A direct numerical simulation analysis of localised forced ignition in turbulent slot jets of CH 4 /CO 2 blends

Abstract

The early stages of flame evolution following successful localised forced ignition of different CH $_{4}$ /CO $_{2}$ blends in a slot jet configuration have been analysed using three-dimensional direct numerical simulations. The simulations have been conducted for three different concentration levels of CO $_{2}$ in the fuel blend composed of CH $_{4}$ and CO $_{2}$ (ranging from 0% to 20% by volume). The effects of CO $_{2}$ concentration have been analysed based on five different energy deposition scenarios which include situations where the mean mixture composition at the ignitor varies, but not its location in space, whereas other cases represent the scenarios where the mean mixture composition within the energy deposition region remains constant, but its spatial location changes with CO $_{2}$ concentration. The most favourable region for successful flame development following thermal runaway, from the mixture composition standpoint (i.e. the highest flammability factor), has been found to be displaced close to the nozzle with an increase in CO $_{2}$ concentration. The flame development following thermal runaway exhibits initial growth of hot gas kernel followed by downstream advection and eventual flame propagation along with radial expansion with a possibility of flame stabilisation irrespective of the level of CO $_{2}$ concentration. The triple flame propagation has been found to play a key role in the upstream flame propagation and eventual stabilisation. The orientation of the local flame normal plays a key role in the flame stabilisation. The lift off height has been found to increase with increasing CO $_{2}$ concentration which also adversely affect flame stabilisation for high levels of CO $_{2}$ concentration.

Keywords

Localised forced ignition jet flame edge flame speed scalar dissipation rate direct numerical simulations

Introduction

The localised forced ignition (spark, laser) of flammable mixtures is a fundamental problem in combustion science and safety analysis,^1,2 that deals with a transient process from flame kernel formation to its expansion and transition to full burning state. The ignition process in a gas turbine is complex^1,3,2 and is influenced not only by the global quantities such as minimum ignition energy (MIE) and critical flame radius but also depends strongly on the local and non-local turbulence characteristics, fuel properties and burner configuration, making it challenging to analyse either numerically or experimentally. Yet, a thorough understanding is needed for safety standards as well as in the design of efficient and reliable ignition systems in automotive engines or gas turbines used in power generation or aerospace applications, in which misfire causes ineffective combustion, or where means of restarting the engine must be provided.

Furthermore, despite growing interest and a concerted effort by the transportation industry and government to move to renewable and eco-friendly solutions to meet increasingly stringent emissions and pollutions standards,⁴ internal combustion engines and gas turbines are likely to remain predominant in the near future for high energy density engineering applications while transitioning to the burning of e-fuels and carbon neutral alternatives. One of these alternative is the use of biogas, whose major components are methane and carbon dioxide (up to $5 % - 40$ % by volume of the fuel blend). However, challenges remain in its production in industrial quantities with a fixed composition due, in part, to its biological origin.⁵ Previous experimental studies^6–9,5 have shown that variations in CO $_{2}$ content can influence the localised forced ignition (e.g. spark and laser ignition) performance potentially affecting the subsequent flame propagation. It has further been reported^10,6,11,9,12 that the CO $_{2}$ acts as a heat sink, which results in an MIE increase as the amount of CO $_{2}$ concentration increases. A similar trend was highlighted by Larsson, Berg and Bonaldo¹³ who experimentally investigated the MIE required for successful ignition of hydrocarbon fuels mixed with inert gas in an effort to address the industry demands to ignite gas turbines using the fuel available on offshore plants. A large modification of the reaction zone⁷ and hindrance to flame kernel formation were also found by Mordaunt and Pierce¹⁴ and Lafay et al.⁷ for biogas/air mixtures in a gas turbine configuration, while the presence of CO $_{2}$ was observed to lead to flame extinction (lean blow-out),¹⁴ both being partly attributed to the lower burning velocity of biogas/air mixtures in comparison to methane/air mixtures. The above experimental studies indicate that an increase in the amount of CO $_{2}$ is detrimental towards successful thermal runaway and subsequent transition to self-sustained combustion, directly highlighting the role of the fuel mixture (fuel and diluent) composition.

Over the last decade, direct numerical simulation (DNS) has greatly contributed to the fundamental understanding and subsequent modelling of the complex phenomena encountered in combustion. Pera et al. ¹⁵ demonstrated that mixture homogeneity induced by residual burnt gas has a significant influence on early flame propagation and cyclic variability in the case of forced ignition. The analysis of the forced ignition of homogeneous and inhomogeneous partially-premixed biogas mixtures using DNS data by the present authors,^{16,17,12,18,19} building upon previous work by Chakraborty and co-workers on methane/air mixtures,^20–26 confirmed the experimental findings of the effects of CO $_{2}$ concentration in the fuel blend on the MIE, kernel growth and ignition success. However, these studies considered homogeneous decaying turbulence such that the understanding of CO $_{2}$ concentration in the fuel blend on the forced ignition, kernel development and subsequent transition to self-sustained propagating or stabilised flame in shear-generated turbulence and inhomogeneous mixtures that leads to the co-existence of numerous flame regimes has received relatively limited attention.

Thanks to its simple geometrical configuration, jet flow ignition offers the possibility of exploring this physical process further through the analysis of the forced ignition of sheared stratified mixtures where experimental measurements of ignition probability, kernel growth, flame propagation and potentially flame stabilisation can be made. Pioneering work by Birch et al. ²⁷ found a correlation of successful kernel initiation with the probability of finding a mixture within the flammability limits in a turbulent free jet. A more recent investigation of the spark and flow parameters on the ignition of a round jet has been performed by Ahmed and Mastorakos²⁸ who reported ignition probability maps and provided flame base speed and position measurements. Ahmed and Mastorakos²⁸ also demonstrated the importance of edge flame propagation on the flame development and stabilisation, which has been confirmed by various DNS studies of lifted jet flames.^29–33 This experimental database has also been extensively used to assess the validity of various ignition and flame propagation models within the Large Eddy Simulation (LES)^34–37 and Reynolds-Averaged Navier-Stokes frameworks.^38,39 Previous numerical studies looked at ignition of shear layers and highlighted the importance of edge flame on the kernel growth using DNS^40,41 as well as the kernel position in mixture fraction space using LES.^42,43A recent DNS analysis by the present authors focused on the localised forced ignition of a methane fuelled turbulent planar jet³¹ and analysed the effect of the energy deposition region on the kernel growth and its transition towards a fully stabilised flame.

This present study aims at bridging the aforementioned gap in the literature by investigating the effects of fuel composition (i.e. CO $_{2}$ concentration in the fuel blend) on the kernel growth and its transition towards a fully stable flame resulting from the forced ignition of a turbulent jet. This is achieved by extending the present authors’ previous DNS analysis of the localised forced ignition of a turbulent planar jet³¹ to the use of various compositions in the fuel stream. The effects of CO $_{2}$ concentration in the fuel blend on the reaction kinetics are accounted for by using a two-step chemical mechanism.^44,45 The main objectives of this work are to understand the effects of CO $_{2}$ concentration in the fuel blend on (a) the success or failure of the jet ignition and (b) the three stages of flame evolution outlined experimentally by Ahmed and Mastorakos,²⁸ that is, (i) kernel growth, (ii) downstream and radial propagation, and (iii) potential upstream flame propagation and/or stabilisation. In the following sections, the necessary mathematical background is presented, followed by a description of the DNS database. Subsequently, the temporal evolution of the resulting flame and its propagation behaviour will be presented and discussed along with the edge flame statistics pertaining to the kernel growth and flame stabilisation. Finally, the main findings will be summarised in the last section.

Mathematical background

Thermo-chemistry

The single-step chemistry used in numerous previous studies of forced ignition of homogeneous and inhomogeneous mixtures^46,24,26,25 is unable to account for the effects of CO $_{2}$ concentration in the fuel blend on the flame characteristics, and thus a different mechanism has been used here. The two-step chemical mechanism of Westbrook and Dryer⁴⁴ recently applied in the analysis of the localised forced ignition of turbulent biogas/air mixing layers¹⁶ is also considered in the present study. This mechanism accounts for the presence of $6$ species (CH $_{4}$ , O $_{2}$ , CO $_{2}$ , CO, H $_{2}$ O and N $_{2}$ ) and represents an intermediate step towards more complex chemistry while remaining affordable. It is listed in the form of three steps as shown in Table 1.

Table 1.

Reaction rate constants for the two-step chemical mechanism of Westbrook and Dryer⁴⁴ with the coefficients of Bibrzycki and Poinsot.⁴⁵

	$A$ (cgs)	$β$ (cgs)	$E_{a}$ (cal/mol)
CH $_{4}$ + $1.5$ O $_{2}$ $\Rightarrow$ CO + $2$ H $_{2}$ O	$2.00 \times 10^{15}$	$0.0$	$35.0 \times 10^{3}$
$n_{{CH}_{4}} = 0.9$ , $n_{O_{2}} = 1.1$
CO + $0.5$ O $_{2}$ $\Rightarrow$ CO $_{2}$	$2.00 \times 10^{9}$	$0.0$	$12.0 \times 10^{3}$
CO $_{2}$ $\Rightarrow$ CO + $0.5$ O $_{2}$	$8.11 \times 10^{10}$	$0.0$	$77.194 \times 10^{3}$

The rate constants are expressed using an Arrhenius law as follows:

k_{j} = A_{j} {\hat{T}}^{β_{j}} \exp (- \frac{E_{a, j}}{R \hat{T}})

(1)

where

A_{j}

β_{j}

and

E_{a, j}

are the pre-exponential constant, temperature exponent and activation energy of reaction

j

,⁴⁷ respectively, and

\hat{T}

is the dimensional temperature. The reaction progress rates are then estimated by

Q_{j} = k_{j} \prod_{l = 1}^{N} {[X_{l}]}^{n_{l j}}

(2)

where

N

is the number of species involved in reaction

j

n_{l j}

is the stoichiometric coefficient of species

l

in reaction

j

and

[X_{l}]

is the molar concentration of species

l

. The coefficients retained in this work follow those proposed for the CERFACS 2s_CM2 mechanism^48,45 and are detailed in Table 1.

This mechanism accurately estimates the laminar burning velocity of lean CH $_{4}$ /CO $_{2}$ /air mixtures but fails to do so for fuel-rich mixtures. To improve the accuracy of the mechanism for these mixtures, the pre-exponential adjustment (PEA) proposed by Bibrzycki and Poinsot⁴⁵ has been retained and tailored such that the laminar burning velocity of rich CH $_{4}$ /air mixtures matches closely with the GRI-mech 3.0⁴⁹ predictions.

The combustion of CH $_{4}$ /CO $_{2}$ blends can be represented by the following global stoichiometric equation

{CH}_{4} + 2 (O_{2} + 3.76 N_{2}) + a Diluent \Rightarrow {CO}_{2} + 2 H_{2} O + 7.52 N_{2} + a Diluent

(3)

where the diluent is CO

_{2}

in this work. Following Galmiche et al.,¹⁰ the fuel stream composition is defined using the dilution percentage

ψ_{r}

that represents the molar fraction of CO

_{2}

in the reactants

ψ_{r} = \frac{a}{a + 1 + 2 + 7.52} = \frac{a}{a + 10.52}

(4)

An alternative definition of the fuel stream composition relies on the dilution percentage

ψ_{f}

that represents the molar fraction of the CO

_{2}

in the fuel blend (i.e. CH

_{4}

+ CO

_{2}

), such that

ψ_{f} = \frac{a}{a + 1}

(5)

Note that only the fuel based definition will be used in this work such that

ψ \equiv ψ_{f}

in the reminder of this article.

The validation of the chemistry is presented in Figure 1, where the unstrained laminar burning velocity of a CH $_{4}$ /CO $_{2}$ /air mixture normalised by the laminar burning velocity of a stoichiometric undiluted mixture (i.e. CH $_{4}$ –air, $s_{l}^{0}$ ) at different values of the equivalence ratio and dilution parameter is compared with reference results obtained with Cantera⁵⁰ and the GRI-mech 3.0.⁴⁹ Figure 1 also highlights the effect of the PEA applied for fuel-rich mixtures. Overall, the equivalence ratio dependence of the laminar flame speed is correctly captured across the whole flammability range. However, the flame speed of diluted mixtures is systematically over-estimated by the present mechanism compared to GRI-mech 3.0 or the experimental data of Galmiche et al. ¹⁰ and this overestimation increases with $ψ$ . As $s_{l}$ is overestimated by the 2-step chemistry adopted in the present analysis for $ψ > 0.2$ , the present analysis has been conducted for $0 \leq ψ \leq 0.2$ .

Figure 1.

Laminar burning velocity of CH $_{4}$ /CO $_{2}$ /air mixtures (left) as a function of $ϕ$ and $ψ$ computed with (plain lines) Senga+ using the 2s_CM2 mechanism with PEA and (dashed line) Senga+ using the unmodified 2s_CM2 mechanism (symbols) Cantera using the GRI-mech 3.0 mechanism and (right) for stoichiometric mixtures as a function of $ψ$ .

Following previous numerical investigations of the ignition and flame kernel development in inhomogeneous mixtures,^{51,24,26,21,22,40} several assumptions are made in this work. All the species are considered as calorically perfect gas with equal heat capacities at constant pressure ( $C_{p}$ ). Moreover, the Lewis number of all species is taken to be unity.

The mixture composition is locally described in terms of the mixture fraction $ξ$ , which is defined following Bilger⁵² as

ξ = \frac{β - β_{o}}{β_{f} - β_{o}}

(6)

with

β = 2 Y_{C} / W_{C} + 0.5 Y_{H} / W_{H} - Y_{O} / W_{O}

β_{o}

and

β_{f}

being the values of

β

in the pure oxidiser and fuel streams, respectively, and

Y_{k}

and

W_{k}

denote the atomic mass fractions and weights, respectively. The stoichiometric mixture fraction is thus a function of the fuel blend composition and is given by

ξ_{s t} = - \frac{β_{o}}{β_{f} - β_{o}}

(7)

It is noted that the current chemistry modelling is not designed to capture quantitatively autoignition, however, it can do so qualitatively. Furthermore, autoignition is not expected to be one of the driving mechanism of the flame kernel growth and potential flame stabilisation. Similarly, the chemical scheme adopted here is expected to capture only qualitatively the occurrence of quenching but quantitative differences with a detailed chemical mechanism are expected. In the case of forced ignition, the temperature attained because of external energy deposition is high enough to result in a very small value of ignition delay time. Therefore, any inaccuracies in the ignition delay prediction do not significantly affect the results presented in this paper because the subsequent flame kernel development is principally determined by flame propagation. Recently, it has been demonstrated by Papapostolou et al.¹² that the minimum ignition energy variation with turbulence intensity for the stoichiometric homogeneous mixed biogas–air mixture using this chemical mechanism provides excellent qualitative and quantitative agreements between DNS^46,12 and experimental^53,54 results. Moreover, the flame structure resulting from localised forced ignition of turbulent gaseous mixing layer and droplet-laden mixtures using single-step Arrhenius-type chemistry^24,55,56 are qualitatively similar to those obtained from detailed chemistry simulations.^57,58

It was demonstrated by Turquand d’Auzay et al.³¹ that the edge flame propagation statistics for forced ignition of turbulent mixing layers for the chemical and transport models used in this article have been found to be qualitatively similar to the detailed chemistry DNS studies.^59–61 It is worth noting that pioneering analytical studies^62–66 on ignition have been conducted for single-step irreversible Arrhenius type chemistry. Furthermore, it has been found that the statistical behaviours of the edge flame displacement speed including its strain rate, curvature, and scalar gradient dependencies from simple single-step chemistry DNS data^{24,67,40,29,30} have been found to be qualitatively consistent with detailed chemistry DNS^59–61 and experimental⁶⁸ findings. The current analysis adopts a chemical mechanism which is more detailed in comparison to several previous analyses.^{24,67,40,29,30} Moreover, the seminal DNS analysis on autoignition by Mastorakos et al.⁶⁹ was conducted in the context of single-step Arrhenius type chemistry, and the findings of that analysis (e.g. negative correlation between reaction rate magnitude and scalar dissipation rate) were subsequently confirmed in detailed chemistry DNS studies.⁷⁰ However, in the future the results of the current analysis will need to be validated using a more detailed chemical mechanism, which is capable of predicting both ignition delay and flame propagation characteristics accurately.

Finally, despite the assumptions made in this work but thanks to the presence of the $CO + 0.5 O_{2} ⇋ {CO}_{2}$ reaction in the mechanism,⁷¹ the adiabatic temperature of both stoichiometric and off-stoichiometric mixtures corresponds approximately to those of the full GRI-mech 3.0 mechanism. As a result, the expansion of the flow around the flame as well as its effect on the mixing field are expected to be appropriately accounted for.

Ignition modelling

The effects of the localised forced ignition are accounted for by the addition of a source term to the energy conservation equation ( $q^{‴}$ ) taken to follow a Gaussian distribution in the radial direction from the ignition centre^{65,66,21,23,25,58,72,26,24,46,16} and expressed as

q^{‴} = A_{s p} \exp (- r^{2} / 2 R_{s p}^{2})

(8)

where

r

is the distance from the ignitor centre and

R_{s p}

is the characteristic width of the energy deposition. The constant

A_{s p}

is determined by a volume integration leading to the total ignition power

\dot{Q}

given by

\dot{Q} = \int_{V} q^{‴} d V = a_{s p} ρ_{0} C_{p} τ T_{0} (\frac{4}{3} π {(δ_{z}^{0})}^{3}) [\frac{H (t) - H (t - t_{s p})}{t_{s p}}]

(9)

where

a_{s p}

is a parameter determining the total energy deposited by the ignitor,^{21,23,25,58,72,20,46,16}

τ = (T_{a d}^{0} - T_{0}) / T_{0}

is the heat release parameter (

T_{0}

and

T_{a d}^{0}

are the reactants and stoichiometric adiabatic undiluted biogas/air flame temperatures, respectively),

δ_{z}^{0}

is the Zel’dovich thickness of the stoichiometric undiluted methane/air premixed flame (

δ_{z}^{0} = D_{0} / s_{l}^{0}

where

D_{0}

is the reactants mass diffusivity) and

H (t)

and

H (t - t_{s p})

are Heaviside functions that ensure that the ignitor is only active until

t = t_{s p}

. The energy deposition duration

t_{s p}

is expressed as

t_{s p} = b_{s p} t_{f}

where

b_{s p}

is the energy deposition parameter and

t_{f} = δ_{z}^{0} / s_{l}^{0}

is a characteristic chemical time scale. The suggested range for optimal duration of energy deposition is

0.2 ⩽ b_{s p} ⩽ 0.4

.⁷³ For a shorter duration, strong shock waves that dissipate energy can be formed, and for a longer one, the temperature is wastefully dissipated outside of the energy deposition region. The details of the spark formation (momentum modification contribution, plasma and shock wave formation) are not considered in this DNS database for the purposes of simplicity and computational economy.

Edge flame identification, displacement and speed

In previous experimental,^{32,74,28,75–77} numerical^59,70,61,78 and analytical^79–84 studies, the formation and propagation of a triple flame structure has been shown to play a significant role in the localised forced ignition of inhomogeneous mixtures. Following previous authors,^85,30,29 the edge flame is defined as the intersection of a mixture fraction iso-surface and a fuel mass fraction iso-surface. In this case, the mixture fraction iso-value chosen corresponds to the stoichiometric mixture fraction ( $ξ_{s t}$ , which depends on the fuel blend composition), while the region of the fuel mass fraction isosurface was selected by the zone defined by having a heat release larger than $0.5 {\dot{ω}}_{T, max}$ , where ${\dot{ω}}_{T, max}$ is the maximum heat release obtained using a 1D laminar stoichiometric premixed flame having parameters (fuel blend composition, reactant temperature, etc.) similar to the jets considered in this study.

The movement of the intersection depends on a combination of the three following mechanisms: flow velocity (i.e. $u$ ), motions relative to the flow of the mixture fraction and fuel mass fraction iso-surfaces (i.e. $s_{ξ}$ and $s_{d}$ , respectively). Following Karami et al.,^30,29 a coordinate system moving with the flow velocity is defined as indicated in Figure 2. Two vectors normal to the mixture fraction ( $N_{ξ}$ , pointing towards the oxidiser) and fuel mass fraction ( $N_{y}$ , pointing towards the reactants) iso-surfaces are defined as

N_{ξ} = - \nabla ξ / | \nabla ξ |

(10)

N_{y} = \nabla Y_{f} / | \nabla Y_{f} |

(11)

The vector

T_{1}

, tangent to both the mixture fraction and fuel mass fraction iso-surfaces is used to define the vector

T_{2}

that is both normal to

T_{1}

(pointing towards the reactants) and tangent to the mixture fraction iso-surface, as follows

T_{1} = (N_{y} \times N_{ξ}) / | N_{y} \times N_{ξ} |

(12)

T_{2} = N_{ξ} \times T_{1}

(13)

The fuel mass fraction and mixture fraction iso-surfaces move with respective velocities

s_{y}

and

s_{z}

with respect to the background fluid motion. These velocities are defined following the existing literature^67,41,77

s_{y} = - \frac{1}{ρ | \nabla Y_{f} |} ({\dot{ω}}_{f} + \nabla \cdot (ρ D \nabla Y_{f}))

(14)

s_{ξ} = \frac{1}{ρ | \nabla ξ |} \nabla \cdot (ρ D \nabla ξ)

(15)

The overall velocity of an edge point on the

ξ_{s t}

iso-surface is denoted

V_{e}

and can be decomposed into the

(N_{ξ},, T_{2})

ortho-normal coordinate system, such that

V_{e} = s_{ξ} N_{ξ} + s_{e} T_{2}

,^29,30 where

s_{e} = V_{e} \cdot T_{2}

is the projection of

V_{e}

into the plane tangential to the mixture fraction iso-surface. By further defining

k = N_{y} \cdot N_{ξ}

, and taking the dot product

V_{e} \cdot T_{2}

, the edge flame speed

s_{e}

is expressed as

s_{e} = \frac{s_{y} - k s_{ξ}}{\sqrt{1 - k^{2}}}

(16)

Accounting for the local flow velocity (

u

), the edge flame propagation speed in the laboratory reference frame is

W_{e} = s_{e} + u \cdot \frac{T_{2}}{| T_{2} |}

(17)

and its streamwise component along

x

W_{e}^{x} = W_{e} (T_{2} \cdot e_{x})

(18)

where

e_{x}

is the unit normal vector along

x

pointing downstream.

Figure 2.

Example of the coordinate system moving with the flow velocity along the edge flame – (orange) $Y_{f}$ and (blue) $ξ_{s t}$ iso-surfaces.

Following Ruetsch et al. ⁷⁹ and accounting for the CO $_{2}$ concentration, the predicted theoretical laminar edge flame speed can be expressed as

s_{e}^{t, ψ} = s_{l}^{0, ψ} \sqrt{ρ_{0} / ρ_{b}^{ψ}}

(19)

where

s_{l}^{0, ψ}

and

ρ_{b}^{ψ}

are the unstrained laminar stoichiometric burning velocity and the burned gas density of the mixture with dilution level

ψ

, respectively.

Flame diagnostics

Some parameters used for the flame diagnostics that will be used in the remainder of the paper to analyse the flame behaviour are introduced here.

First of all, the scalar dissipation rate (SDR) which measures the rate of molecular mixing has been shown to play a key role in the development of flame kernel^16,46 in homogeneous and inhomogeneous mixtures as well as edge flames.^{24,30,85,83,82}It is defined as

χ = 2 D {(\nabla ξ)}^{2}

(20)

In this work, the SDR will be normalised by its extinction value⁸⁶ which is defined as

χ_{e x t} = ξ_{s t}^{2} {(1 - ξ_{s t})}^{2} {(s_{l}^{0, ψ})}^{2} / α_{0}

(21)

where

α_{0}

is the reactants thermal diffusivity.

The degree of mixing between the fuel and oxidiser can be measured using the flame index (FI) proposed by Yamashita et al.,⁸⁷ normalised by Domingo et al.,^88,89 Bray et al. ⁹⁰ and modified by Briones et al. ⁹¹ to allow the separation of rich and lean premixed flames, such that

FI = \frac{1}{2} \frac{ξ - ξ_{st}}{| ξ - ξ_{st} |} (1 + \frac{\nabla Y_{{CH}_{4}} \cdot \nabla Y_{O_{2}}}{| \nabla Y_{{CH}_{4}} | \cdot | \nabla Y_{O_{2}} |})

(22)

where in region of non-negligible heat release,

FI = 0

indicates a non-premixed burning and

FI = + / - 1

indicate, respectively, the rich and lean premixed modes of combustion.

Configuration

The simulations have been carried out using the three-dimensional DNS compressible code Senga+^{92,16,46,67,41} which solves the standard species, momentum and energy transport equations for reactive mixtures. The spatial and time differentiations are carried out using high-order finite difference ( $10$ th-order central difference for the internal points that reduces to second-order one-sided at the non-periodic boundaries) and third-order low-storage explicit Runge-Kutta schemes.

To reduce the computational cost, a slot jet configuration has been considered in this analysis as opposed to the axisymmetric jet configuration experimentally analysed by Ahmed and Mastorakos.²⁸ However, the values chosen for the main ratios (e.g. the ratio of thermal flame thickness to slot width (or jet diameter, $h / δ_{t h}^{0} \approx 11$ ), the ratio of jet velocity to laminar flame speed ( $20 ⩽ U_{j} / s_{l}^{0} ⩽ 80$ ) and the ratio of burned to unburned density ( $ρ_{b} / ρ_{0} \approx 0.14$ )²⁸) are kept as similar as possible. Furthermore, to keep the cost within the available computational time it was not possible to match the jet Reynolds number used in the experiments ( $3000 ⩽ {Re}_{j} ⩽ 9800$ ).

The planar jet simulated here has a slot width of $h = 7.8 δ_{t h}^{0}$ (where $δ_{t h}^{0} = (T_{a d}^{0} - T_{0}) / max (| \nabla \hat{T} |_{L})$ is the thermal flame thickness of the stoichiometric laminar undiluted methane/air flame, with $T_{a d}^{0}$ , $\hat{T}$ and $T_{0}$ being the adiabatic flame temperature of the undiluted stoichiometric methane/air mixture, instantaneous dimensional temperature and unburned gas temperature, respectively). The computational domain is long and wide enough to accommodate both the non-reacting and reacting flow developments with dimensions $L_{x} \times L_{y} \times L_{z} = 37 h \times 19 h \times 4.3 h$ and is shown in Figure 3.

Figure 3.

Flow configuration schematic.

An extensive parametric analysis in terms of spatial location of the ignitor centre and the mixture composition at the ignitor location for different CO $_{2}$ blends has been conducted, of which only a subset is considered here, thus some compromise have been made to limit the computational cost. For example, periodic boundary conditions have been used in the transverse direction due to the much longer domain and flow-through time than those considered in other similar analyses reported in the literature.^{30,29,67,93,94}A partially non-reflecting outflow was applied in the streamwise direction using the Navier-Stokes characteristic boundary conditions method.

Note that, given the periodicity and confinement in the spanwise direction, the configuration is more akin to an infinite row of jets in the spanwise direction. It has been demonstrated elsewhere³¹ that the non-reacting flow statistics in terms of mean and root-mean-square (rms) values of mixture fraction $ξ$ and axial velocity $u$ indicate a fully developed flow regime starting around $x^{+} \approx 9 - 10$ (where $x^{+} = x / h$ ) up to $x^{+} \approx 30$ , where confinement effects begin to manifest. However, this confinement does not manifest in the velocity and mixing profiles below $x^{+} = x / h \approx 30$ .³¹ It has been shown in the Supplemental Material of Turquand d’Auzay and Chakraborty³¹ that the mean velocity, jet width and turbulent velocity fluctuation statistics in non-reacting jet matches well with self-similar solutions and experimental findings,⁹⁵ which inspires confidence in the fidelity of these simulations.

The domain is discretised using a uniform Cartesian grid of $1920 \times 990 \times 225$ cells thus ensuring $52$ cells across the jet width and at least seven grid points within $δ_{t h}^{0}$ . This also ensures that the Kolmogorov length scale is equivalent to $1.2$ grid spacings at the nozzle lip.

A fuel/air mixture is injected through the slot with a bulk mean velocity $U_{j} = 23.5 s_{l}^{0}$ and issues in a coflow of pure air with a velocity of $U_{c} = 0.1 U_{j} = 2.35 s_{l}^{0}$ . This yields a jet Reynolds number of ${Re}_{j} = ρ_{0} U_{j} h / μ = 650$ . The coflow is assumed to be laminar without any velocity fluctuations, whereas the velocity fluctuations at the slot inlet are imposed by scanning a plane through a frozen periodic turbulent channel flow solution with a scanning velocity $U_{j}$ . It is worth noting that the fluctuating velocities of the channel flow corresponding to a frictional Reynolds number ${Re}_{τ} = 395$ are scaled with respect to the mean bulk velocity and these scaled fluctuations are used the inflow. The choice of $R e_{τ} = 395$ for the turbulent inflow specification is motivated by the fact that profiles of turbulent velocity fluctuations normalised by bulk mean velocity reach self-similar pattern for $R e_{τ} \geq 395$ . It is important to appreciate that there is a degree of numerical assumption involved in the specification of turbulent inflow. A fully developed channel flow solution profile is used here to specify realistic velocity fluctuations at the inlet rather than specifying artificial noise, digital filtering method or scanning a plane through homogeneous isotropic turbulence. All of the above methods involve some degree of arbitrariness, and the approach adopted in this paper is probably less arbitrary in comparison to the other alternative methodologies because the length scale and turbulent velocity fluctuations are representative of a fully developed channel flow. Moreover, the flame development following localised ignition takes place sufficiently away from the inlet and thus the inflow turbulent boundary condition does not affect the flame dynamics presented here.

The oxidiser mass fraction in air (coflow) is $Y_{O_{2} \infty} = 0.233$ , while the mixture injected through the slot is composed of $27.5 %$ of CH $_{4}$ /CO $_{2}$ blend and $72.5 %$ of air per volume. Three different CH $_{4}$ /CO $_{2}$ blends are considered in this analysis with $ψ$ varying between $0$ and $0.2$ , where the upper limit is chosen based on experimental observations made on a replica of the Cambridge burner used in²⁸ reporting that for $ψ ⩾ 0.25 - 0.3$ no flame stabilisation was possible.⁹⁶ It was further reported that for such dilution levels, obtaining a kernel required a very large energy input and that the flame kernels thus obtained were not necessarily self-sustained once the spark was switched off.⁹⁶ The values of flame speed, $ξ_{s t}$ , $ξ_{l}$ and $ξ_{r}$ corresponding to the different fuel mixtures investigated are summarised in Table 2.

Table 2.

Jet composition, flammability limits (lean $ξ_{l}$ and rich $ξ_{r}$ ) and laminar burning velocity for the different fuel blends considered.

$ψ$	$Y_{{CH}_{4}, j}$	$Y_{{CO}_{2}, j}$	$ξ_{s t}$	$ξ_{l}$	$ξ_{r}$	$s_{l}^{0, ψ} / s_{l}^{0}$
$0.0$	$0.174$	$0.000$	$0.317$	$0.19$	$0.49$	$1.00$
$0.1$	$0.152$	$0.046$	$0.357$	$0.22$	$0.55$	$0.95$
$0.2$	$0.131$	$0.090$	$0.405$	$0.25$	$0.61$	$0.89$

Both the jet and the coflow temperatures are taken to be $T_{0} = 415$ K yielding a heat release parameter of $τ = (T_{a d}^{0} - T_{0}) / T_{0} = 4.5$ . Standard values have been chosen for the Prandtl number ( $\Pr = 0.7$ ) and the ratio of specific heats ( $γ = 1.4$ ).

A precursor non-reacting flow DNS has been conducted for about two flow-through times (where $t_{f t} = 2 L_{x} / (U_{j} + U_{c})$ ) such that the initial transients disappear and that realistic fluctuations of both the mixture composition and velocity fields are obtained. This field is taken to be the initial condition for the reacting flow simulations.

The ignitor energy, deposition duration and characteristic width are kept constant throughout this study and comparable to previous work^{16,46,31,24,40} with $a_{s p} = 10$ , $b_{s p} = 0.34$ and $R_{s p} = 2.45 δ_{z}^{0}$ . Different locations have been simulated as shown in Figure 4 along with the distributions of mean mixture fraction ( $\tilde{ξ}$ ) and flammability factor ( $F$ ). The flammability factor is defined by Birch et al. ²⁷ as the probability of finding a flammable mixture at a given point and measured as $F = \int_{ξ_{l}}^{ξ_{r}} P (η) d η$ (where $P (η)$ is the probability density function of the mixture fraction extracted from the non-reacting DNS data). The normalised coordinates of the ignitor centre along with $\tilde{ξ}$ , $\tilde{ξ} / ξ_{s t}$ and $F$ values are summarised for the different locations in Table 3. It can be seen from Table 3 that the spatial location of the ignitor centre has been modified which gives rise to the variations of the Favre-averaged mixture fraction value $\tilde{ξ}$ and flammability factor $F$ at the ignitor location. This means altogether eight different cases have been simulated and the simulations $01$ , $02$ and $03$ have been discussed elsewhere³¹ and are not discussed in this article. The current work only considers a subset composed of the cases $00$ , $10$ , $11$ , $20$ and $21$ , while the interested reader is directed to the previous work by Turquand dAuzay and Chakraborty³¹ for details on the spark location effects on the probability of successful ignition and subsequent flame development.

Figure 4.

(Left panel) flammability factor ( $F$ ) and (right panel) mean mixture fraction ( $\tilde{ξ}$ ) for (a) $ψ = 0.0$ , (b) $ψ = 0.1$ and (c) $ψ = 0.2$ – (plain line) ${\tilde{ξ}}_{s t}$ , (dashed line) ${\tilde{ξ}}_{r}$ and (dash-dotted line) ${\tilde{ξ}}_{l}$ – the white dots denote the energy deposition locations.

Table 3.

Energy deposition locations and local mean mixture conditions – the italic denotes the cases considered in this work.

Pos	$x^{+}$	$y^{+}$	$z^{+}$	$\tilde{ξ}$	$\tilde{ξ} / ξ_{st}$	$F$
$ψ = 0.0$
$00$	$15$	$0$	$0$	$0.53$	$1.68$	$0.37$
$01$	$10$	$- 1$	$0$	$0.43$	$1.35$	$0.28$
$02$	$7.5$	$0$	$0$	$0.82$	$2.60$	$0.08$
$03$	$22.5$	$0$	$0$	$0.44$	$1.40$	$0.68$
$ψ = 0.1$
$10$	$15$	$0$	$0$	$0.53$	$1.49$	$0.50$
$11$	$11.7$	$0$	$0$	$0.60$	$1.68$	$0.30$
$ψ = 0.2$
$20$	$15$	$0$	$0$	$0.53$	$1.32$	$0.64$
$21$	$9.7$	$0$	$0$	$0.68$	$1.68$	$0.31$

Looking at the cases $00$ , $10$ and $20$ (ignitor on the jet centreline at $x^{+} = 15$ ), it can be seen that an increase of the CO $_{2}$ content in the fuel blend leads to a corresponding increase of the value of $F$ and brings $\tilde{ξ}$ closer to $ξ_{s t}$ suggesting that more favourable ignition conditions are found. By contrast, it appears necessary to move the ignitor closer to the nozzle to maintain a constant value of $F$ as $ψ$ increases, which could detrimentally affect the ignition due to larger velocity and scalar gradients amongst others.

The reacting flow simulations are run for about $t / t_{s p} = t^{+} \approx 82$ which also corresponds to $t \approx 58 t_{j}$ , where $t_{j} = h / U_{j}$ .

Results and discussion

Flame development

Using the selected deposited energy, a successful thermal runaway (the maximum temperature value either reaches or exceeds the adiabatic flame temperature during or shortly after the energy duration deposition period) is observed at all ignitor locations from Table 3 considered in this study. This constitutes the kernel establishment phase of the flame kernel development following the nomenclature outlined by Mastorakos.^1,2 Additionally, for all these cases, a successful self-sustained growth of the kernel (i.e. without the assistance of the external energy deposition) is observed followed by indications of flame stabilisation.

It is worth noting that the outcome of the local energy deposition is highly stochastic in nature, but this aspect is not detailed further here since it was addressed in detail in³¹ for the cases with $ψ = 0.0$ , and a qualitative alteration is not expected with CO $_{2}$ concentration in the fuel blend, even though the resulting ignition probability maps will be different.

The temporal evolution of the non-dimensional temperature $T^{+} = (\hat{T} - T_{0}) / (T_{a d}^{0} - T_{0})$ field in a $x^{+}$ – $y^{+}$ plane is shown in Figure 5 at different times for the ignitor position $21$ . The behaviour presented here has been found qualitatively similar for all the other cases, i.e. the different stages of the flame development that are (i) initial kernel formation, (ii) kernel downstream advection and radial/transverse growth, (iii) upstream flame propagation, described experimentally²⁸ and in LES studies,^35,36,34,38 are also observed here.

Figure 5.

Time evolution of the non-dimensional temperature in a $x^{+} - y^{+}$ plane for the ignitor position $21$ overlaid with black mixture fraction iso-lines (plain: $ξ_{s t}$ , dashed: $ξ_{r}$ and dash-dotted: $ξ_{l}$ ) – insets (i)–(v) show the $ξ / ξ_{s t}$ field overlaid with similar black iso-lines for the normalised mixture fraction and green iso-lines corresponding to $10$ %, $40$ % and $70$ % of ${\dot{ω}}_{T, \max}^{ψ}$ in the $x^{+} - y^{+}$ plane for (i)-(ii) and (iv)-(v) and $x^{+} - z^{+}$ plane for (iii) – the green dot marks the energy deposition location – the black arrowheads highlight some edge flames.

In Figure 5, mixture fraction iso-lines for $ξ_{l}$ , $ξ_{s t}$ and $ξ_{r}$ are superimposed onto the temperature field. Insets present the field of $ξ / ξ_{s t}$ overlaid with green lines indicating $10 %$ , $40 %$ and $70 %$ of the maximum heat release rate ( ${\dot{ω}}_{T, \max}^{ψ}$ ) of the unstretched laminar premixed stoichiometric flame with dilution $ψ$ .

The high-temperature kernel resulting from the thermal runaway remains roughly spherical during the initial flame establishment stage, for a duration that depends on the local conditions. Due to the higher turbulence levels found at the energy deposition location, the kernel loses its sphericity after roughly $t^{+} \approx 5$ for the cases $11$ and $21$ , and retains it up to $t^{+} \approx 8 - 10$ for the cases $10$ and $20$ with an energy deposition at $x^{+} = 15$ . The hot gas kernel is advected downstream (compare its most upstream position between Figure 5(a) and (b) and the energy deposition position) while the flame grows in the spanwise direction (i.e. $z^{+}$ ).

From $t^{+} \approx 7$ , a triple-flame structure develops around the $ξ = ξ_{s t}$ iso-surface in the shear layer (see black arrow in inset (i)). Subsequently, the volume of the kernel increases rapidly thanks to the downstream edge flame propagation supported by the large velocity magnitude at the jet core (see edge flame in inset (ii) and in Figure 5(c)). Turbulent flow-induced advection along with moderate values of $| \nabla ξ |$ and the availability of flammable mixture gives rise to the growth of the hot gas region in the transverse direction (see inset (iii)) that eventually spans over the whole width of the jet for $t^{+} ⩾ 25$ . Consequently, a second triple-flame structure forms along the $y^{+} ⩾ 0$ mixing layer (inset (iv)) while several other similar structures become apparent as the flame development progresses and the flame sheet spans the whole jet width.

The third stage of the ignition process in jet (i.e. $t^{+} ⩾ 30$ ) relies heavily on edge flames that propagate within the flammability limits. The propensity of finding $ξ = ξ_{s t}$ and flammable mixtures (i.e. $ξ_{l} ⩽ ξ ⩽ ξ_{r}$ ) increases with CO $_{2}$ concentration in the fuel blend for a given axial location of the energy deposition (up to $x^{+} \approx 25$ ), but in the case of location $21$ (where the local $\tilde{ξ} / ξ_{s t}$ was kept similar to $00$ ), this also comes at the cost of significantly higher scalar and velocity fluctuations. Downstream of the reactive layer spanning across the jet, turbulence decays rapidly due to heat release while the jet spreads (e.g. from $Δ y^{+} \approx 7 - 8$ at $t^{+} = 30$ to $Δ y^{+} = 14$ at $t^{+} \approx 80$ ) as a result of thermal expansion. Flow acceleration resulting from the thermal expansion is particularly visible between $t^{+} = 14$ and $t^{+} = 41$ in the fast counter-clockwise rotation of a large pocket of hot gases moving upstream against the slower moving coflow in the left shear layer. Large pockets of unburned mixture can still be found downstream of the flame base (e.g. Figure 5(e)) as the fuel-rich jet turbulent core mixes with either the surrounding fresh air or the hot oxygen-rich products resulting from lean combustion. This pocket penetration phenomenon reduces considerably as $ψ$ increases as the mean equivalence ratio becomes lower for a given axial position (see Figure 4). For all cases, the flame base starts to exhibit indications of stabilisation around $55 ⩽ t^{+} ⩽ 80$ driven by the edge flames finding favourable conditions to propagate upstream (inset (v)) up to a point where a balance between their local propagation speed and the downstream fluid motion may be found.

Flame propagation behaviour

Due to the major contribution of edge flame propagation on the kernel development and the kernel-to-flame transition, more discussion is warranted regarding its behaviour. During the kernel expansion (e.g. up to $t^{+} \approx 15 - 20$ ), regions of high heat release are mostly found around the low $| \nabla ξ |$ regions in the vicinity of $ξ = ξ_{s t}$ . The flame sheet expands into both fuel-lean and fuel-rich mixtures, but upon finding too lean regions (i.e. $ξ ⩽ ξ_{l}$ ) local extinctions can be observed. At later times the edge flame structures are found to be sustained even for high values of $| \nabla ξ |$ at which point, consistent with previous findings,^17,16,24 the lean premixed branch collapses onto the diffusion branch under high magnitudes of mixture fraction gradient. Under this condition, the premixed branches tend to disappear and a single branch at around $ξ = ξ_{s t}$ is obtained. In this case, a rich premixed branch may still be observed, while the lean branch collapses completely.

This is further shown in Figure 6 that presents the respective heat release rate contribution of each branch of the triple flame (i.e. conditional integration of heat release rate conditioned on the flame index values corresponding rich- and lean-premixed and diffusion combustion modes) over the flame evolution. At every stage, except for during the early stage, the rich premixed branch is responsible for the major part of the heat release, with the lean branch contribution being lower but non negligible. On the other hand, the heat release due to the diffusion branch remains negligible with less than $5$ % on average. The CO $_{2}$ concentration in the fuel stream does not seem to affect this balance, although at late time, the rich premixed branch contribution slightly increases with increasing $ψ$ . This seems to appear as a consequence of the upward movement of the flame base towards regions where a much wider fuel-rich core surrounded by the shear layers, in which the flames are stabilised, may be found. Thus, with increasing distance from the nozzle, more flame surface can be found within the jet core compared to the surface existing on the lean side of the mixing layers, explaining the growing imbalance of heat release between the lean and rich branches of the triple flames.

Figure 6.

Temporal evolution of (left) heat release rate contribution of (unfilled symbols) non-premixed, (filled symbols) lean and (lines without symbols) rich premixed flame branches and (right) percentage flame elements (selected by ${\dot{ω}}_{T} ⩾ 0.05 {\dot{ω}}_{T, \max}^{ψ}$ ) propagating into richer mixture (front-supported).

It is also important to understand how the kernel expands within the stratified mixture, which can be achieved by analysing the relative alignment of the mixture fraction and progress variable iso-surface normals, quantified through the angle between $\nabla ξ$ and $\nabla c$ ,

\cos (θ) = \frac{\nabla ξ \cdot \nabla c}{| \nabla ξ | | \nabla c |}

(23)

where the progress variable is defined based on the oxidiser mass fraction as

\frac{c = (1 - ξ) Y_{O_{2} \infty} - Y_{O_{2}}}{(1 - ξ) Y_{O_{2} \infty} - \max (0, (ξ_{s t} - ξ) / ξ_{s t}) Y_{O_{2} \infty}}

(24)

A distinction may be drawn between flame elements propagating towards leaner mixture, usually referred to as back-supported flames (i.e.

\cos (θ) > 0

), and those propagating towards richer mixture or front-supported (i.e.

\cos (θ) < 0

). Figure 6(b) also presents the temporal evolution of the percentage of flame elements (selected by

{\dot{ω}}_{T} ⩾ 0.05 {\dot{ω}}_{T, \max}^{ψ}

) being front supported for the different energy deposition locations and dilution parameter values. This is consistent with the observations made from Figure 5 for case 21, which shows that the flame expands into the fuel-rich zone as the time progresses from the initial hot gas kernel phase and the results shown in Figure 6(a) and (b) reveal that the same qualitative behaviour has been observed in other cases considered here. Moreover, the predominant heat release rate in the rich premixed mode (see Figure 6(a)) is also a consequence of dominant heat release contribution arising from front-supported flame propagation (see Figure 6(b)). The observations made from Figure 6(a) and (b) further reveal that the level of CO

_{2}

concentration in the fuel blend does not significantly affect the qualitative nature of initial flame development following successful localised forced ignition.

Evolution of the flame height

The temporal evolutions of the flame base height, measured by the lower axial position at which $T^{+} = 0.1 (T_{a d}^{ψ})^{+} = 0.1 ({\hat{T}}_{a d}^{ψ} - T_{0}) / (T_{a d}^{0} - T_{0})$ or ${\dot{ω}}_{T} = 0.1 {\dot{ω}}_{T, \max}^{ψ}$ are shown in Figure 7. It can be seen from Figure 7 that the evolutions are qualitatively similar for all cases considered here, which is consistent with previous findings^{31,97,36,34,38} for self-sustained flame propagating following ignition at the jet centreline. It can be seen from Figure 7 that the upstream edge is advected downstream following the end of the energy deposition, which is consistent with the observations of Figure 5, and the extent of the downstream advection is dependent on the local flow velocity and mixture composition. Indeed, between $t^{+} = 1$ and $t^{+} = 7 - 8$ , it can be seen that the speed at which the kernel moved downstream increases while the energy deposition gets closer to the nozzle and encounter larger mean streamwise velocities. Once the edge flame structure is fully established, the flame edge starts to propagate upstream and this appears to happen sooner ( $t^{+} = 7 - 8$ against $t^{+} = 20$ ) as the energy deposition is closer to the nozzle, which could be linked with the larger mixture fraction intermittency that allows for the flame to encounter more easily a locally stoichiometric mixture. The time to establish the edge flame structure also appears independent of the amount of CO $_{2}$ , as can be observed by comparing energy deposition $00$ , $10$ and $20$ . This marks the onset of the third phase of flame development, as reported earlier in experimental²⁸ and numerical studies.^97,36,34,38 The duration of this phase depends on the local conditions but also on the $ξ / ξ_{s t}$ and $F$ values at the ignitor location. The flame base may move further downstream due to low-frequency oscillations as shown in Figure 7 for the cases $00$ , $10$ and $20$ between $t^{+} = 20$ to $t^{+} \approx 45 - 55$ . This movement is controlled by the edge flame propagation statistics in response to the large scale unsteady motion and subsequent propagation or extinction into unfavourable conditions such as regions with equivalence ratio outside the flammability range or high values of $| \nabla ξ |$ .

Figure 7.

Temporal evolution of the flame base height based on two different criteria (lines) $T = 0.1 (T_{a d}^{ψ})^{+}$ and (symbols) ${\dot{ω}}_{T} = 0.5 {\dot{ω}}_{T, \max}^{ψ}$ for the different ignition cases.

Both way of measuring the flame base height may lead to slightly different estimations, as temperature can be advected upstream by low-frequency large scale motions that may be amplified by heat release effects and this can happen irrespective of the local mixture composition. On the other hand, when the flame height is measured using the heat release metric, it may change rapidly due to local extinctions and its evolution becomes temporarily uncoupled from the temperature field. This effect may be seen at $t^{+} \approx 20 - 40$ for case $21$ in Figure 7, and in more details in Figure 5(b) to (d). Similar decoupling happens for case $11$ between $t^{+} = 20 - 40$ , and for both $11$ and $21$ for $t^{+} ⩾ 70$ .

It can further be seen from Figure 7 that the distance between the flame base and the jet inlet increases monotonically for an increase in $ψ$ , when the ignitor is physically placed at the same spatial location. This is attributed to the lower flame speed of the stoichiometric diluted mixture, which directly affects the resulting edge flame speed (see equation (19)) and the height at which an equilibrium can be found between the flame and flow movements in a mean sense as detailed below. In particular, the onset of stabilisation was reported by Turquand dAuzay and Chakraborty³¹ for the undiluted mixture, but a clear indication of flame stabilisation cannot be ascertained for the cases with $ψ = 0.1$ and $ψ = 0.2$ . This also suggests that the probability of obtaining flame stabilisation for CH $_{4}$ /CO $_{2}$ jet flames is likely smaller than that in the case of undiluted methane-air jets.

Edge flame propagation

In the present analysis, the statistics of the edge flame elements propagating upstream are of relevance from the point of view of kernel growth and the onset of flame stabilisation.³¹ The corresponding edge flame elements are selected using the value of the fuel mass fraction iso-surface normal projection on the streamwise direction axis (x), that is, $N_{y, x} ⩽ - 0.5$ (where $N_{y, x}$ is the x-component of $N_{y}$ ). The mean values conditional upon the flame elements characterised by $N_{y, x} ⩽ - 0.5$ are denoted by $⟨ \dots ⟩_{f}$ and are referred to as upstream facing edge flames.

It is well known that the mixture fraction scalar dissipation rate ( $χ = 2 D | \nabla ξ |^{2}$ or SDR) influences the edge flame propagation rate.^{30,29,93,24,85,82,83} Therefore, the temporal evolutions of the mean SDR conditional upon $N_{y, x} ⩽ - 0.5$ normalised by its extinction value (i.e. $⟨ χ ⟩_{f} / χ_{e x t}^{ψ}$ ) are shown in Figure 8(a) for the different CH $_{4}$ /CO $_{2}$ /air mixtures injected while the temporal evolution of the normalised SDR PDF of case $21$ is exemplarily shown in Figure 8(b) and is qualitatively similar to the other cases. The purpose of the background colour in Figure 8 and subsequent figures indicate the frequency of samples and scatter around the mean value. It can be seen from Figure 8(a) that the magnitude of $⟨ χ ⟩_{f} / χ_{e x t}^{ψ}$ at a given time instant remains comparable for different values of $ψ$ and that the temporal evolution of the normalised mean value is not significantly affected by the CO $_{2}$ concentration in the fuel blend, even though a slight increase of $⟨ χ ⟩_{f} / χ_{e x t}^{ψ}$ can be seen with an increase in $ψ$ . The local conditions of mixture fraction and its gradient induced by the realisation of the turbulent mixing rate and turbulent strain rate can lead to significantly different values of $⟨ χ ⟩_{f} / χ_{e x t}^{ψ}$ between different cases at early times but the values of $⟨ χ ⟩_{f} / χ_{e x t}^{ψ}$ relax to comparable magnitudes at later times. Furthermore, $⟨ χ ⟩_{f} / χ_{e x t}^{ψ}$ remains smaller than unity throughout the flame evolution which suggests that flame extinction due to high scalar dissipation rates are unlikely in the conditions considered here. By comparing Figure 8(a) with Figure 7, it can be remarked that the upstream movement of the flame base that occurs after the initial downstream advection starts when $⟨ χ ⟩_{f} / χ_{e x t}^{ψ}$ reaches low values (i.e. $⟨ χ ⟩_{f} / χ_{e x t}^{ψ} < 0.1$ ) independent of the $ψ$ value. From then, the mean levels of normalised scalar dissipation rate observed at the flame base remain low and roughly constant which allow for its upstream propagation during the third stage of ignition³¹ apart from the occasional fluctuations due to the large scale unsteadiness. Figure 8(b) indicates that even though the mean level of normalised SDR remains mostly moderate during the flame evolution, local non-uniformities can be observed along the upstream propagating elements for all CO $_{2}$ concentration levels considered. At early times ( $t^{+} ⩽ 20$ ), large variations are observed between the flame elements during the kernel growth, with local values of $χ / χ_{ext}$ larger than unity indicative of local extinctions leading to the formation of some of the holes in the reacting sheet^93,85,98 that can be observed from the volume rendering animations. Past $t^{+} \approx 25$ , when the edge flames are well anchored in the shear layers, the local variations are much smaller and most of the flame elements propagate in region with moderate values of the normalised SDR.

Figure 8.

Temporal evolution of (left) the mean normalised scalar dissipation rate for the different ignition cases (right) the normalised scalar dissipation rate PDF for the ignition case $21$ , where the (while solid) line denotes $⟨ χ ⟩_{f} / χ_{e x t}$ – statistics measured on the upstream facing edge flames elements ( $N_{y, x} ⩽ - 0.5$ ).

The temporal variations of the edge flame speed value $s_{e}^{ψ}$ conditioned on the upstream facing elements normalised by the laminar freely propagating burning velocity at the corresponding CO $_{2}$ concentration in the fuel blend (i.e. $s_{l}^{0, ψ}$ ) are presented in Figure 9(a) for the different ignition locations and CO $_{2}$ concentration levels considered. After the initial transients, the values of $⟨ s_{e} ⟩_{f} / s_{l}^{0, ψ}$ oscillate around a mean value of about $6.0 - 7.0$ , whereas the theoretical laminar edge flame speed is about $2.35 s_{l}^{0, ψ}$ (i.e. $s_{e}^{t, ψ} \approx 2.35 s_{l}^{0, ψ}$ ) for all cases, leading to $⟨ s_{e} ⟩_{f} / s_{e}^{t, ψ} \approx 3$ . Despite moderate values of Reynolds number, the edge flame speed shows a significant amount of scatter around the most probable value. It can be seen from Figure 9(b) that the edge flame speed shows a significant scatter around the expected laminar value. This scatter is contributed by the strain rate, curvature and scalar dissipation rate fluctuations induced by turbulent motion. This can further be substantiated by the fluctuation of the scalar dissipation rate around its mean value in Figure 8. The integral effect of turbulence is manifested in the much higher value of $⟨ s_{e} ⟩_{f}$ than the theoretical laminar edge flame speed (i.e. $⟨ s_{e} ⟩_{f} / s_{e}^{t, ψ} \approx 3.0$ ). As the edge flame propagation plays a key role in flame stabilisation, the flame evolution following thermal runaway in the cases studied in this paper is considerably different from laminar conditions despite moderate values of Reynolds number of the simulations. The edge flame values shown in Figure 9(a) and (b) correspond to a fluctuation of $⟨ s_{e} ⟩_{f}$ around $2.5 - 3.0 U_{c}$ in this configuration or $0.25 - 0.3 U_{j}$ , indicating that the magnitude of the flame base propagation rate can be significantly larger than the coflow velocity. Thus, the flame orientation (propagation along $T_{2}$ ) may play a key role in the anchoring of the flame base wherever a local equilibrium is obtained between the flow velocity and the edge flame speed. The high mean values of $⟨ s_{e} ⟩_{f} / s_{t}^{0, ψ}$ occur due to the local non-linear stretch rate and scalar gradient dependencies of the edge flame speed, which have been demonstrated and discussed extensively elsewhere^{41,85,30,29,67,77,83,82,17,60,61} for canonical configurations and similar qualitative trends has been observed here but will not be discussed further for the sake of brevity. Figure 9(b) shows the temporal evolution of the normalised edge flame speed PDF of case $21$ . The most probable value of $s_{e}^{ψ}$ at any point in time remains close to the theoretical prediction, while the probability of finding negative value is mostly negligible with small fluctuations. These are indicative of the flame locally reaching unfavourable areas mostly characterised by high SDR values. On the other hand, the probability of finding edge flame speed larger than the theoretical estimate remains non negligible up to $s_{e} / s_{l}^{0, ψ} \approx 10 - 15$ as previously observed by Heeger et al.⁶⁸ The fuel blend composition as well as the ignition location have only a marginal effect on the temporal evolution of the PDF, but noticeably, the probability of finding negative values of $s_{e} / s_{l}^{0, ψ}$ increases with increasing $ψ$ and decreasing distance from the nozzle (i.e. increasing mean SDR). This is mostly visible at very early times here (i.e. $t^{+} ⩽ 20$ ) where the probability of having a negative edge flame speed is dominant before decreasing as the flame base height increases.

Figure 9.

Temporal evolution of (left) the mean normalised edge flame speed for the different ignition cases (right) the normalised edge flame speed PDF for the ignition case $21$ , where the (solid orange) line denotes $⟨ s_{e} ⟩_{f} / s_{l}^{0, ψ}$ , the (chain-dotted white) line denotes the maximum value of the PDF and the (cyan dashed) line denotes $s_{e}^{t, ψ}$ – statistics measured on the upstream facing edge flames elements ( $N_{y, x} ⩽ - 0.5$ ).

The propensity of the edge flame to propagate upstream is demonstrated in Figure 10(a) through the temporal evolution of the normalised mean edge flame propagation speed streamwise component (i.e. $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ ) measured along the upstream facing flame elements (i.e. $N_{y, x} ⩽ - 0.5$ ). A positive value of $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ denotes a downstream movement of the flame elements and vice versa. The large positive values of $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ observed during the initial phase (e.g. $t^{+} ⩽ 10$ ) for all cases is indicative of the rapid kernel downstream advection (first development phase of the kernel to flame transition) and is the highest for case $21$ , where the ignitor is the closest from the nozzle, and thus where the mean streamwise velocity is the highest. The positive value of $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ then starts to decrease (e.g. $t^{+} \approx 15$ ) when the edge flame structure is established. Over the interval $15 ⩽ t^{+} ⩽ 25$ , the flame propagation velocity $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ decreases down to almost zero suggesting a local equilibrium between the edge flame movement and the local flow field in a mean sense. Following that, $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ stabilises to comparable small positive values during $25 ⩽ t^{+} ⩽ 45$ as the flame base height continues to decrease for the cases $00$ , $10$ and $20$ (see Figure 7). Finally, $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ starts to increase again for $t^{+} ⩾ 45$ which is consistent with the increase in flame base height with time (see Figure 7) and this increase is more pronounced for higher values of $ψ$ . A comparison between Figure 9(a) and Figure 10(a) reveals that $⟨ s_{e} ⟩_{f} / s_{e}^{t, ψ}$ (this is also valid for $⟨ s_{e} ⟩_{f} / s_{l}^{0, ψ}$ because $s_{l}^{0, ψ}$ is a constant multiplier of $s_{e}^{t, ψ}$ ) does not change significantly over the duration where $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ increases rapidly. This implies that the upstream propagation of the mean flame base (small or negative values of $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ ) needs the local flame orientation (and thus $T_{2}$ ) to have a large negative streamwise component. This would correspond to the edge flame being at 3 o’clock following the notation introduced by Karami et al.,³⁰ or at the outer-most point of the large eddy where the flame net motion is upstream and inwards, which also correlated with Figure 6 where there are indications of a net motion towards richer mixtures. Finally, the evolution of $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ exhibits another period of decay, which can be associated with large scale fluctuations of the flame base height but can also be attributed to turbulence decay as the mean axial velocity decreases considerably ( $\sim 40 %$ ) by the height where the onset of flame stabilisation eventually starts (e.g. $x^{+} \approx 18$ for $ψ = 0.0$ ). As $s_{l}^{0, ψ}$ decreases with increasing $ψ$ , the local equilibrium between the fluid velocity and the streamwise edge flame speed component is likely to occur at an increasing height from the nozzle, and the flame stabilises further away from the jet inlet for an increase in CO $_{2}$ concentration in the fuel blend. By looking further at Figure 10(b) which exhibits the temporal evolution of $W_{e}^{x} / s_{l}^{0, ψ}$ PDF, one may remark that even if the mean value is positive at almost all time for the different cases, local regions of the PDF show a high probability of finding a null or slightly negative value of $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ indicating the flame elements can locally propagate upstream, on which the flame base depends to stabilise, while the majority of the other elements do not contribute to the stabilisation.

Figure 10.

Temporal evolution of (left) the mean normalised edge flame propagation speed for the different ignition cases (right) the normalised edge flame propagation speed PDF for the ignition case $21$ , where the (solid orange) line denotes $⟨ W_{e}^{x} ⟩_{f} / s_{l}^{0, ψ}$ , the (chain-dotted white) line denotes the maximum value of the PDF and the (cyan dashed) line denotes $W_{e}^{x} / s_{l}^{0, ψ} = 0$ – statistics measured on the upstream facing edge flames elements ( $N_{y, x} ⩽ - 0.5$ ).

Summary and concluding remarks

The localised forced ignition and subsequent flame propagation of a turbulent CH $_{4}$ /CO $_{2}$ jet issuing in ambient air have been analysed using DNSs. A simplified two-step chemical mechanism has been considered to account for the CO $_{2}$ blending effects on the flame speed and temperature, while retaining sufficient accuracy for the present range of operating conditions. Three different CO $_{2}$ concentration levels ranging from $0$ % (i.e. pure CH $_{4}$ fuel composition) to $20$ % have been considered while the energy deposition characteristics (energy, radius, duration) remain unchanged. To investigate the effect of CO $_{2}$ concentration on the jet ignition success, five different energy deposition scenarios have been considered, three cases where the mean mixture composition at the ignitor varies, but not its location in space, and two other where the mean mixture composition within the energy deposition region remains constant, but where its spatial location varies with CO $_{2}$ concentration in the fuel blend. The effects of CO $_{2}$ concentration in the fuel blend on the mean field have been reported, and it was shown that the addition of CO $_{2}$ would lead the most favourable region (from the flammability factor standpoint) for the flame development to be displaced closer to the nozzle. However, moving the ignitor closer to the nozzle could detrimentally affect the ignition due to the larger velocity and scalar gradients encountered locally.

The temporal evolution of the kernel to flame development was found to follow the experimentally reported behaviour with the three stages of development qualitatively reproduced here, from the kernel initial growth and its downstream advection by the mean flow field to the final upstream propagation and onset of stabilisation through the radial expansion and downstream propagation stage. The qualitative evolution was found to be insensitive to CO $_{2}$ concentration in the fuel blend (within the range investigated), although it must be noted that a large energy input has been used. Irrespective of the CO $_{2}$ level concentration in the fuel blend, and similarly to our previous work,³¹ the flame base dynamics was shown to be governed by edge flame propagation in the two shear layers found on either side of the nozzle, while its propensity to stabilise was dictated by the ability of the edge flame to locally propagate faster than the flow field. The effects of CO $_{2}$ concentration on the flame lift-off become noticeable only a long time after the ignition, when both the edge flame structures and the reacting front spanning the jet width are well established. From this point onward, the lift-off height increases monotonically with the CO $_{2}$ levels which is attributed in particular to the lower flame speed of the diluted stoichiometric mixture, while the probability of obtaining flame stabilisation for the cases with $ψ = 0.1$ and $0.2$ appears smaller than that in the case of undiluted mixture.

The heat release contributions to the overall heat release of the different branches of these triple flame revealed that the non-premixed branch contribution is negligible with less than $5$ % of the total heat release on average, while the rich-premixed branch contributes the most with a contribution of at least $60$ % and no effect from the CO $_{2}$ content was found on these.

Lastly, the edge flame statistics were investigated for the different scenarios, with a focus on the edge flame elements being orientated towards the jet, that is upstream facing. An increase in CO $_{2}$ concentration in the fuel blend leads to small albeit noticeable increase of the mean normalised (by its extinction value) scalar dissipation rate experienced by the edge flame, although the long term averaged value stabilises well below the extinction value with low frequency oscillations driven by the large scale motions. After the initial kernel growth driven by the large temperatures resulting from the energy deposition, it was found that the mean edge flame speed oscillates around a value which is larger than the coflow velocity with $⟨ s_{e} ⟩_{f} > 2.5 U_{c}$ and non-negligible with respect to the jet velocity with $⟨ s_{e} ⟩_{f} / s_{l}^{0, ψ} \approx 0.25 U_{j}$ . It was further shown that the orientation of the local fuel mass fraction normal plays a significant role in the stabilisation of the flame by controlling the direction in which it propagates, that is, when it displays a large negative streamwise component, the flame is able to locally propagate upstream. The mean value of $s_{e} / s_{l}^{0, ψ}$ appears only marginally affected by the CO $_{2}$ concentration, although it was found that the probability of finding a negative edge flame speed increased with increasing $ψ$ . In particular, the differences in strain rate history between different levels of CO $_{2}$ concentration leading to decreasing normalised edge flame speed observed by Turquand dAuzay et al.¹⁷ do not seem to play a major role here, which may be attributed to the large scale motion of the flow resulting in new edge flames with short history driving the flame base movement.

This leads to the two main conclusions. The first being that the effects of CO $_{2}$ concentration in the fuel blend on the flame is mostly quantitative, in that the flame speed and heat release (and thus flow expansion around the edge flame, and adiabatic temperature) decrease with increasing CO $_{2}$ concentration in the fuel blend, and the qualitative nature of the flame development does not appear to be affected. The second main conclusion is that combustion models developed for conventional fuels (e.g. CH $_{4}$ /air mixtures) are likely to hold in the case of partially premixed CH $_{4}$ /CO $_{2}$ /air turbulent combustion, provided that the effects of $ψ$ on the laminar burning velocity and adiabatic flame temperature are correctly accounted for. It is noted that future detailed chemistry and transport DNS should be conducted to confirm and refine the findings of the present work, especially during the early growth of the kernel. Additionally, future work should include shear mixing layers (similar in configuration to the LES results presented by Wawrzak and Tyliszczak^42,43) with various amount of shear such that its influence can be isolated.

Footnotes

Acknowledgements

The authors are grateful to the British Council, EPSRC (EP/R029369/1), Rocket (HPC facility at Newcastle University), Cirrus and ARCHER for financial and computational support respectively. The practical help provided by Dr. Khalil Abo-Amsha (Newcastle University) is gratefully acknowledged.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by British Council, EPSRC (EP/R029369/1).

ORCID iD

Nilanjan Chakraborty

Supplemental material

Supplemental material for this article is available online.

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