5.1. The SPL and volume-integrated HRR spectra

Figure 4. Measured SPL for DLR-A, a non-premixed jet (marked as a) (Singh, Frankel & Gore Reference Singh, Frankel and Gore2004) flames and a piloted premixed Bunsen (marked as b) (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009). Here, $\mathrm {SPL}(f) = 10\log (\hat {\wp }/p^2_{{ref}})$ , where $p_{{ref}} = 20\, \mu\rm Pa$ .

Figure 5. Time series $\dot {Q}'$ and $\dot {q}'$ normalised by their respective time averages for cases DLR-A, DLR-B, DLR-C, BBF and JICF. The typical local HRR data are shown for arbitrary probe locations indicated.

The SPL and the volume-integrated HRR spectra were shown to be correlated for open turbulent premixed (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009) and non-premixed (Ihme et al. Reference Ihme, Bodony and Pitsch2006) flames. However, in confined flames, the PSD of pressure is not directly related to volume-integrated HRR only as demonstrated by (2.6). The Green’s function is the pressure generated in response to an impulse of rate of heat addition. Since the flame is generally compact and the mixing is fast, the acoustic impedance of the combustor is nearly constant in the axial direction. Additionally, since the combustor geometry is typically axially uniform, the Green’s function does not vary much in the axial direction. At low frequencies, the higher contributions to Green’s function from the lower mode numbers are compensated by the lower contributions from the higher mode numbers. The trend reverses at higher frequencies (Liu et al. Reference Liu, Dowling, Swaminathan, Morvant, Macquisten and Caracciolo2014) and therefore yields a Green’s function spectrum whose frequency variation is typically within one decade. Therefore, the typical Green’s function spectrum $\vert \hat {G} (\boldsymbol{y},\boldsymbol{x},f )\vert ^2$ does not vary much with $f$ or axial distance, as shown for an industrial combustor by Liu et al. (Reference Liu, Dowling, Swaminathan, Morvant, Macquisten and Caracciolo2014). This suggests that the SPL and the volume-integrated HRR spectrum may indeed be correlated even in confined flames. Hence it is of fundamental interest to assess the effect of partial premixing and confinement on this correlation. Before analysing the HRR signal, the measured SPL for DLR-A is compared with results from past works of unconfined turbulent premixed (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009) and non-premixed (Singh et al. Reference Singh, Frankel and Gore2004) jet flames in figure 4. In order to make meaningful comparisons, the spectra are shifted appropriately to match the peak SPL in both cases. The spectra show an approximate power-law dependence of $f^2$ at low frequencies for all cases. This is consistent with a past study (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009) which showed $\hat {\wp } (\boldsymbol{x},f ) \sim f^2 \psi ^+_{\dot {Q}}$ . It will be shown next that the volume-integrated HRR spectrum $\psi _{\dot {Q}}$ is flat in this frequency range. The high-frequency part ( $f\gt f_p$ ) falls off as $f^{-5}$ for DLR-A and the turbulent non-premixed jet (Singh et al. Reference Singh, Frankel and Gore2004) flames as compared to $f^{-2.2}$ for the premixed piloted flame (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009). While the low-frequency spectra are approximately comparable for all cases, the high-frequency spectra show a substantial difference between the cases. The reasons for this difference can be understood by studying the HRR spectrum.

First, the temporal behaviour of $\dot {Q}'$ and $\dot {q}'$ is studied before analysing their frequency spectra. These results are shown in figure 5, normalised using the respective time averages. Note that $\langle \dot {Q} \rangle = P_{{th}}$ , as given in table 1. The local HRR, $\dot {q} (\boldsymbol {y},t )$ , is driven by a small-scale phenomenon, hence it is highly intermittent, resulting in $\vert \dot {q}'\vert \gt \langle \dot {q} \rangle$ as observed in many past studies (Swaminathan et al. Reference Swaminathan, Xu, Dowling and Balachandran2011b ; Liu & Echekki Reference Liu and Echekki2015). The local HRR vanishes ( $\dot {q}=0$ ) for the partially premixed flames whenever the local mixture fraction value is not within the flammability limits, hence the negative fluctuations of $\dot {q}$ are limited to $\dot {q}' = -\langle \dot {q} \rangle$ . Since $\dot {Q} (t )$ is a volume-integrated quantity that is mostly driven by large-scale flow structures, it varies smoothly with $t$ .

Figure 6. The variation of $\psi ^{+}_{\dot {Q}}$ with (a) $f$ and (b) $f/f_p$ .

Figure 6(a) shows the variation of $\psi ^+_{\dot {Q}}$ (see (2.7)) with $f$ for the eight cases listed in table 1. The normalised PSDs for all the cases, except for JICF, fall in the same frequency range. In the JICF case, variations are across a broader and higher frequency range ( $10^3 \lt f \leq 10^5$ ) due to the higher-frequency hydrodynamics present in this flow. Additionally, this may be attributed to the faster burning rates of the $\textrm {H}_2/\text{N}_2$ fuel mixture, as discussed in the next subsection. Figure 6(b) shows the variation of $\psi ^+_{\dot {Q}}$ with $f/f_p$ , where $f_p$ is given by a convective scaling (Winkler et al. Reference Winkler, Wäsle and Sattelmayer2005; Rajaram & Lieuwen Reference Rajaram and Lieuwen2009) listed in table 2. Such a scaling denotes the time scale for large-scale fluctuations convected over characteristic length scale $L$ by the bulk-mean velocity $U_b$ . The bulk-mean velocity in the partially premixed cases is obtained as $U_b = \dot {m}/\rho _a A_{{in}}$ , where $\dot {m} = \dot {m}_a (1+ (\dot {m}_f/\dot {m}_a )_{{st}}\phi _g )$ , $\rho _a$ is the air density, and $A_{{in}}$ is the cross-sectional area of the combustor inlet. The quantity $ (\dot {m}_f/\dot {m}_a )_{{st}}$ is the stoichiometric fuel–air mass ratio, and typically $ (\dot {m}_f/\dot {m}_a )_{{st}}\phi _g \ll 1$ . The characteristic length is taken to be the flame length, which is defined as the distance between the flame root and the streamwise location of the maximum time-averaged reaction rate of progress variable $\langle \overline {\dot {\omega }_c}\rangle$ , for the DLR flames. For the BBF case, it is approximated as the length of the combustor since the flame brush extends outside the combustor. The flame length for the JICF case is defined as the curvilinear distance between the jet exit and the point along the jet centreline where the maximum $\langle \overline {\dot {\omega }_c}\rangle$ occurs. The DLR-B flame is thermoacoustically unstable, therefore the threshold frequency beyond which $\psi ^{+}_{\dot {Q}}$ decays is given by the frequency of thermoacoustic oscillation $f_p = f_T \approx 250 \, \textrm {Hz}$ . The variation of $\psi ^+_{\dot {Q}}$ with $f$ for $f\lt f_p$ is negligible, as has been observed in past studies (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009), and the collapse in the normalised frequency space is reasonable. The convective scaling ( $U_b/L$ ) for $f_p$ listed in table 2 agrees well with the results seen in figure 6(a).

At high frequencies ( $f\gt f_p$ ), the normalised PSD for most cases decays at an identical rate of approximately $f^{-5}$ , which is quite remarkable. The JICF case decays at a slightly lower rate for $f\gt f_p$ , which is the result of the high-frequency dynamics discussed later in § 5.2. This high-frequency decay rate $f^{-5}$ is considerably higher than the previously measured (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009) ( $f^{-2.2}$ ) or theoretically deduced ( $f^{-2.5}$ ) (Clavin & Siggia Reference Clavin and Siggia1991) rates. The high-frequency spectral decay rate for $\dot {Q}'$ marked in figure 6(b) is identical to the SPL fall-off rate shown in figure 4. This strongly suggests that the Green’s function frequency spectrum (see (2.6)) has little variation with $f$ , as observed by Liu et al. (Reference Liu, Dowling, Swaminathan, Morvant, Macquisten and Caracciolo2014). Therefore, the strong correlation between $\psi ^{+}_{\dot {Q}}$ and SPL observed for premixed unconfined turbulent flames (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009) holds for confined– both premixed and partially premixed– flames also. It is remarkable that these flames with different complexities show almost the same spectral decay $f^{-5}$ . It is instructive to analyse the spectra of $\psi _{\dot {q}}$ to gather more insights. The local spectrum is particularly important for cases when there is significant spatial and frequency variation of the Green’s function. For example, cases with effusion cooling, non-compact flames, multi-stage combustion and the presence of azimuthal modes (Liu et al. Reference Liu, Dowling, Swaminathan, Morvant, Macquisten and Caracciolo2014) result in significant spatial and frequency variation of the Green’s function. In such cases, the sound pressure spectrum cannot be directly related to $\psi _{\dot {Q}}$ .

5.2. Local HRR spectra

Figure 7. The variation of $\psi ^{+}_{\dot {q}}$ with $f$ at ten probe locations for the five cases. The insets in the DLR cases show the normalised PSD of velocity magnitude, $\psi ^{+}_{\vert \boldsymbol{u} \vert }$ , for the same probe locations. The magenta line shows the geometric mean of $\psi ^{+}_{\vert \boldsymbol{u} \vert }$ at the ten probe locations. The vertical red dotted line marks the characteristic frequency $f^*$ .

The local instantaneous HRR $\dot {q} (\boldsymbol {y},t )$ data are collected for ten probe locations shown in figures 1– 3. The normalised PSD $\psi ^{+}_{\dot {q}} (f )$ at these ten locations for the five cases listed in table 2 are shown in figure 7 along with the velocity spectra for the DLR flames. The PSDs can be segregated into two regimes, namely, a low-frequency regime $f\lt f^*$ and a high-frequency regime $f\gt f^*$ , where $f^*$ is the characteristic frequency marked using vertical red dotted lines. The frequency beyond which the PSD starts to fall off is defined to be the characteristic frequency $f^*$ here. The PSD $\mathcal {\psi }^{+}_{\dot {q}}(f)$ is more or less constant, with a small amount of scatter across the spatial points for all the cases when $f \leq f^*$ . A peak near the PVC frequency is evident in the inset showing $\psi ^+_{\vert \boldsymbol {u}\vert } (f )$ . This is marked as PVC in figure 7, and it is seen only for probe locations close to this flow structure. The PVC is a distinct coherent structure modulating the flame surface in these cases (Chen et al. Reference Chen, Langella, Swaminathan, Stöhr, Meier and Kolla2019a ; Massey et al. Reference Massey, Langella and Swaminathan2019a ), and is responsible for the high spectral content for frequencies close to the PVC frequency. Therefore, the characteristic frequency $f^*$ for the local HRR spectra is equal to the PVC frequency in these cases. The spectra decay for $f\gt f^*$ , and there is some small spatial variation of $\psi ^{+}_{\dot {q}}$ . The PSD of the BBF case in figure 7 shows a slightly different spectral behaviour, although the overall qualitative trend across the frequency space is similar to the DLR flames. The low-frequency behaviour has only a small variation similar to DLR flames, as seen in the figure. The spectral roll-off rate for all the locations is similar for $f\gt f^*$ up to approximately 3 kHz. The spectral decay is faster for higher $f$ , and is also moving downstream, as indicated by lighter colour curves. The decay exponent varies from 1.38 for the most upstream locations to 5 for the most downstream locations, as shown in figure 7. The normalised PSD of HRR for the JICF flame has several similarities and differences compared to the other cases. It is similar to the other spectra since it qualitatively captures the same trend. However, there are at least two key differences: (i) there is a significant variation of the normalised PSD across different locations at high frequencies with a discernible narrow-band peak; and (ii) the entire spectral content is shifted to a higher (by one order of magnitude) frequency regime. The decay rates show a substantial spatial variation similar to the BBF case beyond $f\approx 27\, \textrm {kHz}$ . However, unlike the DLR cases, identifying a physically meaningful threshold frequency is challenging and is sought out next using DMD analyses.

Figure 8. DMD mode amplitude of $\overline {\dot {\omega }_c}$ and $\vert \boldsymbol{u} \vert$ for the BBF case, and the spatial distribution of three dominant modes.

Peaks are observed in neither $\psi ^{+}_{\dot {q}}$ nor $\psi ^{+}_{\vert \boldsymbol{u} \vert }$ (not shown) for the BBF case, suggesting that no globally unstable hydrodynamic mode is excited. It is therefore more challenging to identify a physically motivated characteristic frequency $f^*$ for this case. To proceed further, a simultaneous DMD of the reaction rate and velocity magnitude is performed. The results are shown in the mid $x$ – $z$ plane to recognise dominant coherent structures and their modal energy, which could assist in the identification of a suitable $f^*$ . A total of 200 DMD modes $m$ , where each mode corresponds to a particular frequency, are used for this analysis, and this number of modes is deemed sufficient since at least 80 % of the energy content is captured. Figure 8 shows the DMD mode amplitudes $\vert b \vert$ normalised by their peak values, and the mode shapes for the three most dominant modes. The mode shapes of the reaction rate of progress variable and velocity magnitude ( $\textrm{Re} \{\widehat {\overline {\dot {\omega }_c}}\}/\textrm{Re} \{\widehat {\overline {\dot {\omega }_c}}\}_{{max}}$ and $\textrm{Re} \{\hat {\boldsymbol {u}}\}/\textrm{Re} \{\hat {\boldsymbol {u}}\}_{{max}}$ ), shown in figure 8 are essentially the real parts of the eigenvectors of the Koopman operator $\boldsymbol{K}$ (see (4.2)) obtained using DMD. The velocity modes do not reveal distinct coherent structures; however, the HRRs show a greater degree of coherence as seen from the large areas of either blue or red colour in figure 8, especially in the shear layers where the flame is stabilised. The three most dominant modes recognised using DMD show an antisymmetric (sinuous) pattern about $z=0\,\rm mm$ for the HRR, which is most evident at $f = 288\,\rm Hz$ . The lack of substantial coherence and a subsequent globally unstable mode is most likely due to the high turbulence intensity (22 %) in this case. Since the modal amplitudes are high in the frequency range $100\lt f\lt 300\,\rm Hz$ before substantial decay occurs, as seen in the top row of figure 8, the spectral characteristic frequency can be conveniently identified in this range. The precise value of the characteristic frequency in this range is not critical for the analysis since it will be shown to have negligible impact on modelling aspects discussed later in § 5.3.

Figure 9. The DMD mode amplitude of $\overline {\dot {\omega }_c}$ and $\vert \boldsymbol{u} \vert$ for the JICF flame, and the spatial distribution of three modes.

Figure 9 shows the DMD results for the reacting JICF obtained using 350 modes, which sufficiently capture over 80 % of the energy content. The DMD reveals that the zero-frequency mean flow is the most dominant mode that features two branches for the real part of the reaction rate. The second most dominant mode, at $f = 3960\,\rm Hz$ , resembles an elliptic short-wavelength (about 5 times the jet exit diameter) instability of the CVP in the jet region similar to past studies (Laporte & Corjon Reference Laporte and Corjon2000; Ilak et al. Reference Ilak, Schlatter, Bagheri and Henningson2012). This instability is most evident in the region $20\lt x\lt 40\,\rm mm$ and $10\lt z\lt 20\,\rm mm$ where an antisymmetric mode structure in $\widehat {\overline {\dot {\omega }_c}}$ is clearly visible, and this is a characteristic of the short-wavelength elliptic instability of the vortex pairs. This instability results in short waves convecting along the jet trajectory. Furthermore, this mode shows the upright wake vortices being distorted by the background flow in the wake region ( $10\lt x\lt 40\,\rm mm$ and $0\lt z\lt 10\,\rm mm$ ) as is evident in both the velocity and the reaction rate mode structures. These structures also convect along the cross-flow similarly to the short waves on the CVP. This second mode showing the elliptic instability and wake vortices results in a wide region of coherence, hence the corresponding frequency is a reasonable choice for the characteristic frequency in this case. An additional DMD mode can be recognised at $f = 27\,070\,\rm Hz$ as shown in figure 9, which is close to the narrow-band peak observed in figure 7 for the JICF case. Note that the spatial window used for the DMD is much wider than the spatial extent of this mode. Therefore, the mode amplitudes that optimise the faithful modal reconstruction of the entire flow field are heavily weighted against modes that have a short spatial extent in comparison to the entire DMD window. As a result, extremely low amplitudes (note the scale on the $y$ -axis) are seen for this mode, which is active only in a small region close to the jet shear layer. This mode structure closely resembles the typical shear layer roll-up due to KH instability, as seen clearly from the antisymmetric mode structure for velocity magnitude. The reaction rate mode structure shows a complex structure for $0\lt x\lt 10$ mm, possibly from the distortions of the flame surface caused by the shear layer vortices. However, at $x\gt 10\,\rm mm$ the mode structure changes to alternating cusp-like structures before quickly diminishing in strength. These coherent structures significantly increase the spectral content of $\psi ^+_{\dot {q}}$ and explain the narrow-band peak seen in figure 7 for the JICF case.

Figure 10. The PDFs of $\tau _c$ for three spatial zones marked in the right-hand column showing $\langle \overline {\dot {\omega }_c}\rangle /\langle \overline {\dot {\omega }_c}\rangle _{{max}}$ . The dashed vertical lines indicate the mean $\tau _c^{-1}$ in each zone.

The spatially varying decay rates seen for cases BBF and JICF in figure 7 may be understood by analysing the chemical time scale defined as $\tau _c = \overline {\rho }/\overline {\dot {\omega }_c}$ . The increasing decay rates for $\psi ^+_{\dot {q}}$ with downstream locations in the BBF case suggest that the local chemical time scale variation identified by Langella et al. (Reference Langella, Swaminathan and Pitz2016) may play a role. Figure 10(a) shows the probability density function (PDF) of $\tau _c^{-1}$ for the three different regions shown in the right-hand column, depicting the spatial variation of time-averaged reaction rate $\overline {\dot {\omega }_c}$ in the mid $x$ – $z$ or $x$ – $y$ plane. It is clear that $\tau _c$ increases, and smaller time scales become less probable, with downstream distance, as shown in figure 10(a). Therefore, a multi-regime combustion occurs in this case (Langella et al. Reference Langella, Swaminathan and Pitz2016). Fast time scales (high frequencies) have lower spectral content since large reaction rates do not occur at downstream locations. This explains the influence of the spatially varying chemical time scale on the subsequent high-frequency ( $f\gt 3 \, \textrm {kHz}$ ) spectral decay rates seen in figure 7 for the BBF case. Similarly, zone S3 in figure 10 has the largest chemical time scale and subsequently highest spectral decay rate in figure 7 for the JICF case. In contrast, zone S1 has the lowest chemical time scale, and as a result it also has the lowest decay rate, while zone S2, which is on the leeward side of the jet exit, takes intermediate values. The PDFs of chemical time scales in the swirling flow cases are much narrower, and the distributions do not differ substantially between different spatial locations, as seen in figure 10(c) for the DLR-B flame. Similar behaviour is observed for other DLR flames listed in table 2. Therefore, the spectral decay rate of $\psi ^+_{\dot {q}}$ for the swirl-stabilised flames are quite similar across different locations.

In light of the results above, it is interesting to compare the characteristics of $\psi ^+_{\dot {q}}$ and $\psi ^+_{\dot {Q}}$ . The low-frequency spectral content in both these spectra shows a near constant trend below a characteristic frequency ( $f_p$ for $\psi ^+_{\dot {Q}}$ , and $f^*$ for $\psi ^+_{\dot {q}}$ ) for all cases. However, these characteristic frequencies differ considerably for all the cases except JICF. The characteristic frequency in the local spectra generally corresponds to a hydrodynamic instability, as discussed earlier in this subsection. The nearly flat spectral content at $f \lt f^*$ suggests that the local fluctuations are coherent at time scales close to those induced by the coherent motions of the hydrodynamic mode. The frequency $f_p$ for $\psi ^+_{\dot {Q}}$ depends on the convective scaling $U_b/L_f$ for all the cases except DLR-B (thermoacoustically unstable flame), as discussed in the previous subsection. The characteristic frequencies for the JICF case compare well between the normalised PSDs of local and global HRR. Since $f^*$ is dictated by the hydrodynamic modes of convective nature (short-wave elliptic instability and wake vortices) with large spatial extent, it is expected that the global PSD also would have similar characteristic frequency. In contrast, the swirl-stabilised flames are influenced locally by the PVC, which is not a convective mode aligned with the bulk flow. Therefore, one can expect a significant difference between $f_p$ and $f^*$ , for instance, $f_p = 510\,\rm Hz$ whereas $f^* = 1400\,\rm Hz$ for the DLR-A flame.

The high-frequency spectral decay rates show a vast difference between the normalised PSDs of local and global HRR. The spectral decay ranges from $f^{-1}$ to $f^{-6}$ for $\psi ^+_{\dot {q}}$ , whereas the spectral decay for $\psi ^+_{\dot {Q}}$ varies mostly as $f^{-5}$ . The difference in spectral decay rates between the global and local HRR can be assessed using the framework described in Lieuwen (Reference Lieuwen2012). According to this framework, the global HRR spectrum is related to the following physical factors: (i) the local HRR spectra, which depend on velocity fluctuations (Clavin & Siggia Reference Clavin and Siggia1991) and equivalence ratio fluctuations as in this work; (ii) spectral variation of correlation volume ( $V_{corr} (f )$ ); and (iii) convection of fluctuations due to tangential flow along the flame (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009; Lieuwen Reference Lieuwen2012). Therefore, the difference between the local and global HRR spectral decay rates must arise from the spectral characteristics of the correlation volume and/or the phase cancellation due to tangential convection of fluctuations along the flame front. Lieuwen (Reference Lieuwen2012) shows that the global and local heat release spectra are related to each other using the correlation length scale, also known as the coherence length scale (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009). The three-dimensional analogue of this relation is given by

(5.1) \begin{equation} \frac {\psi ^+_{\dot {Q}}}{V_F^2} \approx \begin{cases} \psi ^+_{\dot {q}}, & \mathcal {O} (V_{cor}) \sim \mathcal {O} (V_F), \\ 8\psi ^+_{\dot {q}}\,\dfrac {V_{cor} (f)}{V_F} + \mathcal {O}\left (\dfrac {V_{cor} (f)}{V_F}\right )^2, & \mathcal {O} (V_{cor}) \ll \mathcal {O}(V_F) , \end{cases} \end{equation}

where instead of correlation length, $l_{cor}$ , correlation volume $V_{cor}$ is used, and $V_F$ is the flame volume. Since $V_{cor}$ decays with increasing frequency (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009; Lieuwen Reference Lieuwen2012), it can be seen from (5.1) that at frequencies where $\mathcal {O} (V_{cor} )\ll \mathcal {O} (V_F )$ , the decay rate of $\psi ^+_{\dot {Q}}$ will exceed that of $\psi ^+_{\dot {q}}$ . The correlation volume is proportional to the product of the correlation length scales in the three directions (Wäsle et al. Reference Wäsle, Winkler and Sattelmayer2005). The correlation length scale in each direction can be computed by fitting the decay of the coherence to the following equation (Rajaram & Lieuwen Reference Rajaram and Lieuwen2009):

(5.2) \begin{equation} \gamma ^2\left (x,x+\Delta x,f\right ) = \exp \left (-\frac {\Delta x}{l_{cor}\left (f\right )}\right )^2, \end{equation}

where $\Delta x $ is the separation between an arbitrary point of interest within the flame region and the location of maximum heat release. The coherence $\gamma ^2$ is given by

(5.3) \begin{equation} \gamma ^2\left (x,x+\Delta x,f\right ) = \frac {\vert \check {\psi }_{\dot {q}}\left (x,x+\Delta x,f\right )\vert ^2}{\check {\psi }_{\dot {q}}\left (x,x,f\right )\,\check {\psi }_{\dot {q}}\left (x+\Delta x,x+\Delta x,f\right )}, \end{equation}

where $\check {\psi }_{\dot {q}}$ is the PSD of the transversely integrated HRR given by

(5.4) \begin{equation} \check {\psi }_{\dot {q}} = \int \int \psi _{\dot {q}} \, \mathrm {d}y\, \mathrm {d}z. \end{equation}

The correlation length scales in the other directions ( $y$ and $z$ ) are computed similarly.

The results for the spectrum of correlation volume for three representative cases are shown in figure 11(a). The correlation volume decays as frequency increases for all cases, with the highest decay rate for case DLR-B, which approaches $f^{-3}$ . The comparatively faster decay of $V_{cor}$ for the partially premixed cases is likely due to the mixture inhomogeneities. The slower decay rate of $V_{cor}$ for the JICF case is likely due to the presence of high-frequency hydrodynamic instabilities that maintain a large level of coherence and ensure strong mixing.

Figure 11. (a) The frequency spectrum of the correlation volume normalised by the flame volume $V_F$ . (b–e) Contours of cross-spectrum phase $\varphi _{\dot {q}}$ of transversely integrated HRR along $(y,z)$ for a synthetic signal and cases DLR-B, JICF and BBF. Grey contour lines denote the mark $\varphi _{\dot {q}} = \pi /2$ .

Additionally, consider the cross-spectrum phase of transversely integrated HRR, $\varphi _{\dot {q}} (\unicode{x1D6E5} ,f ) = \arg \{\check {\psi }_{\dot {q}} (x,x+\unicode{x1D6E5} ,f )\}$ , which is useful to discern the contribution of phase cancellation due to convective phenomena. The phase cancellation due to convection of fluctuation along the flame front also results in the decay of global HRR spectrum proportional to $f^{-2}$ at high frequencies. In the presence of convective phenomena, it is possible to identify a convective velocity or phase velocity $v_{\varphi }$ defined as

(5.5) \begin{equation} v_{\varphi } = \left \lvert \frac {\partial ^2\varphi _{\dot {q}}}{\partial f\, \partial \Delta x}\right \rvert ^{-1}. \end{equation}

Figure 11(b) shows the representative cross-spectrum for an arbitrary synthetic signal with $\varphi = (\pi /10 )f \unicode{x1D6E5}$ and a constant phase velocity $v_{\varphi } = 10/\pi \, {\text{m}\,\text{s}}^{-1}$ . The contours of the cross-spectrum phase for the three cases are shown in figures 11(c–e). It can be seen that the contours of $\varphi _{\dot {q}}$ for case DLR-B are qualitatively similar to figure 11(b), suggesting the presence of convective phenomena. However, there are at least two quantitative differences in the phase characteristics of the synthetic signal and case DLR-B. The phase velocities in the case DLR-B are not exactly constant throughout the $ (f,\Delta x )$ space (not shown), and unlike in the synthetic signal, $f=0$ is not an asymptote in the case DLR-B, possibly due to the inadequate low-frequency resolution of the LES. The other cases, namely JICF and BBF, do not show similar smooth hyperbolic contours and a smooth transition from in-phase oscillations ( $\varphi _{\dot {q}}\lt \pi /2$ ) to out-of-phase oscillations ( $\varphi _{\dot {q}}\gt \pi /2$ ) as $f$ and $\Delta x$ increase. Therefore, there are no discernible convective phenomena in these cases.

Figure 12. Instantaneous $\overline {\dot {\omega }_c}$ in the mid $x$ – $z$ plane, and the PSD of the area-integrated $\overline {\dot {\omega }_c}$ in the three regions marked for (a) BBF and (b) JICF flames.

The high-frequency spectral decay rate $\psi ^+_{\dot {Q}}$ for the partially premixed DLR case and premixed BBF case are identical despite the difference in decay rates of $V_{cor}$ . This is because the decay rate $\psi ^+_{\dot {q}}$ increases with increasing downstream distance, as discussed previously in figure 7, and this compensates for the higher level of coherence in the BBF case. This feature of the BBF is further exemplified by showing the normalised PSD of area-integrated $\overline {\dot {\omega }_c}$ in three different regions, as shown in figure 12(a). The higher level of coherence in the JICF flame at high frequencies results in a slightly lower spectral decay rate for $f/f_p\gt 1$ , as seen in figure 6(b). The lower spectral decay at high frequencies can also be due to finite rate chemistry effects of pure $\textrm {H}_2$ . A past study showed that the syngas mixtures had lower decay rates in comparison with syngas and methane blends due to finite-rate chemistry effects (Klein & Kok Reference Klein and Kok1999). Therefore, presence of methane should result in larger $\tau _c$ values. This is consistent with the results in figure 10, where $\tau _c$ for the JICF case, which operates on $\textrm {H}_2$ fuel, is lower compared to the BBF or DLR-B cases operating with pure methane fuel. Therefore, the presence of hydrogen and its inherent high burning rate may also contribute to the lower spectral decay rate of $\psi ^+_{\dot {Q}}$ for the JICF flame. The complementary roles played by chemical kinetics and hydrodynamics on $\psi ^+_{\dot {Q}}$ is further exemplified by computing the area-integrated $\overline {\dot {\omega }_c}$ for three different regions shown in figure 12(b). The area-integrated reaction rate spectra show different spectral decay rates for the three locations, as seen in figure 12(b). The region enclosed by the blue box is close to the shear layer roll-up, which occurs at approximately $27 \, \textrm {kHz}$ , hence the spectral decay rate is elevated in this region. To further understand the competing effects of chemical kinetics and hydrodynamics on the spectral decay rates requires studying the JICF flame with different fuels, such as methane, which is a future study.

In summary, the combination of decaying $V_{cor}$ spectrum and phase cancellation due to convection of perturbations along the flame front is responsible for the vast disparity in decay rates between the local and global HRR spectra for the DLR cases. While there is no convective phase cancellation in the JICF and BBF cases, the spatially decaying local HRR spectra along with the decay of the $V_{cor}$ spectra result in a similar disparity of decay rates between the local and global HRR spectra for these cases. It is also noted that the decay of the $V_{cor}$ spectra in the different cases is dependent on the presence of mixture inhomogeneities, the presence of hydrodynamic instabilities, and finite-rate chemistry effects.

5.3. Modelling aspects

Figure 13. Geometric mean normalised spectra of local HRR and velocity magnitude for the respective cases. The vertical red dotted line marks the characteristic frequency $f^*$ .

For modelling purposes, it is interesting to compare the normalised PSDs of HRR and velocity magnitude. For the sake of simplicity, it is important to consider a meaningful mean PSD that closely approximates the spatial variation of $\psi ^+_{\dot {q}}$ shown in figure 7. The DLR cases show negligible spatial variation and are hence insensitive to the way the mean PSD is obtained. However, the BBF and JICF flames showing a large spatial variation demand an appropriate way to calculate a representative mean PSD. An arithmetic mean may not be suitable for this purpose since it would heavily favour locations with large spectral content. Therefore, a geometric mean is used to obtain a representative PSD for the BBF and JICF flames.

The geometric mean of normalised $\psi ^+_{\dot {q}}$ and $\psi ^+_{\vert \boldsymbol{u} \vert }$ is shown in figure 13 for all the cases. This geometric mean is denoted using $\Psi$ with appropriate subscripts. The decay rate for $\Psi ^+_{\dot {q}}$ ranges from 1 to 3.5 for all cases, whereas the range is from 1.8 to 3.8 for $\Psi ^+_{\vert \boldsymbol{u} \vert }$ . The PSDs for the DLR-A flame remain nearly constant up to $f^* \approx 1400\,\rm Hz$ , which is identical to the PVC frequency. At $f\gt f^*$ , the spectra decays at the rates 1.2 and 1.8 for the HRR and velocity PSDs, respectively. In contrast to the DLR-A flame, the thermoacoustically unstable DLR-B flame has two characteristic frequencies – thermoacoustic frequency $f_T$ , and PVC frequency $f^*$ . The characteristic frequency for $\Psi ^+_{\vert \boldsymbol{u} \vert }$ is identical to the thermoacoustic frequency $f_T$ . It is surprising to note that the characteristic frequency for $\psi ^+_{\dot {Q}}(f_p)$ is equal to $f_T = 250 \, \textrm {Hz}$ (see figure 6 a), whereas it is equal to the PVC frequency $f^* = 400\,\rm Hz$ for $\Psi ^+_{\dot {q}}$ . These results can be explained using a past LES study focusing on the flame leading edge movement in this case (Chen et al. Reference Chen, Langella, Swaminathan, Stöhr, Meier and Kolla2019a ). It was shown that the azimuthal movement of the flame leading point is strongly correlated with the PVC frequency, suggesting that the local HRR would be strongly influenced by the PVC, resulting in a small peak at $f^*$ , as seen in figure 13 for the DLR-B flame. Even though the DLR-B flame has thermoacoustic oscillations, the local HRR – unlike the global HRR– is influenced both by local activity such as the local mixing, PVC, etc., and by the thermoacoustic oscillation. However, the velocity fields are substantially modulated only by the thermoacoustic oscillation, hence the high-frequency spectral decay rates for HRR and velocity magnitude PSDs are more disparate in the DLR-B flame compared to other cases. Furthermore, the cumulative spectral content for $f\gt f_T$ has to compensate for the accumulation of spectral content at $f=f_T$ due to the thermoacoustic oscillation resulting in a higher spectral decay rate for $\Psi _{\vert \boldsymbol{u}\vert }$ .

The DLR-C flame shows a trend similar to that of the DLR-A flame in figure 13, with slightly larger decay rates $1.6$ and $2.5$ for the HRR and velocity magnitude PSDs, respectively. The characteristic frequency $f^*$ for the DLR-C flame is equal to the PVC frequency $400\,\rm Hz$ . In the BBF case, the spectral decay rates for $\Psi ^+_{\dot {q}}$ and $\Psi ^+_{\vert \boldsymbol{u} \vert }$ in the approximate range $1000 \lessapprox f \lessapprox 3000\,\rm Hz$ are $1.6$ and $1.9$ , respectively, as seen in figure 13. At approximately $3 \, \textrm {kHz}$ , the spectral decay rate increases to $3.3$ for HRR and $3.8$ for $\Psi ^+_{\vert \boldsymbol{u}\vert }$ . The increase in the decay rate is due to the increase in $\tau _c$ as discussed previously. The JICF PSD shows a characteristic frequency $f^* \approx 3500\,\rm Hz$ that corresponds to the combined wake-vortex and elliptic instability mode of the CVP. The spectral decay rate is identical for the HRR and velocity magnitude PSDs, and is equal to $1.3$ in the approximate range $3.5 \lessapprox f \lessapprox 27 \, \textrm {kHz}$ , and $3.5$ for $f \gtrapprox 27 \, \textrm {kHz}$ . The shear layer vortex roll-up results in a narrow-band peak at approximately $27 \, \textrm {kHz}$ as discussed previously in figure 12.

Figure 14. Comparison of model spectrum (Hirsch et al. Reference Hirsch, Wäsle, Winkler and Sattelmayer2007) for $\Psi ^+_{\dot {q}}$ with LES results for the DLR flames and BBF case. The vertical red dotted line denotes $f^*$ .

A key observation from figure 13 is that the spectral decay rates of HRR and velocity magnitude are not the same, and it is consistently lower for the former for all cases except JICF. This observation highlights the limitations of existing models (Hirsch et al. Reference Hirsch, Wäsle, Winkler and Sattelmayer2007) for HRR spectra deduced using the velocity spectra. Furthermore, the velocity spectra do not exactly follow the Kolmogorov $-5/3$ law ( $-5/2$ in frequency space; Clavin & Siggia Reference Clavin and Siggia1991) for such complex cases, as is often assumed in existing models.

Before proceeding further it is necessary to gauge the performance of existing models, such as that proposed by Hirsch et al. (Reference Hirsch, Wäsle, Winkler and Sattelmayer2007), which is described in Appendix A. The quantity $\Psi ^+_{\dot {q}}$ estimated using this model for the DLR flames and BBF case are shown in figure 14 alongside the velocity magnitude PSD. Two variations of this model are shown, where $\Psi ^{+H1}_{\dot {q}}$ is the original model (Hirsch et al. Reference Hirsch, Wäsle, Winkler and Sattelmayer2007) and $\Psi ^{+H2}_{\dot {q}}$ uses the LES velocity spectrum instead of the one-dimensional model spectrum by Tennekes & Lumley (Reference Tennekes and Lumley1972). The modelled $\Psi ^{+H1}_{\dot {q}}$ shows severe inaccuracies in the low, characteristic and high frequency characteristics for all cases. The modified model ( $\Psi ^{+H2}_{\dot {q}}$ ) is identical to the velocity spectrum due to normalisation. While the modified spectrum performs better than $\Psi ^{+H1}_{\dot {q}}$ in capturing the qualitative trends, it does not account for the disparity in the spectral decay rates between $\Psi ^+_{\vert \boldsymbol{u} \vert }$ and $\Psi ^+_{\dot {q}}$ discussed previously in this subsection. Another conceptual limitation of this model is the assumption that the scalar dissipation rate of reactive progress variable can be modelled using the linear relaxation model used for passive scalars (Pierce & Moin Reference Pierce and Moin1998). Several past studies have shown that this assumption is problematic in turbulent premixed flames where the flame dilatation has a significant contribution to the scalar dissipation rate that the linear relaxation model fails to capture (Swaminathan & Bray Reference Swaminathan and Bray2005; Dunstan et al. Reference Dunstan, Minamoto, Chakraborty and Swaminathan2013). While the model is strictly applicable only to premixed flames, an approximation for laminar flame speed is obtained based on the global equivalence ratio ( $\phi _g$ ) for the partially premixed swirl stabilised flames. Such a model is also inapplicable in cases where the global equivalence ratio is well below the lower flammability limit, as in the case of the JICF flame. The values of the various parameters involved are listed in table 3 for all cases shown in figure 14.

Table 3. Description and values of parameters in Hirsch’s model (Hirsch et al. Reference Hirsch, Wäsle, Winkler and Sattelmayer2007) for cases DLR-A, B, C and BBF.

To account for the difference in spectral decay rates between velocity magnitude and HRR PSD, the following model is proposed in this work:

(5.6) \begin{equation} \Psi ^{+}_{\dot {q}} \approx \Psi ^{+}_{\vert \boldsymbol {u}\vert } \left (1+ \left (\frac {f}{f^*}\right )^l\right )^{1/2}, \end{equation}

where $f^*$ and $l$ are the characteristic frequency for $\Psi ^{+}_{\dot {q}}$ and scaling exponent, respectively. This model is considerably simpler compared to that in Hirsch et al. (Reference Hirsch, Wäsle, Winkler and Sattelmayer2007), with only two parameters, and depends on $\Psi ^+_{\vert \boldsymbol{u} \vert }$ obtained from reacting flow LES.

Figure 15. Comparison of computed (see figure 13) and modelled (see (5.6)) $\Psi ^+_{\dot {q}}$ for the respective cases.

Figure 16. (a) Comparison of mean reacting (R) and non-reacting (NR) $\Psi ^+_{\vert \boldsymbol{u} \vert }$ , and (b) comparison of LES results and model (5.6), using normalised NR velocity magnitude PSD for DLR-C, BBF and JICF. Here, AM means arithmetic mean, and GM means geometric mean.

The comparison of this modelled and computed $\Psi ^+_{\dot {q}}$ is shown in figure 15. The model compares well with the LES spectrum for all the cases. Since no clear unstable hydrodynamic mode was identifiable in the BBF case, the characteristic frequency is chosen in the range $100\lt f\lt 300\,\rm Hz$ as discussed previously, and weak sensitivity of the PSD is seen for the choice of $f^*$ . This sensitivity is seen only for $f \leq 200\,\rm Hz$ for the BBF case. The scaling exponent is $l=2$ except in the DLR-B and JICF flames. The higher scaling exponent for the DLR-B flame is because of its thermoacoustic instability, which resulted in a larger disparity in the decay rates between $\Psi ^+_{\dot {q}}$ and $\Psi ^+_{\vert \boldsymbol{u} \vert }$ as seen in figure 13. The scaling exponent is much lower and negative for the JICF flame. This is because of the high-frequency dynamics associated with hydrodynamic instabilities, which results in nearly identical decay rates for $\Psi ^+_{\dot {q}}$ and $\Psi ^+_{\vert \boldsymbol{u} \vert }$ as seen in figure 13. The HRR is not an easy quantity to measure in experiments, but $\psi ^+_{\vert \boldsymbol{u} \vert }$ may be measured relatively easily. Hence (5.6) can conveniently model $\Psi ^+_{\dot {q}}$ . This model has several advantages over that suggested by Hirsch et al. (Reference Hirsch, Wäsle, Winkler and Sattelmayer2007). First, this model uses reacting velocity spectra instead of Kolmogorov’s model spectra for homogeneous isotropic turbulence, which is most often inapplicable in practical flows. Second, the model captures well the considerable difference in the spectral decay rates seen for all the cases. Third, this model is physically motivated using the characteristic frequencies associated with hydrodynamic modes for aerodynamically stabilised flames. Finally, this model is fairly robust across the cases and uses only one parameter, the exponent $l$ . A limitation of this model, however, is its reliance on computationally expensive reacting flow simulations to obtain the velocity magnitude PSD $\Psi ^+_{\vert \boldsymbol{u} \vert }$ . It is therefore interesting to investigate whether the non-reacting flow is sufficient to model the HRR spectra. Figur e 16(a) compares the reacting and non-reacting $\Psi ^+_{\vert \boldsymbol{u}\vert }$ for three representative cases (DLR-C, BBF and JICF). The results for DLR-A are expected to be qualitatively similar to the DLR-C flame. This model is inappropriate for case DLR-B since the non-reacting flow is inherently missing the two-way coupling between flame and acoustics necessary for this unstable case. The reacting flow-based model works well, as shown in figure 15, since this coupling is present in the reacting flow velocity spectra. The non-reacting $\Psi ^+_{\vert \boldsymbol{u}\vert }$ is obtained as the geometric mean of $\psi ^+_{\vert \boldsymbol{u}\vert }$ gathered for ten probe locations in the respective non-reacting cases. The peak at approximately $400\,\rm Hz$ in figure 16(a) for the DLR-C case corresponds to the PVC frequency, which is nearly identical between the reacting and non-reacting flows. In the case of BBF, the reacting and non-reacting counterparts are identical up to $3 \, \textrm {kHz}$ when using the geometric mean. The comparisons improve using the arithmetic mean, which tends to favour higher spectral content of the upstream locations. The discrepancy between the arithmetic and geometric means arises because the velocity spectra decay at a faster rate with downstream distance for the non-reacting case compared to the reacting case. This is possibly a result of flame dilatation increasing the spectral content in the reacting case (Bilger Reference Bilger2004; Kolla et al. Reference Kolla, Hawkes, Kerstein, Swaminathan and Chen2014). The reacting and non-reacting $\Psi ^+_{\vert \boldsymbol{u} \vert }$ spectra for the partially premixed cases are not affected much by this effect (Bilger Reference Bilger2004; Knaus & Pantano Reference Knaus and Pantano2009). The reacting and non-reacting spectra are similar in their broad-band nature and the high-frequency range for the JICF case. The shear layer roll-up is dominant in both reacting and non-reacting conditions, resulting in the local narrow-band peak. The interaction between the flame, wake vortices and the short-wave instability of the CVP is strong enough to result in a peak at $f \approx 3500 \, \textrm {Hz}$ in the reacting velocity spectra for this case. However, the absence of such an interaction in the non-reacting flow at these time scales does not result in a prominent narrow-band peak in the local non-reacting $\Psi ^+_{\vert \boldsymbol{u} \vert }$ . These results suggest that the $\Psi ^+_{\vert \boldsymbol{u} \vert }$ from non-reacting flow represents $\Psi ^+_{\dot {q}}$ based on (5.6), although some differences may be expected.

Figure 16(b) compares $\Psi ^+_{\dot {q}}$ obtained using (5.6) with non-reacting flow $\Psi ^+_{\vert \boldsymbol{u}\vert }$ from the corresponding LES for the three cases discussed previously. The comparisons are reasonable for DLR-C, and the spectra show a negligible sensitivity to the method of obtaining the mean. The results for the other DLR flames is also expected to be similar to the DLR-C flame. Figure 16(b) for BBF shows that the arithmetic mean captures the HRR spectra with improved accuracy especially at very high frequencies ( $f\gt 2\, \textrm {kHz}$ ), for reasons discussed previously. The comparisons for the JICF case are reasonable with discrepancies in the low-frequency regime ( $f\lt 10 \, \textrm {kHz}$ ) and at the narrow-band peak at $f \approx 27 \, \textrm {kHz}$ . The discrepancy in the narrow-band peak is because of the stronger peak in the velocity spectra compared to the HRR spectra. The discrepancy in the low-frequency range is due to the differences seen in the comparisons between the normalised reacting and non-reacting velocity magnitude PSDs in figure 16(a) for the JICF. The comparisons are improved significantly by using the velocity spectra from a location away from the shear layer roll-up (point 2 in figure 3). The improvements are obtained because of the lower spectral content at high frequencies ( $f\gt 20 \, \textrm {kHz}$ ) for this location, which lowers the discrepancy in the narrow-band peak at $f \approx 27 \, \textrm {kHz}$ observed previously. To summarise, the non-reacting velocity spectra, but not the Kolmogorov spectra, are sufficient to model the HRR spectra using (5.6) since the parameters $f^*$ and $l$ remain approximately unchanged between the reacting and non-reacting flows as they are related to the hydrodynamics.