Abstract
Combustion, one of the main forms of energy production nowadays, is a complex phenomenon in which turbulent flow, chemical reactions, different phases and heat transfer phenomena can interact. A better understanding of these interactions is essential to improve existing combustion systems and develop new ones. The aim of this work was to study the interaction of turbulent combustion with thermal radiation with the aid of threedimensional numerical simulation. Using the CORBA computational tool, a code for Large Eddy Simulation (LES) of combustion was coupled with a radiative heat transfer code. This technique allows information to be exchanged between the two codes without major changes in their structure, so that advantage can be taken of the different characteristic time of each phenomenon in a highperformance parallelcomputing environment. This study is the continuation of an earlier twodimensional study. The results show a change in the temperature and species fields as well as in the flame dynamics when thermal radiation is taken into account.
Introduction
Air pollution is one of the main problems facing society nowadays. It is responsible not only for local effects, such as the damage it causes to people’s health in large cities, but also for global changes, including global warming (caused by greenhouse gases). Combustion processes are the main contributors to this problem, as around 90 % of the world’s energy is produced by these processes, which are responsible for emissions of CO, NO_{ x }, soots, polycyclic aromatic hydrocarbon (PAH) and CO_{2}, the gases associated with global warming.
To face this challenge, new standards or agreements, such as the Kyoto protocol, which was ratified by more than 170 countries, seek to achieve the “stabilization of greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system” by a global reduction in greenhouse gas emissions, especially in developed countries. Cleaner combustion processes are therefore required in a wide range of industries. To achieve this, a detailed knowledge of combustion processes and new tools to help design and make improvements to combustion systems are essential.
Advances in combustion science require a better understanding of all the physical processes that control a flame: mixing processes, turbulence, chemistry, radiative heat transfer, etc.
This work focuses on radiative heat transfer in combustion, a phenomenon is of crucial importance, as it:

controls charge heating in furnaces;

is a key mechanism in thermal heat losses and wall heat fluxes;

causes minor pollutant species to form;

interacts with soot;

controls the propagation of largescale fires, such as pool fires and forest fires.
Combustion and radiation are very different phenomena. Combustion is controlled by local exchanges and is numerically described through balance over small volumes (finitevolume framework), whereas radiative heat transfer involves longdistance interaction. Hence, taking radiation into account in numerical simulations of combustion leads to various difficulties, such as the very expensive CPUtime requirements of radiation solvers or the need to know spatially and timeresolved information about the reacting flow field, which is usually not available in Reynoldsaveraged Navier–Stokes (RANS) approaches. Notwithstanding all these difficulties, the interaction between turbulence and radiation exists and can have a significant effect on flames [7].
Several authors have already attempted to perform numerical simulations of combustion while taking into account radiative heat transfer, but because of limited computational resources were unable to integrate highly accurate physical models of the aerodynamics, combustion chemistry and radiation [3–6, 11, 13, 27, 41, 47]. For this reason radiation has long been neglected or treated with simple models in numerical simulations of turbulent reacting flows.
Many studies focus on the interaction between turbulence and radiation in combustion, and the SANDIA methane / air turbulent jet diffusion flame, Flame D [1], has been used to validate a range of studies [5, 6, 11, 27]. The simulations were all twodimensional but used different approaches both for the turbulence / chemistry interactions (presumed [11] and transported [27] probability density functions and the Reynolds stress method [5]) and the radiation model (the discrete ordinates method (DOM) [5, 6], ray tracing [5] and the discrete transfer method (DTM) [11]). Threedimensional simulations can also be performed, but in general, because of limited computer resources, a standard turbulence model is combined with a P1Gray or P1FSK model for radiation as in [47], where a threedimensional simulation of an oxygenenriched turbulent nonpremixed jet flame is described.
Silva et al. [41] performed a twodimensional numerical RANS simulation of turbulent nonpremixed combustion of methane and air in a cylindrical chamber where thermal radiation exchanges in the combustion chamber were computed by the zonal method, and the gas absorption coefficient dependence on wavelength was resolved using the weighted sum of graygases model. The results show that the greatest effect exerted by thermal radiation was on heat transfer and the temperature field. In [4] the same case was studied taking into account the radiation effect of nongray gases with two different WSGG models, and DOM was used to solve the radiative transfer equation. The authors showed that modeling radiative heat transfer and radiative properties is very important in combustion predictions. In [3] the same authors included turbulence radiation interactions (TRI) in their analysis of the same combustion chamber and concluded that the inclusion of TRI yielded results even closer to the experimental data, although the effect of including radiation (even without TRI) was more important. Hartick et al. [13] used a 20 kW axisymmetric confined swirling combustion system to study the complex phenomenon of TRI. They proposed a new approach to confined diffusion flames and concluded that TRI can be simulated efficiently without the use of tedious stochastic approaches.
Compared with RANS, unsteady simulations, such as direct numerical simulations (DNS) or large eddy simulations (LES), give access to the instantaneous spatial distribution of fresh and burnt gases, key information for radiation models [37]. Direct numerical simulations were combined with DOM [50] and DTM [48] to investigate twodimensional sooting flames in fires. Wu et al. [49] implemented a photon Monte Carlo method for solving the radiative transfer equation in a turbulentcombustion DNS code to study the turbulence/radiation interaction, and Zhang et al. [52] used DNS and the Monte Carlo transfer method to study the effects of radiation on the boundary layer structure in a turbulent channel flow. The combination of LES and DOM was first used by Desjardin and Frankel [9] to simulate a twodimensional nonpremixed acetyleneair planar jet flame including soot formation and radiation. The same approach was used for threedimensional simulation of a gas turbine [18] and a combustion chamber [31, 32].
In the present work, the new methodology initially described in the thesis of Lecanu [25] and used in [38] to perform LES of turbulentcombustion flows with radiative heat transfer is used to couple a threedimensional LES combustion code with a threedimensional radiative heat transfer code and produce results in a complex geometry with a mesh containing more than 4.7 million cells.
Our approach is based on two dedicated, independent solvers linked through a specialized framework known as CORBA [14] and takes advantage of the different characteristic time of each phenomenon. Highly efficient solvers can then be used for each phenomenon to take advantage of new massively parallel supercomputers. In this study, we used the AVBP LES solver developed by CERFACS and IFP (see, for example, [40]) and the radiation solver based on the parallel threedimensional DOM code known as DOMASIUM [17, 19–21] developed by the RAPSODEE research center at the École des Mines d’AlbiCarmaux.
The goal here was to study the interactions in turbulent combustion with and without thermal radiation by threedimensional numerical simulation using detailed models for combustion and thermal radiation and to compare the results with experimental findings.
The subgrid scales for turbulence radiation interactions were not used. This subject has been studied by various authors, and more detail can be found in [35, 42]. The experimental setup is the same as that used by [31, 32, 51] and also by the present authors in a previous study [38] in which twodimensional simulation was used.
First, the combustion and radiation code are described in Sect. 2. Then the details of the numerical modeling (mesh, boundary conditions and combustion and radiation models) are explained in Sects. 3 and 4. This is followed by Sect. 5, in which comparisons are made between coupled cases, noncoupled cases and experimental data. Finally, some conclusions are drawn in Sect. 6.
Combustion and radiative heat transfer solvers
Combustion solver: AVBP
The AVBP LES solver developed by CERFACS and IFP is used here to solve the threedimensional compressible equations for momentum, massfraction species and energy on hybrid (structured and unstructured) meshes [40].
The flow solver is based on the finitevolume and finiteelement cellvertex methods, with an artificial viscosity sensor and explicit time integration.
The Lax–Wendroff scheme (a finitevolume secondorder Runge–Kutta timeintegration and central secondorder spatialdiscretization scheme) [15] was used for the simulation in this work, since the aim was to investigate the feasibility of LES of reacting flows including thermal radiation.
The discretization schemes are combined with Navier–Stokes characteristic boundary conditions (NSCBC) [30] and walllaw boundary conditions [39]. The code is parallelized by domain splitting using the MPI library, a convenient approach when solving balances over small volumes. AVBP has been proved to be very efficient on massively parallel machines, reaching a speedup factor of 4,078 (i.e., running 4,078 times faster than on one processor) when using 4,096 processors in an IBM BlueGene/L machine [44].
The time step is determined by the minimum value between the convective and diffusive time step when using the thickening flame model (the chemical time step is only considered for flamelet models, which is not our case) [33]. The convective time step is calculated using the Courant–Friedrichs–Lewy condition (CFL), a stability criterion related to the time required for an acoustic wave to go from one node to the following one at a speed \( \ \overrightarrow{u} \  + c\) (\(c\) is the sound speed). The diffusive time step is calculated using a Fourier criterion and is proportional to the diffusion time required to cross a cell.
Radiative heat transfer solver: DOMASIUM code
DOMASIUM (Discrete Ordinates Method Applied with Spectral Integration on Unstructured Meshes) [17, 19–21], the radiation solver used in this work, was developed by the RAPSODEE research center at Ecole des Mines d’AlbiCarmaux to solve the radiative transfer equation on complex threedimensional meshes using DOM. Much effort was spent optimizing the performance of the code, particularly in terms of computational time and memory usage, as these were identified as critical issues at the beginning of the study.
The radiative properties of the gas are described by SNBCK (statistical narrowband cumulativek) or CK [43] models, and the DOMASIUM code is parallelized based on the spectral band and direction of integration. The technique of band number reduction was used in the coupled calculations and yielded a significant reduction in computational time.
As part of this study, tools were specially developed so that the same grid files used with the AVBP LES code could be used with DOMASIUM, which was originally written for unstructured grids using tetrahedral cells. Temperature and massfraction fields were taken directly from an AVBP results file. A new option was added to DOMASIUM to enable it to receive the data field from a given AVBP mesh and, with the use of a connectivity table, perform the radiation heat transfer on a coarser mesh and project the final result onto the original mesh.
Different kinds of angular quadrature can be used, especially \(S_N\) [2], \(T_N\) [46] and [28]. The \(S_4\) quadrature scheme, which has 24 directions and is one of the most popular schemes because of its computational efficiency, was chosen here. Specially for this work, the DOMASIUM code was also parallelized based on the direction of integration, which is the best choice for codes that use DOM according to Gonçalves and Coelho [12].
Special care was taken with the ordering of data for each direction resulting from the optimization of the sweep order. Use of the reordered data as a function of the pathway leads to improved code performance as a result of optimal use of the machine’s cache memory. The main increase in performance is in the loops over all the mesh cells. The sweeping order forces the machine to look for the information outside the memory cache for each step in the loop. By reordering the data set, the machine searches for the information in a highorder memory cache, which can be more than eight times faster than searching for the same information in RAM memory, for example, thus optimizing the calculation time. The same kind of strategy was used by, among others, Elias and Coutinho [10].
The radiative transfer equation was discretized using a finitevolume approach, and three different spatial differencing schemes were considered: “exponential” [36], “step” [28] and “diamond mean flux” [45]. The “diamond mean flux” method was chosen for the simulations.
The parallelization of the code was done using MPI, and very good scalability was achieved. We managed to use more than 300 processors in the coupled simulations without degrading the performance because of the exchange of information between processors. Memory optimization allows the use of meshes with more than 3 million cells in machines such as the Zahir IBM SP4 supercomputer at IDRIS (France) or the Linuxbased cluster at the EM2C laboratory. These mesh sizes are compatible with meshes used in LES of combustion.
Experimental setup and configuration used in the numerical simulations
The geometry used in the numerical simulations is an experimental configuration investigated by Knikker et al. [22–24] and is shown in Figs. 1 and 2. This is the same experimental setup used in previous twodimensional simulations [38]. A premixed propane/air flow is injected into a rectangular combustor. This simple geometry is a first step toward modeling and understanding, for example, the combustion inside a small helicopter gas turbine. The height, depth and length of the combustion chamber are 50, 80 and 300 mm, respectively. The lateral walls are transparent artificial quartz windows that allow the inside of the whole chamber to be viewed. The upper and lower walls are made of thick ceramic material for thermal isolation and include two narrow windows used to introduce laser sheets. A stainless steel triangular flame holder (height 25 mm), corresponding to a 50 % blockage ratio, is embedded in the lateral windows. A Vshaped turbulent flame stabilized by the hot gases recirculating behind the flame holder is investigated. Among the various cases available in [23, 24], the case of a fixed propane/air flow rate of 20 g s^{−1}, corresponding to an upstream mean velocity of about 5 m s^{−1} (turbulence level about 5 %) and an equivalence ratio \(\phi\) = 1.0, was chosen for simulation.
The computational domain (Fig. 3) starts 10 cm upstream of the flame holder and continues until 60 cm downstream. A coflow with a coarse grid was added after the combustion chamber to reduce the influence of the outlet boundary condition on the combustion results.
The LES mesh contains about 4.7 million cells (tetrahedrons), and the grid size downstream of the flame holder is \(\Delta\) = 1.0 mm until 15 cm downstream (Fig. 4), when it increases. In the following simulations, the AVBP time step is about \(\Delta t_{\rm {LES}} \approx\) 0.19 μs. The radiation code uses a different mesh to reduce the computational time and machine memory required. Using a connectivity table, the physical properties (CO_{2} and H_{2}O mass fractions and temperature field from AVBP, and radiation power from DOMASIUM) can be transferred from one mesh to another in both directions. The radiation mesh contains about 3.3 million cells (tetrahedrons). The grid size of both meshes downstream of the flame holder is the same until 12 cm downstream, where the recirculation zone is. This should be well described because it is responsible for flame stabilization. Upstream of the flame holder and in the coflow region, a coarser grid can be used as the power radiation is small with no large variations.
The NSCBC were used in the combustion code to impose both the static pressure at the outlet and the velocity components, static temperature and species mass fractions at the inlets smoothly. Consequently, the ingoing waves are calculated as being proportional to the difference between the actual state at the boundary nodes and the set reference. There are two inlets to the domain; the first is the inlet for the premixed propane/air mixture upstream of the flame holder (equivalent ratio \(\phi\) = 1.0; velocity = 5 m s^{−1} and T _{2} = 300 K), and the second is the coflow inlet, where nitrogen is injected at 22 m s^{−1} and 1,900 K.
A boundary condition defined as a wall with heat loss using a wallfunction approach with zero normal velocity (slip) is imposed in the combustion code; the thermal resistance is estimated to be \(R_{\rm th}\) = 0.096 Km^{2} W^{−1} in the upper and lower ceramic walls and \(R_{\rm th}\) = 0.086 Km^{2} W^{−1} in the lateral artificial quartz walls. A thermal resistance equal to \(R_{\rm th}\) = 120 Km^{2} W^{−1} is used for the triangularshaped aluminum flame holder. All the other walls are assumed to be adiabatic slip walls.
In the radiation code, an emissivity \(\varepsilon\) is specified for these walls. This is assumed to be constant for the ceramic walls (\(\varepsilon = 0.91\)), while the real emissivity of the material as a function of wavelength is used for the quartz walls (semitransparent at short wavelengths becoming opaque at longer wavelengths, Fig. 5). According to [16], typical stainless steels lightly oxidized at 1,000 K have an emissivity of \(\varepsilon = 0.4\). This figure is used for the triangularshaped flame holder, while the coflow walls are assumed to have an emissivity of \(\varepsilon = 0.7\). The other walls are assumed to be cold black bodies with \(\varepsilon =1\), i.e., all rays reaching the walls are absorbed (no reflection and no transmission). The radiation fluxes do not heat the walls and transmission through the walls is neglected.
Combustion LES and radiation modeling
Unresolved turbulent fluxes are modeled here using the classical Smagorinsky model based on a subgridscale viscosity [29]. Turbulent premixed combustion is described by the dynamically thickened flame model [8, 26] (DTFLES). In this formulation, the thickening factor \(F\) is not a constant but increases to \(F_{\rm {max}}\) in flame zones and decreases to unity in nonreactive zones. This model is more suitable for calculating the radiative power than the thickened flame model (TFLES), since it does not change the species or thermal diffusion outside the flame region.
The chemical reaction between propane and air is represented by a onestep global chemical mechanism [40]:
The reaction rate is given by:
The chemical parameters used are summarized in Table 1. These values lead to a laminar flame speed 48 m s^{−1} for an equivalent ratio \(\phi = 1.0\). Again, this combustion model is quite simple but is appropriate here since only perfectly premixed combustion is considered.
In the version of AVBP used here, the chemical scheme that describes the methaneair or propaneair chemistry uses constant but different Schmidt numbers for each species so that the laminar flame speed can be reproduced as accurately as possible. These different Schmidt numbers introduce preferential diffusion in the flame front, potentially leading to large variations in the CO_{2} to H_{2}O ratio, which should be almost constant (Fig. 6). This effect is greatly amplified when the Thickened Flame Model is used. Figure 7 shows the CO_{2} to H_{2}O ratio for the dihedral flame when different Schmidt numbers are used. While in the burned gases the ratio is accurately reproduced, in the front behind the flame holder the ratio is about 1.4, which is unrealistic. While this effect may be negligible when focusing on flame dynamics, it leads to a large error when dealing with radiation, as CO_{2} and H_{2}O are the two major absorbing species in the flame and the ratio between them is crucial when estimating the radiative power of the flame accurately.
Hence, when dealing with radiative heat transfer, the Schmidt numbers must be set to the same value for all the species, as was done here.
In the radiation code the CK distribution method was used with a reduced number of spectral bands (28 for H_{2}O and 8 for CO_{2}) corresponding to the wavelengths that make the most important contributions to the global radiative power. As mentioned before, the Diamond Mean Flux Scheme (DMFS) and a fourthorder quadrature were adopted, resulting in 24 spectral directions [17, 21]. According to [19], these models are the best compromise between calculation time and accuracy. Radiation heat transfer is assumed to be controlled by resolvedscale motions only (and the burnt gas zones), and no subgrid scale radiation model is taken into account.
Results and discussion
First the results with and without radiative heat transfer are compared, and these are then compared with the experimental data.
Calculations without radiative heat transfer using only the AVBP code were carried out to provide a reference case (identified as NR in the following discussion). A previous twodimensional study [38] showed that updating the radiative source term every 100 interactions is a good compromise between accuracy and computational time. This coupling frequency was also used for the threedimensional simulations. Correct load balancing is then achieved by devoting 50 % of the processors to the AVBP LES code and 50 % to the DOMASIUM radiative heat transfer code. Typical runs were conducted using 192 processors (96 for each code).
Coupled simulations were started at \(t_0\) = 0.152 s of a computation without radiative heat transfer, after which simulations with and without radiation were conducted. This procedure allows both flow fields to be compared at the same instant. The timeaveraged results used specially for comparison with the experimental data were calculated based on 760,000 interactions (\(\sim\)0.144 s) or about 8.6 characteristic periods of the flow.
The influence of radiation in the threedimensional combustion simulation
In this section, we will present the coupled combustion and radiation results for the threedimensional simulations and compare the cases without and with radiative heat transfer. Figure 8 shows instantaneous isosurfaces of temperature without and with radiative heat transfer at the same physical time \(t_2\) = 0.28 s. With radiation, the flow structures are modified and the turbulence structures appear to increase several centimeters downstream of the flame holder. This can also be observed in Figs. 9 and 10, which show temperature cuts in the center plane (z = 0) and symmetrical side planes (z = −20 mm and z = 20 mm), respectively. The temperature distribution changes if radiation is taken into account, when maximum values are lower. The highest temperature peak is found in the central plane of the burner.
Figures 11 and 12 compare the mean resolved temperature in the center plane (z = 0) and symmetrical side planes (z = −20 mm and z = 20 mm) without and with radiative heat transfer, respectively. As expected, the maximum mean temperature is about 80 K lower when radiative heat transfer is taken into account because of the increase in energy transfers from hot zones to cold zones and walls as a result of radiation. Differences in temperature begin just a few centimeters downstream of the flame holder, as illustrated in Figs. 13 and 14, where the profile of the mean resolved temperature along the \(x\) axis in the center plane is shown from the flame holder until the end of the combustion chamber for three different values of \(y\).
These temperature differences directly impact the reaction products. For example, the mean CO_{2} mass fraction (Fig. 15) decreased slightly when radiation was included, in spite of the simple onestep global chemical mechanism used here. These observations are confirmed by Fig. 16, which shows longitudinal cuts of the CO_{2} mass fractions at \(y\) = −0.01 m and \(y\) = 0.01 m for \(z\) = 0. Radiative heat transfer can be expected to have a more significant impact when using a more detailed chemical mechanism that includes pollutants such as carbon monoxide (CO) or nitric oxides (NO_{ x }), whose formation is known to be very sensitive to local temperature.
Figure 17 shows the mean heat release rate field in the central plane (z = 0) without and with radiative heat transfer. The flow and temperature modifications when radiation is taken into account directly impact the reaction rate, corroborating the observation above about CO_{2} mean mass fractions. Lower values of heat release are found in the first quarter of the combustion chamber with radiative heat transfer.
To take a deeper look at the temperature modifications caused by radiative heat transfer, we plotted in Fig. 18 the difference between the mean resolved temperature without and with radiative heat transfer, \(\bigtriangleup \langle \widetilde{T} \rangle = \langle \widetilde{T}_{NR} \rangle  \langle \widetilde{T}_{R100} \rangle\). The small values of \(\bigtriangleup \langle \widetilde{T} \rangle\) (less than 25 K) just downstream of the flame holder can be explained in part by the high wall temperatures found (about 1,900 K), which reduces heat exchange by radiation between the obstacle wall and the hot burnt gas. Near the end of the combustion zone, the flame interacts with the combustion chamber walls, resulting in a temperature difference of more than 150 K. It should be remembered that radiative heat flux is not taken into account in the wall boundary condition used in the combustion code. The validity of this approximation should be carefully investigated in zones where the flame directly interacts with the wall.
The length of the recirculation zone is about 7 cm and is not greatly affected by radiative heat transfer, as shown in Fig. 19. However, the mean temperature field changes when radiative heat transfer is included, especially in the recirculation zone downstream of the flame holder (Fig. 13), which plays a key role in the flame stabilization mechanism. Changes in the temperature field in this region may induce major changes in the behavior of the flame and its interaction with turbulence motions
.
The filtered RMS (root mean square) temperature illustrates more quantitatively how the flame dynamics change when radiative heat transfer is taken into account, as shown in Fig. 20, which compares the filtered RMS temperature profiles in the central plane of the burner without and with radiation. In spite of the similarity between the profiles, the values of RMS temperature are higher downstream of the recirculation zone behind the obstacle when radiative heat transfer is taken into account, as confirmed by Figs. 21 and 22, where profiles of RMS temperature are shown along the axis \(x\) at z = 0 for different values of \(y\). Differences of more than 80 K can be observed in some regions, indicating stronger flame movements. These findings confirm the influence of radiative heat transfer despite its limited contribution to energy transfer (less than 2 % of the overall heat released).
To aid analysis of the mean resolved temperature changes and corresponding RMS temperature, Fig. 23 shows the transverse profiles of these two quantities at \(x\) = 0.11 m and z = 0 without and with radiative heat transfer in a zone about 0.04 m downstream of the recirculation zone. Mean temperature is always lower with radiation, and peak temperature is about 50 K lower in the central region. The RMS temperature profiles have higher values in the central and peak regions in the simulation with radiative heat transfer.
In Figs. 24 and 25 a different plane of cut has been used, this time parallel to the top and bottom combustion chamber wall in the central zone (\(y\) = 0). The profiles are extracted at \(x\) = 0.13 m, about 0.06 m downstream of the recirculation zone. The mean resolved temperatures are always lower with radiative heat transfer. A smaller difference is found at the boundaries, and the maximum mean temperature is about 50 K lower with radiation. The RMS temperatures in Fig. 25 are generally higher when radiative heat transfer is taken into account, and the maximum RMS value is increased by about 40 K. However, a zone where the RMS temperature is higher without radiation can be observed between the peaks and the axis of the chamber and should be investigated in a future study.
The same explanation put forward in the twodimensional study [38] can be used here to explain the increase in the resolved RMS temperature: the lower temperature of the burnt gas as a result of radiation heat losses leads to reduced thermal expansion and a decreased flame speed \(S_L^0\), in turn leading to a higher value of \(N_{\rm B} = u' / (\tau S_L^0)\) [\(u'\) is the turbulence intensity, \(\tau = (T_{\rm b} / T_{\rm u})  1\) is the heat release factor, \(T_{\rm u}\) and \(T_{\rm b}\) are the fresh and burnt gas temperatures, respectively, and is the laminar flame speed] and increasing the flame’s sensitivity to turbulent motions (larger flame wrinkling).
Figure 26 shows the instantaneous radiative power field. The large positive central zone corresponds to the radiation power emitted by the hot burnt gases. A small absorption zone is visible upstream of the flame front. These findings are confirmed by Fig. 27, which compares profiles of radiative power (in W m^{−3}) at z = 0 and different values of \(x\) downstream of the flame holder. Figure 28 shows mean radiative power along the central axis of the combustion chamber (\(y\) = 0 and z = 0); the maximum occurs at about \(x\) = 0.12 m. The global radiative power calculated with DOMASIUM for this flame is about 1 kW, which represents less than 2 % of the total flame power. Nevertheless, radiative heat transfer has a major effect on the flame dynamics.
Comparison with experimental data
As discussed in the previous twodimensional study [38], comparison of numerical simulations and experimental data is challenging as, to our knowledge, few experiments that are well characterized both in terms of reacting flow fields and radiative heat transfer are available in the literature. One such experiment, however, is described in a recent study by Rankin et al. [34]. Here, we again attempt to compare numerical simulations and experimental data using the experimental data collected in [23, 24]. Planar laserinduced fluorescence (PLIF) of OH radicals was performed in the center plane of the burner shown in Fig. 1. The progress variable \(c\), which follows the temperature (\(c\) = 0 in fresh gases and 1 in fully burnt gases), is extracted from binarized images assuming an infinitely thin flame front. Instantaneous massfiltered \(\widetilde{c}\) progress variables are then calculated by applying a Gaussian filter of size \(\Delta = F \delta _L^0 = 6.2\) mm, corresponding to the thickness of the resolved threedimensional flame front. Ensemble averages are extracted by processing 200 images.
The numerical timeaveraged filtered progress variable, \(\langle \widetilde{c} \rangle\), and the corresponding resolved variance, \(\langle \widetilde{c}^2\rangle  {\langle \widetilde{c} \rangle }^2\), are reconstructed from the instantaneous filtered temperature fields obtained in the simulation (the operator \(\langle \rangle\) denotes time averages) to extract variables that can be compared with experimental data. These variables are reconstructed from experimental fields under three assumptions: (1) infinitely thin instantaneous flame fronts, (2) twodimensional images and (3) a Gaussian filter of size \(\Delta = F \delta _L^0.\)
Figure 29 compares the transverse profiles of the mean progress variable \(\langle \widetilde{c} \rangle\) and corresponding resolved variance \(\langle (\widetilde{c})^2 \rangle  {\langle \widetilde{c} \rangle }^2\) extracted from the experiment and the numerical simulations without (NR) and with (R) radiative heat transfer for three downstream locations. The mean progress variable profiles are quite similar with and without radiative heat transfer. However, the top of the curve is wider in the experimental data, probably because the parameters used in the simulation result in a slower reaction rate than in the experimental study or because of the threedimensional effects neglected when the experimental data are processed. The resolved variance profiles display better agreement with the experimental values when radiative heat transfer is included, especially 11 cm downstream of the flame holder. Upstream, the numerical simulations tend to overestimate the flame brush thickness and resolved variance both without and with radiation (see profiles at \(x\) = 7 cm in Fig. 29). The downstream location \(x\) = 7 cm corresponds roughly to the end of the recirculation zone downstream of the flame holder. In the recirculation zone, the resolved flame thickness \(F \delta _L^0 = 6.2\) mm is of the order of half the effective flow section and is too large to yield reliable results.
Conclusion
Threedimensional simulations were performed in which an LES combustion code was coupled with a radiation solver. A turbulent premixed propane/air flame stabilized downstream of a triangular obstacle, which had previously been studied in twodimensional simulations, was used as a test case. The results show the threedimensional character of the flame, the importance of the boundary conditions and that when radiative heat transfer is taken into account there is a change in the flame dynamics and better agreement with experimental data.
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Acknowledgments
Rogério Gonçalves dos Santos was the recipient of a scholarship from CNPq (National Council for Scientific and Technological Development, Brazil). The project was integrated in the European Union program DEISA (Distributed European Infrastructure for Supercomputing Applications) in the framework of the 6th PCRD (FOCUS project) and was supported by the French Agence Nationale de la Recherche (ANR) through the CORAYL project (ANR05CICG012). Simulations were carried out on the IBM SP4 zahir supercomputer provided by IDRIS (Institut de Développement et de Ressources en Informatique Scientifique) with the assistance of IDRIS (especially J.M. Dupays, G. Grasseau and D. Girou) for CORBA coupling and CERFACS (Y. Sommerer) for AVBP. Experimental data used for comparison with numerical simulations were processed from the laserinduced fluorescence images collected by Dr. R. Knikker during his PhD.
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Technical Editor: Francisco Ricardo Cunha.
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dos Santos, R.G., Ducruix, S., Gicquel, O. et al. A study of threedimensional LES of turbulent combustion with radiative heat transfer. J Braz. Soc. Mech. Sci. Eng. 38, 33–48 (2016). https://doi.org/10.1007/s4043001503228
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Keywords
 Turbulent combustion
 Radiative heat transfer
 Large eddy simulation
 Code coupling