2.1. Problem formulation

The continuity, momentum, total energy and species equations are listed as follows:

(2.1) \begin{align} \frac {\partial {\rho }}{\partial {t}} + {\boldsymbol{\nabla }} \boldsymbol{\cdot }\left ( {{\rho } {\boldsymbol{u}} } \right ) &= 0, \end{align}

(2.2) \begin{align} \frac {\partial {\rho \boldsymbol{u}}}{\partial {t}} + \boldsymbol{\nabla }\boldsymbol{\cdot }( \rho \boldsymbol{u} \boldsymbol{u}) &= \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau } - \boldsymbol{\nabla }p, \end{align}

(2.3) \begin{align} \frac {\partial {\rho e_t}}{\partial {t}} + \boldsymbol{\nabla }\boldsymbol{\cdot }( \rho \boldsymbol{u} e_t) &= -\boldsymbol{\nabla }\boldsymbol{\cdot }(\boldsymbol{u} p) - \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{q} + \boldsymbol{\nabla }\boldsymbol{\cdot }(\boldsymbol{\tau }\boldsymbol{\cdot }\boldsymbol{u}) , \end{align}

(2.4) \begin{align} \frac {\partial {\rho Y_{k}}}{\partial {t}} + \boldsymbol{\nabla }\boldsymbol{\cdot }( \rho \boldsymbol{u} Y_k) &= -\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{J}_{k} + \dot {\omega }_k , \end{align}

where $\rho$ is the flow density, $\boldsymbol{u}$ is the velocity vector, $Y_k$ is the mass fraction of the $k$ th species and $e_t$ is the total energy. On the right-hand side of the above equations, $\boldsymbol{\tau }$ is the stress tensor, $p$ is pressure, $\boldsymbol{q}$ indicates thermal diffusive flux and $\boldsymbol{J}_{k}$ is the species diffusive flux, together with the reaction rate $\dot {\omega }_k$ .

To facilitate the analysis, we first introduce the non-dimensional quantities $(\tilde{\cdot })$ with reference scales. The reference length is the domain size $L$ in the wall-normal direction, the reference velocity is the laminar flame speed $S_L^0$ and the reference time is $L/S_L^0$ . Progress variable $c$ is defined upon the mass fraction of a reactant $Y_R$ as $(Y_{R,u}-Y_R)/(Y_{R,u}-Y_{R,b})$ , where subscripts $u$ and $b$ denote the unburnt state and burnt state, respectively. Temperature $T$ is non-dimensionalised as $\tilde {T}=(T-T_u)/(T_{\textit{ad}}-T_u)$ , with $T_{\textit{ad}}$ representing the adiabatic flame temperature of the given flow. The wall heat flux $Q_w$ is non-dimensionalised by the flame power $\rho _u C_{p,u} S_L^0 ( T_{\textit{ad}} - T_{u} )$ . The heat release rate $\dot \omega _T$ and the thermal diffusive flux $\boldsymbol{q}$ are non-dimensionalised by $\rho _u C_{p,u} S_L^0 ( T_{\textit{ad}} - T_{u} )/L$ . For all the involved thermal-physical properties, i.e. heat capacity $C_p$ , density $\rho$ , thermal conductivity $\lambda$ and diffusivity for species $D_k$ , reference values are these of the corresponding unburnt gas. For convenience, the tilde sign for non-dimensional quantities is omitted hereafter without explicit mention. To test the consistency of FWI physics under different working conditions, such as the chemical mechanism and boundary conditions, there are two DNS cases for comparison, including the wall-normal flushing flame and inclined sweeping flame case.

Figure 1.

Spatial distribution of: (a) instantaneous hydrogen-based progress variable field; (b) instantaneous temperature field together with the flame isosurface defined by $c=0.8$ .

Figure 1 Long description

Panel A: A 3D visualization showing the spatial distribution of the instantaneous hydrogen-based progress variable field. The visualization is a volumetric rendering with a color scale ranging from blue to red, indicating values from 0 to 1. The inflow is marked with arrows on the top left corner, and the wall is labeled on the right side. All laterals are indicated as outflow. The axes are labeled as x1, x2, and x3. Panel B: A 3D visualization showing the spatial distribution of the instantaneous temperature field together with the flame isosurface defined by c=0.8. The visualization is a volumetric rendering with a color scale ranging from blue to red, indicating values from 0 to 1. The axes are labeled as x1, x2, and x3.

2.2. Wall-normal flushing flame

As shown in figure 1, the fresh reactant as inflow is fed into the domain against the wall-normal direction $x_1$ and flows out through the lateral boundaries along $x_2$ and $x_3$ . At the statistically stationary state, the mean flame–wall distance is strain rate-dependent. In a series of recently finished studies, both the simplified single-step reaction mechanism with constant thermal-physical properties (Zhao et al. Reference Zhao, Wang and Chakraborty2018) and multi-step kinetics with realistic thermal-physical properties (Zhao et al. Reference Zhao, Pérez, Guo, Im and Wang2022) have been investigated. It is found that the multi-step chemistry leads to more complex flame characteristics, including local quenching, flame thickening and various flame topologies.

In the present work, a premixed hydrogen–air reactive flow is solved at atmospheric pressure (Pérez et al. Reference Pérez, Mukhadiyev, Xu, Sow, Lee, Sankaran and Im2018; Desai et al. Reference Desai, Kim, Song, Luong, Pérez, Sankaran and Im2021). The fresh premixed reactant with stoichiometric ratio $\phi =0.7$ and temperature of $300\,\textrm{K}$ is injected along the $x_1$ direction. The mean inlet velocity is $12\,\textrm{m s}^{-1}$ and the fluctuating magnitude is $6.75\,\textrm{m s}^{-1}$ , generated by scanning a prescribed isotropic homogeneous turbulence field (Rogallo Reference Rogallo1981). The wall boundary is isothermal and inert with a temperature of $450\,\textrm{K}$ . To ensure zero species mass flux and zero diffusion velocity, the Navier–Stokes characteristic boundary condition (NSCBC) is applied to prevent non-physical wave reflections. A 23-step reaction mechanism of the hydrogen–air combustion (Burke et al. Reference Burke, Chaos, Ju, Dryer and Klippenstein2012) and the mixture-average transport properties are adopted. In a cubic domain $3.84\,\textrm{mm}$ long on each side, an equally spaced $256^3$ mesh ensures both turbulence and flame resolution, i.e. 10–20 grid points across the thermal flame thickness, and a grid spacing smaller than the Kolmogorov scale.

The flame front is identified as the hydrogen-defined progress variable $c=0.8$ isosurface, corresponding to a peak heat release rate under laminar conditions. In the following, by default, the progress variable $c$ is hydrogen defined. In addition, extinct regions on $c=0.8$ are excluded if the local $\text{OH}$ mass fraction on the flame front falls below $10\,\%$ of the maximum $\text{OH}$ mass fraction (Wang et al. Reference Wang, Chen, Luo, Hawkes, Chen and Fan2021). There are some other flame definitions as well, for instance, the oxygen-defined isosurface $c_{\textrm{O}_{2}}=0.64$ . The distance from the identified position to the wall is defined as the flame-wall distance $\delta _f$ .

Figures 1(a) and 1(b) present the overall flow structure, including the hydrogen-based progress variable and the temperature field together with the flame front. Overall, the spatial distribution of the progress variable is similar to that of the temperature, since both quantities follow similar governing equations. However, a clear distinction appears in the vicinity of the cold wall because of their different wall boundary conditions and the differential diffusion effect. To ensure statistical convergence, we collect $15$ snapshots over $2$ to $3$ through-pass times at the statistically stationary state for data analyses.

2.3. Inclined sweeping flame

Another set-up is the premixed flame flashback over a horizontal flat plate (Chen et al. Reference Chen, Wang, Gruber, Luo and Fan2023). As shown in figure 2, in a box domain with the wall-normal along $x_1$ , the premixed reactant comes into the domain from the left inlet along $x_2$ . After ignition, the flame travels back and interacts with the turbulent boundary layer. As can be seen in figure 2, the flame zone expands along the main stream, forming an overall inclination angle.

Figure 2.

Spatial distribution of an instantaneous DNS temperature field superimposed with a hydrogen-based progress variable $c=0.8$ isosurface.

The DNS parameters are listed as follows. At the left inlet, a lean premixed hydrogen–air mixture with an equivalence ratio $\phi =0.8$ and inlet temperature $500 \, \textrm{K}$ sweeps over a flat plate, which is isothermal at $500 \, \textrm{K}$ in the front part ( $0\,\textrm{mm}\lt x_2\lt 15\,\textrm{mm}$ ) and adiabatic in the rear part ( $15\,\textrm{mm}\lt x_2\lt 20\,\textrm{mm}$ ) to stabilise the flame. The background pressure remains at $2 \, \textrm{atm}$ . The Lewis number of each species is constant, from the mixture-averaged transport calculation. A 19-step mechanism (Li et al. Reference Li, Zhao, Kazakov and Dryer2004) is adopted for the combustion chemistry. The numerical solver (Chen et al. Reference Chen2009) adopts a fourth-order Runge–Kutta scheme for time advancing (Kennedy & Carpenter Reference Kennedy and Carpenter1994) and an eighth-order finite difference scheme for spatial discretisation. The domain size along $(x_1,x_2,x_3)$ is $(L_{x1} =10\,\textrm{mm}, L_{x2}=20\,\textrm{mm}, L_{x3} = 15\,\textrm{mm})$ with grid number $(N_{x1}=480, N_{x2}=1000, N_{x3} =750)$ . Specifically, grids in the $x_2$ and $x_3$ directions are uniform with more than $10$ grids within the thermal flame, while the local refinement in the wall-normal direction $x_1$ ensures more than ten points within $y^+\lt 10$ , which is sufficient to resolve the turbulent boundary layer. The boundary condition is periodic along the $x_3$ direction and non-reflecting for other outflow surfaces. The inlet velocity is carefully fabricated by using the real-time velocity at a given streamwise slice of a running non-reacting turbulent boundary layer DNS flow, whose free-stream velocity is $40 \, \textrm{m s}^{-1}$ . The species NSCBC boundary condition at the wall ensures zero species mass flux and zero diffusion velocity. Consistent with the wall-normal flushing flame case, the flame front is defined by the same $c$ isosurface.

3.1. Laminar wall stagnation flame

First, the quasi one-dimensional laminar wall stagnation flame is studied using Cantera (Goodwin et al. Reference Goodwin, Moffat, Schoegl, Speth and Weber2023) as a reference. Similar to the wall-normal flushing flame set-up, a stagnation premixed hydrogen–air flame is injected from the inlet along $x_1$ . Under atmospheric pressure the equivalence ratio is $\phi =0.7$ , the inlet gas temperature is $300\,\textrm{K}$ and the wall temperature is constant at $450\,\textrm{K}$ . At different inlet strain rates $\varepsilon =u_{\textit{in}}/L$ , where $u_{\textit{in}}$ is the inlet velocity, the laminar flame is anchored at different wall distances. The chemical kinetics includes $9$ species and $23$ elementary reactions (Burke et al. Reference Burke, Chaos, Ju, Dryer and Klippenstein2012).

Figure 3 shows the structure of the hydrogen mass fraction $c$ defined flame varying with $\varepsilon$ . The flame zone is the reaction part between the left and right edges, where the progress variable gradient is negligibly small, e.g. $\boldsymbol{\nabla }c = 0.01\boldsymbol{\nabla }c_{max}$ . For comparison, results from both the mixture-averaged species transport property (in solid lines with subscript 1) and the unity Lewis number (in dashed lines with subscript 2) are presented. The distance from a spatial point to the wall $\delta$ is normalised by the Zel’dovich flame thickness $\delta _z=D_{th}/S_L^0$ , in which $D_{th}$ is the unburnt gas thermal diffusivity. Here, $\delta _u$ and $\delta _b$ denote the wall distance from the flame zone edge at the unburnt side and burnt side, respectively. As the strain rate increases, the flame approaches the wall boundary at $\delta =0$ ; meanwhile, the maximum temperature decreases. For the present chemical mechanism, the reaction zone is clearly thicker than $\delta _z$ ( $\sim 10\delta _z$ ). A sharp difference between the two result sets is that the near-wall heat release rate $\dot \omega _T$ from the unity Lewis number transport model decreases gradually, but that from the mixture-averaged transport property reaches a local peak, especially when the inlet strain rate is larger. Due to the high diffusivity of $\text{H}_2$ , the flame zone from the mixture-averaged property is larger, while $\delta _{{b1}}$ and $\delta _{{b2}}$ are almost identical.

Figure 3.

Distributions of the flame temperature, hydrogen mass fraction defined progress variable and the heat release rate distributions with respect to the non-dimensional wall distance $\delta /\delta _z$ under different non-dimensional inlet strain rates $\varepsilon$ . Results from the mixture-averaged transport property and the unity Lewis number are shown by the solid and dashed lines, respectively; (a) $\varepsilon = 1.55$ , (b) $\varepsilon = 4.50$ , (c) $\varepsilon = 5.97$ , (d) $\varepsilon = 8.92$ .

From the temperature equation along $x_1$

(3.1) \begin{equation} \rho u_1 C_p\frac {\partial T}{\partial x_1} = \frac {1}{RePr}\frac {\partial }{\partial x_1}\left (\lambda \frac {\partial T}{\partial x_1}\right ) -\frac {1}{ReSc}\sum _k J_k\frac {\partial h_k}{\partial x_1} + \dot {\omega }_T , \end{equation}

the relative contributions of the temperature diffusion term (the first term on the right-hand side) and the species enthalpy diffusion term (the second term on the right-hand side) can be compared. Here, $h_k$ is the enthalpy of the $k$ th species, with a reference value of $C_{p,u}(T_{\textit{ad}}-T_{u})$ . Here, $Re$ , $Pr$ and $Sc$ are the Reynolds number, Prandtl number and Schmidt number, respectively. Figure 4 demonstrates that, regardless of the transport model selected, the species enthalpy diffusion term remains relatively negligible.

Figure 4.

Comparison of the non-dimensional diffusion terms in the temperature equation at $\varepsilon =8.92$ . The solid and dashed lines indicate the results from the mixture-averaged transport property and the unity Lewis number, respectively.

3.2. Flame and velocity interaction: from space to wall boundary

The spatial flame and turbulence interaction has been extensively studied using the strain rate tensor. Basically, the most compressive eigenvector dominates species diffusion along the flame normal, while the extensive eigenvector is relevant to the flame corrugation.

With presence of the wall boundary, we introduce the vector

(3.2) \begin{equation} \boldsymbol{F} = \left [0,\left .\frac {\partial u_{2}}{\partial N}\right |_w,\left .\frac {\partial u_{3}}{\partial N}\right |_w\right ], \end{equation}

where $u_{2},u_{3}$ are two wall-tangential velocity components, and $N$ is the wall normal towards the fluid. In parallel, the velocity gradient tensor then reduces to a second-order tensor ${{F}}_{i,j}$ , where ${{F}}_{i,j}=\partial {{F}}_i/\partial x_j\ (i,j\in \{2,3\})$ . One can decompose ${F}_{i,j}$ as

(3.3) \begin{equation} {F}_{i,j} = {S}_{ij} + A_{ij}, \end{equation}

where ${S}_{ij}=( {1}/{2}) (( {\partial {F}_i}/{\partial x_j} )+( {\partial {F}_j}/{\partial x_i}) )$ and ${A}_{ij}=({1}/{2}) (( {\partial {F}_i}/{\partial x_j} )- ( {\partial {F}_j}/{\partial x_i}) )$ represent the symmetric part and antisymmetric part, respectively. From the antisymmetric tensor, a friction vorticity $\omega _F$ can be defined as

(3.4) \begin{equation} \omega _F = \cfrac {\partial F_3}{\partial x_2} - \cfrac {\partial F_2}{\partial x_3}. \end{equation}

Regions with large $\omega _F$ magnitude are defined as the friction vortex structure. The relation between such friction vorticity and the spatial vorticity is analogous to that between the friction line and streamline. The reference value for ${S}_{ij},\, {A}_{ij}$ and $\omega _F$ is $S_l^0/L^2$ .

The joint probability density function (PDF) between $\omega _F$ and the flame curvature represented by the $\textrm{H}_2$ contour lines are shown in figure 5. Here, the flame curvature is quantified as $\kappa =\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{n}$ , where $\boldsymbol{n}$ is the flame normal pointing toward the unburnt side. By definition, the curvature is positive when the flame is convex towards the unburnt side. For comparison, results for the $\textrm{O}_{2}$ contour lines are also shown in figure 6. Since on the wall boundary a small progress variable is relatively rare, samples need to be collected in relatively broader ranges, including $0.65\lt c\leqslant 0.8$ and $0.8 \lt c\leqslant 0.95$ . It can be seen that all the joint PDFs peak at the origin, where both the curvature magnitude and friction vorticity magnitude are low. Large curvature magnitude is inclined to have a small friction vorticity magnitude. From figure 5(a, b), i.e. across the flame front with $c=0.8$ , the distribution of $\omega _F$ is more concentrated because combustion weakens the friction vorticity $\omega _F$ , as well as the spatial vorticity. Overall, joint PDFs are skewed towards the more negative curvature side because of the hydrodynamic instability. A subtle difference between the two configurations is that the friction vorticity of negative curvature samples is more distributed in the inclined sweeping flame, since the near-wall flame under stronger convection is less influenced by diffusion. The statistical results for the oxygen defined flame are shown in figure 6, where $0.49\lt c_{\textrm{O}_{2}}\leqslant 0.64$ and $0.64 \lt c_{\textrm{O}_{2}}\leqslant 0.79$ represent two different sides of the flame front. Compared with figure 5, the qualitative tendencies are consistent. However, the large negative curvature part becomes more pronounced because the smaller diffusivity of oxygen promotes larger curvature formation, which can be better viewed from the following visualisation.

Figure 5.

Joint PDFs between $\textrm{H}_2$ contour line defined curvature $\kappa _{\rm {H}_2}$ and $\omega _F$ for the normal flushing flame: (a) $0.65\lt c\leqslant 0.8$ ; (b) $0.8 \lt c\leqslant 0.95$ , and for the inclined sweeping flame: (c) $0.65 \lt c\leqslant 0.8$ ; (d) $0.8\lt c\leqslant 0.95$ .

Figure 5 Long description

Panel A: A scatter plot representing the relationship between H2 contour line defined curvature and omega F for the normal flushing flame with 0.65 < c < 0.8. The horizontal axis represents omega F times delta z squared over S l 0, ranging from -0.10 to 0.05. The vertical axis represents kappa H2 times delta z, ranging from -0.5 to 0.5. The plot shows a concentration of data points around the center, with a color gradient indicating the density of points. Panel B: A scatter plot representing the relationship between H2 contour line defined curvature and omega F for the normal flushing flame with 0.8 < c < 0.95. The horizontal axis represents omega F times delta z squared over S l 0, ranging from -0.10 to 0.05. The vertical axis represents kappa H2 times delta z, ranging from -0.5 to 0.5. The plot shows a similar concentration of data points around the center, with a color gradient indicating the density of points. Panel C: A scatter plot representing the relationship between H2 contour line defined curvature and omega F for the inclined sweeping flame with 0.65 < c < 0.8. The horizontal axis represents omega F times delta z squared over S l 0, ranging from -0.04 to 0.04. The vertical axis represents kappa H2 times delta z, ranging from -0.8 to 0.4. The plot shows a concentration of data points around the center, with a color gradient indicating the density of points. Panel D: A scatter plot representing the relationship between H2 contour line defined curvature and omega F for the inclined sweeping flame with 0.8 < c < 0.95. The horizontal axis represents omega F times delta z squared over S l 0, ranging from -0.04 to 0.04. The vertical axis represents kappa H2 times delta z, ranging from -0.8 to 0.4. The plot shows a concentration of data points around the center, with a color gradient indicating the density of points.

Figure 6.

Joint PDFs between $\textrm{O}_{2}$ contour line defined curvature $\kappa _{\textrm{O}_2}$ and $\omega _F$ for the normal flushing flame: (a) $0.49\lt c_{\textrm{O}_{2}}\leqslant 0.64$ ; (b) $0.64 \lt c_{\textrm{O}_{2}}\leqslant 0.79$ ; and for inclined sweeping flame: (c) $0.49 \lt c_{\textrm{O}_{2}}\leqslant 0.64$ ; (d) $0.64\lt c_{\textrm{O}_{2}}\leqslant 0.79$ .

Figure 7 presents the surface distribution of $\omega _F$ superimposed with the skin friction vector, where the arrow size indicates its magnitude. The black and white lines denote the hydrogen contour line $c=0.8$ and oxygen contour line $c_{\textrm{O}_{2}}=0.64$ , respectively. For both cases, the oxygen contour lines exhibit larger curvature than the hydrogen contour lines. The same tendency also appears for the flame away from the wall (Zhao et al. Reference Zhao, Pérez, Guo, Im and Wang2022). Quantitatively, along an arbitrarily selected hydrogen contour line ( $c=0.8$ ), curvatures of both $c$ and $c_{\textrm{O}_{2}}$ isolines are plotted with respect to the arclength, as shown in figure 8 for both configurations. It can be observed that the $c_{\textrm{O}_{2}}$ curvature varies in a larger range. Since the convection term is negligible in the wall vicinity, for any progress variable $c_i$ , the chemical source and diffusion term $\boldsymbol{\nabla }\boldsymbol{\cdot }(\rho D_i\boldsymbol{\nabla }c_i)$ are balanced. After decomposition

(3.5) \begin{equation} \boldsymbol{\nabla }\boldsymbol{\cdot }(\rho D_i\boldsymbol{\nabla }c_i)=-\rho D_i \kappa |\boldsymbol{\nabla }c_i|+\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }(\rho D_i\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }c_i). \end{equation}

From the stoichiometry of the species mass consumption ratio, the diffusion term associated with the hydrogen progress variable $c$ approximately balances that of the oxygen progress variable $c_{\textrm{O}_{2}}$ . Because the diffusion coefficient of hydrogen is apparently larger, naturally its isoline curvature needs to be smaller to form smoother contours, as shown in figures 7 and 8. Figure 7 also shows interestingly the friction vortex pair structure, where a positive friction vorticity $\omega _F$ region and a negative $\omega _F$ region are closely located. Moving from the weak to the strong friction vortex region, either positive or negative, the contour line curvature (either $c$ or $c_{\textrm{O}_{2}}$ ) starts to increase until the curvature magnitude reaches a maximum somewhere between a friction vortex pair. In summary, such a mechanism, consistent with the statistical joint PDFs in figures 5 and 6, is a natural outcome of the flame and velocity field interaction on the wall.

Figure 7.

Contour plots of friction vorticity $\omega _F$ superimposed with the skin friction vector $\boldsymbol{F}$ (arrow size indicates its magnitude), where the black contour denotes the $c_{\textrm{H}_2}=0.8$ contour line and the white contour represents the $c_{\textrm{O}_{2}}=0.64$ contour line for different configurations: (a) wall-normal flushing flame, (b) inclined sweeping flame.

Figure 7 Long description

Panel A: A heat map with superimposed vectors representing friction vorticity and skin friction for a wall-normal flushing flame. The x-axis is labeled x2 in millimeters, ranging from 0.5 to 3.5. The y-axis is labeled x3 in millimeters, ranging from 0.5 to 3.5. The color scale ranges from -0.04 to 0.04, with blue indicating lower values and red indicating higher values. The black contour denotes the cH2=0.8 contour line, and the white contour represents the cO2=0.64 contour line. The vectors indicate the direction and magnitude of skin friction. Panel B: A heat map with superimposed vectors representing friction vorticity and skin friction for an inclined sweeping flame. The x-axis is labeled x3 in millimeters, ranging from 0 to 14. The y-axis is labeled x2 in millimeters, ranging from 0 to 14. The color scale ranges from -0.04 to 0.04, with blue indicating lower values and red indicating higher values. The black contour denotes the cH2=0.8 contour line, and the white contour represents the cO2=0.64 contour line. The vectors indicate the direction and magnitude of skin friction.

Figure 8.

The curvature distribution based on hydrogen and oxygen along the arclength $s$ of the $c_{\textrm{H}_2}=0.8$ contour line in figure 7 for (a) wall-normal flushing flame, and (b) inclined sweeping flame.

It is generally believed that in turbulence the scalar gradient tends to align with the most compressive direction in space (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987). With reference to the boundary free flame property, effects of the flame normal strain rate $S_n=n_in_j{S}_{ij}$ and flame tangential strain rate $S_t=(\delta _{ij}-n_in_j){S}_{ij}$ ( $\delta _{ij}$ is the Kronecker delta) are also discussed.

From the normal strain rate results in figure 9, it can be seen that, because of thermal expansion, the positive $S_n$ is predominant, especially in figures 9(a), 9(b) and 9(c). Meanwhile, such an expansion effect is much weakened on the constant temperature wall boundary, compared with boundary free results. For samples with large curvature magnitude, the normal strain rate $S_n$ is close to zero. Consistently, when $S_n$ is strongly negative, the flame front is likely flat with almost zero curvature. Dependence of the flame curvature on the flame tangential strain rate is shown in figure 10. Overall, there exists a weak negative correlation, i.e. the flame curvature is smaller if the flame is more stretched, while the curvature tends to be strongly positive under the action of compressive tangential strain rate.

Figure 9.

Joint PDFs between curvature $\kappa _{\textrm{H}_2}$ and flame normal strain rate $S_n$ for normal flushing flame: (a) $0.65\lt c\leqslant 0.8$ ; (b) $0.8 \lt c\leqslant 0.95$ ; and for inclined sweeping flame: (c) $0.65 \lt c\leqslant 0.8$ ; (d) $0.8\lt c\leqslant 0.95$ .

Figure 10.

Joint PDFs between curvature $\kappa _{\textrm{H}_2}$ and flame tangential strain rate $S_t$ for the normal flushing flame conditional for: (a) $0.65\lt c\leqslant 0.8$ ; (b) $0.8 \lt c\leqslant 0.95$ , and inclined sweeping flame conditional for: (c) $0.65 \lt c\leqslant 0.8$ ; (d) $0.8\lt c\leqslant 0.95$ .

Figure 10 Long description

Panel A: A scatter plot showing the relationship between curvature of H2 and flame tangential strain rate for the normal flushing flame with 0.65<c<0.8. The horizontal axis represents flame tangential strain rate multiplied by the square of the fluctuation in the mixture fraction divided by the initial mixture fraction, with values ranging from -0.1 to 0.3. The vertical axis represents curvature of H2 multiplied by the fluctuation in the mixture fraction, with values ranging from -0.6 to 0.6. The plot shows a concentration of data points around the origin, with a color gradient indicating density, and a visible trend line. Panel B: A scatter plot showing the relationship between curvature of H2 and flame tangential strain rate for the normal flushing flame with 0.8<c<0.95. The horizontal axis represents flame tangential strain rate multiplied by the square of the fluctuation in the mixture fraction divided by the initial mixture fraction, with values ranging from -0.1 to 0.3. The vertical axis represents curvature of H2 multiplied by the fluctuation in the mixture fraction, with values ranging from -0.6 to 0.6. The plot shows a concentration of data points around the origin, with a color gradient indicating density, and a visible trend line. Panel C: A scatter plot showing the relationship between curvature of H2 and flame tangential strain rate for the inclined sweeping flame with 0.65<c<0.8. The horizontal axis represents flame tangential strain rate multiplied by the square of the fluctuation in the mixture fraction divided by the initial mixture fraction, with values ranging from -0.02 to 0.04. The vertical axis represents curvature of H2 multiplied by the fluctuation in the mixture fraction, with values ranging from -0.15 to 0.15. The plot shows a concentration of data points around the origin, with a color gradient indicating density, and a visible trend line. Panel D: A scatter plot showing the relationship between curvature of H2 and flame tangential strain rate for the inclined sweeping flame with 0.8<c<0.95. The horizontal axis represents flame tangential strain rate multiplied by the square of the fluctuation in the mixture fraction divided by the initial mixture fraction, with values ranging from -0.02 to 0.04. The vertical axis represents curvature of H2 multiplied by the fluctuation in the mixture fraction, with values ranging from -0.15 to 0.15. The plot shows a concentration of data points around the origin, with a color gradient indicating density, and a visible trend line.

For more statistical details, the wall surface is partitioned into three regions ( $A$ , $B$ and $C$ ) based on the progress variable gradient magnitude, with the non-reactive region (if the wall heat flux $Q_w\lt 10^{-5}$ ) excluded. Specifically, regions $A$ , $B$ and $C$ correspond to high gradient (top $10\,\%$ of samples), medium gradient ( $80\,\%$ of samples in the middle) and low gradient region (bottom $10\,\%$ of samples), respectively. Here, $\theta _2$ is defined as the acute angle between the progress variable gradient $\boldsymbol{\nabla }c$ and the most compressive eigenvector $\boldsymbol{e}_2$ of the tensor $S_{ij}$ . By definition, $\theta _2=0$ means $\boldsymbol{\nabla }c$ aligns perfectly with the most compressive strain direction, while $\theta _2=\pi /2$ means a perfect alignment between $\boldsymbol{\nabla }c$ and the most extensive strain direction. As shown in figure 11, for both cases there is a strong alignment between $\boldsymbol{\nabla }c$ and $\boldsymbol{e}_2$ in region A. Moving away from the flame zone with decreasing progress variable gradient magnitudes, such alignment weakens.

Figure 11.

The PDFs of the acute angle $\theta _2$ between the progress variable gradient $\boldsymbol{\nabla }c$ and the eigenvector $\boldsymbol{e}_2$ conditioned in different regions for: (a) wall-normal flushing flame; (b) inclined sweeping flame.

3.3. Species alignment relation

Wall interference will also largely change the flame structure in terms of species alignment. If the flame is infinitely thin as a limiting case, the normal directions of all species isosurfaces are the same as the flame normal, which, however, cannot be expected for the near-wall flame. We tentatively introduce a newly defined alignment index as

(3.6) \begin{equation} G_{\textit{align}} = \frac {\sum _{P(i,j)} |\boldsymbol{\nabla }Y_i\boldsymbol{\cdot }\boldsymbol{\nabla }Y_j|}{\sum _{P(i,j)} |\boldsymbol{\nabla }Y_i| |\boldsymbol{\nabla }Y_j|}. \end{equation}

Here, $P(i,j)$ represents pairing between any two species $i$ and $j$ , whose contributions to the overall alignment are weighted by the gradient magnitudes $\boldsymbol{\nabla }Y_i$ and $\boldsymbol{\nabla }Y_j$ . Inclusion of the weighting factor is effective in excluding the noisy influence from minor species. Clearly, if all species (anti-)align perfectly, $G_{\textit{align}} = 1$ , while for purely random structure, analytically $G_{\textit{align}}$ is $0.5$ .

Figure 12 shows for both configurations the joint PDF between $G_{\textit{align}}$ and the flame–wall distance $\delta _f$ in different zones. For the wall-normal flushing flame, the quenching zone, influence zone and free flame zone are quantified as $\delta _f/\delta _z \lt 2.66$ , $2.66 \leqslant \delta _f/\delta _z\leqslant 8$ and $\delta _f/\delta _z \gt 8$ , respectively (Zhao et al. Reference Zhao, Wang and Chakraborty2018). For the inclined sweeping flame set-up with a much lower wall-normal strain rate, the quenching zone and influence zone are differently quantified as $\delta _f/\delta _z \lt 6$ (Bellenoue et al. Reference Bellenoue, Kageyama, Labuda and Sotton2003; Jainski et al. Reference Jainski, Rißmann, Böhm, Janicka and Dreizler2017; Kosaka et al. Reference Kosaka, Zentgraf, Scholtissek, Bischoff, Häber, Suntz, Albert, Hasse and Dreizler2018) and $6\leqslant \delta _f/\delta _z\leqslant 10$ , respectively. The mean alignment index consistently approaches $1.0$ in the free flame zone. Moving towards the influence zone (marked by the green dashed line) and quenching zone (marked by the red dashed line), different species gradients start to misalign from each other, i.e. $G_{\textit{align}}$ continues to decrease as the flame is increasingly distorted under the joint influence from the wall boundary and differential diffusion.

Figure 12.

Joint PDFs between the flame–wall distance and flame alignment index $G_{\textit{align}}$ , for: (a) wall-normal flushing flame; (b) inclined sweeping flame. The vertical red and green dashed lines mark the quenching zone and the influenced zone, respectively.

Figure 12 Long description

Two scatter plots depict the relationship between flamewall distance and flame alignment index for different flame configurations. Panel A: A scatter plot shows the joint probability density function (PDF) between the flamewall distance and flame alignment index for a wall-normal flushing flame. The horizontal axis represents the flamewall distance normalized by delta f over delta z, and the vertical axis represents the flame alignment index. The color scale on the right indicates the probability density values. The vertical red and green dashed lines mark the quenching zone and the influenced zone, respectively. The black dashed line represents the direct numerical simulation (D N S) mean. Panel B: Another scatter plot shows the joint PDF between the flamewall distance and flame alignment index for an inclined sweeping flame. The horizontal axis represents the flamewall distance normalized by delta f over delta z, and the vertical axis represents the flame alignment index. The color scale on the right indicates the probability density values. The vertical red and green dashed lines mark the quenching zone and the influenced zone, respectively. The black dashed line represents the D N S mean.

In more detail, the PDFs of $G_{\textit{align}}$ are further examined. Figure 13 shows that, closer to the wall with intensified boundary influence, the PDF is more distributed with a decreasing expectation, i.e. the flame structure becomes more disorganised. Especially on the wall boundary, where species convection and reactions become negligible, the dominant differential diffusion leads to stronger species misalignment. Small differences between figures 13(a) and 13(b) may arise from the different species transport models, i.e. mixture-averaged transport property for the wall-normal flushing flame and constant Lewis number assumption for the inclined sweeping flame. The newly proposed alignment index $G_{\textit{align}}$ provides an approach for examining the complexity of the near-wall flame.

Because of the disorganised structure of the thickened near-wall flame, the movement of different $c$ -isosurfaces is also disorganised. In such a sense, it is not suitable to use the conventional flame speed (e.g. displacement speed or the diffusion speed) to describe the flame movement in FWI. Instead, the so-called flame zone speed $S_{\textit{zone}}$ is introduced (Zhao et al. Reference Zhao, Wang and Chakraborty2018) as

(3.7) \begin{equation} S_{\textit{zone}} = \frac {\int S_{\rm d} {\rho }|{\boldsymbol{\nabla }} c| \textrm{d} \boldsymbol{n}}{{\rho }_u \int \left |{\boldsymbol{\nabla }} c\right | \textrm{d} \boldsymbol{n}}. \end{equation}

Here, $S_{\textit{zone}}$ represents the ${\boldsymbol{\nabla }} c$ weighted displacement speed $S_d$ along the flame normal across the entire flame zone.

Figure 13.

The PDFs of the flame alignment index $G_{\textit{align}}$ in different zones and on the wall, for: (a) wall-normal flushing flame, (b) inclined sweeping flame.

3.4. Near-wall heat transfer

According to the relative orientation between the flame normal $\boldsymbol{n}$ and the wall normal $\boldsymbol{N}$ (pointing toward the fluid), the near-wall flame can be classified (Zhao et al. Reference Zhao, Wang and Chakraborty2018) as head-on if $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{N}\gt 0$ or entrained if $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{N}\lt 0$ . Figure 14 shows that the dominant part of both flow set-ups is head-on, with $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{N}$ close to 1. Even for the inclined sweeping flame in stronger turbulence, the entrained part is still negligibly small. Therefore, in the following we focus on the head-on flame analysis.

Figure 14.

Joint PDFs between $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{N}$ and flame burnt boundary to wall distance $\delta _b/ \delta _z$ , for both configurations: (a) wall-normal flushing flame, (b) inclined sweeping flame.

Since statistically the flame normal aligns with the wall normal, it is expected that thermal energy transfer along the wall normal, including the diffusion term $q_{n,{diff}} = ({1}/{C_p})( {1}/{RePr} )({\partial ( \lambda ( {\partial T}/ {\partial x_1}) ) }/ {\partial x_1})$ and convection term $q_{n,{conv}} = \rho u_1 ({\partial T}/{\partial x_1})$ , dominates over the wall-parallel part $q_l$

(3.8) \begin{equation} q_l = \frac {1}{C_p}\frac {1}{RePr} \left [\frac {\partial \left ( \lambda \frac {\partial T}{\partial x_2} \right ) } {\partial x_2} + \frac {\partial \left ( \lambda \frac {\partial T} {\partial x_3} \right ) } {\partial x_3} \right ]- \rho u_2 \frac {\partial T}{\partial x_2} - \rho u_3 \frac {\partial T}{\partial x_3}. \end{equation}

To quantify the influence of $q_l$ on each flame element, we consider within the burnt region the integrals $\int q_{n,{diff}} \textrm{d}x_1$ , $\int q_{n,{conv}} \textrm{d}x_1$ and $\int q_l \textrm{d}x_1$ . Since $\int q_{n,{diff}} \textrm{d}x_1\sim Q_w/{C_p}$ , it is meaningful to check the ratio between $Q_l = \int C_p q_l \textrm{d}x_1$ and $Q_w$ . As presented in figures 15(a) and 15(b), because of the isothermal wall conditions, $Q_l/Q_w$ vanishes as $\delta _b$ approaches zero. Away from the wall, although the $Q_l/Q_w$ magnitude increases slightly, the conditional mean remains negligibly small. For the inclined sweeping flame results in figure 15(b), $Q_l/Q_w$ increases with $\delta _b$ because of a large-scale flame front inclination angle, but overall the conditional mean of $Q_l/Q_w$ is still small.

Figure 15.

Relative importance of the lateral heat transfer $Q_l$ to $Q_w$ with respect to $\delta _b/ \delta _z$ for: (a) wall-normal flushing flame; (b) inclined sweeping flame.

4.1. Model construction

In the vicinity of the wall boundary at $x_1=0$ , a local piece of the flame front is shown in figure 16. The flame segment consists of a preheat zone and a reaction zone ranging from $n_{u}$ to $n_{b}$ , inclined with an angle $\theta$ relative to the wall. The temperature field and the velocity field are respectively considered.

4.1.1. Temperature field

From the non-dimensional temperature and progress variable equations

(4.1) \begin{align} {\rho }\frac {\partial {T}}{\partial {t}} + {\rho } {\boldsymbol{u}} \boldsymbol{\cdot }{\boldsymbol{\nabla }} {T} &= \frac {1}{C_p}\frac {1}{RePr} {\boldsymbol{\nabla }} \boldsymbol{\cdot }\left ({\lambda } {\boldsymbol{\nabla }}{T}\right ) + \frac {{\dot {\omega }}_T}{{C}_p}, \end{align}

(4.2) \begin{align} {\rho }\frac {\partial c}{\partial {t}} + {\rho } {\boldsymbol{u}} \boldsymbol{\cdot }{\boldsymbol{\nabla }} c &= \frac {1}{ReSc} {\boldsymbol{\nabla }} \boldsymbol{\cdot }\left ({\rho }{D} {\boldsymbol{\nabla }}c \right ) + {\dot {\omega }}_c, \end{align}

and integrating from the burnt side $n_{b}$ to the unburnt side $n_{u}$ along the local flame normal direction $\boldsymbol{n}$ , we obtain

(4.3) \begin{align} \int _{n_{b}}^{n_{u}} {\rho }\frac {\partial {T}}{\partial {t}} \, \textrm{d} n + \dot {m}_T \int _{n_{b}}^{n_{u}} {\boldsymbol{\nabla }} {T} \boldsymbol{\cdot }\textrm{d}\boldsymbol{n} &= \int _{n_{b}}^{n_{u}} \frac {1}{{C}_p}\frac {1}{RePr} {\boldsymbol{\nabla }} \boldsymbol{\cdot }\left ({\lambda } {\boldsymbol{\nabla }}{T}\right ) \textrm{d} n + \int _{n_{b}}^{n_{u}} {\frac {\dot {\omega }_T}{{C}_p} } \, \textrm{d} n, \end{align}

(4.4) \begin{align} \int _{n_{b}}^{n_{u}} {\rho }\frac {\partial {c}}{\partial {t}} \, \textrm{d} n + \dot {m}_c \int _{n_{b}}^{n_{u}} {\boldsymbol{\nabla }} {c} \boldsymbol{\cdot }\textrm{d}\boldsymbol{n} &= \int _{n_{b}}^{n_{u}} \frac {1}{ReSc} {\boldsymbol{\nabla }} \boldsymbol{\cdot }\left ({\rho }{D} {\boldsymbol{\nabla }}c \right ) \textrm{d} n + \int _{n_{b}}^{n_{u}} {\dot {\omega }_c} \, \textrm{d} n. \\[10pt] \nonumber \end{align}

Here, a temperature-based mean mass flux across the flame zone $\dot {m}_T \equiv \int _{n_{b}}^{n_{u}} \rho \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }T \, \textrm{d} n / {\int _{n_{b}}^{n_{u}} \boldsymbol{\nabla }T \boldsymbol{\cdot }\textrm{d}\boldsymbol{n}}$ , and in parallel a progress variable-based mean mass flux $\dot {m}_c \equiv \int _{n_{b}}^{n_{u}} \rho \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }c \, \textrm{d} n / {\int _{n_{b}}^{n_{u}} \boldsymbol{\nabla }c \boldsymbol{\cdot }\textrm{d}\boldsymbol{n}}$ are introduced. In model implementation, $\dot {m}_T$ and $\dot {m}_c$ are in situ input parameters. Typically, for freely propagating thin flames, $\dot {m}_T=\dot {m}_c$ . However, for the near-wall thickened flame, these two mass fluxes are not necessarily identical.

Figure 16.

Schematic of simplified FWI scenarios for the head-on flame case.

Figure 16 Long description

Panel A: The diagram shows a flame-wall interaction scenario divided into two zones: the preheat zone on the left and the reaction zone on the right. The preheat zone is labeled with n_u, and the reaction zone is labeled with n_b. A curve representing the reaction rate is shown in the reaction zone. Panel B: The diagram depicts a flame interacting with a wall at an angle. The flame is shown in a gradient of colors, indicating different temperatures or reaction rates. The wall is represented by a vertical line labeled O. The angle between the flame and the wall is labeled theta. The normal vector to the wall is labeled N, and the normal vector to the flame is labeled n_r. The horizontal axis is labeled x_1, and the vertical axis is labeled x_r.

In the unburnt upstream region, the temperature and the progress variable remain constant as for the fresh incoming condition, i.e. $T_u = 0$ and $c_u = 0$ at $n_{u}$ . In the region between the wall and the burnt side of the flame, the gradient of the species mass fraction is negligible because of the weak reaction and zero-gradient condition on the wall, i.e. the species diffusion coefficient does not appear in the modelling. Therefore, (4.3) and (4.4) can be written as

(4.5) \begin{align} \int _{n_{b}}^{n_{u}} {\rho }\frac {\partial {T}}{\partial {t}} \, \textrm{d} n - \dot {m}_T T_{b} &= -\frac {1}{\overline {C}_p}\frac {1}{RePr} \left . \left ( \lambda \frac {\partial T}{\partial n} \right ) \right |_{n_{b}} + \int _{n_{b}}^{n_{u}} {\frac {\dot {\omega }_T}{{C}_p}} \, \textrm{d} n, \\[-28pt] \nonumber \end{align}

(4.6) \begin{align} \int _{n_{b}}^{n_{u}} {\rho }\frac {\partial {c}}{\partial {t}} \, \textrm{d} n - \dot {m}_c c_{b} &= \int _{n_{b}}^{n_{u}} {\dot {\omega }_c} \, \textrm{d} n, \\[-12pt] \nonumber \end{align}

where $T_b$ is the temperature at $n_{b}$ and $c_b$ is the progress variable at $n_{b}$ ; $\overline {C}_p$ represents a mean $C_p$ from flame zone averaging. Although the temperature change is large, for typical fuel–air mixtures, $C_p$ varies weakly over a wide temperature range (Poinsot & Veynante Reference Poinsot and Veynante2005). Therefore, $\overline {C}_p$ and local $C_p$ are only moderately different. The integrated source terms in (4.5) and (4.6) are equal in the sense that, once some amount of species is consumed, accordingly the same amount of thermal energy (in the non-dimensional form) must be released. Therefore, after cancellation one obtains

(4.7) \begin{equation} \int _{n_{b}}^{n_{u}} \left({\rho }\frac {\partial {T}}{\partial {t}} - {\rho }\frac {\partial {c}}{\partial {t}}\right) \textrm{d} n - \dot {m}_T T_{b} = -\frac {1}{\overline {C}_p}\frac {1}{RePr} \left . \left ( \lambda \frac {\partial T}{\partial n} \right ) \right |_{n_{b}} - \dot {m}_c c_{b}. \end{equation}

The temperature gradient typically aligns with the flame normal, i.e. $( {\partial T}/{\partial n}) = -( {\partial T}/{\partial \textit {x}_1})( {1}/{\cos {\theta }})$ . Consequently,

(4.8) \begin{equation} \dot {m}_T T_{b}-\int _{n_{b}}^{n_{u}} \left({\rho }\frac {\partial {T}}{\partial {t}} - {\rho }\frac {\partial {c}}{\partial {t}}\right) \textrm{d} n = -\frac {1}{\overline {C}_p}\frac {1}{RePr} \left . \left ( \lambda \frac {\partial T}{\partial \textit {x}_1} \frac {1}{\cos {\theta }} \right ) \right |_{x_{1,b}} + \dot {m}_c c_{b}, \end{equation}

where $x_{1,b}$ is the $x_1$ projection of $n_{b}$ .

In the region between the wall and burnt side, the chemical source is absent and the temperature equation along the wall-normal direction $x_1$ is

(4.9) \begin{equation} \rho \frac {\partial T}{\partial t} + \rho u_1 \frac {\partial T}{\partial x_1} = \frac {1}{C_p}\frac {1}{RePr} \frac {\partial \left ( \lambda \frac {\partial T} {\partial x_1} \right ) } {\partial x_1} + q_l, \quad \text{for}\quad x_1 \geqslant x_{1,b}, \end{equation}

where $q_l$ is the wall-parallel heat transfer term defined in (3.8). Integrating the temperature equation from $x_{1,b}$ to the wall, we obtain

(4.10) \begin{equation} \int _{x_{1,b}}^0 \rho \frac {\partial T}{\partial t} \, \textrm{d} x_1 + \frac {1}{\overline {C}_p}Q_w = -\overline {\rho u_1}(T_w - T_{b}) - \frac {1}{\overline {C}_p}\frac {1}{RePr} \left . \left ( \lambda \frac {\partial T}{\partial x_1} \right ) \right |_{x_{1,b}} + \int _{x_{1,b}}^0 q_l \, \textrm{d} x_1. \end{equation}

Here, $\overline {\rho u_1}$ is the mean $\rho u_1$ along the integral path. The wall heat flux $Q_w$ is a function of various flowing quantities. We denote $\langle Q_w \rangle |_{x_{1,b},T_{b}}$ as the mean heat flux conditional on $x_{1,b}$ and $T_{b}$ and $\langle Q_w \rangle |_{x_{1,b}}$ as the mean heat flux conditional on $x_{1,b}$ . By definition, $\langle Q_w \rangle |_{x_{1,b}}$ is the $p{\rm d}f(T_{b})$ (PDF of $T_{b}$ ) weighted average, i.e.

(4.11) \begin{equation} \left \langle Q_w \right \rangle |_{x_{1,b}}=\int \left \langle Q_w \right \rangle |_{x_{1,b},T_{b}}pdf(T_{b}){\rm d}T_{b}. \end{equation}

Statistically, we are interested in $\langle Q_w \rangle |_{x_{1,b}}$ . The temporal integral terms in (4.8) and (4.10) are dominated by turbulent fluctuations. In the sense of statistical average, they are assumed to be negligible. Consequently, $\langle Q_w \rangle |_{x_{1,b},T_{b}}$ satisfies

(4.12) \begin{equation} \begin{aligned} \left \langle Q_w\right \rangle |_{x_{1,b},T_{b}} &= -{\overline {C}_p}(T_w - T_{b}) \left \langle \overline {\rho u_1}\right \rangle |_{x_{1,b},T_{b}} - \frac {1}{RePr} \left \langle \lambda \frac {\textrm{d} T}{\textrm{d} x_1} \right \rangle |_{x_{1,b},T_{b}} \nonumber\\ &\quad +\, {\overline {C}_p}\left \langle \int _{x_{1,b}}^0 q_l \, \textrm{d} x_1\right \rangle |_{x_{1,b},T_{b}}. \end{aligned} \end{equation}

The solution of $\langle Q_w \rangle |_{x_{1,b},T_{b}}$ from (4.12) can equally be obtained from the following model equations:

(4.13) \begin{equation} \rho U_1 \frac {\textrm{d} T}{\textrm{d} x_1} = \frac {1}{\overline {C}_p}\frac {1}{RePr} \frac {\textrm{d} \left ( \lambda \frac {\textrm{d} T} {\textrm{d} x_1} \right ) } {\textrm{d} x_1} + q_l, \quad \text{for}\quad x_1 \geqslant x_{1,b}, \end{equation}

i.e. (4.9) using a $U_1$ defined velocity without the time derivative, and

(4.14) \begin{equation} \dot {m}_T T_{b} = -\frac {1}{\overline {C}_p}\frac {1}{RePr} \left . \left ( \lambda \frac {\partial T}{\partial \textit {x}_1} \frac {1}{\cos {\theta }} \right ) \right |_{x_{1,b}} + \dot {m}_c c_{b}, \end{equation}

i.e. (4.8) without time derivatives. The velocity $U_1$ here is interpreted as the conditional mean of the turbulent velocity $u_1$ .

4.1.2. Velocity field

We decompose the velocity sufficiently away from the wall into a wall-tangential part and a wall-normal part. The tangential part, which is flow configuration-dependent, contributes to the wall-parallel heat transfer $q_l$ . The wall-normal part’s induced velocity in the flame upstream region ( $x_1 \leqslant x_{1,u}$ ) can be treated as a potential flow velocity, with the $x_1$ component $U_{1,p}$ and the radial component $U_{r,p}$ .

Physically, $U_{1,p}$ equals to the averaged flame zone speed $S_{\textit{zone}}$ , i.e. $\langle S_{\textit{zone}} \rangle$ , at $x_1=x_{1,b}$ . Thus $U_{1,p}$ and $U_{r,p}$ are

(4.15) \begin{equation} U_{1,p}\left (x_1\right )=-2 {\varepsilon }\left [{x}_1-\left ({x}_{1,b}+\frac {\left \langle S_{\textit{zone}} \right \rangle }{2 {\varepsilon }}\right )\right ], \quad U_{r,p}\left ({x}_r\right )={\varepsilon }{x}_r, \end{equation}

where $\varepsilon$ is the strain rate (normalised with $S_L^0/L$ ). As elaborated in Zhao et al. (Reference Zhao, Wang and Chakraborty2018), the mean flame zone speed $\langle S_{\textit{zone}} \rangle$ can be modelled as

(4.16) \begin{equation} \left \langle S_{\textit{zone}} \right \rangle =c_1+c_2 S_L(T_b). \end{equation}

Here, $S_L$ is the freely propagating flame speed at burnt temperature $T_b$ . The parameters $c_1$ and $c_2$ satisfy $c_1+c_2=1$ , implying that the movement of the near-wall flame is under the joint control of free propagation and wall confinement. Since $\langle S_{\textit{zone}} \rangle$ describes the near-wall flame movement, this quantity is almost independent of the flow configuration away from the wall. Its DNS fitting is used as the model input.

In the region between the flame boundary and wall ( $x_1 \geqslant x_{1,b}$ ), the viscous effect is important and $\boldsymbol{U}$ can be treated as a Homann flow (Homann Reference Homann1936) with variable density. Specifically, $U_r$ is linearly proportional to $x_r$ and $U_1$ is $x_r$ -independent. Therefore pressure $P$ satisfies $({\partial }/{\partial x_r})( {\partial P}/{\partial x_1}) = 0$ , i.e. the pressure gradient $\partial P /\partial x_r$ is a constant along $x_1$ . It can be further determined that $\partial P /\partial x_r=-\rho _u \varepsilon ^2 x_r$ via the potential flow boundary condition. Therefore, the momentum equation along $x_r$ is

(4.17) \begin{align} {\rho } {U}_1 \frac {\partial {U}_r}{\partial {x}_1} + {\rho } {U}_r \frac {\partial {U}_r}{\partial {x}_r} = \rho _u \varepsilon ^2 x_r + \frac {\partial }{\partial x_1} \left ({\mu } \frac {\partial {U}_r}{\partial {x}_1}\right ) + \frac {\partial }{\partial {x}_r}\left ({\mu }\frac {\partial {U}_r}{\partial {x}_r}\right ) + \frac {{\mu }}{x_r}\frac {\partial U_r}{\partial x_r} - {\mu }\frac {U_r}{x_r^2}.\\[-22pt]\nonumber \end{align}

For the variable density case, we introduce a stretched coordinate $\eta$ after the Levy–Lees transformation (Lees Reference Lees1956) as

(4.18) \begin{equation} {\eta }=\left (\frac {2 {\rho }_u {\varepsilon }}{{\mu }_u}\right )^{\tfrac {1}{2}} \int _0^{{x}_{1}}\left (\frac {{\rho }}{{\rho }_u}\right ) \textrm{d} \zeta ={\eta }({x}_1). \end{equation}

Therefore, the streamfunction $\psi$ , defined as ${\rho } {U}_1(x_1)=-( {1}/{{x}_r}) ({\partial \psi }/{\partial {x}_r})$ and $\rho {U}_r(x_1,x_r)=( {1}/{{x}_r})( {\partial \psi }/{\partial {x}_1})$ , can be written as

(4.19) \begin{equation} \psi = \sqrt {{\rho }_u{\mu }_u{\varepsilon } /2} x_r^2 f(\eta ). \end{equation}

Consistently with the assumption in non-reactive near-wall heat transfer analyses (Lees Reference Lees1956; Libby & Sepri Reference Libby and Sepri1968), $\rho \mu$ is assumed as constant, or unity $({\rho } {\mu })/({\rho }_u {\mu }_u)$ , because qualitatively the dependencies of viscosity and density on the temperature are opposite. Then, the momentum (4.17) becomes

(4.20) \begin{equation} {f}^{\prime \prime \prime } + {f} {f}^{\prime \prime }+\frac {1}{2}\left (\frac {{\rho }_u}{{\rho }}-{f}^{\prime 2}\right )=0. \end{equation}

Under the isobaric condition $\rho = 1/(1 + \tau {T})$ , where the heat release parameter $\tau = (T_{\textit{ad}} - T_u )/T_{\textit{ad}}$ , (4.20) can be simplified as

(4.21) \begin{equation} {f}^{\prime \prime \prime } + {f} {f}^{\prime \prime }+\frac {1}{2}\left (1+\tau {T}-{f}^{\prime 2}\right ) =0, \end{equation}

together with the following boundary conditions:

(4.22) \begin{equation} \begin{aligned} f(x_{1,b}) &= -\rho _b \left \langle S_{\textit{zone}} \right \rangle \cos \theta /\left (2 {\rho }_u {\mu }_u {\varepsilon }\right )^{{1}/{2}}, \\ f(0) &=0 ,\\ f^{\prime }(0) &=0. \end{aligned} \end{equation}

The function $f$ is coupled with the temperature in (4.13). Under the given boundary conditions, including (4.14) for temperature, (4.22) for velocity and a prescribed wall temperature, the governing equations for $f$ and $T$ can be numerically solved by iteration, until the variations of $f$ and $T$ are below a convergence criterion. The wall heat flux can then be quantified.

4.2. Model validation

Although developed for turbulent flames, the model is applicable for laminar FWI as well. For validation under broader working conditions, the proposed wall heat flux model is first tested using a quasi-one-dimensional laminar flame, as described in § 3.1. There are three input parameters, namely $\dot {m}_T$ , $\dot {m}_c$ and $\delta _b$ , i.e. the distance between $n_{b}$ and the wall. As aforementioned, in the near-wall region, the flame zone speed $\langle S_{\textit{zone}} \rangle$ is primarily flow configuration-independent. The relation between $\langle S_{\textit{zone}} \rangle$ and temperature can be predetermined.

We examine both the unity Lewis number results in figure 17(a) and the mixture-averaged transport results in figure 17(b). It can be seen that the agreement is satisfactory across a large variation of the inlet strain rate, even using detailed chemical kinetics. Moreover, despite the non-zero near-wall heat release rate from the mixture-averaged transport property, the model prediction will not be clearly influenced because most of the heat is still released from the flame zone.

Figure 17.

Comparison of the modelled wall heat flux at various non-dimensional inlet strain rates $\varepsilon$ with the Cantera calculation: (a) results from the unity Lewis number; (b) results from the mixture-averaged species transport property.

4.2.1. Wall-normal flushing flame test

In a turbulent flow field, the flame front can be spatially folded. Starting from a wall point and moving along the normal direction, the first detected flame patch is identified as the one corresponding to the wall point. To evaluate the wall heat flux from the model, the mean flame zone speed $\langle S_{\textit{zone}} \rangle$ parameters $c_1$ and $c_2$ in (4.16) for head-on flames are set as $c_1=0.2$ and $c_2=0.8$ from DNS database fitting. To test the importance of lateral heat transfer $q_l$ , in the following, model results both with DNS determined $q_l$ and without $q_l$ will be compared. Figure 18 shows the predicted mean wall heat flux with respect to the flame–wall distance $\delta _{f}$ , together with the DNS joint PDF and its conditional mean. It can be seen that the DNS result and model predictions with $q_l$ correction and without $q_l$ almost collapse, justifying the model performance in this flushing flame configuration. Consistent with the small mean value of wall-parallel heat transfer in figure 15(a), the influence of $q_l$ on the wall heat flux is insignificant.

Figure 18.

Model predicted wall heat flux with respect to the flame–wall distance, both with $q_l$ correction and without $q_l$ , in comparison with the DNS joint PDF and its conditional mean.

Figure 18 Long description

A scatter plot representing the relationship between flame-wall distance (delta f/delta z) on the horizontal axis and wall heat flux (Q w) on the vertical axis. The plot includes hundreds of data points with a color gradient indicating density, ranging from blue (low density) to red (high density). Three different model predictions are shown: DNS mean (black dashed line), Model mean (red dotted line), and Model mean with q l (blue solid line). The data points form a diagonal band from the top left to the bottom right, indicating a negative correlation between flame-wall distance and wall heat flux. The DNS mean line closely follows the center of this band, while the Model mean and Model mean with q l lines show slight deviations.

4.2.2. Inclined sweeping flame test

In this inclined sweeping flame quenching scenario, the flame zone is not aligned with the wall boundary. According to (4.14), to obtain the model solution, the orientation of the local flame elements has to be considered. For the present inclined sweeping FWI case, the elevated ambient pressure leads to the change of chemical reaction rates, which then the mean flame zone speed $\langle S_{\textit{zone}} \rangle$ fitting will slightly change to $c_1=0.3$ and $c_2=0.7$ .

Figure 19(a) shows the model predictions. When the flame is close to the wall ( $\delta _f/\delta _z \lt 100$ ), the modelled wall heat flux is a bit overpredicted, while when the flame is away from the wall, the agreement is satisfactory. It is also interesting to test the model robustness with possible simplifications. Figure 19(b) presents the model result without correction from the flame element orientation, i.e. setting $\cos {\theta } = 1$ in (4.14) and (4.22). It can be seen that the result is almost invariant, implying the weak influence of the flame element orientation, since the dominant part of the wall heat flux aligns with the wall normal. Furthermore, if the wall-parallel heat transfer $q_l$ in (4.13) is also excluded, the model result is shown in figure 19(c). The difference is still small. Only a slight deviation exists at the far end of the solution curve. Figure 19(d) collectively shows different simplification results of the model. In summary, the model is capable of capturing the dominant contribution from the wall heat flux with good robustness, even without correction from the local flame element orientation and wall-parallel heat transfer.

Figure 19.

Wall heat flux predictions for the inclined sweeping FWI case, in comparison with the DNS joint PDF and its conditional mean from: (a) complete model solution; (b) model solution without the influence from the local flame element orientation $\cos \theta$ ; (c) model solution without influences from flame element orientation $\cos \theta$ and the wall-parallel heat transfer $q_l$ ; (d) conditional means of the DNS and different simplified model solutions.