Combustion Theory and Modelling - Chemical Engineering

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A quantitative method for a priori evaluation of combustion reaction models

James C. Sutherland a; Philip J. Smith a; Jacqueline H. Chen b a Department of Chemical & Fuels Engineering, University of Utah, Salt Lake City, UT, USA b Combustion Research Facility, Sandia National Laboratories, Livermore, CA, USA Online Publication Date: 01 April 2007 To cite this Article: Sutherland, James C., Smith, Philip J. and Chen, Jacqueline H. (2007) 'A quantitative method for a priori evaluation of combustion reaction models', Combustion Theory and Modelling, 11:2, 287 - 303 To link to this article: DOI: 10.1080/13647830600936969 URL: http://dx.doi.org/10.1080/13647830600936969

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Combustion Theory and Modelling Vol. 11, No. 2, April 2007, 287–303

A quantitative method for a priori evaluation of combustion reaction models JAMES C. SUTHERLAND∗ †, PHILIP J. SMITH† and JACQUELINE H. CHEN‡ †Department of Chemical & Fuels Engineering, University of Utah, Salt Lake City, UT, 84098, USA ‡Combustion Research Facility, Sandia National Laboratories, Livermore, CA, 94551, USA (Received 12 May 2006; in final form 18 July 2006) In recent years, direct numerical simulations have been used increasingly to evaluate the validity and performance of combustion reaction models. This study presents a new, quantitative method to determine the ideal model performance attainable by a given parameterization of the state variables. Data from direct numerical simulation (DNS) of unsteady CO/H2 –air jet flames is analysed to determine how well various parameterizations represent the data, and how well specific models based on those parameterizations perform. Results show that the equilibrium model performs poorly relative to an ideal model parameterized by the mixture fraction. The steady laminar flamelet model performs quite well relative to an ideal model parameterized by mixture fraction and dissipation rate in some cases. However, at low dissipation rates or at dissipation rates exceeding the steady extinction limit, the steady flamelet model performs poorly. Interestingly, even in many cases where the steady flamelet model fails (particularly at low dissipation rate), the DNS data suggests that the state may be parameterized well by the mixture fraction and dissipation rate. A progress variable based on the CO2 mass fraction is proposed, together with a new model based on the CO2 progress variable. This model performs nearly ideally, and demonstrates the ability to capture extinction with remarkable accuracy for the CO/H2 flames considered. Keywords: Combustion modelling; direct numerical simulation; manifold; progress variable

1. Introduction The primary challenge in turbulent combustion lies in the large range of time scales that are coupled through interactions between thermochemistry and fluid dynamics. In some problems, however, the fluid-dynamics length and time scales overlap with only a subset of the thermochemical time scales. Thus, some degree of decoupling may be possible. Indeed, many combustion models rely on the assumption that many chemical timescales are significantly faster than the fluid-dynamic scales of interest and can be decoupled. This class of models represent the entire thermochemical state by a very small set of parameters which we call reaction variables. Accordingly, these models are only valid when the thermochemical state for the system being modelled is well approximated by a manifold in the lower dimensional space defined by the reaction variables [1]. Then, only the controlling variables for

∗ Corresponding

author. E-mail: [email protected]

Combustion Theory and Modelling c 2007 Taylor & Francis ISSN: 1364-7830 (print), 1741-3559 (online) http://www.tandf.co.uk/journals DOI: 10.1080/13647830600936969


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J. C. Sutherland et al.

the chemical processes which couple with the fluid dynamics need to be transported, and all other thermochemical variables may be represented in terms of these controlling reaction variables. Given a single-phase reacting system with Ns species, one must specify Ns +1 variables (e.g., Ns −1 mass fractions, temperature and pressure) to uniquely specify the entire thermochemical state, φ, of the system [2–4]. Reaction models parameterize φ by Nη Ns + 1 parameters called reaction variables, which we represent collectively as η. A reaction model then provides a unique mapping from η to φ, i.e., each φi is represented by an Nη -dimensional surface in η-space. In general, however, the thermochemical state of the underlying physical system is (Ns +1)-dimensional, and a unique surface does not necessarily exist in the lower-dimensional space parameterized by η. This raises two questions: (i) How well can a given choice of η parameterize the thermochemical state, i.e., what is the ideal performance attainable by a model parameterized by a specific choice of η? (ii) How well does a particular reaction model parameterized by η perform relative to an ideal model parameterized by η? Obviously, the answer to these questions is problem dependent. The goal of this paper is to provide a method to quantitatively address both of these questions using Direct Numerical Simulation (DNS) or experimental data. Previous work has proposed a method to determine the error incurred in parameterizing φ by a given set of η based on a correlation analysis [5]. This study adopts a slightly different approach and, additionally, compares actual model performance to the theoretical performance achievable by a given parameterization. In a Reynolds-Averaged Navier–Stokes (RANS) or Large-Eddy Simulation (LES) calculation, mean or filtered state state variables are needed as functions of the mean (or filtered) reaction variables. This is typically achieved through φ¯ i =

η Nη

···

η1

φi∗ (η) P η1 · · · η Nη dη1 · · · dη Nη ,

(1)

where φi∗ (η) is the reaction model, which provides state variables, φ, as unique functions of the set of reaction variables, η, and P(η1 · · · η Nη ) is the joint probability density function (PDF) of all reaction variables. Clearly, in a RANS or LES computation, the error in φ¯ has contributions from the reaction model as well as the model used to approximate the joint PDF. This study examines only the error due to the reaction model, φi∗ (η). Many reaction models have been proposed for non-premixed combustion modelling in the literature (see [6] for a recent review), and there have been several evaluations of such models using data from DNS using simple [7–13] and more complex [14] chemistry. One thing that has been absent from these studies is a quantification of the error in a given model. This study provides a framework for evaluating reaction models using data from direct numerical simulation (DNS) or appropriate experiments.

2. Reaction models Before describing the DNS calculations and the method for evaluating various models, we first describe the models that will be evaluated.



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2.1 The mixture fraction As many of the results will use the mixture fraction as a reaction variable, it is briefly defined here. Elemental mass fractions, Z , may be defined in terms of the species mass fractions, Yi , as Z =

Ns a,i W i=1

Wi

Yi ,

(2)

where Ns is the number of species, a,i is the number of atoms of element in species i, W is the molecular weight of element , and Wi is the molecular weight of species i. The mixture fraction, f , may be written in terms of coupling functions, β, as [2] f =

β − β0 , β1 − β0

(3)

where β1 and β0 are constants evaluated in the fuel and oxidizer streams, respectively. The coupling function, β, is defined in terms of the elemental mass fractions as β=

Ne

γ Z =

=1

Ne =1

γ

Ns a,i W Yi i=1

Wi

,

(4)

where γ are weighting factors. The γ are not unique; for this study Bilger’s definition [15] of the mixture fraction is adopted, for which γC = 2/WC , γH = 1/(2WH ), γO − 1/WO , γN = 0. The stoichiometric mixture fraction, f st , is determined from equation (3) with β = 0. The dissipation rate is defined as χ = 2D∇ f · ∇ f , with D obtained from Le = λ/(ρc p D). In this study, the Lewis number of the mixture fraction was assumed to be unity. 2.2 The equilibrium model By assuming that all chemical reactions are in equilibrium (i.e. chemistry is reversible and infinitely fast), we may represent the entire thermochemical state by the mixture fraction. See [16] for a detailed discussion of chemical equilibrium. The equilibrium model accounts for variation in stoichiometry, but ignores any effect of diffusion or transient flame behaviour. The state of the system is parameterized by the mixture fraction, η = f alone. The equilibrium model will be referred to as EQ through the remainder of this document. All equilibrium calculations performed in connection with this work are based on the STANJAN program [17]. 2.3 Steady laminar flamelet model The laminar flamelet model proposed by Peters [18–20] is an example of a reaction model which accounts for stoichiometry and diffusion simultaneously. The steady laminar flamelet model (SLFM) is defined by steady-state solutions to the equations ∂Yi 1 χ ∂ 2 Yi + Wi ωi , = 2 ∂t 2 ∂f ρ n ∂T 1 χ ∂2T − h jWjωj = ∂t 2 ∂f 2 ρc p j=1

(5)

where χ is the dissipation rate, defined as χ = 2D(∇ f · ∇ f ),

(6)


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and D is a suitable diffusivity for the mixture fraction. Following Peters [18, 20], we define the dissipation rate in equations (5) as* χ ( f ) = χmax exp[−2(erf−1 [2 f − 1])2 ],

(7)

where erf−1 denotes the inverse of the error function. The SLFM model, thus, parameterizes the state variables by the mixture fraction ( f ), and the dissipation rate (χmax ). 2.4 Low-dimensional manifold models Manifold models stem from the observation that fast chemical time scales act to bring the chemical state onto a manifold whose trajectory is dictated by the slow chemical time scales. This trajectory naturally terminates at thermochemical equilibrium. Intrinsic lower-dimensional manifold (ILDM) models were originally proposed by Maas and Pope [1]. ILDM models are based on an eigenvector analysis of the reaction rate Jacobian matrix [1, 21–23]. In principle, an error tolerance may be set and the dimensionality of the ILDM may be determined such that the error tolerance is achieved. However, this leads to different manifold dimensions (i.e. a different number of reaction variables) in different areas of state space. A much simpler strategy is to set the dimensionality of the manifold a priori. While the techniques for obtaining intrinsic manifolds have been theoretically established, many other techniques are in use for generating manifolds. Most commonly, a simple reactor configuration is selected and solutions are obtained over a wide range of parameter space by varying the natural reactor parameters. The solutions are then re-parameterized by whatever choice of reaction variables seems most appropriate. In this way, the manifold variables (i.e. reaction variables) do not need to be specified a priori. For example, a batch reactor may be selected as a reactor configuration and then the two natural parameters of a batch reactor (residence time and initial reactant composition) may be varied. The solutions obtained from the batch reactor, however, may be parameterized by mixture fraction (to parameterize the initial composition) and a product concentration (to parameterize the residence time). A discussion of manifold techniques applied to batch reactors may be found in [24]. Alternatively, more complex reactor configurations such as a freely propagating premixed flame or a counterflow geometry may be used [25–28]. Opposed-jet and freely propagating premixed flame configurations naturally incorporate a diffusive-reactive balance, while ILDM methods must account for the effects of diffusion on the manifold explicitly [1, 21, 26]. Manifolds constructed from more complex reactor configurations provide parameterizations that may be useful over a wider range of state space at the expense of more computer time required to obtain solutions over a wide range of the natural parameters for the reactor. A requirement of any manifold method is that the reaction variables must parameterize the source term for any reaction variables which are not conserved. For example, if mixture fraction and YCO2 are chosen as reaction variables, then the source term ωCO2 must be parameterized by η = ( f, YCO2 ). 2.4.1 A new progress-variable model. The progress variable model proposed here is a two-variable model based on the mixture fraction and ηCO2 , a progress variable derived from the CO2 mass fraction, although the technique presented is applicable to any choice of the progress variable. The model is generated by re-parameterizing the solution to the flamelet *

Sometimes χst (dissipation rate at f st ) is used rather than χmax (which occurs at f = 0.5 in equation (7)). These are easily related through equation (7) as χmax F(0.5) = χst F( f st ).



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Figure 1. Temperature (a) and ωCO2 (b) as functions of ηCO2 (top axis) and χ0 (bottom axis) at f st .

equations, (5) by ( f, ηCO2 ). The effects of extinction may be incorporated by calculating the transient extinction process beginning with the steady solution at χmax . The solutions are then tabulated as functions of ( f, ηCO2 ), with ηCO2 defined as ηCO2 =

YCO2 − β , α−β

(8)

where α is the maximum attainable value conditioned on mixture fraction, α = max(YCO2 | f ) and β is the minimum attainable value conditioned on mixture fraction, β = min(YCO2 | f ). Note that with this definition of ηCO2 , a different definition can be considered for each model. For example, if a model were constructed where extinction were not included, then α and β would be different, and thus the definition of η would change. Figure 1 shows the model-predicted values of the temperature and CO2 reaction rate, as functions of ηCO2 and χo at f st . Figure 2 shows the model-predicted values of ωCO2 coloured by temperature for the entire range of ( f, χo ) and ( f, ηCO2 ). Steady extinction occurs at χmax = 313 s−1 , which corresponds to T = 1293 K and ηCO2 = 0.478, as can be deduced from figure 1. ηCO2 > 0.478 corresponds to the steady burning solutions, while ηCO2 < 0.478 corresponds to transient extinction. The steady extinction limit is demarcated in figure 2b. Comparing

Figure 2. Surface plots of ωCO2 as functions of f and ηCO2 (a) and ηCO2 (b). Colour scale is temperature. The solid line represents the steady extinction limit.


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the ( f, χ ) parameterization with the ( f, ηCO2 ) parameterization, it is clear that the ( f, ηCO2 ) parameterization captures the transient extinction process without difficulty. 2.4.2 Model implementation. Introduction of the parameter ηCO2 begs the question: should YCO2 or ηCO2 be transported in an application code? If ηCO2 were transported, then the source term and diffusive fluxes for ηCO2 may be difficult to obtain, particularly since they are functions of the mixture fraction through equation (8). Therefore, the transport equation for ηCO2 would involve terms such as ∂α/∂ f and ∂β/∂ f . A simpler approach may be to transport an equation for YCO2 . The source term and the remainder of the state variables (which are tabulated as a function of f and ηCO2 ) could be obtained via a procedure such as the following: (i) Given f , determine the table entry in the ηCO2 -dimension corresponding to a CO2 mass fraction which matches the value determined from the YCO2 transport equation. (ii) Obtain the full thermochemical state, φ, as well as ωCO2 , α, and β from the table. (iii) Ensure that YCO2 lies between β and α. In the case where there are many ηi , this procedure may be extended to a multidimensional root-finding problem. Algorithms such as bisection or a Newton-based method (if the Jacobian ∂φi /∂η j is available) may be used to aid in this process. In reality, the reaction model is not used alone in most CFD applications; a model accounting for the unresolved turbulent fluctuations must also be used. While this is not discussed here, a procedure similar to the one outlined above could be employed. 3. Reaction model evaluation method Consider a set of reaction variables η, by which we wish to parameterize the thermochemical state, φ, of the system. We may project a DNS dataset into η-space and determine a mean surface that the DNS data occupies by φ|η, the average value of the state variables conditioned on a given value of the reaction variables. This procedure is directly applicable to any dimensionality of η, provided that there is sufficient data to yield a statistically meaningful representation of φ in η-space, φ|η. These concepts are illustrated in figure 3. Data points representing realizations of the temperature (φ = T ) from a DNS dataset are plotted against the mixture fraction (η = f ). The thick solid line represents the conditional mean of T in mixture fraction space, T | f , while the thick dashed line represents the temperature obtained if the system were in thermochemical equilibrium and will be discussed shortly. The thin lines in figure 3, together with additional details of the DNS dataset, will be provided shortly.

Figure 3. Results from DNS case A: temperature (φ = T ) projected into mixture fraction (η = f ) space, showing the data (points) and conditional mean, φ|η (solid line). Also shown is the equilibrium solution (dotted line).



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Given the projected data in η-space (points in figure 3), and the conditional mean (thick solid line in figure 3), we may determine higher-order moments to quantify the degree to which the DNS departs from the mean surface in η-space. Specifically, the standard deviation of φ from its mean in η-space may be expressed as DNS DNS 2 σφi (η) = − φi η η , (9) φi where φi |η represents all values of the ith state variable which correspond to the given values η, and indicates an average. Note that while this measure is local in η-space, it is non-local in physical space. It is also non-unique in physical space, i.e. there may be many points in physical space which have the same η. Thus, σφi is a statistical measure of the ability of η to parameterize φ. If the η uniquely parameterize φi , then σφi defined by equation 9 is identically zero. Thus, by definition, σφi = 0 for η = (Y1 , Y2 , . . . , Y Ns −1 , T, P), since all state variables are unique functions of temperature, pressure, and composition. If, however, a set of η was chosen such that the mapping were not unique (as in figure 3), then equation (9) would be non-zero for at least some values of η. Thus, for an arbitrary choice of η, σφ as defined by equation (9) provides a measure of how well φ may be parameterized by η. From the data shown in figure 3, it is clear that η = f does not uniquely parameterize φ = T . Thus, σT is non-zero, and provides a measure of the accuracy with which T may be parameterized by f . The thin solid line in figure 3 shows σφ as a function of f . Comparing the solid lines in figure 3, we see that the data deviates from an ideal model by approximately 70 K at f = 0.43, corresponding to a 4% deviation. The dashed lines in figure 3 will be discussed next. By definition, a reaction model parameterizes φ by η; it defines a unique surface for each φi in η-space, φi∗ (η), where the asterisk denotes a model-predicted state variable. The dashed line in figure 3 represents the temperature predicted by the equilibrium model, which is a unique function of the mixture fraction for an adiabatic system. The deviation of the DNS data from the surface defined by the model may be defined as 2 DNS σφ∗i (η) = φi η − φi∗ (η) , (10) where φiDNS |η is a realization of the DNS data conditioned on a specific value set of η, and φi∗ (η) is the ith state variable as given by the model. The thin dashed line in figure 3 shows the deviation of the equilibrium-predicted temperature from the DNS temperature, σT∗ , for the equilibrium model. Several important observations can be made from equations (9) and (10): (i) σφ∗i ≥ σφi for all values of η. In other words, σφi obtained from equation (9) provides a quantitative measure of the best possible performance a given model parameterized by η can achieve relative to the DNS data. We will refer to σφi given by equation (9) as the ideal model performance, while σφ∗i given by equation (10) will be referred to as the actual model performance. (ii) The actual model performance relative to the DNS data is measured by σφ∗i from equation (10). Thus, by comparing σφi and σφ∗i , we obtain a measure of how optimally the given model performs. If σφ∗ = σφ , then φ ∗ (η) = φ|η, and the model performs ideally (though not necessarily accurately). Furthermore, if σφ∗i > σφi , this is due only to incorrect representation of the mean φi surface in η-space, i.e. φiDNS |η >= φi∗ (η). (iii) Comparing σφ∗i to σφi provides a quantitative means to determine where in η-space and how much improvement is possible for a given model parameterized by η. This method of a priori testing provides valuable insight into how well a model can perform. It also provides a way to quantify the performance of any given reaction model. This method


294


allows not only an absolute error in a given model to be determined, but (equally importantly), its performance relative to that of an ideal model. This provides direct input on whether a given model can be improved or not. What is not clear from a priori testing is how significant the shortcomings in a model are to the result of the coupled simulation. In other words, the effect of model error feedback to the simulation cannot be evaluated a priori. This issue can be fully addressed only with a posteriori testing, and is not addressed in this study. Nevertheless, the techniques presented herein can be applied directly to DNS or experimental data to determine the potential performance of a model. Further analysis utilizing a posteriori testing can then be applied. The analysis techniques described above will be applied to DNS datasets of spatiallyevolving, two-dimensional ‘turbulent’ CO/H2 /N2 –air jet flames. The next section provides a brief description of the DNS calculations which will be used in this study.

4. Numerical configuration The DNS code employed for this study solves the compressible, reacting Navier–Stokes equations using eighth-order, explicit finite-differences [29] with a fourth-order Runge–Kutta method in conjunction with a temporal error controller [30]. Explicit, tenth-order spatial filtering is performed at each time step to eliminate aliasing errors [29]. Thermal radiation, body force terms, barodiffusion, and terms involving the thermal diffusion coefficient (Soret and Dufour effects) are neglected. Mixture-averaged transport is employed, with transport coefficients calculated from the CHEMKIN TRANSPORT package[31]. The scalar boundary conditions at the inlet are constant in time. The mixture fraction profile is specified using a hyperbolic tangent function, and a steady flamelet solution (see section 2.3) is then used to specify temperature and species mass fractions. The mean velocity profile is specified as a hyperbolic tangent. Coherent velocity fluctuations conforming to the two-dimensional von Karman–Pao kinetic energy spectrum [32] are superimposed on the velocity field in the fuel stream. The spanwise and outlet boundaries are non-reflecting, with improvements to allow flames to pass through the computational boundaries [33]. Further details of the computational configurations as well as characterization of the results can be found elsewhere [34]. Two different DNS cases will be considered here, as summarized in table 1. All simulations are two-dimensional. Future investigation will consider three-dimensional turbulent jet-flame simulations with skeletal CO/H2–air chemistry [35]. 4.1 Case A The fuel stream is composed of (in mole %) 45% CO, 5% H2 , 50% N2 at 300 K, and the oxidizer is air at 300 K. These streams yield a stoichiometric mixture fraction of f st = 0.437. The kinetic mechanism employed for CO/H2 oxidation includes 12 species and 33 reactions[36, 37].

Table 1. Description of DNS cases.

Case A B

f st

ujet /ucoflow (m s−1 )

|u o | (m s−1 )

Inlet χmax

nx × ny

Lx × Ly (cm)

0.4375 0.4375

50/1 60/10

2.0 3.0

25 125

2160 × 720 2432 × 1024

12 × 4 11.88 × 5



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Figure 4. Mixture fraction (a) and dissipation rate (b) fields for case A. Stoichiometric isocontour is overlayed.

The mean jet velocity was 50 m s−1 with a co-flow velocity of 1 m s−1 . Coherent velocity fluctuations with rms |u o | ≈ 2.0 m s−1 and a turbulence integral scale of 0.3 cm are superimposed in the fuel stream at the inlet boundary. The Reynolds number based on the fuel stream properties, the jet width and jet velocity is 4600. Figure 4 shows the mixture fraction and dissipation rate fields, with the stoichiometric isocontour overlayed, for case A. Additional details may be found in [34]. 4.2 Case B Figure 5 shows the mixture fraction and dissipation rate fields, with the stoichiometric isocontour overlayed, for case B. There are two differences between case A and case B:

r The dissipation rate and shear rate at the inlet are higher in case B. While the inlet χmax for case A is 25 s−1 , the inlet χmax for case B is 125 s−1 .

r The turbulence intensity is larger in case B, with |u | ≈ 3.0 m s−1 . o

However, the different dissipation rate at the inlet also implies a different species and temperature profile through the flame zone at the inlet because SLFM is used to impose the reacting scalar profiles on the inlet boundary. The result of the changes to the inlet boundary condition is that the flame in case B is closer to extinction at the inlet. Figure 6 shows the temperature field, with the stoichiometric isocontour overlayed, and illustrates that there is significant extinction present in case B. 5. Results and discussion We now apply the concepts outlined in section 3 to the DNS data described in section 4. We will consider three different parameterizations, all of which are commonly used in combustion modelling: η = f , η = ( f, χ ), and η = ( f, CO2 ). For each of these respective parameterizations, the specific models described in sections 2.2, 2.3, and 2.4.1 are evaluated. The averages in the equations presented in section 3 are computed using a number average and using several realizations of the DNS data.

Figure 5. Mixture fraction (a) and dissipation rate (log scale, b) fields for case B. Stoichiometric isocontour is overlayed.


296


Figure 6. Temperature field for case B. Stoichiometric isocontour is overlayed.

The following procedure outlines the approach used to perform the a priori tests of reaction models. (i) (ii) (iii) (iv)

Determine all required state variables, φDNS , from the DNS data. Calculate all reaction variables, η, from the DNS data. Obtain the model-predicted state, φi∗ (η), at each η. Define discrete intervals in each ηi over which conditional means will be taken and calculate σφi and σφ∗i from equations (9) and (10), respectively.

In the remainder of this section, each DNS case described in section 4 will be discussed in connection with three modelling approaches outlined in section 2. 5.1 Case A As described in section 4.1, case A shows no signs of extinction, and the stoichiometric surface is not highly curved in most areas. 5.1.1 Models parameterized by f . Figure 3 shows the ability of a model parameterized by f alone to represent the temperature. Figure 7 shows the scaled deviations, σT /T DNS | f and σT∗ /T DNS | f , with σT and σT∗ given by equations (9) and (10) respectively. Figures 3 and 7 show that an ideal model parameterized by f alone can perform quite well, with errors in temperature of 4% (70 K) at stoichiometric conditions. The EQ model (shown by the dashed line), however, performs quite poorly, giving approximately 26% (440 K) deviation

Figure 7. Scaled temperature deviations from case A: σT /T DNS | f (solid line) and σT∗ /T DNS | f (dashed line).



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Figure 8. DNS realizations of YOH (points); YOH | f (bold solid line); EQ T ( f ) (bold dashed line); DNS σT (thin solid line); EQ σT∗ (thin dashed line).

near f st . This analysis shows that the EQ model is not a good choice to represent the temperature and that there is significant improvement possible without adding additional reaction variables. Figure 8 shows the realizations of OH mass fraction projected into f -space. Also shown are the ideal model results (thick solid line) and the EQ model prediction (thick dashed line). Interestingly, an ideal model parameterized by f alone could represent even intermediate species reasonably well, with the DNS deviating from an ideal model by approximately 10% on average at f st (i.e. σYOH /YOH | f st ≈ 0.1). However, the EQ model strongly underpredicts the OH concentration. 5.1.2 Models parameterized by ( f , χ). Figure 9 shows the results of an ( f, χ ) parameterization of temperature for an ideal model as well as the SLFM model described in section 2.3. Comparing figures 3 and 9, it is clear that addition of χ as a second parameter allows significantly better representation of the data, with maximum errors only around 3% and 9% for ideal and SLFM models respectively at f st .

Figure 9. Results of parameterizing temperature by ( f, χ ) for case A, conditioned on f st . Results for an ideal model, as well as the SLFM model are shown.


298


Figure 10. Results of an ( f, ηCO2 ) parameterization of the temperature for case A. Results are shown at f st .

5.1.3 Models parameterized by ( f , η CO2 ). Figure 10 shows the results of an ( f, ηCO2 ) parameterization for temperature. The progress variable ηCO2 is defined in section 2.4.1, and is based on the CO2 mass fraction. Note that, while the ideal model does not perform significantly better than the ideal ( f, χ ) model, the model proposed in section 2.4.1 performs nearly ideally across the entire range of ηCO2 . The SLFM model, on the other hand, deviates from an ideal ( f, χ) model at both low and high χ , as shown by figure 9. Another noteworthy observation is that the data does not occupy the entire range [0, 1] on ηCO2 . Recall from section 2.4.1 that the range of ηCO2 is defined by the model, not by the DNS data. The model in this case includes extinction as well as the equilibrium state (χ → 0). This same model will be applied to case B, where extinction is present. As discussed in section 2.4.1, the progress variable model must also parameterize the source term – in this case, the source term appearing in the transport equation for the CO2 mass fraction. Figure 11 shows the performance of the ideal and actual models for parameterizing ωCO2 , and indicates that the model proposed in section 2.4.1 performs very nearly ideally. While, the error in the CO2 source term is relatively large at high ηCO2 , the source term is approaching zero as the system approaches equilibrium. The result of errors in ωCO2 on the coupled simulation cannot be determined by a priori evaluation. Ultimately, the utility of the model must be determined via a posteriori tests.

Figure 11. Results of an ( f, ηCO2 ) parameterization of the CO2 reaction rate for case A. Results are shown at f st .



299

Figure 12. Results of parameterizing temperature by mixture fraction for case B. Results for an ideal model, as well as the equilibrium model are shown.

5.2 Case B As described in section 4.2, case B employs the same fuel and oxidizer compositions as case A. The primary difference is the turbulence intensity and strain rates, which cause significant extinction in case B. Figure 12 shows results for a model parameterized by the mixture fraction alone. Clearly, the thermochemical state is not well-parameterized by the mixture fraction alone, as would be expected for a system with more active chemical time scales. 5.3 Models parameterized by ( f , χ) Figure 13 shows the results for an ( f, χ ) parameterization of the data, along with the performance of the SLFM model. As expected, the SLFM model performs poorly above the steady extinction limit, as it predicts fully-extinguished states while the DNS allows transient extinction. More interestingly, however, is the observation that the SLFM model deviates from the ideal model more at low χ than was observed for case A. Figure 14 shows the probability of flame interaction for a single realization of the DNS dataset for case B. For the purposes of this analysis, flames are considered to be interacting when two stoichiometric isosurfaces are within a characteristic reaction zone thickness of one another. Evident in figure 14 is that at low dissipation rate there is much more likelihood of two stoichiometric surfaces interacting. This is due to flame-folding, where the dissipation

Figure 13. Results of an ( f, χ ) parameterization of temperature. Plots show results conditioned on f st .


300


Figure 14. Probability of stoichiometric isosurface interaction for one realization of the DNS data for case B.

Figure 15. Results of parameterizing temperature by ( f, ηCO2 ) for case B. Results for an ideal model, as well as the model proposed in section 2.4.1 are shown.

Figure 16. Results of parameterizing ωCO2 by ( f, χ) for case B. Results for an ideal model, as well as the SLFM model are shown.



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Figure 17. Results of parameterizing ωCO2 by ( f, ηCO2 ) for case B. Results for an ideal model, as well as the model proposed in section 2.4.1 are shown.

rate can decrease rapidly. This also explains the behaviour observed in figure 13 where, in dissipation space, the state can move rapidly from high dissipation rate to low dissipation rate with the chemistry having insufficient time to respond. 5.4 Models parameterized by ( f , η CO2 ) Figure 15 shows that ( f, ηCO2 ) parameterizes temperature extraordinarily well, with 5% maximum error at f st . Comparing with an ( f, χ ) parameterization as shown in figure 13, where even an ideal model would yield up to 40% error, it is quite remarkable that the ( f, ηCO2 ) parameterization provides 5% maximum error at f st . Additionally, the simple model proposed in section 2.4.1 performs nearly ideally over all ηCO2 . The reason that the model appears to perform better than the ideal model at low ηCO2 in figure 15 is due to numerical error caused by the discrete interval sizes used to obtain averages. In reality, σ ∗ ≥ σ as discussed in section 3. Choosing smaller discrete intervals for calculating σ and σ ∗ results in σ ∗ ≥ σ ∀ η, but also results in noisier data. Figures 16 and 17 show the parameterization of the CO2 source term at f st for the ( f, χ ) and ( f, ηCO2 ) parameterizations, respectively. Clearly, the ( f, ηCO2 ) parameterization is superior to the ( f, χ) parameterization in representing the CO2 source term. Furthermore, the model proposed in section 2.4.1 performs nearly ideally over the entire range of ηCO2 . In the region ηCO2 < 0.3, which corresponds to the transient extinction region, the ( f, ηCO2 ) parameterization begins to break down. However, given that ωCO2 is small in this region, the errors may not be terribly important.

6. Conclusions A new method was proposed for a priori model evaluation. This method defines a theoretical model performance, and allows quantification of the performance of a particular model relative to a theoretical ideal model for a given set of data. The data can be from DNS or experiment. The new technique also provides an error distribution in parameter space for a given model, which could prove useful in improving the model. In the case where an experimental dataset was available, this method can also be used to extract the ideal model from available data, if desired.


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DNS data of CO/H2 /N2 –air combustion was analysed to demonstrate the proposed technique and evaluate the performance of the equilibrium, steady laminar flamelet, and a newly proposed model. In the absence of extinction, ideal models parameterized by the mixture fraction alone can represent many state variables fairly well. However, the equilibrium model performs very poorly as a particular model parameterized by f . Using a solution from SLFM at a moderate value of χo would probably be significantly better than using the EQ model. As expected, adding the dissipation rate as a second parameter significantly increases the accuracy with which state variables can be represented. While the steady laminar flamelet model performs well relative to an ideal model ( f, χ )-parameterization over a moderate range of χ, it does not perform well at low and high dissipation rates. At high dissipation rates, this is due to the inability of SLFM to describe the transient extinction process. At low dissipation rates, however, SLFM is likely failing because it does not account for the highly transient behaviour of the flame as χ is rapidly reduced due to flame interaction. A new model parameterized by mixture fraction and a progress variable based on the CO2 mass fraction is proposed and evaluated. This two-parameter model shows great promise for capturing the thermochemical state of CO/H2 –air flames even in the presence of extinction, with very significant improvement over parameterizations based on the mixture fraction and dissipation rate. A requirement of this model is that it represent the CO2 source term with reasonable accuracy. Results show that the model performs nearly ideally across all parameter space. The model based on CO2 parameterizes the oxidation process very well for CO/H2 –air flames. However, in the case of hydrocarbon combustion, where the carbon oxidation pathway leads to many stable intermediates, a simple choice of CO2 as the progress variable may be insufficient. Linear combinations of species or more model parameters may be required in that case. Future work will focus on a posteriori evaluation of the model proposed herein, together with standard SLFM and EQ models. This will help address the open question as to the adequacy of the source term parameterization by the progress variable model.

Acknowledgements This work is supported by the Division of Chemical Sciences, Geosciences, and Biosciences, the Office of Basic Energy Sciences, the US Department of Energy. We also gratefully acknowledge the Scalable Computing Research and Development department at Sandia National Laboratories, the National Energy Research Scientific Computing Center, and the Center for Computational Sciences at Oak Ridge National Laboratory, which provided computational resources for the calculations described herein. References [1] Maas, U. and Pope, S.B., 1992, Implementation of simplified chemical kinetics based on intrinsic lowdimensional manifolds. Proceedings of the Combustion Institute, 24, 103–112. [2] Williams, F.A., 1985, Combustion Theory, 2nd edn (Cambridge: Perseus Books). [3] Felder, R.M. and Rousseau, R.W., 1986, Elementary Principles of Chemical Processes, 2nd edn (New York: John Wiley). [4] Smith, J.M., Van Ness, H.C. and Abbott, M.M., 2001, Introduction to Chemical Engineering Thermodynamics, 6th edn (Boston: McGraw-Hill). [5] Maas, U. and Thévenin, D., 1998, Correlation analysis of direct numerical simulation data of turbulent nonpremixed flames. Proceedings of the Combustion Institute, 27, 1183–1189. [6] Veynante, D. and Vervisch, L., 2002, Turbulent combustion modeling. Progress in Energy and Combustion Science, 28, 193–266.



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