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Abstract

Due to the incurred damages to the combustors, large-amplitude self-sustained thermoacoustic oscillations are unwanted in many propulsion systems, such as liquid/solid rocket motors and aero-engines. To suppress these thermoacoustic oscillations efficiently, the mechanism of thermoacoustic instability needs to be clarified. Following the previous experimental work, the transitions to instability in a Rijke-type thermoacoustic system with an axially distributed heat source are studied numerically in this paper. The URANS numerical method is utilized and verified by means of a mesh sensitivity analysis. The influences of the axially distributed heater length, the heater location, and the mean flow velocity on the nonlinear dynamic behaviors of thermoacoustic oscillations are evaluated. To explore the corresponding mechanism behind these influences, the principle of acoustic energy conservation has been applied. The acoustic energy gains from the thermal-acoustic coupling are quantified via Rayleigh’s integral, and their phase differences are calculated by the cross-correlation function. The acoustic damping induced by the vortex dissipation is qualitatively analyzed by the characteristics of the flow fields in the Rijke tube. Finally, as the heater length, the heater location, or the mean flow velocity is varied, three mechanisms of the transitions to instability in a Rijke-type thermoacoustic system are identified.

Keywords

Acoustic energy gain axially distributed heat source thermoacoustic instability vortex dissipation

Introduction

Energy conversion from heat to sound is desirable in thermoacoustic engines (TAEs)^1–4 due to its advantages in fewer moving parts and non-exotic materials compared to internal combustion engines, steam turbines, etc. Nevertheless, it is unwanted in aero-engines and land-based gas turbines^5–8 for the incurred large-amplitude thermoacoustic oscillations which would cause severe damages, such as jet panel ablation and structural vibration. The heat to sound energy conversion is generally caused by the coupling between the heat release rate pulsation and the internal acoustic field in the combustion chamber.^9–12 With the increasing stringent requirement of low-emission and high combustion efficiency, combustion instability becomes more popular in the new generation of gas turbines. To suppress and predict thermoacoustic oscillations in combustion systems,^13,14 it is essential to figure out the mechanism of thermoacoustic instability.

Considering the fact that the actual energy conversion from heat to sound is complex by the interactive coupling between flow, temperature, and acoustic field in real combustion systems, direct investigation of the mechanism of thermoacoustic instability would encounter long periods and expensive costs. It is believed that the study on the generation mechanism of the Rijke-type thermoacoustic instability could facilitate the understanding of thermoacoustic instability occurring in the more complex practical system.^15–19 Based on a bifurcating Rijke-type combustor, Zhao evaluated the performance of the Helmhotlz resonator²⁰ and a multi-input/single-output tuning strategy²¹ on damping self-sustained thermoacoustic oscillations, and the amplitudes of the limit cycle oscillations were simultaneously reduced in the two outlet branches. In addition, Zhao²² introduced a temperature-variable heat exchange into the Rijke-type thermoacoustic system which utilized an ammonia/hydrogen combustion system as a heater source. After evaluating the damping capability of the heat exchange on the thermoacoustic oscillations, it was found that this approach was not a potential way to deal with the unfavorable oscillation in the practical engines.

Previous work generally chose some compact heater to provide the unsteady heat release rate. Matveev²³ designed a thin nickel-chromium alloy screen to predict the stability boundary of the horizontal Rijke tube thermoacoustic system in terms of the heater power and air flow rate. Li²⁴ introduced the classic n-τ model via the Dirac function to represent the compact energy source, analyzed the influence of the flow-acoustic coupling strength and the averaged temperature ratio before and after the combustion zone, and explained how these factors affect the response of the Rijke tube. Chow²⁵ evaluated the stability of the system in terms of a hydrodynamic region adjacent to the flame holder and concluded that the frequency of thermoacoustic oscillations strongly depends on the location of the hydrodynamic region. Balasubramanian²⁶ studied the influence of non-normality and nonlinearity on thermoacoustic responses, in which the heat release rate of the heating element is modeled by a modified King’s Law and introduced via the Dirac function. Based on the same theoretical model, Subramanian²⁷ obtained the nonlinear bifurcation diagrams in terms of heater power and heater location, and found that increasing the heater power would make the thermoacoustic system more prone to large-amplitude oscillation, and the unstable heater location for the compact heater source is around 0.15∼0.4. Zhao^28,29 took the temperature distribution caused by laminar premix flame into account and predicted the most “dangerous” position by analyzing the maximum transient growth rate of acoustic energy.

However, it is worth noting that most previous researches on Rijke-type thermoacoustic systems have been limited to an infinitely thin heat source. In theoretical analysis, the unsteady heat release is regarded as a single concentrated plane and is usually represented by a form of the Dirac delta function. In the experimental studies, the heat source can be provided by thin flakes with small thicknesses. But in the real propulsion system, the combustion process in the combustor chamber is distributed over a nontrivial length of the combustor. For example, in a solid rocket engine with a large amount of aluminum propellant, the metal fuel cannot burn out on the thin combustion surface initially, and it would distribute near the centerline of the combustor and take reaction in a large region in the axial direction. The distributed combustion process would couple with the acoustic waves and be one of the main energy sources for driving thermoacoustic oscillations.³⁰ In aero-engine annular combustors³¹ and liquid rocket engines,³² the combustion process involves a series of physical and chemical processes, including injection, atomization, evaporation, premixing, and chemical reaction, which elongates the reaction zone in the longitudinal and tangential directions. Therefore, in real thermoacoustic systems, the combustion region is always distributed.

In recent years, the effects of distributed heat sources on thermoacoustic instability have attracted more and more attention. Li^33,34 studied the effect of non-uniform circumferential and axial distribution of heat release rate on a thermoacoustic system by establishing a three-dimensional thermoacoustic model. It was found that the non-uniform circumferential distribution has a significant effect on the oscillation frequency and growth rate. The longer the heat source axially distributed interval is, the greater the maximum transient growth rate is, thus increasing the possibility of triggering instability. Du³⁵ conducted an experimental study on tangential heat source distribution in the Rijke tube, and the experimental data proved that there was a dangerous spacing to maximize the stability boundary and the amplitude of thermoacoustic oscillations. Dowling³⁶ applied the linear stability theory to study the effect of an axially distributed heat source on the longitudinal instability in a tube with the inlet choked and the outlet opened, and found that the length of the axial heat source has a significant effect on the frequency of thermoacoustic oscillation when the temperature ratio across the heater is larger. Li³⁷ designed a coiled-wire heater source to represent the axially distributed heat source and studied the transition to instability in a Rijke tube. The influences of the characteristics of heat source, mass flow rate, and tube materials on the nonlinear dynamic behaviors and stability boundaries are evaluated. To reproduce the nonlinear thermoacoustic oscillation numerically in the open-open and closed-open Rijke tube, Hantschk^38,39 proposed a banded heater with multiple-concentric constant-temperature rings as a heater source and chosen a thermal conductivity which is 10 times higher than the reasonable value in a Rijke tube. Zhao⁴⁰ used the similar heater source in a convection-driven T-shaped thermoacoustic system model and studied the nonlinear coupling between unsteady heat release and acoustic disturbances.

Following our previous experimental investigations in Ref. 37, this paper utilizes the method of numerical simulation to explore the internal mechanism for the large-amplitude pressure oscillation in the Rijke-type thermoacoustic system with an axially distributed heat source. Firstly, the geometric configuration and the numerical methodology are described Secondly, the effects of the axially distributed heater length, heater location, and the mean flow velocity on the stability of the thermoacoustic system as well as the related mechanisms are discussed. Finally, the main conclusions are summarized.

Numerical methodology

Geometric configuration

A Rijke-type thermoacoustic system with an axially distributed heat source is studied. The configuration is shown in Figure 1. It is comprised of a straight tube with both ends opened and a heating element inside. The tube has an overall length of 1 m and a diameter of 50 mm, and it is placed horizontally to work as an acoustic resonator. An axis-symmetric annular heater band with a blockage ratio of 0.46 is placed at $x = x_{h}$ from the inlet, and its length is represented by the symbol $L_{h}$ . The heat source is designed to be convenient to adjust the location and axial length of the heat source.

Figure 1.

Schematic of a Rijke-type thermoacoustic system and the computational grid in the region of the heating element. A 2D axis-symmetric model is applied.

Governing equations

In this paper, 2D numerical investigations of the unstable pressure oscillations and heat oscillations in the Rijke-type thermoacoustic system are conducted by the URANS (unsteady Reynolds Averaged Navier–Stokes) method.⁴¹ For compressible flows, the mean fluid variables can be represented as $\tilde{φ} = \bar{ρ φ} / \bar{ρ}$ via the density-weighted time average (Favre-average), and the overbar denotes the operator of the Reynolds average. Then the instantaneous fluid variable $φ$ can be decomposed to be $φ = \tilde{φ} + φ^{″}$ , where $\tilde{φ}$ and $φ^{″}$ are the mean and fluctuating parts according to the Favre decomposition. Substituting the fluid variables of these forms into the equations of conservation of mass, momentum, and energy yield

\frac{\partial \bar{ρ}}{\partial t} + \frac{\partial (\bar{ρ} {\tilde{u}}_{i})}{\partial x_{i}} = 0

(1)

\frac{\partial (\bar{ρ} {\tilde{u}}_{i})}{\partial t} + \frac{\partial (\bar{ρ} {\tilde{u}}_{i} {\tilde{u}}_{j})}{\partial x_{j}} = - \frac{\partial \bar{p}}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} ({\tilde{τ}}_{i j} - \bar{ρ} \tilde{u_{i}^{″} u_{j}^{″}})

(2)

\frac{\partial}{\partial t} (\bar{ρ} {\tilde{e}}_{t}) + \frac{\partial}{\partial x_{i}} (\bar{ρ} {\tilde{u}}_{i} {\tilde{h}}_{t}) = \frac{\partial}{\partial x_{i}} (λ \frac{\partial \bar{T}}{\partial x_{i}} - \bar{ρ} \tilde{u_{i}^{″} T^{″}} + 2 {\tilde{τ}}_{i j} {\tilde{u}}_{j} - \bar{ρ} \tilde{u_{i}^{″} u_{j}^{″}} {\tilde{u}}_{j})

(3)

where the subscripts

i

and

j

are indices of the coordinates.

λ

is the thermal conductivity of the fluid.

e_{t}

represents the total energy per unit mass, and it is related to the internal energy

e

via

e_{t} = e + u_{i} u_{i} / 2

h_{t}

denotes the total enthalpy per unit mass, and

h_{t} = e_{t} + p / ρ

. The viscous stress tensor

τ_{i j}

is given as

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) - \frac{2 μ}{3} \frac{\partial u_{j}}{\partial x_{j}} δ_{i j}

(4)

where

δ_{i j}

is the Kronecker delta function.

μ

is the dynamic viscosity. The above governing equations are not closed due to the unknown Reynolds stress tensor

u_{i}^{″} u_{j}^{″}

in equation (2) and the turbulent heat vector

\tilde{u_{i}^{″} T^{″}}

in equation (3).

To model the Reynolds stresses, a Boussinesq hypothesis⁴² is used, that is

\tilde{u_{i}^{″} u_{j}^{″}} = - \frac{μ_{t}}{μ} {\tilde{τ}}_{i j} + \frac{2}{3} k δ_{i j}

(5)

where

μ_{t}

is the turbulence viscosity, and

k

stands for turbulence kinetic energy. The turbulent heat vector is modeled by

\tilde{u_{i}^{″} T^{″}} = - \frac{μ_{t}}{σ_{T}} \frac{\partial \bar{T}}{\partial x_{i}}

(6)

where

σ_{T}

denotes the turbulent Prandtl number. Considering that the unsteady heat release rate is determined by the process of the heat transfer between the heating element and surrounding air, SST k-ω was selected for the turbulence model, for its advantage in modeling the wall heat transfer and separation flow.

The transport equations for the SST k-ω model are given as

\frac{\partial}{\partial t} (ρ k) + \frac{\partial}{\partial x_{i}} (ρ k u_{i}) = \frac{\partial}{\partial x_{j}} (Γ_{k} \frac{\partial k}{\partial x_{j}}) + G_{k} - Y_{k}

(7)

\frac{\partial}{\partial t} (ρ ω) + \frac{\partial}{\partial x_{i}} (ρ ω u_{i}) = \frac{\partial}{\partial x_{j}} (Γ_{ω} \frac{\partial ω}{\partial x_{j}}) + G_{ω} - Y_{ω} + D_{ω}

(8)

where

G_{k}

denotes the generation of turbulence kinetic energy due to mean velocity gradients.

G_{ω}

is the generation of ω.

Γ_{k}

and

Γ_{ω}

denote the effective diffusivity of k and ω, respectively.

Y_{k}

and

Y_{ω}

represent the dissipation of

k

and

ω

due to turbulence.

D_{ω}

represents the cross-diffusion term.

Numerical algorithm and boundary conditions

To solve the above governing equations, the finite volume method is utilized. The Back-Euler difference method is employed for the temporal difference. The second-order central difference scheme and second-order upwind scheme are used in spatial difference to discretize the diffusion terms and convection terms, respectively. The PISO algorithm for the pressure-linked equations is adopted to solve the pressure field.

The boundary conditions are listed in Table 1. The acoustic resonator tube is set as a wall boundary, and it is characterized by a non-slip adiabatic condition. The inlet and outlet ends of the tube are treated to be acoustically opened. In another word, both ends work as pressure nodes for standing waves formed in the tube. To model the acoustically opened inlet and outlet, the pressure at both ends is set as constant, and the pressure difference between both ends is varied to control the mean flow velocity inside the tube. The heater band is configured to be with constant temperature.

Table 1.

Boundary conditions.

Boundary	Type
Inlet: $x = 0$	Pressure inlet: the relative pressure is fixed to be 0 Pa
Inlet: $x = 0$	Inlet temperature: $T_{i}$ = 300 K
Outlet: $x = L$	Pressure outlet: the relative pressure is negative value
Tube	Non-slip and adiabatic wall
Heater	Wall, the temperature is fixed to be $T_{h}$ = 2000 K

During the calculation, a better initial flow field was obtained by steady-state calculation first, and then transient calculation was carried out. The time step was set as $1 \times 10^{- 5} s$ .

Model dependence study

To monitor the system response within the Rijke tube, nine evenly distributed points are set up along the Rijke tube, as is shown in Figure 2. Here P1 and P9 are placed at the inlet and outlet of the Rijke tube, respectively.

Figure 2.

Distribution of monitoring points along the tube.

The grid adopts regular quadrilateral structures in the 2D model. The numerical method is verified by means of a mesh sensitivity analysis. Six different computational meshes, ranging from 100,000 to 320,000 node numbers, are evaluated for two-dimensional grid verification. The thermoacoustic system is first configured as follows: the axial heat source location is $x_{h} = L / 4$ , the axial heat source length is $L_{h} = 20 mm$ , the heat source temperature is $\bar{T} = 2000 K$ , and the mean flow velocity is around $\bar{U} = 1.51 m / s$ . Such a case is taken to analyze in detail, and the comparison results in terms of the amplitude of the most exciting pressure oscillation at P2 are shown in Figure 3. $p^{'}$ represents the acoustic pressure. By comparison, the maximum difference among the last five mesh configurations is less than 0.44% in acoustic pressure amplitude. Therefore, the fourth configuration with 220,000 grids is believed to be sufficient to describe the behaviors of acoustic pressure oscillations.

Figure 3.

Comparison of the most excited pressure amplitude $p_{2}^{'}$ at the fifth monitor point, as $x_{h} = L / 4$ , $L_{h} = 20 mm, \bar{T} = 2000 K$ , and $\bar{U} = 1.51 m / s$ .

Results and discussion

The parameters involved in this paper mainly include: (1) the axial heat source location x_h, (2) the axial heat source length L_h, and (3) the mean flow velocity $\bar{U}$ . The control variable method is adopted to study the influence of one parameter on the stability of the Rijke-type thermoacoustic system, when the other two parameters remain constant.

Figure 4 shows the time traces of acoustic pressure via the previous numerical model, and both of the exponential growth and large-amplitude pressure oscillation are observed. Figure 5 shows the acoustic pressure mode shape and corresponding acoustic velocity mode shape when the heater length is varied. It can be seen that the acoustic pressure is maximized near the center of the tube (Figure 5(a)), and the acoustic velocity is maximized at both ends of the tube. Due to the occurrence of the heat source and the flow disturbance, the acoustic velocity node is shifted to the upstream part.

Figure 4.

Time evolution of the self-sustained acoustic oscillation.

Figure 5.

(a) The non-dimensional acoustic pressure mode shape $p^{'} / \bar{p}$ ; (b) The non-dimensional acoustic velocity mode shape $u^{'} / u_{\max}^{'}$ .

Effects of the axial heater location

Then, the effects of the axially distributed heat source location on the stability of the Rijke-type thermoacoustic system are investigated in Figure 6, as the heat source length is $L_{h} = 20 mm$ . It can be seen that the curve exhibits a parabolic shape within $0.1 < x_{h} / L < 0.5$ , and there is a sensible axial heat source location ( $x_{h} / L = 0.25$ ) that makes the thermoacoustic oscillation become more intense.

Figure 6.

Amplitude of thermoacoustic oscillation under different heat source locations, as $L_{h} = 20 mm$ , $\bar{U} = 2.55 m / s$ , and $\bar{T} = 2000 K$ .

To explore the mechanism behind the variation trends further, the principle of energy conservation is applied. Here, the acoustic energy gain from the coupling between the unsteady heat release rate and acoustic pressure can be evaluated by the Rayleigh’s criterion,⁴³ and it has been mathematically derived^40,44,45 and explicitly expressed as

∆ E_{a d d} = \frac{γ - 1}{p_{o} γ} \int_{V} \int_{t}^{t + t_{T}} p^{'} {\dot{Q}}^{'} d t d V

(9)

where

p^{'}

represents the acoustic pressure, and

{\dot{Q}}^{'}

denotes the unsteady heat release rate.

γ = 1.4

is the specific heat,

p_{o}

denotes the mean ambient pressure, and

t_{T}

represents the time period.

Based on the cross-correlation function, the phase difference between acoustic pressure oscillation and heat release rate oscillation of a thermoacoustic system can be obtained via the following expression

θ (t) = \cos^{- 1} (\int_{0}^{t} \frac{p^{'} (Θ) Q^{'} (Θ) d Θ}{\sqrt{\int_{0}^{t_{T}} p^{' 2} (Θ) d Θ} \sqrt{\int_{0}^{t_{T}} Q^{' 2} (Θ) d Θ}})

(10)

Figure 7(a) evaluates the cross-correlation function $R (τ)$ between acoustic pressure oscillation and heat release rate oscillation, where $R (τ) = \int_{0}^{t_{T}} p' (t) Q' (t + τ) d t$ . It can be found that the correlation function oscillates with a period of the fundamental acoustic waves in the Rijke tube, and finally converge to zeros. This variation is satisfied with the stochastic dynamic theory that the larger the time lag is, the weaker the correlation between the two physical processes becomes. From Figure 7(b), it can be observed that the acoustic pressure is almost in phase with the heat release oscillation, indicating a positive coupling between the unsteady heat release rate and acoustic pressure. In addition, the phase difference between acoustic pressure and heat release rate oscillation of the thermoacoustic system is calculated via equation (10), and it is around $θ = 36 °$ which is obviously less than 90°.

Figure 7.

(a) The cross-correlation function for the acoustic pressure and unsteady heat release rate; (b) Acoustic pressure pulsation and heat release rate pulsation, as $x_{h} = L / 4$ , $L_{h} = 40 mm$ , $\bar{U} = 2.65 m / s$ , and $\bar{T} = 2000 K$ .

Table 2 shows the acoustic energy gains within one cycle for the excited oscillations and the corresponding phase differences. It can be seen that a large acoustic energy gain corresponds to a small phase difference, which is consistent with Rayleigh’s criterion. The acoustic damping is examined via the flow field in Figure 8. For the heating element with a fixed heater length, as the location of the distributed heating element is continuously varied, the amount of the vortex behind the heater responds nonlinearly, and the location of

x_{h} = L / 4

is observed to have the least and smallest vortex. Such variation of the flow field is not consistent with that of cold flow. It is worth to be noticed that the distributed heater with the heater location of

x_{h} = L / 4

has maximum acoustic energy gains and pressure oscillation amplitude, and these trends agree well with the thermoacoustic system having the thin heater in the previous experimental and analytical research.^46,47 Therefore, for the distributed heater source, as the heater location is varied, the acoustic field is dominant in modifying the response of a thermoacoustic system, and the flow field is forced to be modified by acoustic disturbances.

Table 2.

Acoustic energy gains and phase difference under different axial heat source locations.

$x_{h}$	$∆ E_{a d d}$	$θ$
3L/16	9.14 × 10⁻⁴	53.82°
L/4	2.33 × 10⁻³	46.73°
5L/16	1.05 × 10⁻³	47.52°

Figure 8.

The time-averaged temperature contour and time-averaged flow field near the heating element under different axial heat source locations, as $L_{h} = 20 mm$ , $\bar{U} = 2.55 m / s$ , and $\bar{T} = 2000 K$ . (a) $x_{h} = 3 L / 16$ ; (b) $x_{h} = L / 4$ ; (c) $x_{h} = 5 L / 16$ .

For the effects of the axial heat source location on the frequency of thermoacoustic oscillations, Figure 9 shows that the thermoacoustic oscillation frequency is decreased, as the heater is moving downstream. It can be seen from Figure 10 that the hot region is decreasing, when the heat source moves downstream, which results in a decrease in the mean temperature and mean sound speed. Therefore, the resonant frequency of the acoustic oscillation in the tube declines.

Figure 9.

Thermoacoustic oscillation frequency under different axial heat source locations, as $L_{h} = 20 mm$ , $\bar{U} = 2.55 m / s$ , and $\bar{T} = 2000 K$ .

Figure 10.

The time-averaged temperature contour under different axial heat source locations, as $L_{h} = 20 mm$ , $\bar{U} = 2.55 m / s$ , and $\bar{T} = 2000 K$ . (a) $x_{h} = L / 8$ ; (b) $x_{h} = L / 4$ ; (c) $x_{h} = 3 L / 8$ .

Effects of the axial heater length

Second, the effect of the axial heat source length on the stability of the Rijke-type thermoacoustic system is studied numerically in Figure 11. The amplitude of pressure oscillation in the tube increases first and then decreases gradually, when the axial heat source length is increased. There is an optimal axial heat source length, which makes the thermoacoustic oscillation being most intense.

Figure 11.

Amplitude of thermoacoustic oscillation under different axial heater lengths, as $x_{h} = L / 4$ , $\bar{U} = 2.65 m / s$ , and $\bar{T} = 2000 K$ .

Furthermore, the acoustic energy gain and phase difference are evaluated and shown in Table 3, respectively. It can be seen that the shortest heater source with

L_{h} = 10 mm

has a phase difference larger than 90°, which leads to a negative acoustic energy gain. Therefore, no oscillation occurs, and this prediction is consistent with the stable case in Figure 11. For the other three cases, the phase difference is within 90°, and the acoustic energy gain is positive. By comparison, with the increase of the heater length, the acoustic energy gain is decreasing, that is,

∆ E_{20 - a d d} > ∆ E_{40 - a d d} > ∆ E_{60 - a d d}

, but the phase difference is decreased too. To explore the reason and explain the thermoacoustic system response, acoustic damping must be taken into account. Considering that the adiabatic boundary condition is applied in the modeling, the heat loss to the wall is ignored, and only viscous dissipation is mainly studied. The temperature and flow field distributions near the heating element within one period are examined in Figure 12. From the temperature field, it can be seen that the mean temperature downstream is increased with the increase of the heater length, which indicates more heat power is transferred to the flow. From the flow field, some recirculation zones are found to be formed behind the heating element. More vortices are observed in Figure 12(a). Therefore, the shortest heater source with

L_{h} = 10 mm

may not provide enough heat power to overcome the large acoustic damping and destabilize the system. Meanwhile, a large-scale eddy is observed near the centerline of the tube, and the size of the largest vortices is increased, as the heater length is increased from 20 mm to 60 mm. This trend is similar to that of cold flow, and therefore thermoacoustic interaction between the heat release rate and acoustic waves has trivial influence on the flow field. However, large vortices correspond to intense acoustic damping caused by the vortex dissipation and lead to small amplitude of acoustic pressure. Compared with the thermoacoustic interaction between the unsteady heat release rate and acoustic waves, as the heater length is changed, the influence of the flow field is dominant in the response of the thermoacoustic system, and it may modify the phase difference.

Table 3.

Acoustic energy gains and phase differences under different axial heat source lengths.

$L_{h}$	$Δ E_{a d d}$	$θ$
10	−9.36 × 10⁻⁶	96.10°
20	2.00 × 10⁻³	49.01°
40	1.43 × 10⁻³	36.00°
60	1.50 × 10⁻⁴	34.02°

Figure 12.

The time-averaged temperature contour and time-averaged flow field near the heating element under different axial heat source lengths, as $x_{h} = L / 4$ , $\bar{U} = 2.65 m / s$ , and $\bar{T} = 2000 K$ . (a) $L_{h} = 10 mm$ ; (b) $L_{h} = 20 mm$ (c) $L_{h} = 40 mm$ ; (d) $L_{h} = 60 mm$ .

Then, the frequency of the thermoacoustic oscillation is evaluated in Figure 13. The curve shows that the frequency of pressure oscillation in the Rijke tube is increased, as the axial heat source length is increased. The reason behind this is explored in Figure 14, where the temperature contours are illustrated in the whole tube. It can be seen that, when the airflow passes through the heat source, the mean gas temperature increases obviously, and then decreases to a relatively stable value. With the increase of the length of the axial heat source with constant temperature, more heat energy is transferred, resulting in an average temperature rise downstream of the heater source. Finally, a larger sound speed ( $a = \sqrt{γ R T}$ ) is obtained, and the corresponding longitudinal acoustic frequency is increased.

Figure 13.

The frequency of thermoacoustic oscillation under different axial heat source lengths, as $x_{h} = L / 4$ , $\bar{U} = 2.65 m / s$ , and $\bar{T} = 2000 K$ .

Figure 14.

The time-averaged temperature contour under different axial heat source lengths, as $x_{h} = L / 4$ , $\bar{U} = 2.65 m / s$ , and $\bar{T} = 2000 K$ . (a) $L_{h} = 10 mm$ ; (b) $L_{h} = 20 mm$ ; (c) $L_{h} = 40 mm$ ; (d) $L_{h} = 60 mm$ .

Effects of the mean flow velocity

Last, the influence of the mean flow velocity on the stability of the Rijke-type thermoacoustic system is studied in Figure 15, as the axial heat source location is configured at $x_{h} = L / 4$ , the axial heat source length is $L_{h} = 40 mm$ , and the heat source temperature is $T = 2000 K$ . The numerical curve shows that the system is stable when the mean flow velocity is small. When the mean flow velocity is larger than a critical value, the system becomes unstable, and large-amplitude thermoacoustic oscillation appears. With the increase of the mean flow velocity, the amplitude of thermoacoustic oscillation increases firstly and then decreases. This indicates that the thermoacoustic system becomes unstable and exhibits self-excited oscillations in a limited interval of mean flow velocity, and there also exists an optimal mean flow velocity to make the thermoacoustic oscillation of the system oscillate most intensively.

Figure 15.

Amplitude of thermoacoustic oscillation in different mean flow velocities, as $x_{h} = L / 4$ , $L_{h} = 40 mm$ , and $\bar{T} = 2000 K$ .

Furthermore, the influence of the axial heater length on the stability range of the Rijke-type thermoacoustic system is studied in Figure 16. It can be seen that the unstable velocity interval is decreased with the increase of the heat source length, and the unstable velocity interval for the longer heat source length is trapped in that for the shorter heat source length. These trends are similar with the experimental measurement in Ref. 37. The main difference is the configuration with $L_{h} = 20 mm$ , for two unstable regions are observed.

Figure 16.

Variation curves of the amplitude of thermoacoustic oscillation with mean flow velocity under different axial heat source lengths, as $x_{h} = L / 4$ and $\bar{T} = 2000 K$ .

Six representative mean flow velocities, as denoted in Figure 16, are chosen to evaluate the stability mechanism with the heater length of

L_{h} = 20 mm

. Table 4 gives the calculated acoustic energy gain and the corresponding phase difference. When the mean flow velocity is smaller, that is,

\bar{U} < 2 m / s

. The acoustic energy gain is negative for Case 2, and a stable thermoacoustic system is obtained in Figure 16.

Table 4.

Acoustic energy gains and phase difference under different flow velocities.

Case	$\bar{U} (m / s)$	$∆ E_{add}$	$θ$
1	1.67	8.39 × 10⁻⁴	34.56°
2	1.72	-2.28 × 10⁻⁵	127.58°
3	1.92	4.53 × 10⁻⁴	45.54°
4	2.34	1.94 × 10⁻³	41.98°
5	2.55	2.11 × 10⁻³	46.73°
6	2.76	1.69 × 10⁻³	53.06°

Then, the effects of the mean flow velocity on the acoustic model shape and the mean temperature distribution are further studied. It can be seen from Figure 17 that increasing the mean flow velocity would decrease the acoustic pressure value inside the heater location where the acoustic intensity is maximum in the axial direction. In addition, when $\bar{U}$ =1.72 m/s, the acoustic velocity inside the heater and the amount of the vortices are maximum. Therefore, the large acoustic damping from velocity disturbances stabilizes the system. For Case 1 and Case 3, it can be seen that a larger acoustic energy gain corresponds to a smaller phase difference, which agrees with the Rayleigh’s criterion. The vortex formed behind the heating element in Figure 18(b) and (d) is almost similar in Case 1 and Case 3. Therefore, the flow velocity disturbances dominate the stability of the thermoacoustic system.

Figure 17.

The effects of the mean flow velocity on acoustic pressure and acoustic velocity for the low mean flow velocity.

Figure 18.

The time-averaged temperature and time-averaged flow field near the heating element under different incoming flow velocities, as $L_{h} = 20 mm$ , $x_{h} = L / 4$ , and $\bar{T} = 2000 K$ . (a) Case 1, $\bar{U} = 1.67 m / s$ ; (b) Case 2, $\bar{U} = 1.72 m / s$ ; (c) Case 3, $\bar{U} = 1.92 m / s$ .

When the mean flow velocity is increased to a large value, that is, $\bar{U} \geq 2 m / s$ , the acoustic energy gains of Cases 4, 5, and 6 are around one order larger than that of the previous three cases. Table 4 shows that the acoustic energy gain is increased first and then decreased. Figure 19 shows that increasing the mean flow velocity has trivial influence on the acoustic pressure mode shape, but leads to a monotonic decrease in the acoustic velocity within the heater location and an increase in the amount of vortices. Figure 20(a) shows that the maximum acoustic energy gain corresponds to the largest mean temperature within the heater location where the acoustic intensity is maximum. Therefore, the system response is mainly determined through the mean temperature distribution, as the large flow velocity is increased.

Figure 19.

The effects of mean flow velocity on the acoustic pressure and acoustic velocity for the large mean velocity.

Figure 20.

The time-averaged temperature contour and time-averaged flow field near the heating element under different incoming flow velocities, as $L_{h} = 20 mm$ , $x_{h} = L / 4$ , and $T = 2000 K$ . (a) Case 4, $U = 2.34 m / s$ ; (b) Case 5, $U = 2.55 m / s$ ; (c) Case 6, $U = 2.76 m / s$ .

Conclusions

In this paper, a Rijke-type thermoacoustic system with an axially distributed heat source is studied numerically and experimentally. It has been found that the response of the thermoacoustic system is mainly determined by the characteristics of the acoustic waves as the axially distributed heater location is varied. Comparative, the increase of the heater length would lead to large vortices and large acoustic damping, and characteristics of the flow field determine the system response. Furthermore, the influence of the mean flow velocity is further evaluated, and found that, as the mean flow velocity is small, the response of the thermoacoustic system is dominated by the thermoacoustic interaction. However, as the mean flow velocity is large enough, the variation in the flow field and the mean temperature field is dominant, and the phase difference responds nonlinearly as the acoustic energy gain is increased. Last, there are some sensible length and location of the axially distributed heat source and mean flow velocity to incur maximum amplitudes of thermoacoustic oscillation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by Young Scientists Fund (Grant number: 52006012).

ORCID iD

Xinyan Li

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Investigation on the stability of the Rijke-type thermoacoustic system with an axially distributed heat source

Abstract

Keywords

Introduction

Numerical methodology

Geometric configuration

Governing equations

Numerical algorithm and boundary conditions

Model dependence study

Results and discussion

Effects of the axial heater location

Effects of the axial heater length

Effects of the mean flow velocity

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References