The influence of fuel Lewis number \({\mathrm{Le}}_{F}\) on the statistical behaviour of wall heat flux and flame quenching distance has been analysed using direct numerical simulation (DNS) data for the turbulent V-shaped flame-wall interaction in a channel flow configuration corresponding to a friction velocity-based Reynolds number of 110 for fuel Lewis number, \({\mathrm{Le}}_{F}\), ranging from 0.6 to 1.4. It has been found that the maximum wall heat flux magnitude in turbulent V-shaped flame-wall interaction increases with decreasing \({\mathrm{Le}}_{F}\) but just the opposite trend was observed for 2D laminar V-shaped flame-wall interaction and 1D laminar head-on quenching cases. This behaviour has been explained in terms of the correlation of temperature and fuel reaction rate magnitude with local flame surface curvature for turbulent flames due to the thermo-diffusive effects induced by the non-unity Lewis number. The wall heat flux magnitude and wall shear stress magnitude are found to be negatively correlated for all cases considered here. Moreover, their mean variations in the streamwise direction are qualitatively different irrespective of \({\mathrm{Le}}_{F}\), although the magnitudes of wall heat flux and wall shear stress increase with decreasing \({\mathrm{Le}}_{F}\). Furthermore, the flame alignment relative to the wall also affects the wall heat flux and it has been found that local occurrences of head-on quenching can lead to higher magnitudes of wall heat flux magnitude. It has been found that \({\mathrm{Le}}_{F}\) also affects the evolution of the flame quenching distance in the streamwise direction with the progress of flame quenching for different flame normal orientations with respect to the wall. This analysis shows that the effects of fuel Lewis number on flame orientation, correlations of reaction rate and temperature with local flame curvature and coherent flow structures within the turbulent boundary layer ultimately affect the wall heat transfer and flame quenching distance. Thus, the thermo-diffusive effects arising from the non-unity Lewis number need to be taken into account for accurate modelling of wall heat transfer during flame-wall interaction in turbulent boundary layers.
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1 Introduction
Flame-wall interaction (FWI) is critically important for determining the cooling load and avoiding the thermal fatigue of combustor walls. Thus, it is important to analyse and understand the heat transfer characteristics in turbulent boundary layers during FWI. Several experimental (Huang et al. 1988; Jainski et al. 2017, 2018; Jarosinski 1986; Mann et al. 2014; Rißmann et al. 2017; Vosen et al. 1985) and numerical (Ahmed et al. 2018, 2020, 2021a, b, c; Ghai et al. 2022a, c; Konstantinou et al. 2021; Lai and Chakraborty 2016; Lai et al. 2018, 2022; Zhao et al. 2018a, b, 2021, 2023) studies focussed on different aspects of FWI but the heat transfer characteristics during FWI in turbulent boundary layers received limited attention (Alshaalan and Rutland 2002; Ghai et al. 2022a; Gruber et al. 2010). It has been found experimentally (Huang et al. 1988; Jarosinski 1986; Vosen et al. 1985) that in the case of one-dimensional head-on quenching configuration for hydrocarbon-air flames the typical quenching distance remains of the order of 1.0 mm and the typical magnitudes of the maximum wall heat flux can vary between 20–40% of the flame power. These values have subsequently been confirmed by direct numerical simulations (DNS) based on simple chemistry (Lai and Chakraborty 2016; Poinsot et al. 1993; Zhao et al. 2018a) and detailed chemistry (Gruber et al. 2010; Lai et al. 2018; Zhao et al. 2023). It has been demonstrated previously based on DNS studies (Lai and Chakraborty 2016, 2023; Lai et al. 2018; Poinsot et al. 1993; Zhao et al. 2018a, 2021) that the maximum wall flux magnitude during turbulent premixed FWI can be significantly greater than the corresponding value obtained under laminar flow conditions. The augmentation of flame surface area under turbulence increases the rate of heat transfer, which is reflected in the higher magnitudes of wall heat flux under turbulent conditions. The flame area generation, flame temperature and flame quenching distance are affected by the characteristic Lewis number of the flame and thus the wall heat flux magnitude is also significantly affected by it. However, most of these DNS analyses (Ahmed et al. 2018; Dabireau et al. 2003; Jiang et al. 2019; Lai and Chakraborty 2016; Lai et al. 2018; Poinsot et al. 1993; Sellmann et al. 2017; Zhao et al. 2018a, b, 2021, 2023) were conducted in the canonical configuration of head-on quenching under decaying turbulence or flame impinging on inert isothermal walls but relatively limited attention has been devoted to FWI in turbulent boundary layers where the interaction of the flame surface with near-wall coherent structures could potentially affect the flame area generation and determine the flame location with respect to the wall (Ahmed et al. 2020, 2021c; Alshaalan and Rutland 1998, 2002; Bruneaux et al. 1996; Bruneaux et al. 1997; Ghai et al. 2022a, b, c; Gruber et al. 2010, 2012; Kitano et al. 2015). In most engineering devices, FWI takes place within turbulent boundary layers and the corresponding wall heat transfer characteristics ultimately determine the cooling load and thermal fatigue of the combustor walls. As modern combustors are made smaller in size when compared to their earlier incarnations, the higher surface-to-volume ratio of new smaller combustors makes the flame quenching more likely unless the wall heat transfer is adequately optimised. This calls for an improved fundamental understanding of heat transfer characteristics of FWI in turbulent boundary layers, which serves as the motivation behind the current analysis. In this paper, the heat transfer and flame quenching statistics have been analysed using three-dimensional DNS data for the statistically stationary configuration corresponding to oblique-wall quenching of a V-shaped premixed flame in a turbulent channel flow due to its interaction with inert isothermal channel walls. In order to analyse the thermo-diffusive effects induced by non-unity Lewis numbers, simulations have been conducted for three different fuel Lewis numbers ranging from 0.6 to 1.4 (i.e. \({\mathrm{Le}}_{F}=0.6\), 1.0 and 1.4). In this respect, the main objectives of the current analysis are:
(a)
To provide the heat flux statistics in statistically stationary oblique wall quenching of a V-shaped flame due to its interaction with an inert isothermal wall in a turbulent channel flow for different fuel Lewis numbers.
(b)
To demonstrate and explain the dependence of wall heat flux on the local flame orientation and near-wall coherent flow structures at the location of flame quenching.
The remainder of the paper is organised in the following manner. The mathematical background and numerical implementation relevant to the current analysis are discussed in the next section. This is followed up by the presentation of the results and the subsequent discussion. The main findings are summarised, and conclusions are drawn in the final section of this paper.
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2 Mathematical Background and Numerical Implementation
The flame quenching and wall heat transfer in FWI are characterised in terms of normalised wall heat flux \({\Phi }_{w}\) and Pectlet number \(\mathrm{Pe}\) (Huang et al. 1988; Jarosinski 1986; Lai and Chakraborty 2016; Poinsot et al. 1993; Vosen et al. 1985), which are defined as:
where \({{q}_{w},\rho }_{0}\), \({c}_{p0}, {S}_{L},{T}_{0}\) and \({T}_{ad}\) are the wall heat flux, unburned gas density, unburned gas specific heat, unstretched laminar burning velocity, unburned gas temperature, and adiabatic flame temperature, respectively. In Eq. 1, \(y\) is the wall-normal distance of the non-dimensional temperature \({\theta }^{*}=({T}^{*}-{T}_{0})/({T}_{ad}-{T}_{0})\) isosurface so that \({T}^{*}\) corresponds to the temperature in the unstretched laminar premixed flame for which the maximum heat release rate occurs, \({\delta }_{z}={\alpha }_{T0}/{S}_{L}\) is the Zel’dovich flame thickness with \({\alpha }_{T0}=\lambda /{\rho }_{0}{c}_{p0}\) being the thermal diffusivity of the unburned gas and \({q}_{w}\) is given by: \({q}_{w}=-\lambda {\left(\partial T/\partial y\right)}_{y=0}\) with \(T\) being the instantaneous dimensional temperature and \(\lambda\) is the thermal conductivity at the wall. It can be appreciated from Eq. 1 that the minimum value of the Peclet number provides the measure of the flame quenching distance \({\delta }_{Q}\) in the following manner:
For hydrocarbon-air flames, the maximum heat flux magnitude \(|q_{w} |_{max}\) can be scaled as: \(|q_{w} |_{max} \sim \lambda \left( {T_{ad} - T_{0} } \right)/\delta_{Q}\), which suggests:
The scalar fields in FWI are characterised in terms of reaction progress variable \(c\) and non-dimensional temperature \(\theta\), which are defined as:
where \(Y\) is the suitable species mass fraction (which is taken to be the fuel mass fraction for the current analysis) and the subscripts 0 and \(\infty\) refer to the values in the unburned gas and fully burned products, respectively. The effects of differential diffusion due to non-unity Lewis number are augmented in the presence of flame curvature (Chakraborty and Cant 2005; Konstantinou et al. 2021; Rutland and Trouvé 1993; Trouvé and Poinsot 1994) and its impact on \({\Phi }_{w}\) will be analysed in Sect. 3 of this paper. The local flame curvature \({\kappa }_{m}\) for a given reaction progress variable \(c\) isosurface can be defined as:
where \({N}_{i}=-(\partial c/\partial {x}_{i})/|\nabla c|\) is the ith component of the flame normal vector. According to the convention used in this paper, the flame surface elements, which are convex (concave) to the reactants, have positive (negative) curvature values.
The chemical reaction is simplified in this analysis by a single-step irreversible reaction given by: \(1.0\,\text{unit mass of Fuel}+s\,\text{unit mass of Oxidiser}\to (1+s) \) unit mass of Products in the interest of a parametric analysis in terms of fuel Lewis number \({\mathrm{Le}}_{F}\) where \(s\) is the stoichiometric ratio of oxidiser to fuel by mass. The values of \({\mathrm{Le}}_{F}\), stoichiometric air–fuel ratio and fuel–air mixtures are summarised in Table 1 where \({\mathrm{Le}}_{F}\) is estimated using \(1/({\mathrm{Le}}_{F} )={\chi }_{1}/({\mathrm{Le}}_{1} )+{\chi }_{2}/({\mathrm{Le}}_{2} ),\) where \({\chi }_{i}\) being the volumetric fractions of the fuel species and \(i=\mathrm{1,2},\dots .{n}_{s}\) is the number of species in the fuel mixture (Dinkelacker et al. 2011). For all cases, the heat release parameter \(\tau =({T}_{ad}-{T}_{0})/{T}_{0}\) is taken to be 2.3 (i.e. \(\tau =2.3\)) for the purpose of comparison of the effects of the fuel Lewis number \({\mathrm{Le}}_{F}\) in isolation. The value of \(\tau =2.3\) corresponds to different unburned gas temperatures for different values of \({\mathrm{Le}}_{F}\). The adiabatic flame temperatures for all the cases lie in the range of 2400–2450 K. Therefore, the difference in the values of the unburned gas temperature is not very high. The values of the unburned gas temperature corresponding to \(\tau =2.3\) are 735 K, 730 K and 740 K for the \({\mathrm{Le}}_{F}\) = 0.6, 1.0 and 1.4 cases, respectively. Standard values are taken for Prandtl number \(\mathrm{Pr}\), ratio of specific heats \(\gamma\) (i.e. \(\mathrm{Pr}=0.7\) and \(\gamma =1.4\)) and a stoichiometric mixture is considered for the current analysis.
Table 1
The values of fuel Lewis Number \({\mathrm{Le}}_{F}\), stoichiometric air–fuel ratio and composition of fuel–air mixture for the cases considered here
It is important to note that several previous studies utilised single-step chemistry for head-on quenching simulations of premixed turbulent flames under initially isotropic turbulence conditions (Ahmed et al. 2018; Lai and Chakraborty 2016; Poinsot et al. 1993; Sellmann et al. 2017), head-on quenching in a turbulent channel flow (Bruneaux et al. 1996), and also in a V-flame configuration (Alshaalan and Rutland 1998, 2002). It has been found that including a detailed chemical mechanism with variable transport properties does not affect the qualitative nature of flame dynamics (Lai et al. 2018, 2022) or the behaviour of flame-turbulence interaction in the presence of inert isothermal walls (Ahmed et al. 2018). The models proposed for the flame surface density (FSD) and scalar dissipation rate (SDR) based on simple chemistry DNS data are found to be valid for detailed chemistry DNS in the case of FWI (Lai et al. 2018, 2022), and wall heat flux and wall Peclet number obtained from simple chemistry DNS have been found to be in good agreement with experimental findings (Huang et al. 1988; Jarosinski 1986; Vosen et al. 1985). The fluid-dynamical aspects of oblique-wall quenching of turbulent V-shaped premixed flames based on simple chemistry DNS data (Alshaalan and Rutland 1998, 2002) are found to be consistent with detailed chemistry results (Gruber et al. 2010). A recent experimental analysis (Jainski et al. 2017) revealed that the models developed using single-step chemistry DNS can represent global features of the near-wall FSD profiles obtained in experiments.
The detailed chemistry-based DNS analyses (Gruber et al. 2010; Lai et al. 2018, 2022; Zhao et al. 2023) revealed that low-temperature chemical reactions originating from H, HO2 and H2O2 can give rise to heat release at the chemically inert wall despite flame quenching. These effects are not captured by simple chemistry, but they can be strong for high H2 content fuels, while these effects are weak in the case of hydrocarbon fuels (Gruber et al. 2010; Lai et al. 2018, 2022; Zhao et al. 2023). However, a recent detailed chemistry DNS-based FWI analysis of H2-air flames by Zhao et al. (2023) revealed that the wall-normal temperature profile and wall heat flux do not change significantly between the cases with the absence and presence of the wall heat release. As the wall normal temperature and heat flux are not significantly affected by the wall heat release, it can be expected that the conclusions of the present work, which focuses on wall heat transfer and flame quenching statistics in wall-bounded flows, will remain valid at least qualitatively despite the simplifications associated with single-step chemistry.
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The simulations have been conducted using a compressible DNS code called SENGA+ (Jenkins and Cant 1999) where the conservation equations of mass, momentum, energy and species are solved in non-dimensional form. In SENGA+, all the spatial derivatives for the internal grid points have been calculated using a 10th-order central difference scheme but the order of accuracy gradually decreases to a one-sided 2nd-order scheme at the non-periodic boundaries. The time advancement has been carried out using a low-storage 3rd order Runge–Kutta scheme. A non-reacting turbulent plane channel flow subjected to a constant streamwise pressure gradient (i.e. \(-\partial p/\partial x=\rho {u}_{\tau ,NR}^{2}/h\) where \(p\) is the pressure, \({u}_{\tau ,NR}=\sqrt{\left|{\tau }_{w.NR}\right|/\rho }\) is the friction velocity for the non-reacting channel flow, \({\tau }_{w,NR}\) is the mean wall shear stress for the non-reacting channel flow and \(h\) is the channel half height) has been simulated to specify the initial condition for the reacting flow simulation and also for specifying the inlet boundary condition. The friction velocity based Reynolds number \({\mathrm{Re}}_{\tau }={\rho }_{0}{u}_{\tau ,NR}h/{\mu }_{0}\) is 110 for this non-reacting flow simulation which corresponds to a bulk Reynolds number of \({\mathrm{Re}}_{b}={2\rho }_{0}{u}_{b}h/{\mu }_{0}=3285\) where \({u}_{b}=1/2h{\int }_{0}^{2h}udy\) is the bulk mean velocity, \({\rho }_{0}\) is the unburned gas density and \({\mu }_{0}\) is the unburned gas viscosity. The ratio of unstretched laminar burning velocity to the non-reacting flow friction velocity (i.e. \({S}_{L}/{u}_{\tau ,NR}\)) is taken to be 0.7, and the Mach number based on \({u}_{\tau ,NR}\) (i.e. \(Ma={u}_{\tau ,NR}/{a}_{0}\) where \({a}_{0}\) is the acoustic speed in the unburned gas) is \(3\times {10}^{-3}\) for all cases considered here. The non-reacting flow simulation results in terms of mean velocity and fluctuating velocity statistics have been validated with respect to previous DNS results (Tsukahara et al. 2005) and an excellent agreement was obtained (Ahmed et al. 2021b, c). For the channel flow configuration, the longitudinal integral length scale \({L}_{11}\) scales with \(h\) at \({\mathrm{Re}}_{\tau }=110\), and the root-mean-square velocity fluctuation scales with \({u}_{\tau ,NR}\) (Ahmed et al. 2021a). These values lead to a Damköhler number \(\mathrm{Da}={L}_{11}{S}_{L}/{u}^{^{\prime}}{\delta }_{th}\) of 15.80 and a Karlovitz number \(\mathrm{Ka}={\left({u}^{^{\prime}}/{S}_{L}\right)}^{3/2}{\left({L}_{11}/{\delta }_{th}\right)}^{-1/2}\)) of 0.36, which are representative of the corrugated flamelet regime of combustion (Peters 2010).
For the current study, the simulation domain is taken to be \({L}_{x}\times {L}_{y}\times {L}_{z}=22.22h\times 2h\times 4h\), which is discretised by an equidistant Cartesian grid of \(4000\times 360\times 720\). This grid spacing ensures that 8 points reside within the thermal flame thickness \({\delta }_{th}=({T}_{ad}-{T}_{0})/{\mathrm{max}\left|\nabla T\right|}_{L}\) and the maximum value of \({y}^{+}\) for the grid points adjacent to the wall remains smaller than 0.6. For the V-flame simulation, the flame holder is placed within the fully developed channel flow at a location such that the centre of the flame holder is located at a streamwise distance of \({x}_{h}=0.83h\) from the inlet and a wall-normal distance of \(h\) (i.e., corresponds to \({y}^{+}=110\) for the non-reacting flow) from the walls. The radius of the flame holder is considered to be \(0.5{\delta }_{th}\) and the species mass fractions, velocity components and temperature at the flame holder are specified using a Gaussian function following Dunstan et al. (2010) and further details are provided elsewhere (Ahmed et al. 2021b). A turbulent inflow with specified density and velocity components from non-reacting flow simulations and a partially non-reflecting outflow boundary are specified for the boundaries in the streamwise direction which will henceforth be referred to as the \(x\)-direction. The walls are taken to be chemically inert and under the isothermal condition with the same temperature as the unburned gas temperature (i.e. \({T}_{y=0}={T}_{y=2h}={T}_{0})\). The walls are impenetrable suggesting that the wall-normal component of the mass flux disappears at the wall and the wall tangential velocity components vanish due to the no-slip condition. The boundaries in the spanwise direction (i.e. \(z\)-direction) are considered to be periodic. All the non-periodic boundaries are specified using an improved version of the Navier–Stokes Characteristic Boundary condition (NSCBC) (Yoo and Im 2007). It is important to note that to date all DNS studies on FWI under turbulent conditions have been carried out either for isothermal (Ahmed et al. 2018; Alshaalan and Rutland 1998, 2002; Gruber et al. 2010; Gruber et al. 2012; Lai and Chakraborty 2016; Lai et al. 2018, 2022; Poinsot et al. 1993; Sellmann et al. 2017) or adiabatic (Ahmed et al. 2021b) walls. As the heat transfer statistics is the subject of the current study, the isothermal wall boundary condition is chosen for this analysis following previous studies. In the real scenario, the wall conditions are going to be somewhere between isothermal and adiabatic wall boundary conditions and the condition prevailing at the wall is determined by the conjugate heat transfer. However, the spatial variation of the temperature on the wall surface is not expected to be very large. The temperature of the wall surface in the real stationary gas turbine combustors is usually maintained in the range of 700–800 K for power generation up to 400 MW, which is close to the wall temperature considered in the present simulations. Thus, at least the qualitative nature of the heat transfer statistics presented in this paper are unlikely to be altered in the presence of conjugate heat transfer.
All the simulations have been conducted for \(2.0\) flow-through times based on bulk velocity (i.e., \(2.0{L}_{x}/{u}_{b}\)), and statistics have been extracted after one flow through time once the initial transience has decayed. The Reynolds/Favre averaged quantities are evaluated by time-averaging and subsequently by spatial averaging in the statistically homogeneous \(z\)-direction.
3 Results and Discussion
3.1 Effects of \(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}\) on Global Features of FWI
The isosurfaces of the reaction progress variable \(c=0.5\) after reaching a quasi-steady state for the oblique quenching of the turbulent V-flames for \({\mathrm{Le}}_{F}=0.6, 1.0\) and 1.4 are shown in Fig. 1a along with the distributions of normalised vorticity magnitude \(\mathrm{W}=\sqrt{{w}_{i}{w}_{i}}\times h/{u}_{\tau ,NR}\) at \(z/h=2.0\) where \({w}_{i}\) is the ith component of vorticity. The vortical flow structures close to the wall, seen from Fig. 1a, affect the flame wrinkling and the interaction of the flame surface with the wall leads to eventual flame quenching due to heat loss through the cold isothermal wall. The distributions of Favre-averaged temperature \(\widetilde{\theta }\) in the central midplane with \(\widetilde{c}=\mathrm{0.1,0.5}\) and 0.8 superimposed are shown in Fig. 1b for \({\mathrm{Le}}_{F}=0.6, 1.0\) and 1.4 cases. It can be seen from Fig. 1 that the flame starts to interact with the wall at \(x/h>12\) for the cases considered here but Fig. 1b suggests that the flame in the \({\mathrm{Le}}_{F}=0.6\) case interacts with the wall closer to the flame holder than in the \({\mathrm{Le}}_{F}=1.0\) case, whereas just the opposite trend is observed for the \({\mathrm{Le}}_{F}=1.4\) case. This can be substantiated by Fig. 2a, which shows the variation of streamwise distance from the flame holder \((x-{x}_{h})/h\) of the intersection point of the \(\widetilde{c}=0.5\) contours with the wall for \({\mathrm{Le}}_{F}=0.6, 1.0\) and \(1.4.\) It can further be seen from Fig. 2b, where instantaneous views of \(c=0.75\) isosurface (this isosurface is chosen because the maximum fuel reaction occurs close to \(c=0.75\) for the thermochemistry used) for the bottom flame branch of the V-flame coloured by the wall-normal distance \(y/h\) are shown, that the flame interacts with the wall farther downstream even in the instantaneous sense for higher values of LeF. The upstream movement of the onset of FWI with a decrease in \({\mathrm{Le}}_{F}\) is a consequence of the increase in turbulent burning velocity with a decrease in fuel Lewis number, which can be substantiated by the values of normalised volume-integrated burning rate \(\Lambda ={\int }_{V}|{\dot{\omega }}_{F}|dV/[{\rho }_{0}{S}_{L}{h}^{2}\left({Y}_{F0}-{Y}_{F\infty }\right)]\) listed in Table 2 for the cases considered here. Several previous analytical (Clavin and Joulin 1983; Williams et al. 1985) and numerical (Chakraborty and Cant 2005; Haworth and Poinsot 1992; Konstantinou et al. 2021; Lai and Chakraborty 2016; Ozel-Erol et al. 2021; Rutland and Trouvé 1993; Trouvé and Poinsot 1994) studies reported increasing trends of turbulent burning velocity and flame surface area with a decrease in Lewis number without the influence of walls.
Fig. 1
a Isosurfaces of \(c=0.5\) (shown in pink) after reaching a statistically steady state for oblique wall-interaction of turbulent V-flames with \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\). The distributions of normalised vorticity magnitude \(\mathrm{W}=\sqrt{{w}_{i}{w}_{i}}\times h/{u}_{\tau ,NR}\) in the central mid-plane are also shown. b Fields of Favre-averaged temperature \(\widetilde{\theta }\) with \(\widetilde{c}=\mathrm{0.1,0.5},\) and 0.8 (shown in red) superimposed for oblique wall-interaction of turbulent V-flames with \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\)
Fig. 2
a Streamwise distance from the flame holder \((x-{x}_{h})/h\) of the intersection point of \(\widetilde{c}=0.5\) contour with the wall and b isosurfaces of \(c=0.75\) coloured by wall normal distance \(y/h\) after reaching quasi-steady state, for bottom branches of oblique wall-interaction of turbulent V-flames with \({\mathrm{Le}}_{F}=0.6\), \(1.0\) and \(1.4\)
Table 2
The values of normalised volume-integrated burning rate \(\Lambda ={\int }_{V}|{\dot{\omega }}_{F}|dV/[{\rho }_{0}{S}_{L}{h}^{2}\left({Y}_{F0}-{Y}_{F\infty }\right)]\) for the cases considered here
\(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}=0.6\)
\(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}=1.0\)
\(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}=1.4\)
\(\Lambda =1.55\)
\(\Lambda =1.33\)
\(\Lambda =1.28\)
×
×
The focussing of fuel diffusion is faster into the reaction zone than the rate at which the defocussing of thermal diffusion takes place in the flame regions which are convex to the reactants for \({\mathrm{Le}}_{F}<1\). This leads to the simultaneous presence of high temperatures and high fuel concentrations in these regions. By contrast, the defocussing of fuel takes place at a faster rate than the focussing of heat in the flame elements which are concave to the reactants for LeF < 1. This gives rise to a combination of low fuel concentration and temperature in the wrinkles which are concave to the reactants in the \({\mathrm{Le}}_{F}=0.6\) case. As a result, in the \({\mathrm{Le}}_{F}=0.6\) case, the flame elements, which are convex to the reactants (i.e. \({\kappa }_{m}>0\)), grow due to high reaction rates in these regions, whereas the wrinkles that are concave to the reactants (i.e. \({\kappa }_{m}<0\)) persist due to weaker flame propagation rates in these regions. Therefore, the extent of flame wrinkling and overall burning rate (e.g. \(\Lambda\)) are greater in the \({\mathrm{Le}}_{F}=0.6\) case than that in the \({\mathrm{Le}}_{F}=1.0\) case. By contrast, in the \({\mathrm{Le}}_{F}=1.4\) case, the focussing of fuel diffusion is slower into the reaction zone than the rate at which the defocussing of heat diffusion takes place in the regions which are convex to the reactants, whereas just the opposite behaviour is obtained in the regions which are concave to the reactants. Thus, the flame elements, which are concave (convex) to the reactants, have the simultaneous presence of high (low) temperature and high (low) fuel concentration for \({\mathrm{Le}}_{F}>1\). These combinations act to reduce the extent of burning rate in the regions which are convex to the reactants and high propagation rates in the concavely curved zones reduce the curvature of the wrinkles for \({\mathrm{Le}}_{F}>1\) and as a result the flame surface area for the \({\mathrm{Le}}_{F}=1.4\) case is smaller than that in the \({\mathrm{Le}}_{F}=1.0\) case. The mechanisms described above are responsible for the increasing trend of \(\Lambda\) with decreasing \({\mathrm{Le}}_{F}\) in Table 2.
The extent of flame wrinkling can be quantified in terms of turbulent flame surface area \({A}_{f}={\int }_{V}\left|\nabla c\right|dV\) normalised by the corresponding value (i.e. \({A}_{L}\)) at the same \(x/h\) in the 2D laminar V-shaped flames with the same centreline velocity as that of the turbulent cases. It can be seen from Fig. 3 that \({A}_{f}/{A}_{L}\) exhibits an increasing trend with decreasing \({\mathrm{Le}}_{F}\) when the flame is sufficiently away from the wall (e.g., \(x/h\le 10\)). The opposite trend is obtained when the flame starts to interact with the wall for \(x/h\ge 12\). The magnitude of \(\left|\nabla c\right|\) drops as flame quenching progresses irrespective of the fuel Lewis number (for explanations please refer to Lai and Chakraborty (2016) and Ahmed et al. (2021b)) and this leads to a decreasing trend of \({A}_{f}/{A}_{L}\) with decreasing \({\mathrm{Le}}_{F}\) for \(x/h\ge 12\), as the \({\mathrm{Le}}_{F}=0.6\) (\({\mathrm{Le}}_{F}=1.0\)) case remains at a more advanced stage of quenching than LeF = 1.0 (LeF = 1.4) at these streamwise locations.
Fig. 3
Variations of turbulent flame surface area \({A}_{f}\) normalised by the corresponding value (i.e. \({A}_{L}\)) for flame-wall interaction of 2D laminar V-shaped flames for \({\mathrm{Le}}_{F}=0.6, 1.0\) and \(1.4\) at \(x/h=6, \mathrm{8,10}, 12, 14\) and \(16\)
×
Here, the term ‘stage of quenching’ can be quantified in terms of the drop of normalised flame surface area \({A}_{f}/{A}_{L}\) with increasing \(x/h\) for \(x/h\ge 12\) because of flame extinction. For example, in the \({\mathrm{Le}}_{F}=0.6\) case, \({A}_{f}/{A}_{L}\) drops by 34% at \(x/h=14\) in comparison to the value obtained at \(x/h=12\), whereas this number increases to 61% at \(x/h=16.\) Similarly in the \({\mathrm{Le}}_{F}=1.0\) (\({\mathrm{Le}}_{F}=1.4\)) case, \({A}_{f}/{A}_{L}\) drops by 10% (0.01%) at \(x/h=14\) in comparison to the value obtained at \(x/h=12\), whereas this number increases to 40% (20%) at \(x/h=16.\) Therefore, the location \(x/h=16\) is at a more advanced stage of quenching when compared to \(x/h=14\) for all the cases considered here. This can further be seen from the larger thermal boundary layer thickness for the \({\mathrm{Le}}_{F}=0.6\) case at a given value of \(x/h\) for the region given by \(x/h\ge 12\) in Fig. 1b and the intersection of \(\widetilde{c}=0.8\) with the wall at a smaller value of \(x/h\) in Fig. 1b.
The aforementioned differential diffusion effects due to \({\mathrm{Le}}_{F}\ne 1.0\) can be substantiated from the isosurfaces of \(c=0.75\) coloured by the local non-dimensional temperature \(\theta =(T-{T}_{0})/({T}_{ad}-{T}_{0})\), reaction rate magnitude of fuel \(\left|{\dot{\omega }}_{F}\right|\times h/{\rho }_{0}{u}_{\tau ,NR}\) and normalised flame curvature \({\kappa }_{m}\times {\delta }_{th}\) values in Figs. 4 and 5 for turbulent V-flame cases with \({\mathrm{Le}}_{F}=0.6\), 1.0 and 1.4. It can be seen from Figs. 4 and 5 that the high values of \(\theta\) (i.e. \(\theta >c\)) on a given \(c\) contour/isosurface are associated with positively curved locations, whereas the negatively curved locations have small \(\theta\) (i.e. \(\theta <c\)) values in the \({\mathrm{Le}}_{F}=0.6\) case where the flame is away from the wall. Just the opposite behaviour is observed for the \({\mathrm{Le}}_{F}=1.4\) case where the relatively high (low) temperatures on a given \(c\) contour/isosurface are associated with negative (positive) curvature locations. This behaviour is consistent with previous simple (Chakraborty and Cant 2005; Konstantinou et al. 2021; Ozel-Erol et al. 2021; Rutland and Trouvé 1993) and detailed (Aspden et al. 2011) chemistry results. It is worth noting that the range of Lewis numbers considered in this analysis is not small enough to exhibit cellular instabilities. Cellular instability for the present thermochemistry is expected for fuel Lewis numbers smaller than 0.3 (Pelce and Clavin 1982) and thus the flame curvature is induced purely due to turbulence in the cases considered here.
Fig. 4
Isosurfaces of \(c=0.75\)a coloured by non-dimensional temperature \(\theta\) (left) and the flame curvature (right) and b coloured by normalised reaction rate magnitude of fuel, \(\left|{\dot{\omega }}_{F}\right|\times h/{\rho }_{0}{u}_{\tau ,NR}\) after reaching a quasi-steady state for bottom branches of oblique wall-interaction of turbulent V-flames with \({\mathrm{Le}}_{F}=0.6\), \(1.0\) and \(1.4\). Positive and negative curvature \({\kappa }_{m}\) locations with respect to the bottom wall are shown in the inset of Fig. 4a
Fig. 5
Isosurfaces of \(c=0.75\) coloured by local values of normalised curvature \({\kappa }_{m}\times {\delta }_{th}\) after reaching a quasi-steady state for bottom branches of oblique wall-interaction of turbulent V-flames with \({\mathrm{Le}}_{F}=0.6\), \(1.0\) and \(1.4\). Insets of each individual subfigure are also shown for the normalised streamwise distance range of \(8\le x/h\le\) 9
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The flame curvature \({\kappa }_{m}\) dependence of non-dimensional temperature \(\theta\) gives rise to high (low) values of \(|{\dot{\omega }}_{F}|\) at the positive (negative) values of \({\kappa }_{m}\) on a given \(c\) isosurface away from the wall for the \({\mathrm{Le}}_{F}=0.6\) case. By contrast, relatively high (low) values of \(|{\dot{\omega }}_{F}|\) are associated with the negative (positive) values of \({\kappa }_{m}\) on a given \(c\) isosurface away from the wall for the \({\mathrm{Le}}_{F}=1.4\) case. For low Mach number, unity Lewis number and adiabatic conditions \(\theta =c\) is maintained and therefore both \(\theta\) and \(|{\dot{\omega }}_{F}|\) do not exhibit any spatial variation on the \(c=0.75\) isosurface when the flame is away from the wall in the \({\mathrm{Le}}_{F}=1.0\) case (see Figs. 4, 5). However, this coupling between \(\theta\) and \(c\) (and also between \(|{\dot{\omega }}_{F}|\) and \(c\)) is lost close to the wall during FWI because of the different boundary conditions for \(c\) and \(\theta\) for isothermal walls (i.e., Neumann condition for \(c\) and Dirichlet condition for \(\theta\)).
A comparison between Figs. 4 and 5 reveals that there is no correlation between \(\theta\) and \({\kappa }_{m}\) when the flame is in the vicinity of the wall and begins to quench. Therefore, the curvature dependence of \(|{\dot{\omega }}_{F}|\) becomes weak in the vicinity of the wall where the flame quenching initiates. This is found to be consistent with recent findings (Kaddar et al. 2022) in the context of detailed chemistry-based DNS of premixed V-flame-wall interaction. As the heat release rate in the context of single-step chemistry is directly proportional to \(|{\dot{\omega }}_{F}|\), the correlations of heat release rate with curvature for these cases are going to be identical to the correlation between curvature and \(|{\dot{\omega }}_{F}|\), as discussed here.
3.2 Effects of \(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}\) on Wall Heat Flux and Flame Quenching Statistics
The curvature dependences of temperature, fuel reaction rate magnitude and heat release rate affect the wall heat flux and the flame quenching distance during FWI. The maximum magnitudes of wall heat flux \({\Phi }_{w,max}\) and the quenching Peclet number \({\mathrm{Pe}}_{Q}\) for turbulent OWQ V-flame, 2D laminar V-flame and 1D head-on quenching of laminar flame are shown in Fig. 6 for different values of \({\mathrm{Le}}_{F}\) where the Peclet number is evaluated based on the wall-normal distance of \({\theta }^{*}=0.75\) isosurface. The peak heat release rate in the unstretched laminar flame is obtained at \({\theta }^{*}\approx 0.75\) for the fuel Lewis numbers considered here and thus \({\theta }^{*}=0.75\) isosurface is used for evaluating the quenching distance. The value of \({\rho }_{0}{c}_{p0}{S}_{L}({T}_{ad}-{T}_{0})\) does not change appreciably for the mixtures considered here (e.g. 1.0% variation) and thus the variation of \({\Phi }_{w}\) occurs principally due to the variation of the wall heat flux magnitude \({|q}_{w}|\). It can be seen from Fig. 6 that \({\Phi }_{w,max}\) in turbulent V-flame-wall interaction increases with decreasing fuel Lewis number \({\mathrm{Le}}_{F}\), whereas the minimum Peclet number \({\mathrm{Pe}}_{min}\) for turbulent V-flame-wall interaction decreases with decreasing \({\mathrm{Le}}_{F}\). However, just the opposite trend in terms of \({\mathrm{Le}}_{F}\) has been observed for both 1D head-on quenching and 2D V-flame-wall interaction. As the thermal diffusion rate is faster than the fuel diffusion rate for \({\mathrm{Le}}_{F}>1.0\), \({\theta }^{*}=0.75\) isosurface reaches closer to the wall before quenching in the 1D laminar head-on quenching case with \({\mathrm{Le}}_{F}=1.4\) than in the corresponding unity Lewis number case. By contrast, \({\theta }^{*}=0.75\) isosurface remains farther from the wall while quenching in the 1D laminar head-on quenching case with \({\mathrm{Le}}_{F}=0.6\) than in the corresponding unity Lewis number case due to a weaker thermal diffusion rate than the diffusion rate of fuel. The values of \({\Phi }_{w,max}\) and \({\mathrm{Pe}}_{min}\) in the 1D planar laminar flame head-on quenching and oblique quenching of the 2D V-shaped laminar flame with the same centreline velocity remain comparable for a given value of \({\mathrm{Le}}_{F}\). The shear rate induced by the wall alters the values of \({\Phi }_{w,max}\) and \({\mathrm{Pe}}_{min}\) in the 2D V-shaped laminar flame in comparison to the 1D planar laminar flame head-on quenching case and slightly higher values of \({\mathrm{Pe}}_{min}\) are obtained for the 2D V-shaped laminar flame than the 1D planar laminar flame head-on quenching case and accordingly \({\Phi }_{w,max }\sim 1/{\mathrm{Pe}}_{min}\) (Lai and Chakraborty 2016; Lai et al. 2018) assumes lower values in the 2D V-shaped laminar flame than in the 1D planar laminar flame head-on quenching case. In the case of turbulent V-flames, the positively curved flame elements for a given value of \(c\) have higher temperatures in the \({\mathrm{Le}}_{F}=0.6\) case than for the corresponding \({\mathrm{Le}}_{F}=1.0\) case, as shown in Fig. 4, and thus these elements are more resistant to flame quenching due to wall heat loss and reach closer to the wall than that in the \({\mathrm{Le}}_{F}=1.0\) case. This gives rise to a higher value of \({\Phi }_{max}\) and a lower \(P{\mathrm{e}}_{min}\) in the turbulent V-flame case with \({\mathrm{Le}}_{F}=0.6\) than for the corresponding \({\mathrm{Le}}_{F}=1.0\) case. Although it was shown in Figs. 4 and 5 that the negatively curved flame elements for a given \(c\) value have higher temperatures in the turbulent V-flame case with \({\mathrm{Le}}_{F}=1.4\) than for the corresponding \({\mathrm{Le}}_{F}=1.0\) case, the negatively curved zones for the \({\mathrm{Le}}_{F}=1.4\) case remain farther away from the wall than the positively curved regions with lower temperatures and thus the regions with \({\kappa }_{m}>0\) quench more readily than in the \({\mathrm{Le}}_{F}=1.0\) case. This leads to a smaller value of \({\Phi }_{max}\) and greater value of \({\mathrm{Pe}}_{min}\) in the turbulent V-flame case with \({\mathrm{Le}}_{F}=1.4\) than for the corresponding \({\mathrm{Le}}_{F}=1.0\) case.
Fig. 6
Variations of the quenching Peclet number \({\mathrm{Pe}}_{min}\) (left) and the maximum magnitudes of wall heat flux \({\Phi }_{w,max}\)(right) for \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\) for 1-D HOQ of laminar flame (i.e. HOQL), 2-D laminar OWQ V-flame (i.e. OWQL), and 3-D turbulent OWQ V-flame (i.e. OWQT). The Peclet number is evaluated based on the wall-normal distance of \({\theta }^{*}=0.75\) isosurface
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The instantaneous distributions of normalised wall heat flux magnitude \({\Phi }_{\mathrm{w}}\) at the time instant when its maximum value is obtained are shown in Fig. 7a for \({\mathrm{Le}}_{F}=\mathrm{0.6,1.0}\) and 1.4 cases, respectively, which show that there are significant spatial variations of \({\Phi }_{\mathrm{w}}\) for all cases considered here and the FWI mostly takes place for \(x/h>12.0\) in all cases. This can further be substantiated by the variation of mean values of normalised wall heat flux \({\overline{\Phi } }_{w}={|\overline{q} }_{w}|/[{\rho }_{0}{c}_{p0}{S}_{L}\left({T}_{ad}-{T}_{0}\right)]\) with the streamwise direction in Fig. 7b, which shows that the maximum value of \({\overline{\Phi } }_{w}\) marginally increases with decreasing \({\mathrm{Le}}_{F}\) for the range of \({\mathrm{Le}}_{F}\) considered here.
Fig. 7
a Instantaneous distributions of normalised wall heat flux magnitude \({\Phi }_{w}\) at the time instant when its maximum value is obtained for oblique wall-interaction of turbulent V-flames with \({\mathrm{Le}}_{\mathrm{F}}=\)\(0.6\)\(1.0\) and \(1.4\). b Distribution of the mean wall heat flux magnitude \({\overline{\Phi } }_{w}\) with the normalised streamwise distance \(x/h\) for oblique wall-interaction of turbulent V-flames with \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\)
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In order to explain this behaviour, the probability density functions (PDFs) of the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and \(16\) for different values of \({\mathrm{Le}}_{F}\) are shown in Fig. 8a. The schematic diagram of the samples on the temperature iso-surface, \({\theta }^{*}=0.75\) representing the locations based on which Peclet number, \(\mathrm{Pe}\) is evaluated is shown in Fig. 8b. The choices of \(x/h=\) 12, 14 and \(16\) are driven by the fact that \(\overline{\Phi }\) assumes significant non-zero values for \(x/h>12\) and the reaction rate on the \({\theta }^{*}=0.75\) isosurface tends to decrease before vanishing altogether, as the flame completely quenches for \(x/h\ge 16\). It can be seen from Fig. 8a that the probability of finding smaller values of \(\mathrm{Pe}\) increases with decreasing \({\mathrm{Le}}_{F}\) in the initial stages of FWI (e.g., \(x/h=12)\). With the progress in FWI, with increasing downstream distance from the flame holder, the flame starts to quench and the probability of finding smaller values of \(\mathrm{Pe}\) increases with increasing Lewis number (e.g., \(x/h=16)\). It is worthwhile to consider the curvature distribution at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and \(16\) for different values of \({\mathrm{Le}}_{F}\) to explain the variation of \(\mathrm{Pe}\) with \({\mathrm{Le}}_{F}\) shown in Fig. 8a. The PDFs of \({\kappa }_{m}\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and \(16\) for different values of \({\mathrm{Le}}_{F}\) are shown in Fig. 9a, which shows that the most probable curvature remains close to zero but the PDFs show a long negative tail as flame quenching progresses in the downstream streamwise locations. This can further be verified by increasing probabilities of negative curvatures with downstream distance at Fig. 9b where probabilities of finding \({\kappa }_{m}>0\) and \({\kappa }_{m}<0\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and \(16\) are shown for different values of \({\mathrm{Le}}_{F}\). This trend is more prevalent for the \({\mathrm{Le}}_{F}=0.6\) (\({\mathrm{Le}}_{F}=1.0\)) case than in the \({\mathrm{Le}}_{F}=1.0\) (\({\mathrm{Le}}_{F}=1.4\)) case because \({\mathrm{Le}}_{F}=0.6\) (\({\mathrm{Le}}_{F}=1.0\)) case remains at a more advanced stage of quenching than the \({\mathrm{Le}}_{F}=1.0\) (\({\mathrm{Le}}_{F}=1.4\)) case for a given value of \(x/h\) (see Fig. 2).
Fig. 8
a Probability density functions (PDFs) of the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames for different values of \({\mathrm{Le}}_{F}\) at \(x/h=12, 14\) and \(16\), b Schematic diagram of the sample points on the temperature iso-surface, \({\theta }^{*}=0.75\) representing the locations where Peclet number, \(\mathrm{Pe}\) is evaluated. The statistics shown in Figs. 9, 10, 13, 14, 17, 19 and 20 are evaluated at the same locations shown in b
Fig. 9
a PDFs of normalised curvature \({\kappa }_{m}\times {\delta }_{th}\) and b Probability of finding positive and negative values of \({\kappa }_{m}\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames for different values of \({\mathrm{Le}}_{F}\) at \(x/h=\)\(12, 14\) and \(16\). The green vertical line in a shows \({\kappa }_{m}\times {\delta }_{th}=0\)
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The PDFs \(\Omega ={\left|{\dot{\omega }}_{F}\right|}_{\theta =0.75}/{\left| {\dot{\omega }}_{F}\right|}_{\theta =0.75,L}\) (where \({\left|{\dot{\omega }}_{F}\right|}_{\theta =0.75,L}\) is the fuel reaction rate magnitude at \(\theta =0.75\) for the unstretched 1D laminar premixed flame) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and \(16\) are shown in Fig. 10 for different values of \({\mathrm{Le}}_{F}\). Figure 10 shows that there is a higher probability of finding high values of \(\Omega\) at a given value of \(x/h\) for non-unity \({\mathrm{Le}}_{F}\) cases than in the case with \({\mathrm{Le}}_{F}=1.0\) at early stages of flame quenching (e.g. \(x/h=12\)) and this behaviour arises due to relatively higher reaction rate magnitudes at positive and negative curvature locations in the \({\mathrm{Le}}_{F}=0.6\) and 1.4 cases, respectively. However, this trend weakens with the progress of flame quenching (see \(x/h=14\) in Fig. 10) and the probability of high values of \(\Omega\) decreases with decreasing \({\mathrm{Le}}_{F}\) at \(x/h=16\) because the \({\mathrm{Le}}_{F}=0.6\) case remains at a more advanced stage of quenching than the \({\mathrm{Le}}_{F}=1.0\) case at this location (see Figs. 1b, 2a). The predominance of \({\kappa }_{m}<0\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface also suggests that these regions of relatively small values of \(\left|{\dot{\omega }}_{F}\right|\) in the \({\mathrm{Le}}_{F}=0.6\) case (see Fig. 4b) quenches more readily than the case with \({\mathrm{Le}}_{F}=1.0\). By contrast, relatively high values of \(\left|{\dot{\omega }}_{F}\right|\) are associated with negatively curved locations in the \({\mathrm{Le}}_{F}=1.4\) case (see Fig. 4b) and thus these regions can resist flame quenching for longer and reach closer to the wall than in the case with \({\mathrm{Le}}_{F}=1.0\). Thus, the probability of small values of \(\mathrm{Pe}\) in Fig. 8 increases with increasing \({\mathrm{Le}}_{F}\) in the region where a considerable amount of FWI takes place, and eventually, the flame starts to quench (e.g. \(x/h=16.0\)). High values of \(\left|{\dot{\omega }}_{F}\right|\) at the positively curved regions for the turbulent V-flame case with \({\mathrm{Le}}_{F}=0.6\) acts to increase both the maximum values of \({\Phi }_{\mathrm{w}}\) in comparison to the corresponding \({\mathrm{Le}}_{F}=1.0\) case. By contrast, relatively small values of \(\left|{\dot{\omega }}_{F}\right|\) at the positively curved regions for the turbulent V-flame case with \({\mathrm{Le}}_{F}=1.4\) acts to decrease the maximum value of \({\Phi }_{\mathrm{w}}\) in comparison to the corresponding \({\mathrm{Le}}_{F}=1.0\) case. The flame quenching distance is mostly greater (smaller) in the \({\mathrm{Le}}_{F}=0.6\) (\({\mathrm{Le}}_{F}=1.4\)) case than in the \({\mathrm{Le}}_{F}=1.0\) case, and this acts to counter the influence of the correlation between \(\left|{\dot{\omega }}_{F}\right|\) and \({\kappa }_{m}\) arising out of non-unity \({\mathrm{Le}}_{F}\) to yield mostly comparable values of mean normalised wall heat flux magnitude \({\overline{\Phi } }_{w}\).
Fig. 10
PDFs of \(\Omega ={\left|{\dot{\omega }}_{F}\right|}_{\theta =0.75}/{\left| {\dot{\omega }}_{F}\right|}_{\theta =0.75,L}\) (where \({\left|{\dot{\omega }}_{F}\right|}_{\theta =0.75,L}\) is the fuel reaction rate magnitude at \(\theta =0.75\) for the unstretched 1D laminar premixed flame) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for different values of \({\mathrm{Le}}_{F}\) at \(x/h=12, 14\) and 16\(.\) The PDF values are shown in log-scale for the ease of comparison
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3.3 Influence of \(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}\) on the Wall Heat Flux Dependencies on Local Wall Shear Stress
The foregoing analysis indicates that \({\mathrm{Le}}_{F}\) has a significant impact on the maximum and mean values of wall heat flux and the flame quenching distance. Therefore, it is worthwhile to analyse the impact of \({\mathrm{Le}}_{F}\) on the wall heat flux dependencies on local wall shear stress, coherent flow structures and flame orientation. The variation of the normalised mean wall shear stress magnitude \(|{\overline{\tau }}_{w}|/{\rho }_{0}{u}_{\tau ,NR}^{2}\) with the streamwise distance along with the mean wall heat flux \({\overline{\Phi } }_{w}\) variation are shown in Fig. 11, which shows that both mean wall heat flux and mean shear stress increase with decreasing \({\mathrm{Le}}_{F}\) because of the strengthening of thermal expansion effects as a result of the higher likelihood of having locally high temperatures in the burned gas regions. However, the variation of \(|{\overline{\tau }}_{w}|/{\rho }_{0}{u}_{\tau ,NR}^{2}\) is qualitatively different from \({\overline{\Phi } }_{w}\) for all cases considered here. The joint PDF contours between \({\Phi }_{w}\) and \({|\tau }_{w}|/{\rho }_{0}{u}_{\tau ,NR}^{2}\) are shown in Fig. 12 for the streamwise distance range given by \(12\le x/h\le 16\) for the different values of \({\mathrm{Le}}_{F}\) considered here. Figure 12 shows a negative correlation between \({\Phi }_{w}\) and \(|{\tau }_{w}|/{\rho }_{0}{u}_{\tau ,NR}^{2}\) for all cases but the strength of the correlation changes with \({\mathrm{Le}}_{F}\) variation, which suggests that wall heat flux and wall shear stress are related but the distributions of their mean values are qualitatively different. This further suggests that it is not only the local velocity gradients but also the non-local flow features such as the coherent flow structures and flame orientation with respect to the wall can influence the local variations of wall heat flux.
Fig. 11
Variations normalised mean wall shear stress magnitude \(|{\overline{\tau }}_{w}|/{\rho }_{0}{u}_{\tau ,NR}^{2}\) (solid line) along with the mean wall heat flux magnitude, \({\overline{\Phi } }_{w}\) (broken line) with the streamwise distance for turbulent V-flames with \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\)
Fig. 12
Joint PDF contours between \({\Phi }_{w}\) and \(|{\tau }_{w}|/{\rho }_{0}{u}_{\tau ,NR}^{2}\) at \(x/h=\)\(12, 14\) and \(16\) for turbulent V-flames with (1st row) \({\mathrm{Le}}_{F}=0.6\), (2nd row) \({\mathrm{Le}}_{F}=1.0\) and (3rd row) \({\mathrm{Le}}_{F}=1.4\)
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3.4 Effects of \(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}\) on Heat Flux Statistics Conditional Upon Coherent Flow Structures
The coherent flow structures in turbulent boundary layers are characterised by outward interactions (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}>0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}>0\)), ejection (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}<0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}>0\)), inward interactions (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}<0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}<0\)), and sweep (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}>0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}<0\)) where \({u}_{i}^{{^{\prime}}{^{\prime}}}\) is the ith component of the Favre fluctuation of velocity. The probabilities of finding different coherent structures at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and \(16\) are shown in Fig. 13 for \({\mathrm{Le}}_{F}=0.6, 1.0\) and 1.4, respectively. It can be seen from Fig. 13 that both ejections and sweep remain the predominant coherent structures at the region of the turbulent boundary layer where the onset of flame quenching takes place. The PDFs of \({\Phi }_{w}\) for outward interactions (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}>0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}>0\)), ejection (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}<0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}>0\)), inward interactions (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}<0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}<0\)), and sweep (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}>0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}<0\)) events at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and \(16\) are shown in Fig. 14 for \({\mathrm{Le}}_{F}=0.6, 1.0\) and 1.4 cases, respectively. Figure 14 suggests that the PDFs of \({\Phi }_{w}\) remain comparable for all the coherent structures at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface. Although the prevailing coherent structure at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) at a given \(x/h\) does not have a significant impact on \({\Phi }_{w}\) for a given streamwise location, this does not exclude the possible non-local effects of coherent structures on wall heat flux magnitude, but this aspect is not straightforward to extract and consequently is beyond the scope of the current investigation.
Fig. 13
Probabilities of occurrences of different coherent structures at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\mathrm{12,14}\) and 16 for turbulent V-flames with \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\)
Fig. 14
PDFs of \({\Phi }_{w}\) for outward interactions (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}>0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}>0\)), ejection (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}<0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}>0\)), inward interactions (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}<0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}<0\)), and sweep (i.e. \({u}_{1}^{{^{\prime}}{^{\prime}}}>0\) and \({u}_{2}^{{^{\prime}}{^{\prime}}}<0\)) events at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\) 12,14 and 16 for turbulent V-flames with \({\mathrm{Le}}_{F}=0.6\), \(1.0\) and \(1.4\). The PDF values are shown in log-scale for the ease of comparison
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It is important to note that Fig. 13 shows the probability of finding different coherent structures at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface but the probabilities for occurrences of different coherent structures in the region given by distance \(\mathrm{Pe}\ge y/{\delta }_{z}\ge 0\) can be different to the distributions shown in Fig. 13. This can be substantiated by Fig. 15 where the probability of the occurrences of different coherent structures in the region given by distance \(\mathrm{Pe}\ge y/{\delta }_{z}\ge 0\) (where \(\mathrm{Pe}\) is the normalised wall normal distance of the \({\theta }^{*}=0.75\) isosurface) at \(x/h=\mathrm{12,14}\) and 16 are shown in Fig. 15 for \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\). It can be seen from Fig. 15 that ejections and sweeps remain predominant coherent structures in the region given by \(\mathrm{Pe}\ge y/{\delta }_{z}\ge 0\) for \({\mathrm{Le}}_{F}=\)\(1.0\) and \(1.4\) but the \({\mathrm{Le}}_{F}=0.6\) case exhibits an increased probability of outward accelerations at the expense of the reductions of occurrences of ejections and inward accelerations. The increased occurrences of outward accelerations within the zone where the flame is likely to quench in the \({\mathrm{Le}}_{F}=0.6\) case is a consequence of stronger thermal expansion as a result of a higher burning rate than in \({\mathrm{Le}}_{F}=\)\(1.0\) and \(1.4\) cases (see Table 2). These differences in probabilities in coherent structure in the quenching zone in response to \({\mathrm{Le}}_{F}\) will be reflected in the Reynolds stress statistics, which is beyond the scope of the current analysis and will not be discussed here.
Fig. 15
Probabilities of occurrences of different coherent structures in the region given by distance \(\mathrm{Pe}\ge y/{\delta }_{z}\ge 0\) where \(\mathrm{Pe}\) is the normalised wall normal distance of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\mathrm{12,14}\) and 16 for turbulent V-flames with \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\)
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3.5 Effects of \(\mathbf{L}{\mathbf{e}}_{{\varvec{F}}}\) on Heat Flux Statistics Conditional on Flame Orientation
Finally, it is worthwhile to consider the effects of flame orientation on the wall heat flux. The flame orientation with respect to the wall can be characterised by \(cos\beta =\overrightarrow{N}\bullet {\overrightarrow{n}}_{w}\) where \({\overrightarrow{n}}_{w}\) is the wall normal vector. For a head-on quenching one obtains \(cos\beta =-1.0\) and for a pure side wall quenching \(cos\beta =0\) is obtained, whereas \(cos\beta >0\) refers to entrained flame elements. The schematic diagrams for the different orientations of flame configurations are shown in Fig. 16. The entrained flame elements, as shown in Fig. 16c, can occur due to vortical motion at the tip of the quenched flame which can entrain the burned products closer to the wall than the unburned reactants, which was mentioned in a recent analysis (Kaddar et al. 2022). The probabilities of \(cos\beta\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{Z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\mathrm{12,14}\) and 16 are shown in Fig. 17 for \({\mathrm{Le}}_{F}=0.6, 1.0\) and 1.4, respectively. It can be seen from Fig. 17 that for all cases there is a considerable probability of head-on quenching, and this probability increases with decreasing \({\mathrm{Le}}_{F}\). However, the probabilities of \(cos\beta >0\) and \(cos\beta <0\) remain comparable in the region given by distance \(\mathrm{Pe}\ge y/{\delta }_{z}\ge 0\), as can be seen from Fig. 18. Thus, the head-on and entrained flame elements remain almost equally likely in the region where the flame quenches in this configuration. Moreover, there is a considerable probability of side wall quenching events for the cases with \({\mathrm{Le}}_{F}\) = \(1.0\) and \(1.4\).
Fig. 16
Schematic of different orientations of flame configuration: a head-on quenching, b side wall quenching and c Entrained flame
Fig. 17
Probabilities of \(cos\beta =\overrightarrow{N}\bullet {\overrightarrow{n}}_{w}\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\)\(12, 14\) and \(16\) for turbulent V-flames with \(\mathbf{a}\)\({\mathrm{Le}}_{F}=0.6\), b\({\mathrm{Le}}_{F}=1.0\) and c\({\mathrm{Le}}_{F}=1.4\)
Fig. 18
Probabilities of \(cos\beta =\overrightarrow{N}\bullet {\overrightarrow{n}}_{w}\) in the region given by distance \(\mathrm{Pe}\ge y/{\delta }_{z}\ge 0\) where \(\mathrm{Pe}\) is the normalised wall normal distance of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=12, 14\) and 16 for turbulent V-flames with \({\mathrm{Le}}_{F}=\)\(0.6\), \(1.0\) and \(1.4\)
×
×
×
The mean values of \({\Phi }_{w}\) conditioned upon \(cos\beta\) values at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\mathrm{12,14}\) and 16 are shown in Fig. 19 for different \({\mathrm{Le}}_{F}\) values. Figure 19 shows that the head-on elements give rise to higher magnitudes of wall heat flux than the entrained flame elements. The entrainment of burned gases and their mixing with the unburned reactants gives rise to a smaller wall normal temperature gradient than the head-on flame elements. The probability of obtaining head-on quenching increases downstream with the progress of flame quenching as the flame reaches the vicinity of the wall (see Fig. 17). The entrained flame elements remain farther from the wall than in the case of head-on quenching at the early stages of flame quenching (e.g. \(x/h=12\)) but this trend weakens with the progress of flame quenching, which can be substantiated by Fig. 20 where the mean values of the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface conditioned upon \(cos\beta\) are shown for \({\mathrm{Le}}_{F}=0.6, 1.0\) and 1.4. However, the mean value of \(\mathrm{Pe}=y/{\delta }_{z}\) conditioned upon \(cos\beta\) increases with the progress of flame quenching with \(x/h\) for the \({\mathrm{Le}}_{F}=0.6\) case but the opposite trend is obtained for the \({\mathrm{Le}}_{F}= 1.0\) and 1.4 cases, although the mean value of \({\Phi }_{w}\) conditioned upon \(cos\beta\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface decreases with increasing \({\mathrm{Le}}_{F}\). The high values of wall heat flux magnitude despite larger values of normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for the \({\mathrm{Le}}_{F}=0.6\) case is a combined result of high reaction rate magnitudes at the positive curvatures (see Fig. 4b) and a higher likelihood of the flame quenching at predominantly negatively curved locations (see Fig. 9b). In the \({\mathrm{Le}}_{F}= 1.0\) and 1.4 cases, the higher mean value of \({\Phi }_{w}\) is associated with a smaller mean value of \(\mathrm{Pe}=y/{\delta }_{z}\) as the smaller flame normal distance of predominantly negatively curved zones induces greater heat flux values. This behaviour is not eclipsed by the burning rate at the positively curved regions in the \({\mathrm{Le}}_{F}= 1.0\) and 1.4 cases because the burning rate is not dependent on curvature for \({\mathrm{Le}}_{F}=1.0\) case and the burning rate is relatively weak at positive values of \({\kappa }_{m}\) for \({\mathrm{Le}}_{F}= 1.4\).
Fig. 19
Mean values of \({\Phi }_{w}\) conditioned on the values of \(cos\beta\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\)\(12, 14\) and \(16\) for turbulent V-flames with \({\mathrm{Le}}_{F}=0.6\), \(1.0\) and \(1.4\)
Fig. 20
Mean values of \(\mathrm{Pe}\) conditioned on the values of \(cos\beta\) at the normalised wall normal distance \(\mathrm{Pe}=y/{\delta }_{z}\) of the \({\theta }^{*}=0.75\) isosurface for turbulent V-flames at \(x/h=\)\(12, 14\) and \(16\) for turbulent V-flames with \({\mathrm{Le}}_{F}=0.6\), \(1.0\) and \(1.4\)
×
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The foregoing discussion indicates that the fuel Lewis number affects the correlations of temperature and fuel reaction rate magnitude with local curvature, which in turn gives rise to changes in the flame quenching distance and wall heat flux magnitude. Moreover, the increased flame wrinkling for small values of fuel Lewis number also alters the flame orientation with respect to the wall, which in turn affects the wall heat flux and quenching distance. This suggests that the thermo-diffusive effects induced by the non-unity Lewis number need to be accounted for in the modelling of premixed FWI to predict the wall heat flux accurately.
4 Conclusions
The effects of fuel Lewis number \({\mathrm{Le}}_{F}\) on the statistical behaviour of wall heat flux and flame quenching distance have been analysed for turbulent V-shaped flames interacting with an isothermal inert wall in a channel flow corresponding to a friction velocity-based Reynolds number of 110 for \({\mathrm{Le}}_{F}\) ranging from 0.6 to 1.4. It has been found that the maximum wall heat flux magnitude in turbulent V-shaped flame-wall interaction (FWI) increases with decreasing \({\mathrm{Le}}_{F}\) but just the opposite trend is observed for 2D laminar V-shaped flame-wall interaction and 1D laminar head-on quenching cases. By contrast, the flame quenching distance decreases with decreasing \({\mathrm{Le}}_{F}\) for turbulent V-flames but the quenching distance is found to increase with decreasing \({\mathrm{Le}}_{F}\) for 2D laminar V-shaped flame-wall interaction and 1D laminar head-on quenching cases. This behaviour has been explained based on the positive (negative) correlation between temperature and fuel reaction rate magnitude with local flame surface curvature for turbulent flames with \({\mathrm{Le}}_{F}<1.0\) (\({\mathrm{Le}}_{F}>1.0\)) due to the thermo-diffusive effects induced by the non-unity Lewis number. It has been demonstrated that the flame curvature values are predominantly negative for the flame elements which are closest to the wall, and this implies that the small values of temperature and reaction rate magnitudes at the negatively curved zones in the \({\mathrm{Le}}_{F}=0.6\) case can quench more readily farther away from the wall than in the corresponding \({\mathrm{Le}}_{F}=1.0\) case. By contrast, the high temperature and reaction magnitudes at the negatively curved flame elements close to the wall in the \({\mathrm{Le}}_{F}=1.4\) case can resist quenching and can reach closer to the wall than in the corresponding \({\mathrm{Le}}_{F}=1.0\) case. The correlations between temperature and fuel reaction rate magnitude with local flame surface curvature for turbulent flames due to the thermo-diffusive effects induced by the non-unity Lewis number have been utilised to explain the influence of fuel Lewis number on wall heat flux magnitude and flame quenching distance in this configuration. It has been found that the wall heat flux magnitude and wall shear stress magnitude are negatively correlated for all cases considered here but their mean variations in the streamwise direction are qualitatively different irrespective of the value of the fuel Lewis number. However, the magnitudes of mean values of wall heat flux and wall shear stress increase with decreasing \({\mathrm{Le}}_{F}\). Moreover, the nature of the coherent flow structure at the wall-normal distance where the flame begins to quench does not have a major influence on wall heat flux values. The flame alignment relative to the wall also affects the wall heat flux and it has been found that the local head-on flame elements can lead to higher magnitudes of wall heat flux but the flame quenching distance for different flame orientations is affected by the fuel Lewis number \({\mathrm{Le}}_{F}\). This analysis shows that the effects of fuel Lewis number on flame orientation, correlations of reaction rate and temperature with local flame curvature and coherent flow structures within turbulent boundary layers ultimately affect both wall heat transfer rate and flame quenching distance. Thus, the thermo-diffusive effects arising from the non-unity Lewis number need to be included in the modelling of wall heat transfer during FWI in turbulent boundary layers. Although several previous analyses (Lai et al. 2018; Zhao et al. 2023) reported good qualitative and quantitative agreements between wall heat transfer magnitudes between simple and detailed chemistry DNS results and also with experimental data (Huang et al. 1988; Jarosinski 1986; Vosen et al. 1985), it is still necessary to confirm the statistics presented in this paper when detailed chemistry and transport are employed along with higher values of Reynolds number. Furthermore, conjugate heat transfer between the solid wall and the reacting gases needs to be considered for accurate prediction of wall heat transfer in combustors. These aspects will make the foundation for further analyses in the future.
Acknowledgements
The authors are grateful for the financial and computational support from the Engineering and Physical Sciences Research Council (Grant: EP/V003534/1 and EP/R029369/1), CIRRUS, and ROCKET (HPC facility at Newcastle University).
Declarations
Conflict of interest
The authors do not have any competing interests to declare that are relevant to the content of this article.
Ethics Approval
This study does not involve any research with human participants and/or animals, so no ethical approval was required.
Informed Consent
This study does not involve any research with human participants animals, so no informed consent was required.
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