# Investigation of Harmonic Response in Non-Premixed Swirling Combustion to Low-Frequency Acoustic Excitations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{f}). Low-frequency disturbance induced notable responses, while high-frequency oscillations had minimal influence. Using the flame describing function, Pan [10] studied nonlinear heat release response, finding that flame length was sensitive to excitation frequency and amplitude, and the flame exhibited low-pass filtering characteristics. Palies [6,11] highlighted the interference between vortex shedding and flame angle oscillations, which affected local gain peaks and valleys in the vortex–flame transfer function. At high-frequency excitations, the flame–vortex interactions at the flame tip led to phase-shifted heat release oscillations, resulting in a weak global flame response.

## 2. Experimental Setup

_{4}) fuel, which mixes with the air in the annular duct through 12 symmetrical orifices located in the burner head. Each orifice has a diameter of 1.5 mm. The swirling flow is generated by the 12 axial swirler blades positioned at an angle of $\beta =45\xb0$. The estimated swirl number is $\mathrm{S}\text{}\approx \text{}\frac{2}{3}tan\beta \text{}\approx \text{}0.67$. The combustion chamber has dimensions of 300 mm in length and 80 mm in width. In addition, the air tube has an internal diameter of 38 mm, and the fuel tube has an external diameter of 14 mm. For ease of analysis, a three-dimensional coordinate system was established with the center of the burner inlet ring as the origin. In this coordinate system, the z-axis was aligned with the axis of the bluff body, while the x- and y-axes were in the plane of the inlet ring.

## 3. Numerical Approach

#### 3.1. LES Model

_{i}is the velocity components in the x

_{i}direction; ρ is density; p is pressure; and ν is the kinematic viscosity. Quantities with overbars are spatially filtered. In the governing equations, unknown terms appear as a result of spatial filtering. These unknown terms account for the effect of the filtered turbulence scales; hence, they are known as the residual stress tensor τ

_{ij}. According to the Boussinesq assumption,

_{sgs}is the subgrid-scale viscosity and $\stackrel{-}{Sij}$ is the filtered strain-rate tensor for the resolved scale. To fully describe the governing equations, the subgrid-scale viscosity, υ

_{sgs}, is modeled using the wall-adapting local eddy-viscosity (WALE) model [24]. This model eliminates the need for additional wall or damping functions, making it suitable for turbulent transition regimes. The subgrid-scale viscosity of the model is defined as follows:

_{sgs}is the subgrid-scale kinetic energy, and S

_{ij}

^{d}is the velocity gradient tensor. C

_{k}= 0.094 and C

_{W}= 0.325, which are constants.

#### 3.2. Computational Domain

^{6}cells in the computational domain. To ensure simulation accuracy, the mesh was refined in areas such as the swirler, nozzle, and shear layers while gradually coarsening downstream to improve computational efficiency. The minimum mesh size near the wall was set to 0.07 mm, ensuring a y+ value less than 1.0 to accurately capture flow details near the wall [25].

_{0}, which was obtained from a precursor RANS simulation employing the realizable k-ε model [26]:

_{0}value. Celik et al. [27] suggested that in order to resolve 80% of the turbulent kinetic energy using LES, approximately 5 or more cells over the integral length scale l

_{0}would be required. The ratio between the turbulent integral length scale l

_{0}and the cubic root of the cell volume Δ is characterized by the defined parameter L

_{grid}:

_{grid}, where values exceeding 5 had been clipped so that the well-resolved areas did not appear, and the not-so-well-resolved regions could be identified easily. The most critical regions for mesh refinement were represented by the inner and outer shear layers at the burner outlet. It could be observed that the mesh met the computational requirements for LES in almost all regions. Therefore, this mesh was used for further calculations.

#### 3.3. Numerical Setup Details

_{in}and V

_{0}represent the inlet velocities with and without acoustic excitation, respectively. Additionally, A and f denote the amplitude and frequency of the excitation, respectively. In this study, a uniform excitation amplitude of 0.3 was employed, while the excitation frequency varied among three values: 20 Hz, 50 Hz, and 150 Hz.

^{−5}s. This approach guaranteed that the global Courant number remained below the threshold of 0.4 at all times.

#### 3.4. Description of Monitoring Points

#### 3.5. Validation

## 4. Proper Orthogonal Decomposition (POD) Method

_{i}is the temporal coefficient of mode i, and φ characterizes the spatial distribution features of mode i. The data sets are chronologically organized to form a matrix as follows:

_{i}corresponds to the one-dimensional matrix of data at time i. By employing the singular value decomposition (SVD) method [31], matrix A can be decomposed as follows:

_{n}signifies the nth-order spatial mode, v

_{n}represents the nth-order time series, and σ

_{i}denotes the eigenvalue corresponding to each mode. These eigenvalues are indicative of the proportion of energy carried by each mode within the system. Since the eigenvalues obtained by SVD are arranged in descending order, the modes obtained using the POD method are ranked according to their energy. An analysis of the higher-energy modes reveals the main variations in the data.

## 5. Results and Discussion

#### 5.1. Flame Response

_{com}), as well as the pulsation characteristics of acoustic pressure and heat release rate. To reveal the dynamics of the forced flame, additional POD analysis was performed on the flame images to explore their spatial structural characteristics and local spectral features.

_{com}was observed with increasing excitation frequencies, measured at 18.2 mm, 17.1 mm, and 16.0 mm, respectively. The flame Strouhal number (St

_{f}), based on the H

_{com}and the mean inlet jet velocity U

_{j}(1.105 m/s), was calculated to be 0.329, 0.774, and 2.172. The St

_{f}represents the number of perturbations induced by acoustic excitations along the H

_{com}and relates the flame oscillation frequency to the upstream fluid dynamics [32,33].

_{i}and the spatial structures φ

_{i}obtained from the POD analysis are presented in Figure 7.

_{com}: two regions for the 50 Hz excitation and three regions for the 150 Hz excitation.

#### 5.2. Flow Dynamics

#### 5.2.1. Spatial Response Characteristics

^{5}was used here to describe the vortex structures. Additionally, the influence of the vortex structures on the reactant mixing process was assessed by color-coding the vortex structures according to the equivalence ratio.

#### 5.2.2. Axial Velocity Spectra in the Shear Layer

#### 5.2.3. Vortex Motion Characteristics

_{com}) measured under the respective excitation conditions experimentally, and the black solid line shows the expected flame shape under stoichiometric fraction. Significant differences in vortex generation and axial convective velocity were observed under the influence of harmonic responses to different excitation frequencies. Under 150 Hz excitation, a single vortex was formed at the inlet that underwent complete evolution, including growth, detachment, and downstream propagation. The coexistence of multiple vortices downstream of the nozzle corresponded to folds in the local equivalence ratio distribution. The expected flame height was relatively stable during the excitation period. However, under 50 Hz excitation, a larger outer vortex and fragmented smaller vortices were discernible within the excitation period. This phenomenon could be attributed to the interference of the 100 Hz harmonic response on the flow field’s reaction to the 50 Hz excitation. The motion of the larger outer vortex aligned with the fragmentation of the equivalence ratio field, and a fracture even appeared in the expected flame shape (2/8T). Under 20 Hz excitation, the vortex structure within the period became more intricate, indicating the presence of stronger harmonic responses. Compared to 50 Hz, the axial stretching and fragmentation of the equivalence ratio field became more pronounced, leading to a more complex, expected flame shape. To analyze the vortex convective transport process, the axial convective wavelength λ [46] was defined as the axial height at which the vortex propagated downstream within one excitation period. The axial convective wavelengths of the vortex under different excitations are annotated in Figure 11. By calculating the ratio of the excitation period t to the axial convective wavelength λ, the axial convective velocity U

_{e}of the vortex could be determined, which played a crucial role in calculating the flame delay time. The vortex axial convective velocity decreased with increasing excitation frequencies, with respective values of 0.803 m/s, 0.783 m/s, and 0.496 m/s. Further calculations for the ratio of vortex axial convective velocity to the mean inlet jet velocity U

_{j}(1.105 m/s) revealed that under excitation frequencies of 20 Hz, the axial vortex convective velocity is slightly less than the jet velocity. This supported the assumption that the vortex convective velocity was often regarded as the jet velocity [6,47]. Future research is necessary to explore the influence of flame thermal expansion on vortex convective velocity under reacting conditions. Specific values can be found in Table 3.

_{com}corresponding to the 150 Hz excitation, the coexistence of three vortices was observed for the majority of the time; similarly, the coexistence of two vortices was predominant for the H

_{com}corresponding to the 50 Hz excitation. The number of coexisting vortices at these excitation frequencies was consistent with the number of oscillation regions revealed by the first-mode POD results. As for the 20 Hz excitation, the observation predominantly showed one vortex existing within the H

_{com}, which was consistent with the frequency-flashing mode of the first-mode POD results. The comparative analysis between LES and POD results revealed that the oscillation modes in non-premixed combustion were primarily influenced by the spatial distribution characteristics of vortices within the shear layers. In the non-reacting flows, the vortices within the shear layers under acoustic excitations exhibited motion patterns that are susceptible to harmonic interference, with the strength of this harmonic motion becoming more pronounced as the excitation frequency decreased. These acoustically induced vortices affected the mixing process of reactants, potentially driving the harmonic response in the flame’s heat release rate.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic representation of the burner (Green arrow represents the mixed flow). (

**b**) Three-dimensional view of the burner.

**Figure 2.**(

**a**) Computational domain mesh. (

**b**) Contour of mesh quality according to the criterion (Equation (10)).

**Figure 4.**Temporal evolution of vortices obtained from LES (left) and schlieren imaging (right) under 50 Hz excitation.

**Figure 5.**The time-averaged distributions of CH* chemiluminescence under (

**a**) 20 Hz, (

**b**) 50 Hz, and (

**c**) 150 Hz excitations.

**Figure 6.**Spectral plots of acoustic pressure and heat release rate under (

**a**) 20 Hz, (

**b**) 50 Hz, and (

**c**) 150 Hz excitations.

**Figure 7.**First four POD modes of the CH* images of the flame under (

**a**) 20 Hz, (

**b**) 50 Hz, and (

**c**) 150 Hz excitations.

**Figure 8.**Spatial response characteristics of the flow field under (

**a**) 20 Hz, (

**b**) 50 Hz, and (

**c**) 150 Hz excitations.

**Figure 9.**Axial velocity spectra of the monitoring points in the OSL and ISL under (

**a**) 20 Hz, (

**b**) 50 Hz, and (

**c**) 150 Hz excitations.

**Figure 10.**Evolution of axial velocity and acceleration at point A equivalence ratio at point B3 and C3, and vorticity at points D and E under (

**a**) 20 Hz, (

**b**) 50 Hz, and (

**c**) 150 Hz excitations. (

**d**) Spectral plot of acceleration at point A.

**Figure 11.**Evolution of vorticity and equivalence ratio under (

**a**) 20 Hz, (

**b**) 50 Hz, and (

**c**) 150 Hz excitations.

Point | X/mm | Y/mm | Z/mm |
---|---|---|---|

A | −14.5 | 0 | 0 |

B1 | −20.5 | 0 | 5 |

B2 | −25.5 | 0 | 15 |

B3 | −32.5 | 0 | 25 |

C1 | −12.5 | 0 | 5 |

C2 | −18 | 0 | 15 |

C3 | −24.5 | 0 | 25 |

D | −19 | 0 | 0.5 |

E | −10 | 0 | 0.5 |

f_{excited} (Hz) | f_{response} (Hz) | Normalized Amplitude (×100%) | |||||
---|---|---|---|---|---|---|---|

B1 | B2 | B3 | C1 | C2 | C3 | ||

20 | 10 | 1.74 | 6.41 | 13.77 | 14.03 | 11.61 | 12.86 |

20 | 100.00 | 110.69 | 36.64 | 109.69 | 90.75 | 75.13 | |

40 | 30.36 | 46.45 | 18.18 | 29.27 | 15.30 | 43.76 | |

60 | 5.90 | 12.21 | 17.29 | 21.25 | 6.24 | 42.13 | |

80 | 13.08 | 20.56 | 9.31 | 53.53 | 21.14 | 4.15 | |

100 | 10.44 | 16.28 | 2.31 | 41.53 | 13.81 | 8.27 | |

120 | 7.36 | 3.65 | 0.64 | 15.26 | 10.29 | 5.50 | |

50 | 25 | 4.76 | 2.55 | 9.50 | 13.57 | 7.46 | 12.68 |

50 | 100.00 | 33.41 | 42.48 | 70.78 | 48.19 | 33.31 | |

100 | 19.02 | 28.13 | 5.21 | 8.48 | 21.91 | 1.38 | |

150 | 75 | 6.56 | 7.78 | 3.65 | 4.06 | 16.91 | 25.19 |

150 | 100.00 | 55.21 | 28.03 | 73.59 | 42.96 | 25.50 | |

300 | 21.19 | 1.53 | 0.37 | 1.83 | 0.64 | 0.86 |

f_{excited} (Hz) | 20 | 50 | 150 |

λ (mm) | 40.163 | 15.656 | 3.309 |

t (s) | 0.050 | 0.020 | 0.007 |

U_{e} (m/s) | 0.803 | 0.783 | 0.496 |

U_{e}/U_{j} | 0.726 | 0.708 | 0.449 |

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## Share and Cite

**MDPI and ACS Style**

Bao, J.; Ji, C.; Pan, D.; Zong, C.; Zhang, Z.; Zhu, T.
Investigation of Harmonic Response in Non-Premixed Swirling Combustion to Low-Frequency Acoustic Excitations. *Aerospace* **2023**, *10*, 812.
https://doi.org/10.3390/aerospace10090812

**AMA Style**

Bao J, Ji C, Pan D, Zong C, Zhang Z, Zhu T.
Investigation of Harmonic Response in Non-Premixed Swirling Combustion to Low-Frequency Acoustic Excitations. *Aerospace*. 2023; 10(9):812.
https://doi.org/10.3390/aerospace10090812

**Chicago/Turabian Style**

Bao, Jinrong, Chenzhen Ji, Deng Pan, Chao Zong, Ziyang Zhang, and Tong Zhu.
2023. "Investigation of Harmonic Response in Non-Premixed Swirling Combustion to Low-Frequency Acoustic Excitations" *Aerospace* 10, no. 9: 812.
https://doi.org/10.3390/aerospace10090812