In-situ Flame Particle Tracking Based on Barycentric Coordinates for StudyingLocal Flame Dynamics in Pulsating Bunsen Flames

Thorsten Zirwesa,b,∗, Feichi Zhangb, Yiqing Wangc, Peter Habisreutherb, Jordan A. Deneva, Zheng Chenc, HenningBockhornb, Dimosthenis Trimisb

aSteinbuch Centre for Computing, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen,Germany

bEngler-Bunte-Institute, Division of Combustion Technology, Karlsruhe Institute of Technology, Engler-Bunte-Ring 1, 76131 Karlsruhe, GermanycSKLTCS, CAPT, BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China

Abstract

Flame particles (FP) are massless, virtual particles which follow material points on the flame surface. This workpresents a tracking algorithm for FPs which utilizes barycentric coordinates. The methodology can be used withany cell shape in the computational mesh and allows computationally fast spatial interpolation as well as efficientdetermination of the intersection of FP trajectories with iso-surfaces. In contrast to previous flame particle tracking(FPT) approaches, the code is fully parallelized and can therefore be used in-situ during the simulation. It alsoincludes fully parallelized computation of flame consumption speed by integrating reaction rates along a line normalto the flame surface at each FP position. Direct numerical simulations of laminar pulsating premixed hydrogen-airBunsen flames serve as validation cases and showcase the added value of tracking material points for studying localflame dynamics. Exciting the inlet flow harmonically with frequencies equal to the inverse flame time scale leads toa pulsating mode where the flame front is corrugated. Ten times higher frequencies nearly resemble the steady statesolution. The FPs are seeded along the flame surface and are used to track the unsteady diffusive, convective andchemical contributions at arbitrary points on the flame front over time. Their trajectories reveal a phase shift betweenthe unsteady flame stretch rate and local flame speed of the order of 0.1 flame time scales for rich hydrogen flames.This is caused by a time delay between straining and stretch due to curvature. The reason is that diffusive processesfollow the time signal of curvature while chemical processes are most strongly affected by the straining rate, whichdominates the high Lewis number hydrogen flames investigated. This time history effect may help to explain the largescattering in the correlation of local flame speed with flame stretch found in turbulent flames.

Keywords: Flame Particle Tracking, Barycentric Coordinates, Oscillating Flames, Laminar Flames, DirectNumerical Simulation

Supplementary Material: yesWord count: 6200 words

∗Corresponding author: Thorsten ZirwesEmail address: thorsten.zirwes@kit.edu (Thorsten

Zirwes)

Preprint submitted to Proceedings of the Combustion Institute May 13, 2020

Manuscript Click here to view linked References

1. Introduction

The interaction of flames with turbulent flows is ahighly complex phenomenon due to the large numberof relevant time and length scales of many physicalprocesses. Therefore, it is often described by globalquantities like integral length and time scales or tur-bulent flame speed. When it comes to local flame dy-namics, however, statistical descriptions are usually uti-lized, like the correlation between local flame speedand flame stretch. These often show a large scattering,although the same correlation in laminar steady-stateflames yields a unique relation between each value offlame stretch and flame speed [1].

In order to study local flame dynamics in more detailand gain more information from direct numerical simu-lation (DNS), a Lagrangian viewpoint can be employed.In this work, we apply the flame particle tracking (FPT)methodology [2]. FPT is a Lagrangian method, whereinfinitely small, massless, virtual particles, that do notinterfere with the flame or flow, are tracked during thesimulation. These particles, called flame particles (FP),follow material points on the flame surface [3]. Theirtrajectories give insight into the time evolution of localflame dynamics. The method known as FPT was intro-duced by Chauduri [2, 4] and Uranakara et al. [5]. Theyused FPs to study the correlation of local displacementspeed with curvature and strain rates on temperature iso-surfaces using data from DNS of turbulent flames. Thesame methodology was used by Dave et al. [6], but intheir work, the particles are tracked backwards in time.In this way, the FPs can be used to track the evolutionof flame surface area. Chauduri et al. [7] studied Bach-elor’s pair dispersion law for FPs in turbulent flames.Uranakara et al. [8] used FPT to create ensemble statis-tics of displacement speed and curvature in ignition ker-nels in turbulent flows. These works all were limited inthe number of FPs, because the FPT methodology wasimplemented as a post-processing step. For example,in [2], Matlab is utilized for the post-processing of theDNS data. This requires to store large amounts of datawith time steps in high frequency and also limits the uti-lization of parallel computer hardware.

Other authors used Lagrangian methods for studyinglocal flame dynamics recently as well: Hamlington [9]studied fluid particles moving with the fluid flow to in-vestigate residence times and path lengths of the parti-cles in turbulent flames. Another study involving fluidparticles was done by Towery et al. [10] who trackedchemical modes in highly turbulent flames. Scholtisseket al. [11] used Lagrangian tracking to recover un-steady flamelet histories from turbulent flames. Zheng

et al. [12] tracked the evolution of flame surface area el-ements in turbulent flames to study the statistics of prin-cipal curvature and surface area ratio evolution. Simi-larly, Yang et al. [13] investigated the evolution of La-grangian surfaces in turbulent flows and Oster et al. [14]tracked the deformation of flame surface surface micro-patches. In an even earlier work, Sripakagorn et al. [15]used flame element tracking to gain insight into reigni-tion events.

Because all previous FPT approaches were imple-mented as a post-processing step, we present a fullyparallelized FPT implementation in this work, whichgains additional computational efficiency and flexibil-ity due to the use of barycentric coordinates, and canbe used in-situ during the simulation. In all aforemen-tioned works, local flame speed has been evaluated interms of the displacement speed. In some cases, theconsumption speed represents a physically more mean-ingful quantity because it measures the consumption offuel across the whole thickness of the flame front. Thedisplacement speed can be calculated at each position inthe flame front based on the local mixture composition.The consumption speed, however, requires the integra-tion of reaction rate of e.g. the fuel species along a linenormal to the flame front. Doing this on a large num-ber of points is computationally expensive. Therefore,a fully parallelized in-situ computation of the consump-tion speed is presented as well. The implementation de-tails are given in section 2.

The main focus of this work lies on introducing thenovel, computationally efficient flame particle track-ing (FPT) algorithm along with its mathematical basicsand implementation, which is then further validated bymeans of oscillating laminar flames similar to a previouswork [1]. The flame dynamics of highly transient lam-inar flames can help to explain and interpret the morecomplex phenomena in turbulent flames. Therefore,DNS of pulsating laminar Bunsen flames are carried outin section 4 to validate the implemented approach andstudy local time history effects. Additionally, the useful-ness of the FPT method for studying local flame dynam-ics is demonstrated by tracking chemical, diffusive andconvective contributions along trajectories of materialpoints on the flame surface together with the differentcomponents of flame stretch and displacement speed,which is not possible without the use of FPs. Choos-ing a laminar setup allows to study transiently stretchedflames in a controlled manner with specific oscillationfrequencies instead of the broad-band frequency spec-trum present in turbulent flames. Nevertheless, first re-sults for FPT on an iso-surface of a turbulent flame aregiven in S10 in the supplementary materials.

2

2. Tracking Algorithm

Flame particles (FPs) track material points on theflame surface. The flame surface is defined as an iso-surface of a scalar quantity, e.g. temperature or speciesmass fraction. Once a FP is seeded onto an iso-surface,it will stay on that iso-surface over time. By only look-ing at iso-surfaces representing the flame front at differ-ent times, it is not immediately clear which point on aniso-surface at time t1 corresponds to which other pointon an iso-surface at a later time t2. The iso-surface ofthe flame front moves due to two effects: convectionwith the fluid velocity ~u and a velocity due to the burn-ing of the fuel, the displacement speed sd, which movesthe iso-surface in its normal direction ~n. The absolutemovement velocity of the iso-surface ~w can therefore beexpressed as

~w = ~u + sd~n = ~ut + ~un + sd~n (1)

using the convention that ~n points towards the freshgases. This method describes the movement of ma-terial points exactly and without assumptions and allquantities can be recovered from DNS data. The move-ment in normal direction is caused by ~un + sd~n, and themovement in tangential direction to the iso-surface isdue to ~ut, where ~un = (~u · ~n)~n is the fluid velocity innormal direction and ~ut = ~u − ~un in tangential direc-tion. This is illustrated in Fig. 1a. Therefore, FPs movewith ~w, whereas the more commonly known fluid parti-cles move only with ~u.

In some cases, it is advantageous to compute the unitnormal vector ~n from

~n = ∇

(Φ − Φiso

|∇Φ|

)∣∣∣∣∣∣Φ=Φiso

(2)

If evaluated on the iso-surface, this is equivalent to themore common definition ~n = ∇Φ/|∇Φ|. However, usingEq. (2) ensures that ∇ × ~n = 0. This makes computa-tion of some stretch contributions easier, because it isguaranteed that ∇t · ~at = ∇ · ~at, where ∇t is the tangen-tial gradient and ~at an arbitrary vector tangential to theflame surface (see also the supplementary material).

In the most simple case, the position ~p of a ma-terial point on the new iso-surface can be found by~p(t1 + ∆t) ≈ ~p(t1) + ~w∆t, where ∆t is the time step.This methodology is the same for laminar and turbu-lent flame cases. However, especially for turbulentflames, the particle movement should be computed us-ing a predictor-corrector step, Runge-Kutta integrationor special higher order interpolation methods to achievesufficient accuracy. A more detailed discussion on thistopic can be found in [5, 16, 17].

Figure 1: (a) Iso-surfaces at two different times t1 < t2. A point on theiso-surface can be tracked by using a FP, which is shown as red sphereand moves with the velocity ~w. (b) Consumption speed is computedby tracking two additional virtual particles in positive and negativenormal direction, summing up the reaction rates along the line.

Because the FP stays on the iso-surface of an arbitraryscalar Φ it fulfills the conditions

∂Φ

∂t+ ~w · ∇Φ = 0, sd = −

1|∇Φ|

DΦ

Dt(3)

where D/Dt = ∂/∂t + ~u · ∇. If a specie mass frac-tion Φ = Yk is chosen for the iso-surface, the dis-placement speed can equivalently be computed as sd =

(∇ · ~jk − ωk)/|ρ∇Yk |, where ~jk is the diffusive flux ofspecie k and ωk its reaction rate.

In addition to tracking the FPs, another particle typeis introduced for the computation of the consumptionspeed sc in terms of the fuel species F:

sc = −1

ρu(YF,b − YF,u)

∫ωF dn (4)

Here, u and b denote the fully unburnt and burnt state.At each position on the flame front, where a FP is lo-cated, two additional virtual particles are generated (yel-low spheres in Fig. 1b). These are tracked in positiveand negative normal direction to the flame front over adistance of the laminar flame unstretched thickness δ0

thin each direction. Because this is done in-situ, thesevirtual particles can hit the boundary between processesin the decomposed computational domain, so that theyhave to be transferred from one process to another. Thetracking and movement between processes is done con-currently for all virtual particles, enabling the computa-tion of sc on a large number of positions on the flamefront without reducing the performance of the simula-tion significantly.

In this implementation, all virtual particles aretracked in barycentric coordinates. The position of eachparticle is given by the four-component vector ~λ =

(λ0, λ1, λ2, λ3)T . First, each cell in the computationaldomain is decomposed into tetrahedra, where one ver-tex of the tetrahedra is always the cell center, and the

3

Figure 2: (a) Arbitrarily shaped cell with cell center and FP inside.(b) Decomposition into tetrahedra. (c) Correction of the FP positiononto the iso-surface.

other three are cell corner points. One advantage is, thatany polyhedral shape can be uniquely decomposed intotetrahedra, so that the methodology also works on ar-bitrary unstructured grids. Figure 2 illustrates this: inFig. 2a, a polyhedral cell is shown with the cell centerin the middle and a FP in the cell. In Fig. 2b, the cellis decomposed into tetrahedra, highlighting the tetrahe-dron made up by the points a, b, c, d containing the FP.The barycentric coordinates ~λ represent the weights atthe tetrahedron vertices, e.g. vertex a is represented by(1, 0, 0, 0)T , b by (0, 1, 0, 0)T and so on. The barycentricweights fulfill the constraint∑3

i=0 λi = 1 (5)

For an in depth discussion of barycentric coordinates,see [18]. Another advantage of the barycentric coor-dinate representation is, that interpolation of any quan-tity ϕ, e.g. temperature, at the FP position φp can bedone in a computationally efficient way:

ϕp =∑3

i=0 λp,iϕi (6)

where λp,i are the barycentric coordinates of the FP andϕi are the values of the scalar ϕ at the four tetrahedronvertices. Because the Lagrangian tracking of the FP issubject to numerical errors, the FP will not be moved ex-actly on the iso-surface at the next time step after track-ing. In order to avoid accumulation of numerical errors,the position of the FP is corrected after each time stepso that it is positioned exactly on the iso-surface. Inprevious works [2, 5], this was achieved by computingthe iso-surface globally and finding the intersection ofthe iso-surface with the FP trajectory via ray tracing,which is computationally demanding. In this work, theiso-surface is only determined within the tetrahedron,where the FP is located in. The intersection of the iso-surface with the FP trajectory can be determined analyt-ically without the need for ray tracing. The FP positionis corrected by ~n∆c from Eq. (7), which is the shortestdistance between the iso-surface and the FP.

∆c = (T · ~λp) · ~n − (T · ~λedge) · ~n (7)

T is the 3 × 4 matrix (~a ~b ~c ~d) containing the Carte-sian position vectors of the tetrahedron corners and ~λp isthe position of the FP in barycentric coordinates. ~λedgeis found by combining Eq. (5) with the description ofthe iso-surface Φiso =

∑i λiΦi according to Eq. (6).

By identifying an edge of the tetrahedron that inter-sects the iso-surface (e.g. edge b–c in Fig. 2c) and set-ting the two components of λedge to zero correspond-ing to the tetrahedron corners not part of that edge (e.g.~λedge = (0, λ1, 0, λ3)T ), the remaining two unknownsλ1, λ3 can be found with the help of the two equa-tions Eq. (5) and Eq. (6) and the correction vector ∆c

can be computed. If the intersection point lies in an-other tetrahedron, the particle is iteratively moved by∆c = ε(Φiso−Φ)∇Φ|∇Φ|−2 evaluated at the particle posi-tion until it enters the correct tetrahedron. ε is an under-relaxation factor set to 0.9.

The algorithm as described here is implementedin the open-source tool OpenFOAM [19], which pro-vides a framework for particle tracking in barycentriccoordinates. In a medium-scale DNS of a turbulentmethane/air flame on a grid with 64 million cellsusing 500 CPU cores, 6 million FPs were added andtracked during the simulation. The overall computationtime was increased by about 20 %. An animation ofthe turbulent flame case is included in the supple-mentary materials. The code for the in-situ trackingof the FPs, calculating ~n and ~w, position correctiononto the iso-surface, interpolation of arbitrary quan-tities at the FP position and parallel computationof the consumption speed can be downloaded from(vbt.ebi.kit.edu/index.pl/specialtopic/DNS-Links). Note that the code is written in a generalway to perform the particle tracking for any kind ofiso-surface, not just for reactive flows.

3. Physical Interpretation of FP Movement

Surface flame stretch K is usually expressed as

K =1A

dAdt

= ∇t · ~u + sd∇ · ~n (8)

where A is the area of an infinitesimal surface elementon the flame front. An equivalent expression is [20]:

K =1A

dAdt

= ∇t · ~w (9)

This is the same ~w from Eq. (1). Therefore, the move-ment of material points and by definition the trajecto-ries of FPs’ are directly linked to flame stretch. FPsmoving closer to each other signifies negative flame

4

Figure 3: Computational domain with boundary conditions, dimen-sions, inlet profiles, grid resolution and the flame front as red line.

stretch and FPs moving away from each other positivestretch. Therefore, FPs tend to move toward regionswith negative flame stretch which act as sink for flamesurface area. This means that regions with strong nega-tive stretch act as attractors for FPs. This effect can beobserved in the animation of the turbulent flame casein the supplementary materials. In [6], this effect isshown by starting out with equally spaced FPs, whichafter some time collect in regions with negative curva-ture or negative flame stretch, respectively. When takingensemble probability density functions (pdf), this effectshould be considered because otherwise the pdf will beskewed toward the negative stretch range.

4. Pulsating Bunsen Flames

The implemented FPT methodology is employedin DNS of pulsating premixed hydrogen-air Bunsenflames. The aim is to show the added value of employ-ing FPT by studying time history effects on the localflame dynamics, even in the context of laminar flames.

Pulsating laminar flames have been investigated bymany authors in the past [21–28]. Many of the earlierworks, however, used oscillating counterflow flames,which are not stretched due to curvature. In this work,all stretch components are included. Both lean (φ = 0.5)and rich (φ = 4) hydrogen-air flames are considered andthe oscillation frequency is set to f = 0.1/τc, f = 1/τc

and f = 10/τc, where τc is the time scale of the flame,determined from τc = δ0

th/s0l .

The domain is 2D with rotational symmetry (Fig. 3).A center inlet for fresh gas with T = 300 K, p = 1 barand prescribed equivalence ratio φ, and a pilot with theburnt gas at the same φ in chemical equilibrium is used.Both inlets have parabolic velocity profiles. All dimen-sions of the domain are set as multiples of the flamethickness δ0

th = (max(T ) − min(T ))/max(∂T/∂n) andflame speed s0

l , both of the unstretched freely propa-gating flame at the inlet conditions. Therefore, as theequivalence ratio changes in different simulations, thephysical dimensions of the domain change as well. Themaximum velocity is chosen to be umax = 6s0

l , result-ing in Reynolds numbers between 500 and 1000 withrespect to the fresh gas inlet for the investigated cases.The height of the domain is approximately 3 flameheights h f and the radius of the inlets are multiples of δ0

thas well. This has several advantages: flames with differ-ent equivalence ratio experience stretch rates of compa-rable Karlovitz number Ka = K δ0

th/s0l ; the resolution

of the mesh is equidistant ∆x = ∆y = δ0th/20, ensuring

that the flame is resolved with about 20 cells. Comparedto the setup employed in a previous work [1], this setuphas several advantages: only half a flame is included dueto the symmetry, making the simulation more computa-tionally efficient; radial symmetry instead of Cartesiansymmetry causes higher stretch rate due to the curvaturein circumferential direction; the hot pilot stabilizes theflame to allow for higher stretch rates.

This setup has several advantages over common lam-inar flame setups: The flame is stretched both positivelyand negatively and is also curved both positively andnegatively. In the oscillating case, the flame is also tran-siently stretched by one frequency f . This is achievedby letting the velocity profiles at the inlet oscillate withuy = uy,inlet(1 + 0.5 sin(2π f t)). Hydrogen has been cho-sen as fuel because it exhibits a strong Lewis numbereffect and therefore high sensitivity to flame stretch forlean mixtures (Le � 1) and rich mixtures (Le � 1).

During the simulation, FPs are seeded onto the flamesurface, shown in Fig. 3 as red sphere. As explained inthe previous section, material points on the flame sur-face will generally move from positive stretch regionsto negative stretch regions. In this configuration, the tipis negatively stretched and the flame base positively, sothat the FP move toward the flame tip. This can alsobe seen from the components of ~w, shown in Fig. 3: Inthe stationary case, the FP does not move in the normaldirection, because for any stationary flame, ~un = −sd~n.The remaining part of ~w is purely tangential, which isthe tangential fluid velocity component pointing towardthe flame tip, causing a movement of the FP upward.

The simulation is performed with an in-house DNS

5

Figure 4: Cutting plane in the back shows heat release rate Q of theφ = 4 flames excited by f = 0, f = 0.1/τc, f = 1/τc and f =

10/τc (left to right). In the foreground, 3D reconstructed iso-surfacesof YH2 = YH2 ,iso are depicted colored by temperature. For the f =

0.1/τc, heat release contours in the background show the flame inits highest and lowest position during the oscillation. See also theanimation in the supplementary materials.

extension for OpenFOAM [19], which includes detailedchemical reaction mechanisms, in this case a hydrogenmechanism by Li et al. [29], and uses Cantera [30] tocompute detailed diffusion coefficients. For more infor-mation, see [1, 31, 32]. The code uses the finite vol-ume method, second order implicit time discretizationand fourth order interpolation methods for spatial dis-cretization. The total number of cells in the computa-tional mesh is approximately 350 000.

Figure 4 shows snapshots from the simulation of theφ = 4 case. At f = 0 (left), the flame is in steady state.At the flame tip, the heat release rate Q is largest dueto the tip being negatively stretched and Le > 1. Atf = 0.1/τc, the flame oscillates as a whole, here de-picted at the time with the largest flame height as 3Dreconstructed hydrogen mass fraction iso-surface in theforeground and the heat release field at the highest andlowest point during the oscillation in the background(see also the animation in the supplementary materials).At f = 1/τc, the flame front becomes corrugated, asshown by the varying temperature values on the hydro-gen iso-surface. For f = 10/τc, the flame cannot fol-low the excitation anymore and the flame resembles thesteady-state solution. This behavior is the same for all φ.

For the lean case (φ = 0.5) and f = 1/τc in Fig. 5,the wrinkles become more pronounced due to Le � 1.The flame shows the typical tip opening and quenchingregions where the flame is strongly negatively curved.There is even island formation at the flame tip (anima-tion in the supplementary materials). Yellow spheresdepict the FP which track material points on the iso-surface. The black line on the right shows the iso-surface that is tracked by the FP, which in this work isalways defined as the value of the hydrogen mass frac-tion YH2,iso, where the heat release rate has its maximumin an unstretched flame at the same φ.

Figure 5: Case φ = 0.5 and f = 1/τc. Left: snapshot of the 3Dreconstructed iso-surface of YH2 = YH2 ,iso. Right: Black line showsthe YH2 = YH2 ,iso iso-line. The FP are illustrated by yellow spheres.See also the animation in the supplementary materials.

0 2t/τc

−50

−25

0

25

Bal

ance

Ter

ms

(kg/

m3/s

)

y

t

−∇ ·~jH2

ωH2

0 1 2t/τc

y

t

−∇ ·~jH2

ωH2

Figure 6: Time signal on the steady-state flame (left) and f = 1/τc(right) for φ = 0.5. Sub-figures in the lower right corner show they position of the FP over time, which moves upward from the flamebase to the tip. Along its trajectory, the contribution of the chemicalsource term ( ), (negative) convection ( ) and diffusion ( ) aredepicted over time. At steady-state (left), their sum ( ) is zero.

An example of the recorded trajectory of a FP track-ing the YH2,iso iso-surface is shown in Fig. 6. The sub-figures in the lower right corner show the height (y) ofthe FP. It is seeded near the root of the flame and movesalong the iso-surface to the tip over time. The contri-bution terms to the mass fraction balance equation ofhydrogen ∂(ρYH2 )/∂t + ∇ · (ρ~uYH2 ) = −∇ · ~jH2 + ωH2

are recorded over time for one material point, where∇ · (ρ~uYH2 ) is the convective term (conv) and the twoterms on the rhs are diffusion (diff) and chemical sourceterm (RR). At steady-state (left of Fig. 6), the flamestretch becomes more negative as the FP approaches theflame tip, leading to a reduced chemical reaction rateuntil the flame extinguishes because Le < 1 at φ = 0.5.Since the case is in steady-state, the sum of the threecontributions is zero. For the oscillating case (right ofFig. 6), the sum of the three contributions is not zero.

Figure 7 shows time signals of different quantitiesfrom a single FP moving along the flame surface of anoscillating H2/air Bunsen flame at φ = 4, f = 1/τc (third

6

0.75

1.00

1.25

1.50

s/s0 l

a)

0.0

0.5

1.0

1.5

Ka

s∗dsc

Ka

0

1

Ka

b)KasKac

Ka

0.6

0.8

1.0

s∗ d,R

R/s

0 l

c)

0.1

0.2

0.3s∗ d,d

iff/s

0 l

s∗d,diff

s∗d,RR

0 1 2 3 4t/τc

−1000

−500

0

RR

,co

nvO

2(k

g/m

3/s

)

d)

−60

−40

−20

diff

O2

(kg/

m3/s

)

RR O2

conv O2

diff O2

Figure 7: Time signals of one single FP moving along the flame to thetip for the case φ = 4 and f = 1/τc showing: a) s∗d = sd

ρρu

, sc and Ka.b) Total flame stretch and its components. c) Reactive and diffusivecontribution to sd . d) Balance terms for the transport equation of YO2 .

from the left in Fig. 4). Fig. 7a) shows time historiesof the flame speed (consumption speed sc and displace-ment speed s∗d) as well as the normalized total flamestretch Ka. Blue dashed lines mark the times wherepeaks in Ka occur and red dashed line shows peaks ofthe flame speed. It can be clearly seen that there is a timedelay of about 0.25τc to 0.3τc between the flame stretchand the response of the flame in terms of sc and s∗d whilestaying in a moderately low stretch regime. In a previ-ous work [1], this analysis was limited to the movementof the flame tip because it is the only point on the flamesurface that can be tracked without FP. There, phaseshifts between the local flow and the flame movementare of the order of O(0.1/τc) for rich hydrogen flames,which is consistent with the new findings.

In Fig. 7b), the normalized flame stretch Ka is splitinto its components Kac = sd∇ ·~nτc and Kas = ∇t · ~uτc.Peaks of s∗d = sdρ/ρu and sc coincide with peaks of theaerodynamic straining Kas in time, while peaks of the

flame stretch due to curvature Kac are shifted in time.This explains the time delay between the total flamestretch Ka and the flame speed s∗d and sc from Fig. 7a).Because the flame’s curvature is negative for t/τc > 1,the black dashed lines mark the negative peaks of Kac,although they are strictly speaking half an oscillationperiod ahead compared to the other dashed lines. s∗dcan be further split into s∗d,RR = −ωH2/(ρu|∇YH2 |) ands∗d,diff = s∗d − s∗d,RR, where s∗d,RR accounts for chemicalreactions and s∗d,diff for diffusion. Fig. 7c) shows thatthe time signal of s∗d,RR follows most closely the timesignal of Kas, while s∗d,diff follows Kac (this is also truefor the tangential and normal components, not shownhere). Because s∗d,RR is the largest component of s∗din this flame case, the time shift of the flame speed inFig. 7a) can be explained. A similar result is foundin Fig. 7d), where time signals of the convective, dif-fusive and chemical terms of the oxygen mass frac-tion balance equation are depicted. Here, the convec-tive and chemical term follow the time signal of Kas,while the diffusion term follows Kac more closely. Thediffusion decreases with increasing negative curvaturedue to the defocusing of diffusive reactant fluxes. Be-cause the reactive term is much larger than the diffu-sive contribution in this flame at the chosen iso-surface,convection mostly balances the chemical source termsand therefore shares the same time history. In conclu-sion, the time delay between total stretch rate and ob-served local flame speed (both in terms of sc and s∗d)is caused by a time delay between Kas and Kac dueto the different time scales of diffusion and chemistry.The flame’s curvature most strongly affects diffusionwhile the chemical source term follows the aerodynamicstraining, which dominates this flame.

5. Conclusions

A computationally efficient flame particle tracking(FPT) implementation based on barycentric coordinatesis introduced for tracking flame particles (FP) whichfollow material points on the flame surface. This al-lows fast interpolation of arbitrary quantities at the FPlocation and efficient position correction of the FP tothe iso-surface. The methodology can also be used onunstructured meshes due to the decomposition of com-putational cells into tetrahedra. In contrast to previousworks, the implementation is fully parallelized and canbe used in-situ. An in-situ implementation of consump-tion speed calculation is included as well, which has notbeen considered in the previous works related to FP.

DNS of pulsating laminar Bunsen flames, includingboth positive and negative stretch, serve as validation for

7

the method and demonstrate its ability to gain additionalinformation by evaluating time-resolved data along tra-jectories of material points on the flame. Tracking ma-terial points in this case reveals that flame stretch andflame speed show a time shift of the order of 0.1τc forthe considered case, so that the history of previously en-countered flame stretch determines the current value offlame speed. This is caused by a time delay betweenthe aerodynamic straining and flame stretch due to cur-vature. It has been shown that the reason for this de-lay is that time signals of chemical processes follow theaerodynamic straining while diffusive processes havethe same time evolution as the flame’s curvature for theinvestigated H2/air case with Le > 1. Applying the FPTmethod to turbulent flames has revealed similar timehistory effects but their quantification is left for futurework. These results can help to shed light on what themajor influences on the local flame dynamics are andlead to a better understanding of global quantities suchas the turbulent flame speed. The presented implemen-tation is available online, has been shown to efficientlyuse the resources of high performance computers andcan be applied to large-scale DNS.

Acknowledgements

This work utilized the computational resourceForHLR at KIT funded by the Ministry of Science, Re-search and the Arts Baden-Wurttemberg.

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