Peter M. Sockol

Glenn Research Center, Cleveland, Ohio

A Radiation Solver for the National Combustion Code

NASA/TM—2015-218815

July 2015
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Peter M. Sockol

Glenn Research Center, Cleveland, Ohio

A Radiation Solver for the National Combustion Code

NASA/TM—2015-218815

July 2015

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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NASA/TM—2015-218815 1 

A Radiation Solver for the National Combustion Code 

Peter M. Sockol 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A methodology is given that converts an existing finite volume radiative transfer method that requires 

input of local absorption coefficients to one that can treat a mixture of combustion gases and compute the 

coefficients on the fly from the local mixture properties. The Full-spectrum k-distribution method is used 

to transform the radiative transfer equation (RTE) to an alternate wave number variable, ݃. The 

coefficients in the transformed equation are calculated at discrete temperatures and participating species 

mole fractions that span the values of the problem for each value of ݃. These results are stored in a table 

and interpolation is used to find the coefficients at every cell in the field. Finally, the transformed RTE is 

solved for each ݃ and Gaussian quadrature is used to find the radiant heat flux throughout the field. The 

present implementation is in an existing cartesian/cylindrical grid radiative transfer code and the local 

mixture properties are given by a solution of the National Combustion Code (NCC) on the same grid. 

Based on this work the intention is to apply this method to an existing unstructured grid radiation code 

which can then be coupled directly to NCC. 

Introduction 

The National Combustion Code (NCC) is a state of the art Computational Fluid Dynamics (CFD) 

program specifically designed for combustion processes (Ref. 1). The code employs an unstructured grid, 

massively parallel computing (Refs. 2 and 3), a dynamic wall function with the effect of adverse pressure 

gradient (Ref. 4), a low Reynolds number wall treatment (Ref. 5), a cubic non-linear k-epsilon turbulence 

model (Ref. 6), a lagrangian liquid phase spray model (Ref. 7), and stiff laminar chemistry integration. 

Recently, viscous low-speed preconditioning (Ref. 8) has been added to improve the low-speed 

convergence of the NCC in viscous regions and the ability to handle multiple sets of periodic boundary 

conditions has also been added. The combination of these features is usually not available in other CFD 

codes and gives the NCC an advantage when computing recirculating, turbulent, reacting spray flows. 

Previously, the NCC has undergone extensive validation studies for simple flows (Ref. 9), complex flows 

(Ref. 10), NOx emissions prediction (Ref. 11) and traditional gas turbine combustor injectors (Ref. 12). 

For most high-temperature combustion problems, thermal radiation plays an important role in the heat 

transfer to the surrounding surfaces and the absorption and emission of radiation can also have a 

significant effect on the temperature distribution throughout the domain. Hence, if a designer wanted to 

use NCC to aid in the design of a jet engine combustor or a furnace combustion chamber, it would be 

very useful to be able to compute the radiative transfer as part of the process. In this work we have taken a 

significant step in that direction. 

At present two codes exist for solving the radiative transfer equation (RTE). The first is a stand-alone 

cartesian or cylindrical grid code (Ref. 13) that solves in 1D, 2D, or 3D. The second is an unstructured 

grid code (Ref. 14) that could be coupled with NCC. Both of these codes require input of the local 

absorption coefficients. Hence the user of either code needs a means of coming up with at least 

approximate values of the absorption coefficients as a function of gas properties and wave number. 

A gas emits and absorbs radiation only at wave numbers in the vicinity of spectral lines. Since the 

main participating gases (CO2 and H2O) each have more than one million absorption/emission lines, it 

would be impractical to solve the RTE at enough wave numbers to resolve this physics. Modest and 

coworkers have developed a methodology for handling this behavior by transforming the RTE to an 
NASA/TM—2015-218815 2 

alternate wave number variable space where the coefficients vary smoothly (Ref. 15). We currently have 

a Spectral Radiation Calculation Software package (Ref. 16) (SRCS) which includes source code and data 

sets necessary to run the method as part of another application. 

In this work we use the existing cartesian/cylindrical radiation code modified to solve in the 

transformed wave number space. We also use the Modest full-spectrum k-distribution method and adapt 

the SRCS package to run with the modified cartesian/cylindrical radiation code. NCC is run on a cartesian 

grid to obtain local temperatures and species mole fractions. SRCS is used to obtain a table of absorption 

coefficients at discrete values of temperature and participating species mole fractions where these 

properties span the range of values from the NCC solution. We next interpolate within the table to obtain 

cell center values of k for every cell in the grid. Finally, we run the modified cartesian/cylindrical 

radiation code with these k values to obtain radiation intensity and heat transfer throughout the field. The 

expectation is that this same methodology will work quite well with the unstructured grid radiation code 

and thus provide a solver that can be coupled with NCC. 

Full-Spectrum k-Distribution Method 

Over the years Modest and coworkers have added significant refinements to the basic method 

(Refs. 17 to 21). These increase the accuracy for inhomogeneous gas mixtures, but they also increase the 

complexity of the method. Since the turbulence-chemistry modeling used in NCC is still undergoing 

significant development, it was decided to use the basic full-spectrum k-distribution (FSK) method in this 

work (Ref. 15). In addition, since radiation scattering is not considered important for combustion gases, 

this effect is neglected (Ref. 22). Also, while the presence of soot affects emission and absorption, 

scattering by soot is typically negligible. 

We begin with the spectral radiative transfer equation for an absorbing/emitting medium: 

ఎܫ݀

݀ݏ ൌ ߢఎ ቀ߶, ߟቁ ൣܫ௕ఎሺܶሻ െ ܫఎ൧ , (1) 

where ܫఎ is the spectral intensity varying along a path s, ߟ is the wavenumber, ߢఎ is the spectral 

absorption coefficient, ܫ௕ఎ is the Planck function and ߶ ൌ ൫ܶ, ݌, ݔ൯ is an array of state variables. Here ݔ is 

the mole fraction vector. We reorder the absorption coefficient in Equation (1) by multiplying by the delta 

function ߜ ൬݇െߢఎ ቀߟ,଴߶ቁ൰,	and integrating over the entire spectrum. Here ߢఎ ቀߟ,଴߶ ቁ is the absorption 

coefficient evaluated at some reference state ߶଴ ൌ ൫ܶ଴,݌଴, ݔ଴൯. This gives 

௞ܫ݀

݀ݏ ൌ	݇∗ ቀ߶, ݇ቁ ቂ݂ ቀܶ, ߶଴, ݇ቁ ܫ௕ሺܶሻ െ ܫ௞ቃ , (2) 

provided that at every wavenumber across the spectrum where ߢఎ ቀ߶଴, ߟቁ has one and the same value 

݇,	we also have a unique value for ߢఎ ቀ߶, ߟቁ ൌ ݇∗ ቀ߶, ݇ቁ. In Equation (2) ܫ ௞and ݂	are defined as 

ܫ ௞ൌ න ܫఎߜ ൬݇െߢఎ ቀ߶଴, ߟቁ൰ ஶ

଴

 (3) ,ߟ݀

݂ ቀܶ, ߶଴, ݇ቁ ൌ 1

௕ܫ

න ܫ௕ఎሺܶሻߜ ൬݇െߢఎ ቀ߶଴, ߟቁ൰ ݀ߟ.

ஶ

଴

(4) 
NASA/TM—2015-218815 3 

In terms of the cumulative k-distribution, 

݃ ቀܶ, ߶଴, ݇ቁ ൌ න ݂ ቀܶ, ߶଴, ݇ቁ ݀݇ ൌ න ݂ ቀܶ, ߶, ݇∗ቁ ݀݇∗ ൌ ݃ ቀܶ, ߶, ݇∗ቁ ,

௞∗

଴

௞

଴

(5) 

the RTE becomes: 

௚ܫ݀

݀ݏ ൌ ݇∗ ቀܶ଴, ߶, ݃ቁ ൣܽሺܶ, ܶ଴, ݃ሻܫ௕ሺܶሻ െ ܫ௚൧, (6) 

where 

ܫ ௚ൌ ܫ௞ൗ݂ ቀܶ଴, ߶଴, ݇ቁ , (7) 

ܽሺܶ, ܶ଴, ݃ሻ ൌ ݂ ቀܶ, ߶଴, ݇ቁൗ݂ ቀܶ଴, ߶଴, ݇ቁ , (8) 

and ݇∗ ቀܶ଴, ߶, ݃ቁ is the ݇ vs. ݃ distribution as given by Equation (5), with the absorption coefficient 

evaluated at the local conditions ߶ and the Planck function evaluated at the reference temperature ܶ଴. 

The reordered spectral intensity ܫ௚can be obtained by solving the modified RTE, Equation (6), and the 

spectrally integrated intensity is then evaluated as 

ܫ ൌ න ܫఎ݀ߟ ൌ න I௞݂ ቀܶ଴, ߶଴, ݇ቁ ݀݇ ൌ න ܫ௚݀݃.

ଵ

଴

ஶ

଴

ஶ

଴

(9) 

Since ܫ ௚is a smooth function of ݃, the integral in Equation (9) can be evaluated by Gaussian quadrature 

with values of ܫ௚computed and stored at the quadrature points. Similarly the integrated heat flux in any 

direction can also be found by Gaussian quadrature. 

Use of the SRCS package 

The SRCS package (Ref. 16) can be used to calculate the ݇ versus ݃ distribution as a function of the 

local state variables ߶. However, to do this at every cell in a large 3D computation would be prohibitively 

expensive. Instead we construct a multidimensional table using discrete values of temperature and 

participating species mole fractions that span the values from the NCC solution. The SRCS software is 

used to populate the table with values of ݇௤ versus ݃௤ where the ݃௤ are the Gaussian quadrature points. 

We then interpolate within the table to obtain cell center values of ݇௤ for each ݃௤ at every spatial cell in 

the field. Note the same process is used to obtain ܽ௤ for each ݃௤ at every cell. 

We begin by calculating the reference state ߶଴. Following Modest (Ref. 15) we choose a volume 

averaged mole-fraction and a Planck-mean temperature: 

1 ൌ଴ ݔ

ݔ න ܸ

௏

ܸ݀, (10) 
NASA/TM—2015-218815 4 

ߢ ௉ቀܶ଴, ݔ଴ቁ ൌ 1

௉ߢ න ܸ

௏

൫ܶ, ݔ൯ܫ௕ሺܶሻܸ݀, (11) 

and it is assumed that the pressure variation across the domain is negligible. For combustor 

simulations, the volume V for these averages would be restricted to regions with significant 

concentrations of combustion products (i.e., regions downstream of fuel injectors). 

Next, for each ൫ܶ, ݔ൯ grouping, we obtain the	݇ ቀܶ଴,߶, ݃ቁ distribution using the SRCS software, 

interpolate within the distribution to obtain ݇௤ for each ݃௤ and store the ݇௤ in the table. Also, for each ܶ

in the table, we obtain ܽሺܶ, ܶ଴, ݃ሻ, interpolate to get ܽ௤ for each ݃௤ and store the ܽ௤ in a second table. 

Since the absorption coefficients are expected to be more sensitive to ܶ than to ݔ, for the first table we 

construct cubic spline fits for ݇௤ versus ܶ for each ݔ and ݃௤. We also construct cubic spline fits for ܽ௤

versus ܶ for each ݃௤. 

Once this preliminary work has been completed, we proceed to compute ݇௤ and ܽ௤ for each cell in 

the field by interpolating within the two tables. Since for the present work we consider only two 

participating species (CO2 and H2O), the spline fit is used to find ݇௤ for the local ܶ at the four corners of a 

box containing the local ݔ. Bilinear interpolation is then used to find the local value of ݇௤. For three or 

four participating species this is readily extended to trilinear or quadralinear interpolation. We use the 

spline fit in the second table to find ܽ௤ at the local ܶ. 

Solution of the Radiative Transfer Equation 

The cartesian/cylindrical solver used in this study (Ref. 13) uses a finite volume method to solve the 

transformed RTE, Equation (6). First note that ܫ ௚depends on both the spatial coordinates and the 

radiation direction (polar angle ߠ and azimuthal angle ߶). We integrate Equation (6) over a control 

volume ߂ܸ and a control angle ߗ߂ and apply the divergence theorem to obtain 

න න ܫ௚ሺ࢙ො ∙ ࢔ෝሻ݀ܣ݀ߗ ൌ 	 න න ݇∗ ቀܶ଴, ߶, ݃ቁ ൣܽሺܶ, ܶ଴, ݃ሻܫ௕ሺܶሻ െ ܫ௚൧ܸ݀݀ߗ

௱ఆ ௱஺ ௱ఆ ௱௏

(12) 

where ࢙ො is the unit vector in the radiation direction and ࢔ෝ is the outward unit normal along the control 

surface ܣ߂	. If the intensity is assumed constant over the control volume ߂ܸ	௜, then Equation (12) can be 

rewritten as 

௝௜ܫ ෍

௟

௝

௝௜ܦ௜ܤെ ൌ௜௝ ܣ߂ ௟

௟

௜ܫ

௟ ൅ ܵ௜௟ , (13) 

where the summation is over the faces of ߂ܸ ௜, ܫ௝௜

௟ is the intensity at face ܣ߂ ௝௜of ߂ܸ ௜in direction ࢙ො௟

 , 

௝௜ܦ

௟ ൌ න൫ݏ̂ ∙݊ො௜௝൯݀ߗ

௱ఆ೗

(14) 

௜ܤ

௟ ൌ ݇௜

∗

௟ߗ߂௜ܸ߂

, (15) 

ܵ௜

௟ ൌ ݇௜

∗

௟ߗ߂௜ܸ߂௜ሻ௕ܫሺ௜ܽ

, (16) 
NASA/TM—2015-218815 5 

and the subscript g has been dropped to simplify the notation. The face intensities ܫ௝௜

௟ are obtained from 

the step scheme. For ܦ௝௜

௟ ൐ 0 (an outflow of radiant energy), we use ܫ௜௟

 the center value of ߂ܸ	௜. For ܦ௝௜

௟ ൏

0 (an inflow of radiant energy) we use ܫ௜ᇱ

௟ the center value of the adjacent control volume ߂ܸ௜ᇱ except 

where ܣ߂ ௝௜lies on a domain boundary and a boundary condition is used to obtain ܫ௝௜

௟ . Finally, the 

directional space is defined by taking constant spacing for ߠ	and ߶, where ߠ is measured relative to ݁̂

௭ and 

߶ is measured relative to ݁̂

௫ in the ݁̂௫ െ ݁̂௬ plane. 

For a cartesian grid the solution of Equation (13) is straightforward. Start at one of the eight corners 

of the 3D domain and select that octant of the total directional space for which all the directions point into 

the interior of the grid. For each of these directions we sweep through the grid in such a way that all of the 

radiant inflow to a control volume comes either from adjacent control volumes that have been updated 

during the current sweep or from the adjacent boundary condition. Next move to another corner and 

repeat the process for a different octant of the directional space. After all eight corners have been used, 

the radiation intensity ܫ௜

௟

 will have been updated for every control volume ߂ܸ ௜and control angle ߗ߂௟ in 

the entire domain. 

Sample Cases 

Hot Gas Cylinder 

Before we present results from a combustion problem, we consider a simple case to show the kinds of 

results that can be obtained from the current cartesian/cylindrical radiation code. We choose a cylinder of 

hot gas with cold walls. The gas temperature ܶ௚ is set to 1000 K, the pressure ܲ௚ to 1 atm. and the mole 

fractions ݔ ௚௣of the participating gases to 0.2 and 0.02 for CO2 and H2O, respectively. The walls are set to 

0 K. Symmetry conditions are used to reduce the domain to radial distance r = 0 to rmax and axial 

distance z = 0 to 4 rmax with symmetry at r = 0 and z = 0. Even though the problem is axisymmetric, the 

azimuthal distance goes from ߮ = 0 to 2ߨ. The number of spatial grid cells is taken as 20߮ ൈ 40ݎ ൈ 100ݖ

with hyperbolic tangent stretching used to cluster the cells near the walls in r and z. The grid spacing in ߮

is kept uniform. The directional discretization uses 8ߠ ൈ 20߶ with uniform spacing in both directions. 

Finally we note that we solve the RTE for each of 10 Gaussian quadrature points ݃௤. The final 

distributions of ݍ	௥, ݍ௭	, ܩ	ܽ݊݀	݀݅ݒሺݍሻ are then found by Gaussian quadrature. 

For a first calculation we set rmax = 1.0 m. The resulting distributions are shown in Figure 1, where 

each of the variables is divided by an appropriate reference value: ݎݍ݂݁ ൌ 	ߪܶ௚

ସ, ݎܩ݂݁ ൌ

	ݎݍ݂݁ ߨ,			݀ݎݒ݂݁ ൌ 	 ݎݍ݂݁ ݎ݉ܽݔ ⁄ ⁄ and ߪ is the Stefan-Boltzman constant. In Figure 1(a) the radial heat 

flux rises gradually from zero at the centerline to a maximum at r = rmax, in Figure 1(b) the axial heat 

flux is close to zero until the end wall is approached at z = 4 rmax, in Fig. 1(c) the mean radiative 

intensity is largest at the centerline and decreases gradually as the cold walls are approached and finally in 

Figure 1(d) the radiative heat source is negligible until very near the cold walls. Thus this is an optically 

thick case. 

As a second calculation we set rmax = 0.01 m. The resulting distributions are shown in Figure 2, 

where the reference values are the same as in Figure 1 except that ݀ݎݒ݂݁ is 100 times as large. The results 

shown in Figure 2(a) to (c) are similar to those in Figure 1. However, Figure 2(d) shows that the radiative 

heat source, while negligible near most of the centerline, gradually becomes substantial as the cold walls 

are approached. Thus this case could be considered optically thin. 

Sandia Flame D (Ref. 23) 

This is one of a series of piloted CH4/air flames that were set up to provide data to test combustion 

models. In this case a mixture of 25% CH4 and 75% dry air enters at a temperature of 294 K and mean 

velocity of 49.6 m/sec through the main jet located on the centerline with a diameter d = 7.2 mm. The jet 
NASA/TM—2015-218815 6 

exit Reynolds number is 22400. The annular pilot, which surrounds the jet, has an outer diameter of 

18.2 mm. Burnt gas from the pilot enters at a mean temperature of 1880 K and a mean velocity of 

11.4 m/sec. Outside the pilot, dry air enters at a temperature of 291 K and a mean velocity of 0.9 m/sec. 

The entering composition of the pilot gas and the velocity and turbulence profiles, which were measured 

separately (Ref. 24), are given in the above reference. 

This flame was simulated with NCC on a domain with radial distance r = 0 to 0.045 m and axial 

distance z = 0 to 0.375 m. A 2D-axisymmetric grid is used with 120 cells in r clustered at the outer edge 

of the jet, and 100 cells in z gradually stretching from the inlet to the exit. Axisymmetric flow was 

assumed. The calculations used a ݇-epsilon turbulence model with only linear terms retained in the 

Reynolds stress. The chemistry model used finite rate chemistry with 12 species and 10 reactions and no 

turbulence-chemistry interaction. The resulting distributions are shown in Figure 3. Figure 3(a) shows hot 

temperatures reaching the centerline very close to the jet exit in contrast to the experimental data 

(Ref. 23) where large temperatures along the centerline occur at z/d > 15. Figure 3(b) and (c) show that 

CO2 and H2O mass fractions also reach large values on the centerline very close to the jet exit. The poor 

prediction for the flame location is likely a result of the lack of turbulence-chemistry interaction in these 

calculations. Since the purpose of this work is to demonstrate the capability of the radiation solver with 

input from NCC, this combustion simulation is still relevant. 

For the cartesian/cylindrical radiation solver, r and z are taken from the NCC grid and, although the 

problem is axisymmetric, the azimuthal distance ߮ again goes from ߮ = 0 to 2ߨ. The radiation solver 

spatial grid is taken as 20߮ ൈ 120ݎ ൈ 100ݖ and the directional grid is taken as 8ߠ ൈ 20߶ control angles. 

The local transformed absorption coefficents ݇∗ are calculated from the local values of temperature and 

mole fractions of the two participating species (CO2 and H2O) as obtained from the NCC solution. As in 

the above case we again solve the RTE for each of 10 Gaussian quadrature points and obtain the final 

distributions of ݍ	௥, ݍ௭	, ܩ	ܽ݊݀	݀݅ݒሺݍሻ by Gaussian quadrature. The resulting distributions are shown in 

Figure 4. Note the reference values are as defined for the Hot Gas Cylinder above with ܶ௚ set to 2000 K 

and rmax replaced by the jet diameter d = 0.0072 m. Since there are no solid walls, the temperatures on 

the domain boundaries are taken from the values at the adjacent grid cell and the emissivity on these 

boundaries is taken as 1.0. As a result Figure 4(a) and (b) shows that the radial and axial heat fluxes are 

relatively uninteresting. Figure 4(c) shows the radiant intensity increasing with z as the temperature 

increases. Figure 4(d) shows the radiative heat source largest in the interior of the flame along the 

centerline. 

Finally it should be noted that for this case the computer cost for the two codes is comparable. In 

particular, on an HP Z400 workstation, the time per iteration per cell was 1.810–4 sec for NCC while for 

the cartesian/cylindrical radiation code this time was 1.510–4 sec per spatial cell for a single solution of 

the RTE. Since in general it takes far fewer iterations to converge the RTE than it does for NCC, it would 

be quite practical to couple the two codes by running the radiation code every 10 to 100 iterations of NCC 

and then updating the radiation source term in the energy equation of NCC. This would allow the 

temperature distribution to adjust to the presence of the radiation coming from the flame. 

Concluding Remarks 

This paper presents a methodology for converting an existing finite volume radiative heat transfer 

method with input absorption coefficients to one that can treat a mixture of combustion gases and 

compute the coefficients on the fly from the local mixture properties. The Full- spectrum k-distribution 

method is used to transform the radiative transfer equation (RTE) to an alternate wave number variable, 

݃. A Spectral Radiation Calculation Software package is used to calculate the transformed absorption 

coefficients at discrete values of temperatures and participating species mole fractions that span the values 

of the problem for each value of ݃. These results are stored in a table and interpolation is used to obtain 

cell centered values of the coefficients for every cell in the field. Finally, the transformed RTE is solved 
NASA/TM—2015-218815 7 

for each ݃ and Gaussian quadrature is used to find the radiant heat transfer, mean radiant intensity and 

radiant heat source throughout the field. 

In the present work the methodology is implemented in an existing cartesian/cylindrical grid radiation 

code and local mixture properties are obtained from a solution of the NCC on the same grid. The method 

is readily extendable to unstructured grids and provides the foundation for coupling radiation heat transfer 

with the flow solver in NCC. 

References 

1. Stubbs, R.M. and Liu, N.-S., (1997), “Preview of the National Combustion Code,” AIAA-97-3114, 

33rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 6-9, Seattle, WA, USA. 

2. Quealy, A., Ryder, R., Norris, A. and Liu, N.-S., (2000), “National Combustion Code: Parallel 

Implementation and Performance,” NASA/TM—2000-209801. 

3. Quealy, A., (2002), “National Combustion Code Parallel Performance Enhancements,” 

NASA/CR—2002-211340. 

4. Shih, T.-H., Povinelli, L.A. and Liu N.-S., (2003), “Application of Generalized Wall Function for 

Complex Turbulent Flows,” Journal of Turbulence, Vol. 4, No. 15, pp 1-16. 

5. Chien, K.Y., (1982), “Prediction of Boundary Layer Flows with a Low-Reynolds-Number 

Turbulence Model,” AIAA J., Vol. 20, No.1, pp 33-38. 

6. Shih, T.-H., Chen, K.-H. and Liu, N.-S., (1998), “A Non-Linear k-epsilon Model for Turbulent 

Shear Flows,” AIAA-98-3983, 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & 

Exhibit, July 13-15, Cleveland, OH, USA. 

7. Raju, M.S., (2004), “LSPRAY-II: A Lagrangian Spray Module,” NASA/CR—2004-212958. 

8. Choi, Y.-H. and Merkle, C. L., (1993), “The Application of Preconditioning in Viscous Flows,” 

J. Comput. Phys., Vol. 105, pp 207-223. 

9. Chen, K.-H., Norris, A.T., Quealy, A. and Liu, N.-S., (1998), “Benchmark Test Cases for The 

National Combustion Code,” AIAA-98-3855, 34th AIAA/ASME/SAE/ASEE Joint Propulsion 

Conference & Exhibit, July 13-15, Cleveland, OH, USA. 

10. Iannetti, A., Tacina, R., Jeng, S.-M. and Cai, J., (2001), “Towards Accurate Prediction of Turbulent, 

Three-Dimensional, Recirculating Flows with the NCC,” NASA/TM—2001-210761, AIAA-2001-

0809. 

11. Shih, T.-H., Norris, A., Iannetti, A.C., Marek, C.J., Smith, T.D., Liu, N.-S. and Povinelli, L.A., 

(2001), “A Study of Hydrogen/Air Combustor Using NCC,” AIAA-2001-808, 39th Aerospace 

Sciences Meeting & Exhibit, Jan. 8-11, Reno, NV, USA. 

12. Iannetti, A.C. and Chen, K.-H., (2000), “An Initial Comparison of National Combustion Code 

Simulations Using Various Chemistry Modules with Experimental Gas Turbine Combustor Data,” 

AIAA-2000-0330, 38th Aerospace Sciences Meeting, Reno, NV, USA. 

13. Chai, J.C., Lee, H.S. and Patankar, S.V., (1994), “Finite Volume Method for Radiation Heat 

Transfer,” Journal of Thermophysics and Heat Transfer, Vol. 8, No. 3, pp 419-425. 

14. Moder, J.P., Kumar, G.N. and Chai, J.C., (2000), “An Unstructured-Grid Radiative Heat Transfer 

Module for the National Combustion Code,” AIAA 2000-0453, 38th Aerospace Sciences Meeting, 

Reno, NV, USA. 

15. Modest, M.F., (2003), “Narrow-Band and Full-Spectrum ݇-Distributions for Radiative Heat Transfer 

– Correlated-݇ vs. Scaling Approximation,” Journal of Quantitative Spectroscopy & Radiative 

Transfer, Vol. 76, pp 69-83. 

16. Modest, M.F., (2008), “Spectral Radiation Calculation Software,” copyrighted package provided by 

the copywriter, Michael F. Modest. 

17. Modest, M.F. and Riazzi, R.J., (2005), “Assembly of Full-Spectrum k-Distributions from a 

Narrow-Band Database; Effects of Mixing Gases, Gases and Nongray Absorbing Particles, and 

Mixtures with Nongray Scatterers in Nongray Enclosures,” Journal of Quantitative Spectroscopy & 

Radiative Transfer, Vol. 90, pp 169-189. 
NASA/TM—2015-218815 8 

18. Wang, L. and Modest, M.F., (2005), “Narrow-Band Based Multiscale Full-Spectrum k-Distribution 

Method for Radiative Transfer in Inhomogeneous Gas Mixtures,” Journal of Heat Transfer, 

Vol. 127, pp 740-748. 

19. Pal, G., Modest, M.F. and Wang, L., (2008), “Hybrid Full-Spectrum Correlated k-Distribution 

Method for Radiative Transfer in Nonhomogeneous Gas Mixtures,” Journal of Heat Transfer, 

Vol. 130, pp 082701-082708. 

20. Pal, G. and Modest, M.F., (2009), “A Multiscale Full-Spectrum k-Distribution Method for Radiative 

Transfer in Nonhomogeneous Gas-Soot Mixtures with Wall Emission,” An International Journal of 

Computational Thermal Sciences, Vol. 1, pp 1-22. 

21. Pal, G. and Modest, M. F., (2010), “A Narrow Band-Based Multiscale Multigroup Full-Spectrum 

k-Distribution Method for Radiative Transfer in Nonhomogeneous Gas-Soot Mixtures,” Journal of 

Heat Transfer, Vol. 132, pp 0233071-0233079. 

22. Viskanta, R. and Menguc, M.P., (1987), “Radiation Heat Transfer in Combustion Systems,” 

Prog. Energy Combust. Sci., Vol. 13, pp 97-160. 

23. Barlow, R. and Frank, J., (2007), “Piloted CH4/Air Flames C, D, E, and F – Release 2.1,” 

www.sandia.gov/TNF/DataArch/FlameD/SandiaPilotDoc21.pdf. 

24. Schneider, Ch., Dreizler, A., Janicka, J. and Hassel, E. P., (2003), “Flow Fie-ld Measurements of 

Stable and Locally Extinguishing Hydrocarben-Fuelled Jet Flames,” Combustion and Flame, 

Vol. 135, pp 185-190. 
NASA/TM—2015-218815 9 

Figure 1.—Hot gas cylinder with cold walls: (a) radial heat flux, (b) axial heat flux, (c) 

mean radiant intensity and (d) heat flux divergence, optically thick case, ݎ݉ܽݔ ൌ 1.0݉.
NASA/TM—2015-218815 10 

Figure 2.—Hot gas cylinder with cold walls: (a) radial heat flux, (b) axial heat flux, 

(c) mean radiant intensity and (d) heat flux divergence, optically thin case, 

0.01݉. ൌ ݔ݉ܽݎ
NASA/TM—2015-218815 11 

Figure 3.—Sandia Flame D—NCC results: (a) temperature distribution, (b) CO2 mass 

fraction distribution, (c) H2O mass fraction distribution, (d) CH4 mass fraction distribution. 
NASA/TM—2015-218815 12 

Figure 4.—Sandia Flame D—radiation results: (a) radial heat flux, (b) axial heat flux, (c) mean radiant 

intensity and (d) heat flux divergence.

z/d

r/d

0 10 20 30 40 50

-0.15 -0.1 -0.05 -0.02 0 0.02 0.05 0.1 0.2

(a) qr/qref

z/d

r/d

0 10 20 30 40 50

-0.4 -0.2 0 0.2 0.4 0.6 0.8 (b) qz/qref

z/d

r/d

0 10 20 30 40 50

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

(c) G/Gref

z/d

r/d

0 10 20 30 40 50

-0.002 0.002 0.006 0.01 0.014 0.018 (d) div(q)/dvref