Numerical Investigation of Local Heat-Release Rates and Thermo-Chemical States in Side-Wall Quenching of Laminar Methane and Dimethyl Ether Flames

In the DC simulation the laminar flow field is described by the conservation equation for mass and momentum. Neglecting body forces, the equations readwhere \(\rho \) is the density, u the flow velocity, p the pressure and \(\nu \) the kinematic viscosity obtained through Sutherland’s law (Saksena and Saxena 1963). The laminar flow is treated as incompressible using the low Mach number assumption. Besides these two equations, a transport equation for each species considered in the reaction mechanism has to be solved. Assuming unity Lewis number for all species, the balance equation readswhere \(Y_{k}\) and \({\dot{\omega }}_k\) are the mass fraction and the species source term for species k, and D is the diffusion coefficient, respectively. Finally, the transport equation of the enthalpy h as sum of sensible and enthalpy of formation readsFor the simulations, the GRI 3.0 mechanism (Smith et al. 1999) (53 species and 325 reactions) and the Zhao mechanism (Zhao et al. 2008) (55 species and 290 reactions) are used for the methane-air and the DME-air flame, respectively.

Three models for the description of mixing-chemistry interaction, namely Flamelet-generated Manifolds (FGM), Quenching Flamelet-generated Manifolds (QFM) and Reaction-Diffusion Manifolds (REDIM), are considered. The manifolds are stored in so-called flamelet look-up tables (FLUTs) based on two control variables: the progress variable and the enthalpy. The latter is important since heat transfer to the wall must be accounted for. Thereby, the thermo-chemical quantities, the source term of the progress variable, and the computed signals are parameterized as a function of enthalpy h and progress variable \(Y_\mathrm {c}\)The progress variable \(Y_c\) for methane and DME is chosen to be the mass fraction of \( {\text{CO}}_{{\text{2}}} \), which is consistent with previous studies of flame-wall interactions (Ganter et al. 2017, 2018). The construction of FGM, QFM and REDIM is described in Sects. 2.2.1,  2.2.2 and 2.2.3, respectively.

In a coupled simulation using FLUTs the flow field is described by the mass and momentum transport, see Eqs. (1) and (2), respectively. To account for the chemical processes, additional transport equations for the control variables are solved: The equation for the enthalpy is given in Eq. (4), while for the progress variable the balance equation readsAfter solving the transport equations, the progress variables can be used to access the thermo-chemical state stored in the FLUT. All other thermo-chemical quantities, the reaction rate and computed signals (see Sect. 2.3) are retrieved from the tabulated manifold. Figure 1 shows the look-up procedure during a coupled simulation using chemistry manifolds.

For the DC simulations, as well as the reduced manifolds, the \(\mathrm {OH}\)-LIF and \( {\text{CH}}_{{\text{2}}} {\text{O}} \)-LIF signals are computed based on the numerical thermo-chemical state. For the chemistry manifolds, the signals are incorporated in the FLUT as a function of progress variable and enthalpy. \(\mathrm {OH}\) (Popp et al. 2015; Hunger et al. 2017) and \( {\text{CH}}_{{\text{2}}} {\text{O}} \)-LIF (Popp et al. 2015) computed signals were used previously. Within these studies a detailed description of the underlying method is given. In this section, the method is briefly introduced.

The \(\mathrm {OH}\)-LIF signal for the \(Q_1(6)\) transition of the \(A^2\varSigma ^+ \leftarrow X^2\prod ^+\) band is calculated for the linear LIF regime with the excited-state decay rate that is dominated by collisional quenching (Kosaka et al. 2019),with \(N_\mathrm {OH}\) being the \(\mathrm {OH}\) number density, \(f_\mathrm {J}\) the Boltzmann fraction population of the absorbing state with rotational quantum number J. The spontaneous emission rate is described by A; Q is the total collisional quenching rate of the excited state, which is the sum of the quenching rates for each collisional partner and is dependent on the composition and the local temperature. The total collisional quenching rate is given bywith \(N_\mathrm {tot}\) being the total number density. The parameters \(k_\mathrm {B}\), \(X_i\), \(\sigma _i\) and \(\mu _i\) are the Boltzmann constant, the mole fraction, the quenching cross-section and the reduced mass of the species i, respectively. The quenching cross sections are obtained from Tamura et al. (1998).

The calculation of the \( {\text{CH}}_{{\text{2}}} {\text{O}} \)-LIF signal is very challenging since it involves excitation of overlapping transitions and complex temperature and species-dependent cross sections. The \( {\text{CH}}_{{\text{2}}} {\text{O}} \)-LIF signal is calculated following the methodology described by Kosaka et al. (2019), which goes back to the procedure proposed in Coriton et al. (2015) and Popp et al. (2015). The \( {\text{CH}}_{{\text{2}}} {\text{O}} \)-LIF signal is obtained using Eqs. (8) and (9) with an additional term accounting for the temperature dependence of the quenching cross-sections and the populations of overlapping transitions. The temperature dependence is considered using the same polynomial fit as in the experiments by Kosaka et al. (2019) that is based on spectral simulations using the simulation code AsyrotWin (Birss and Ramsay 1985; Judge and Clouthier 2001). Within the simulations, the transitions near the \(339~\mathrm {nm}\) excitation of the \({\widetilde{A}}^{1} A_{2} \leftarrow {\widetilde{X}}^{1} A_{1}\) system are considered.

In the study by Kosaka et al. (2019), a correlation based on the product of the normalized LIF signals of \(\mathrm {OH}\) and \({\text{CH}}_{{\text{2}}} {\text{O}} \) is used to predict the HRR. Based on the product of the measurement signals a normalized local HRR of the flame can be definedwith \(\langle S_{\mathrm{OH}{\text{-}}\mathrm{LIF}} \rangle \) and \( \langle S_{{{\text{CH}}_{{{2}}} {\text{O-LIF}}}} \rangle \) being the normalized averaged measured signal intensity of \(\mathrm {OH}\) and \({\text{CH}}_{{\text{2}}} {\text{O}} \), respectively. The product is normalized using the maximum value along the flame front in the lower part of the measurement domain (\(z_q < -3~\mathrm {mm}\)) corresponding to an unstretched flame region. To allow a direct comparison between the DC simulations and the experiments, the above HRR definition is used in both. In the DC simulation the HRR is calculated using computed signals (see Sect. 2.3). The HRR definition was validated in Kosaka et al. (2019) using three different flame configurations (1) an unstretched planar flame (2) a curved and stretched flame and (3) a HOQ configuration. For the first two configurations, the results agree qualitatively with the normalized HRR defined aswhere \({\dot{\omega }}_i\) is the source term and \(h_{f,i}\) is the enthalpy of formation of the species i. For the unsteady HOQ configuration, the HRR correlations agree reasonably well until the point where the flame is quenched at the wall. During the quenching process the radicals in the vicinity of the wall are consumed faster than the HRR declines and the correlation deteriorates (Kosaka et al. 2019). In this work, the validation is extended to the steady SWQ scenario, which will be discussed further for the methane-air flame next.

In "Appendix" the flame structure and the specifics of the methane-air and DME-air flame are discussed. Thereby, the fundamental flame characteristics of the two fuels are outlined. In the following, we focus on the local HRR during SWQ. Figure 3 shows the local HRR over the wall-normal direction y and the relative quenching height \(z_q\) for the methane case. The experimental results are displayed in subfigure (a), while (b) and (c) correspond to the DC simulation with the HRR definition based on Eqs. (10) and (11), respectively. Additionally, the white line in the Fig. 3c displays the isoline of \(\mathrm {HRR}_\mathrm {num}=0.05\). The HRR definition used in the experiments \( {\text{HRR}}_{{{\text{exp}}}} \) is in very good agreement with \( {\text{HRR}}_{{{\text{num}}}} \) considering HRR values greater than 5% (see Fig. 3b, c). At the edges of the reaction zone, however, the formaldehyde vanishes and the prediction based on \({\text{HRR}}_{{{\text{exp}}}} \) deteriorates. These results are consistent with the HOQ scenario analyzed in Kosaka et al. (2019) and extend those findings to the more complex SWQ configuration. Similar results are obtained for the DME-air flame that are not shown here for brevity.

For a consistent comparison with the experimental data, \({\text{HRR}}_{{{\text{exp}}}} \) is used. The HRR prediction of the DC simulation in Fig. 3b shows an overall good agreement with the measurements shown in Fig. 3a. Nevertheless, the experimental results show a broader reaction zone. This is particularly evident in the lower part of the flame where the flame structure should correspond to an unperturbed laminar flame. Potential reasons for these differences are (1) the measurement resolution, (2) the measurement uncertainties caused by the image intensifiers, and (3) fluctuations in the instantaneous flame position and flame angle of the laminar flame due to Helmholtz resonances originating from the plenum of the burner. Even though these fluctuations are accounted for in the post-processing of the measurements by shifting the individual images vertically on the average quenching point, uncertainties regarding the exact flame position and flame angle remain (Jainski et al. 2017a, b; Kosaka et al. 2018, 2019).

To further investigate potential reasons for the differences in the numerical simulation and the experimental findings, the HRR based on Eq. (10) is analyzed along wall-normal profiles that are shown as black solid lines in Fig. 3a. The obtained HRR profiles are normalized such that the integral of the numerical simulations and the experimental data match. This allows a comparison of the shape of the HRR distribution along the wall normal lines. Then, artificial measurement uncertainties are superimposed onto the simulation results using a box filter with a variable filter width \(\varDelta \). Since the measurement uncertainties cannot be specified, the use of a box filter presents a worst-case estimation of errors originating from the measurement uncertainties. Figure 4 displays the corresponding profiles of the methane case in a coordinate system relative to the HRR peak. While in the undisturbed part of the flame the HRR peak position differs by over \(200~\upmu \mathrm {m}\), in the near-wall region the shift vanishes. Therefore, the thermo-chemical state during quenching is unaffected by this shift.

The filtering of the signals leads to a broadening of the HRR profile in the y-direction. In the unperturbed part of the flame (\(z_q=-3~\mathrm {mm}\)), a filter width of \(\varDelta >300~\upmu \mathrm {m}\) is necessary to match the width of the experimental profile. With decreasing distance from the quenching height, however, the required filter width decreases. In the vicinity of the quenching height (\(z_q=-0.5~\mathrm {mm}\)), a filter with of \(\varDelta =200~\upmu \mathrm {m}\) is sufficient. Assuming remaining uncertainties in the flame angle in the post-processed measurement results, the measurement uncertainty would increase with increasing distance from the quenching point, since the spatial distortion increases proportionally with the distance. This leads to an increasing filter width with increasing distance to the quenching height.

Figure 5 shows the corresponding HRR profiles of the DME case. In contrast to the methane-air flame, the DC simulation of the DME case predicts the thickness of the HRR profile in the undisturbed part of the flame satisfactorily using a filter width of \(\varDelta \approx 250~\upmu \mathrm {m}\). Additionally, the position of the HRR peak in the numerical simulations of the DME-air flame is shifted between \(60~\upmu \mathrm {m}\) and \(140~\upmu \mathrm {m}\) to larger wall distances over the whole flame front. Other than in the methane case, the shift does not decrease in the near-wall region (\(\varDelta _y=140~\upmu \mathrm {m}\)). The thermo-chemical state that is discussed in Sect. 5 shows a similar shift. Even though great care has been taken to determine the exact distance to the wall in the experiments, thermal disturbances and beam steering, that affect the determination of the horizontal position of the wall, cannot be avoided. The resulting uncertainties lie in the range of the deviations.

Compared to the methane-air flame (Fig. 4), the DME-air flame shows a thinner HRR zone in the undisturbed part of the flame. This is related to the flame thickness of a stochiometric DME-air flame compared to a methane-air flame and can already be observed in a simple freely propagating flame, see "Appendix" for further detail. During quenching, the HRR peak decreases, while the width of the HRR zone is unaffected by the quenching process. It is interesting to note that the different flame thickness of the methane-air and DME-air flame during quenching can be explained by fuel characteristics that can already be observed in very simple flame configurations, like a freely propagating flame.

Finally, Fig. 6a, b display the 2D profiles of the unfiltered and filtered HRR using a filter width of \(\varDelta =300~\upmu \mathrm {m}\) and \(\varDelta =250~\upmu \mathrm {m}\) for the methane-air and DME-air flame, respectively. The filtered and unfiltered signals agree reasonably well with the measurement data. Based on the discussion above, filtered signals result in a thickened HRR profile in the undisturbed part of the flame which better reflect the measurement results. This finding, however, does not imply that the measurement resolution is only \(\varDelta \), rather we believe that the potential reasons (1)-(3) are responsible for the experimentally observed broadening. As mentioned above, the agreement between the experiment and simulation deteriorates with increasing distance from the quenching height for the methane case. This could be related to the fluctuations in the instantaneous flame position.

While in the previous section DC simulations for the HRR were compared to experimental data, in the following the suitability of manifold-based approaches to predict the near-wall HRR, is investigated. Therefore, the manifold-based approaches are compared to their corresponding DC simulations. The HRR definition used for the comparison is based on Eq. (11). Figure 7 displays the obtained results for the methane-air (top) and DME-air (bottom) flame along wall-normal lines. For both fuels, FGM, QFM and REDIM compare favorably with the DC results. This is not a trivial finding, since within the near-wall region the enthalpy level in the flame varies due to heat transfer to the wall. These different enthalpy levels in the flame are considered in the manifolds by using the enthalpy as a control variable of the FLUT. The agreement of the manifolds with the DC simulations reveals, that the manifolds, even though they use different approaches to obtain the thermo-chemical state, are suitable to predict the HRR in the near-wall region and thereby to account for the enthalpy losses to the wall. This is consistent with previous findings that showed a good prediction quality for the tabulation strategies regarding global flame properties in methane-air flames (Ganter et al. 2017, 2018; Efimov et al. 2019).

The numerical simulations of the methane-air flame conducted here, confirm the observations published in the previous studies and, therefore, are not shown for brevity. In the following we focus on the DC simulation of the DME-air flame by first comparing with the available experimental data. Afterwards, the suitability of the chemistry manifolds to describe the near-wall thermo-chemical state is analyzed. The analysis is performed using FGM, QFM, an extension of FGM, and REDIM. The latter two have been specifically designed to model the thermo-chemical state during flame-wall interactions (Strassacker et al. 2018a, b; Efimov et al. 2019). Note, as stated above, REDIM are not very sensitive with respect to the gradient estimation.

In the experimental study by Kosaka et al. (2018) (see also Sect. 3) the thermo-chemical state of the flame was characterized using the CO concentration from CO-LIF measurements and the temperature based on CARS. Figure 8 shows the thermo-chemical state for the DME-air flame at wall distances of \(y=0.1~\mathrm {mm}\), \(y=0.3~\mathrm {mm}\) and \(y=0.5~\mathrm {mm}\) along a wall-normal line at quenching height. The experimental data is shown as scatter. The conditional mean as well as the standard deviation is included, separated into a CO formation (blue dots) and a CO oxidation (black dots) branch, based on the CO-T probe volume position. As mentioned previously, the flame tip is not stationary in the experiments, but it fluctuates around the quenching point reported to be in the range of \(\pm 200~\upmu \mathrm {m}\) (Kosaka et al. 2018). These flame tip fluctuations are not present in the DC simulations. To allow a comparison with the experimental data, the CO-T state of the DC simulation is plotted along wall-parallel lines. Thereby, the flame fluctuations are captured, since both, the CO formation, as well as the oxidation branch, are crossed. The thick solid part of the lines corresponds to the area \(\pm 200~\upmu \mathrm {m}\) around the quenching height. Even though the majority of the points lies within this range, it seems that the measurement uncertainty of the flame tip position is higher than anticipated in the near-wall region (\(y=0.1~\mathrm {mm}\) and \(y=0.3~\mathrm {mm}\)). Similar observations were made in a recent numerical study of the SWQ burner (Zirwes et al. 2019). Therefore, the numerical results are plotted also over the entire simulation domain.

Additionally, as discussed in the previous section, the analysis of the local HRR reveals a deviation between the measurement data and the simulations results regarding the wall-normal position of the HRR peak of around \(140~\upmu \mathrm {m}\). This difference in the flame position effects the thermo-chemical state of the flame. To asses the sensitivity of the wall-normal flame distance to the wall on the thermo-chemical state, the extraction lines of the DC simulations are shifted in wall-normal direction by \(100~\upmu \mathrm {m}\) and \(150~\upmu \mathrm {m}\). While the region far away from the wall (\(y=0.5~\mathrm {mm}\)) shows a small sensitivity, the near-wall region is strongly affected by uncertainties in the wall position. This demonstrates the high complexity of the near-wall diagnostics, since on the one hand, the thermo-chemical state is very sensitive to the wall-normal position, on the other hand, the measurements very close to the wall are especially challenging. After correcting for the observed shift, the DC results achieve good agreement with the experimental data in the near-wall region (\(y=0.1~\mathrm {mm}\) and \(y=0.3~\mathrm {mm}\)).

In the following, near-wall \(\mathrm {CO}\) predictions for the reduced chemistry manifolds, namely FGM, QFM and REDIM, are compared and analyzed. Thereby, the DC simulations serve as a reference solution. While FGM are based on a series of freely propagating flames with different enthalpy levels, QFM and REDIM are based on a transient 1D HOQ simulation, see Sect. 2. Diffusive effects in enthalpy direction that are not captured in the FGM tabulation can be accounted for. Figure 9 shows the CO mass fraction in the enthalpy progress variable space for the chemistry manifolds. While QFM and REDIM do not show significant differences, FGM predict a lower CO concentration in the near-wall region (left boundary). It is interesting to note that QFM as well as REDIM lead to a similar thermo-chemical state, even though the table generation procedure differs between the two methods.

In Fig. 10, the tabulated chemistry simulations are compared with the DC simulation results for different wall distances. In the undisturbed region of the flame (\(y>0.3~\mathrm {mm}\)) the FGM prediction is of comparable accuracy to REDIM and QFM. All manifolds over-predict the CO concentration of the flame slightly. In the near-wall region, however, the FGM prediction, deteriorates with decreasing wall-normal distance. This is caused by diffusive effects in enthalpy direction that are not captured by the FGM creation procedure described in Sect. 2.2.1. Similar observations were made for methane-air flames in previous studies (Strassacker et al. 2018b; Ganter et al. 2017, 2018; Efimov et al. 2019). As mentioned above, QFM and REDIM are able to capture these effects and, hence, show a significantly improved prediction of the CO concentration at the wall. While, the peak CO concentration is still over-predicted. This is evident in the near-wall region and the undisturbed part of the flame. Efimov et al. (2019) showed (for methane-air) that even though the species diffusion in the direction of the enthalpy gradient is included in an HOQ configuration, its rate might be over-predicted leading to an increased CO concentration in the vicinity of the wall. This is caused by the rate of heat loss to the wall which is up to a factor of two lower in the SWQ scenario compared to HOQ. To model these effects, a third control variable accounting for the varying heat transfer rate to the wall could be introduced in future REDIM tabulations or QFM (Efimov et al. 2019).