Assessing the impact of multicomponent diffusion in direct numerical simulations of premixed, high-Karlovitz, turbulent flames
Aaron J. Fillo, Jason Schlup, Guillaume Blanquart, Kyle E. Niemeyer
AAssessing the impact of multicomponent diffusion indirect numerical simulations of premixed,high-Karlovitz, turbulent flames
Aaron J. Fillo a , Jason Schlup b , Guillaume Blanquart c , Kyle E. Niemeyer a, ∗ a School of Mechanical, Industrial, and Manufacturing Engineering, Oregon StateUniversity, Corvallis, OR 97331, USA b Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, USA c Mechanical Engineering, California Institute of Technology, Pasadena, USA
Abstract
Implementing multicomponent diffusion models in numerical combustion stud-ies is computationally expensive; to reduce cost, numerical simulations com-monly use mixture-averaged diffusion treatments or simpler models. How-ever, the accuracy and appropriateness of mixture-averaged diffusion has notbeen verified for three-dimensional, turbulent, premixed flames. In this studywe evaluated the role of multicomponent mass diffusion in premixed, three-dimensional high Karlovitz-number hydrogen, n -heptane, and toluene flames,representing a range of fuel Lewis numbers. We also studied a premixed,unstable two-dimensional hydrogen flame due to the importance of diffu-sion effects in such cases. Our comparison of diffusion flux vectors revealeddifferences of 10–20 % on average between the mixture-averaged and multi-component diffusion models, and greater than 40 % in regions of high flamecurvature. Overall, however, the mixture-averaged model produces small dif- ∗ Corresponding author:
Email address: [email protected] (Kyle E. Niemeyer)
Preprint submitted to Combustion and Flame October 19, 2020 a r X i v : . [ phy s i c s . f l u - dyn ] O c t erences in diffusion flux compared with global turbulent flame statistics. Toevaluate the impact of these differences between the two models, we com-pared normalized turbulent flame speeds and conditional means of speciesmass fraction and source term. We found differences of 5–20 % in the meannormalized turbulent flame speeds, which seem to correspond to differencesof 5–10 % in the peak fuel source terms. Our results motivate further studyinto whether the mixture-averaged diffusion model is always appropriate forDNS of premixed turbulent flames. Keywords:
DNS, Turbulent flames, Diffusion, Multicomponent, Mixtureaveraged
1. Introduction
Mass, heat, and momentum diffuse simultaneously in turbulent reactingflows, affecting local transport and consumption of chemical species at smalltime and length scales [1, 2]. This coupling of turbulent mixing and heat re-lease during the combustion process can locally impact the flame’s structure,curving it and forming steep, multi-directional gradients in the temperatureand scalar fields [3]. In these regions of high flame curvature, mass diffusiontransport is most accurately represented by the multicomponent diffusionmodel, which uses a dense matrix of coupled diffusion coefficients to eval-uate the relative transport of each chemical species against the remainingspecies in the mixture [4]. The Maxwell–Stefan multicomponent diffusionmodel comes from Boltzmann’s kinetic theory of gases [4–13], and is themost rigorous model for mass diffusion in reacting-flow simulations.However, modeling full multicomponent mass diffusion transport in a di-2ect numerical simulation (DNS) can be computationally expensive, causedboth by the cost of calculating the diffusion coefficients and the memoryrequired to store the multicomponent diffusion coefficient matrix at every lo-cation [14]. As a result, researchers typically use simplified diffusion modelsto reduce the computational costs associated with calculating the diffusioncoefficients [15, 16]. These include, in order of increasing complexity and ac-curacy, the unity Lewis number, constant non-unity Lewis number [17], andmixture-averaged diffusion assumptions [1]. These models approximate thefull multicomponent diffusion coefficient matrix as a constant scalar, a con-stant vector, and a non-constant diagonal matrix, respectively, reducing thehigh computational expense associated with numerical combustion studies[1, 4, 16, 18, 19]. In addition, several approaches further reduce the system’scomplexity by approximating multicomponent diffusion processes in terms ofequivalent Fickian processes, such as those used by Warnatz [20] and Coltrinet al. [21]. While these assumptions may be computationally efficient, to ourknowledge, the accuracy and appropriateness of the physics they model hasnot been evaluated against full multicomponent mass diffusion for DNS ofthree-dimensional turbulent flames at moderate-to-high Karlovitz numbers(e.g., 140–210).Although few results exist from three-dimensional reacting flow simula-tions with multicomponent transport, several studies have investigated theeffects of multicomponent transport in simpler configurations. These stud-ies include one-dimensional [15, 22–25] and two-dimensional flames [26, 27]of various unburnt conditions. These works compared the multicomponentmodel with various diffusion and transport property models (from constant3ewis number to mixture-averaged properties). In general, these studieshighlighted the importance of differential diffusion effects but only investi-gated simplified flame configurations where these effects are relatively small,such as unstretched laminar flames.For example, in evaluating five simplified diffusion models, Coffee andHeimerl [22] observed that laminar flame speed and species profiles are moresensitive to the input values of individual species transport properties thanthe specific model used, using simulations of one-dimensional, steady, lam-inar, premixed hydrogen flames. They noted that their findings do not in-dicate that transport phenomenon or model selection are unimportant, butrather that even low-complexity models can be calibrated by carefully select-ing the species transport properties to improve accuracy.Focusing more on the underlying physics of differential diffusion, Ernand Giovangigli [24] demonstrated that both methane and hydrogen coun-terflow flames are sensitive to multicomponent transport. Specifially, neglect-ing multicomponent effects can lead to overpredicting the extinction strainrate, especially in rich hydrogen flames. Similarly, Charentenay and Ern [26]demonstrated that multicomponent transport only moderately affects globalflame properties in two-dimensional, low Karlovitz number, premixed hydro-gen/oxygen flames, thanks to the smoothing induced by turbulent fluctua-tions. However, when in highly curved flames or flames with local quenching,such as at moderate-to-high Karlovitz numbers, they concluded that the suf-ficiently large impact of multicomponent transport justifies its inclusion inaccurate DNS.Despite this evidence that multicomponent transport may impact the ac-4uracy of turbulent premixed DNS, studies of three-dimensional turbulentflames continue to rely on simplified diffusion models and do not considertheir accuracy relative to multicomponent diffusion, in complex configura-tions. Prior evaluations of diffusion models in three-dimensional simulationscompared the unity Lewis number, constant but non-unity Lewis number,and mixture-averaged approximations. For example, Lapointe and Blan-quart [18] compared the relative accuracy of the unity and non-unity Lewisnumber assumptions for n -heptane, iso-octane, toluene, and methane flames.The flames simulated using the non-unity Lewis number approximation havelower turbulent flame speeds than similar flames simulated with the unityLewis number assumption. They attributed these differences to reducedfuel-consumption rates caused by differential diffusion effects [18]. Similarly,Burali et al. [16] compared the non-unity Lewis number assumption to themixture-averaged diffusion for lean, unstable hydrogen/air flames and lean,turbulent n -heptane/air flames. They demonstrated that using the unityLewis number assumption underpredicts by 50 % or more the conditionalmeans of the fuel mass fraction and source term, but using the non-unityLewis number assumption results in much smaller differences, on the order of3 % or less; both were compared with simulations using the mixture-averagedassumption [16]. Moreover, Burali et al. [16] demonstrated that the relativedifference associated with the non-unity Lewis number assumption can beminimized by carefully selecting the Lewis-number vector for a wide rangeof flames, including non-premixed turbulent configurations.These results reinforce previous conclusions that differential-diffusion ef-fects can impact flame dynamics. However, there has not been a detailed5nvestigation of the accuracy and appropriateness of the mixture-averageddiffusion model relative to full multicomponent diffusion for turbulent react-ing flows. For high-pressure, non-reacting systems, Borchesi and Bellan [28]developed and analyzed multicomponent species mass flux and turbulentmixing models for large-eddy simulations. They focused on turbulent mixingof a five-species combustion-relevant mixture of n -heptane, oxygen, carbondioxide, nitrogen, and water. Their multicomponent transport model sig-nificantly improves the accuracy and fidelity of the solution throughout themixing layer. However, as this study was restricted to non-reacting flows,it did not assess the impact of multicomponent transport on the chemistryinherent in turbulent combustion.Motivated by the observed differences between the mixture-averaged andsimpler diffusion models, several groups have developed affordable multi-component transport models. Ern and Giovangigli [12, 24, 29] developed thecomputationally efficient Fortran library EGLIB for accurately determiningtransport coefficients in gas mixtures. Ambikasaran and Narayanaswamy [30]proposed an efficient algorithm to compute multicomponent diffusion veloc-ities, which scales linearly with the number of species. Both methods reducethe computational cost of inverting the dense matrix associated with theStefan–Maxwell equations [1, 31, 32]. Most recently, Fillo et al. [14] proposeda fast, semi-implicit, low-memory algorithm for implementing multicompo-nent mass diffusion, which we use here with the DNS code NGA. As a pre-liminary demonstration of their method, Fillo et al. [14] simulated lean, pre-mixed, three-dimensional turbulent hydrogen/air flames at moderate-to-highKarlovitz numbers using the mixture-averaged and multicomponent diffusion6odels. In these flames, the mixture-averaged diffusion model underpredictsthe peak mean source term and normalized turbulent flame speed by 5.5 %and 15 %, respectively [14].In addition to mass diffusion, several groups have also investigated theimpact of multicomponent Soret and Dufour thermal diffusion effects. In par-ticular, studies have examined the importance of including thermal diffusionin a wide range of flame configurations [19, 22, 24, 25, 27, 29, 33–35]. For ex-ample, Giovangigli [27] demonstrated that multicomponent Soret effects sig-nificantly impact a wide range of laminar hydrogen/air flames: laminar flamespeeds in flat flames and extinction stretch rates in strained premixed flames.Using a mechanistic approach, Yang et al. [33] observed that Soret diffusion ofhydrogen radical (H) in premixed hydrogen flames actively modifies its con-centration and distribution in the reaction zone. This effect was especiallyevident in symmetric, twin counter-flow, premixed hydrogen flames, whereSoret diffusion increases and decreases individual reaction rates in lean andrich mixtures, respectively. Performing a similar mechanistic approach ex-amining planar and stretched premixed n -heptane and hydrogen flames, Xinet al. [34] demonstrated that these chemical kinetic effects result from Soretdiffusion diluting or enriching the reactant concentrations in the reactionfront, and could substantially impact fuel burning rates—especially in highlystretched flames. Han et al. [35] recently examined Soret diffusion in turbu-lent non-premixed hydrogen flames, comparing DNS using mixture-averageddiffusion with and without Soret effects. They found that it significantlyaffects H and OH profiles in the flame but negligibly modifies H .Finally, Schlup and Blanquart [19] examined the impact of multicompo-7ent thermal diffusion in DNS of turbulent, premixed, high-Karlovitz hydro-gen/air flames. They observed that simulations using the mixture-averagedthermal diffusion assumption underpredict flame speeds compared with sim-ulations using full multicomponent thermal diffusion. In addition, they ob-served that including multicomponent thermal diffusion increases local pro-duction rates in in regions of high positive curvature [19]. These observed dis-crepancies in similar flame simulations with different diffusion models warranta detailed investigation of the fundamental transport phenomena involved.However, while thermal diffusion can be important in some fuel/air mixtures,in this article we focus on mass diffusion, and direct interested readers to thework of Schlup and Blanquart [19], for example, for an investigation of theseeffects.The primary objective of this study is to evaluate the accuracy and ap-propriateness of the mixture-averaged diffusion assumption for use in DNSof premixed unsteady laminar and turbulent flames. This objective will berealized via an a priori analysis of the orientation and magnitude of themixture-averaged diffusion flux vector, relative to that of the multicompo-nent model, for a range of flame configurations. We will further analyzedifferences between the diffusion models by considering a posteriori resultsof turbulent flame structures (i.e., species mass fraction and source termprofiles). Finally, we will compare the time history and average normal-ized turbulent flame speeds of hydrogen/air, n -heptane/air, and toluene/airflames as a global measure of the differences between the multicomponentand mixture-averaged diffusion models.The paper is organized as follows: Section 2 describes the governing equa-8ions, diffusion models, and flow configurations for the simulations. Then,Section 3 presents and discusses the results from a priori, a posteriori, tur-bulent flame speed, and chemical pathway analyses. Finally, in Section 4 wedraw conclusions from the comparisons of the diffusion models.
2. Numerical approach
This section describes the governing reacting-flow equations and flowsolver used, and briefly discusses the diffusion models to be studied. It alsopresents the two- and three-dimensional flow configurations used.
We solve the variable-density, low Mach number, reacting flow equationsusing the finite-difference code NGA [36, 37]. The complete conservationequations are ∂ρ∂t + ∇· ( ρ u ) = 0 , (1) ∂ρ u ∂t + ∇· ( ρ u ⊗ u ) = −∇ p + ∇· τ + f , (2) ∂ρT∂t + ∇· ( ρ u T ) = ∇· ( ρα ∇ T ) + ρ ˙ ω T − c p N (cid:88) i c p,i j i · ∇ T + ραc p ∇ c p · ∇ T , (3) ∂ρY i ∂t + ∇· ( ρ u Y i ) = −∇· j i + ˙ ω i , (4)where ρ is the mixture density, t is time, u is the velocity, p is the hy-drodynamic pressure, τ is the viscous stress tensor, f represents volumetricforces, T is the temperature, α is the mixture thermal diffusivity, c p,i is theconstant-pressure specific heat of species i , N is the number of species, c p isthe constant-pressure specific heat of the mixture, and j i , Y i , and ˙ ω i are the9iffusion flux, mass fraction, and production rate of species i , respectively.In Eq. (3), the temperature source term is given by ˙ ω T = − c p N (cid:88) i h i ( T ) ˙ ω i , (5)where h i ( T ) is the specific enthalpy of species i as a function of temperature.The density is determined from the ideal gas equation of state.NGA solves Eqs.(1)–(4) using a numerical scheme second-order accuratein both space and time [36, 37], via a semi-implicit Crank–Nicolson timeintegration method [38]. It uses the third-order Bounded QUICK scheme(BQUICK) [39] for scalar transport. We discuss the diffusion solver in moredetail next in Section 2.2. The diffusion fluxes are calculated using the semi-implicit scheme devel-oped by Fillo et al. [14] with either mixture-averaged or multicomponent[1, 4] models, both of which are based on Boltzmann’s equation for the ki-netic theory of gases [4, 5]. For this study, we neglect both baro-diffusionand thermal diffusion (Soret and Dufour effects). The baro-diffusion term iscommonly neglected in reacting-flow simulations under the low Mach numberapproximation [40]. We also neglect thermal diffusion because our objectiveis to investigate the impact of mass diffusion models; Schlup and Blanquartpreviously explored the effects of thermal diffusion modeling [19].The species diffusion flux for the mixture-averaged diffusion model (ab-breviated by MA hereafter) is related to the species gradient by a Fickianformulation, and is expressed as j MA i = − ρD i,m Y i X i ∇ X i + ρY i u (cid:48) c , (6)10here X i is the i th species mole fraction, D i,m is the i th species mixture-averaged diffusion coefficient as expressed by Bird et al. [1]: D i,m = 1 − Y i (cid:80) Nj (cid:54) = i X j / D ji , (7)where D ji is the binary diffusion coefficient of species i and j , and u (cid:48) c is thecorrection velocity used to ensure mass continuity: u (cid:48) c = N (cid:88) i D i,m Y i X i ∇ X i . (8)Alternatively, the multicomponent diffusion model (abbreviated as MChereafter), as presented by Bird et al. [1], calculates the species diffusion fluxas j MC i = ρY i X i W N (cid:88) j W j D ij ∇ X j , (9)where W is the mixture molecular weight, W j is the molecular weight ofthe j th species, and D ij is the ordinary multicomponent diffusion coefficientbetween species i and j , which we compute here using the MCMDIF subroutineof CHEMKIN II [41] with the method outlined by Dixon–Lewis [42].Though Fillo et al. [14] provide further details on how these methods areimplemented in NGA, we will summarize the key aspects here. First, wemodified the treatment of mass-diffusion terms in the semi-implicit schemeof Savard et al. [37], using the mixture-averaged diffusion coefficient matrixto precondition the diffusion source term. Furthermore, the multicomponentimplementation uses a dynamic memory algorithm to reduce the numberof times the multicomponent diffusion coefficient matrix must be evaluated.Fillo et al. showed the stability and accuracy of this semi-implicit scheme [14],11nd that the computational costs of the mixture-averaged and multicompo-nent diffusion models scale linearly and quadratically with the number ofspecies in the chemical kinetic model. Due to the efficient memory algo-rithm, most of the cost comes from the Chemkin II [41] routines used tocalculate the diffusion coefficients.
We used three flow configurations in this work. The first is a one-dimensional, unstretched (flat), laminar, hydrogen/air flame with an unburnttemperature of 298 K, pressure of 1 atm, and equivalence ratio of φ = 0 . . Toensure the flame remained centered in the computational domain, comprisedof 720 grid points where ∆ x = S L = − (cid:82) ρ ˙ ω H dxρ u Y H ,u , (10)where ρ u is the unburnt mixture density and Y H ,u is the unburnt fuel massfraction. We selected the grid spacing to ensure at least 20 points through thelaminar flame, with the thickness defined using the maximum temperaturegradient: l F = ( T max − T min ) / |∇ T | max . Schlup and Blanquart [19] used anidentical configuration to investigate the impact of Soret and Dufour thermaldiffusion effects.For the second and third configurations, we selected multidimensionalcases where diffusion modeling may be particularly important. The secondconfiguration considered is a two-dimensional domain used to study unsteady,freely propagating lean hydrogen/air flames [16, 19]. The third configura-tion is a doubly-periodic, turbulence-in-a-box configuration we used to study12 able 1: Parameters of the two- and three-dimensional simulations. ∆ x is the grid spacing, η u is Kolmogorov length scale in the unburnt gas, ∆ t is the simulation time step, φ is theequivalence ratio, p is the thermodynamic pressure, T u is the temperature of the unburntmixture, T peak is the temperature of peak fuel consumption rate in the one-dimensionallaminar flame, S L is the laminar flame speed, l F = ( T b − T u ) / |∇ T | max is the laminarflame thickness, l = u (cid:48) /(cid:15) is the integral length scale, u (cid:48) is the turbulence fluctuations, (cid:15) isthe turbulent energy dissipation rate, Ka u is the Karlovitz number in the unburnt mixture,Re t is the turbulent Reynolds number in the unburnt mixture, ν u is the unburnt kinematicviscosity, A force is the turbulent forcing coefficient used in NGA [43], and Le F is the fuelLewis number, where D F is the fuel diffusion coefficient from the mixture-averaged model. H (2D) H n -C H C H CH MA MC MA MC MA MC MA MCDomain L × L L × L × L L × L × L L × L × LL x x x x Grid ×
472 1520 × ×
190 1408 × ×
128 1408 × × x [m] 4.24 × −5 × −5 × −5 × −5 η u [m] – 2.1 × −5 × −6 × −6 ∆ t [s] 5 × −6 × −7 × −7 × −7 φ p [atm] 1 1 1 1 T u [K] 298 298 298 298 T peak [K] 1190 1180 1190 1180 1270 1230 1420 1420 S L [cm/s] 23.0 22.3 23.0 22.3 35.1 37.3 34.3 34.4 l F [mm] 0.643 0.651 0.643 0.631 0.390 0.385 0.410 0.420 l/l F – 2 2.04 1.1 1.1 u (cid:48) /S L – 18 18.6 18 16.9 17 16.9Ka u = τ F /τ η – 149 151 220 207 200 204Re t = ( u (cid:48) l ) /ν u – 289 190 175 A force [1/s] – 973.05 4730 4333Le F = α/D F H = 0 . , Le C H CH = 2 . , and Le C H = 2 . . This allows us toevaluate if the relative strength of mass diffusivity relative to thermal diffu-sivity affects a flame’s sensitivity to multicomponent diffusion. For example,the low Lewis number of hydrogen can result in differential diffusion effects,which cause the instabilities found in lean hydrogen/air flames. Further, weselected the turbulence timescales at the high Karlovitz numbers consideredto match the order of magnitude of the diffusion time scales, such that diffu-sion may interact with turbulence. All three configurations have been usedin previous studies, and so we provide only a brief overview here. The two-dimensional analysis is performed using a hydrogen/air mixturewith a nine-species, 54-reaction chemistry model from Hong et al. [44–46](forward and backward reactions are counted separately). The domain hasinlet and convective outlet boundaries in the streamwise direction and pe-riodic boundaries in the spanwise direction. The inlet velocity boundarycondition is fixed at the mean effective burning velocity, such that the unsta-ble flame remains statistically stationary in the domain. The mean effectiveburning velocity, S D eff , is defined as S D eff = − (cid:82) A ρ ˙ ω H dAρ u Y H ,u L , (11)where L is the spanwise dimension of the computational domain. This veloc-ity boundary condition allows the simulation to run for an arbitrary lengthof time to collect statistics. 14 igure 1: Temperature contour for the two-dimensional freely propagating unsteady hy-drogen/air flame obtained with the multicomponent diffusion model. Table 1 includes details of the computational domain. The unburnt mix-ture has an equivalence ratio of φ = 0 . , temperature of T u =
298 K, andpressure of p o = x F, = E + A (cid:88) i =1 , cos (cid:16) πk i yH (cid:17) (12)where x F, is the initial flame position, E is the average flame position, A = 10 − m is the amplitude, k = 20 and k = 30 are two coprime modes, y is the vertical coordinate, and H is the height of the domain [16]. Schlup etal. [19] and Burali et al. [16] used the same set of disturbance parameters toto initially perturb the flame asymmetrically and trigger Darrieus–Landauinstabilities. Figure 1 shows an example temperature contour with a repre-sentative unsteady flame clearly visible. Three fuel/air mixtures are simulated in the three-dimensional configura-tion: φ = 0 . hydrogen/air, φ = 0 . n -heptane/air, and φ = 0 . toluene/air.15he hydrogen/air mixture uses the same chemical-kinetic model as in thetwo-dimensional case [44–46]. The n -heptane/air mixture uses the reducedkinetic model described by Savard et al. [47, 48] consisting of 35 speciesand 217 reactions. Finally, the toluene/air mixture uses the 47-species, 290-reaction kinetic model of Bisetti et al. [49].Table 1 gives the details of the computational domains used for the three-dimensional simulations. The domains consist of inflow and convective out-flow boundary conditions in the streamwise direction, and periodic bound-aries in the two spanwise directions. The inflow velocity is the mean tur-bulent flame speed, which keeps the flame statistically stationary such thatturbulent statistics can be collected over an arbitrarily long run time. In theabsence of mean shear, a linear turbulence forcing method [43, 50] maintainsthe production of turbulent kinetic energy through the flame.Table 1 also provides details on the unburnt mixture, corresponding one-dimensional flames, and inlet turbulence. The unburnt temperatures andpressures for all cases are 298 K and 1 atm, respectively. Table 1 gives thedefinitions of the Karlovitz number, Ka u , and turbulent Reynolds number,Re t , where τ F = l F /S L is the flame time scale and τ η = ( ν u /(cid:15) ) / is theKolmogorov time scale of the incoming turbulence.
3. Results and discussion
This section presents a priori results for the one-dimensional flat hydro-gen/air flame and two-dimensional, unsteady, premixed hydrogen/air flame,as well as a priori and a posteriori results for the three-dimensional, turbu-lent, premixed fuel/air flames of hydrogen, n -heptane, and toluene.16 .1. A priori diffusion flux comparison Mass, momentum, and heat diffusion are strongly coupled processes inreacting flows, and as a result isolating the causes of observed effects can bedifficult. To overcome this challenge we compared the diffusion models withan “a priori” analysis that calculates the mass-diffusion flux vectors for eachmethod using identical scalar gradient fields. This analysis highlights thedifferences in diffusion flux vectors from each method before any differencesinfluence the flowfield. By calculating the mass diffusion fluxes in this way, weisolate the effects of the diffusion model on the resulting diffusion vectors fromany time evolution of the reacting flow field. To assess disagreement betweenthe mixture-averaged and multicomponent diffusion models, we evaluate therelative orientation and magnitude of the diffusion flux vectors they produce.
Figure 2 compares the a priori diffusion fluxes for the one-dimensional,flat, hydrogen/air flame relative to the local mixture temperature. As ex-pected, the flux profiles for the mixture-averaged and multicomponent casesmatch in shape and magnitude. However, the mixture-averaged model un-derpredicts the maximum flux magnitude of hydrogen radical (H) by approxi-mately 40 %. Similarly, the mixture-averaged model underpredicts molecularhydrogen (H ) and hydroxyl radical (OH) fluxes by approximately 18 %, andoxygen radical (O) by 16 %. These differences are substantial but agree withprevious results for one-dimensional premixed hydrogen/air flames [24, 26].These differences disrupt mass continuity by locally altering the equiva-lence ratio in regions of high mass-diffusion flux. This effect is clear whenconsidering the correction velocity, which is based on the mole and mass frac-17ions of the species. As a result, it lumps a large portion of the correctionfor mass continuity into the N mass flux, which the mixture-averaged modeloverpredicts by 40 %. The correction velocity is not correcting for the errorsin the mixture-averaged model; rather, it simply ensures mass continuity. − . − . − . − . − . . . . j [ k g/ ( m s ) ] × − (a) N MulticomponentMixture-averaged − . − . . . . . . . × − (b) H − . − . − . − . − . . . × − (c) O − − j [ k g/ ( m s ) ] × − (d) O − − × − (e) OH − . − . − . − . − . − . − . − . − . . × − (f) H
400 600 800 1000 1200 1400 T [K]0 . . . . . . . j [ k g/ ( m s ) ] × − (g) H O
400 600 800 1000 1200 1400 T [K] − − − × − (h) HO
400 600 800 1000 1200 1400 T [K] − − − × − (i) H O Figure 2: A priori comparisons of mass diffusion fluxes vs. temperature in a one-dimensional hydrogen/air flame at φ = 0 . . .1.2. Multi-dimensional flames We next performed an a priori assessment of the species mass diffu-sion fluxes in the multi-dimensional flames. However, because of the addeddegrees-of-freedom in multi-dimensional flows, we now investigate the rela-tive angles of the flux vectors with respect to the species gradient vectorsto assess the relative direction of mass flux, in addition to the flux magni-tudes. The mixture-averaged flux vector for a given species is based on thegradient of that species and, as a result, should align almost perfectly withits gradient and in the opposite direction. However, some misalignment mayalso arise because of the velocity correction term in Eq. (7). In contrast, asshown by Eq. (9), the multicomponent flux of a given species is based onthe net influence of the remaining species (but not itself) and thus may notnecessarily align with its own gradient. Differential diffusion may misalignthe species gradient and multicomponent diffusion flux vectors in regions ofhigh flame curvature where strong gradients can exist in multiple directions.As a qualitative assessment, Figure 3 shows two-dimensional slices offuel mass fraction, fuel mass diffusion flux, and angle between species fluxvector and gradient vector from the turbulent hydrogen/air flame for themixture-averaged and multicomponent models. For this assessment an angleof π means that the species flux and gradient vectors align, while smallerangles show misalignment. To help highlight small differences between thetwo diffusion models, we also present the logarithm of the mass diffusion fluxfield. The location of the flame is indicated by isolines of T = T peak −
300 K(green) and T = T peak +
300 K (blue) included on the fields of fuel massfraction. 19 a) MA Y H (b) MC Y H (c) MA j H [kg/(m s)] (d) MC j H [kg/(m s)](e) MA log ( j H ) (f) MC log ( j H ) (g) MA ∇ Y H ∠ j H (h) MC ∇ Y H ∠ j H Figure 3: Fields of fuel mass fraction (a, b), fuel mass diffusion flux (c, d), log of the fuelmass diffusion flux (e, f), and angle between fuel mass flux and species gradient vectors (g,h) for one time step of the hydrogen-air turbulent premixed flame for the mixture-averaged(MA) and multicomponent (MC) diffusion cases. Shown are domain cross-sections throughthe midplane. The green and blue lines correspond to isosurfaces of T u = T peak −
300 K and T b = T peak + 300 K , respectively, and represent the inflow and outflow surfaces of theflame front. log ( j H ) ≤ −30 at the inlet of the domain correspond to flux magnitudes of1 × −30 kg/(m s) or less. At the far right of Figures 3g and 3h, the relativeangles for both models are roughly constant at π , anti-parallel to the speciesgradient vector. In this region fluxes are small but non-zero as residual fuelis present in small concentrations—as a result, scalar gradients are small.Finally, although the flux angle appears to deviate from π in small regionsthroughout the flame for both the mixture-averaged and multicomponentflames, these deviations correspond to areas where the flux magnitude islocally very small, approaching zero. Furthermore, the relative angle of theflux vector is consistently π in regions of high species gradients, such asthrough the flame front, and agrees well between the two models. The anglesbetween the mass-flux and gradient vectors agree similarly well for all speciesand flame configurations considered.To confirm our qualitative observations of the relative direction of theflux vector, Figure 4 shows the probability density function (PDF) of the an-gles between the fuel species diffusion flux vector and mass fraction gradientvector for the two-dimensional unsteady hydrogen flame, three-dimensionalturbulent hydrogen flame, turbulent n -heptane flame, and turbulent tolueneflame. This quantitatively measures the alignment of the vectors to compare21 pd f (a) H (2D) MulticomponentMixture-averaged (b) H (3D) π π π πθ pd f (c) C H π π π πθ (d) C H CH π
16 31 π π − π
16 31 π π − π
16 31 π π − π
16 31 π π − Figure 4: A priori assessment of the mixture-averaged and multicomponent models, com-paring the PDFs of the angle between species flux vectors and species gradient vectors: θ = ∇ Y F ∠ j F , where F is the fuel in the (a) two-dimensional unsteady hydrogen, (b)three-dimensional turbulent hydrogen, (c) three-dimensional turbulent n -heptane, and (d)three-dimensional turbulent toluene flames. The inset plots use a semi-log scale on thevertical axis. π , anti-parallel to thespecies gradient vector. As expected, this indicates that mass diffusion oc-curs primarily in the direction of negative species gradient (i.e., from high tolow concentration). We attribute small deviations of the mixture-averagedangle away from π to the velocity correction term in Eq. (6).The two-dimensional unsteady flame exhibits negligible differences in theangles separating the species diffusion flux and gradient vectors, but thisagreement does not extend to the three-dimensional turbulent flame. In thiscase, the angle PDF is roughly 50 % higher for the multicomponent model.Although this difference between the two models is large, it is tempered bythe tiny magnitude of the PDFs away from π . For both cases these vectorsshow a clear preferential alignment at π , with the magnitude of the PDFdropping to much less than one for angles smaller than π/ . As seenin Figures 4(c) and 4(d), the three-dimensional turbulent n -heptane/air andtoluene/air flames show similar differences.To better show where the two models deviate in the domain, Figure 5presents contours of the point-wise difference between the fuel diffusion fluxmagnitudes between the models, normalized by the peak multicomponentfuel diffusion-flux magnitude. These contour plots provide a reference for23he physical location of peak differences between the mixture-averaged andmulticomponent models within the flame.Examining Figure 5, in all cases the highest differences between the twomodels occur in regions of high flame curvature where the species gradientfield is strong and highly variable. To more-concisely discuss these differencesacross multiple species and flame configurations, Table 2 presents the meanand standard deviations of the angles between the mixture-averaged andmulticomponent diffusion fluxes, as well as relative L , L , and L ∞ errornorms of the differences in magnitude of these diffusion fluxes. We calculatedthese statistics in regions where species diffusion is strong, i.e., the diffusionflux magnitude is greater than 10 % of the peak diffusion flux magnitude.The relative L , L , and L ∞ error norms measure the modal, mean, andmaximum difference, respectively, for the diffusion flux magnitude: L (cid:0) j MA i (cid:1) = (cid:80) N p n =1 (cid:12)(cid:12)(cid:12)(cid:12) j MA i,n (cid:12)(cid:12) − (cid:12)(cid:12) j MC i,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:80) N p n =1 (cid:12)(cid:12) j MC i,n (cid:12)(cid:12) , (13) L (cid:0) j MA i (cid:1) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:80) N p n =1 (cid:0)(cid:12)(cid:12) j MA i,n (cid:12)(cid:12) − (cid:12)(cid:12) j MC i,n (cid:12)(cid:12)(cid:1) (cid:80) N p n =1 (cid:12)(cid:12) j MC i,n (cid:12)(cid:12) , and (14) L ∞ (cid:0) j MA i (cid:1) = max n ∈ N p (cid:0)(cid:12)(cid:12) j MA i,n (cid:12)(cid:12) − (cid:12)(cid:12) j MC i,n (cid:12)(cid:12)(cid:1) max n ∈ N p (cid:0)(cid:12)(cid:12) j MC i,n (cid:12)(cid:12)(cid:1) , (15)where N p is the number of points in the domain where the diffusion flux isgreater than 10% of the peak magnitude and j i,n indicates the diffusion fluxof the i th species at point n .As observed in Table 2, a majority of the mixture-averaged diffusion fluxvectors match the multicomponent diffusion flux vectors within a mean angle, µ ∠ , of 0.06 rad for the turbulent cases, with negligible differences for the24 a) Two-dimensional H (b) Three-dimensional H (c) n-C H (d) C H CH Figure 5: Local differences between mixture-averaged and multicomponent mass dif-fusion fluxes of the fuel normalized by peak multicomponent mass diffusion flux: (cid:16)(cid:12)(cid:12)(cid:12) j MA (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) j MC (cid:12)(cid:12)(cid:12)(cid:17) (cid:14) max N p (cid:16)(cid:12)(cid:12)(cid:12) j MC (cid:12)(cid:12)(cid:12)(cid:17) . Shown are domain cross-sections through the mid-plane. The green and blue lines correspond to isosurfaces of T u = T peak −
300 K and T b = T peak + 300 K , respectively, and represent the inflow and outflow surfaces of theflame front. able 2: Statistical quantities of the mixture-averaged and multicomponent diffusion mod-els for a representative set of major, radical, and product species: the mean ( µ ∠ ) andstandard deviation ( s ∠ ) of the angles between the mixture-averaged and multicomponentflux vectors, as well as relative L error norms (Eq. (14)). µ ∠ [rad] s ∠ [rad] L (cid:0) j MA i (cid:1) L (cid:0) j MA i (cid:1) L ∞ (cid:0) j MA i (cid:1)
2D unsteady hydrogenH × −5 × −4 × −2 × −2 × −2 H 5.7 × −5 × −6 × −2 × −2 × −2 OH 2.4 × −4 × −5 × −2 × −2 × −3 H O 7.4 × −3 × −4 × −2 × −2 × −2
3D hydrogenH × −2 × −2 × −1 × −1 × −1 H 2.5 × −2 × −3 × −1 × −1 × −1 OH 7.1 × −4 × −4 × −1 × −1 × −1 H O 1.5 × −1 × −3 × −1 × −1 × −1 n -heptane n -C H × −2 × −2 × −1 × −1 × −1 H 2.7 × −2 × −2 × −1 × −1 × −1 OH 6.0 × −2 × −2 × −1 × −1 × −1 CO × −3 × −3 × −1 × −1 × −1
3D tolueneC H CH × −2 × −2 × −1 × −1 × −1 H 1.4 × −2 × −2 × −1 × −1 × −1 OH 8.6 × −3 × −2 × −1 × −1 × −1 CO × −3 × −3 × −1 × −1 × −1 O for thehydrogen/air turbulent flame with a mean angle of 0.12 rad. As expected,the turbulent cases show larger (albeit still small) values of µ ∠ . Additionally,the standard deviations of the angle between the diffusion fluxes, s ∠ , aresmall and generally the same order as the mean angles themselves.Finally, Table 2 shows that the magnitudes of the diffusion fluxes agreethroughout much of the domain. The L error norms indicate the averagedifference in the mixture-averaged flux magnitude is on the order of 20 %.However, by definition, the L norm is sensitive to outliers, which increasethese errors when present. Alternatively, the L error norm weights all pointsin the domain equally, providing a measure of the modal error. Comparingthese two values, we can see the differences are smaller than 20 % through-out the domain for most species, with the largest differences occurring inthe three-dimensional flame configurations. Finally, examining the L ∞ errornorms, the observed differences in Figure 5 correspond to large differences,on the order of 30–50 %, in regions of high flame curvature for each of thethree-dimensional flame configurations. The largest differences occur in the n -heptane/air and toluene/air flames.The differences between the two models seem to increase proportionallyto the magnitude of the driving species gradient. In Figure 5a, showing thelean, two-dimensional, unsteady, laminar, hydrogen/air flame, the mixture-averaged model matches the multicomponent model within 2 % for the fulldomain. For this two-dimensional case the species gradient vectors are pri-marily aligned in the flow-wise direction and roughly constant across the do-main. As a result, the scalar gradient fields locally vary a small amount and27he mixture-averaged diffusion model matches the multicomponent modelwell; this is true even near thermal instabilities. In contrast, the lean, three-dimensional, turbulent hydrogen air flame shows significantly larger differ-ences between the models. The equivalence ratio between these two flamesmatches ( φ = 0 . ) and so any local increases in the species gradient field comefrom increases in local flame curvature caused by a higher dimensionality andturbulent mixing. In these regions of high-flame curvature, the scalar gra-dient field is steep, highly variable, and not strictly aligned in the flow-wisedirection; as a result, mass can diffuse in directions other than the directionof the species gradient. Comparing the definitions of the mixture-averagedand multicomponent diffusion fluxes in Eq. (6) and Eq. (9), respectively, thestrict alignment of the mixture-averaged diffusion flux with its own gradientmay overvalue the impact of that gradient and overpredict the mass fluxwhen the gradient vector is large, such as in regions of high flame curvature.This overprediction in the mixture-averaged diffusion flux is most evidentin the turbulent, n -heptane/air and toluene/air flames where, in additionto turbulent mixing, the equivalence ratio is higher than in the turbulenthydrogen/air flame ( φ = 0 . compared with φ = 0 . ). Although it is diffi-cult to compare these flames one-to-one due to differences in chemistry, theincreased equivalence ratio relative to the lean hydrogen/air flames causessteeper species gradients through the flame, since the unburnt fuel massfractions are much higher in these flames: 0.056 and 0.063 for n -heptane/airand toluene/air, respectively, compared with 0.012 for hydrogen/air. Thisincrease in the unburnt fuel mass fraction, coupled with the presence of tur-bulent mixing, increases the magnitude of the species gradients through the28ame front relative to the three-dimensional hydrogen flame. Furthermore,the larger differences in the n -heptane and toluene flames over the hydro-gen/air flames supports the theory that high species gradients can lead tothe mixture-averaged model overpredicting diffusion flux.Finally, although the reported differences in the predicted flux magnitudesfor the mixture-averaged over the multicomponent model may seem signifi-cant, it is important to examine these values from the perspective of turbulentscaling. The relative magnitude of both the mixture-averaged and multicom-ponent mass diffusion fluxes are small: on the order of 5 × −3 kg/(m s) orsmaller on average for all four simulations, compared to total mass fluxeson the order of 1 kg/(m s) or greater. Moreover, the multicomponent andmixture-averaged fluxes consistently show the same order of magnitude fora given species in a point-wise comparison, for most points in the domain.In other words, when comparing any given point in the domain, the ex-pected fluxes have the same order of magnitude for the mixture-averaged andmulticomponent models. This suggests that the differences in the mixture-averaged flux may be small compared to global turbulent flame statistics. To further compare the mixture-averaged and multicomponent diffusionmodels, we present a posteriori statistics for the three turbulent flame simu-lations using both mixture-averaged and multicomponent diffusion. In otherwords, how much does model choice impact global and statistical quantitiesin a simulation? For this analysis, we started simulations using the mul-ticomponent and mixture-averaged diffusion models from the same initialconditions, and allowed them to evolve in the domain until any initial tran-29ients had convected through the domain: approximately six eddy turnovertimes ( τ eddy = k/(cid:15) , where k is the turbulent kinetic energy) . We then raneach simulation for an additional τ eddy to provide a representative sampleof instantaneous flame speeds and collect turbulent statistics.Figure 6 shows the time-history of the turbulent flame speed, S T , normal-ized by the laminar flame speed, S L . The turbulent flame speed is definedas S T = − (cid:82) V ρ ˙ ω F dVρ u Y F ,u L , (16)where ˙ ω F and Y F are the fuel source term and mass fraction respectively, ρ is the density, L is the span-wise domain width, and V is the volume ofthe domain. Table 3 presents the mean turbulent flame speeds of the threefuels, based on the collected samples; we report these values to provide a sim-ple, single metric to identify global differences between the models. Whilethe mixture-averaged model seems to lower the normalized turbulent flamespeeds by 13 % and 5 % for the hydrogen and toluene flames, it causes a 20 %higher normalized flame speed for the n -heptane flame. These trends do notseem to correlate to the Lewis number for each fuel—recall that Le H = 0 . ,Le C H CH = 2 . , and Le C H = 2 . . The n -heptane/air and toluene/airflames have similar Lewis numbers but show opposite trends in the differ-ences of turbulent flame speeds between the diffusion models. However, thedifferences are small between the models for toluene, and may be attributedto statistical error. In contrast, the differences between the models are larger In practice we determine τ eddy with the turbulence forcing scheme: τ eddy = 1 / A force ,where A force is the turbulent forcing coefficient imposed as a parameter in the simulation,developed by Carroll et al. [43]. S T / S L (a) H MulticomponentMixture-averaged S T / S L (b) C H . . . . . . . . . t/τ eddy S T / S L (c) C H CH Figure 6: Time histories of the normalized turbulent flame speed from the turbulent (a)hydrogen/air, (b) n -heptane/air, and (c) toluene/air cases for both diffusion models. n -heptane flames (13% and 20%, respectively); thesevalues are similar to the spread in mean turbulent flame speed ( ∼ n -heptane flames with varying equivalence ratio and chemical model shownby Lapointe and Blanquart [18]. Table 3: Mean turbulent flame speed normalized by unstretched laminar flame speed( S T /S L ) for three-dimensional turbulent hydrogen/air, n -heptane/air, and toluene/airmixtures, comparing the impact of mixture-averaged and multicomponent diffusion mod-els. MC MA DifferenceH H H CH To better understand the observed differences in the turbulent flamespeed, we assess the impact of the diffusion models on flame chemistry viathe average fuel mass fraction and source term. As demonstrated in previousstudies, differential diffusion can modify the local equivalence ratio in re-gions of high flame curvature [3, 18, 51]. We have demonstrated this effect inTable 2, by showing that the mixture-averaged diffusion assumption overpre-dicts the magnitude of the mass diffusion flux in these regions. The increasein mass flux into these regions of high flame curvature may impact localchemistry and modify the fuel source term. Lapointe and Blanquart [18]previously suggested that the normalized turbulent flame speed is propor-tional to the product of turbulent flame area and the mean fuel source termconditioned on flame temperature: S T S L ∝ A T A (cid:104) ˙ ω F | T (cid:105) ˙ ω F, lam , (17)32here A T is the turbulent flame area, A is the cross-sectional area of the do-main, and ˙ ω F, lam is the fuel source term in the laminar flame. Moreover, theydemonstrated that the area ratio ( A T /A ) controls large-scale fluctuations inthe flame speed on the order of their mean values, while the normalized meansource term ( (cid:104) ˙ ω F | T (cid:105) / ˙ ω F, lam ) controls smaller-scale fluctuations [18].To evaluate if the observed difference in the diffusion mass fluxes cor-relate to the observed differences in the normalized turbulent flame speedsbetween the two models, Figures 7 (a) and (c) present the means of the fuelmass fractions and source term, conditioned on temperature and shown nor-malized by their respective adiabatic flame temperatures, T ad . Figures 7 (b)and (d) show the percent differences between the conditional means of fuelmass fraction and source term, respectively, with respect to the multicompo-nent model results; only differences corresponding to normalized conditionalmeans from the multicomponent model above 0.01 and 0.05 are shown, re-spectively. The calculated conditional means of the fuel mass fractions differby negligible amounts for most of the domain: less than 1 % for most points,with only larger differences (i.e., greater than 5 %) between small values ofthe mean fuel mass fraction. This strong agreement between the conditionalmeans of the fuel mass fraction suggests that although local differences inthe diffusion mass flux fields do exist between the two models, they do notappear to significantly affect the averaged distribution of fuel in the flame.In contrast, we observe more differences in the conditional means of thefuel source term in Figures 7 (c) and (d). At their peaks, the mean sourceterms for the multicomponent flames are 5.5 % and 1.6 % higher than themixture-averaged models for the hydrogen and toluene flames, respectively,33 . . . . . . h Y F | T i / Y M C F , m a x H C H C H CH (a) MulticomponentMixture-averaged P e r ce n t d i ff e r e n ce H C H C H CH (b) . . . . . . . T/T ad . . . . . . . . . h ˙ ω F | T i / ˙ ω M C F , p e a k H C H C H CH (c) . . . . . . . T/T ad P e r ce n t d i ff e r e n ce H C H C H CH (d) Figure 7: Turbulent flame structure for the three fuel/air mixtures, showing conditionalmeans (left) and percent differences (right) of the mixture-averaged values with respectto the multicomponent values, for (a–b) fuel mass fraction and (c–d) fuel source term, asfunctions of temperature T normalized by T ad . All values are normalized by their peaksfrom one-dimensional flat flames using the multicomponent model. n -heptane flame. Away from the peak locations, theconditional means disagree by about 1–35 % for the hydrogen/air flame, andby up to about 24 % for the n -heptane/air flame. However, most differencesoccur where the values of mean source term are small. For the toluene/airflame, the mixture-averaged and multicomponent cases agree within 3.5 % atall locations.For the hydrogen/air flame, we see differences of 5–35 % between themixture-averaged and multicomponent in the super-adiabatic regions, forthe conditional means of both the fuel mass fraction and source term. Theseregions, also called “hot spots”, result from differential diffusion, and havebeen predicted both in theoretical studies [52] and in numerical analyses oflean hydrogen/air mixtures [53–55].Comparing the observed differences in the conditional means of the fuelsource term with the mean normalized turbulent flame speeds, the resultsagree well with the proportional relation in Eq. (17) given by Lapointe andBlanquart [18]. Specifically, the peak normalized source term is 5.5 % higherfor the multicomponent hydrogen/air flame over mixture-averaged, resultingin a 13 % higher normalized flame speed. Alternatively, the peak normalizedsource term is 9.4 % lower for the multicomponent n-heptane/air flame com-pared to mixture-averaged, resulting in a 19 % lower normalized flame speed.In all three cases, the differences in normalized turbulent flame speed appearproportional to the differences in normalized conditional mean by approx-imately a factor of two. This strong proportional relationship agrees withsimilar results observed by Lapointe and Blanquart [18]. Moreover, our re-sults demonstrate that, relative to the multicomponent model, the observed35ifferences in the mixture-averaged diffusion flux vectors do impact a poste-riori flame statistics. These results raise questions about the appropriatenessof the mixture-averaged diffusion assumption for simulations with high flamecurvature. Mass diffusion plays an important role in premixed flames, since diffusionof radical species can alter elementary reaction rates in the reaction zone.In this section, we examine how the choice of diffusion model impacts therelative contribution of major reactions to fuel consumption.We performed additional one-dimensional, unstretched (flat), laminarflame simulations using the
FreeFlame
Cantera v2.5.0 [56] for each of thethree fuel/air mixtures, corresponding to the unburnt conditions given in Ta-ble 1. We used the freely-propagating adiabatic flat flame solver (
FreeFlame )with grid refinement criteria for both slope and curvature set to 0.01 and arefinement ratio of 2.0.In the laminar simulations, these reactions account for more than 98 % ofthe overall fuel consumption rate in the hydrogen flame: H + O ←−→ H + OH (18) H + OH ←−→ H + H O , (19)in the n -heptane flame: n -C H + H −−→ H + H (20) n -C H + O −−→ H + OH (21) n -C H + OH −−→ H + H O , (22)36nd in the toluene flame: C H CH + H ←−→ C H + CH (23) C H CH + H −−→ C H CH + H (24) C H CH + OH −−→ C H CH + H O . (25)Tables 4, 5, and 6 list the percentage contributions of the primary reactions tothe overall fuel consumption rate of the corresponding hydrogen, n -heptane,and toluene flames, respectively. In the hydrogen and toluene flames, wefocus on the reactions that contribute most to fuel consumption only, andleave out reactions that re-form the fuel. Table 4: Percentage contribution of major reactions to the overall fuel consumption ratein the hydrogen flames. “MC” and “MA” represent cases using the multicomponent andmixture-averaged diffusion models, respectively.
Reaction Model 1D TurbulentH + O ←−→ H + OH MC 11.5 17.5MA 11.5 17.7H + OH ←−→ H + H O MC 86.5 82.5MA 86.5 82.3These results show that the choice of diffusion model does not significantlymodify the primary reaction pathways for fuel consumption. In the one-dimensional laminar flames, the percentage contributions of each reaction arenearly identical for all fuels (i.e., within 0.1 %). In the turbulent flames, the n -heptane flame shows differences of about 2 % between the diffusion models,37 able 5: Percentage contribution of major reactions to the overall fuel consumption rateof the n -heptane flames. “MC” and “MA” represent cases using the multicomponent andmixture-averaged diffusion models, respectively. Reaction Model 1D Turbulent n -C H + H −−→ H + H MC 59.5 53.5MA 59.3 55.4 n -C H + O −−→ H + OH MC 7.78 8.80MA 7.83 8.95 n -C H + OH −−→ H + H O MC 31.7 37.0MA 31.8 34.7
Table 6: Percentage contribution of major reactions to the overall fuel consumption rateof the toluene flames, neglecting fuel re-formation reactions. “MC” and “MA” representcases using the multicomponent and mixture-averaged diffusion models, respectively.
Reaction Model 1D TurbulentC H CH + H ←−→ C H + CH MC 22.4 20.5MA 22.4 20.7C H CH + H −−→ C H CH + H MC 42.1 36.3MA 42.0 36.1C H CH + OH −−→ C H CH + H O MC 35.1 43.0MA 35.2 43.038ith even smaller differences in the hydrogen and toluene flames. Overall,the choice of diffusion model impacts reaction pathways less than turbulence.In fact, in the case of toluene, turbulence changes the ordering of primaryreactions; both diffusion models equally capture these effects. Our resultsagree with and extend those of Lapointe et al. [51], who previously showedthat using two Lewis-number models (unity and non-unity Lewis number)did not substantially change the primary reaction pathways in high-Karlovitzturbulent premixed n -heptane/air flames, like those studied here.
4. Conclusions
This article compares the mixture-averaged and multicomponent massdiffusion models for premixed two-dimensional, unsteady hydrogen/air andthree-dimensional, turbulent flames, considering hydrogen, n -heptane, andtoluene fuel/air mixtures. We compared the methods using both a prioriand a posteriori assessments of differences.The a priori analysis indicated that the mixture-averaged model accu-rately reproduces the relative direction and magnitude of the flux vectorsthrough much of the domain. However, in the turbulent cases, we foundaverage differences of 10–20 % in the magnitude of the diffusion flux vectorfor all three fuels, and differences greater than 40 % in regions of high flamecurvature.Our a posteriori analysis indicated that using the mixture-averaged modeldoes affect turbulent statistics, such as conditional means of the fuel massfraction and consumption rates. The impact on flame statistics is relativelysmall: the mixture-averaged model results in differences of up to 20 % in39ormalized turbulent flame speed, 10 % in conditional mean of fuel sourceterm, and 1 % in the conditional mean of fuel mass fraction. Thus, thelarger differences we observed in instantaneous diffusion fluxes seem to lead tosmaller—though nonzero—differences in average quantities. The differencesbetween the diffusion models also do not significantly impact the relativecontributions of reactions to fuel consumption. Due to differences betweenthe three turbulent flames examined, notably the varying Karlovitz num-bers and flame stability, we are unable to draw firm conclusions on the rootcauses of the observed differences. These results warrant further investiga-tion into the causes of these differences and the appropriateness of using themixture-averaged diffusion model in the DNS of three-dimensional, premixedturbulent flames at moderate-to-high Karlovitz numbers. Acknowledgments
This material is based upon work supported by the National ScienceFoundation under Grant Nos. 1314109, 1761683, and 1832548. This re-search used resources of the National Energy Research Scientific ComputingCenter, a DOE Office of Science User Facility supported by the Office ofScience of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, as well as the Extreme Science and Engineering Discovery Envi-ronment (XSEDE), which is supported by National Science Foundation grantnumber 1548562. 40 ppendix A. Availability of material
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