Versteeg H., Malalasekra W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method

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12.30 AN EXAMPLE 407

well predicted. The mean radial velocity measurements, which are much

smaller in magnitude, are also well reproduced at all near-ﬁeld locations.

There is some deviation at locations further downstream (not shown here).

The mean radial velocity measurements show some scatter, which may be

attributed to the inaccuracies in the radial component measurements, as

discussed in Dally et al. (1998b).

Figure 12.19 shows the radial mixture fraction proﬁles at four axial locations.

The mixture fraction proﬁles are in excellent agreement with measurements

in the near ﬁeld (x/D < 0.9). Here D is the bluff-body diameter. Further

Figure 12.19 Mean mixture fraction proﬁles:

•

, measurements; , calculations

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downstream (x/D > 1.3), the mixture fraction near the axis is underpredicted

(not all the results are shown here). This error can be attributed to the known

overestimation of the spreading rate of the round jet by the k–

turbulence

model as explained earlier. Clearly, the decay rate is not well represented

even after modifying the constant C

. Radial proﬁles of the mixture fraction

variance are shown in Figure 12.20. Mixture fraction variance is slightly

overpredicted in the near ﬁeld (x/D < 1.3). However, the location of the peak

in the near ﬁeld is well reproduced.

408 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.20 Mean mixture fraction variance proﬁles:

•

, measurements; , calculations

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12.30 AN EXAMPLE 409

Radial temperature proﬁles are shown in Figure 12.21. The mean tem-

perature distribution is fairly well predicted by the laminar ﬂamelet model.

Further downstream results show a slight overprediction. It is worth men-

tioning that a signiﬁcant overprediction at x/D = 0.26 was also observed by

Dally et al. (1998a). They noted that this might not be due to shortcomings

of the simulation, but caused by averaging effects in the temperature meas-

urements as a result of the intermittency in the ﬂame at these locations.

Figure 12.21 Mean temperature proﬁles:

•

, measurements; , calculations

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Radial distributions of the mass fractions of major species CO

and

O are shown in Figures 12.22 and 12.23 respectively. Again reasonable

agreement with the experimental data is obtained. As mentioned earlier, the

inclusion of detailed chemistry in the laminar ﬂamelet calculation allows us

to predict minor species such as OH and CO. Radial proﬁles of the mass frac-

tion of OH, which is formed by dissociation in the high-temperature region

of the ﬂame, are shown in Figure 12.24. The experimental data show that the

reaction zone of OH is thin in the near ﬁeld. The thickness of the reaction

zone increases further downstream. This illustrates that the ﬂamelet model

410 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.22 Mean CO

mass fraction proﬁles:

•

, measurements; , calculations

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12.30 AN EXAMPLE 411

is able to capture the general trends of increasing thickness of the OH reac-

tion zone and the rate of decay of OH in the downstream direction.

Prediction of NO

We have also applied the thermal NO prediction procedure explained in

section 12.29. Figure 12.25 shows the NO source terms N

/4 as a function

of mixture fraction for different stretch conditions. The ﬁgure shows that the

source term is very sensitive to the scalar dissipation rate. At

= 0.064 s

−1

Figure 12.23 Mean H

O mass fraction proﬁles:

•

, measurements; , calculations

ANIN_C12.qxd 29/12/2006 04:44PM Page 411

the source term is negative in the fuel-rich zone (

> 0.055), indicating con-

sumption of NO. At

= 0.428 s

−1

, the negative zone of the source term

almost vanishes, but the peak value remains constant. At scalar dissipation

rates higher than

= 0.428 s

−1

, the decrease of temperature is more promin-

ent and this reduces the source term rapidly. As the scalar dissipation rate

increases further to

= 77.01 s

−1

the formation of NO has almost completely

ceased.

412 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.24 Mean OH mass fraction proﬁles:

•

, measurements; , calculations

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12.30 AN EXAMPLE 413

The comparison between the predictions and the measurements for the

radial mass fraction of NO is shown in Figure 12.26. The ﬂamelet model

seems to underpredict the NO proﬁles, but it should be noted that we have

only considered thermal NO formed through the Zel’dovich mechanism.

Even though Chen and Chang (1996) have shown that the Zel’dovich mech-

anism is the dominant pathway for the production of NO in a turbulent

jet ﬂame, it appears that calculations conducted with the thermal

NO mechanism alone do not give correct NO levels in this ﬂame. This is

further illustrated in this example by considering a more elaborate NO

mechanism. Shown by dashed lines in Figure 12.26 is a repeat of the NO

calculation using the GRI 2.11 mechanism, which consists of 49 species and

279 reactions. This mechanism includes reactions to represent both thermal

and prompt NO

chemistry. It can be seen that the dashed lines are in better

agreement with the experimental data than the thermal-only predictions

(solid lines). The trend is also well predicted. There is still some degree of

over-prediction further away from the burner exit. This could be due to dis-

crepancies in temperature predictions seen earlier. Given that the NO mass

fractions are very small, the degree of agreement from the GRI 2.11 predic-

tions seen here could considered to be very good (see Murthy et al., 2006, for

further details).

In summary, we have illustrated how the laminar ﬂame model can be used

to predict important ﬂame properties such as temperature, major species,

minor species and NO. The predictions were compared against reported

experimental data. In the laminar ﬂamelet model, the mean temperature,

density and composition in the turbulent ﬁeld are obtained by appropriate

averaging of the ﬂamelets. The concentration of NO, however, is calculated

by solving its own transport equation with a source term obtained from the

ﬂamelet library. The ﬂow ﬁeld is reasonably well reproduced in the calcula-

tion. The temperature and concentration of major and minor species are

also well reproduced by the ﬂamelet model with unity Lewis number. The

prediction of NO is not as good as other species, and it is suggested that

inclusion of other NO production mechanisms can improve NO predictions.

Figure 12.25 Flamelet proﬁles

of the NO source term for

different scalar dissipation rates

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Further details of this example and the effects of using differential diffusion

(non-unity Lewis numbers) can be found in Hossain and Malalasekera

(2003) and Hossain (1999). The quality of the results presented in this exam-

ple is very similar to that of Kim and Huh (2002), who used the conditional

moment closure (CMC) method. Overall, laminar ﬂamelet model predic-

tions are very good, and the ability to incorporate detailed chemistry at an

acceptable cost allows us to predict minor species and pollutants, which is a

major advantage over simpler models such as SCRC.

414 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.26 Mean NO mass fraction proﬁles:

•

, measurements; , predictions (thermal NO only), ,

predictions including prompt NO

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12.32 MODELLING OF PREMIXED COMBUSTION 415

There are several other modelling concepts available for non-premixed

combustion modelling. Among them, the conditional moment closure (CMC)

model, pdf transport models and ﬂame surface density models have been

demonstrated to be successful in predicting turbulent combustion. It is

beyond the scope of this chapter to describe these models in detail. The

interested reader should consult the relevant literature. For the CMC model,

Bilger (1993), Smith et al. (1992), Klimenko and Bilger (1999) and Kim and

Huh (2002); for pdf transport models, Pope (1985, 1990, 1991) and Dopazo

(1994); and for the ﬂame surface density models, Marble and Broadwell (1977),

Blunsdon et al. (1996), Beeri et al. (1996) and Veynante and Vervisch (2002)

provide details and example applications. There is also growing interest in

large eddy simulation (LES) models for turbulent combustion. Details can

be found in Poinsot and Veynante, (2005), DesJardin and Frankel (1998),

Cook and Riley (1998), Branley and Jones (2001) and Selle et al. (2004).

As mentioned in the introduction to this chapter, fuel and air are mixed prior

to combustion in premixed combustion. The strength of the mixture may be

expressed by the equivalence ratio. During combustion in a premixed ﬂame,

the ﬂame front propagates at a certain speed and leaves burnt products

behind the ﬂame front. In premixed combustion, laminar and turbulent

ﬂame speeds and a parameter known as the reaction progress variable are

used to formulate models. If we deﬁne T

as the temperature of unburnt gas,

as the temperature of burnt gas and T as the ﬂame temperature, then the

reaction progress variable c is deﬁned as

c = (12.203)

Sometimes the same reaction progress variable (c) is deﬁned as

c = (12.204)

where Y

, Y

and Y

are local, unburnt and burnt fuel mass fractions,

respectively. With these deﬁnitions the value of the reaction progress vari-

able is zero where the mixture is unburnt and unity where the mixture is fully

burnt. It can be shown that, like mixture fraction in non-premixed combus-

tion, the reaction progress variable c in premixed combustion is governed by

the following transport equation (see Veynante and Vervisch, 2002):

c +

c =

D + M (12.205)

Special variants of the main combustion models for diffusion ﬂames

(presumed-pdf, laminar ﬂamelet, ﬂame surface density, eddy break-up model

etc.) have been formulated for premixed combustion. As we mentioned

at the beginning of this chapter, we do not intend to describe the details of

premixed combustion models here, and the interested reader is referred,

among others, to Peters (1986), Veynante and Vervisch (2002) and Poinsot

and Veynante (2005) for further details.

∂

− Y

T − T

− T

Modelling

of premixed

combustion

12.32

Other models

for non-premixed

combustion

12.31

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At the start of this chapter we introduced some basic thermodynamic and

chemical kinetics concepts useful for the CFD combustion modeller. We

have shown in detail how the fast chemistry assumption can be used to

simplify combustion calculations, and demonstrated and its direct applica-

tion to a laminar ﬂame was demonstrated. The basics of chemical kinetics

and reaction rates were introduced, and the advantage of using reduced

mechanisms over detailed chemical mechanisms was discussed. The com-

plexities arising from turbulent non-premixed combustion modelling were

discussed, and the advantages of employing Favre-averaged equations were

explained. The probability density function approach for the calculation of

mean quantities in turbulent combusting ﬂows was discussed in detail. We

have also shown how the fast chemistry assumption can be used in conjunc-

tion with the pdf approach to calculate mean temperatures and species mass

fractions. Some other approaches, such as the eddy break-up model, equi-

librium models etc., were brieﬂy discussed, and the need to incorporate some

degree of detailed chemistry in order to be able to predict minor species was

highlighted. Finally, we introduced the concepts of the laminar ﬂamelet

model, which allows the use of detailed chemistry. We discussed in some

detail the methods for generating laminar ﬂamelet libraries and calculating

mean variables in the laminar ﬂamelet method and showed how pollutant

concentrations could be evaluated. A complete example illustrating the use

of the laminar ﬂamelet model was presented. Other models available for

turbulent combustion were mentioned, along with a brief description of the

main features of models for premixed combustion. This chapter should be

useful for novice combustion modellers wishing to gain some overall under-

standing of the basic approach used in CFD. In the interest of brevity many

intermediate details have been omitted in the presentation of various con-

cepts. The reader should follow up appropriate references and gather full

details in order to understand the ﬁner points of various models.

416 CHAPTER 12 CFD MODELLING OF COMBUSTION

Summary12.33

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