Numerical investigation of thermoacoustic instability in a model afterburner with a simplified model for observed lock-in phenomena

Muthaiah Manickam, Indian Institute of Technology-Madras, Chennai-India memuthaiah@gmail.com

Ragul Senthilkumar 1

Indian Institute of Technology-Madras, Chennai-India

Dr. VarunKumar S 2

Indian Institute of Technology-Madras, Chennai-India

ABSTRACT Thermoacoustic oscillations in a gas-turbine afterburner are numerically investigated using CFD. A simplified 2-dimensional axisymmetric afterburner with bluff-body stabilized flame is considered in the investigation. Occurrences of high-frequency thermo-acoustic oscillations in the afterburner chamber is observed at specific fuel flow rates. The flow field from the CFD shows the bluff-body vortex shedding frequency to excite the acoustics of the afterburner chamber during the thermo-acoustic oscillations. The synchronization of the bluff-body wake with the acoustics of the domain results in thermoacoustic coupling. The Proper Orthogonal Decomposition of the flow field revealed the presence of chamber acoustics in the pressure field confirming the coupling. Then a simplified coupled oscillator model based on the van-der Pol and simple harmonic oscillators is attempted to reproduce the observed resonant oscillations. The oscillator model qualitatively captures the observed resonant oscillations in the chamber. This model could be extended to combustors with bluff-body wake in predicting thermoacoustic oscillations

1. INTRODUCTION

At certain operating conditions, large-amplitude pressure oscillations are encountered in some combustion systems. These oscillations are referred to as thermoacoustic instability. As the name implies, these oscillations arise due to the interaction between the heat release and acoustics of the combustion chamber. When the unsteady processes of fluctuations in the heat release rate and the chamber acoustics are in phase, the energy in the acoustic field of the chamber could increase. It

1 ragulsk002@gmail.com

2 varuns@iitm.ac.in

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was first mathematically described by Lord Rayleigh [1]. Many applications ranging from liquid rocket engines, and aircraft gas turbines to industrial burners and furnaces suffer from the sudden occurrence of this instability. These oscillations result in unwanted vibrations/noise to catastrophic failure of the system or its components. Thus analysis and prediction of the instability gain importance. Numerous studies starting from the 1960s have focused on these thermo-acoustic oscillations. These high-pressure oscillations can be classified as low frequency/rumble and high-frequency/screech oscillations. A large body of work is available for low-frequency oscillations with a widely established theory for prediction [2]–[4]. However, the high-frequency counterpart has received relatively low attention, and the theory for prediction is still open. The complexity in the analysis of high-frequency oscillations is due to the participation of transverse modes, hydrodynamics, and non-compact heat release. Accounting for all of these in an analysis/investigation poses great challenges due to the multiscale nature of the processes that constitute the thermoacoustic loop. A review of various acoustic-hydrodynamic-acoustic mechanisms resulting in thermoacoustic oscillations can be found in [5].

Flame holders are used in stabilizing and sustaining the flame in high-speed air streams. They play an essential role in aerospace applications such as afterburners and ramjets. The unsteady wake flame is known to be a part of thermo-acoustic coupling leading to high-frequency oscillations [6]. Thus dynamics of the flame holder stabilized flames have been the focus of high-frequency screech research [7]–[11].

Many experimental studies were reported on the lab-scale setups with a flame holder placed in a rectangular or cylindrical duct [8], [9], [12]. Experiments with premixed inlet conditions showed high amplitude screech oscillations [8], [12]. Studies accounting for variations in inlet density ratios showed differing wake oscillations with a strong dependence on inlet density[13]. Convectively unstable mode with the varicose mode of oscillation is observed for high-density ratio, and absolutely unstable mode with sinuous oscillations for low-density ratio wakes [13].

This study investigates unsteady thermoacoustic oscillations in an axisymmetric afterburner setup with two bluff-body stabilized flames using CFD. High temperature vitiated inlet conditions encountered in the practical afterburner inlet conditions are also accounted for. The fuel injection rates are varied, with the other boundary conditions kept unchanged. The flow field is analyzed for the high-pressure resonant oscillations to identify the thermoacoustic coupling. In addition, the flow field is decomposed with the proper orthogonal decomposition (POD) technique to identify the dynamics of the spatio-temporal structures in the flow field. Then a representative simplified oscillator model based on the Van der Pol and a simple harmonic oscillator is developed. In the next sections, the geometric details of the afterburner setup, computational domain, CFD results and the POD, and the attempt on the simplified model are described.

2. GEOMETRY, COMPUTATIONAL DOMAIN & NUMERICS The 2D afterburner setup used in the present study is shown in Figure 1. The afterburner chamber consists of a jet pipe, flame holder arrangement, a convergent-divergent nozzle, and an extended domain into the atmosphere. The geometry is a simplified version of the afterburner setup based on [14]. The flame-holders are placed about 0.2 m from the inlet of the afterburner at a radius of 0.2 m and 0.32 m, respectively. The flame holders are wedge type having a width of 0.04

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m and a double angle of 30 o . An extended region with an atmosphere after the exhaust nozzle is included in the 2D CFD domain.

Figure 1: Afterburner Schematic

A fully structured grid is made of hexagonal elements with 80k cells (with a specified empty direction for 2D calculations in OpenFOAM). The computational domain is extended beyond the nozzle exit to ensure that the acoustic waves are transmitted away without any reflections. Considering main gas turbine exhaust conditions, vitiated gas at 1273 k with a mass flow rate of 60 kg/s is selected as the inlet condition. Vitiated gas composition on a mass basis is taken as 15% O 2 , 10% CO 2 , 5% H 2 O, and 70 % N 2 based on [9]. The static pressure at the inlet is taken as 2 bar, and the pressure at the exit boundary is considered 50 kPa simulating altitude conditions at 5000 m. Walls are modeled as no-slip walls with adiabatic boundaries. The outlet is modeled as the non-reflecting boundary with wave transmissive boundary conditions. Kerosene fuel is injected 0.1 m upstream of flame holders and tracked in the flow domain as Lagrangian particles. A particle diameter of 100 µm is used, and it is injected at 300 K. Six cases with varying fuel injection rates are studied. These injection rates used are 0.36, 0.72, 0.9, 1.26, 1.8, and 2.0 kg/s giving overall equivalence ratios of 0.1, 0.21, 0.27, 0.375, 0.54 and 0.59 respectively. Hereon these cases are referred to as cases c1 to c6.

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3. RESULTS AND DISCUSSIONS

3.1. Unsteady Results

Pressure oscillations measured by the probes from all 6 CFD calculations are studied for signatures of high-pressure screech oscillations. The compilation of pressure probe data from probe P1 and the corresponding Fourier spectrums are shown in Figures 2 &3. Figure 2 shows an apparent growth in amplitude and stretching in the time period of the pressure oscillations. Cases c5 and c6 show a “beating” phenomenon, i.e., low-frequency modulation of amplitudes of high-frequency oscillations. Beating is generally observed during the presence of 2 closely spaced frequencies or

during a system forced close to its natural frequency. Here, the beating oscillations observed could indicate the afterburner chamber being forced close to one of its natural frequencies.

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Figure 2: Pressure fluctuations at probe P1

From Figure 3, with the increase in fuel injection rate, the dominant frequencies shift gradually towards a frequency of ~2300 Hz. At the same time, the pressure amplitude increased to ~ 2000 Pa indicating resonant oscillations in the chamber.

Figure 3: Spectrum of pressure fluctuations at probe location p1

4000

3.2. Unsteady Flame Oscillations and Elongation

From the unsteady flow field show, the oscillatory flames in the wake due to asymmetric vortex shedding from the flame folders are seen. These structures are due to the B-VK shedding from the flame holder lips. The density ratio of 1.5 observed here is known to be globally unstable with asymmetric vortex shedding [13]. Also, the flame in the wake elongates with the increase in fuel injection rate. For quantifying the flame elongation, the length of the flame in the vortex street is measured. The wake flame lengths L1 and L2 are used, as shown in Figure 4.

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Figure 4: Flame rollup in the wake

Variations in the lengths L1 and L2 with fuel injection rate for both the flame holders are shown in Figure 5. The general increase in the lengths is due to a reduction in the vortex shedding frequency from the flame holders. The increase in the heat release rate decreases the vorticity at the flame holder lip and hence the rate of vortex roll-up and shedding. This elongates the flame due to convection. The lengths L1 and L2 indicate saturation in the flame length for high fuel injection rates. The flame holder near the chamber wall produces a smaller flame than the inner flame holder. The smaller flames possibly occur due to the presence of a wall in the wake’s near field, which decreases the vortices staggering in the wake.

Figure 5: Flame rollup in the wake

0.187 0.16 Length (m) & aS 0.12 0.10 Flame holder-1 T Flame holder-2 T T T T -e- L1 J [ -e- L1 ae 2 =x 12

3.3. Vortex Shedding

The same could also be confirmed with the coefficient of lift measurement from the flame holders. The coefficient of lift values of the flame holder is shown in Figure 6. The dashed lines indicate flame holder 2, and the continuous line represents flame holder 1. The two flame holders show similar behavior in time. Values for both the cases are in phase for the low injection rate cases (c1-c4). For cases c5 and c6, the values go out of phase. The frequency decomposition revealed a decrease in the Strouhal number. This decrease corroborates conclusions from the flame length measurements on reducing vortex shedding rate.

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Figure 6: Coefficient of lift

Now from the analysis of the CFD data, an increase in the amplitude of the pressure oscillations with fuel injection rate is observed. The flame measurements indicate a decrease in vortex shedding from the flame holders. The same could also be confirmed with the coefficient of lift measurement from the flame holder. The wake oscillations could force the afterburner chamber close to a natural frequency causing resonant oscillations. The presence of spatial modes in the pressure oscillations can help in identifying the acoustic modes and confirm the resonant oscillations.

0.25 0.005 Casel -0.255 } i ' J 0.000 0.001 0.002 0.003 _0.004 0.0 0.2 0.4 0.6 0.8 0.25 0.00 ° Case2 -0.25 0.2 | 2 Case3 > Zoo J a § O°? | Case4 0.0 4 | Case5 Case6 ~°59 900 0.001 6.002 0.003 6.004 “0.0 0.2 0.4 0.6 08 Time Strouhal no vwall2 — wwalll

3.4. Proper Orthogonal Decomposition

To identify the spatial structures present in the flow field, the Proper Orthogonal Decomposition (POD) technique is used. The technique is commonly used to identify the dominant spatial structures in the flow field arranged hierarchically by their energy content [15], [16]. The POD modes are spatially orthonormal, with each mode having corresponding time coefficients. An overview of the technique can be found in [17]. The presence of POD modes in the CFD flow field with spatial structure and its time dynamics resembling the chamber acoustic modes can confirm the presence of chamber acoustics in the unsteady flow field. For that, the pressure field from the CFD calculations is sampled at a rate of 10000 Hz for the POD analysis for a flow time of 0.02 seconds (200-time steps). The sampling rate and the time window of the sampling are chosen to contain at least 40 cycles of oscillations at ~2200 HZ observed from the pressure probe data. The mean subtracted CFD field is stacked into a matrix, and the POD algorithm is applied. From the algorithm, spatial modes, energy content, and the corresponding time coefficients are obtained.

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Figure 7: Energy fraction in the POD modes

The cumulative energy fraction of the POD modes from all six calculations is shown in Figure 7. A significant fraction of the energy in the flow field is present in a small number of spatial modes. For cases c1 and c2, the first 10 POD modes account for more than 60 percent of the total energy in the corresponding unsteady field. The POD modes show a greater contribution from the vortex street of the flame holders. In cases c3-c6, POD modes show more variation in individual energy fractions. The first 25 modes contain about 60 percent of the total energy in the corresponding flow field. In these cases, in addition to the vortex structures, the chamber acoustic modes are also observed. The dominant peaks from time coefficients’ Fourier transform also show the same decreasing frequency behavior.

POD modes from the highest fuel injection case (c6) are described here. The first five modes show a larger energy content, with their cumulative energy accounting for more than 40 percent of the total energy in the field. The remaining modes have the rest of the energy distributed among them. The high energy content modes are visualized to understand the spatial pattern in the flow field. The spatial structures are not uniquely identified based on frequencies due to the limitation of

Normalized Energy Fraction Cumulative Energy Distribution oe a a wba" 77 P= AM FF oF J 7 ee j 1 Le ATT. c2 —s c3 | —- 4! — -=- 6 | a a a 25 50 $75 100 125 150 175 200 POD Modes

the POD algorithm in differentiating the time dynamics of the structures. Fourier transform is performed on the POD time coefficients to get the time dynamics of the corresponding mode.

The first 2 POD modes and the FFT of the time coefficients are shown in Figure 8. These show that the unsteady pressure field in the chamber is composed majorly as a combination of the chamber acoustic modes. The spatial structures near the axis of the domain from the POD correspond well with the acoustic modes. FFT of time coefficients also shows that the POD mode frequencies are close to the natural frequencies. This confirms the presence of the chamber acoustics in the CFD pressure field, which acts as the external forcing on the flame holder vortex shedding. POD modes with lower energy content do not show a dominant peak at the screech frequencies observed, and these modes are not shown here.

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POD-1

Figure 8: First 2 POD modes for C6 (Acoustic modes for reference)

4. SIMPLIFIED MODEL A simplified coupled oscillator model for the flame holder vortex shedding and the chamber acoustics is then attempted. An open-loop model of thermo-acoustic interaction is attempted. The model requirements are given as,

● Since the flame holder wake has a globally unstable mode, it acts as an oscillator. The model for the flame holder needs to show self-excited oscillations reaching limit cycle amplitudes.

POD-1

● An unforced Van der Pol (VDP) oscillator satisfies the requirement of being a self-driven oscillator having limit-cycle oscillations. It is used to model the wake oscillations of the flame holder. The VDP oscillator has been used previously in similar modeling approach [18]

● Chamber acoustics at high frequencies have a multitude of frequencies and corresponding natural modes. However, the chamber acoustics is modeled as a damped simple harmonic (SH) oscillator in this simplified modeling.

● Since the one-way open coupling model is proposed, the simple harmonic oscillator of acoustics needs to be coupled with the oscillations from the self-excited wake oscillator.

● This is achieved in the model by including the wake oscillator output to the simple harmonic oscillator of the acoustics using a coupling strength parameter. It helps denote the amount of strength of coupling between the oscillators.

The simplified model is given in the equations 1 & 2. Here, subscript 1 denotes the parameters corresponding to VDP oscillator and 2 denotes the SH oscillator. The value represents the 𝑥 1 pressure oscillation from the flame holder and the pressure oscillation in the chamber. The is 𝑥 2 ϵ

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the non-linearity parameter for the VDP oscillator and is the natural frequency of the ω 𝑛

corresponding oscillator. is the damping factor for the simple harmonic oscillator. is the ζ 𝐾 12 coupling strength. Natural frequency of the SH (acoustic) oscillator is used to nondimensionalize the time variable. Hence the non-dimensional time of 1 unit corresponds to one time period of the SH oscillator. The values of nonlinearity parameters are chosen as an arbitrary value of 0.1. Eight non-dimensional frequency values were used as the natural frequency of the self-excited VDP oscillator. These values were chosen to represent the decrease in the shedding frequency with an increase in the fuel injection. The non-dimensional frequency values used are 0.7, 0.8, 0.9, 0.95, 1.0, 1.05, 1.2. The coupling strength is taken as 1000. Initial conditions of the oscillators are 𝐾 12 chosen as = 0.1, = 0.01 and = 0.0, = 0. 𝑥 1 𝑥 1

˙ 𝑥 2 𝑥 2

˙

The coupled oscillator is numerically integrated till a non-dimensional time of 100 units. The oscillations from the VDP oscillator and the SH oscillator are shown in Figure 9. The VDP oscillator settles to a limit cycle amplitude value of 2 for all the frequency values. The SH oscillator settles to an amplitude depending on the forcing from the VDP oscillator. The amplitude increases gradually as the VDP oscillator’s non-dimensional natural frequency approaches 1.0. The amplitude versus forcing frequency diagram is shown in Figure 10. This response qualitatively matches the response measured from the CFD by showing the resonant oscillations. However, in the model the

natural frequency of the VDP oscillator needs to be adjusted to simulate the reduction in the vortex shedding rate with increase in fuel injection. It is planned to extend the model by including a functional form for the variation in the vortex shedding rate.

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Figure 9: Simplified VDP and damped SH oscillator model

Figure 10: Response of damped SH oscillator with variation in VDP ω 𝑛1

5. CONCLUSIONS Wake flame dynamics and thermo-acoustic oscillations in an afterburner setup are numerically investigated. With the increase in kerosene flow rates, an increase in the amplitude of the pressure oscillations with a decrease in frequency is observed. Investigations on the heat release and the flame structures in addition to the coefficient of lift measurements revealed a reduction in the flame holder shedding rate with fuel injection rate. This increase in wavelength is due to the reduction in the vortex strength at the flame holder lip due to the heat release.

This low-density ratio wake acts as an oscillator in the combustion chamber, creating an open loop of fluid dynamic - heat release - acoustics processes in the chamber. In the higher fuel injection rate cases, the frequency of vortex shedding reduces to ~2300 Hz resulting in a large amplitude pressure response in the chamber. The pressure field data from the flow field is analyzed with the proper orthogonal decomposition technique to identify the coherent spatial structures in the flow field. Also, the spectrum of the POD time coefficients is analyzed to identify the time dynamics of the POD modes. These POD modes of the pressure field are identified as the chamber acoustic modes. The presence of acoustic modes in the POD confirmed the presence of thermoacoustic open-loop in play. Then a simplified mathematical model based on coupled Van der pol - simple harmonic oscillator was made to qualitatively reproduce the observed behavior. The model shows the resonant acoustic oscillations due to forcing from the flame holder wake. The main limitation of the simplified model is that the self-excited oscillator needs to be adjusted to replicate the change in vortex shedding rate that is observed from CFD. Further work is required to include the vortex shedding library in the simplified model and extend it to practical combustors with bluff-body stabilized flames in predicting thermoacoustic oscillations.

6. ACKNOWLEDGMENTS We gratefully acknowledge the Ministry of Human Resources and Development Government of India, and the Indian Institute of Technology-Madras, for the support and the computational facilities.

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