Nonlinear response of laminar premixed flames to dual-input harmonic disturbances

Xiaozhen Jiang School of Astronautics, Beihang University Beijing 102206, China

Jingxuan Li 1

School of Astronautics, Beihang University Beijing 102206, China Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University Ningbo 315100, China

Aimee S. Morgans Department of Mechanical Engineering, Imperial College London South Kensington Campus, London SW7 2AZ, UK

Lijun Yang School of Astronautics, Beihang University Beijing 102206, China Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University Ningbo 315100, China

ABSTRACT The two-way interaction between the flame and acoustic waves leads to combustion instability. To understand and quantify the response of the flame to acoustic waves, studies have typically considered single-harmonic forcing of the flame, which assumes a dynamically linear or weakly nonlinear flame response. This study extends this approach by introducing an additional disturbance at the frequency S t 2 to influence the flame nonlinear response to the first perturbation at the frequency S t 1 to gain further insight into these associated combustion instabilities. The spatial front-tracking of premixed flames were derived from the analytical and numerical solutions of the G-equation model. The nonlinear behavior of flame response was presented and the related mechanism of that was also elucidated. Due to the flame propagating forward normal to itself, from the near-order asymptotic analysis point of view, the third-order nonlinear interaction of the additional and first perturbations induces an external flame response related to S t 1 . It significantly a ff ects the first fundamental frequency response of the flame.

1 jingxuanli@buaa.edu.cn

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

1. INTRODUCTION

High-thrust rocket engines, lean-combustion aero-engines and land-based gas turbines often su ff er from combustion instabilities, causing serious damage [1–4]. To predict combustion instability, it is key to understand the flame response to inflow perturbations [5, 6]. As a simpler alternative to experiments [7] and numerical simulations [8], the flame dynamic model based on the G -equation proposed by Markstein [9] for tracking infinitely thin flame fronts is able to to quantify the premixed laminar or weakly turbulent flame response to harmonic disturbances [5, 10–13]. However, the obtained results of most previous research were limited to the specific case where the flame is disturbed by a pure single harmonic disturbance. The nonlinear response of the the flame can be better characterised by consider perturbations with dual or even multiple harmonics. This phenomena have been confirmed from experimental observations [14–17] and results of numerical simulations [18–20]. In recent research, Han et al [21] numerically investigated the e ff ect of two perturbations at 160 Hz and 320 Hz on the flame nonlinear response and the interaction of a lean premixed flame forced externally by strong input velocity oscillations. The introduction of a higher harmonic significantly changes the heat release rate fluctuation; the level of flame response at the fundamental frequency is reduced by 70% compared to the flame response subjected to only a single frequency perturbation. However, these results are limited to the case that the second frequency is double the fundamental frequency. As shown in the experiments conducted by Lamraoui et al. [14], the second frequency in general is uncorrelated with the first one. Orchini and Juniper [11] presented the computation of a non-static flame dual-input description function (FDIDF) based on the G -equation in the case of a laminar conical flame. The results show that FDIDF improves the stability prediction, which predicts the onset of the Neimark-Sacker bifurcation and determines the frequency of oscillations around the unsteady limit cycle. It should be noticed that, Orchini and Juniper [11] treated the flame module as a “black box” embedded in the thermoacoustic network, which neglected the specific formation mechanism of the flame nonlinear response under two input perturbations. As mentioned above, the introduction of additional perturbation at the frequency S t 2 results in an interaction of the flame nonlinear response at S t 2 and the first perturbation frequency S t 1 . It significantly a ff ects the flame response at S t 1 . The objective of this work is to investigate how additional harmonic perturbations a ff ect the flame response at S t 1 and to further elucidate the corresponding mechanisms in a general case, where the frequency and magnitude of the additional perturbation are independent on those of the first one.

2. FORMULATION

Laminar premixed conical and V-shaped flames considered in this work, and they are anchored on the burner rim as shown in figure 1 ( a ) and ( b ) respectively, The symmetrical flames are subjected to two velocity perturbations u ′ 1 and u ′ 2 with di ff erent frequencies. The flame front kinematics quantified based on the G -equation model is given by

∂ G

∂ t + u · ∇ G = S L |∇ G | (1)

where, u and S L are respectively the local velocity vector and the laminar flame displacement speed. G is a level-set function, where G = 0 indicates the flame front separating the reactants ( G < 0) and products ( G > 0). Equation 1 is validated under following assumptions (it is similar to our previous researches [22,23]): (i) The flame front is infinity thin; (ii) The flame laminar displacement speed is assumed constant; (iii) The flow condition upstream of the flame is pre-defined; (iv) The flame front is considered as a single-value function of the spatial location.

x

x

( ) , r t 

( ) , r t 

r

r

1 u  2 u 

1 u  2 u 

u

u

( a ) ( b )

Figure 1: Schematic of ( a ) the conical flame and ( b ) the V-shaped flame subjected to dual-input harmonic disturbances ( x , r the spatial coordinate; ξ ( r , t ) the flame front location; t the time; u ′ 1 , u ′ 2 the first and second perturbations and ¯ u the mean bulk velocity).

For the cylindrical coordinate system, Equation 1 is rewritten under the x − r − θ coordinate, which follows,

s ∂ G

! 2 ∂ G

! 2 1

! 2 (2)

r ∂ G

r ∂ G

∂ G

∂ t + u ∂ G

∂ x + u r ∂ G

∂ r + u θ

∂θ = S L

∂ x

∂ r

∂θ

where, x , r and θ denote the spatial location; t is the time; u , u r and u θ are the velocity of x − , r − and θ − directions respectively. According to assumption (iv), G ( x , r , θ, t ) is transformed into an explicit form with respect to the instantaneous flame front ξ ( r , θ, t ):

G ( x , r , θ, t ) = x − ξ ( r , θ, t ) (3)

It is possible to Substitute Equation 3 into Equation 2, and neglect the parametric variation in the θ -direction. In addition, the experimental results obtained by Birbaud et al. [24] indicate that velocity fluctuations are almost unchanged in the radial direction and the perturbation term of the radial velocity is omitted. The governing equation for the flame front can be simplified as

s

! 2 (4)

∂ξ

1 + ∂ξ

∂ t − u = − S L

∂ r

For the rest of this paper, all parameters are normalised, i.e., velocities are normalised by the mean bulk velocity ¯ u , spatial coordinates are normalised by the burner radius Ω and time by Ω / ¯ u . The boundary conditions of Equation 4 for di ff erent flames can be given by

ξ C ( r = 1 , t ) = 0; ξ V ( r = Ω b , t ) = 0 (5)

where, subscripts “C” and “V” denote the conical and V-shaped flames respectively; Ω b is the radius of the center body of the V-shaped flame. The velocity field upstream the flame presented in figure 1 is expressed as the superposition of the mean flow and two disturbances:

u ( S t 1 , S t 2 , x , t ) = 1 + ϵ 1 C 1 ( S t 1 , x , t ) + ϵ 2 C 2 ( S t 2 , x , t ) (6)

where,  C 1 ( S t 1 , x , t ) = cos [ S t 1 ( K 1 x − t )]

C 2 ( S t 2 , x , t ) = cos [ S t 2 ( K 2 x − t )] (7)

Figure 2: Dependence of K on S t for di ff erent threshold Λ in di ff erent flames ( Ω b = 0).

where, ϵ means the perturbation amplitude; subscripts “1” and “2” denote properties of the first and additional perturbations; S t = ω Ω / ¯ u is the normalised angular frequency, ω the angular frequency; K 1 and K 2 characterise the speed of perturbation u c and are expressed as the ratio of the mean velocity to u c . The perturbation propagates at the mean bulk velocity or speed of sound, depending on the cases K equals unity or tends to zero. Previous related experimental studies [24, 25] pointed out that the disturbance propagation is significantly a ff ected by the modulation frequency S t . The results of a model relating them are shown in figure 2 (obtained based on the methods of our previous research [22], and in which the variable Λ relating to the perturbation velocity is a cut-o ff value). Results in figure 2 suggest that K decreases monotonically from unity to zero with the increase of S t , and the perturbation speed rises monotonically from the mean bulk velocity to infinity. Due to the di ff erence in the steady flame front tracking and boundary condition expressions in the conical flame and the V-shaped flame, the relation between K and S t di ff ers for di ff erent flames. The relation between K and S t is adopted in the latter part of this paper unless otherwise stated. Λ = 0 . 2 is chosen to match the PIV experimental results obtained by Yang et al. [25].

2.1. Numerical approach Equation 4 can be solved directly according to the numerical method. The discretisation of spatial derivations uses a seventh-order Weighted Essentially Non-Oscillatory (WENO7) scheme [26]. A fifth-stage fourth-order Runge-Kutta method with Strong Stability Preserving (SSP-RK45) [27] is employed for the time integration and Local Lax-Friedrich (LLF) [26] scheme for the improved stability. The grid size of spatial discrete unit is 2 π S t / (200 K ) and time gird size is determined by the Courant-Friedrichs-Levy (CFL) number, i.e., CFL = 0 . 2.

2.2. Near-order asymptotic analysis Asymptotic analysis of the governing equation is e ff ective for gaining insight into the key mechanism by which the additional perturbation a ff ects the first fundamental frequency flame

response. The asymptotic analysis is extended to the third order of ϵ 1 and ϵ 2 , as follows,

ξ ( r , t ) = ξ 0 ( r ) |{z} Steady term + ϵ 1 ξ 1 ( r , t ) + ϵ 2 ξ 2 ( r , t ) | {z } Linear terms + ϵ 1 2 ξ 1 , 1 ( r , t ) + ϵ 2 2 ξ 2 , 2 ( r , t ) + ϵ 1 3 ξ 1 , 1 , 1 ( r , t ) + ϵ 2 3 ξ 2 , 2 , 2 ( r , t ) | {z } Self − nonlinear terms + ϵ 1 ϵ 2 ξ 1 , 2 ( r , t ) + ϵ 1 2 ϵ 2 ξ 1 , 1 , 2 ( r , t ) + ϵ 1 ϵ 2 2 ξ 1 , 2 , 2 ( r , t ) | {z } Mutual − nonlinear terms + O h ( ϵ 1 , ϵ 2 ) 4 i

(8)

Herein, these ten terms included in the asymptotic analysis are categorized into three types: linear terms, self-nonlinear terms and mutual nonlinear terms. Linear and self-nonlinear terms correspond to linear and nonlinear responses of flame front-tracking, respectively, subjected purely to the first or additional perturbation. Mutual-nonlinear terms account for the nonlinear kinematic response of the flame front location in two simultaneous harmonic perturbations. The steady term solutions for di ff erent flames can be easily obtained, as follows,

ξ C , 0 ( r ) = (1 − r ) cot α ; ξ V , 0 ( r ) = r cot α (9)

Substituting Equation 8 into the nonlinear source on the right hand side of Equation 4, and extending the Taylor expansion to the third order, it can be written as s

! 2 ≈ ι 0 + ι 1

! 2 (10)

1 + ∂ξ

∂ξ

∂ r − ∂ξ 0

∂ξ

∂ r − ∂ξ 0

! + 1

2 ι 2

∂ r

∂ r

∂ r

where,

1 / 2 ;

! 2 

ι 0 =  1 + ∂ξ 0

∂ r

− 1 / 2 ;

! 2 

 1 + ∂ξ 0

ι 1 = ∂ξ 0

(11)

∂ s

∂ r

− 1 / 2 − ∂ξ 0

− 3 / 2 .

! 2 

! 2  1 + ∂ξ 0

! 2 

ι 2 =  1 + ∂ξ 0

∂ r

∂ r

∂ r

Substituting Equation 8 and Equation 10 into Equation 4 and matching terms with the same order, the PDE corresponding to each order is determined. The PDEs of di ff erent flames have di ff erent but similar forms. For the sake of brevity, only governing equations of each order of the conical flame front are given. For the linear terms, they are

2 sin 2 α ∂ξ 1

∂ξ 1

∂ t − 1

∂ r = C 1 ( r , t ) (12)

∂ξ 2

2 sin 2 α ∂ξ 2

∂ t − 1

∂ r = C 2 ( r , t ) (13)

The forms of the self-nonlinear terms are

! 2 (14)

∂ξ 1 , 1

2 sin 2 α ∂ξ 1 , 1

2 sin 4 α ∂ξ 1

∂ t − 1

∂ r = − 1

∂ r

! 2 (15)

∂ξ 2 , 2

2 sin 2 α ∂ξ 2 , 2

2 sin 4 α ∂ξ 2

∂ t − 1

∂ r = − 1

∂ r

∂ξ 1 , 1 , 1

2 sin 2 α ∂ξ 1 , 1 , 1

∂ r ∂ξ 1 , 1

∂ r = − sin 4 α ∂ξ 1

∂ t − 1

∂ r (16)

∂ξ 2 , 2 , 2

2 sin 2 α ∂ξ 2 , 2 , 2

∂ r ∂ξ 2 , 2

∂ r = − sin 4 α ∂ξ 2

∂ t − 1

∂ r (17)

and the three mutual-nonlinear terms are

∂ξ 1 , 2

2 sin 2 α ∂ξ 1 , 2

∂ r = − sin 4 α ∂ξ 1

∂ r ∂ξ 2

∂ t − 1

∂ r (18)

∂ξ 1 , 2 , 2

2 sin 2 α ∂ξ 1 , 2 , 2

∂ r ∂ξ 2 , 2

∂ r ∂ξ 1 , 2

∂ r + ∂ξ 2

∂ r = − sin 4 α ∂ξ 1

! (19)

∂ t − 1

∂ r

∂ξ 1 , 1 , 2

2 sin 2 α ∂ξ 1 , 1 , 2

∂ r ∂ξ 1 , 2

∂ r ∂ξ 1 , 1

∂ r = − sin 4 α ∂ξ 1

∂ r + ∂ξ 2

! (20)

∂ t − 1

∂ r

The results corresponding to those equations can be quantified by considering Equation 5. The specific forms of solutions are following. For the linear terms, there are

ξ 1 ( r , t ) = µ 1 ( r ) cos  S t 1 t + χ 1 ( r )  (21)

ξ 2 ( r , t ) = µ 2 ( r ) cos  S t 2 t + χ 2 ( r )  (22)

The forms of self-nonlinear terms are

ξ 1 , 1 ( r , t ) = ξ I , 1 , 1 ( r , t ) + ξ II , 1 , 1 ( r ) = µ I , 1 , 1 ( r ) cos  2 S t 1 t + χ I , 1 , 1 ( r )  + µ II , 1 , 1 ( r ) (23)

ξ 2 , 2 ( r , t ) = ξ I , 2 , 2 ( r , t ) + ξ II , 2 , 2 ( r ) = µ I , 2 , 2 ( r ) cos  2 S t 2 t + χ I , 2 , 2 ( r )  + µ II , 2 , 2 ( r ) (24)

ξ 1 , 1 , 1 ( r , t ) = ξ I , 1 , 1 , 1 ( r , t ) + ξ II , 1 , 1 , 1 ( r , t ) = µ I , 1 , 1 , 1 ( r ) cos  S t 1 t + χ I , 1 , 1 , 1 ( r ) 

+ µ II , 1 , 1 , 1 ( r ) cos  3 S t 1 t + χ II , 1 , 1 , 1 ( r )  (25)

ξ 2 , 2 , 2 ( r , t ) = ξ I , 2 , 2 , 2 ( r , t ) + ξ II , 2 , 2 , 2 ( r , t ) = µ I , 2 , 2 , 2 ( r ) cos  S t 2 t + χ I , 2 , 2 , 2 ( r ) 

+ µ II , 2 , 2 , 2 ( r ) cos  3 S t 2 t + χ II , 2 , 2 , 2 ( r )  (26)

and the three mutual-nonlinear terms are

ξ 1 , 2 ( r , t ) = ξ I , 1 , 2 ( r , t ) + ξ II , 1 , 2 ( r , t ) = µ I , 1 , 2 ( r ) cos  ( S t 1 + S t 2 ) t + χ I , 1 , 2 ( r ) 

+ µ II , 1 , 2 ( r ) cos  | S t 1 − S t 2 | t + χ II , 1 , 2 ( r )  (27)

ξ 1 , 1 , 2 ( r , t ) = ξ I , 1 , 1 , 2 ( r , t ) + ξ II , 1 , 1 , 2 ( r , t ) + ξ III , 1 , 1 , 2 ( r , t ) = µ I , 1 , 1 , 2 ( r ) cos  S t 2 t + χ I , 1 , 1 , 2 ( r ) 

+ µ III , 1 , 1 , 2 ( r ) cos  | 2 S t 1 − S t 2 | t + χ III , 1 , 1 , 2 ( r )  (28)

+ µ II , 1 , 1 , 2 ( r ) cos  (2 S t 1 + S t 2 ) t + χ II , 1 , 1 , 2 ( r ) 

ξ 1 , 2 , 2 ( r , t ) = ξ I , 1 , 2 , 2 ( r , t ) + ξ II , 1 , 2 , 2 ( r , t ) + ξ III , 1 , 2 , 2 ( r , t ) = µ I , 1 , 2 , 2 ( r ) cos  S t 1 t + χ I , 1 , 2 , 2 ( r ) 

+ µ III , 1 , 2 , 2 ( r ) cos  | 2 S t 2 − S t 1 | t + χ III , 1 , 2 , 2 ( r )  (29)

+ µ II , 1 , 2 , 2 ( r ) cos  (2 S t 2 + S t 1 ) t + χ II , 1 , 2 , 2 ( r ) 

where, µ ( µ ≥ 0) and χ are the amplitudes and phases of the flame front wrinkles, respectively. The subscripts “I”, “II”, and “III” (if existent) denote the properties corresponding to the first, second and third disturbance frequencies included in the solution, respectively. For the V-shaped flame, the solution forms from Equation 21 ∼ Equation 29 are the same, but the specific expressions of µ ( r ) and χ ( r ) are di ff erent from those of the conical flame. A notable phenomenon is that the additional perturbation at the frequency S t 2 causes the third order nonlinear interaction of that and the first perturbation at S t 1 . It results in the additional flame response with respect to frequency S t 1 , which can be seen in the first term of right side hand in Equation 29 ( ξ I , 1 , 1 , 2 ). This is an important implication that the flame dynamic response at the first frequency is changed by the additional disturbance at a di ff erent frequency. There is a special case, when the additional frequency is the same as the first frequency, the mutual nonlinear e ff ect ξ I , 1 , 1 , 2 disappears. However, the perturbation amplitude corresponding to the first frequency is enhanced, resulting in an increase in ξ I , 1 , 1 , 1 . This also obviously a ff ects the flame response at the first disturbance frequency.

3. RESULTS AND DISCUSSES

The main objective of the current work is to elucidate the e ff ect of additional perturbation on the flame spatial response at the first perturbation frequency. Based on the previous near-order asymptotic analysis, the first fundamental frequency response of flame front kinematics can be written as

ξ S t 1 ( r , t ) = ϵ 1 ξ 1 ( r , t ) | {z } Linear solution + ϵ 1 3 ξ I , 1 , 1 , 1 ( r , t ) | {z } Self − nonlinear solution + ϵ 1 ϵ 2 2 ξ I , 1 , 1 , 2 ( r , t ) | {z } Mutual − nonlinear solution (30)

where, the subscript “ S t 1 ” denotes the solution of flame front kinematics at the fundamental frequency S t 1 . To determine the accuracy of the solution from Equation 30, comparison of these results with the numerical results are conducted. A Fourier series expansion is used for the flame front expression to get the first fundamental frequency flame response of the numerical results. The flame kinematic response at S t 1 of the numerical solution is obtained. µ N S t 1 ( r ) ( µ N ≥ 0) and χ N S t 1 ( r ) are the amplitude and phase of flame spatial response. The superscript “ N ” used to distinguish the asymptotic solution denotes the numerical result. As mentioned earlier, the magnitude of the flame response from the asymptotic analysis is required. Figure 5 compares the numerical results µ N S t 1 ( r ) and the asymptotic solutions µ S t 1 ( r ) of the first fundamental frequency flame response under disturbances with di ff erent frequencies and amplitudes, where µ S t 1 ( r ) is magnitude of ξ S t 1 ( r , t ). The spatial response amplitude of the flame at S t 1 obtained by the asymptotic method is in good agreement with that using the numerical method. When the input frequency is low or the position is close to the flame holder, the flame responds purely linearly. These characteristics are not a ff ected by the perturbation amplitude and the additional frequency. The magnitude of flame wrinkling generally increases along the flame. These mentioned characteristics were also observed in related experiments [28,29] because the flame propagates along its own normal direction and smooths its wrinkles. It is named as the flame intrinsic nonlinearity [6,10]. To understand how the additional perturbation at the di ff erent frequency a ff ects the spatial flame response at the first perturbation, quantitative results for each component and their mutual relation in Equation 30 are obtained. Figure 4 ( a ) presents the magnitude and phase of each solution of

Figure 3: Comparison of µ N S t 1 ( r ) /ϵ 1 and µ S t 1 ( r ) /ϵ 1 of the conical flame along the radial position r under the disturbance of di ff erent frequencies and amplitudes. ϵ 1 = ϵ 2 = 0.1 in the case of R = 0 . 3 and ϵ 1 = ϵ 2 = 0.2 in the rest cases ( µ S t 1 ( r ) is the magnitude of ξ S t 1 ( r , t ) in Equation 30 and cot α = 3).

( a ) ( b )

Figure 4: ( a ) The magnitude and phase of each term of the conical flame front at S t 1 in Equation 30. ( b ) The magnitude and phase of combined solutions of the conical flame surface at S t 1 . ( S t 1 = 5, S t 2 = 16, R = S t 2 / S t 1 , ϵ 1 = ϵ 2 = 0.2, cot α = 3).

Equation 30 in a general case (modulation frequency a ff ects K ). The magnitude of the self-nonlinear solution is smaller than that of the mutual nonlinear solution. The phases of di ff erent nonlinear terms are the same, whose propagation speed is about 3 times that of the linear solution. It can be explained that the nonlinear solutions are derived from the third-order PDEs. Figure 4 ( b ) shows the magnitude and phase of the flame spatial response, where ξ PL , S t 1 = ϵ 1 ξ 1 , ξ PN , S t 1 = ϵ 1 ξ 1 + ϵ 1 3 ξ I , 1 , 1 , 1 and ξ F N , S t 1 = ξ S t 1 + ϵ 1 3 ξ I , 1 , 1 , 1 + ϵ 1 2 ϵ 2 ξ I , 1 , 2 , 2 . Due to the introduction of nonlinear solutions (both the self-nonlinear and mutual-nonlinear terms), the magnitude and phase of the flame spatial response at the first frequency are corrected based on the linear results. Among them, with the increase of downstream distance, the amplitude of the flame wrinkle decreases, and the phase step phenomenon is suppressed. In fact, considering the relation between K and S t , µ I , 1 , 2 , 2 ( r ) is a ff ected by the additional frequency S t 2 and may be smaller than µ I , 1 , 1 , 1 ( r ), but χ I , 1 , 2 , 2 ( r ) is still equal to χ I , 1 , 1 , 1 ( r ). The relative value H S t 1 ( r ) of the mutual-nonlinear and self-nonlinear amplitudes is defined as µ I , 1 , 2 , 2 ( r ) − µ I , 1 , 1 , 1 ( r ), and corresponding results are shown in figure 5. For the conical flame in figure 5 ( a ) ( b ) ( c ), the mutual-nonlinear amplitude is smaller than that of self-nonlinear within a certain low range of the forcing frequency. With increasing the additional frequency S t 2 , µ I , 1 , 2 , 2 ( r ) monotonically increases first and then decreases resulting in a steady peak. With increasing the first frequency S t 1 , µ I , 1 , 2 , 2 ( r ) is usually larger than µ I , 1 , 1 , 1 ( r ), especially at the spatial location approaching the flame tip. It should be noticed that regardless of S t 1 , the steady peak of the mutual-nonlinear solution always exists and corresponds to the case of S t 2 ≈ 3 . 85, which is determined by the relation between the modulation frequency and K . It can be explained that the mutual nonlinear solution is governed by the modulation frequency and K , monotonically increasing with the development of the modulation frequency or K ( K ∈ [0 , 1]). However, there is a negative correlation between the modulation and K , resulting in a stable peak corresponding to a certain forcing frequency. This characteristic can be further verified in the V-shaped flame. Due to quantitative di ff erences in the relation between modulation frequency and K for di ff erent flames, S t 2 corresponding to the stable peak of the mutual nonlinear solution changes in the V-shaped flame, i.e., S t 2 ≈ 3 . 5 shown in figure 5 ( e ) ( f ). These values exactly match the results obtained for the relation between the modulation frequency and K in figure 2. It can be concluded that the introduction of an additional perturbation at S t 2 produces the mutually non-linear term, which in most cases is more e ff ective than the first perturbation in smoothing flame spatial wrinkles caused by the first perturbation. In addition, the mutual-nonlinear solution has a stable peak with the increase of the additional perturbation frequency, and it is controlled by the relation between S t and K .

4. CONCLUSIONS

This paper analysed the variation of the flame nonlinear response at the first perturbation frequency S t 1 by introducing an additional perturbation at the frequency S t 2 , to quantify its role in the flame response at S t 1 . The nonlinear results for the flame spatial response were derived from near-order asymptotic analysis and numerical methods based on the model framework of the G -equation. Due to the flame propagating forward normal to itself, from the near-order asymptotic analysis point of view, the third-order nonlinear interaction of the additional and first perturbations induces an external flame response relative to the first frequency S t 1 . It directly a ff ects the flame response at S t 1 .

ACKNOWLEDGEMENTS

The authors would like to gratefully acknowledge financial support from the Chinese National Natural Science Funds for National Natural Science Foundation of China (Grant Nos. 11927802 and U1837211), and National Major Science and Technology Projects of China (2017-III-0004-0028). The European Research Council grant AFIRMATIVE (20182023) is also gratefully acknowledged. The authors would also like to thank Dongbin Wang, Jiaqi Nan and Shuoshuo Zhu for their assistance.

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REFERENCES

[1] A. P. Dowling and A. S. Morgans. Feedback control of combustion oscillations. Annual Review of Fluid Mechanics , 37:151–182, 2005. [2] T. Poinsot. Prediction and control of combustion instabilities in real engines. Proceedings of the Combustion Institute , 36(1):1–28, 2017. [3] M. P. Juniper and R. I. Sujith. Sensitivity and nonlinearity of thermoacoustic oscillations. Annual Review of Fluid Mechanics , 50:661–689, 2018. [4] T. Schuller, T. Poinsot, and S. Candel. Dynamics and control of premixed combustion systems based on flame transfer and describing functions. Journal of Fluid Mechanics , 894:P1, 2020. [5] T. Schuller, D. Durox, and S. Candel. A unified model for the prediction of laminar flame transfer functions : comparisons between conical and V-flame dynamics. Combustion and Flame , 134(1- 2):21–34, 2003. [6] T. Lieuwen. Nonlinear kinematic response of premixed flames to harmonic velocity disturbances. Proceedings of the Combustion Institute , 30(2):1725–1732, 2005. [7] T. Schuller, S. Ducruix, D. Durox , and S. Candel. Modeling tools for the prediction of premixed flame transfer functions. Proceedings of the Combustion Institute , 29(1):107–113, 2002. [8] H.J. Krediet, C.H. Beck, W. Krebs, and J.B.W. Kok. Saturation mechanism of the heat release response of a premixed swirl flame using LES. Proceedings of the Combustion Institute , 34(1):1223–1230, 2013. [9] G. Markstein. Nonsteady flame propagation . Oxford: Pergamon Press, 1964. [10] R. Preetham, H. Santosh, and Tim Lieuwen. Dynamics of laminar premixed flames forced by harmonic velocity disturbances. Journal of Propulsion and Power , 24(6):1390–1402, November 2008. [11] A. Orchini and M. P. Juniper. Flame double input describing function analysis. Combustion and Flame , 171:87–102, 2016. [12] V. Acharya and T. Lieuwen. Nonlinear response of swirling premixed flames to helical flow disturbances. Journal of Fluid Mechanics , 896:A6, 2020. [13] V. Acharya and T. C. Lieuwen. Non-monotonic flame response behaviors in harmonically forced flames. Proceedings of the Combustion Institute , 38(4):6043–6050, 2020. [14] A. Lamraoui, F. Richecoeur, S. Ducruix, and T. Schuller. Experimental analysis of simultaneous non-harmonically related unstable modes in a swirled combustor. Proceedings of ASME Turbo Expo 2011 , pages 1289–1299, 2011. [15] A. Albayrak, T. Steinbacher, T. Komarek, and W. Polifke. Convective scaling of intrinsic thermo-acoustic eigenfrequencies of a premixed swirl combustor. J. Eng. Gas Turb. Power , 140(4):041510, April 2018. [16] Y. Guan, V. Gupta, M. P. Wan, and L. K. B. Li. Forced synchronization of quasiperiodic oscillations in a thermoacoustic system. Journal of Fluid Mechanics , 879:390–421, November 2019. [17] A. Roy, S. Mondal, S. A. Pawar, and R. I. Sujith. On the mechanism of open-loop control of thermoacoustic instability in a laminar premixed combustor. Journal of Fluid Mechanics , 884:A2, February 2020. [18] M. Haeringer, M. Merk, and W. Polifke. Inclusion of higher harmonics in the flame describing function for predicting limit cycles of self-excited combustion instabilities. Proceedings of the Combustion Institute , 37(4):5255–5262, 2019. [19] M. Haeringer and W. Polifke. Time-domain bloch boundary conditions for e ffi cient simulation of thermoacoustic limit cycles in (can-)annular combustors. Journal of Engineering for Gas Turbines and Power , 141(12):121005, December 2019.

[20] N. Tathawadekar, N. A. K. Doan, C. F. Silva, and N. Thuerey. Modeling of the nonlinear flame response of a bunsen-type flame via multi-layer perceptron. Proceedings of the Combustion Institute , 38(4):6261–6269, 2021. [21] X. Han, J. Yang, and J. Mao. LES investigation of two frequency e ff ects on acoustically forced premixed flame. Fuel , 185:449–459, 2016. [22] X. Jiang, L. Yang, T. Liu, and J. Li. Nonlinear models of laminar premixed slit flame responses subjected to two-way perturbations. AIAA Journal , 60(2):962–975, 2022. [23] X. Jiang, J. Li, L. Yang, and T. Liu. A nonlinearly kinematic model of the asymmetrically turbulent premixed slit flame subjected to two-way harmonic disturbances. Combustion and Flame , 240:112021, 2022. [24] A.L. Birbaud, D. Durox, and S. Candel. Upstream flow dynamics of a laminar premixed conical flame submitted to acoustic modulations. Combustion and Flame , 146(3):541–552, 2006. [25] Y. Yang, Y. Fang, L. Zhong, Y. Xia, T. Jin, J. Li, and G. Wang. DMD analysis for velocity fields of a laminar premixed flame with external acoustic excitation. Experimental Thermal and Fluid Science , 123:110318, 2021. [26] G.-S. Jiang and D. Peng. Weighted eno schemes for hamilton–jacobi equations. SIAM Journal on Scientific Computing , 21(6):2126–2143, 2000. [27] C. W. Shu. High order weighted essentially nonoscillatory schemes for convection dominated problems. Siam Review , 51(1):82–126, March 2009. [28] S. Shanbhogue, D.-H. Shin, S. Hemchandra, D. Plaks, and T. Lieuwen. Flame-sheet dynamics of blu ff -body stabilized flames during longitudinal acoustic forcing. Proceedings of the Combustion Institute , 32(2):1787–1794, 2009. [29] D.-H. Shin, D.V. Plaks, T. Lieuwen, U.M. Mondragon, C. T. Brown, and V. G. McDonell. Dynamics of a longitudinally forced, blu ff body stabilized flame. Journal of Propulsion and Power , 27(1):105–116, 2011.