temperature correlation are positive for the former gas, as indicated by the signs of the second and third terms of Eq. (11) in Fig. 7, while, for the latter, this last correlation is negative. It should also be noted that the signs of the absorption coefficient-temperature correlation for CO2 and H2O indicated by Fig. 7b agree with the behavior presented in Fig. 3.

6. Conclusion This study presented an analysis of turbulence-radiation interaction in a non-reacting channel flow of a high temperature homogeneous gas, using LES in a low Mach number formulation to generate transient fields based on which TRI effects could be evaluated. The finite volume method was adopted for the angular integration of the RTE, while both the weighted-sum-of-gray-gases and the gray gas models were implemented to resolve the spectral integration. By comparing the transient LES-generated data and results of single-time step simulations initialized with timeaveraged flow and temperature fields, turbulence-radiation interaction did not show a significant impact on the predicted mean wall heat fluxes, in agreement with previous studies on nonreacting problems. Deviations between the mean volumetric radiative heat source obtained while considering and neglecting TRI effects were greater, but these results were considerably influenced by elevated local relative differences found in a small number of cells for which Sr was close to zero; when this influence was removed, the domain-averaged difference dropped to below 0.5%. For simulations adopting the WSGG model, TRI effects were generally greater than for computations with the GG model, indicating that the consideration or not of the spectral variation of radiative properties has an impact on the turbulence-radiation interaction, probably due to the different behavior of individual spectral bands when exposed to temperature fluctuations. The modeling of the mean radiative emission proposed by Snegirev [9] was evaluated in the framework of the WSGG model. Despite small differences being found between the correction factor introduced by the Snegirev approximation and an analytical factor computed as to account for all temperature fluctuations, the model did not provide an improvement in the estimation of mean wall radiative heat fluxes and mean radiative heat sources. This led to the proposal of an adjustment to the approximation, based on an optimization process focused on two parameters of the original expression. After such modifications, a considerably better performance was observed, with domain-averaged differences relative to the solution where the turbulent fluctuations are explicitly resolved of less than 0.1% for the mean wall radiative

673

heat flux, and of about 1% for the mean radiative heat source for almost all studied cases. Although this paper focused on non-reactive cases, the methodology presented here for the evaluation of the Snegirev model and the optimization of its constants could be adopted for reactive problems, for which it is expected a stronger contribution of TRI effects. Fluctuations in the medium composition, neglected in the present work, could be easily accounted for using the WSGG model, since the absorption coefficient for each gray gas in the model has a linear dependence with the concentration of the participating species. However, it should be noted that this methodology may not be directly transferable to low resolution LES of turbulent flames, in which the flamelet temperature remains unresolved and pronounced subgrid temperature gradients are expected.

Conflict of interest The authors declared that there is no conflict of interest. Acknowledgements Author GCF thanks CAPES for a master’s degree scholarship and CNPq for a doctorate scholarship. Author FHRF thanks CNPq for research grants 309961/2013-0 and 476490/2013-8. References [1] S.P. Burns, Turbulence Radiation Interaction Modeling in Hydrocarbon Pool Fire Simulations, Tech. Rep., Sandia National Labs., Albuquerque, NM (US); Sandia National Labs., Livermore, CA (US), 1999. [2] P.J. Coelho, Numerical simulation of the interaction between turbulence and radiation in reactive flows, Prog. Energy Combust. Sci. 33 (4) (2007) 311–383. [3] G. Li, M. Modest, Numerical simulation of turbulence-radiation interactions in turbulent reacting flows, in: B. Sundén, M. Faghri (Eds.), Modelling and Simulation of Turbulent Heat Transfer, WIT Press, 2005. [4] M.F. Modest, Multiscale modeling of turbulence, radiation, and combustion interactions in turbulent flames, Int. J. Multiscale Comput. Eng. 3 (1) (2005) 85–105. [5] V. Kabashnikov, G. Kmit, Influence of turbulent fluctuations on thermal radiation, J. Appl. Spectrosc. 31 (2) (1979) 963–967. [6] J. Hartick, M. Tacke, G. Früchtel, E. Hassel, J. Janicka, Interaction of turbulence and radiation in confined diffusion flames, Symposium (International) on Combustion, vol. 26, Elsevier, 1996, pp. 75–82. [7] M.F. Modest, Radiative Heat Transfer, Academic Press, 2013. [8] G. Cox, On radiant heat transfer from turbulent flames, Combust. Sci. Technol. 17 (1–2) (1977) 75–78. [9] A.Y. Snegirev, Statistical modeling of thermal radiation transfer in buoyant turbulent diffusion flames, Combust. Flame 136 (1) (2004) 51–71. [10] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, RT Edwards, Inc., 2005. [11] P.J. Coelho, A theoretical analysis of the influence of turbulence on radiative emission in turbulent diffusion flames of methane, Combust. Flame 160 (3) (2013) 610–617. [12] F.R. Centeno, C.V. da Silva, F.H. França, The influence of gas radiation on the thermal behavior of a 2D axisymmetric turbulent non-premixed methane–air flame, Energy Convers. Manage. 79 (2014) 405–414. [13] F.R. Centeno, R. Brittes, F.H. França, C.V. da Silva, Application of the WSGG model for the calculation of gas–soot radiation in a turbulent non-premixed methane–air flame inside a cylindrical combustion chamber, Int. J. Heat Mass Transfer 93 (2016) 742–753. [14] G. Krishnamoorthy, A comparison of gray and non-gray modeling approaches to radiative transfer in pool fire simulations, J. Hazard. Mater. 182 (1) (2010) 570–580. [15] G. Krishnamoorthy, A comparison of angular discretization strategies for modeling radiative transfer in pool fire simulations, Heat Transfer Eng. 33 (12) (2012) 1040–1051. [16] R. Yadav, A. Kushari, V. Eswaran, A.K. Verma, A numerical investigation of the Eulerian pdf transport approach for modeling of turbulent non-premixed pilot stabilized flames, Combust. Flame 160 (3) (2013) 618–634. [17] F.R. Centeno, G. Velasco, F.H.R. França, C.V. da Silva, Evaluation of turbulenceradiation interactions model correlations on rans simulations of a 2d axisymmetric turbulent non-premixed methane-air flame, in: International Congress of Mechanical Engineering (COBEM), S ao Paulo, 2013. [18] G.E. Velasco, A.P. Petry, F.H.R. França, Study of turbulence-radiation interaction in participating media using large eddy simulation method, in: 15th Brazilian Congress of Thermal Sciences and Engineering, Belém, PA, Brazil, 2014.

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