Leiyong JiangInstitute for Aerospace Research,

National Research Council Canada,

1200 Montreal Road, M-10,

Ottawa, ON K1A 0RG6, Canada

e-mail: leiyong.jiang@nrc-cnrc.gc.ca

A Critical Evaluation ofTurbulence Modelingin a Model CombustorBased on the previous benchmark studies on combustion, scalar transfer, and radiationmodels, a critical evaluation of turbulence models in a propane-air diffusion flame com-bustor with interior and exterior conjugate heat transfers has been performed. Resultsobtained from six turbulence models are presented and compared in detail with a com-prehensive database obtained from a series of experimental measurements. It is foundthat the Reynolds stress model (RSM), a second moment closure, is superior over the fivepopular eddy-viscosity two-equation models. Although the main flow patterns are cap-tured by all six turbulence models, only the RSM is able to successfully predict thelengths of both recirculation zones and give fairly accurate predictions for mean velocity,temperature, CO2 and CO mole fractions, as well as turbulence kinetic energy in thecombustor chamber. In addition, the realizable k-e (Rk-e) model illustrates better per-formance than four other two-equation models and can provide comparable results tothose from the RSM for the configuration and operating conditions considered in thepresent study. [DOI: 10.1115/1.4023306]

Keywords: Reynolds stress turbulence model, eddy-viscosity turbulence models, combus-tor, flamelet combustion model

1 Introduction

Turbulence modeling is one major factor which affects the pre-cision of current numerical simulations, particularly for reactingflows. The random nature of turbulence involves a wide range oftime and length scales, and it is one of the principal unsolvedproblems in physics today [1]. Despite the rapid development ofcomputing power, direct numerical simulations of turbulent flowsremain practical only at low Reynolds numbers, while large eddysimulations are limited to benchmark cases with relatively simplegeometries [2]. This is particularly true for turbulent reactingflows. Combustion, even without turbulence, is an intrinsicallycomplex process which can involve hundreds of species and thou-sands of element reactions and cause numerical difficulties [3].For the above reasons, it is necessary to utilize turbulence modelsin numerical simulations for the development of advanced practi-cal combustion systems.

Much effort has been devoted to the development of turbulencemodels in the last five decades. Progress has been reviewed by anumber of authors, and was brought up to date for reacting flowsby Jones [4]. Various algebraic, one- and two-equation turbulencemodels were systematically evaluated by Wilcox [2] against anumber of well-documented nonreacting flows, including free-shear, boundary-layer, and separated flows. Some guidelinesregarding applications of these turbulence models were provided.Six eddy-viscosity and two variants of Reynolds stress turbulencemodels were critically assessed by Kim and Rhee [5]. In theircase, the flow field around a ship hull with strong cross-flows andstreamwise vortices was considered. It was found that the twoReynolds stress models were able to reproduce all the salientfeatures of the flow field and the predicted Reynolds stresses andturbulence kinetic energy were in good agreement with the experi-mental results. Turrell et al. [6] found that the RSM was able topredict a vortex core in a high-swirl lean premixed gas turbinecombustor, which was qualitatively supported by the experimentalobservations. In contrast, the standard k-e model failed to predictthis phenomenon.

There is a large number of publications on numerical simula-tions of practical combustion systems such as Refs. [6–9], andtremendous contributions have been made for understandingadvantages and limitations of various turbulence models. How-ever, systematic assessment and validation of turbulence modelsin combustor flow fields against well-defined comprehensiveexperimental results are rare.

To provide a benchmark database for evaluation and develop-ment of various physical models, a series of experiments has beenperformed at the National Research Council of Canada. Measure-ments were made in a propane/air diffusion flame combustorusing advanced measurement techniques [10]. The combustorgeometry was relatively simple compared with practical combus-tion systems, but fundamentally similar and pertinent to themodeling of other complex systems. A three-dimensional laserDoppler anemometer (LDA) was used to measure three velocitycomponents in the downstream region of the combustion chamber.Due to limited optical access, a two-dimensional LDA was usedto obtain axial and circumferential velocities in the upstreamregion. Gas temperatures were acquired using an uncoated,250-lm diameter, type “S” thermocouple mounted in a twin boreceramic tube. Gas species measurements were made with asampling probe connected to a Varian Model 3400 Gas Chromato-graph. The major species measured included CO, CO2, C3H8,and O2.

The present work is a continuation of the previous studies onthis combustor [11–13], where different combustion models, radi-ation models, and the effect of Prandtl/Schmidt number wereassessed against the comprehensive experimental data. It wasfound that the flamelet combustion model illustrated the bestperformance among four combustion models, i.e., the eddy-dissipation, eddy-dissipation-finite-rate, probability density func-tion, and laminar flamelet models [11]. Reference [12] revealsthat the turbulent Prandtl/Schmidt number has significant effectson the predicted temperature and species fields in the combustor.For accurate velocity and scalar field predictions, an optimizedPrandtl/Schmidt number of 0.5 is recommended for the presentconfiguration and flow conditions. As shown in Ref. [13], theeffect of radiation on the flow field is minor, particularly to thevelocity field. However, it has significant effect on the NO field.

Manuscript received February 11, 2012; final manuscript received September 11,2012; published online June 24, 2013. Assoc. Editor: Srinath V. Ekkad.

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Presented and discussed in this paper are the numerical resultsobtained from six turbulence models, i.e., the standard k-e model,realizable k-e (Rk-e) model, renormalization group k-e (RNG)model, k-x model, shear stress transport (SST) model, and RSM.Except for the RSM, these are popular turbulence models in engi-neering applications: the first three have been extensively used ininternal reacting and nonreacting flows such as [Refs. 6–9,14],while the k-x and SST models have been widely used in non-reacting flows such as aircraft, turbo-machinery, etc. such asRefs. [15–17]. By qualitative and quantitative comparisonsbetween the numerical results and experimental database, theadvantages and shortcomings of these turbulence models arerevealed.

2 Numerical Simulations

Axi-symmetric steady incompressible and turbulent reactingflows were considered in the present study, and a commercialsoftware package, Fluent, was used for all numerical simulations.The computation domain, selected physical models, boundaryconditions, and solution methods are described in the followingsub-sections.

2.1 Computational Domain. A schematic diagram of themodel combustor is shown in Fig. 1, including the fuel and airinlets, disk flame-holder, combustion chamber, steel/insulationwalls, and contracted exhaust section (all dimensions are in mm).Air entered the combustion chamber around a bluff body, whilefuel was fed through the center of the disk flame-holder.

The computational domain and mesh are shown in Fig. 2, wherefor clarity only a small portion of nodes are displayed. Since theflow field was axisymmetric, 2D quadrilateral meshes were gener-ated over the flow region (from the fuel/air inlets to the exhaustexit), the interior conjugate heat-transfer region (the flame holderbody within the inlet section) and the exterior conjugate heat-transfer regions (the combustion chamber and insulation walls).Fine grids were laid in the combustion chamber in order to prop-erly resolve the flow recirculation and reacting regions. Fine gridswere also generated in the shear layers between the recirculationregion and the fuel and air jets. Coarse grids were used in theexhaust section and solid wall regions. A total number of 74,100elements were used for all the simulations. The skewness was lessthan 0.2 in the flow-field domain and the aspect ratio was lessthan 12 for 99.5% elements. Effort was made to keep the nondi-mensional wall-distances, yþ, to the desired value of 30. A num-ber of meshes were tested to ensure the mesh independence of thenumerical solutions.

2.2 Turbulence Models. The governing Reynolds-averagedconservation equations for mass, species, momentum and energyare not reproduced here. They can be readily found in the classicliterature, such as Ref. [18].

For closure of the governing equations, Reynolds stresses,q u00i u00j , have to be modeled. Detailed description and formation ofeach turbulence model are beyond the scope of the present paper.However, the main concepts of these models are outlined below.

For the five eddy-viscosity two-equation turbulence models, theBoussinesq hypothesis is adopted to model Reynolds stresses

� q u00i u00j ¼ lt

@Ui

@xjþ @Uj

@xi

� �� 2

3dij lt

@Uk

@xkþ q k

� �(1)

With this approach, the turbulence viscosity of the k-e, Rk-e,and RNG models, lt, for high Reynolds number flows is given bythe expression

lt ¼ qClk2=e (2)

where Cl is a constant, and the values of the turbulence kineticenergy, k, and dissipation rate, e, are calculated from a pair oftransport differential equations. The detailed descriptions of thesek-e models are given in Refs. [19–21], respectively.

The turbulence viscosity of the k-x and SST models for highReynolds number flows is obtained by the following two equa-tions, respectively

lt ¼ qk=x (3)

and

lt ¼�qk

x1

max 1;SF

ax

� � (4)

where the values of the turbulence kinetic energy and the specificdissipation rate, x, are also determined from a pair of transportFig. 1 The model combustor

Fig. 2 Computational domain and mesh

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differential equations [2,22,23], F is equal to one in the near-wallregion and zero in the free shear layer, S is the strain rate magni-tude, and “a” is a constant.

The RSM is a second-moment closure. It is more “general”than the above eddy-viscosity turbulence models, where the linearrelations between Reynolds stresses and mean rate of strainand the isotropic eddy viscosity are assumed, as indicated inEqs. (1)–(4). For the RSM, all Reynolds stress components in theflow field are directly computed from their corresponding trans-port differential equations. In the present axisymmetric case, four

components of Reynolds stresses (q u00i u00j ) are resolved in the com-

bustor flow field. In principle, this model is likely to have a widerrange of applicability than the eddy-viscosity concept models. Toconvert Reynolds stress equations into a closed set of equations,unknown terms must be modeled by mean flow variables and/orReynolds stresses. Details can be found in Refs. [24,25].

2.3 Other Physical Models. Other physical models wereselected based on the previous benchmark studies [11–13]. Forcombustion modeling, the laminar flamelet model [26] was usedto simulate chemical reactions in the flow. A major advantage ofthe flamelet model over other combustion models, such as theprobability density function and eddy-dissipation, is that detailedmore realistic chemical kinetics can be incorporated into turbulentreacting flows. In the present case, the chemical reaction mecha-nism for propane-air flames used by Stahl and Warnatz [27] wasemployed. It consists of 228 chemical reactions and 31 species,including O, O2, OH, H, H2, H2O, H2O2, HO2, N2, CO, CO2, CH,CH2, CH3, CH4, CHO, CH2O, CH2CO, CH3CO, CH3CHO, C2H,C2H2, C2H3, C2H4, C2H5, C2H6, C3H6, C3H8, N*C3H7, I*C3H7,and C2HO.

To account for the radiation heat transfer between the gas mix-ture and flame-holder body and combustion chamber walls, thediscrete ordinates radiation model [28] was employed. Theabsorption coefficient of gaseous mixture was determined fromthe local species mass fractions in the mixture. For the turbulentscalar transfer modeling [12], the optimized turbulent Prandtl/Schmidt number of 0.5 was chosen for all numerical simulations.

Although the effort was made to keep yþ values around 30,there were some local regions where yþ was above or below thisvalue. To account for this, an enhanced wall boundary treatmentwas applied to all wall boundaries for the three k-e models andRSM. In this approach, a two-layer model [29] is enhanced bysmoothly blending the linear (laminar) and logarithmic (turbulent)laws-of-the-wall [30]. Consequently, it can apply to the entirenear-wall region [23].

Polynomials derived from the JANAF tables [31] were used tocalculate the specific heats of species as a function of temperature.For other thermal properties of the mixture such as molecularviscosity and thermal conductivity, the values for air at 900 Kwere used.

2.4 Boundary Conditions. The fuel (propane) mass flowrate was 16.2 g/s, the airflow rate was 550 g/s, and the correspond-ing overall equivalence ratio was 0.46. The Reynolds numberbased on the air entry velocity and flame-holder diameter was1.9� 105. An estimated turbulence intensity of 10% and hydraulicdiameters of the fuel/air inlets (8.4 mm and 36.3 mm, respectively)were used to estimate Reynolds stress components and turbulencedissipation rate at the fuel and air inlets. For both air and fuelflows, the inlet temperature was 293 K.

The exterior wall temperatures were defined based on the ex-perimental measurements and observation, as shown in Fig. 3. Aroom temperature of 293 K was assigned to the walls of the inletsection, and the upstream edges of the combustion chamber andinsulation walls. A linear temperature profile from 293 K to 405 Kwas specified along the outer boundary of the insulation wall,which was a good approximation to the experimental measure-ments. The temperature of the exterior boundary of the exit

section was set to 960 K, as an estimation of the experimentalobservation. Based on the thermal resistance and preliminaryresults, a linear temperature profile from 960 K to 780 K wasassigned to the downstream edge of the combustion chamber wall,while another linear profile from 780 K to 405 K was defined atthe downstream edge of the insulation wall. Finally, the pressureat the combustor exit was set to the atmospheric value.

2.5 Solution Methods. A segregated (pressure-based) solverwith a second-order accurate scheme was used to resolve the flowfield. At convergence, the normalized residuals of flow variableswere about or less than 10�5 in all test cases. The monitored axialvelocities at two points in the shear layer downstream of the flameholder remained unchanged at least for the first four digits. Thisensured that the flow field reached steady conditions. A 4-nodeLINUX cluster providing 64-bit, 2.6 GHz, 8-Core, and 32 GB RAMper node, was used to carry out all simulations.

3 Results and Discussion

The predicted distributions of velocity, temperature, and spe-cies inside the combustor chamber with combustion are presentedin the following sub-sections. By comparing the numerical resultswith the experimental database, the advantages and short-comingsof the five combustion models are revealed.

3.1 Velocity Distributions. The upper halves of six plots inFig. 4 show the numerical results of axial velocity contours andflow path-lines for six turbulence models, respectively, while thelower halves are the experimental data with the zero axial velocity

Fig. 3 Exterior wall temperature profiles of the computationaldomain

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lines specified. For the experimental plots, due to the limited num-ber of data points no flow path-lines are drawn. The main flowfeatures or patterns in the combustion chamber are captured by allmodels. That is, there are two recirculation zones or vorticesbehind the flame-holder. The central recirculation zone (CRZ)

induced by the fuel jet is completely confined within the annularrecirculation zone (ARZ) created by the annular air jet. Thismeans that the transportation of fuel into the main flow field isrealized by laminar and turbulent diffusion only through the ARZ.The zero axial velocity lines divide the recirculation zones intotwo parts: one with flow moving upstream and the other movingdownstream. As expected, another separated flow zone isobserved at the upper left corner of the combustion chamber.

In terms of predicting the reattachment points or lengths of theARZ and CRZ, various degrees of agreement with the experimen-tal data are observed among the six turbulence models. For thestandard k-e model as shown in the first plot of Fig. 4, both ARZand CRZ lengths are significantly under-estimated. This defi-ciency is also observed by other investigators for nonreactingflows [32,33]. The performance of the Rk-e model is superior tothe standard k-e model, where the ARZ length is correctly pre-dicted although the CRZ length is under-predicted. In the case ofthe RNG model, a slight under-prediction of the ARZ length and aconsiderable under-prediction of the CRZ length are observed. Itis a little surprising that the performance of the two k-x modelsis not as good as the Rk-e and RNG models. For the k-x model,the ARZ length is under-estimated, while the CRZ length isover-estimated. For the SST case, the ARZ length is significantlyover-estimated although the CRZ length is captured. As shown inthe last plot of Fig. 3, the RSM model demonstrates the bestperformance, where both ARZ and CRZ lengths are properlypredicted.

Based on the results of the RSM, the ratio of the circulatedmass flow rate in the ARZ versus the airflow input is 5.5%, andthe ratio of the ARZ length and the flame-holder diameter is 1.7.These parameters could be useful to experimentalists in combus-tion stability and emissions.

Figure 5 presents the axial velocity profiles along the combustorcenterline for the six turbulence models. The numerical resultsare compared with the experimental measurements where thenegative axial velocity reaches its peak value (about �10 m/s) atx� 80 mm. Superimposed in the figure are error bars representingan estimated measurement error of 4%. Large deviations areobserved in the upper stream region (10–80 mm) for the k-e andk-x models, and in the downstream region (80–360 mm) for the k-e and SST models. The centerline axial velocities are reasonablywell estimated by the RSM, Rk-e, and RNG models. Overall, theRSM illustrates the best performance among the six turbulencemodels. This is consistent with the fact that only the RSM canadequately predict both recirculation zones. Among the five two-equation models, the Rk-e results show better agreement with theexperimental data than the other models.

Displayed in Fig. 6 are the axial velocity profiles at sevencross-sections from x¼ 20 to 240 mm, including three across therecirculation zones, one near the annular stagnation point, and

Fig. 4 Axial velocity contours and flow path-lines Fig. 5 Axial velocities along the combustor centerline

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three further downstream. The predicted trends and magnitudesare close to the measurement data, except for the k-e model atx¼ 40–200 mm and the SST model at downstream sections,x� 160 mm, where significant deviations are observed. Since theSST model properly predicts the central recirculation zone asshown in Fig. 4, it illustrates best performance at the threeupstream sections. The results from the Rk-e and RNG models arecomparable with those from the RSM. The RSM shows improve-ment over the Rk-e model in the regions of x¼ 120–200 mm andR¼ 30–40 mm.

Figure 7 shows quantitative comparisons between the numeri-cal and experimental results for the turbulence kinetic energy atfour cross-sections from x¼ 60 to 240 mm. Error bars in the figurerepresent 8% of measurement accuracy. It is noted that only theRSM gives encouraging results at all sections. The RNG modelsubstantially over-predicts the turbulence kinetic energy at allsections, and this is also true for the k-e model at the upstream sec-tions. The Rk-e and k-x models show reasonable agreement withthe experimental data, except for the k-x model at sectionx¼ 100 mm. For the SST model, the turbulence kinetic energy is

reasonably well predicted in the central region, but over-predictedaway from the combustor centerline. This is consistent with thefact that the SST model is able to predict the CRZ length, but failsto predict the ARZ length as shown in Fig. 4. The cross-section ofx¼ 60 mm cuts through both recirculation zones, and the twopeaks are where the centers of the two recirculation zones arelocated. Unfortunately, none of the models captures the centralpeak.

In short, in terms of velocity flow-field prediction, the RSMis superior over the five two-equation models, and in general theRk-e model illustrates better performance than the other fourtwo-equation models.

3.2 Temperature Distributions. The temperature contourresults of the six turbulence models are shown in the upper halvesof Fig. 8. Superimposed is the stoichiometric line of the meanmixture fraction (~f ¼ 0:064), which passes through the middle ofthe high-temperature region in the combustion chamber for allcases. The flame starts at the edge of the flame-holder disk andspreads downstream around the annular recirculation zone, wherethe mixture of recirculated hot gases and fuel mixes with the freshair and burns. This agrees with the experimental observation thata carbon deposit forms at the middle of the disk edge. In compari-son with the experimental data in the lower halves, it is foundthat for the RSM and Rk-e models, the size and location of thehigh-temperature or flame region are in good agreement with theexperimental data, and the RSM results are slightly better thanthose from the Rk-e model. For the k-e and RNG models, thehigh-temperature region is under-predicted and shifted upstream.

In contrast, the high-temperature region is significantly over-predicted and shifted downstream by the k-x and SST models.Note that the optimized turbulent Prandtl/Schmidt number of 0.5is used in the simulations as stated earlier. If a default value of0.85 is applied, the predicted high-temperature region will beeven greater. This is because the turbulent Prandtl/Schmidt num-ber represents the ratio of the momentum eddy diffusivity to thescalar eddy diffusivity (heat and mass), and the greater thePrandtl/Schmidt number, the slower the turbulent scalar transpor-tation in comparison with the momentum transfer [12].

The predicted temperature profiles along the combustor center-line are quantitatively compared with the experimental results in

Fig. 6 Axial velocity profiles at cross-sections, x 5 20–240 mm

Fig. 7 Turbulence kinetic energy profiles at sections,x 5 60–240 mm

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Fig. 9, where the measurement error is about 5%. Along the com-bustor centerline (50–350 mm), the predicted trends are close tothe experimental values although the numerical results showpeaks in the middle portion, while the measurements tend to be

flat. In general, the RSM and Rk-e models show better agreementthan other four models. For the k-e and RNG models, the tempera-ture is over-estimated in the upstream region and under-estimatedin the downstream region. In contrast to this, it is under-predictedin the upstream and significantly over-predicted in the down-stream for the k-x and SST models.

In Figs. 8 and 9, it is found that the predicted temperatures arehigher than the measured values in the center region from x¼ 140to 250 mm. The maximum difference is about 150 K. Three possi-ble reasons are expected. First, the temperature was measured bya 0.25 mm diameter thermocouple, as mentioned earlier. Owing tothe radiation and conduction losses from the thermocouple, themeasurement error could exceed 100 K over regions where thegas temperature was high and the flow velocity was low [34].Second, the intrusion of the temperature probe could alter localflow structure, thereby enhancing local turbulent mixing, andresulting in an increase in local temperature [34]. The third possi-ble reason is that the turbulence kinetic energy (Fig. 7) is notaccurately predicted. As a result, the combustion process and tem-perature prediction could be affected.

Figure 10 gives the temperature profiles at seven cross-sectionsfrom x¼ 52 to 353 mm for six turbulence models. The numericalresults agree reasonably well with the experimental results for theRSM and Rk-e models, except for the most upstream section andthe region near the combustor wall. It is also true for the RNGmodel at sections x¼ 82–293 mm. Poor agreement is observed forthe k-e model at upstream sections, x¼ 52–112 mm, the k-xmodel at sections, x¼ 82, 232–353 mm, and the SST model atmost sections.

In brief, similar to the trends for velocity prediction, the tem-perature distributions obtained from the RSM and Rk-e modelsare comparable and consistent with the experimental data ingeneral.

3.3 Species Distributions. As mentioned earlier, the detailedchemical mechanism with 31 spices is employed in the presentstudy, where the intermediate species, reactions and dissociationeffect are considered. Therefore, the results are more relevant andprovide insight into the flow field than those from a simplifiedone-equation combustion model. Figure 11 presents the CO2 molefraction profiles at five cross-sections, x¼ 21–171 mm, for six tur-bulence models, and compares them with the experimental data.For species measurements, the estimated error is about 5%, asindicated by error bars in the figure. To show the main features ofchemical reactions, the locations of the five cross-sections wereselected as: two across the central recirculation zone, two passingthrough the annular recirculation zone, and the last one behind therecirculation region (see Fig. 4).

Fig. 8 Temperature contours and flow path-lines

Fig. 9 Temperature profiles along the combustor centerline

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Carbon dioxide is one of the final major products of propane-aircombustion. As shown in Fig. 11, except for the central portion atx¼ 111 mm and the middle portion at x¼ 81 mm, the Rk-e andRSM results agree fairly well with the measured data. Poor per-formance is observed for the k-e model at x¼ 51 mm, k-x modelat x¼ 81 and 171 mm, and SST model at x¼ 171 mm.

The radial profiles of CO mole fraction, one major immediatespecies in hydrocarbon fuel combustion, are shown in Fig. 12, andare quantitatively compared with the experimental results. Likethe CO2 case, the CO profile at the most upstream section is prop-erly predicted by all models, except for the SST model which hada small bump at r¼ 38 mm. For the RSM and Rk-e models, theresults agree fairly with the experimental data at x¼ 51 and171 mm. At the other two sections, x¼ 81 and 111 mm, it is inter-esting to notice that the CO mole fraction is over-estimated in thecentral region, while the CO2 is under-estimated by these twomodels. However, the sum of CO2 and CO of the numericalresults is close to the sum of the measured CO2 and CO. Thisimplies that the predicted oxidization of CO at these two sectionsis slower than the reality. Again poor performance is observed for

Fig. 10 Temperature profiles at cross-sections, x 5 52–353 mm

Fig. 11 CO2 profiles at cross-sections, x 5 21–291 mm

Fig. 12 CO profiles at cross-sections, x 5 21–291 mm

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the k-e model at x¼ 51 mm, k-x model at x¼ 171 mm, and SSTmodel at x¼ 111–171 mm.

In general, the species predictions are encouraging. Except forsome local regions, the Rk-e and RSM results are consistent withthe measured data, and show better overall performance than theother models.

From the above qualitative and quantitative comparisons ofvelocity, temperature and species distributions inside the combus-tor between the numerical and experimental results, it is clear thatthe second-moment closure model, RSM, is superior over theeddy-viscosity models. This is consistent with the observations byother authors, such as Kim and Rhee [5] for a nonreacting flowand Turrell et al. [6] for a reacting flow as mentioned earlier.Among the five two-equation models, the Rk-e model displaysbetter performance than others. Application of the Rk-e model tonumerical simulations of practical gas turbine combustors, insteadof the RSM, may eliminate some difficulties, such as time-consuming and numerical stability issues.

The SST model has been successfully applied to many non-reacting flows, such as adverse pressure gradient, backward-facing step, and NACA 4412 airfoil flows [22]. This may explainwhy it is able to properly predict the central recirculation zone asshown in Figs. 4, 6, and 7, since the temperature in this region islow as displayed in Fig. 8. However, it significantly over-predictsthe annular recirculation zone and high-temperature region in thecombustor. The fact that the predicted flame region from the SSTmodel is significantly larger than that obtained from the Rk-emodel is also observed for the simulations of a practical gasturbine combustor [35]. The reasons may be two fold. First, thevalidation cases for the SST model [2,22] are isothermal or almostisothermal flows and the effects of combustion and significantthermal expansion may not be properly modeled. Second, the flowwith multirecirculation zones is a typical phenomenon in combus-tion systems, which is more complicated than the flow field with asingle backward-facing step. In particular, for the present case, thecentral recirculation zone is completed confined by the annularrecirculation zone.

4 Conclusion

A propane-air diffusion flame combustor with the interior andexterior conjugate heat transfers was numerically investigatedwith six turbulence models, including the standard k-e, the realiz-able k-e, the renormalization group k-e, the standard k-x, the SST,and the Reynolds stress models. Also used were the laminarflamelet combustion model and the discrete ordinates radiationmodel.

Although the flow features or patterns are captured by all turbu-lence models, in terms of quantitatively predicting the velocity,temperature and species fields, various degrees of agreementwith the experimental data are observed among the six turbulencemodels. The RSM model illustrates the best performance, and it isthe only one that can correctly predict the lengths of both recircu-lation zones and give reasonable prediction on the turbulencekinetic energy distribution in the combustor. Although the RSMmodel uses more computing time than other RANS models, this isnot a hurdle for current affordable parallel computing clusters.

The present study also indicates that the performance of the Rk-e model is better than other four two-equation models, and canprovide similar predictions as those from the RSM for the presentconfiguration and operating conditions considered. Further studieson the k-x and SST models are planned to check the above find-ings, including simulations with yþ value¼ 1 and at isothermalconditions.

Acknowledgment

The author is grateful to Dr. Ian Campbell for his permission touse the valuable experimental database for this benchmarkingwork.

Nomenclature

a ¼ constantCl ¼ constant

f or ~f ¼ mean mixture fractionk ¼ turbulence kinetic energy, (J or m2/s2 per unit mass)r ¼ radial coordinate, (mm or m)S ¼ strain rate magnitude, (/s)T ¼ temperature, (K)U ¼ mean axial velocity, (m/s)Ui ¼ ith mean velocity component, (m/s)ui00 ¼ ith fluctuating velocity component, (m/s)xi ¼ ith Cartesian coordinate, (mm or m)x ¼ coordinate along the combustor axis of symmetry,

(mm or m)y ¼ distance to the wall boundary, (mm or m)

yþ ¼ nondimensional distance of the first node away from a

wall,ffiffiffiffiffiffiffiffiffiffiffiffisw=qf

qy=t

sw ¼ wall shear stress, (kg/s2/m)� ¼ kinematic viscosity, (m2/s)e ¼ turbulence dissipation rate per unit mass, (m2/s3)

x ¼ specific turbulence dissipation rate per unit mass, (/s)q ¼ density, (kg/m3)

q u00i u00j ¼ Reynolds stresses, (kg/m/s2)lt ¼ turbulence viscosity, (N s/m2)dij ¼ Kronecker delta function (dij¼ 1 if i¼ j; dij¼ 0 if i= j)

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