Combustion and Flame 157 (2010) 1316–1324

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Combustion and Flame

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CFD modeling of two-stage ignition in a rapid compression machine:Assessment of zero-dimensional approach

Gaurav Mittal a,*, Mandhapati P. Raju b, Chih-Jen Sung c

a Department of Mechanical Engineering, The University of Akron, Akron, OH 44325, USAb General Motor R&D Tech Center, Warren, MI 48090, USAc Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 September 2009Received in revised form 5 October 2009Accepted 24 February 2010Available online 27 March 2010

Keywords:Rapid compression machineTwo-stage ignitionCFD modeling

0010-2180/$ - see front matter � 2010 The Combustdoi:10.1016/j.combustflame.2010.02.019

* Corresponding author.E-mail address: gaurav@uakron.edu (G. Mittal).

In modeling rapid compression machine (RCM) experiments, zero-dimensional approach is commonlyused along with an associated heat loss model. The adequacy of such approach has not been validatedfor hydrocarbon fuels. The existence of multi-dimensional effects inside an RCM due to the boundarylayer, roll-up vortex, non-uniform heat release, and piston crevice could result in deviation from thezero-dimensional assumption, particularly for hydrocarbons exhibiting two-stage ignition and strongthermokinetic interactions. The objective of this investigation is to assess the adequacy of zero-dimen-sional approach in modeling RCM experiments under conditions of two-stage ignition and negative tem-perature coefficient (NTC) response. Computational fluid dynamics simulations are conducted for n-heptane ignition in an RCM and the validity of zero-dimensional approach is assessed through compar-isons over the entire NTC region. Results show that the zero-dimensional model based on the approach of‘adiabatic volume expansion’ performs very well in adequately predicting the first-stage ignition delays,although quantitative discrepancy for the prediction of the total ignition delays and pressure rise in thefirst-stage ignition is noted even when the roll-up vortex is suppressed and a well-defined homogeneouscore is retained within an RCM. Furthermore, the discrepancy is pressure dependent and decreases ascompressed pressure is increased. Also, as ignition response becomes single-stage at higher compressedtemperatures, discrepancy from the zero-dimensional simulations reduces. Despite of some quantitativediscrepancy, the zero-dimensional modeling approach is deemed satisfactory from the viewpoint of theignition delay simulation.

� 2010 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

Rapid compression machines (RCMs) are extensively used forinvestigating the chemical kinetics of hydrocarbon fuels at low-to-intermediate temperatures and elevated pressures. At lowtemperatures, many hydrocarbons exhibit two-stage ignition re-sponse and negative temperature coefficient (NTC) behavior.When validating chemical kinetic mechanisms against experi-mental data obtained from RCMs, kinetic modeling is usually con-ducted using a zero-dimensional (zero-D) code, such as SENKIN[1], while properly accounting for the heat loss effects (cf. [2–6]). Furthermore, recognizing the effect of the roll-up vortex inRCMs [6–15] and usefulness of the creviced piston in mitigatingsuch an effect [6,7,10,15], various researchers have implementedcreviced piston in their RCMs. Creviced piston enables attainmentof homogeneous temperature field inside the reaction chamber,

ion Institute. Published by Elsevier

and hence is expected to positively influence the validity of thezero-D modeling approach along with an adequate heat lossmodel.

In our previous investigation with H2 ignition in an RCM [16], itwas shown that SENKIN simulations in conjunction with the ap-proach of ‘adiabatic volume expansion’ perform very well in ade-quately predicting the ignition delay, as compared with theresults obtained by computational fluid dynamics (CFD) simula-tions. Such a zero-D modeling approach based on ‘adiabatic vol-ume expansion’, however, has not been assessed for theconditions of two-stage ignition. Under the conditions of two-stageignition, significant heat release may occur during the first-stageignition as well as prior to the final hot-ignition event. Conse-quently, the influence of multi-dimensional effects due to the pres-ence of boundary layer, cooler crevice zone, and non-uniform heatrelease could be significant. Although the use of a heat loss modelin zero-D modeling approach aims to capture the above-men-tioned multi-dimensional effects, it is not clear whether such anapproach can be extended to the cases exhibiting two-stage igni-tion characteristics.

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G. Mittal et al. / Combustion and Flame 157 (2010) 1316–1324 1317

Some investigations into the relationship between zero-D andCFD calculations were reported by Griffiths et al. [8,17]. Griffithset al. [17] implemented a fluid dynamics code in conjunction withthe Shell generalized kinetic model to predict the development ofautoignition in the reaction chamber of an RCM. Griffiths et al.[8] also discussed the differences between the computational re-sults of RCM obtained using homogeneous conditions and CFDsimulations. Differences between the two sets of computationswere attributed to the effect of temperature nonuniformity thatwas accounted for in CFD analysis [8]. However, these simulationsin [8,17] were conducted on a very course mesh with then avail-able computational resources and did not specifically address theresponse within the NTC regime in detail.

This study focuses on the CFD modeling of reactive n-heptane/oxidizer mixtures with detailed chemistry in an RCM for the pur-pose of assessing the validity of the zero-D modeling approach un-der the conditions of two-stage ignition. n-Heptane is taken as thecandidate fuel as it is well known to exhibit strong NTC behaviorand two-stage ignition phenomena. In the following, we sequen-tially describe the specifications of the present numerical simula-tions and the result comparison of CFD and zero-D simulations.

2. Numerical specifications

The specifications of the rapid compression machine simulatedhere have been detailed in [5,6,18]. Briefly, this RCM system con-sists of a driver piston, a reactor piston, a hydraulic motion controlchamber, and a driving air tank. The machine is pneumatically dri-ven and hydraulically stopped. Moreover, the driver cylinder has abore of 127 mm and the reactor cylinder bore is 50.8 mm. Addi-tionally, the machine incorporates an optimized creviced pistonhead design to promote a homogeneous reaction zone, as dis-cussed earlier. Here we simulate RCM experiments for two reactiven-heptane/oxidizer mixtures over a range of compressed tempera-tures spanning the NTC regime. For the mixture of n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922 by mole percentage, com-pressed pressures of approximately 10 and 15 bar are studied.For the mixture with mole percentages of n-heptane/O2/N2/Ar = 0.562/6.184/62.969/30.285, the compressed pressures investi-gated are approximately 15 and 20 bar. For a given mixture com-position, the initial mixture temperature is kept at roomtemperature, and the compressed temperature at the top dead cen-ter (TDC) is varied through a variation of the compression ratio.

Cylinder Wall Crevice

Piston

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Fig. 1. Typical computational grid at the end of the compression stroke.

2.1. Computational fluid dynamics simulations

CFD simulations are conducted for an axisymmetric configura-tion using the FLUENT package [19]. In the actual physical situationin the RCM, the resulting flow may deviate from axisymmetricbehavior and exhibit asymmetrical patterns. However, axisymmet-ric configuration is nevertheless chosen because it considerably re-duces the computational time while capturing the essentials of theunderlying fluid dynamics.

Computations are performed from the beginning of the compres-sion stroke with a compression time of 30 ms. Initially, before com-pression stroke begins, the gas mixture at rest is specified with auniform temperature of 297 K and known pressure. A fixed temper-ature of 297 K and no-slip conditions are specified at the cylinderwall boundary and the piston surface. FLUENT simulations use thesegregated implicit solver with the PISO (Pressure-Implicit Split-Operator) algorithm for the pressure–velocity coupling, the PRESTO(Pressure Staggering Option) scheme for pressure, and the second-order upwind discretization for density, momentum, and species.

The kinetic mechanism of n-heptane and the associated ther-modynamic parameters are taken from the model of Liu et al.

[20] in CHEMKIN [21] format. This skeletal mechanism of Liuet al. [20] consists of 43 species and 185 reactions. It has a conciserepresentation of low-temperature kinetics, including formation ofalkyl peroxy radicals, subsequent isomerization, and addition ofsecond O2. Furthermore, this skeletal mechanism was validated[20] against ignition delay measurements from RCM and shocktube at pressures up to 44 bar, species profiles for n-heptane oxida-tion in a plug flow reactor at 3 atm, and species distribution pro-files in n-heptane–air counterflow diffusion flames.

In the simulations, the piston starts from rest and its motion isgiven in a manner similar to the piston motion in an engine, byspecifying dimensions for the crank radius and the connectingrod length. At the top dead center, the piston comes to rest and re-mains there for the subsequent time-steps. Throughout the calcu-lations, a time-step of 41.667 ls is taken, which corresponds to0.25 crank angle degree of a compression stroke of 30 ms (180crank angle). The choice of this time-step was partially governedby the need for having a consistent computational grid duringthe entire compression stroke. It was found that the time-steps lar-ger than this value result in the failure of deletion/merging of thecells adjacent to the moving piston, and eventually lead to negativecell volumes. Furthermore, we will show and discuss in Section 3.1that for the conditions simulated here this time-step of 41.667 lsis also adequate for the post-compression event prior to finalhot-ignition.

A typical computational grid distribution at the end of compres-sion is shown in Fig. 1. In the axial and radial directions, fine gridsare taken near the walls/surfaces. Independence with respect tothe size of the time-step and grid distribution are ascertained bychecking selected calculations on a finer grid with smaller time-step size yielding identical pressure histories. Representative re-sults for grid-size and time-step independence will be presentedin due course. At every time-step, convergence is ascertained bymonitoring residues to less than 10�3 for the governing equationsof continuity and momentum and less than 10�6 for the energy andspecies conservation equations. It is noted that a much finer time-step size is required to accurately resolve temperature and pres-sure rise during the rapid hot-ignition event. However, the use ofsuch a fine time-step does not alter the solution before the hot-ignition or the value of the overall ignition delay. This is furtherconfirmed by conducting calculations near the hot-ignition eventby using a time-step size smaller by two orders of magnitude. Assuch, the very fine resolution of solutions during the entire hot-ignition process is deemed unnecessary for the presentinvestigations.

All calculations are performed for laminar flow conditions. Bycomparing the CFD calculations with the experimental measure-ments, it was shown [15] that the laminar simulations adequately

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describe the temperature and velocity fields inside the currentRCM, whereas turbulent simulations based on Reynolds-Average-Navier Stokes equations smear out the features of the temperaturefield and fail to predict the experimentally observed temperaturefield.

2.2. Zero-dimensional simulations

In addition to the CFD simulations, zero-D simulations are alsoconducted using SENKIN [1]. As discussed in [6], the SENKIN calcu-lations include the compression stroke and take the effect of heatloss during compression and post-compression period into accountthrough an approach based on volume expansion, which uses theempirically determined heat loss parameters obtained from thenon-reactive pressure history. The non-reactive pressure historycan be used to back calculate the time-dependent ‘effective vol-ume’ of the core region. The heat loss model developed for thepresent RCM is briefly described in the following.

In order to obtain the effective volume, an empirically-deter-mined parameter, Vadd, is added to the actual time-dependent geo-metric volume of the combustion chamber, Vg(t), in order to matchthe simulated pressure at the end of the compression stroke,t = tTDC, with the experimental or desired pressure, PC, value. Here,Vadd simulates the effect of heat loss during compression. After theend of compression, t > tTDC, volume expansion is specified in termsof a polynomial fit, vp(t). The effective volume after compression isthen taken as the product of the effective volume at TDC, Veff(tTDC),and the fitted volume expansion term, vp(t); namely,

Veff ðtÞ ¼ VgðtÞ ¼ Vadd t 6 tTDC ð1ÞVeff ðtÞ ¼ VeffðtTDCÞvpðtÞ t > tTDC ð2Þ

Therefore, Vadd and the polynomial function vp(t) are the keyparameters of the present volume expansion model.

For simulating the RCM experiment of a given reactive mixture,a corresponding non-reactive experiment is first carried out with amixture of the same values of the mixture heat capacity, initialpressure, and initial temperature as for the reactive mixture. Assuch, the parameters required for the present heat transfer model,including Vadd and vp(t), can be determined from the pressure his-tory, P(t), of the non-reactive experiment. The empirically-deter-mined parameters are then used for simulating the correspondingreactive case. Numerical calculations are performed using the San-dia SENKIN code [1] in conjunction with CHEMKIN [21]. For model-ing the RCM experiment, conditions specified in the SENKINcalculations are those of a closed adiabatic system with the time-varying volume prescribed by Veff(t). Implicit in this approach isthe assumption that the heat loss parameters deduced from thenon-reactive run are unchanged even after the first-stage ignition.

In this paper, the primary objective is to compare the CFD cal-culations with the present zero-D model to assess the applicabilityof such a zero-D approach. As done in the experiments, the pres-sure trace obtained from the corresponding non-reactive CFD sim-ulation is used as the reference pressure trace for deriving the heatloss parameters that are used in the subsequent zero-D simulation.The reactive CFD solutions are then compared with those obtainedfrom the reactive zero-D simulations using SENKIN. As such, CFDcalculations represent ‘computational experiments’, which willbe simulated through a zero-D modeling approach.

0 2 4 6 8 10 12 14 16

Time (ms)

Fig. 2. Comparison of solutions obtained from FLUENT (lines) and SENKIN(symbols) for the constant volume, adiabatic calculation using the skeletalmechanism of Liu et al. [20]. Initial pressure = 15 bar, initial temperatures = 750 Kand 850 K, and mixture molar composition: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922.

3. Results

3.1. Computational validations

In order to ensure the predictability of the FLUENT package forreactive simulations, calculations are first conducted for a constant

volume adiabatic system and compared with the solutions obtainedfrom SENKIN following the same approach as in [16]. For these FLU-ENT calculations, a simple two-dimensional grid with adiabatic wallconditions is used and the reactive mixture is at rest initially. Fig. 2shows such a comparison with initial pressure of 15 bar and initialtemperatures of 750 K and 850 K using the mixture of n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922 by mole percentage. At750 K, two-stage ignition is observed, whereas it is single-stage at850 K. In addition, for both FLUENT and SENKIN results an inflectionpoint is noted during the final hot-ignition event perhaps due to thereduced chemistry of the skeletal mechanism employed. Moreimportantly, it is seen that the FLUENT solutions closely follow theSENKIN results and small discrepancy is noted when approachingthe time of hot-ignition. For the conditions investigated, the pre-dicted overall ignition delays of FLUENT are typically 3% shorter thanSENKIN. Here, the overall ignition delay is defined as the time dura-tion from the start of the calculation to the inflection point in thepressure history during the final hot-ignition. Considering the differ-ences in the numerical solvers, a first-order implicit unsteady for-mulation in FLUENT and a higher order formulation in SENKIN, theFLUENT solution is deemed adequate, especially for the predictionof highly temperature sensitive ignition delay.

Furthermore, grid-size and time-step independencies were en-sured. A representative case is discussed in the following. Fig. 3presents CFD simulated RCM pressure traces for the mixture ofn-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922 by mole per-centage, compressed to 14.63 bar and 766.1 K from the initial con-ditions of 0.585 bar and 297 K. Pressure traces are presented for:(a) grid distribution similar to that shown in Fig. 1 and a time-stepof 41.667 ls, (b) grid distribution similar to Fig. 1 and a time-stepsmaller by an order of magnitude (i.e., 4.1667 ls) and (c) grid dis-tribution finer by a factor of 4 than Fig. 1 and a time-step of41.667 ls. All pressure traces clearly depict the phenomenon oftwo-stage ignition. Three curves in Fig. 3 are on top of each otherand it is evident that the grid-size and time-step refinement havenegligible effect on the evolution of the pressure history, for theconditions investigated. Therefore, it is ascertained that the grid-size employed and time-step of 41.667 ls are adequate for thepresent simulations. Further, heat release during the first-stageignition is also adequately captured by the numerical scheme.

3.2. Temperature fields and thermokinetic interactions for reactivesimulations

Features of the temperature fields obtained from the CFDsimulations depend on the compressed gas temperature. In the

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Fig. 3. Demonstration of the grid-size and time-step independence. CFD-simulatedpressure traces for (a) grid distribution of Fig. 1 and time-step = 41.667 ls (circles),(b) grid distribution of Fig. 1 and time-step = 4.1667 ls (line), (c) grid distributionfiner by a factor of 4 than Fig. 1 and time-step = 41.667 ls (triangles). Mixture molarcomposition: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Conditions at TDC(time = 0 ms): PC = 14.63 bar and TC = 766.1 K.

G. Mittal et al. / Combustion and Flame 157 (2010) 1316–1324 1319

following, we present and discuss the evolution of the temperaturefield for two representative cases. Mixture composition is n-hep-tane/O2/N2/Ar = 0.562/10.307/58.209/30.922 by mole percentage,and compressed conditions at TDC are respectively PC/TC = 14.63 bar/766.1 K and 14.1 bar/888 K. Both temperatures liewithin the NTC regime with 766.1 K at the lower end and 888 Kat the upper end of the NTC temperature window. It is noted thatcompression to 888 K required a compression ratio of 16.45, withstroke length of 25.4 cm and clearance of 1.4 cm; whereas for766.1 K, the compression ratio was lower at 10.83, obtained forma shorter stroke of 17.63 cm and a larger clearance of 1.55 cm.Fig. 4 shows and compares pressure traces for both cases revealingignition phenomena of two-stage at 766.1 K and single-stage at888 K. Solid circles, marked a–h in Fig. 4, are the time instants atwhich the temperature fields are presented in Fig. 5.

Fig. 5 presents temperature field snapshots for compressedtemperatures of 766.1 K (a–d) and 888 K (e–h). Specifically,Fig. 5b and c shows the temperature fields before the onset andafter the end of the first-stage ignition. It is noted that for thecompressed temperature of 766.1 K, the effect of roll-up vortex iscompletely absent and temperature evolution proceeds homoge-neously except for the boundary layer and the cool crevice zone.In contrast, in Fig. 5e–h for compressed temperature of 888 K,some effect of roll-up vortex is noted in the form of temperaturenon-homogeneity. In this case, the vortex effect is not completelysuppressed by creviced piston due to the use of a long compressionstroke. A longer compression stroke tends to exacerbate the forma-tion of the roll-up vortex.

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Fig. 4. CFD-simulated pressure traces. Mixture molar composition: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Conditions at TDC (time = 0 ms): PC =14.63 bar/TC = 766.1 K and PC = 14.1 bar/TC = 888 K.

In order to clearly illustrate the temperature non-homogeneityand thermokinetic interactions at 888 K, Fig. 6 presents and com-pares temperature fields for both reactive and non-reactive simu-lation. Unlike actual experiments, for simplicity the non-reactivesimulation is conducted by suppressing the fuel oxidation chemis-try during calculations without changing the mixture composition.Furthermore, for clarity only the temperature field from the max-imum instantaneous temperature (Tmax) to Tmax �60 K is shownand the crevice zone is not shown. The effect of the roll-up vortexis now clearly visible. From Fig. 6, some interesting observationscan be made by comparing the temperature fields for reactiveand non-reactive simulations at any instance of time. It is notedthat the temperature non-homogeneity at any instant of time isgreater for the non-reactive simulations than for the reactive sim-ulations. This behavior is due to the NTC behavior in which exo-thermic reactions proceed at a faster rate at lower temperatureand thereby diminish the vortex induced temperature non-homo-geneity. This observation is consistent with the experimental workof Desgroux et al. [9] and Griffiths et al. [12]. Also, by comparingthe temperature fields for the reactive and non-reactive simula-tions at 15 ms, we note that the maximum instantaneous temper-ature for the reactive conditions occurs in the vortex-affectedregion. Effectively, due to the influence of the NTC related reac-tions, heat release in some portion of the vortex-affected regionhas more than compensated for the temperature non-homogeneity.

Desgroux et al. [9] used fine wire thermocouple for measuringtemperature field for non-reactive and reactive mixtures at differ-ent radial locations in RCM. For the non-reactive mixtures, roll-upvortex induced temperature non-homogeneity persisted aftercompression, whereas for reactive iso-octane mixtures a tempera-ture leveling effect, ascribed to the negative temperature coeffi-cient of reaction rate, was observed [9].

Consistent with the work of Desgroux et al. [9], Griffiths et al.[12] showed that for n-pentane at compressed temperatures closeto the upper end of the NTC region, the reaction in the cooler zonedeveloped faster and the temperature non-homogeneity inducedby the roll-up vortex inside the reaction chamber rapidlysmoothed out. However, Griffiths et al. [12] pointed out that spatialnon-homogeneity of concentrations of intermediates could bethere, which would affect the eventual evolution of spontaneousignition. Present CFD simulations could help us understand theinfluence of such spatial non-homogeneity on ignition.

Fig. 7 shows non-homogeneity in species concentration by pre-senting mole fractions of selected species at 10 ms post-compres-sion. From Fig. 6, it is clearly noted that the extent of thetemperature non-homogeneity at 10 ms for the reactive simulationis smaller than the non-reactive counterpart. Therefore, a closerlook at the species distributions at this particular time can high-light the chemical influences that tend to smooth out the associ-ated temperature non-homogeneity. Only concentrations ofparent reactants (n-heptane and O2), formaldehyde (CH2O), andH2O2 are taken for illustration. CH2O is taken because it forms insubstantial amount in cool flames, and H2O2 is known to initiatethe hot-ignition through its thermal decomposition. In Fig. 7, it isseen that the mole fractions of n-heptane and O2 are of lowest val-ues in the low temperature, vortex-affected region, whereas molefractions of CH2O and H2O2 happen at their highest in that region.This is again due to the influence of the low-temperature chemis-try through peroxidation reactions. At low temperatures,chain-branching through peroxidation and isomerization reactionsdominates, eventually leading to the formation of highly reactiveOH radicals which quickly consume parent hydrocarbon. Due tothese reactions, formaldehyde also accumulates. Furthermore, con-centration of H2O2 builds up through HO2 that is formed via Habstraction by O2 molecule from lower hydrocarbons. The overall

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Fig. 5. Temperature fields (K) at different times for reactive simulations. Mixture molar composition: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Conditions at TDC(time = 0 ms): (a–d) PC = 14.63 bar and TC = 766.1 K; (e–h) PC = 14.1 bar and TC = 888 K.

1320 G. Mittal et al. / Combustion and Flame 157 (2010) 1316–1324

low temperature chain-branching mechanism is highly exother-mic. At higher temperature, non-branching pathway through b-scission dominates which reduces the reactivity. Lower concentra-tions of n-heptane and O2 and higher concentrations of CH2O andH2O2 within the low-temperature, vortex-affected region, asshown in Fig. 7, is therefore due to the domination of the low-tem-perature chemistry.

We further note that due to the NTC behavior, temperature non-homogeneity in the vortex-affected region is reduced and higherconcentration of H2O2 is developed therein. From Fig. 6a at 15 msand Fig. 7d, we note that the maximum temperature occurs withinthe vortex-affected region, and H2O2 concentration also happens tobe higher there. Since hot-ignition is attributed to thermal decom-position of H2O2, we might expect ignition event to ensue in thesame region. Fig. 8 shows the temperature field and the concentra-tion distributions of H2O2 and OH at 20 ms post-compression whenthe maximum instantaneous temperature has reached a value of1060 K. Existence of maximum temperature in the vortex-affectedregion indicates that ignition happens there. Also, the concentra-

tion of H2O2 in the vortex-affected region is now lower due to itsdecomposition through H2O2 + M ? 2OH + M.

Through the discussion of the temperature and concentrationfields, we notice that multi-dimensional effects are indeed presentin the RCM. Possibly, deviation from the zero-D modeling approachmight occur due to the cooler boundary layer and crevice zone aswell as the effect of vortex. In the following, CFD simulated resultsfor ignition delay are compared with the associated zero-D model-ing in order to assess the validity of the zero-D approach.

3.3. Assessment of zero-D modeling

Fig. 9 illustrates an example of comparison of zero-D and CFDsimulated results for non-reactive conditions. Again, all non-reac-tive simulations are conducted by simply suppressing the fuel oxi-dation chemistry during calculations. The mixture, n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922 by mole percentage, is com-pressed to 14.63 bar and 766.1 K. For the CFD simulation, the max-imum instantaneous temperature (Tmax) is plotted. Furthermore, in

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Fig. 6. Temperature fields (K) at different times for (a) reactive simulation and (b) non-reactive simulation. Mixture molar composition: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Conditions at TDC (time = 0 ms): PC = 14.1 bar and TC = 888 K. The non-reactive simulation is conducted by simply suppressing the fuel oxidation chemistry.

(a) n-heptane (b) O2 (c) CH2O (d) H2O2

4.55e-3 4.24e-3 3.93e-3 3.62e-3 3.30e-3 2.99e-3 2.68e-3 2.37e-3 2.06e-3 1.75e-3 1.43e-3

1.03e-1 1.02e-1 1.01e-1 1.00e-1 9.97e-2 9.89e-2 9.81e-2 9.72e-2 9.64e-2 9.56e-2 9.47e-2

2.35e-3 2.12e-3 1.88e-3 1.65e-3 1.41e-3 1.18e-3 9.42e-4 7.06e-4 4.71e-4 2.36e-4 1.12e-5

1.31e-3 1.18e-3 1.05e-3 9.18e-4 7.87e-4 6.56e-4 5.25e-4 3.94e-4 2.63e-4 1.32e-4 6.28e-7

Fig. 7. Mole fractions of selected species at 10 ms post-compression. Conditions: same as Fig. 6a. Initial reactant mole fractions: n-heptane = 0.00562 and O2 = 0.10307.

(a) Temperature (b) H2O2 (c) OH

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1.32e-3 1.19e-3 1.06e-3 9.25e-4 7.93e-4 6.61e-4 5.30e-4 3.98e-4 2.66e-4 1.35e-4 3.01e-6

3.91e-7 3.52e-7 3.13e-7 2.74e-7 2.35e-7 1.95e-7 1.56e-7 1.17e-7 7.82e-8 3.91e-8 0

Fig. 8. Temperature field (K) and mole fractions of selected species at 20 ms post-compression. Conditions: same as Fig. 6a.

G. Mittal et al. / Combustion and Flame 157 (2010) 1316–1324 1321

order to obtain the pressure and temperature traces from zero-Dsimulation, the non-reactive pressure trace from the CFD calcula-tions is first used to back calculate the ‘effective volume’ as de-scribed previously. The effective volume is then used for thezero-D simulation. It is noted that based on this approach not onlythe pressure but the maximum instantaneous temperature is alsoexactly reproduced by the zero-D SENKIN solution, as shown in

Fig. 9. The same degree of agreement between zero-D and CFD sim-ulated non-reactive cases was obtained for all the conditions stud-ied herein.

Fig. 10a and b presents a comparison of the CFD and zero-D-simulated pressure and temperature traces for the reactiveconditions of Fig. 9. Again, for the CFD simulation, the maximuminstantaneous temperature (Tmax) is plotted in Fig. 10b. Both

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Fig. 9. Comparison of CFD (lines) and zero-D (circles) simulations for non-reactiveconditions. Mixture molar composition: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Conditions at TDC (time = 0 ms): PC = 14.63 bar and TC = 766.1 K.The non-reactive simulation is conducted by simply suppressing the fuel oxidationchemistry. For the CFD results, maximum temperature profile is plotted forcomparison.

0

5

10

15

20

25

Pres

sure

(ba

r)

Time (ms)

non-reactive

zero-DCFD

(a)

400

600

800

1000

1200

1400

-30 -20 -10 0 1

-30 -20 -10 0 10

Tem

pera

ture

(K

)

Time (ms)

zero-DCFD(b)

Fig. 10. Comparison of CFD and zero-D simulated (a) pressure traces and (b)temperature traces for reactive conditions. Mixture molar composition: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Conditions at TDC (time = 0 ms): PC =14.63 bar and TC = 766.1 K. For the CFD results, maximum temperature profile isplotted for comparison. The non-reactive simulation is conducted by simplysuppressing the fuel oxidation chemistry.

1322 G. Mittal et al. / Combustion and Flame 157 (2010) 1316–1324

pressure traces reveal the phenomenon of two-stage ignition. Fol-lowing the convention, the first-stage ignition delay (s1) and thetotal ignition delay (s) are defined as the time duration from theend of compression to the inflection point in the pressure historyduring the first-stage ignition and the hot-ignition, respectively.In this case, the CFD simulated total ignition delay (s) of 10.7 msis greater than the zero-D simulated total ignition delay of8.66 ms. In contrast, the predicted first-stage ignition delay (s1)for both simulations is practically the same. This example shows

that although the pressure and temperature traces are adequatelyreproduced by the zero-D approach with ‘adiabatic volume expan-sion’ for non-reactive conditions, the same approach could intro-duce errors in the prediction of the total ignition delay forreactive conditions with two-stage ignition. In Fig. 10a, pressuretraces also reveal that the zero-D-simulated pressure rise duringthe first-stage ignition is higher than the CFD-simulated pressurerise. A comparison of simulated temperature profiles shown inFig. 10b demonstrates that the first-stage ignition leads to slightlylower temperature rise in CFD than in zero-D. Furthermore, for theCFD simulation the mass averaged temperature remains signifi-cantly lower than the maximum instantaneous temperature dueto the cooler boundary layer and crevice zone.

As shown and discussed in Fig. 5, the simulated temperaturefield illustrates complete absence of the roll-up vortex and homo-geneous evolution of combustion except for the cooler boundarylayer and crevice. Therefore, the discrepancy between the CFDand zero-D simulated results cannot be attributed to the effect ofthe vortex. It is then logical to expect that the discrepancy is dueto the multi-dimensional effects during the first-stage ignition,which is not properly captured through the empirically deter-mined heat loss parameters.

During the first-stage ignition under conditions when vortex isalso suppressed (cf. Figs. 5a–d), significant chemical heat releasetakes place in the core region which is maintained at high temper-ature, and practically no heat release occurs in the cooler boundarylayer and crevice. As such, temperature in the crevice is too low toallow for any low temperature heat release through low-tempera-ture chemistry. In addition, no low-temperature reactivity was ob-served in the cooler boundary layer. It is further noted that in thiscase the thermal boundary layer thickness is�0.8 mm at the end ofthe compression and increases gradually thereafter. As a conse-quence of the non-uniform heat release, the pressure and temper-ature rise are lower than what it would have been in the absence ofthe cooler boundary layer and crevice. The zero-dimensional ap-proach of ‘adiabatic volume expansion’ can successfully take intoaccount the effect of heat loss to the wall, and predict the temper-ature and pressure traces of the core regime for non-reactive mix-tures and also for reactive mixtures before there is non-negligibleheat release (e.g., the situation prior to the onset of the first-stageignition). Thus, the model result for the first-stage delay closelymatches with the CFD predicted value. However, the zero-D modelfails to account for the effect of non-uniform heat release duringthe first-stage ignition, and hence predicts higher pressure andtemperature rise due to the first-stage ignition and consequentlyshorter total ignition delay than the prediction of the CFD analysis.

For the situations of single-stage ignition at compressed tem-peratures near the upper end of the NTC temperature window(cf. Figs. 5e–h and 6), heat release due to low-temperaturechemistry can happen in the vortex-affected region before thehot-ignition event and attempt to smooth out the temperaturenon-homogeneity. In this case, multi-dimensional effects couldact in opposite ways. Whereas heat release as a result of low-tem-perature chemistry could accelerate local ignition, the absence ofany heat release in the crevice and boundary layer could delaythe final hot-ignition. Therefore, it is important to ascertain the le-vel of discrepancy between the CFD and zero-D results over the en-tire NTC regime.

In the following, we present a comparison of the CFD and zero-D simulated results over the entire NTC regime with variations inpressure and composition. Fig. 11a compares the results of first-stage ignition delay (s1) and overall ignition delay (s) for the molarcomposition of n-heptane/O2/N2/Ar = 0.562/6.184/62.969/30.285at compressed pressure near 15 bar. It is seen from Fig. 11 thatthe first-stage ignition delays are accurately reproduced by thezero-D simulations. In addition, the zero-D simulated total ignition

0

10

20

30

40

50

725 775 825 875

Igni

tion

Del

ay (

ms)

TC (K)

τ

τ1

(a)

0

5

10

15

20

25

30

725 775 825 875

% E

rror

in I

gnit

ion

Del

ay

TC (K)

τ

τ1

(b)

Fig. 11. (a) Comparison of CFD (circles) and zero-D (triangles + lines) simulatedresults of first-stage ignition delays (s1) and total ignition delays (s); (b) percentageerror in ignition delay prediction, 100 � |sCFD � szero-D|/sCFD. Mixture molar com-position: n-heptane/O2/N2/Ar = 0.562/6.184/62.969/30.285. Pressure at TDC:PC � 15 bar.

0

5

10

15

20

25

30

Igni

tion

Del

ay (

ms)

TC (K)

τ

τ1

(a)

0

5

10

15

20

25

30

775 825 875

775 825 875

% E

rror

in I

gnit

ion

Del

ay

TC (K)

τ

τ1

(b)

Fig. 12. (a) Comparison of CFD (circles) and zero-D (triangles + lines) simulatedresults of first-stage ignition delays (s1) and total ignition delays (s); (b) percentageerror in ignition delay prediction, 100 � |sCFD � szero-D|/sCFD. Mixture molar com-position: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Pressure at TDC:PC � 15 bar.

0

5

10

15

20

25

Igni

tion

Del

ay (

ms)

TC (K)

τ

τ1

(a)

0

5

10

15

20

25

775 825 875 925

775 825 875 925

% E

rror

in I

gnit

ion

Del

ay

TC (K)

τ

τ1

(b)

Fig. 13. (a) Comparison of CFD (circles) and zero-D (triangles + lines) simulatedresults of first-stage ignition delays (s1) and total ignition delays (s); (b) percentageerror in ignition delay prediction, 100 � |sCFD � szero-D|/sCFD. Mixture molar com-position: n-heptane/O2/N2/Ar = 0.562/6.184/62.969/30.285. Pressure at TDC:PC � 20 bar.

G. Mittal et al. / Combustion and Flame 157 (2010) 1316–1324 1323

delays follow the same trend as the CFD results and are consis-tently lower. Furthermore, the discrepancy is larger at lower TC

when the first-stage ignition is also more vigorous. Fig. 11b showsthe percentage error introduced by the zero-D simulations in thefirst-stage and total ignition delays. Percentage error in ignition de-lay is defined as 100 � |sCFD � szero-D|/sCFD. It is noted that the per-centage error in the first-stage delays is negligible. On the otherhand, the error for the total ignition delays is higher and decreasesas compressed temperature is increased beyond the cut-off tem-perature for the existence of the first-stage activity. Nevertheless,the maximum error in total ignition delays for these simulationsis less than 20%.

Fig. 12 shows and compares the results at compressed pressureof �15 bar for a different mixture with the molar composition of n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Here, the samelevel of discrepancy is also observed. In order to assess the effectof compressed pressure on the discrepancy, simulations were fur-ther conducted for a lower compressed pressure of �10 bar and ahigher compressed pressure of �20 bar. Results are shown inFig. 13 for n-heptane/O2/N2/Ar = 0.562/6.184/62.969/30.285 atPC � 20 bar, and in Fig. 14 for n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922 at PC � 10 bar. While the overall behavior in Figs.13 and 14 is similar to that in Figs. 11 and 12, a noteworthy featureis that the percentage error in the total ignition delay is lower atthe higher PC of �20 bar and higher at the lower PC of �10 bar. Thisobservation makes sense because heat loss effects are reduced athigher pressure due to lower thermal diffusivity of the reactingmixture. Furthermore, lower thermal diffusivity implies thinnerboundary layer at higher pressures and consequently decreasedinfluence of multi-dimensional effects and non-uniform heat re-lease. From Figs. 12–14, we also note that the percent error in s1

could increase with increase in TC. At higher compressed tempera-tures, s1 is small and the intensity of first-stage ignition is alsomild. Also, mild heat release during the first-stage ignition has to

overcome concurrent heat loss and pressure drop for a net pressurerise and inflexion point to be observed. Reduced intensity of thefirst-stage reactivity in CFD simulations in comparison to zero-D,

2

10

18

26

34

42

50

58

Igni

tion

Del

ay (

ms)

TC (K)

τ

τ1

(a)

0

5

10

15

20

25

30

35

40

725 775 825 875

725 775 825 875

% E

rror

in I

gnit

ion

Del

ay

TC (K)

τ

τ1

(b)

Fig. 14. (a) Comparison of CFD (circles) and zero-D (triangles + lines) simulatedresults of first-stage ignition delays (s1) and total ignition delays (s); (b) percentageerror in ignition delay prediction, 100 � |sCFD � szero-D|/sCFD. Mixture molar com-position: n-heptane/O2/N2/Ar = 0.562/10.307/58.209/30.922. Pressure at TDC:PC � 10 bar.

1324 G. Mittal et al. / Combustion and Flame 157 (2010) 1316–1324

when coupled with mild first-stage ignition and small value of s1,results in an increase in percent error for the prediction of s1.

Finally, the result that the percent error from zero-D simula-tions decreases under the conditions exhibiting single-stage igni-tion is consistent with our previous work for hydrogen ignition[16]. It was demonstrated in [16] that zero-D simulations yieldedvery small error in comparison to CFD results when the effect ofthe vortex roll-up is adequately suppressed.

4. Conclusions

In this work, CFD simulations are conducted for the ignition ofn-heptane in an RCM with a creviced piston in order to assessthe validity of the zero-dimensional modeling in ignition delayprediction under the conditions of two-stage ignition and NTC re-sponse. From evolution of temperature field during reaction, it isnoted that at compressed temperatures close to the peak of theNTC response, reaction could proceed faster in the vortex inducedlow temperature region, which in turn smoothes out temperaturenon-homogeneity. Based on the comparison of the CFD simulationsand zero-D calculations over the entire NTC regime, it is observedthat the zero-D simulations predict the first-stage ignition delaysaccurately but lead to a higher pressure rise in the first-stage igni-tion and consequently shorter total ignition delay. However, thetrend of the total ignition delays is nicely reproduced by thezero-D approach. Also, the discrepancy by using the zero-D ap-proach is pressure dependent and decreases as compressed pres-sure increases. This work shows that although creviced pistonoffers the advantage of suppression of the vortex, multi-dimen-sional effects are not completely avoided during the conditions of

two-stage ignition. Furthermore, when only single-stage ignitionis present, the error introduced by zero-D simulations is consider-ably smaller.

Since computational fluid dynamics simulations of RCM exper-iments are not always a viable approach because of computationalcost, the extent of agreement between the CFD and zero-D could beimproved by incorporating multiple zones, including core region,boundary layer, and crevice, and allowing for the exchange of massand energy. However, such a model could introduce additionaluncertainties because physical processes of heat loss to the walland pressure drop are coupled with the chemical processes andheat release. It is felt that the benefits derived from introducing acomplex model may not be worth as the expected discrepancyfrom the currently used zero-D approach is not excessive fromthe viewpoint of ignition delay prediction. It is further anticipatedthat the level of discrepancy will be fuel dependent, with higherdiscrepancy for fuels exhibiting stronger first-stage heat releaseand lower for fuels with weaker first-stage heat release. Presentsimulations exhibit reasonably strong first-stage heat release andsimulation conditions cover up to long ignition delays of 50 ms.Therefore, typically we may not expect discrepancy of greater than30%, based on the current investigation. We believe that instead ofresorting to a more complicated simulation model for RCMs, itwould be sufficient if modelers exercise caution that zero-D simu-lated results may not well predict total ignition delays under theconditions of two-stage ignition, as discussed in this work.

Acknowledgments

G.M. was supported by The University of Akron, College of Engi-neering. C.J.S. would like to acknowledge the support of the Na-tional Aeronautics and Space Administration under Grant No.NNX07AB36A.

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