#### Abstract

In order to investigate effects of apex angle (*α*) on chemically reacting turbulent flow and thermal fields in a channel with a bluff body V-gutter flame holder, a numerical study has been carried out in this paper. With a basic geometry used in a previous experimental study, the apex angle was varied from 45° to 150°. Eddy dissipation concept (EDC) combustion model was used for air and propane premixed flame. LES-Smagorinsky model was selected for turbulence. The gird-dependent learning and numerical model verification were done. Both nonreactive and reactive conditions were analyzed and compared. The results show that as *α* increases, recirculation zone becomes bigger, and Strouhal number increases a little in nonreactive cases while decreases a little in reactive cases, and the increase of *α* makes the flame shape wider, which will increase the chamber volume heat release ratio and enhance the flame stability.

#### 1. Introduction

Flames can only be stabilized in high-velocity reactant streams over a certain range of conditions, while a variety of approaches are used to stabilize flames in a combustor, for example, bluff body flame holders such that V gutters are widely used in many modern combustion devices, such as augmenters or after burners in turbojet/turbofan engines and ramjet engines [1]. Combustors with bluff-body flame holders are characterized by a shear layer where vortices are shed due to Kelvin-Helmholtz instability [2], and this shear layer separates the region of high-speed fresh mixtures from the wake region of lower-speed hot products, and due to the boundary layer separation, a V-gutter bluff body creates a recirculation zone that acts as an ignition source for incoming fuel/air mixture by recycling hot products and radicals from the burned mixture.

Both experimental and numerical studies on bluff body flame have been done over years. Fujii et al. [3] and Yue et al. [4] carried out experiments to study nonreactive flow without reaction characteristics behind a flame stabilizer. Nakamura [5] studied vortex-shedding frequency of bluff bodies with different after-body shape, his study showed that Strouhal number increases as the ratio of afterbody length to cross-flow dimension of the bluff body is increased. Sanquer et al. [6] investigated the chemically reacting flow characteristics behind V gutters. Cuppoletti et al. [6] measured the high-frequency combustion instabilities with a radial V gutter. Sjunnesson et al. [7, 8] used LDA and CARS measuring the velocities and turbulence in a bluff body stabilized flame and describe the rig in detail and report preliminary computations of the flow field using a two-step global mechanism solved online with Arrhenius expressions in conjunction with the Magnusen-Hjerthager combustion model. Siewert [9] studied the methane/air premixed flames at high pressure using PIV and OH PILF. In numerical field, Eriksson [10] carried out numerical study of different RANS turbulence models efferent on rig VR-1 flame using CFD software CFX based on Zimont Turbulent Flame Closure (TFC) model, but they did not discuss LES model and vortex shedding. Engdar et al. [11] apply the Level-Set Flamelet Library approach in conjunction with varying 2-equation turbulence models, flame speed models, and flame thickness models. 50 species and 402 elementary reactions were used to build the flamelet library. Giacomazzi et al. [12] used the fractal model as SGS model of LES for turbulence and EDC combustion model to compare experiment dates of VR-1 with LES-FM, RANS k-*ε*, and LES-Smagorinsky model. Wang [13], Fureby [14], and Porumbel and Menon[15] had also performed LES computations on VR-1. Erickson and Soteriou [16] carried out numerical study on reactant temperature effect of a two-dimensional triangular bluff body flame. Park and Ko [17] introduced a G equation to Les subgrid scale combustion model for turbulent premixed flame stabilized by the bluff body.

Although direct numerical simulation (DNS) is able to resolve all turbulence scales, its computational resource requirements are prohibited for practical applications. In recent year, LES and hybrid RANS/LES approaches received great attentions among researchers as it can capture more transient turbulence structures than RANS for flame holder studies as discussed above. In order to investigate the bluff apex angle effects on flow field, 2D numerical study has been carried out using CFD software FLUENT 6.3.26 in this paper, EDC flame model which assumes that chemical reactions take place only at the dissipative scales of turbulence is used for premixed flame simulation. In turbulence, Smagorinsky subgrid LES model is applied.

#### 2. Theoretical Fundamentals of the EDC Model

Most fuels are fast burning, and the overall rate of reaction is controlled by turbulent mixing. In premixed flames, the turbulence slowly convects/mixes cold reactants and hot products into the reaction zones, where reaction occurs rapidly. In such cases, the combustion is said to be mixing limited, and the complex chemical kinetic rates can be safely neglected. Both physical and chemical processes are important in simulating combustion systems. The formation of turbulent structures is a physical process, for which the concept of the eddy cascade model is the basis [19]. The largest turbulent structures have a size, which is of the magnitude of the system’s dimensions. By interaction among themselves, eddies dissipate to smaller ones. The smallest eddies have an extension: where is the Kolmogorov length scale. At these scales, dissipation of turbulent kinetic energy takes place with a rate where is the kinematic viscosity. At a size smaller than no turbulent structures exist, because in those regions molecular diffusion is faster than turbulent transport. From this thought the conception of the EDC was developed. There is a range in which the reactions can be regarded as ideally mixed. Thus, chemical reaction kinetics determines the speed of the process. while outside this range the reactants are not mixed and do not react, it assumes that reaction occurs in small turbulent structures, called the fine scales. The length fraction of the fine scales is modeled as

The volume fraction of the fine scales is calculated as . Species are assumed to react in the fine structures over a time scale: with .

The species transport equation (conservation equation) is written as

The computation of the reaction rate source terms is accelerated with the in-situ adaptive tabulation (ISAT) algorithm [19] embodied in the solver. For simplicity, the propane-air combustion was simulated using a one-step irreversible chemical reaction:

#### 3. Test Case Description and Computational Details

##### 3.1. Case Geometry and Boundary Conditions

Volvo Aero Corporation (VAC) Triangular Bluff Body Stabilized Combustion rig VR-1 [8] has been extensively researched both in terms of experiments and theoretical treatment, and one case of VR-1 has been used in this paper. The flow field size is shown in Figure 1. The air and propane mixture is introduced at an equivalence ratio , the air mass flow rate is 0.6 kg/s, the inlet temperature is 288 K, and the mixture velocity at channel inlet is about 17.2 m/s. The nominal pressure is 1 atm. The Reynolds number based on the bluff-body dimension and the inlet velocity is nearly 50.000.

The governing equations are solved using CFD software ANSYS FLUENT 6.3.2.6. The mass flow inlet and pressure outlet boundary conditions are used. The mass flow rate, total temperature, and species mass fractions are assumed fixed while pressure was extrapolated at inlet. Outlet pressure was 1 atm. No slip and zero normal velocity conditions are imposed at the side and bluff body wall. The flow field of different apex angle is solved in 2D and also been compared with 3D result at . The finite-volume scheme is implicit, second order in time, and second order upwind in space. The integration time step of LES nonreactive field is set to 1e-04s, and 8e-05s for reactive field. In this paper LES-Smagorinsky model has been used. Both nonreactive and reactive flow fields have been calculated and compared. When staring the reactive calculation, the nonreactive result is used as an initialization value. In LES simulation when the flow filed becomes stable, 20 characteristic periods of the flow are sampled for plotting time-mean results.

##### 3.2. Grid-Dependent Learning and Verification

It is important and necessary to carry out grid-dependent study for determining the optimal grid with a good balance of accuracy and speed; on the other hand, LES model needs more grid than RANS model, especially near the walls. The hexahedral grids ranged from approximately 30.950 cells to 71.267 cells are created by ICEM 11.0; the grids near bluff wall are refined which ensures the + values no larger than 2. Three grid size options are shown in Table 1, and the temperature distribution of different grid size cases at cm is shown in Figure 2. After examining the results with different grid sizes, it is found that the optimal grid size is approximately 71.267 cells.

Although there are many studies on flow field using 2D LES problem such as flow past a cylinder or premixed bluff body flame, and 2D LES method could also provide important results as 3D model for basic research, it is important to find out dissimilarity between 2D and 3D models in order to understand difference between numerical simulation and actual result better. So the comparison between 2D and 3D (about 800.000 cells) model has been carried out in this section. From Figure 2 it can be seen that, at low temperature zone, the numerical result agrees well with experiment date, but both 2D and 3D maximum temperatures are little higher than experiment result; this is mainly because endothermic radicals reactions are missing when using one-step reaction and subgrid scale model for LES as discussed in [12]. Figure 3 shows comparisons in recirculation region ( cm); it can be seen that the distributions agree well with experiment dates both in nonreactive and reactive case, but the reverse flow velocity is little higher using 2D model both nonreactive and reactive case. In reactive case, the curve shape is narrower than 3D model and experimental result.

**(a) Nonreactive case**

**(b) Reactive case**

#### 4. Results and Analysis

##### 4.1. Nonreactive Case

Figure 4 shows the mean streamwise velocity () distribution along the center line. It is known that a recirculation region is formed because of vortex shedding; the mean recirculation zone is defined as the region where the mean streamwise velocity is negative (). From Figure 4 it is found that as *α* is increased, the mean length of recirculation zone increases, so does the backflow velocity magnitude. Figure 5 shows the distribution at cm (this location is in recirculation region), and as *α* is increased, the width of curve shape is increased within m~0.02 m; this means that the width of recirculation zone is increased. So according to Figures 4 and 5, it can be concluded that when *α* is increased, the recirculation zone area is increased and the backflow strength is enhanced.

From Figure 4, it can be seen that there are two critical points near mm and cm (marked by dash line), which divide distribution into two parts: before mm it is recirculation zone, and as *α* increases the backflow velocity is increased after mm and before cm, as *α* increases; the of 150° is bigger than other cases, but when cm, is decreased as the flow develops, and 150° is lower than other cases, this is because the recirculation zone will make the transverse flow losses increase. The bigger recirculation zone, the more flow losses, and the flow losses will affect the streamwise velocity and make of 150° that is the lowest case after cm.

Figure 6 shows the time-average streamlines after the bluff body, and it can be seen that when *α* is increased, the recirculation area increases. The increases of *α* (decrease of base angle) mean that the angle between flow direction and base edge of bluff is increased; two limit conditions can be assumed: *α* tends to 0° and *α* tends to 180°; when *α* tends to 0°, the flow mechanism near bluff base edge may correspond to the flow around a long square with angle of attack (AOA) close to 0°, and when *α* tends to 180°, the flow mechanism near bluff base edge may correspond to the flow around a very thin plate with AOA close to 0°; when the fluid flow past a thin plate, the rebound effects of wall are much stronger than long square, so the recirculation zone becomes bigger as Figure 7 shows.

In Figure 8 Strouhal number results in nonreactive and reactive are given. The Strouhal number is defined as
where cm, m/s, and is the vortex shedding frequency, which is calculated through fast Fourier transform. For nonreactive case (), St-2D is 0.252 and 3D is 0.249 (experiment result is 0.25 [12]); for reactive case, St-2D is 0.37 and 3D is 0.364. It can be found that as *α* increases, St increases in nonreactive cases but is almost unchanged in reactive cases. According to many scholars results about flow past body [20, 21] that St will be increased when Re is higher, higher Re means a bigger recirculation zone, so it can be learned that if Re is unchanged, a bluff body which has a bigger recirculation zone that will have a higher St. As Figure 6(a) shows when *α* is increased from 60° to 90°, the recirculation zone length increases more than 45° to 60°, so the sudden increase of St of nonreactive case is easily explicable.

##### 4.2. Reactive Case

The reacting simulations were carried out using the same numerical methodology and mesh utilized for the nonreacting simulation. Therefore, any changes in fluid dynamics can be attributed to the heat release from the combustion model described earlier. The dynamics of the reactive flow can be interpreted as due to coupling between vortex shedding and excitation of acoustic oscillations produced by heat release [18]. Comparing the nonreacting case and the reactive case, it is evident that combustion extended the recirculation region further downstream as Figure 9 shows. It was observed that in the instantaneous flow fields the asymmetric von shedding of coherent vortices no longer exists in the reacting case. Therefore, the reacting case does not possess any large-scale mixing mechanisms which typically reduce the recirculation zone length [18]. Mehta and Soteriou [22] attributed this absence of asymmetric von Karman shedding for reacting conditions to the dilatation effect of the heat release. They also found that baroclinic vorticity augments the impact of dilatation, but its effect is secondary. The recirculation zone is longer and broader in the reactive case, meaning a more gradual dissipation of momentum in the wake region with respect to the nonreactive case, and St is higher than nonreactive as Figure 8 shows.

Three typical locations are selected in order to discuss the temperature distribution against coordinate after the bluff; the three typical locations are cm (in the recirculation zone), 15 cm (after recirculation zone), and 35 cm (near the outlet). Figure 10(a) shows the temperature distribution along the centerline at different *α*, it is found that case 90° has a large temperature fluctuation, the minimum mean temperature is about 1780 K, and the maximum temperature is close to 2050 K. Figures 10(b)–10(d) show the temperature distribution of three typical locations at different location. It can be found from Figure 10(b) that as *α* increases, the maximum temperature is almost unchanged, but the temperature distribution curve becomes wider and 150° is wider than other cases in recirculation zone; the wither curve means that the reaction area in *Y* directions is more bigger. This is because the recirculation zone becomes bigger as *α* increases, the turbulence and flow pulsation are enhanced (see Section 4.1) which will make the combustion more efficiently. As the flow and reaction develop towards outlet, 150° is also the widest case (Figures 10(c) and 10(d)). When near the outlet the maximum temperature of 150° is little lower than other cases, this is because the streamwise velocity becomes lower which makes the turbulence fluctuation weak, and the flame temperature decreases near the outlet as *α* is increased.

**(a) Centerline**

(b) cm |

(c) cm |

(d) cm |

The curve shape difference means different flame shape. Figure 11 gives the mean static temperature contour at different *α*, and Figure 12 summarizes the comparison between three typical conditions of flame shape ( K). There is a minimum width position of the flame shape, and it can be seen that *α* has a strong effect on temperature distribution and flame shape: as *α* is increased, the minimum width of flame shape increases and the heat release increases. This means that, at the same inlet condition, the bigger *α*, the higher combustion chamber volume heat release rate (VHRR). VHRR is equal to the total release heat to burning chamber volume. A higher VHRR indicates that, to release the equal heat, smaller combustion chamber is needed, which is very important for shortening the length of chamber in aircraft using a bluff body to stable the flame [23]. On the other hand, the effects of the turbulence are generally advantageous for the efficiency of the combustion since turbulence enhances the mixing of component chemical species and heat, but adverse effects upon combustion can also occur if the turbulence level is sufficiently high to create flame extinction. In turn, combustion may enhance the turbulence through dilatation and buoyancy effects caused by the heat release; the flame temperature is lower, the heat release is reduced which will make the flame unstable. According to Kim et al. [18] ’s study, in a flame lean blow-out process the temperature distribution discontinuity first happens at the position where the flame is narrowest as Figure 13 shows, so it can be conjectured that the wider flame is more stable than narrow flame, so is more stable than lower *α* cases.

**(a)**

**(b)**

#### 5. Conclusions

LES simulation based on Smagorinsky subgrid scale and EDC combustion model of a premixed bluff body flame is carried out in this paper; different apex angle effects on both nonreactive and reactive flow field have been discussed. The results obtained may be summarized as follows.(1)The recirculation zone area increases as apex angle is increased.(2)Combustion of the fluid could extend the recirculation region length. When apex angle is increased, St increases in nonreactive case, but decreases a little in reactive case. St of reactive case is higher than nonreactive case.(3)A bigger apex angle makes the flame shape wider, increases chamber volume heat release, and enhances the flame stability.

#### Acknowledgments

This work was supported by Scientific Research Fund of Hunan Provincial Education Department under Grant no. 12B023. It is also supported by National Natural Science Foundation of China (nos. 61204039 and 61106029) and Scientific Research Foundation for Returned Scholars, Ministry of Human Resources, and Social Security of the People’s Republic of China.