Large Eddy Simulation of a Bluff Body Stabilised Premixed Flame Using Flamelets

The conservation equations for mass, momentum and total enthalpy (energy) are solved along with two additional equations for combustion. The Favre-filtered form of these equations are used because of strong density variations in the flow arising from the heat release from combustion. The additional equations are for the Favre-filtered reaction progress variable and its SGS variance. These five equations are written as

Mass:

Momentum:

Total enthalpy:

Reaction progress variable:

SGS variance:

The density and velocity are denoted using ρ and U respectively, and the total enthalpy is the sum of the sensible and chemical enthalpies, which is denoted as h. The notation D/Dt = ∂/∂t + U_j∂/∂x_j denotes the substantial derivative. The progress variable, c, can be defined using either temperature or species mass fractions. For this study, the CO and CO₂ mass fractions are used, and hence the progress variable is defined as \(c = ({ Y }_{\mathrm { CO } } + { Y }_{{ \mathrm { CO } }_{2 } } ) / { ({ Y }_{\mathrm { CO } } + { Y }_{{ \mathrm { CO } }_{2 } } ) }_{b }\), where the subscript b denotes the burnt mixture value. This takes a value of zero and unity in the unburnt and burnt mixtures respectively. The symbols α and \(\mathcal { D }\) respectively denote the molecular thermal diffusivity of the mixture and the mass diffusivity of c. The SGS stresses, denoted by the last term of Eq. 2, are modelled using the eddy-viscosity concept [9], where the dynamic Smagorinsky approach [21, 22] is used to model the sub-grid eddy viscosity, ν_T. The sub-grid scalar fluxes in Eqs. 3 to 5 are modelled using gradient hypotheses, where the turbulent Schmidt number, Sc_T, is calculated dynamically [22]. The filtered reaction rate, \(\overline { \dot { \omega } }\), in Eq. 4 and the reaction-related source terms in Eq. 5 for the SGS variance, \({ \sigma }_{c , \mathrm { sgs } }^{2 }\), achieve closure by the combustion modelling described in Section 3.2. The sub-grid scalar dissipation rate, \({ \widetilde { \chi } }_{c , \mathrm { sgs } }\), is influenced by both combustion and turbulence in premixed flames and thus, its modelling should include contributions from these two processes. One such model was developed [23] and tested [24‐26] in past studies and this model is briefly discussed next, along with the combustion modelling used for this study. The importance of transporting the SGS variance and the closure models used within its transport equation have been discussed in previous studies [11, 24] and the behaviour of the SGS variance in bluff body flames is discussed in [11].

The combustion modelling used here is based on the unstrained premixed flamelet concept. The flamelet concept assumes that the flame is thin, so that turbulence cannot affect its inner structure. This allows the thermochemistry to be decoupled from turbulence and the flame can be seen as a series of one-dimensional thin structures, known as flamelets. The profile of the reaction progress variable for each flamelet is identical to the profile of a laminar flame. The wrinkling of the flame is accounted for by a presumed PDF, where the progress variable and its SGS variance are used to track the reaction. Numerous previous studies have tested this approach, which include laboratory scale flames with Reynolds-Averaged Navier-Stokes (RANS) [27‐30] and unsteady RANS (URANS) methodologies for laboratory scale flames [31] and for a practical burner [32, 33]; this model for LES has been developed and tested in [11, 24, 34]. The model is discussed briefly here and a more detailed description can be found in the references cited above.

The filtered reaction rate is given by where \(\widetilde { P } (\zeta ; \widetilde { c } , { \sigma }_{c,\text {sgs}}^{2 } )\) is the density-weighted sub-grid PDF of the reaction progress variable, and its sample space variable is denoted ζ. The flamelet reaction rate and mixture density are denoted using \(\dot {\omega } \left (\zeta \right )\) and ρ respectively. The shape of the sub-grid PDF is assigned using a beta function for given values of c and \({ \sigma }_{c, \mathrm { sgs } }^{2 }\), which are obtained from their respective transport equations, Eqs. 4 and 5.

The modelling of the reaction related term in the SGS variance equation follows a similar procedure and is written as

The above two sources (integrals) are precomputed using an unstrained premixed laminar flame calculation and are tabulated as a function of \(\widetilde { c }\) and \({ \sigma }_{c, \mathrm { sgs } }^{2 }\), which is used in the LES. The sub-grid dissipation rate, which is the fourth term in Eq. 5, is modelled using an algebraic closure [23], which has been tested thoroughly in past studies [11, 24, 35]. This expression is where the function \(\mathcal { F } = 1 - \exp (- 0.75 { {\Delta } }^{+ } )\) ensures that the SGS dissipation rate goes to zero when the filter width approaches zero. The normalised filter width is Δ⁺ = Δ/δ_th, where the laminar flame thickness is δ_th = 0.55 mm, and the laminar flame speed is s_L = 0.26 m/s for the methane-air mixture used in the test case described earlier. The thermochemical parameter is K_c = 0.79τ, where τ is the temperature rise across the flamelet, which is normalised by the unburnt mixture temperature, T_u. The other parameters are defined as \({ C }_{3 } = 1.5 \sqrt { { \mathrm { Ka } }_{{\Delta } } } / (1 + \sqrt { { \mathrm { Ka } }_{{\Delta } } } )\), C₄ = 1.1/(1 + Ka_Δ)^0.4, where \({ \mathrm { Ka } }_{{\Delta } }= { ({ u }_{{\Delta } }^{\prime } / { s }_{L } ) }^{3/2 } { ({ \delta }_{\mathrm { th } } / {\Delta } ) }^{1/2 }\), and \({ \mathrm { Da } }_{{\Delta } } = { s }_{L } {\Delta } / ({ u }_{{\Delta } }^{\prime } { \delta }_{\mathrm { th } } )\). The symbol \({ u }_{{\Delta } }^{\prime }\) is a SGS velocity scale requiring a closure and is modelled using \({ u }_{{\Delta } }^{\prime } = { C }_{q } { \sum }_{j } \vert \widetilde { U }_{j } - \widehat { \widetilde { U } }_{j } \vert \) [36], where \(\widehat { \widetilde { U } }_{j }\) is the velocity field using a Gaussian test filter during the LES. The test filter width is \(\widehat { {\Delta } } \simeq 2 {\Delta }\), where the LES filter width is estimated as \({\Delta } = { \mathcal { V } }^{1/3 }\), with \(\mathcal { V }\) being the volume of the computational cell.

It has been established in past studies that the above parameters and their values for Eq. 8 are closely connected to certain physical aspects of the scalar dissipation rate transport and elaborate detail can be found in [23, 37, 38]. The term \({ \sigma }_{c ,\mathrm { sgs } }^{2 } / { \beta }_{c }\) is related to the influence of flame curvature, which is induced by wrinkling. The parameter β_c is therefore scale dependent, which is evaluated using a dynamic approach described in [25, 35]. Thus, there is no adjustable parameter in this modelling approach, but for the sake of comparison, a static value of β_c = 0.4 is also used for this study.

The Favre-filtered temperature is obtained from the computed total enthalpy according to \(\widetilde { T } = { T }_{0 } + (\widetilde { h } - { \widetilde { {\Delta } h } }_{f }^{0 } ) / { \widetilde { c } }_{p }\), where \({ \widetilde { {\Delta } h } }_{f }^{0 }\) and \({ \widetilde { c } }_{p }\) represent the formation enthalpy and specific heat capacity respectively at constant pressure of the gas mixture, and the reference temperature is T₀ = 298.15 K. The mixture density is computed as \(\overline { \rho } = \overline { p } \widetilde { M } / { \Re }^{0 } \widetilde { T }\), where \(\widetilde { M }\) represents the Favre-filtered mixture molecular mass and R⁰ is the universal gas constant. The three thermochemical quantities for the mixture, \({ \widetilde { {\Delta } h } }_{f }^{0 }\), \({ \widetilde { c } }_{p }\) and \(\widetilde { M }\), are calculated in a manner similar to Eq. 6, described in detail in [39]. These quantities, along with the reaction-related source terms, are tabulated as a function of \(\widetilde { c }\) and \({ \sigma }_{c, \mathrm { sgs } }^{2 }\). This look-up table has 101 and 51 evenly distributed points for \(\widetilde { c }\) and σc,sgs2 respectively for this study. The flamelet is computed using the PREMIX code [40] and the GRI-Mech 3.0 chemical kinetic mechanism for methane-air combustion.

The computational model of the open bluff body burner, discussed in Section 2, is shown in Fig. 1b. As seen in this figure, the computational grid starts at roughly 70 mm upstream of the bluff body base, where a flat velocity profile is prescribed at the inlet to give the required mass flow rate through the burner. This mass flow rate gives the reference bulk mean velocity, U_b, at the base of the bluff body, as marked in Fig. 1a. A cylindrical region is added downstream from the bluff body base, as illustrated in Fig. 1b, to represent the boundary entraining the atmospheric air. A small velocity of U_air = 0.1 m/s is specified at the boundary, marked as “Coflow” in Fig. 1b, to mimic the ambient air entrainment around the bluff body. Adiabatic no-slip wall conditions are imposed on the pipe walls and bluff body, following the previous numerical study [11]. The cylindrical boundary is specified to be a slip wall and for the outlet, the streamwise gradient of all the variables is set to zero.

The variables \(\widetilde { c }\) and \({\sigma }_{c, \text { sgs}}^{2}\) are set to be zero for both the inlet and coflow boundaries and the enthalpies for these boundaries are set to be consistent with their temperature and composition. The effect of the entrainment is captured by transporting a passive fluid marker, \(\widetilde { Z }\), which is set to be unity in the methane-air stream and zero for ambient air. Following earlier studies [24, 34], the thermochemical property, \(\widetilde { \varphi }\), of the mixture is determined using a mixing rule \({ \widetilde { \varphi } }_{\mathrm { mix } } = \widetilde { Z } { \widetilde { \varphi } }_{\mathrm { reac } } + (1 - \widetilde { Z } ) { \varphi }_{ \mathrm { air } }\), where the subscripts “reac” and “air” denote the values of \(\widetilde { \varphi }\) taken from the look-up table and air respectively.

A block structured computational grid is used to discretise the computational volume shown in Fig. 1b. The grid consists of approximately 3.6 million hexahedral cells in total, with refinement near the bluff body and in the regions where the shear layers and filtered flames are expected to be present. The minimum cell size in these regions is around 0.2 mm. The wall boundary layers are resolved by placing two cells within the viscous sub-layer, as recommended in a previous study [11].

The code, PRECISE-MB [41], used for this work solves the reacting flow equations along with the combustion modelling, described in Sections 3.1 and 3.2 respectively, using the finite volume methodology. The spatial gradients are calculated using second-order accurate central difference schemes [42]. The discretised equations are time-advanced using a second-order scheme. The Courant-Friedrichs-Lewy (CFL) number is kept below 0.3 by specifying a constant time step of 5 μ s. The velocity and pressure coupling is maintained using the SIMPLEC algorithm [43].

All the simulations reported in the next section are run using the Cambridge High Performance Computing Cluster, Darwin. Each node of this cluster has two 2.6GHz eight-core Sandy Bridge E5-2670 processors. The simulations were run using 96cores, which required 36hrs of wall clock time for a simulation over a period of sixteen flow through times. The flow through time is defined as ℓ/U_b, where ℓ is the reference length and is taken as ℓ = 150mm. The time-averaged statistics are obtained using the samples collected over the latter half of the simulation time, which are discussed in the next section.

Figure 2 compares the computed and measured axial variation of the streamwise velocity along the centreline. The time-averaged value is normalised using the reference velocity, U_b, and the axial distance is normalised using the bluff body diameter, D. The time-averaged velocity is also averaged azimuthally because of the axisymmetric nature of the averaged flow features and the averaged quantities are denoted using angle brackets in the following discussion; this approach is also used for the flame results, which are discussed in the next subsection. The velocity measurements were obtained using PIV [1]. As shown in this figure, the comparison between the measured and computed values is good. The negative values imply the reverse flow within the recirculation zone and thus, the length of the recirculation zone is given by the x distance of the zero crossing of the normalised velocity. The computed value is 1.15D, which agrees well with the measured value of roughly 1.22D.

The computed radial variations of the time-averaged axial velocity, normalised using U_b, is compared to the measurements in Fig. 3 for three streamwise locations. The comparison seen here is good and suggests that the salient features, such as shear layers and peak velocity values of the flow, are well caputured in the computations. The broad peak seen for the location x/D = 0.2 corresponds to the annular jet region, implying there are two shear layers, namely the inner and outer shear layers. The width of the recirculation zone changes from roughly 0.5D at x/D = 0.2 to around 0.4D at x/D = 0.8. The change in the radial variation with the streamwise distance shows that the width of the shear layers are increasing with x. These variations are captured well in the computations, as seen in Fig. 3. Specifically, the recirculation zone size is directly influenced by the turbulence level near the bluff body base because this zone is established by the momentum diffusion caused by the turbulent diffusivity. It is essential to accurately capture the averaged velocity and turbulence statistics variations near the bluff body base, as they control the recirculation zone attributes that help the flame’s stabilisation. As noted in Section 2, the turbulence is shear driven and it is expected that the root mean square (rms) value of the axial velocity fluctuations, u^′, will be larger in the regions with large \(\partial \langle \widetilde { U } \rangle / \partial r\), which can be clearly seen in Figs. 3 and 4. This figure compares the computed radial variations of u^′, which is normalised using U_b, with measurements for the three axial locations shown in Fig. 3. Both the positions and magnitudes of the two local peaks of u^′/U_b at the location x/D = 0.2 are captured well in the computations. There is an overall increase in the axial rms velocity further downstream, as seen in Fig. 4b, because of the increase in shear driven turbulence, but there is some small under prediction in the computation for this location. The local peaks are less defined as the shear layers become thicker with axial distance because of turbulent diffusion. The measured values for the location x/D = 1.6 are well captured in the computation, as seen in Fig. 4c, and this location is outside the recirculation zone (beyond the downstream stagnation point). The comparisions shown between the computed and measured statistics for isothermal flow suggest that the numerical grid used and the computational model is good and this set-up is used for reacting flow, which is discussed next.

It has been shown that the flame and flow features are well captured by the LES, showing good comparisons with the measurements to a similar standard reported in previous numerical studies using different combustion models [18‐20]. In general, the magnitudes of u^′ and v^′ increase with axial distance from the bluff body base, suggesting that the turbulent kinetic energy production increases with axial distance. The turbulent kinetic energy at the axial locations x/D = 0.2, 0.8 and 1.6 analysed so far is computed using both the experimental data and LES. Since w^′ is not available from the measurements [1], the turbulent kinetic energy is estimated as 〈k〉_exp = 0.5(u^′2 + 2v^′2) by assuming that w^′≃ v^′. This is compared to the turbulent kinetic energy from the LES results, computed as 〈k〉_res = 0.5(u^′2 + v^′2 + w^′2), using only the resolved velocities, and also the total kinetic energy 〈k〉_tot = 〈k〉_res + 〈k〉_sgs by including the SGS kinetic energy \({ k }_{\mathrm { sgs } } = 3 { { u }_{{\Delta } }^{\prime } }^{2 } / 2\). These variations are shown in Fig. 12, where the symbols are for 〈k〉_exp, the black line represents 〈k〉_tot and the dash-dotted black line is for 〈k〉_res. It is worth noting that the PIV measurements may lack the velocity fluctuations that are smaller than the size of the interrogation region, if the interrogation window is larger than several times the Kolmogorov length.¹ Grey lines represent the shear layer edges originating from the bluff body and red lines denote the flame brush, which is marked using \(\langle \widetilde { c }\rangle = 0.1\) and 0.9. The edges of the shear layer are drawn using the contours at 10% of the local peaks of \(\partial \langle \widetilde { U } \rangle / \partial r\).

It is seen that the sharp peak of the total kinetic energy at x/D = 0.2 is located within the inner shear layer, where the flame brush is also located. It should be noted that the other sharp peak is within the outer shear layer and is not considered for the analysis here, as it is outside the flame brush. Since this peak is not present for 〈k〉_res, this means that there are some flame generated velocity fluctuations in this region. These fluctuations can come from two sources; one is flame generated turbulence and the other one is due to intermittent effects from the flame. The current framework for the analysis makes it challenging to ascertain these two mechanisms and to identify a dominant one; this will be explored in a future study. Since the numerical grid used here resolves more than 80% of the turbulence, the SGS turbulence is expected to be small and flame induced effects would be strong, as noted above. Nevertheless, the trends are the same for both 〈k〉_res and 〈k〉_tot. Furthermore, it should be noted that w^′≃ v^′ is assumed for the experimental data and the validity of this approximation is an open question. At x/D = 0.8 and 1.6, it is observed that the peak value of 〈k〉_res is within the shear layer and decays from the product side to the reactant side, for the burner configuration used here. There is a slight increase close to the reactant side for these two locations, which is caused by the outer shear layer resulting from ambient air entrainment. Also, it is observed that there is some interaction between the flame and inner shear layer. Although the entire shear layer is within the flame brush near the flame stabilisation region, this layer moves out of the flame gradually when moving downstream. The contour of \(\langle \widetilde { c } \rangle = 0.9\) moves from the inner to outer side of the shear layer starting from the bluff body base to x/D = 1, as shown in Fig. 12. This contour then moves back into the shear layer in a similar manner to the outer part of the shear layer. On the other hand, the \(\langle \widetilde { c } \rangle = 0.1\) contour continuously moves further away from the inner shear layer from x/D = 0.6 because the reactant stream is unconfined. This relative movement of the shear layer and the flame brush will lead to an unexpected behaviour of turbulent kinetic energy across the flame brush for various downstream positions.

A previous DNS study [44] reported that the turbulent kinetic energy peaks within the flame for the corrugated flamelet regime, but increases across the flame from the reactant to product side of the flame for the thin reaction zones regime. These behaviours are seen in the variations of 〈k〉_tot with \(\langle \widetilde { c } \rangle \) and are depicted in Fig. 13. It is seen that the peak turbulent kinetic energy is at the centre of the flame for x/D = 0.1 and then shifts towards the product side for x/D = 0.2 and x/D = 0.4. However, there is no peak for the location x/D = 0.8, which suggests that there is a transition in the combustion regimes in the region 0.2 < x/D < 0.8. The flame is burning within the corrugated flamelet regime close to the bluff body, which is shown by the flame wrinkling caused by large scale eddies resulting from the Kelvin-Helmholtz instability, and this thin layer structure is seen in Fig. 5. Hence, it could be said that the flame near the stabilisation region is quasi-steady and quasi-laminar. This claim is also supported by the high values of the parameter \(\widehat { \mathrm { Da } }_{{\Delta } }\) (indicating stronger reaction rates) in those regions, as observed in Fig. 5.

To further investigate this quasi-laminar flame observation, the variation of the peak averaged reaction rate, \(\langle \overline { \dot { \omega } } \rangle \), and the flame brush thickness, \({ \delta }_{T } = 1 / { \vert \partial \langle \widetilde { c } \rangle / \partial r \vert }_{\mathrm { max } }\), are studied. If the flame is truly laminar, then the peak reaction rate should scale as ρ_us_L/δ_th, as laminar flame theory suggests. Therefore, the peak averaged reaction rate, normalised using this scaling, should be of order unity if the flame is quasi-laminar, and δ_T/δ_th must also be of order unity. These two normalised quantities are plotted in Fig. 14 as a function of x/D. The strain thinning of the filtered flame, caused by large scale eddies near the flame base, yields the normalised reaction rate to be larger than unity. Furthermore, it is shown that δ_T/δ_th < 1 for x/D ≤ 0.2, which supports the quasi-laminar flame observation.

Figures 15 and 16 show the radial variation of 〈k〉_res, normalised by \({ s }_{L }^{2}\), for three axial locations inside the recirculation zones of confined bluff body stabilised flames; this configuration has been investigated in an earlier study [11]. These two flames are denoted as CF02 and CF22 for the following discussion. For this study, both of these flames have an equivalence ratio of ϕ = 0.8, which is close to the value used for flame A1, as discussed previously. Flame CF02 has a turbulence intensity of approximately u^′/U_b = 2% at the base of the bluff body, whereas flame CF22 has a turbulence intensity of approximately u^′/U_b = 22%. The lengths of the recirculation zones are 1.37D and 0.73D for flames CF02 and CF22 respectively, where it is seen that these recirculation zone lengths are significantly shorter than the length of 2.02D observed for flame A1. As noted in Section 1, the aim of the following discussion is to distinguish the behaviour between flame A1 (open flame) and flames CF02 and CF22 (confined flames). Flame A1 has only shear driven turbulence, with no incoming turbulence, but the other two flames have both shear driven and incoming turbulence.

It is clear that the relative locations of the shear layers and flame brush are very similar near the bluff body for these three flames, as seen when comparing Figs. 12, 15 and 16. In the low turbulence level case in Fig. 15, the initial evolution of the shear layer is similar to that observed for the flame A1 in Fig. 12. However, the \(\langle \widetilde { c } \rangle = 0.9\) contour moves completely out of the shear layer at around x/D = 1.18 which is approximately 86% of the recirculation zone length. It can be suggested that beyond this point, the flame does not experience any shear generated turbulence. This is not the case with flame A1, as the absence of walls allows the flame to remain within the shear layer, whereas the shear layer is pushed further inwards for the confined flames. The turbulent kinetic energy increases across the flame, since the combustion is in the corrugated flamelets regime, as noted by Langella et al. [11]. This behaviour of the turbulent kinetic energy is consistent with many past DNS studies using statistically planar premixed flames.

When the incoming turbulence intensity is increased to 22%, it is shown that part of the shear layer always remains within the flame for the entire recirculation zone length, as illustrated in Fig. 16, which is also the case for flame A1. However, flame CF22 experiences more shear generated turbulence than CF02, as the flame brush and shear layer are thicker, and hence there is more of an overlap for flame CF22. The increasing trend of the turbulent kinetic energy across the contour 〈c〉 = 0.1 for flame A1 is due to the outer shear layer resulting from the air entrainment. The flame stabilisation mechanisms (the relative positions of the shear layer and the flame brush in the vicinity off the bluff body base) are similar for both confined flames as the shear layer is pushed inwards very close to the bluff body from approximately x/D = 0.1 and x/D = 0.05 for flames CF02 and CF22 respectively. This is not the case for flame A1, as both the product and reactant sides of the flame are well aligned with the shear layer until x/D = 0.2. The reactant side of the flame remains aligned with the outer part of the shear layer from this point until x/D = 0.6, as seen in Fig. 12. Thus, the presence of the walls close to the bluff body does affect the relative positions of the flame and shear layer. Although a strained flamelet model may capture this stabilisation with some improved accuracy, the framework of the unstrained flamelet model used here has previously given a better overall performance compared to a strained flamelet model [24] and for a bluff body flame [11].