On combustion in a closed rectangular channel with initial vorticity

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On combustion in a closed rectangularchannel with initial vorticityRohitashwa Kirana, Indrek Wichmana & Norbert Muellera

a Department of Mechanical Engineering, Michigan StateUniversity, 428 S. Shaw Lane, Room 2555, East Lansing, MI 48823,USAPublished online: 03 Apr 2014.

To cite this article: Rohitashwa Kiran, Indrek Wichman & Norbert Mueller (2014): On combustionin a closed rectangular channel with initial vorticity, Combustion Theory and Modelling, DOI:10.1080/13647830.2014.894643

To link to this article: http://dx.doi.org/10.1080/13647830.2014.894643

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Combustion Theory and Modelling, 2014

http://dx.doi.org/10.1080/13647830.2014.894643

On combustion in a closed rectangular channel with initial vorticity

Rohitashwa Kiran∗, Indrek Wichman and Norbert Mueller

Department of Mechanical Engineering, Michigan State University, 428 S. Shaw Lane, Room 2555,East Lansing, MI 48823, USA

(Received 18 July 2013; accepted 27 January 2014)

This article examines the detailed combustion process in a theoretical model withapplicability to combustion in a wave rotor or wave disc engine. The model comprisesa single channel into which an initial loading of methane and air is admitted and ignitedafter all inlet and exit ports have been closed. Combustion takes place at constant volume.However, the initial gaseous mixture in the channel is not at rest. The initial opening andclosing of the ports generates significant vorticity which influences the evolution of thecombustion process. Numerical evaluations are provided for the detailed flame shape forsimplified (one-step) chemistry and a simulation using the detailed 235-step San Diegoscheme. Quantities examined are the evolution of vorticity, pressure fluctuations, massconsumption rate, flame surface area and the influences on combustion of adiabaticand non-adiabatic channel walls. Combustion regimes are identified and compared withsimpler model studies (no initial flow). Pointwise quantities are examined to describethe various stages of burning in the channel. The focus of the study is directed towardsquantities that influence overall burning rate and completeness of combustion.

Keywords: confined channel; premixed; vorticity; transition; detailed chemistry

1. Introduction

In constant volume combustion devices, it is important to minimize the time required forcomplete combustion of a fuel and air mixture. This study was motivated by the need foran assessment of the burning process and the required combustion time in a wave rotor ora wave disc engine. These devices achieve compression of a combustible mixture by thesudden closing of one port and power extraction when another port opens. Details on themechanical and thermal operation of such devices can be found in [1–4].

It is commonly observed that a flame propagating from one end to the other in anarrow channel starts off with a parabolic or ‘mushroom’ shape flame, which evolvesinto a concave (toward the burned combustion products) ‘finger’ shaped flame that laterinteracts in a complicated manner with its self-generated flow to evolve into a ‘tulip’shaped convex flame. The tulip flame has been observed and studied for nearly a centurybeginning with the 1928 work of Ellis [5]. More recently, thorough discussions of theessential research, along with numerous references, are to be found in [6–8]. Kratzel et al.[9] have performed experiments and numerical simulations that demonstrate this tulip flameformation. Gonzalez et al. [10] have carried out numerical simulations of flame propagationin a closed tube ignited at one end. These simulations used laminar flow, one-step chemistryand a computing procedure based on the finite volume technique that was restricted to two-dimensional, compressible, reacting flows.

∗Corresponding author. Email: [email protected]

C© 2014 Taylor & Francis

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The tulip flame phenomenon has, to date, been chiefly attributed to the Darrieus–Landau instability, in which small perturbations of an initially flat flame front lead towrinkling and unstable growth of the flame surface. The subsequent acceleration of theflame is mitigated by the formation of the convex or tulip shape, thus flames propagating inthe tulip stage of combustion display relatively constant velocities and mass consumptionrates. A detailed theoretical examination of the tulip flame is to be found in [11,12]. Recentwork by Hariharan et al. [13,14] has determined that the apparent instability that leads tothe formation of the tulip flame is the consequence of the movement of a stagnation (or,more precisely, saddle) point that forms after ignition (called the mushroom flame stage)and subsequent flame motion (called the finger flame stage), behind the flame. As thisstagnation or saddle point propagates closer to the flame front from the rear of the flame(since the saddle point starts in the combustion product gases behind the propagatingflame front) the flame flattens, which reduces the curvature until it finally becomes zerowhen the flame becomes planar. The saddle point passes through the flame front whenit is exactly planar. Once it has passed through, the saddle point remains in front of thepropagating flame at a fixed distance from it. In this stage of combustion the flame shapebecomes convex toward the ignition side, which yields the classical tulip shape discussedin the literature, see [5–14]. The flame evolution described in [13,14] emphasizes the factthat the process is not one of stability or instability but rather a basic interaction betweenflame and flow morphology. Instead of becoming ‘unstable’, the flame front shape andstructure evolves in response to changing local and global conditions and constraints.

For the practical problem of the wave disc engine, the fundamental considerationsof tulip shapes and possible instabilities and their evolution are not as important as isthe reduction of burn time while maintaining completeness of combustion. In the actualoperation of an engine, the initial conditions prior to the ignition of a combustible mixturein a wave disc engine cylinder (i.e. a constant volume combustion device) are anything butquiescent. The opening and closing of the inlet and exhaust ports, along with the ingestionof fresh oxidizer and fuel, produces an abundance of pressure waves which propagaterapidly back and forth through the channel, serving dramatically to alter the structure ofthe initial spark flame kernel and the subsequent flame front.

The subject of premixed flame and flow interaction has a lengthy and complicatedhistory. In areas related to the current investigation, Markstein examined the generationof vortices when a weak shock passes over a premixed flame in a shock tube [15]. Thevortices thus formed interact with the flame to alter its shape further. Renard et al. [16] pre-sented a review of the progress in theoretical, experimental and numerical investigations offlame/vortex interactions. Picone et al. [17] have discussed the theory behind the generationof vorticity in great detail. Other important work in this area is attributed to Batley et al.[18], Lee and Tsai [19], and Ju et al. [20]. We should note here that the work of Hariharanet al. [13,14] is also, essentially, a flame–vortex interaction study since the stagnation pointthat is instrumental to the evolution of the flame shape cannot exist, or even translate alongthe axis of symmetry in the channel, if the requisite fundamental vortices were not presentin the flow. Until these counter-rotating vortices are generated and modify the character ofthe local flow field near the flame through the formation of local nodes and saddle pointsin the flow field, the flame front shape cannot be substantively altered.

The present study examines numerical simulations of combustion in a closed rectangularchannel with and without an initial pressure field generated by opening and closing of the endwalls. The end walls serve as inlet and outlet ports for the engine. The opening and closingprocess pre-loads the channel with distributed vorticity that influences the subsequent flamepropagation and burning rate process.

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Combustion Theory and Modelling 3

Figure 1. Geometry used in numerical simulations. The channel is rectangular, serving as a modelfor the actual engine channels, which are curved.

The equations and initial/boundary conditions are specified in Section 2. In Section 3 thenumerical solutions are presented. In Section 4 these solutions are analysed and discussedin fundamental terms. Finally, Section 5 presents a set of conclusions for this study as wellas recommendations for future work.

2. Physical problem and equations

2.1. Physical problem

A rectangular channel of length 5 cm and width 0.5 cm (Figure 1) was chosen as thecomputational domain for the numerical simulations. This choice was made not becausechannels of this size were used in the engine (the actual engine channels were larger) butprimarily because the establishment of a numerical solution grid required that the channeldomain should not be extremely large if the required O(10) aspect ratio (note: AR =5/(0.5) = 10) was retained and efficient computability was required. In the initial state,the channel was bounded on all sides by adiabatic walls and contained air at high pressure(7 atm) and high temperature (2200 K). This initial state can be considered to correspondto a fully combusted methane–air mixture from the previous cycle.

Actual engine channels in prototype wave rotor and wave disc engines employ a curvedchannel structure for combustion. These channels are narrower at one end than the otherand hence they are not, strictly speaking, rectangular. The rectangular channel configurationemployed here is for modelling and examining the important processes occurring in thecombustion segment of the engine cycle. The following events occur in sequence.

(1) Outlet port CD opens. The right boundary CD changes from a wall to a pressureoutlet. Outside the channel is air at ambient temperature and pressure. As theair inside the channel rushes out (Figure 2(a)) a leftward-propagating expansionwave is generated which moves towards boundary AB while lowering the pressurebehind it (Figure 2(b)). Once this expansion wave strikes the wall AB, it turnsand propagates oppositely towards the right as the pressure continues to drop.Just before this reflected expansion wave has reached CD, the channel pressure isbelow the ambient pressure (Figure 2(b)). The lowest pressure inside the channelis approximately 0.4 atm below ambient.

(2) Inlet port AB opens. Once the expansion wave has decreased the pressure insidethe channel below the ambient pressure by approximately 40%, boundary AB ischanged into a pressure inlet. The stagnation pressure at this boundary is maintainedat 1.1 atm along with temperature at 300 K. The gas entering the channel from thisinlet is premixed stoichiometric methane–air. The pressure gradient created byevents (1) and (2) forces the fresh methane–air mixture into the channel at a high

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Figure 2. Steps in the loading of the channel with a pressurised, stoichiometric, combustible mixture.The colourmaps show contours of static pressure in pascals.

velocity of approximately 150 m s−1, hence a first estimate for the time required tofill the channel is approximately 5 cm/150 m s−1 = 3.3e − 4 s. A direct computationof the fill-up time gives 5.3e−4 s because the inlet gases do not flow uniformlyinto the channel with velocity 150 m s−1. The effective fill-up velocity is therefore94.3 m s−1.

(3) Outlet port CD closes. The outlet port CD at the far end of the channel is changedfrom a pressure outlet to a wall just prior to the instant at which the methane–airmixture reaches CD, see Figure 2(c). It is advantageous to close the wall early tocapture the high momentum of the flow leaving CD. The ram effect of the suddenstoppage of the flow produces an instantaneous pressure rise at the right end ofthe channel. If the incoming fluid momentum is sufficiently high, this can producea shock akin to the rupturing of a diaphragm in a shock tube. A pressure wavesubsequently travels left toward the near wall, AB.

(4) Inlet port AB closes. The inlet port AB is changed from a pressure inlet boundaryto a wall just before the pressure wave generated by event (3) reaches AB. The com-bustible mixture is confined inside the closed (and now constant-volume) chamberat an elevated pressure of approximately 1.3 atm. The elevated pressure helps toproduce faster burning of the trapped reactant mixture. The total time required forprocesses (1)–(4) is 0.57 ms.

(5) Trapped mixture is ignited. The computational domain contains a circle of radius1 mm, which is of the order of one laminar flame thickness, δL. This region, which isalong the channel centreline with its centre at exactly 2 mm from the inlet port AB, is

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Combustion Theory and Modelling 5

employed in this article as the ignition source for the premixed combustible gas. Inthis article the spark is understood as a region of high temperature that is expected todeposit a locally significant quantity of energy to the combustible gas surrounding it.Thus, after events (1)–(4) are completed, the circumference of this ring is instantlyconverted into a heat source by raising its temperature to 2500 K along with a highheat conduction coefficient (k ∼ 1e + 6 W/m-K), see Figure 2(d). It is importantto understand that only the heat transfer from the spark domain boundary to thefluid next to it is subjected to this high conductivity. The subsequent heat transferto the next layer of numerical cells takes place with the ordinary conductivity ofthe fluid. This thermal process simulates a spark kernel having energy depositionka(2πδL.w)�T/δL ∼ 1.1 kW, where w is the depth of the ring, taken here asunity, and where ka is the thermal conductivity of the gas mixture (taken as air)having average temperature 1650 K (= {[2500+800]/2}), ka = 10.25e−2 W/m-K.In addition, �T = 2500 − 800 = 1700 K is the temperature difference betweenthe fictitious surface and the 800 K pre-heated reactant mixture. The spark timeis approximately 1 ms, which yields a total spark energy of approximately 1 J. Inresponse to this concentrated application of energy, the immediate vicinity of thespark attains a high temperature, and subsequently a chemical reaction is initiated.A flame is generated which travels in a highly stochastic manner unlike a laminarflame in a still medium. The time history of propagation of this flame described inthe following sections is seen to be influenced by events (1)–(4).

2.2. Equations and numerical scheme

The commercial computational fluid dynamics (CFD) code Ansys Fluent was used to anal-yse the two-dimensional channel problem. The compressible Navier–Stokes equations weresolved using a pressure–velocity coupling scheme, with first-order accuracy in time andspace. Species transport equations were solved for the gas mixing and the chemical reac-tions of combustion in the channel. A finite-rate model was used to compute the chemicalsource terms in the species transport equations using Arrhenius kinetic expressions, andignoring the influences of turbulent fluctuations on the reaction rates. The gas viscosity wasmodelled using the Sutherland law. The governing transport equations were solved untiltheir error residuals decreased below 1e−3 at every time step.

One detailed simulation was performed using the 235-step San Diego chemical kineticmechanism (see [21,22]) for CH4 combustion in air. Since we had to resolve the influencesof compressibility (the propagation of pressure waves) as well as to solve 235 transportequations at each time step for the different chemical species, this was a computationallyintensive simulation. This detailed simulation allows the resolution of the flame front,with approximately five grid points over the flame thickness, which was understood to beapproximately O(1 mm) on average. The shear layers formed at the side walls by events (1),(2) and (3) of Section 2.1 are approximately three cells wide. In addition, and perhaps ofgreater value to the investigation, the results from the 235-step simulation can be used as areference for comparing results obtained from simpler and shorter reduced mechanisms withand without the resolution of pressure waves and associated compressibility phenomena.There are approximately 500 grid points in the longitudinal direction and 50 cells in thetransverse direction. This gives a mesh size of approximately 0.01 mm2. The mesh sizeremains constant throughout the simulations.

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Figure 3. Flame propagation in regime 1 using CH and OH as markers. Note the formation ofthe C-shaped fronts. The OH appears to track the flame front and the high-temperature combustionproducts.

3. Numerical simulation results

Regime 1 (ignition and initial flame propagation)

After the spark is ignited, the mixture around it burns, forming a C-shaped flame frontextending toward CD. This ballooning front is often referred to as a mushroom shape in thequiescent initiation case that leads to a tulip flame [13, 14]. The location of the flame frontis delineated by tracking the contours of the radical species CH and OH (see Figure 3).The temperature distribution in the channel shows that a high burned-gas temperature ofapproximately 2000 K is attained within this C-shaped boundary.

Pressure waves arising from the ignition event proceed outward from the spark. Thesepressure waves are reflected by the walls on the opposite ends of the channel. As theypropagate rapidly back and forth, the pressure in the channel rises. In this regime the speedof sound in the amalgamated gaseous mixture of reactants and combustion products isapproximately 550 m s−1 so that a complete traversal of the pressure wave from any fixedpoint back to itself (one complete period) requires 10 cm/(550 m s−1) = 2e − 4 s. In contrastwith the results obtained for an initially quiescent mixture [13,14], the shape of the initialflame front is found to be dictated by the velocity field that pre-exists in the channel andby the placement of the spark kernel inside the channel. Thus, the combustion processis dependent on the detailed inflow and outflow processes described in Section 2.1. Oursimulation shows that the arms of the C-shaped flame front stretch forward as time passes,and that they subsequently coalesce into one coherent front. The coalescence event heraldsthe end of regime 1 burning.

Regime 2 (flame propagation of statistical appearance)

The coherent flame front at the end of regime 1 splits into many small flame fragmentsat approximately 1 ms. Contours of CH and OH radicals in this phase are shown inFigures 4(a) and 4(b). There is no distinct flame front and the flame appears as though it isturbulent. Although the closing of the ports prior to the initial spark causes the bulk flowto cease, we believe that the high initial flow-through with large Reynolds number of order(150 m s−1)(0.5e − 2 m)/(1.5e − 5 m2 s−1) ∼ 5e + 4 is sufficient to produce turbulence inthe channel, which generates a flame that presents as a statistical process. This value of Reis 12.5 times as large as the turbulent transition value of Re in a duct (Retrans � 4e + 3).

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Combustion Theory and Modelling 7

Figure 4. Flame propagation in regime 2, which ends when the separate flames have coalesced. Inregime 2 the flame brush is highly fragmented.

The visible flame propagates to half the channel length by 2.8 ms after ignition, which isapproximately 40% of the total burn time of about 7 ms.

Regime 3 (coalesced flame front propagation)

In regime 3, which begins at approximately t ∼ 1.5 ms, the irregular or statistical combustionprocess of regime 2 subsides as the various combusted regions coalesce into one singleblock of burnt gas. In this regime, combustion is found to occur for the 235-step schemein the vicinity of the right end of the channel near CD. This phenomenon arises becausehigh-temperature combustible gas that was trapped after closing the exit port CD is finallyself-ignited, producing a flame that propagates leftward into the unburned mixture from theright end of the channel. Pressure waves continue to traverse the channel due to reflectionfrom the end walls, causing the flame front to oscillate in the horizontal direction. Betweent ∼ 1.5 ms and t ∼ 5 ms the speed of sound in the composite mixture is approximately565 m s−1.

The visual combustion process proceeds more slowly from the mid-point onwards: itrequires slightly more than 7 ms for complete combustion of the entire mixture in thechannel, and the difference between the beginning of this regime (at approximately 2.8 ms)and extinction (at approximately 7 ms ) is about 4.2 ms or 60% of the total burn time.Figures 5(a) and 5(b) show CH and OH radical concentrations in the immediate vicinity ofthe flame front during the latter part of regime 3.

Figure 5. Flame propagation in regime 3. In this stage of burning, the flame front is coherent.

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Figure 6. Volume average of CH4 in the channel with time. The slope changes near 0.75 ms and at1.5 ms.

Regime 4 (extinction)

In this stage of combustion, which begins for the 235-step scheme at approximately t ∼5.5 ms, the flame has nearly reached the far (right hand) wall CD and is being extinguishedby heat losses (for the case of the isothermal wall) and by reactant depletion. The extinctionregime is especially apparent in some of the simulations.

4. Analysis

In this section we will discuss two sets of numerical simulations. One set, Section 4.1,examines the case with initial flow-through as described in Section 2.1. Both multi-stepand one-step chemistry are examined. The second set of simulations, Section 4.2, examinesthe case without any initial flow-through, i.e. an initially quiescent zero-initial-vorticitysituation.

4.1. Flow-through case with initial vorticity

The different combustion regimes described in Section 3 do not concern only the variationin flame shape and flame position. More importantly, the rate at which fuel is consumeddiffers in each regime. Figure 6 shows the temporal variation of methane mass fraction YCH4

in the channel. In regime 1, which is a short regime, the magnitude of dYCH4/dt is large(−18 s−1) and the CH4 is consumed rapidly. There follows a fast transition to regime 2,where the CH4 consumption rate remains essentially identical (−22.5 s−1). Centred around1.5 ms, there is change in the slope of the CH4 consumption rate. This coincides with thetime when the various stretched flamelets recombine. Combustion then proceeds at a slowerrate in regime 3 (−5.3 s−1) as indicated by the visual location of the flame and principallyby the diminished slope of the methane mass fraction versus time.

4.1.1. Vorticity and flame area

The mechanism by which the flame shape and morphology change [13, 14] and the mannerin which the flame is sheared and split apart, and later recombined, is related to the creationand destruction of vorticity in the channel. Figure 7 shows a plot of the temporal evolutionof the volume integral of vorticity magnitude in the channel. There is a sharp increase in the

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Figure 7. Volume integral of vorticity magnitude in the channel with time. Shown are the threeregimes of combustion.

overall vorticity in the channel immediately after ignition. The higher vorticity ushers inthe ‘chaotic’ regime 2 where fuel is burned at a fast rate; dYCH4/dt � −18 s−1 in regime 1,and dYCH4/dt � −22.5 s−1 in regime 2, as stated previously. The slope of this plot attains alower value starting around 1.5 ms in regime 3. The integrated or total vorticity magnitudehas a fluctuating component that is superposed onto a decreasing trend. This fluctuatingamplitude correlates with the pressure waves that travel back and forth in the longitudinaldirection after reflection from the end walls.

The generation and evolution of vorticity is linked to the initial steps for loading thechannel described in Section 2. As the hot combusted mixture leaves the channel (steps 1and 2 of Section 2.1), the velocity at the far end is very high. The sudden closing of theoutlet (step 3 of Section 2.1) causes the flow to turn on itself, which creates large vortices.In addition, intense vorticity is generated in the shear layers near the walls of the channel inthe flow-through stages 1 and 2 of Section 2.1. Once the mixture is ignited, the behaviourof the ‘churned’ fluid inside the channel primarily determines the resultant flame shape.Figure 8 shows the velocity vectors and CH radical concentration around the spark. Thereare two large vortices above and below the spark which stretch the flame into a C-shape.The apparent ‘chaos’ of regime 2 is in part caused by vorticity generation, predominantlythrough the baroclinicity of the fluid near the flame. The pressure gradient always pointsin the longitudinal direction due to the reflections from the end walls. The density gradient

Figure 8. Comparison of velocity vectors and CH radical concentrations which track the flamefront.

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Figure 9. Comparison of pressure and density contours near the spark indicate a strong baroclinicityat the flame surface after regime 1. Notice that thee pressure and density gradients are aligned onlyover very short segments of the flame front, and that the symmetry produces counter-rotating vortices.

points toward diverse directions around the flame, from the burnt to the unburned gas. Thepressure and density contours in Figure 9 show this baroclinicity. The gradient mismatchgenerates vorticity through �∇ρ × �∇p. Many small vortices are created which stretch andeventually break the front from a coherent front into several parts that form the bulk offlame ‘brush’ propagation in regime 2.

The breakup of the flame front leads to a rise in the flame area, producing fastercombustion. The intermediate species CH exists where combustion occurs, i.e. at the flamelocation. We have taken the concentration of CH in the channel as a measure of flame area.Figure 10 shows the volume average of CH mass fraction in the channel versus time. Whenthe mixture ignites, a large quantity of CH is created. As the flame is sheared and fracturedin regime 2, the CH concentration assumes an approximate value of 8e − 8 until regime3 is entered, where the average value is approximately 1.2e − 8, about 20/3 times smaller.In regime 3, the flame is almost flat and has a near constant area, thus the burn rate iscorrespondingly slower.

Figure 10. Volume average of CH in the channel with time. The ratio of average values is (8e −8)/(1.2e − 8) � 6.7.

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4.1.2. Relationship between mass consumption rate and flame area

It is important to attempt to quantify the difference between flame area and mass consump-tion rate, if any. Ideally, the two concepts are interchangeable, but in reality, although thereis significant overlap, the correspondence is not complete. To see this, we first evaluatethe mass consumption rate, noting that dmCH4/dt = mdYCH4/dt , where the total mass inthe channel is constant throughout combustion. From Figure 6, dYCH4/dt = −22.5 s−1 inregime 2 and dYCH4/dt = −5.3 s−1 in regime 3 (Figure 6). This yields a ratio of 4.2 for theburn rate of CH4 in the two regimes. We compare this to the flame surface area computation,which yields a mass fraction of CH = 7.93e − 08 in regimes 1 and 2 and 1.20e − 08 inregime 3. We treat the CH mass fraction as an indicator of the flame area, and the ratioacross the two regimes is 6.6, which is of the order of magnitude of the CH4 consumptionrate ratio of 4.2. Although an increased flame area is correlated with a higher burn rate,correspondence between these two measures is not one to one. Possible causes of discrep-ancy are inaccurate determinations of the flame surface area using the CH mass fractionmeasure or the CH4 mass consumption rate, or both. Regardless of the actual reasons forthe discrepancy, it is sufficient to notice that in practice the specification of ‘flame surfacearea’ is a difficult and elastic concept that is unlikely to constitute a viable combustionmeasure in any but the simplest problems. The mass consumption rate of reactant (CH4)appears as a much more reliable and robust measure of the overall reaction rate.

4.1.3. Vorticity scaling

The vorticity field in the channel is examined in order to determine its characteristic scalingand also to find where the regions of high vorticity are primarily concentrated. The vorticityin the channel is generated in two dominant ways, first by the initial high-speed flow-through and the creation of two intense shear layers during the loading of the channelwith combustible mixture and, second, by the flame sheet once ignition has taken place.In this simplified scaling analysis only magnitudes, not directions, of the vorticity will bediscussed.

The surface- or boundary-layer-generated vorticity is | �ωflow| = | �∇ × �V | ∼ V/δ, whereδ is the average boundary layer thickness in the channel during flow-through. The total inte-grated vorticity in the channel becomes

∫vol | �ωflow| dτ = ∫

vol | �∇ × �V | dτ ∼ O[(V/δ)τflow],where τ designates volume. Since the volume occupied by the boundary layers on theupper and lower surfaces of the channel is 2δLw, we find for the integrated flow vor-ticity

∫vol | �ωflow| dτ ∼ O[2V Lw]. Substituting V = 150 m s−1 for the initial flow-through

velocity, L = 5e−2 m for the length of the channel and w = 1 m (unit depth) gives∫vol | �ωflow| dτ ∼ 15 m3 s−1.

The scaling of the flame-generated vorticity is achieved by examining the equationD �ω/Dt = ( �∇ρ × �∇p)/ρ2 to obtain

| �ωflame| ∼ (t/ρ2)(�ρ/δF )(�p/δF ), (1)

where the density and pressure gradients are calculated across the flame sheet of thicknessδF (to be specified later). In our scaling we estimate �ρ ∼ ρ0, �p ∼ ρ0S2, ρ ∼ ρ0,where S is the flame speed and ρ0 is a reference value of the gas density, to find | �ωflame| ∼tS2/δ2

F . Integrating this result over the entire channel volume yields∫

vol | �ωflame| dτ ∼O[(tS2/δ2

F )τflame], where τ flame = δFAflame, and the flame area Aflame scales with the channelcross section, hw, so we write Aflame = λhw, where the factor λ accounts for the fact that

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Figure 11. Pressure and vorticity magnitude at a point in the flow field. Between t = 2.6 and 3.3 msthere are eight vorticity oscillations giving the period T � 0.875 ms. Between 2.65 and 3.35 ms thereare four pressure oscillations giving T � 0.4375 ms, for a ratio of 2.0.

the actual flame area is usually approximately O(10) times larger than the cross section hw.We take the characteristic time scale in this expression as the time required for a particle totraverse the flame sheet, t ∼ δF/S. The result is

∫vol | �ωflame| dτ ∼ O[λShw]. Consequently,

the ratio of total flow-to-flame-generated vorticity in the channel scales as follows:

≡∫

vol | �ωflow| dτ∫vol | �ωflame| dτ

∼ 2L

λh

V

S. (2)

Using the values for L and h, V = 150 m s−1, S ∼ 5 m s−1 – because the mixture burnsin O(10 ms) over a 5 cm channel length – and λ ∼ 10 in Equation (2) yields ∼ 60. Sincethe flow-generated total vorticity was estimated as 15 m3 s−1, the flame-generated vorticityis estimated to scale as 0.25 m3 s−1. We take this to be one fluctuation, hence the totalamplitude for the flame-generated vorticity fluctuation is approximately twice this value, or0.5 m3 s−1, which is in general agreement with the order of magnitude of the fluctuationsshown in Figure 11.

These estimates of the scale of the vorticity in the channel during combustion canbe used to filter the total vorticity numerically as follows. The estimated value of theinstantaneous, local vorticity produced by the shear flow is | �ωflow| ∼ V/δ , where δ ∼0.21L/Re is the average turbulent boundary layer thickness (which is derived using thewell-known turbulent boundary layer correlation δ/x = 0.382(Rex)−1/2 for Rex < 1e +7). Our numerical parameters yield | �ωflow| ∼ 1.3e + 5 s−1 as a characteristic magnitudeof the vorticity in this stage. If we use the simpler correlation δ/x = 5(Rex)−1/2 we find| �ωflow| ∼ 2.2e + 5 s−1. For the flame vorticity we use | �ωflame| ∼ tS2/δ2

F ∼ S2/α and findan order of magnitude | �ωflame| ∼ 8e + 4 s−1. Thus the local flow-generated vorticity isonly about three times as large as the local flame-generated vorticity. This should becomeapparent in a filtering process in which smaller cut-offs are used to visualize the localvorticity levels in the channel, see Figures 12(a)–12(d).

4.1.4. Tracking physical properties at a point in the flow field

The centre point of the channel was selected in order to track several flow field propertiesas the flame passes over it. The value of p follows a cyclical rise and fall correspond-ing to the reflecting pressure waves from the opposite side walls of the channel. There

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Combustion Theory and Modelling 13

Figure 12. Four different filters applied to the vorticity field to show where the different scales ofvorticity magnitude are located.

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Figure 12. Continued.

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Combustion Theory and Modelling 15

Figure 13. The burnt and unburnt mixture at different entropy levels. The y-axis shows the log of aquantity θ which is supposed to remain constant in an isentropic process. Two such values of θ arechosen, namely p/ργ and Tγ /(γ − 1)/p. The quantities T∗, p∗ and ρ∗ represent T/Tref, p/pref and ρ/ρref,respectively.

is, however, a general and continuous rise in p as more mixture gets burned and theoverall temperature in the channel rises. The quantities ρ and T occupy two differentsteady values before and after the flame front has passed over the test point. The spe-cific entropy at a point in the flow field is constant before and after the flame spreadsacross it. From elementary thermodynamics, the quantities p/ργ and Tγ /(γ − 1)/p are con-stant when entropy is constant. Figure 13 shows these quantities at a point plotted againsttime as the flame brush passes over it. The entropy increases from one nearly constantvalue to a higher constant value. From these figures the entropy jump is given by s2 −s1 = R[ln (θ2/θ1)], θ = Tγ /(γ − 1)/p, which leads to s2 − s1 = 0.287 kJ/kg-K[ln(θ2/θ1)]T .Using this value we see that for the entire mixture of mass m = 2.8e − 4 kgwe have the heat generation Q = T �S = mT (s2 − s1) = mRT ln(θ2/θ1) = (2.8e −4 kg)(0.287 kJ/kg-K)([2250 K + 800 K, ]/2) ln(1, 316, 770/36, 909) = 0.44 kJ, which isclose to the value Q = 0.37 kJ obtained from the global energy balance in Section 4.1.9.

It is interesting to note the variation of p and the vorticity magnitude at this fixed point(Figure 11). Between t = 2.5 and 3.5 ms the average period of vorticity fluctuations is0.09 ms, while the average period of pressure fluctuations is 0.18 ms, giving a numericalpressure-to-vorticity frequency ratio of 2. From the vorticity equation we see that D �ω/Dt ∝(1/ρ2)( �∇ρ × �∇p) and hence for fluctuations in which density is proportional to pressurein the gas and D �ω′/Dt ∝ (1/ρ2

0 )( �∇ρ ′ × �∇p′) for the fluctuating components we see thatthe substitutions p′ ∝ exp(iωt) and ρ ′ ∝ exp(iωt) give ω′ ∝ exp(2iωt) so that the vorticityoscillation is twice as fast as the pressure oscillation. It is evident from this computationthat in this burning stage the vorticity is generated by the baroclinic torque term.

4.1.5. Vorticity and the mass burning rate relationship

The rapid (V = 150 m s−1) initial flow-through and the sudden closure of the end walls ABand CD (Figure 1) produce an initial total or integrated vorticity level of approximatelyO(20 m3s−1) in the channel (see Figure 7). Then the ignition event and the subsequentbaroclinic torque produce an additional approximate 4–5 m3 s−1 (Figure 7 at ∼0.75 ms).This series of events, which we refer to as regime 1 (ignition) occurs between approximately0 and 0.75 ms, see Figure 7, during which time interval the integrated total vorticity increasesfrom about 18 to 22.5 m3 s−1. After 0.75 ms the total vorticity monotonically decreasesexcept for the small-amplitude oscillations, and any distinctions between regimes 2 and 3are not evident. In fact, the decay of total vorticity follows an exponential decay, which can

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be described by the function 4.8 exp [− 0.49 × (t − 1.5)] where t is in milliseconds fort > 1.5 ms.

By contrast, the mass burning rate shown in Figure 6 produces three distinctstages in which the CH4 consumption rate follows three distinct burning rates. Inregime 1, dYCH4/dt = −18 s−1, in regime 2, dYCH4/dt = −22.5 s−1, and in regime 3,dYCH4/dt = −5.3 s−1. Functionally, total vorticity increases while total CH4 massdecreases in regime 1, thus the relationship is of inverse kind. In regimes 2 and 3, however,both total vorticity and total CH4 mass decrease, hence the relationship is of direct kind.The decay of total vorticity is exponential, whereas the decay of CH4 follows two distinctstraight-line slopes (excepting the oscillations for both sets of curves). Consequently thefunctional relation, though direct, cannot be one of proportionality: mass consumptionrate and total vorticity are not proportional to one another, not in the ‘decaying turbulencestage’ (regime 3) and especially not in the ‘growing turbulence stage’ (regime 1) wherethe former decreases while the latter increases. We believe this result, in regime 3 at least,is partly the consequence of pre-loading the channel with intense flow vorticity prior tocombustion. This pre-loading (whose level is arbitrary, since V can be altered) destroysany potentially simple relationship between total vorticity and mass consumption.

4.1.6. One-step chemistry

Additional simulations were performed using a one-step model for CH4 combustion. Thesame distinct phases are observed as for the multi-step chemistry case. Since this is a one-step model, no intermediate radicals can be displayed. The major quantitative difference inthese results from those in Section 3 is that there is no flame proceeding from the right sidetoward the left in the channel. This can be explained by the absence of reactive radicalsnear the end CD. This increases the time for complete combustion of the reactants in thechannel. The time for complete combustion is now 20 ms, approximately three times aslong as for the multi-step chemistry case.

4.1.7. Correspondence of one-step and multi-step chemistry

We believe it is necessary to establish, if possible, a level of correspondence between theone-step and multi-step chemistry simulations. A complete correspondence is of coursenot possible since there are various features of the multi-step model that are inaccessibleto the single-step simplified model. One of the most prominent of these is the absence inthe one-step model of pressure-induced far-field chemical reactions that ignite and sustaina flame in the multi-step case (see Figure 5(a) and the discussion in Section 3). However,we believe that several global or overall parameters and measures should exhibit a strongcorrespondence. Two of these are the overall mass consumption rate and the vorticity level,both normalised to the maximum values appropriate to each.

Shown in Figure 14 is a plot of YCH4 versus normalised combustion time t = t/tmax,where tmax is the maximum burn time and is different for the one-step and multi-step cases(by a factor of 3). Similarities in the figure are the overall slopes in regimes 1 and 2 tothose in regime 3, and essentially identical normalised transition times t � 0.2 betweenregime 2 and regime 3 combustion. It is not only the oscillations that are diminished but themagnitude of the vorticity that is also diminished for the one-step chemistry case relativeto the multi-step chemistry case. The multi-step chemistry case produces a more detailedstructure of the flame that in effect generates a larger flame area, more vorticity and higherfluctuations as reactions and their reverses occur in the channel. These phenomena are

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Combustion Theory and Modelling 17

Figure 14. Plots of normalised (with maximum value) CH4 mass fraction variation versus nor-malised time scale t/tmax for one-step and multi-step chemistry. The variations are much closer thanthey are with respect to absolute time scales, which differ by a factor of three.

smoothed with one-step chemistry. The smoother flame front leads to smaller local andnet baroclinic torque, therefore not only fluctuation values but integrated volume vorticityvalues are affected. Shown in Figure 15 is a plot of the normalised volume integral ofthe vorticity magnitude, where normalisation is with the maximum value, versus t . Asdiscussed above, the overall trend is very similar except for the lack of oscillations in theone-step case.

These plots indicate that even though various details may differ, there are neverthelesspoints of agreement that permit the discussion of the one-step mechanism and render itspredictions adequate for the purposes of understanding and describing various features ofthe problem. Thus, one-step chemistry reproduces various global features and should forthat reason alone not be abandoned as a research tool.

4.1.8. Adiabatic and constant-temperature walls

The simulations shown so far have assumed adiabatic walls, thus there was no heat lossfrom the domain. A single one-step simulation was performed with the walls at constanttemperature 800 K. The constant-temperature walls do not change the overall burn time bya significant amount. The present research did not address complicated chemistry duringthe flame–wall interaction process since flame–wall interaction is a major research problemin its own right. The authors believe this problem deserves greater attention than it hasreceived to date, see for example [23–25].

Figure 15. Plots of normalised (with maximum value) volume integral of vorticity magnitude versusnormalised time scale t/tmax for one-step and multi-step chemistry.

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Figure 16. Comparison of one-step chemistry burning rate of CH4 in the channel for the caseswith constant-temperature and adiabatic walls. Except for a small middle portion between 0.006 and0.01 s, the CH4 mass consumption rates are essentially identical.

Figure 16 compares the two boundary conditions of constant-temperature walls andadiabatic walls. The two curves are generally similar with respect to the location of thethree burning regimes. The only substantial difference appears between t ∼ 6 ms and t ∼11 ms.

4.1.9. Global (first law) energy balance

The first law, Q = �U, gives the heat released by the combustion reaction as Q =m

∫ 21 cv(T ) dT = m[uair(2250 K) − uair(800 K)], where we use the total mass m = 2.8e −

4 kg and the internal energy values for air as a reasonable proxy for the burned and unburnedgases. This yields Q ∼ 2.8e − 4 kg(1921.3 − 592.3) kJ/kg = 0.37 kJ, which is reasonablyclose to the value estimated using the entropy balance in Section 4.1.4. We are satisfied thatthe overall heat release during the combustion event is approximately 0.4 kJ, which is about400 times larger than the spark ignition energy of 1 J (see Section 2.1). We conclude that thespark ignition event has no influence on the overall energetics of the combustion process,although it may alter the geometric evolution of the flame, as will be discussed later in thisarticle. Our strategy of keeping the spark on for 1 ms ensures that a suitably small quantityof energy is imparted by the spark to the premixed gas in the channel. The quantity of heatconducted to the gas (about 1 J) is reasonable and is of the order of magnitude of a normalspark used in previous work [1–4].

4.2. End walls permanently closed (no initial vorticity)

A separate set of simulations was performed without the opening and closing of end walls asmentioned in Section 2. The San Diego mechanism was used with all parameters remainingthe same as before. Once again, four distinct regimes were observed. In this case, however,the overall combustion was much slower.

4.2.1. Mass consumption rate and flame area

Figure 17 shows the variation with elapsed time of the volume average of the massfraction of CH4 in the channel. In the fast regime we have dYCH4/dt � −11.6 s−1. Theslope of this plot shows two separate temporal regimes of fast and slow burning. After

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Combustion Theory and Modelling 19

Figure 17. Volume average of CH4 in the channel with time (walls always closed). Uses the SanDiego mechanism (full chemistry) to describe the flame chemistry.

about 5 ms, the slope of the graph becomes low and the time to burn the remaining CH4

in the channel is greater than 10 ms. In the slow regime we find dYCH4/dt � −3.3 s−1.Thus, the ratio of CH4 consumption rates in the fast to the slow regimes is −11.6/−3.3 �3.5, which is close to the value 4.2 calculated for the initial flow-through case (see Section4.1.2). The plots of CH (Figure 18) and volume integral of vorticity magnitude (Figure 19),respectively, versus time, shed light on this behaviour. As for the flow-through case, theignition process initiates pressure waves which reflect off the channel walls AB and CD andwherever there are misaligned pressure and density gradients in the channel, vortices arecreated which twist the flame front and split it into many parts. This is apparent in the rise invorticity magnitude as well as mass fraction of CH in the channel. The vorticity magnitude,however, never attains the high values observed in the previous simulations (maximummagnitude ∼23 m3 s−1 in Figure 7). In Figure 19 the maximum integrated vorticity isapproximately 4.5 m3 s−1 which is smaller than the flow-through case by approximately thefactor 23/4.5 ∼ 5. Note that the value of ∼4 m3 s−1 is of the order of the rise in vorticitybetween t = 0.5 and 0.75 ms in Figure 7, suggesting that ∼4 m3 s−1 is the integratedvorticity associated with flame initiation.

Figure 18. Volume average of CH in the channel with time (walls always closed). An approximateaverage value for 0 < t < 3 ms is 1.23e − 7, for t > 3 ms it is � 1.71e − 08 for a ratio of 1.23/1.7 �7.2.

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Figure 19. Volume integral of vorticity magnitude in the channel with time (walls always closed).Uses full multi-step methane–air San Diego mechanism.

4.2.2. Vorticity scaling

Analogous to the flow-through scaling of Section 4.1.3, the flame-generated baroclinictorque vorticity scales as Equation (1), where the time t is now measured from the start ofcombustion. We scale the various quantities in Equation (1) as follows: �ρ∼ρ0, �p∼ρ0S

2,ρ∼ρ0, δF∼(α/S). Here α is the gas thermal diffusivity and S is the flame speed in the initialburning regime. The result is | �ωflame|∼t(S4/α2). In order to obtain the total or integratedvorticity we now evaluate the quantity

∫vol | �ωflame| dτ∼t(S4/α2)δF hw = t(S3/α)hw. This

enables writing the integrated initial vorticity variation in Figure 19 as | �ωflame| · τflame =|�ωflame|0 · τflame + t(S3/α)hw, where the first term on the RHS is the initial value of the totalvorticity produced by the ignition of the spark: from Figure 19 it has the value 2 m3 s−1. Notethat, except for the fluctuations, this variation is linear in t. From the preceding relationship,a mathematical expression for the flame speed is deduced:

S =[

α

hw

d

dt(| �ωflame| · τflame)

]1/3

. (3)

In Figure 19, the slope of the linear rise is d [| �ωflame| · τflame] /dt =[(3.752) m3 s−1]/(2.9e − 3 s) = 600 m3 s−2. Therefore using α = 3e − 4 m2 s−1, h = 0.5e −2 m and w = 1 m (unit depth) in Equation (3) gives S∼3.3 m s−1. Multiplying this by theelapsed time of 2.9e − 3 s in this stage indicates that the flame has propagated a distanceof approximately 1 cm. The total burn time is 20 ms, hence the ratio 5 cm ∗ (3 ms/20 ms)yields a flame propagation distance of approximately 0.75 cm, which is of the same orderof magnitude as the scaling estimate of 1 cm.

4.2.3. Vorticity decay

The growth of vorticity in the channel cannot continue indefinitely. During the entiregrowth process, viscous decay is present according to ∂ �ω/∂t ∼ ν∇2 �ω which simplifies to∂| �ω|/∂t ∼ −ν| �ω|/l2 for a homogeneous turbulence. Here l is a characteristic vorticity decaylength scale, which can be estimated from the solution | �ω|/| �ω0| = exp[−(ν/l2)(t − t0)]where t0 = 2.9 ms and | �ω|0L · h · w = 3.9 m3/s. The vorticity decay approximately followsthe function 3.9 exp [ − 0.187 × (t − 2.9)] where t is in milliseconds. The vorticity scaleis therefore a fraction of a millimeter.

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5. Conclusions

The purpose of this study has been to investigate the nature of combustion of a fuel–air mixture confined to a rectangular chamber or channel with aspect ratio AR = 10.Based on our numerical simulations, we can make some remarks about the importantphenomena involved in the combustion process. First, we are able to identify the distinctcombustion regimes 1–4. Second, we have demonstrated that the parameter which enablesthe combustion regimes to be clearly identified is the mass consumption of fuel, whichmay or may not correspond to the nominal value of the flame surface area. Events andprocesses which lead to an increase in flame area serve to increase the mass consumptionrate of fuel in the chamber although the correspondence has been shown in Section 4.1.2not to be one to one, since dYCH4/dt , which measures the mass consumption rate, does notcorrelate exactly with the flame surface area (as measured by the total mass fraction of CH).Between regimes 2 and 3, for example, the mass consumption rate decreases by a factor of4.2 whereas the surface area (measured by the CH mass fraction) decreases by a factor of6.6. The numerical correspondence is therefore 64%.

In this problem, the only way to increase the flame area is by distorting the flame front.The distortion of the flame front is a result of vorticity that exists prior to ignition andadditional vorticity generation inside the channel by the flame front: the former is nearly anorder of magnitude larger than the latter. Vortices created inside the channel help in swirlingthe interface between the burnt and unburnt gases and breaking the flame into several smallflame fragments. This, along with the vorticity produced by the initial flow-through shearlayers, leads to the period of fastest burning in regimes 1 and 2. As fuel is consumed, and theflame structure (brush) becomes aware of the far wall CD and the distinctly finite volumeavailable for combustion, the flamelets recombine and a single flame front is formed. Thisis the start of regime 3 of combustion in the channel. The vortices inside the channel settledown (in regime 3) with time, and the mass consumption rate of fuel attains a nearly steady,lower value. The mass consumption rate in this slower stage of burning (regime 3) is foundto be about four times slower than that in the initial combustion regimes 1 and 2.

Vorticity generation happens for three primary reasons. The first is before ignition andoccurs due to the opening and closing of end walls prior to flow-through. As the fluidmoving at a high velocity (about 150 m s−1) strikes a suddenly closed wall, it reflectsand large vortices are created. The shear layers at the channel side walls generate concen-trated vorticity, which we refer to herein as flow vorticity. The third way in which vorticity isgenerated, after ignition, is by misaligned local pressure and velocity gradients at the flamesurface. The pressure gradient is longitudinal and fluctuates as pressure waves move backand forth across the length of the channel with a period of approximately 3.3e−4 s. Thedensity gradient points from the unburnt to the burnt fluid. Thus, the flame surface is subjectto baroclinic torques acting in different directions at various locations along the flame frontin the channel. This contribution to the total vorticity is referred to as flame vorticity.

When the flame becomes stable and assumes the form of a single front, the pressureand density gradients are aligned, therefore the above mechanism for vorticity generation issuppressed. The density gradient points from left to right. Pressure waves also travel backand forth from left to right and, thus, barely any additional vorticity is generated after theinitial stochastic regime of combustion. Indeed, regime 3 is one of largely continual andmonotonic decay of overall vorticity.

Concerning the vorticity generated by the shear layers near the wall, we numericallyexamined the velocity profile in the gas in order to examine the dependence of the stream-wise velocity on the vertical coordinate, y. Although there are high velocity gradients near

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the upper and lower walls, there appear to be sufficient cells that the computation of thevorticity via ∂v/∂x − ∂u/∂y is reasonable. We note in addition that the very high near-wallvorticity in the two shear layers is important only for the first few instants of the problem, ineffect only for a fraction of regime 1 since this vorticity is caused by the initial flow-through.Once the end walls of the channel are closed the outlet port is closed (step (3) in Section2.1) flow-through ends and the intense near-wall vorticity is considerably diminished.

There is one feature of the solution generated in this article that does not align withprevious studies of the ‘tulip’ flame phenomenon, see [6–14]. In our numerical solutions,even when the channel walls are completely closed for the entire numerical experiment(Section 4.2), significant levels of vorticity are generated in the flow field and, in addition,the flame never evolves into the classical tulip shape during the latter part of the burn. Sincethe only difference between this analysis and the others is spark placement, we hypothesizethat the heat generated by the 1 mm spark, the centre of which is located 2 mm from thewall AB (see Figure 1), generates the initial flow that produces the C-shaped fronts whichlater morph into the flame brush. Consequently, ignition is a crucially important processthat can, and does, alter the fundamental dynamic structure of the propagating flame.

In the channel it is clear that when there is a high level of mixing the rate of burning ishighest. By contrast, once the mixture has ‘settled’ the rate of fuel consumption decreases toapproximately one-fourth of its maximum value in regime 2 (see Figure 6). This diminish-ment is found even when the two end walls are always closed and the mixture has no initialvorticity generated by flow-through (see Figure 17). This work confirms the long-known factthat in order to burn fast the combustible mixture in the channel should be as well mixed aspossible. This work serves in part as an indication of what happens to the burning rate whenthe level of mixedness is not maintained at its initially high value and instead diminishes. Ifthe mixture were to burn entirely at the initial rate, in which the first half of the mixture burnsat four times the rate of the second half, the overall combustion time would decrease by60%. We note also that the burning rate, though usually strongly dependent upon mixing andvorticity, is not proportional to the latter. In fact, the relationship is not one of proportional-ity even when the two quantities obey the same general trend because they follow differentfunctional forms. This was discussed in Section 4.1.5 and is illustrated in Figures 6 and 7.

We have demonstrated that the ignition energy is approximately 1 J whereas the totalenergy released by combustion ( ∼0.4 kJ) is 400 times larger. It is unlikely that the energeticsof ignition influences the subsequent course of the combustion event. As discussed above,however, the placement of the ignition source is crucial for the flow that is generated inthe channel. Work on this topic at Michigan State University and elsewhere suggests thatmultiple sparks and varied spark placements in the combustion chamber can dramaticallyalter combustion behaviour and time.

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[24] F. Dabireau, B. Cuenot, O. Vermorel, and T. Poinsot, Interaction of flames of H2 + O2 withinert walls, Combust. Flame 135 (2003), pp. 123–133.

[25] R. Owston, V. Magi, and J. Abraham, Interactions of hydrogen flames with walls: Influence ofwall temperature, pressure, equivalence ratio, and diluents, Int. J. Hydr. Energy 32(12) (2007),pp. 2094–2104.

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