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ABSTRACT
The localized forced ignition and subsequent flame propagation have been analyzed for stoichiometric biogas-air mixtures ( blend) with different spatial distributions (uniform, Gaussian and bimodal) and mean levels of
dilution (i.e. mole fraction of
) for different flow conditions (e.g. quiescent laminar condition and different turbulence intensities) using three-dimensional Direct Numerical Simulations. A two-step chemical mechanism with sufficient accuracy is used to capture the effects of
dilution on the laminar burning velocity of the
. A parametric analysis in terms of the mean, standard deviation and integral length scale of the initial Gaussian and bimodal distributions of spatial
dilution in the unburned gas has been conducted. Qualitatively similar behavior has been observed for all three initial spatial distributions of
. A departure from uniform conditions was found to increase the range of reaction rate magnitudes of
, which impacts the burned gas volume and, in turn, increases the variability of the outcomes of the ignition event (successful thermal runaway and subsequent self-sustained propagation or misfire). Turbulence intensity and the mean level of dilution were found to have significant impacts on the outcome of the localized forced ignition, and an increase of either quantity acts to reduce the burned gas volume irrespective of mixture composition due to the enhancement of heat transfer from the hot gas kernel, and the heat sink effects of
, respectively. An increase of the integral length scale
for the spatial distribution of
dilution increased the probability of a successful outcome of the ignition event. A uniform initial spatial distribution was found to be optimal, and in the case of a departure from non-uniform conditions, larger variability in the outcome are obtained for a Gaussian initial distribution of
dilution than a bimodal one. The variance of the
dilution distribution has been found to have a negligible impact on the outcomes observed, as it was dominated by the effects arising from turbulence intensity, nature of initial
distribution and integral length scale of
dilution.
INTRODUCTION
Interest in alternative fuel solutions has increased over the years, and the transportation industry and governments are making a concerted effort to move to renewable and eco-friendly solutions to meet ever more stringent emissions and pollution standards (Koonaphapdeelert, Aggarangsi, Moran Citation2020; Korberg, Skov, Mathiesen Citation2020; Mustafi and Agarwal Citation2020; Patterson et al. Citation2011; Rafiee et al. Citation2021). Despite electric vehicles being heralded as the future of the industry, the use of the internal combustion engine is expected to play a dominant role for the upcoming decades due to existing investment in such technologies along with its widespread use. Additionally, in remote locations and developing countries, the infrastructure required to be able to efficiently utilize electric cars is still lacking. In many of these places, the cost of conventional fossil fuels is high, and thus there is a demand for solutions that are cost-effective, renewable, efficient and functional under a wide range of operating conditions. One solution which has been gaining traction in recent years in Singapore (Koonaphapdeelert, Aggarangsi, Moran Citation2020; Mustafi and Agarwal Citation2020), Middle East (IRENA Citation2018), Eastern Europe (Papacz Citation2011) and Nordic (Persson Citation2007) countries, is the alteration of internal combustion engines operating on fossil fuels to operate them using Biogas-air mixtures for personal and light commercial vehicles (Barik and Murugan Citation2014; Crookes Citation2006; Jatana et al. Citation2014; Shah et al. Citation2017). The modifications required for such an alteration are straightforward, cheap, and allow the vehicle not only to run using solely fossil fuels as originally designed, but also to operate using a mixture consisting primarily of biogas mixed with the existing fuel (EU and Bluesky Citation2018; Surataa et al. Citation2014). One of the problems with the use of biogas in internal combustion engines for personal vehicles is that the efficiency and stability of the biogas-fueled engine are worse than that obtained for the fuel with which the engine was originally designed to operate. This is further amplified by inconsistencies and impurities in the fuel blend (Barik and Murugan Citation2014; EU and Bluesky Citation2018; Jatana et al. Citation2014; Ryckebosch, Drouillon, Vervaeren Citation2011; Shah et al. Citation2017; Surataa et al. Citation2014).
Biogas is primarily composed of and
, but the production of industrial quantities of biogas with a fixed composition is hard due to its biological origins (Rasi, Veijanen, Rintala Citation2007; Ryckebosch, Drouillon, Vervaeren Citation2011). Depending on the production method used, biogas can be a carbon-neutral fuel and due to existing natural gas infrastructure, it can easily be stored and transported whilst also being used in conjunction with natural gas for power generation and transportation. Additional advantages of biogas are that it can be produced locally whilst offering an eco-friendly solution to disposing of sewage and agricultural waste. For example, sewage can be used to produce the biogas to power the waste collection trucks (https://www.csmonitor.com/Environment/Living-Green/2010/0107/California-garbage-trucks-fueled-by-garbage). Due to its organic nature, its storage is significantly less risky than that of fossil fuels, making it an attractive energy source for smaller countries looking to become energy independent and move to renewable energy production. In-depth reviews (Koonaphapdeelert, Aggarangsi, Moran Citation2020; Korberg, Skov, Mathiesen Citation2020; Mustafi and Agarwal Citation2020; Patterson et al. Citation2011; Rafiee et al. Citation2021) of biogas utilization ranging from instructional to a high technical level have been published to aid the efficiency of biogas utilization. These reviews indicate that biogas utilization and upgrading is still a developing field, with a strong demand on both micro (personal) and macro (governmental and institutional) levels, for the numerous reasons mentioned above. Another advantage of using biogas blended with fossil fuels is the significant reduction of emissions and soot formations, whilst engine performance is not severely impacted (Crookes Citation2006; EU and Bluesky Citation2018; Jatana et al. Citation2014; Shah et al. Citation2017; Surataa et al. Citation2014). However, these findings have been obtained under laboratory conditions where the fuel composition and mixing are fully controlled. Of particular interest is the preprocessing of the biogas to reduce sulfur and other trace species impurities (Rasi, Veijanen, Rintala Citation2007; Ryckebosch, Drouillon, Vervaeren Citation2011; Surataa et al. Citation2014), along with the perfect premixing of the combustible mixture, which is achieved under controlled laboratory conditions. The impurities present in the biogas mixture fall outside of the scope of this study, however the perfect mixing achieved under laboratory conditions is difficult to achieve in practice, and thus the ignition and subsequent combustion of biogas-air mixtures can take place under inhomogeneous conditions in real-life applications. Thus, the mixture can have significant variations in the composition encountered at different locations in the combustion chamber, which can lead to a reduction of performance or even misfires. Hence, the forced ignition of inhomogeneous biogas-air mixtures has significant applications in direct-injection engines in the automotive industry, and energy production methods involving biogas.
Previous studies (Forsich et al. Citation2004; Lafay et al. Citation2007; Lieuwen et al. Citation2008; Mulla et al. Citation2016; Rasi, Veijanen, Rintala Citation2007; Ryckebosch, Drouillon, Vervaeren Citation2011; Vasavan, De Goey, van Oijen Citation2018) have shown that variations in content can influence the localized forced ignition (e.g. spark and laser ignition) performance, leading to adverse effects on the subsequent flame propagation. Existing studies (Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013), which investigated forced ignition of biogas-air mixtures, have focussed on perfectly premixed biogas-air mixtures with a fixed level of
content. This, in turn, leads to a uniform spatial distribution of the
content leading to homogeneous mixtures being encountered at all points of the combustion chamber. However, the effects of the non-uniform composition of biogas encountered locally have not been analyzed to date. Galmiche et al. (Citation2011) reported that
acts as a heat sink, and an increase in the amount of
dilution leads to an increase in the energy required for a successful ignition, and that successful ignition was possible for mixtures with up to 40% (by volume)
dilution. A large modification of the reaction zone (Lafay et al. Citation2007) and a hindrance to flame kernel formation were found by both Mordaunt and Pierce (Citation2014) and Lafay et al. (Citation2007) for biogas/air mixtures in a gas turbine configuration. The lower burning velocity of biogas-air mixtures in comparison to methane-air mixtures leads to a slower combustion process (Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013). The presence of
in the fuel can potentially lead to flame extinction (lean blow-out) (Mordaunt and Pierce Citation2014). Industrial demand to ignite gas turbines using the fuel available on offshore plants led Larsson, Berg, and Bonaldo (Citation2013) to experimentally investigate the minimum ignition energy (MIE) required for successful ignition across hydrocarbon fuels mixed with inert gases. Nitrogen and carbon dioxide were separately used as the diluents, and mixed with methane and propane mixtures individually, in order to find the maximum level of diluent which could be present in each gas composition, whilst also being able to be successfully ignited. The above experimental studies indicate that an increase in the amount of
is detrimental toward successful thermal runaway and subsequent self-sustained combustion, indirectly highlighting the importance of the fuel composition in terms of diluent that the flame kernel encounters.
Ever more detailed Direct Numerical Simulations (DNS) of localized forced ignition events in gaseous mixtures have become feasible due to the increase in computational power over the last decade. These advancements have provided improved physical insights into the ignition process and subsequent flame propagation. Pera, Chevillard, and Reveillon (Citation2013) demonstrated that the mixture inhomogeneity induced by residual burnt gas has significant influence on early flame propagation and cyclic variability in the case of forced ignition using DNS. Further numerical investigations were carried out by Patel and Chakraborty (Citation2014; Citation2015, Citation2016a-c) to analyze the effects of turbulence intensity on forced ignition of stratified mixtures. It was found that an increase of turbulence intensity with a fixed length scale has a detrimental effect on successful ignition. Patel and Chakraborty (Citation2016c) also focussed on the effects of initial mixture distribution (i.e. variance and length scale of equivalence ratio), where the equivalence ratio was initialized using both Gaussian and bimodal distributions, to investigate the effects of different initial distributions on the localized forced ignition of globally stoichiometric stratified mixtures. An increase in the burning rate was observed when the initial equivalence ratio distribution was Gaussian as opposed to bimodal. However, for the Gaussian initial distribution a higher possibility of flame extinction was observed for large values of turbulence intensity, in comparison to the bimodal distribution. Three-dimensional simple chemistry DNS was also used by Patel and Chakraborty (Citation2016a-c) to investigate the influence of turbulence intensity, fuel Lewis number and mixture fraction gradient on ignition success and subsequent flame propagation characteristics in turbulent stratified mixtures. In a recent DNS analysis (Turquand d’Auzay et al. Citation2019a) by the present authors, the variations of turbulence intensity and dilution levels on the localized forced ignition of turbulent mixing layers of for different extents of
dilution with
air mixtures have been investigated, which demonstrated that an increase in
dilution leads to reductions of burning rate and the probability of obtaining self-sustained combustion once the ignitor is switched off. Interested readers are referred to Mastorakos (Citation2009) for a comprehensive review of different aspects of forced ignition for both homogeneous and inhomogeneous mixtures.
Existing computational investigations (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty and Mastorakos Citation2008; Patel and Chakraborty Citation2014, Citation2015, Citation2016a-c; Turquand d’Auzay et al. Citation2019b) focused primarily on single fuels such as methane or hydrogen but not compound fuels such as biogas. To be more specific, the effects arising from the spatial distribution of the diluent of choice (i.e for biogas) on the localized forced ignition and subsequent flame propagation have not yet been investigated either experimentally or computationally to the best of the authors’ knowledge. Thus, a systematic parametric study has been carried out across a range of different turbulence intensities, concentrating on the effects of the nature of the spatial distribution (uniform, Gaussian and bimodal) of
dilution and their characteristics (i.e. mean, variance and integral length scale of
dilution) in the case of localized forced ignition of stoichiometric biogas-air mixtures (assumed to be
+
-air mixtures). Thus, the objectives of this paper are:
(a) to demonstrate the effects of different spatial distributions (uniform, Gaussian and bimodal) of dilution on the localized forced ignition of stoichiometric biogas-air mixtures.
(b) to provide physical explanations for the observed influence of different initial distributions on the burning rate characteristics of stoichiometric biogas-air mixtures.
The mathematical background and numerical framework pertaining to the current analysis are presented in the next two sections. This is followed by the presentation of results along with their discussion. The main findings are summarized, and conclusions are drawn in the final section of this paper.
MATHEMATICAL BACKGROUND
The following global equation represents the stoichiometric combustion of biogas represented by a mixture:
and the level of dilution with present in the biogas-air mixture is quantified in terms of the unburnt mixture according to the definition presented by Galmiche et al. (Citation2011):
where represents the dilution percentage which is the molar fraction of the diluent (
) in the unburned gases and
represents the stoichiometric coefficient, as shown in EquationEq. 1
(1)
(1) . The level of dilution present can also be defined based on the dilution in the
mixture as:
where and
are mass fractions of
and
respectively and
and
are the molecular weights of
and
respectively. Thus,
represents the molar fraction of the diluent present in the
mixture. To account for the effects of dilution with
and to also allow for the study to be computationally feasible (in terms of cost), a two-step chemical mechanism (Bibrzycki and Poinsot Citation2010; Westbrook and Dryer Citation1981) has been used for the present analysis. It should be noted that the cases considered for the present analysis constute a large parametric analysis involving more than 350 different three-dimensional simulations. It becomes impractical to carry out such a parametric analysis based on detailed chemistry because the computational cost becomes exorbitant (Chen et al., Citation2009). Typically, a detailed chemistry simulation for a skeletal mechanism of methane-air combustion involving 16 species and 25 reactions (Smooke and Giovangigli Citation1991) is roughly 20 times more expensive than one of the simulations conducted for this analysis. Thus, it is impossible to carry out the analysis conducted here using even a skeletal chemical mechanism. Therefore, the chemical representation in this analysis is simplified here by a two-step mechanism which involves incomplete oxidation of
into
, and an equilibrium reaction between oxidation of
to
and dissociation of
into
. It is worth noting that pioneering analytical studies (Champion et al. Citation1986; Espí and Liñán Citation2001, Citation2002; He Citation2000; Sibulkin and Siskind Citation1987) on ignition have been conducted for single-step irreversible Arrhenius type chemistry.
Several assumptions are made in the present analysis for the purpose of simplicity, and to also allow for the simulations to be computationally feasible. The Lewis number of all species is taken to be equal to unity, and all species in the gaseous phase are taken to be ideal gases, and thus the ideal gas law is used for compressible flow DNS. The species included are CH4, CO, CO2, N2, H2O and O2 whose Lewis numbers are bounded by and thus, the unity Lewis number assumption does not significantly introduce any inaccuracy in a qualitative sense. This was also confirmed in previous analytical (He Citation2000; Sibulkin and Siskind Citation1987) and DNS (Chakraborty, Hesse, Mastorakos Citation2010; Patel and Chakraborty Citation2016a) studies. The ratio of specific heats (
, where
and
are the gaseous specific heats at constant pressure and volume, respectively) and Prandtl number (
, where
is the dynamic viscosity and
is the thermal conductivity of the gaseous phase) take standard values, respectively. Fick’s law of diffusion is used to account for the species diffusion velocity. The unburned gas mixture is preheated to
to reduce the error made by using the assumption of equal specific heat for all species as the heat capacities of
and
become equal to each other at
. For lower temperatures (e.g.
), the difference in heat capacity is as high as 40%. For higher temperatures, (
) the difference of the heat capacity between the two species is at most 6%. Furthermore, as the gaseous mixture is predominantly composed of
for small values of
, the heat capacity of the gaseous mixture would be very close to that of
. This, in turn, means that the errors introduced by the assumption of equal specific heat are small. In this analysis, the specific heat at constant pressure
, dynamic viscosity
, thermal conductivity
and the density-weighted mass diffusivity
are considered to be constant for the sake of simplicity following previous analyses (Chakraborty and Mastorakos Citation2006; Hesse, Chakraborty, Mastorakos Citation2009; Wandel, Chakraborty, Mastorakos Citation2009; Hesse, Malkeson, Chakraborty Citation2012; Papapostolou et al. Citation2019, 2021; Turquand d’Auzay et al. Citation2019a; Turquand d’Auzay, Papapostolou, Chakraborty Citation2021), which ensures that the Lewis number does not change within the flame. The constant thermophysical properties do not alter the mixing statistics in the unburned gas. For simulations with temperature-dependent thermophysical properties, the increased viscosity can lead to faster decay of turbulence which limits the extent of flame-turbulence interaction that can be analyzed using DNS because of moderate values of turbulent Reynolds number. Moreover, the mass diffusivity increases within the flame more rapidly in the simulations with temperature-dependent thermophysical properties than that obtained in the case of constant thermophysical properties. This implies that the mixture inhomogeneity decays faster in the case of temperature-dependent thermophysical properties than that in the case of constant thermophysical properties, and this limits the extent to which mixture inhomogeneity effects can be studied using DNS for moderate values of turbulent Reynolds number. Poinsot, Echekki, and Mungal (Citation1992) observed that the stretch rate dependence of flame displacement speed for constant thermophysical properties is qualitatively similar to that obtained for temperature-dependent thermophysical properties for simple chemistry, but quantitative differences do exist. It was also demonstrated by Aspden (Citation2017) that the constant viscosity assumption does not affect the qualitative nature of the flame morphology and vorticity distribution across the statistically planar turbulent premixed flames. However, the analyses by Poinsot et al. (2019) and Aspden (Citation2017) are not done in the context of localized forced ignition of biogas-air mixtures, and therefore further analyses on forced ignition of biogas-air mixtures in the presence of temperature-dependent thermophysical properties will be needed to ascertain the extent of qualitative similarities between variable and constant thermophysical properties. It is worth noting that the the Chapman-Rubesin variable transport model can potentially offer a realistic dependence of transport properties on temperature without increasing the computational cost but its application to DNS of localized forced ignition is beyond the scope of current investigation. However, this aspect deserves further investigation in the future.
The two-step chemical mechanism along with the transport model selected for this analysis has been validated against other chemical mechanisms (GRI-Mech 3.0 (Smith et al. Citation1995) and 2s_CM2 (Bibrzycki and Poinsot Citation2010; Selle et al. Citation2004)) and experimental results (Galmiche et al. Citation2011) for both and
mixtures under all the above simplifying assumptions. For large values of
(e.g.
), the two-step chemical mechanism overestimates the laminar burning velocity, and thus the maximum level of mean dilution investigated in the present study is
, for which flame speed predictions are sufficiently accurate (Turquand d’Auzay et al. Citation2019a). For further details regarding the performance and validation of the mechanism, interested readers are directed to recent work by Turquand d’Auzay et al. (Citation2019a).
It is important to note that the chemical mechanism used in this analysis has been benchmarked for flame propagation and simplifications for the transport quantities are made in the interest of a detailed parametric analysis. In the case of forced ignition, the temperature attained during the external energy deposition duration is high enough to result in a very small value of ignition delay time. Therefore, 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 the flame propagation. Moreover, the seminal DNS analysis by Mastorakos, Baritaud, and Poinsot (Citation1997) with single-step Arrhenius type chemistry revealed that the reaction rate magnitude of fuel and scalar dissipation rate are negatively correlated in the case of autoignition, which was found to be qualitatively similar in subsequent detailed chemistry DNS studies (Im, Chen, Law Citation1998). Recently, it has been demonstrated by the authors (Papapostolou, Turquand d’Auzay, Chakraborty Citation2020) that the MIE variation with turbulence intensity for the stoichiometric homogeneous mixed biogas-air mixture using this chemical mechanism provides excellent qualitative and quantitative agreements between DNS and experimental (Cardin et al. Citation2013a, Citation2013b) results. Moreover, the flame structure resulting from localized forced ignition of turbulent gaseous mixing layer and droplet-laden mixtures using single step Arrhenius type chemistry (Chakraborty and Mastorakos Citation2006, Citation2008; Papapostolou et al. Citation2019) are qualitatively similar to those obtained from detailed chemistry simulations (Ray et al., Citation2001; Neophytou, Mastorakos, Cant Citation2010). Turquand d’Auzay, Papapostolou, and Chakraborty (Citation2021) demonstrated that the edge flame propagation statistics for forced ignition of turbulent mixing layers for the chemical and transport models used in this paper are qualitatively similar to the results obtained from detailed chemistry DNS studies (Echekki and Chen Citation1998; Im and Chen Citation1999, Citation2001). Furthermore, it has been found that the statistical behaviors of the edge flame displacement speed including its strain rate, curvature, and scalar gradient dependences from simple single-step chemistry DNS data (Chakraborty, Hesse, Mastorakos Citation2010; Chakraborty and Mastorakos Citation2006; Hesse, Chakraborty, Mastorakos Citation2009; Hesse, Malkeson, Chakraborty Citation2012; Karami et al. Citation2015, Citation2016) have been found to be qualitatively consistent with detailed chemistry DNS (Echekki and Chen Citation1998; Im and Chen Citation1999, Citation2001) and experimental (Chung Citation2007; Heeger et al. Citation2009; Ko and Chung Citation1999) findings. In addition, the current analysis adopts a chemical mechanism which is more detailed in comparison to several previous analyses (Chakraborty and Mastorakos Citation2006; Hesse, Chakraborty, Mastorakos Citation2009; Hesse, Malkeson, Chakraborty Citation2012; Karami et al. Citation2015, Citation2016).
Due to the large computational cost associated with DNS and the large parameter space covered in this work, a canonical configuration which considers solely the volume surrounding the energy deposition location (i.e. the ignition site) has been used. This approach was followed in several previous studies on ignition (Bansal and Im Citation2011; Hawkes et al. Citation2006; Pera, Chevillard, Reveillon Citation2013; Sankaran et al. Citation2005; Yu and Bai Citation2010) and the same approach was followed in this study. Due to the canonical nature of the simulation domain in DNS, the results obtained from these analyses are mostly qualitative in nature and therefore the simplifications made in this analysis do not significantly affect the conclusions made from this analysis.
The mixture fraction is used to describe the mixture composition and is defined using Bilger’s definition (Bilger Citation1980) as
where and
are the values of
in the oxidizer and fuel streams respectively, with
, and
and
denoting the atomic mass fractions and molecular weights. The stoichiometric mixture fraction value is defined as (Bilger Citation1980)
A reaction progress variable that increases monotonically from zero in the unburned mixture to unity in the products is used to measure the extent of the completion of chemical reactions, and in the present study is defined based on the fuel mass fraction as (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty, Hesse, Mastorakos Citation2010; Chakraborty and Mastorakos Citation2008; Turquand d’Auzay et al. Citation2019a)
where is the mass fraction of fuel (
) in the pure fuel stream, whilst
within brackets indicates values that correspond to that level of
dilution. The reaction progress variable can also be defined in terms of oxidizer mass fraction, however in the present study the choice of the reaction progress variable definition does not affect the qualitative nature of the findings.
Following previous studies (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty and Mastorakos Citation2008; Chakraborty, Hesse, Mastorakos Citation2010; Turquand d’Auzay et al. Citation2019a,b; Wandel, Chakraborty, Mastorakos Citation2009) on localized forced ignition, only the thermal effects have been accounted for and the details of the spark formation (momentum modification contribution, plasma and shock wave formation) are not considered in this analysis for the purpose of simplicity and computational economy. An additional source term, with
being the distance from the ignitor center, which is taken to be the geometric center of the domain, and
represents the characteristic width of energy deposition (Espí and Liñán Citation2001, Citation2002)) is added to the energy conservation equation to account for the effects of the localized forced ignition (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty and Mastorakos Citation2008; Chakraborty, Hesse, Mastorakos Citation2010; Turquand d’Auzay et al. Citation2019a,b; Wandel, Chakraborty, Mastorakos Citation2009). The constant
in the source term
is determined by a volume integration which leads to the total ignition power
in the following manner (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty and Mastorakos Citation2008; Chakraborty, Hesse, Mastorakos Citation2010; Turquand d’Auzay et al. Citation2019a,b; Wandel, Chakraborty, Mastorakos Citation2009)
where is a parameter determining the total energy deposited by the ignitor,
is the heat release parameter (where
and
are the adiabatic flame temperature of the undiluted stoichiometric
-air mixture and unburned gas temperature respectively),
is the Zel’dovich flame thickness of the undiluted stoichiometric methane-air mixture (
, where
is the unburned mass diffusivity of the reactants and
is the unstrained laminar burning velocity of the undiluted stoichiometric methane-air mixture), and
, and
are Heaviside functions, which ensure that the ignitor is only active during
. The energy deposition duration
is expressed as
, where
is a characteristic chemical timescale of the undiluted methane-air mixture, and the energy deposition parameter
is given a value of
in the present study, which falls within its optimal range (
) for spark-ignition (Ballal and Lefebvre Citation1975, Citation1977). In the present study,
are considered following a previous analysis (Turquand d’Auzay et al. Citation2019a) and these values remain unaltered throughout this analysis.
NUMERICAL IMPLEMENTATION
A well-known DNS code SENGA+ (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty and Mastorakos Citation2008; Chakraborty, Hesse, Mastorakos Citation2010; Wandel, Chakraborty, Mastorakos Citation2009; Patel and Chakraborty Citation2014, Citation2015, Citation2016a-c; Turquand d’Auzay et al. Citation2019a,b) was used for all the simulations. A cubic computational domain of size 66 was used, and discretized using a Cartesian grid of
, with a uniform grid spacing
This discretization ensures
, where
is the Kolmogorov length scale (e.g.
in the case of the highest initial
considered here,i.e.
), along with ensuring that 10 grid points are present across the undiluted thermal flame thickness of the stoichiometric undiluted methane-air mixture
where
is the instantaneous dimensional temperature. A high order finite-difference scheme (10th order central difference scheme for the internal grid points, with the order of accuracy gradually reducing to a one-sided 2nd order scheme at non-periodic boundaries) is used for the spatial differentiation, whilst a Runge-Kutta (3rd order low storage) scheme is used for explicit time advancement. An incompressible, homogeneous, isotropic field with prescribed values of root-mean-square (rms) values,
, and integral length scale,
, according to the Batchelor-Townsend spectrum (Bachelor and Townsend Citation1948), is used to initialize the turbulent velocity fluctuations Dt~u'lt.
The initial normalized integral length scale () is kept unaltered for all cases. As the geometry of a combustion chamber is fixed, the integral length scale is kept unchanged. Additionally, the variation of the integral length scale of turbulence is not expected to influence the qualitative nature of the findings of the current analysis, as for a given
, an increase in
augments turbulent diffusivity, which scales as Dt~u'lt This, in turn, increases the heat transfer rate from the hot gas kernel, leading to a reduction in the volume of the burned gas. Due to the numerous parametric variations which are investigated in the present study (
,
,
,
and initial distribution) giving rise to 120 cases in total, each additional parameter increases the number of DNS cases required. For this reason, the variation of
is not addressed here due to the limited additional insight that would be gained, whilst incurring a significant additional computational cost.
In the present analysis, the initial Gaussian spatial distributions of dilution (
) are specified using the same pseudo-spectral method (Rogallo Citation1981) utilized to generate the initial turbulent velocity fluctuations. By contrast, a pseudo-spectral method proposed by Eswaran and Pope (Citation1988) was used to generate the initial bimodal distribution of
dilution (
). The initial fields of
and
are generated in such a manner that they remain completely independent and uncorrelated to each other. Following EquationEqs. 1
(1)
(1) –Equation3
(3)
(3) , the mass fractions of
and
can be recovered for the stoichiometric mixture once the dilution field is initialized. For each different initial spatial realization (uniform, Gaussian and bimodal), three different mean values of dilution
have been considered alongside the undiluted homogeneous stoichiometric
mixtures. The species mass fractions in the unburned gas which correspond to the different levels of
and
investigated in this study are provided in .
Table 1. Values of and
and corresponding CH4, CO2 and O2 mass fraction in the unburned gas mixture.
Different values of standard deviations () and integral length scales (
for both the Gaussian and bimodal initial
dilution distributions have been considered for a given set of values of
The Gaussian distribution follows the well-defined bell-shaped curve, with 99% of the samples falling within
, whilst the bimodal distribution displays two symmetrical peaks located at
, centered around
. A summary of the dilution levels, turbulence parameters, integral length scales, standard deviations, and initial distributions are presented in , which also shows the total number of cases investigated for each of the parameters. The equivalence ratio is kept constant at
and is not varied as the burning rate will predictably decrease for fuel-lean and fuel-rich mixtures.
Table 2. Summary of simulation parameters and number of simulations considered for the present analysis.
The initial realizations of the field for
and
with
and
are exemplarily shown in , and their characteristics are detailed in . All the aforementioned mixtures have been considered under laminar conditions and for initial
and
under decaying turbulence. All the simulations have been carried out till at least
, which remains comparable to that used in several previous studies (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty and Mastorakos Citation2008; Chakraborty, Hesse, Mastorakos Citation2010; Turquand d’Auzay et al. Citation2019a,b; Wandel, Chakraborty, Mastorakos Citation2009). By this time, the value of
had decayed by 25% and 38% in comparison to its value at
for the initial
and
cases, respectively. It is worth noting that DNS is computationally expensive and thus it is not feasible to carry out a detailed parametric analysis in a configuration representative of industrial combustors. Furthermore, it is particularly difficult to match the ratio of the integral length scale to flame thickness in IC engines in typical DNS simulations. Therefore, previous DNS studies by other authors, who focussed on autoignition representative of homogeneous charge compression ignition engines (Bansal and Im Citation2011; Hawkes et al. Citation2006; Sankaran et al. Citation2005) and cyclic variability in the case of spark ignition engines (Pera, Chevillard, Reveillon Citation2013), considered decaying turbulence under canonical configurations (e.g. rectangular box which considers solely the volume surrounding the energy deposition location under homogeneous isotropic turbulence) to enable a detailed parametric variation in order to obtain fundamental physical insights into the relevant processes. The same procedure of numerical experimentation is adopted in this analysis. However, there is experimental evidence that in an IC engine cylinder the turbulence intensity increases following ignition (Boree and Miles Citation2014). Under sustained/increased turbulence, the turbulent diffusivity is likely to increase in comparison to decaying turbulence cases, which is likely to increase the heat transfer rate from the hot gas kernel and thus more ignition energy than in the case of decaying turbulence needs to be supplied to achieve the self-sustained combustion.
Figure 1. Variations of at
(i.e. initial conditions), for the cases with
and
, for
and 1
(left to right), for both bimodal and Gaussian initial distributions (top to bottom) as indicated by the respective labels.
![Figure 1. Variations of ψf at t=0 (i.e. initial conditions), for the cases with ψmeanf=0.356 and σfψf=0.05, for lψ=0.5lt,1.0lt and 1.5lt (left to right), for both bimodal and Gaussian initial distributions (top to bottom) as indicated by the respective labels.](/cms/asset/8e423c39-4111-41e4-ab8b-c9e1730c68b3/gcst_a_1975688_f0001_oc.jpg)
The mean dilution level considered here falls within the range considered in previous experimental analyses (Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013). A diffusion timescale can be estimated as tdiff ~λ2Ψ/D (where is the Taylor-micro scale of
inhomogeneity) varies between
for
to
, respectively. This indicates that the
inhomogeneity survives during the course of the simulation for all cases considered here.
The initial bimodal and Gaussian distributions of the mixture composition have been chosen in such a manner to mimic real-life biogas combustors, and engine modifications widely used in eastern Europe, India and Bangladesh to utilize biogas instead of petroleum-based fuels due to its availability and low cost (Koonaphapdeelert, Aggarangsi, Moran Citation2020; Korberg, Skov, Mathiesen Citation2020; Mustafi and Agarwal Citation2020; Patterson et al. Citation2011; Rafiee et al. Citation2021). The overall composition of the biogas may be controlled (i.e. ) but the fuel composition supplied to the fuel injectors might not be perfectly homogeneous (i.e.
) giving rise to different initial ‘dilution distributions’ which the present study attempts to mimic.
RESULTS & DISCUSSION
It is important to discuss the scope of this study prior to interpreting the results. As mentioned in the introduction, extensive experimental and computational studies have been carried out into the investigation of premixed (uniform spatial distribution) biogas mixtures. As existing studies have focused on the variation of the level of present in the mixture, the effects arising from an increase in the
content of the fuel mixture are well-known (Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013). An increase of the level of
serves to decrease the burning rate of the mixture, and the maximum values of temperature and fuel reaction rate magnitude, and is detrimental toward obtaining successful combustion. The present study principally focuses on the effects arising from the variation of the initial spatial distributions (uniform, bimodal and Gaussian) and not the effects arising from the variation of mean carbon dioxide content and initial turbulence intensity. This analysis aims to reveal how the variation of the spatial distribution of
can affect the evolution of the hot gas kernel initiated by thermal runaway by comparing the differing initial spatial distributions of
(Gaussian and bimodal distribution). Thus, it is instructive to first inspect the variation in the mixture composition encountered between the different initial spatial distributions under initial conditions (see ) and also its evolution when the flame kernel is established as a result of localized forced ignition. For this purpose the isosurfaces of
at
for initial
and
for the homogeneous mixture and inhomogeneous bimodal and Gaussian distribution cases corresponding to initial
and
are exemplarily presented in . The translucent pink sphere in indicates the energy deposition volume with a radius of
. To allow for inspection of the mixture composition, which is encountered by the flame kernel, planes in all three directions at a perpendicular distance of 11
from the center of the domain colored by
are shown. As presents the behavior at
, turbulence has been able to aid the mixing of
. Despite the high value of initial turbulence intensity, the mixture composition remains significantly different between the cases shown and the influence of the inhomogeneous
distribution is still visible, even at this advanced time. However, despite the significantly different mixture compositions encountered by the hot gas kernel across the different initial spatial distributions, the shapes of the isosurfaces remain similar but have some minor differences. The most noticeable of these is present for the initial bimodal distribution, as the tendril of the isosurface closest to the spark volume, has separated from the rest of the hot gas kernel for these cases whilst this behavior is not present for the other cases. This behavior is attributed to the dominant effects arising from the high initial turbulence intensity, which primarily governs the shape of the hot gas kernel and the same initial turbulent velocity field is used for all these cases. Secondly, the amount of energy being deposited into the mixture is the same across all initial distributions, and is also significantly higher than the corresponding MIE for the perfectly premixed diluted mixture (Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013; Turquand d’Auzay et al. Citation2019a). Thus, the deposited energy is enough to ensure a successful thermal runaway for this level of mean dilution. The dominant effects of flame wrinkling and mixing driven by initial turbulence intensity alongside the deposited energy being greater than the MIE give rise to similar flame shapes across the different initial distributions shown in .
Figure 2. Isosurfaces of c at
for initial
and
for the perfectly homogeneous (1st column), inhomogeneous spatial distributions of CO2 dilution following initially Gaussian (2nd column) and bimodal (3rd column) distributions with initial.
. The translucent pink sphere indicates the energy deposition volume. The surfaces showing the local distribution of
are situated at a perpendicular distance of 11
from the center of the domain.
![Figure 2. Isosurfaces of c = 0.85 at t=6.0tsp for initial u ′/sl0=8.0 and ψmeanf=0.356 for the perfectly homogeneous (1st column), inhomogeneous spatial distributions of CO2 dilution following initially Gaussian (2nd column) and bimodal (3rd column) distributions with initial. lψ=1.5lt and σfψf=0.07. The translucent pink sphere indicates the energy deposition volume. The surfaces showing the local distribution of ψf are situated at a perpendicular distance of 11δz0 from the center of the domain.](/cms/asset/e7867964-24c4-4cd1-a631-22ede166a40b/gcst_a_1975688_f0002_oc.jpg)
The temporal evolutions of the maximum values of nondimensional temperature (i.e.
) and the normalized maximum methane reaction rate magnitude
(referred to as
for brevity) for all the laminar cases with
and
0.07, across all initial spatial distributions are shown in 3b, respectively. The cases shown below are representative of the behavior which is observed across all the parametric variations investigated in the present study, and the trends outlined here hold across all values of parameters investigated. As mentioned earlier, increases in initial turbulence intensity and mean level of dilution act to decrease
and
and thus are not explicitly shown here and interested readers are directed to Turquand d’Auzay et al. (Citation2019a) for further detail on these effects. It can be observed from that a near collapse between the profiles of the maximum temperature and reaction rate for all three different initial distributions are present but there are some subtle differences.
Figure 3. Temporal evolution of (top row) and for
(bottom row) for initial
and 1
(a-c) as indicated by the respective labels, under laminar conditions for cases with
and initial
. Green indicates Gaussian, blue indicates bimodal, and red indicates Uniform, initial spatial distributions of
, respectively. Magenta indicates the corresponding undiluted mixture (
). Circle markers indicate cases with non-uniform spatial distribution, whilst square markers indicate cases with uniform spatial distribution (i.e. perfectly homogeneous mixture), to help with readability.
![Figure 3. Temporal evolution of Tmax (top row) and for ω˙CH4max+(bottom row) for initial lψ=0.5lt,1.0lt and 1.5lt (a-c) as indicated by the respective labels, under laminar conditions for cases with ψmeanf=0.212 and initial σfψf=0.07. Green indicates Gaussian, blue indicates bimodal, and red indicates Uniform, initial spatial distributions of CO2, respectively. Magenta indicates the corresponding undiluted mixture (ψf=0.0). Circle markers indicate cases with non-uniform spatial distribution, whilst square markers indicate cases with uniform spatial distribution (i.e. perfectly homogeneous mixture), to help with readability.](/cms/asset/e6b58e02-a418-47d5-b8b1-f59511a7a2c6/gcst_a_1975688_f0003_oc.jpg)
The profiles for the Gaussian initial distributions remain near identical to that of the homogeneous mixture regardless of the length scale, however the bimodal distribution displays slightly smaller values of maximum temperature as the length scale increases, with the difference becoming more visible as time advances. However, the normalized reaction rates exhibit an opposite trend to that observed for the temperature profiles, as the maximum normalized reaction rate observed increases from the uniform to the bimodal to the Gaussian initial distribution. To understand why such a behavior is observed, it is useful to refer back to , which indicate that under the inhomogeneous distribution of
the hot gas kernel can locally encounter pockets of undiluted mixture (pure
), giving rise to the higher maximum value of
than obtained in the case of uniform distribution of diluent. Moreover, heat is transferred from the zones of high fuel reaction rate magnitude to the neighborhood zones with smaller reaction rate magnitude for inhomogeneous distributions of
, which also acts to reduce the corresponding
value. It is worth noting that the maximum values presented in provide information at a given point in the domain, and thus they have limited significance on the overall burning behavior. Additionally, as can be seen from , the differences in the profiles observed are small, and for the turbulent cases, which are not shown here, the profiles completely collapse upon one another. As the effects arising from the variation of the initial distribution as a whole cannot be interpreted from the variations shown in , it is useful to inspect the probability density functions (PDFs) of
for the region corresponding to
(i.e. reaction zone for the present thermo-chemistry) at
for all the cases with
and
with initial
0.07, which are shown in .
Figure 4. PDFs of for the region corresponding to
extracted at
for initial
and
(top to bottom), and initial
and 1
(left to right) as indicated by the respective labels, for the cases with initial
, with the solid lines indicating
and the dashed lines indicating
. Green indicates Gaussian, blue indicates bimodal, and red indicates Uniform, initial spatial distributions of
, respectively.
![Figure 4. PDFs of ω˙CH4+ for the region corresponding to 0.5≤c≤0.95 extracted at t=5.0tspfor initial u ′/sl0=0,4.0 and 8.0 (top to bottom), and initial lψ=0.5lt,1.0lt and 1.5lt (left to right) as indicated by the respective labels, for the cases with initial σfψf=0.07, with the solid lines indicating ψmeanf=0.212 and the dashed lines indicating ψmeanf=0.356. Green indicates Gaussian, blue indicates bimodal, and red indicates Uniform, initial spatial distributions of CO2, respectively.](/cms/asset/60ba43bf-9182-40f5-b4e1-54d12dbd524b/gcst_a_1975688_f0004_oc.jpg)
An increase in (solid to dashed lines) leads to a decrease of the probability of finding a large magnitude of
in accordance with pevious findings (Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013; Turquand d’Auzay et al. Citation2019a). Moving from left to right, implying an increase of
, reveals that the PDFs of
are only marginally affected by the variation of
within the parameter range considered here. It was also found that the variation of
had no substantial effect and thus those cases are not shown in for the sake of conciseness. The similarity of the PDFs with the variations of both
and
are consistent with the findings from . Of particular interest, however, is the spread and distribution of the values of
across the different initial spatial distributions. Differences in the profiles of the PDFs across the Gaussian and bimodal distributions are observed, specifically at the tails of the PDFs. Across all cases, a larger spread of values of
are observed for the initial Gaussian distributions of
dilution compared to the initial bimodal distributions, and this arises due to the nature of the distributions. As can be seen from , the range of values of
present for the Gaussian initial distribution is larger than that for the bimodal initial distribution, which drives the larger spread of values of
observed for the Gaussian distribution. As expected, the homogeneous distribution displays the smallest range of
values and a range of values of
in these cases is obtained due to the variations of
and
within the range of reaction progress variable considered in . For both Gaussian and bimodal distributions, the kernel can also locally encounter mixtures with
which in turn give rise to low values of
. As the range of values of
, which can be encountered locally under the bimodal conditions, is more restricted than those which can be found for the initial Gaussian distribution of
dilution, sharper peaks of the PDFs of
are observed for bimodal cases than in the corresponding initial Gaussian distributions. These peaks are not present for the uniform mixture, as for the homogeneous mixture cases with
.
In order to fully explain the behavior of the peaks of the PDFs for , and the similarity of the profiles of the PDFs for
for both Gaussian and bimodal distributions of
dilution, the PDFs of
in the unburned mixture (i.e.
) and preheat zone (i.e.
) at three time instants (
,5.0 and 8.0
) have been exemplarily shown in , respectively for
in the case of initial
. The same qualitative behavior has been observed for the other values of mean dilution and variance across both the Gaussian and bimodal initial distributions of
dilution, and thus they are not shown here for the sake of clarity. Across all cases, the PDFs of
become increasingly narrow with peaks converging around their respective
values as time advances due to mixing. Mixing is primarily driven by the effects of turbulence, and also by the integral length scale of the spatial distribution of
. It is worthwhile to start by inspecting the laminar cases to allow for the effects arising from the increase of
in the unburnt mixture to be highlighted, due to the absence of turbulence. For
, both Gaussian and bimodal distributions of
dilution converge toward a distribution with a peak at their respective mean values, and this behavior is more visible for the bimodal distribution. However, for
, both Gaussian and bimodal distributions retain some resemblance to their initial profiles throughout the duration of the simulation time. Lastly, for
, the PDFs of
broaden indicating that a wider range of values of
can potentially be encountered in the unburned gas for both Gaussian and bimodal distributions. The rate of mixing increases with increasing initial
and thus the distribution of
in the unburned gas becomes increasingly narrow with a peak at
as time advances with an increase in
. Although the mixture is stoichiometric (i.e.
) everywhere in the domain, both
and
exhibit spatial variations due to the inhomogeneous distribution of
. Thus, the integral length scale of
provides a measure of the scalar dissipation rate of mixture fraction
such that a small value of
yields large values of
(Patel and Chakraborty Citation2014, Citation2016a). As a result, a decrease in
acts to increase the mixing of
in the unburned gas. For the large value of
, the mixing rate of
dilution is relatively small and thus the diffusion of
produced as a result of the chemical reaction into the unburned gas acts to slightly increase the range of value of
encountered in the unburned mixture, as time advances. This is responsible for the observed broadening of
PDFs with time in for initial
but the rate of mixing of
is faster for smaller values of
and thus the PDFs of
narrow with time for initial
and
cases. The trend described above in terms of
for laminar cases is also present in the turbulent cases, however it is not as visible due to the mixing induced from the effects of turbulence. As initial turbulence intensity increases, so too does the rate of mixing, as evidenced by the faster narrowing of the PDFs around the initial mean
values.
Figure 5. PDFs of evaluated for the unburned mixture (i.e.
) at
(solid lines),
(dashed lines) and
(dotted lines with circle markers) for both initial Gaussian (green) and bimodal (blue) distributions for
with initial
for initial
and
(top to bottom), and initial
and 1
(left to right) as indicated by the respective labels. The vertical dashed lines indicate
.
![Figure 5. PDFs of ψf evaluated for the unburned mixture (i.e. T≤0.01) at t=2.0tsp (solid lines), t=5.0tsp (dashed lines) and t=8.0tsp (dotted lines with circle markers) for both initial Gaussian (green) and bimodal (blue) distributions for ψmeanf=0.356 with initial σfψf=0.07 for initial u ′/sl0 =0, 4.0 and 8.0 (top to bottom), and initial lψ=0.5lt, 1.0lt and 1.5lt (left to right) as indicated by the respective labels. The vertical dashed lines indicate ψmeanf=0.356.](/cms/asset/36256c47-8827-441c-9132-06f9984de49a/gcst_a_1975688_f0005_oc.jpg)
Figure 6. PDFs of evaluated for the preheat zone (i.e.
) at
(solid lines),
(dashed lines) and
(dotted lines with circle markers) for both initial Gaussian (green) and bimodal (blue) distributions for
with
for initial
and
(top to bottom), and initial
and 1
(left to right) as indicated by the respective labels. The vertical dashed lines indicate
.
![Figure 6. PDFs of ψf evaluated for the preheat zone (i.e. 0.05≤T≤0.45) at t=2.0tsp (solid lines), t=5.0tsp (dashed lines) and t=8.0tsp (dotted lines with circle markers) for both initial Gaussian (green) and bimodal (blue) distributions for ψmeanf=0.356 with σfψf=0.07 for initial u ′/sl0 =0, 4.0 and 8.0 (top to bottom), and initial lψ=0.5lt, 1.0lt and 1.5lt (left to right) as indicated by the respective labels. The vertical dashed lines indicate ψmeanf=0.356.](/cms/asset/d1c62568-e9d5-44df-9207-53db4559e066/gcst_a_1975688_f0006_oc.jpg)
At this point, it is worth recalling that demonstrate that the differences between the results obtained for bimodal and Gaussian distributions of dilution are marginal, despite the significant differences in both initial conditions and the mixture composition encountered by the kernels during its evolution. In order to understand this behavior, it is worthwhile to consider the PDFs shown in .
The PDFs of suggest that the most probable mixture composition in the preheat zone of the hot gas kernel is almost the same for both Gaussian and bimodal distributions of dilution, and this tendency strengthens as time advances. At
, the PDFs of
for both initial Gaussian and bimodal distribution cases become similar due to turbulent and molecular mixing occurring in the unburned mixture, as explained previously. For the homogeneous
distribution, the hot gas kernel always interacts with mixtures with a dilution level of
. However, in the case of non-uniform spatial
distributions, the probabilities of finding
increase for
. The samples corresponding to
include different values of
. The mixture composition corresponding to
burns faster and attains higher burned gas temperatures and higher
values. Thus, the low temperature slow burning regions (e.g.
) predominantly include samples from high
dilution leading to a mean
, which is greater than
. This also explains why the cases with a non-uniform spatial distribution of
exhibit different
statistics to that in the case of the homogeneous mixtures (see ), but also as to why similar behavior is observed in spite of the significantly different initial conditions arising from the Gaussian and bimodal initial distributions of
dilution. It can further be seen from Fig, 6 that the range of values of
which the flame kernel encounters are much more limited for the bimodal distribution than those with an initial Gaussian initial
distribution. Thus, the probability of finding small values of
is higher for the case of initial Gaussian distribution than in the case of initial bimodal distribution.
In this analysis, three statistically independent realizations for the parameters listed in were used to analyze the stochastic effects. Due to a large number of cases, only the cases with initial for
and 0.356 are presented for the sake of conciseness. In order to illustrate the effects arising from stochastic effects, the temporal evolutions of the mean volume of the region with
(i.e.
), normalized by
are shown in , with the error bars indicating the standard deviation based on all three realizations, at that given time. demonstrates that
decreases with increasing
(solid to dashed lines) for a given
, which is consistent with previous experimental (Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013) and computational (Turquand d’Auzay et al. Citation2019a) findings. An increase in initial turbulence intensity acts to increase the turbulent eddy diffusivity Dt~u'lt and thus enhances the heat transfer rate from the hot gas kernel, and acts to reduce the burned gas volume, which is consistent with several previous analyses (Chakraborty, Cant, Mastorakos Citation2007; Chakraborty and Mastorakos Citation2008; Chakraborty, Hesse, Mastorakos Citation2010; Wandel, Chakraborty, Mastorakos Citation2009; Patel and Chakraborty Citation2014, Citation2015, Citation2016a-c; Turquand d’Auzay et al. Citation2019a,b). The growth of the hot gas kernel depends on the competition between the heat release rate and heat transfer rate from the hot gas kernel. An increase in
dilution reduces the burning rate and thus heat release also decreases, which leads to a reduction in
with increasing
.
Figure 7. Temporal evolutions of mean with initial
for
(solid lines) and
(dashed lines), for initial
and
(top to bottom) and initial
and 1
(left to right) as indicated by the respective labels. Blue indicates bimodal, green indicates Gaussian, and red indicates uniform initial spatial distributions of
, respectively. The lines indicate the mean value of
at that time instant across all three realizations investigated, whilst the error bars indicate the standard deviation. All cases shown in this figure have an initial value of
.
![Figure 7. Temporal evolutions of mean Vc≥0.9/Vsp with initial σfψf=0.07 for ψmeanf=0.212 (solid lines) and ψmeanf=0.356 (dashed lines), for initial u ′/sl0=0,4.0 and 8.0 (top to bottom) and initial lψ=0.5lt,1.0lt and 1.5lt (left to right) as indicated by the respective labels. Blue indicates bimodal, green indicates Gaussian, and red indicates uniform initial spatial distributions of CO2, respectively. The lines indicate the mean value of Vc≥0.9/Vsp at that time instant across all three realizations investigated, whilst the error bars indicate the standard deviation. All cases shown in this figure have an initial value of σfψ=0.07.](/cms/asset/8bbe72d6-43f1-4936-b0db-c895bddaa458/gcst_a_1975688_f0007_oc.jpg)
To be able to fully understand the influence of the multiple parameters which play a role in determining the outcome of the localized forced ignition, it is instructive to inspect the two different levels of separately. For the cases with
(solid lines) the standard deviation of
, as indicated by the error bars, is significantly smaller than the standard deviation observed for the
cases. The MIE for biogas increases with increasing
(Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013) and thus for a given amount of input energy, the ratio of the deposited energy to the corresponding MIE is greater in the
cases than in the
cases. Observing the uniform distributions (red lines) for
(bottom row), it can be seen that for
(solid lines), successful propagation (defined as a positive temporal derivative of
at the end of the simulation) is obtained in a mean sense. This is not the case for the equivalent
(dashed lines) cases, as the profiles are indicative of flame quenching in a mean sense. An increase in initial turbulence intensity has similar effects to those outlined for an increase of
and the same reasoning regarding the relative magnitude of the deposited energy in comparison to the corresponding MIE applies here due to the increasing trend of the MIE with an increase in
(Galmiche et al. Citation2011; Larsson, Berg, Bonaldo Citation2013).
An increase of (moving left to right), is found to be beneficial for obtaining self-sustained propagation for the hot gas kernels resulting from localized ignition. The slow mixing rate for higher values of
increases the probability of finding
and the rate of heat release is higher for mixtures with smaller
dilution levels. Thus, the heat release rate from the hot gas kernel counters the heat transfer from it more effectively for a higher value of
. This is best highlighted by the turbulent cases (bottom two rows) with
(dashed lines) where quenching of the flame kernel occurs at a later time as
increases for both the bimodal and Gaussian initial distributions.
Additionally, an increase of also serves to increase the variability of the outcomes observed (i.e. larger standard deviations), and this can best be seen for the laminar cases (top row) with
(solid lines). As outlined in the context of , a decrease in the value of
serves to increase the rate of mixing, and thus the effects arising from the differing initial distributions of
dilution between the different realizations disappear earlier for smaller values of
due to the faster rate of mixing.
Lastly and most importantly, the effects of the different initial dilution distributions need to be understood. As expected, the homogeneous
distribution has the least amount of variability in its outcomes, and the variability in the outcome in these cases is induced solely by the different turbulent realizations. A comparison of the outcomes for initially Gaussian and bimodal spatial distributions of
reveals that the variability of the outcome for the bimodal distribution is smaller than that in the case of Gaussian distribution. This occurs due to the smaller range of
values in the bimodal distribution as opposed to the Gaussian distribution, as shown previously in . Additionally, a Gaussian distribution is expected to increase the chances of a successful outcome compared to the initial bimodal distribution, as highlighted by the higher mean values of
at later times. It can be surmised from the above findings that a bimodal distribution will allow for more repeatable results in comparison to the Gaussian distribution when the deposited energy is significantly greater than the corresponding MIE value. However, an initial Gaussian distribution increases the likelihood of a successful self-sustained flame propagation following thermal runaway compared to the initial bimodal distribution in cases where successful flame propagation is not ensured due to high levels of
. To summarize, if a departure from a non-uniform initial spatial distribution of
is to occur for a given value of
, it is beneficial to have a combination of large values of
, small values of
and a Gaussian initial distribution of
in order to increase the probability of a successful self-sustained flame propagation subsequent to thermal runaway as a result of external energy addition. It is important to appreciate that three statistically independent realizations are used in the current analysis as a compromise because of the high computational cost of DNS (altogether 354 simulations are conducted here). However, despite the absence of statistical convergence, these statistically independent realization results show a clear trend, which indicates that high values of
, small values of
for an initially Gaussian distribution of
increase the chance of successful self-sustained flame propagation following thermal runaway induced by the external energy addition.
CONCLUSIONS
The localized forced ignition and subsequent flame propagation have been analyzed for stoichiometric biogas-air mixtures with three different initial spatial distributions of dilution (uniform, bimodal and Gaussian) under different flow conditions (initial
and 8.0 in addition to a quiescent laminar flow condition). The initial spatial distributions of
dilution are characterized by Gaussian and bimodal PDFs with specified values of mean (
and
), standard deviation (
= 0.05, 0.07) and integral length scale
(
). Additionally, perfectly premixed uniformly distributed undiluted (
) and diluted stoichiometric methane-air mixtures (equivalent to the uniform initial conditions) have been considered for each different initial flow condition for the purpose of comparison with the ignition characteristics of stoichiometric biogas-air mixtures.
It has been found that non-uniform dilution has a marginal effect on the shape of the flame kernel, and on the temporal evolutions of the maximum values of burned gas temperature and fuel (i.e.
) reaction rate magnitude in biogas-air mixtures, irrespective of the nature of the initial spatial distribution of
dilution. Additionally, a departure from uniform
dilution leads to a larger range of values of the consumption rate of the fuel (i.e.
) than that observed for the homogeneous
dilution distribution. A Gaussian initial distribution gives rise to higher values of the consumption rate of the fuel compared to the bimodal initial distribution of dilution. The variations of
and
(i.e. the characteristics of the initial distribution) have been found to have marginal influence on the consumption rate of the fuel. The increases of
and
have been found to have detrimental effects on the consumption rate of the fuel.
The variations of considered here have been found to influence the level of mixing in the unburned gas, with a decrease in
acting to increase the rate of mixing. An increase of initial
has also been found to aid the rate of mixing in the unburned gas. The initial bimodal distribution of
has been found to mix more rapidly and converge toward a uniform dilution distribution corresponding to
faster than the initial Gaussian
distribution for the unburned mixture. The initial Gaussian distribution of
dilution leads to a non-negligible possibility of the kernel encountering mixtures with
, unlike in the initial bimodal
distribution. Thus, the cases with initial Gaussian distribution of
dilution exhibit higher likelihood of obtaining a successful self-sustained flame propagation than the bimodal initial distribution. The variation of
within the parameter range considered in the present study has been found to have a marginal influence due to the dominant effects arising from
and
.
The stochastic effects of different initial turbulence and dilution distributions have also been analyzed in detail. An increase in
and
was found to increase the variability of the burned gas mass, with an increase of
also increasing the variability of a successful thermal runaway and subsequent self-sustained propagation being obtained across all initial spatial distributions. Although the variation of
for a given value of
has been found to have a marginal influence, the non-zero values of
and
(i.e. departure from uniform conditions) have been found to be detrimental for burning rate and decrease the chances of a successful ignition and subsequent self-sustained flame propagation. Due to the multiple parameters investigated in this study and their interaction with each other, it is useful to list them in terms of their importance. The most significant effects were found to arise from the variations of
and initial
. An increase of both parameters led to a higher variability of the outcome observed, along with a decrease in the probability of a successful event being observed. The variations of the initial distributions of
and
have been found to have secondary effects on the localized forced ignition of biogas-air mixtures. The ignition outcomes in the case of homogeneous
distributions have been found to be the most repeatable, and these cases exhibit a high likelihood of successful ignition events. Both the bimodal and Gaussian distributions of
dilution in the unburned gas have been found to increase the variability of the outcomes, leading to a higher probability of flame extinction. However, it was found that the Gaussian distribution performed better than the bimodal distribution in terms of successful outcomes of the ignition event. An increase of
leads to a higher probability of obtaining successful events in a mean sense, and the physical reasons for this behavior have been explained in terms of the effects of
on the rate of mixing of
. Lastly, the variations of
were found to have a negligible effect on the rate of burning for the range of parameters considered here. The present findings indicate that if a departure from a perfect premixing (i.e. homogeneous distribution) is to occur, a roughly Gaussian profile (e.g. diffusion from a point source) of
dilution has been found to increase the probability of a successful self-sustained flame propagation subsequent to thermal runaway induced by external energy addition in comparison to that a bimodal
dilution distribution for a given set of values of
,
and
.
It is important to note that the present analysis has been conducted for a two-step chemical mechanism in the interest of computational economy for a detailed parameteric analysis. Moreover, the simulations have been conducted in a canonical configuration and thus the results obtained here are of qualitative significance. Therefore, future analyses using three-dimensional DNS with detailed chemistry and transport, which is capable of predicting both the flame propagation and ignition delay accurately, will be necessary for further validation. Moreover, the present analysis has been conducted for decaying turbulence but this may not be true in actual IC engines and thus this analysis should also be conducted under forced turbulence to ascertain if any aspects of the current findings alter in the presence of forced turbulence.
Acknowledgments
The financial support of the British Council and EPSRC (EP/R029369/1) and the computational support of Rocket and ARCHER are gratefully acknowledged.
Correction Statement
This article has been republished with minor changes. These changes do not impact the academic content of the article.
Additional information
Funding
References
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