## Abstract

Premixed flames are sensitive to flow disturbances, which can arise from acoustic or vortical fluctuations. For transverse instabilities, it is known that a dominant mechanism for flame response is “injector coupling,” whereby pressure oscillations associated with transverse waves excite axial flow disturbances. These axial flow disturbances then excite heat release oscillations. The objective of this paper is to consider another mechanism—the direct sensitivity of the unsteady heat release to transverse acoustic waves—and to compare its significance relative to the induced axial disturbances, in a linear framework. The rate at which the flame adds energy to the disturbance field is quantified using the Rayleigh criterion and evaluated over a range of control parameters, such as flame length and swirl number. The results show that radial modes induce heat release fluctuations that always add energy to the acoustic field, whereas heat release fluctuations induced by mixed radial-azimuthal modes can add or remove energy. These amplification rates are then compared to the flame response from induced axial fluctuations. For combustor-centered flames, these results show that the direct excitation mechanism has negligible amplification rates relative to the induced axial mechanism for radial modes. For transverse modes, the fact that the nozzle is located at a pressure node indicates that negligible induced axial velocity disturbances are excited; as such, the direct mechanism dominates. For flames that are not centered on pressure nodes, the direct mechanism for mixed modes dominates for certain nozzle locations and flame angles.

## Introduction

Combustion instabilities are a common issue in gas turbines, presenting themselves in various forms, including low-frequency longitudinal acoustic modes in can combustors [14], low-frequency transverse acoustic modes in annular combustors [5,6], and high-frequency transverse acoustic modes in can combustion systems [7] and jet engine augmentors [812]. Combustion instabilities occur because heat release oscillations can add energy to the acoustic field, as given by [13]
$1Ea∂Ea∂t=1Ea(γ−1ρ0c02)∫Vp1q˙1dV$
(1)
This expression shows that the growth or decay of acoustic energy for a given mode ($Ea$) in the system is governed by the relationship between unsteady heat release ($q˙1$) and unsteady pressure ($p1$). This growth rate is proportional to the Rayleigh integral [14]
$RI=∫t∫Vp1q˙1dVdt$
(2)
Over the last two decades, significant work has been performed on low-frequency longitudinal instabilities [1,1517]. Typically, gas turbines operate with flame lengths that are small relative to this long acoustic wavelength (i.e., acoustically compact), and so Eq. (2) reduces to
$RI=∫tp1(t)∫Vq˙1dV︸Q˙1(t)dt=∫tp1(t)Q˙1(t)dt$
(3)

In this case, it is the relative phase of the spatially integrated heat release that controls the growth/decay rate. For this reason, the spatially integrated heat release, characterized with the flame transfer function (FTF), was often used to characterize low-frequency thermoacoustics and to quantify the sensitivity of $Q˙1(t)$ to various coupling pathways, namely disturbances in velocity, equivalence ratio, and pressure.

Several of these velocity coupling pathways are illustrated in Fig. 1. There are several direct and indirect pathways through which flame excitation occurs. First, the acoustic motions can directly excite unsteady heat release, indicated as $FT$. Second, the transverse acoustic motions can introduce axial acoustic motions ($FTL$) in the nozzle–combustor juncture due to the oscillating pressure field [18,19]. The direct and induced motions both excite vortical disturbances ($FTω,FLω$) due to interactions with flow instabilities that also cause a flame response at the convective length-scale.

Fig. 1
Fig. 1
Close modal

Work by Acharya et al. [20] has clearly shown that, for compact flames, the $FT$ pathway is negligible, as the FTF scales as $kLf$ (where $Lf$ is a characteristic flame length), which by assumption is a small parameter. However, a key distinction of high-frequency transverse instabilities is that the flame is not acoustically compact and thus the simplification from Eq. (2) to Eq. (3) does not apply. Thus, the FTF is not the correct marker for system stability. A simple example of this is the stability of the first transverse (1-T) mode. The heat release distribution that maximizes the growth rate for this mode is one where the left and right halves of the unsteady heat release are out of phase with each other; i.e., even though $Q˙1(t)$ and FTF are identically zero, the growth rate of the mode is positive.

The role of the $FT$ path relative to the other pathways for noncompact flames requires further analysis. Recent phenomenological works by Sattelmayer and coworkers [2125] and Méry [26] suggest that transverse modes may be intrinsically unstable due to this direct acoustic velocity coupling mechanism. That is, there are no islands of growth or decay, but rather this mechanism is inherently destabilizing due to the phasing between unsteady pressure and heat release. This result should be contrasted with typical experience for axial instabilities. A typical feature of axial instabilities is the nonmonotonic dependence of system stability upon control parameters. Thus, growth/decay occurs in “islands,” such as in discrete regions of the convective time delay or combustor length parameter space. There is no such thing as an intrinsically stable injector or mixer; rather, the same injector can be stable or unstable, depending upon the larger system in which it is placed.

Using a rigorous representation of premixed flame sheet space–time dynamics, prior work by the authors has further explored the $FT$ pathway for several configurations: (a) an axisymmetric flame in a combustor axis centered nozzle [27], (b) an axisymmetric flame with a nozzle offset from the combustor axis [28], and (c) a nonaxisymmetric flame [29]. In each case the normalized growth/decay rate (RI in Eq. (1)) was evaluated. In case (a), results indicated that purely radial modes had $RI>0$ values for all cases and, thus, were destabilizing in all instances—this corroborated prior arguments that some transverse modes are always destabilized by the $FT$ route. These results also showed that there were regions in the control parameter space where, for example, the first radial-azimuthal (1, 1) mode had $RI<0$. Further expanding this analysis to case (b), the authors showed that radial and azimuthal offsets had strong effects on RI depending on the mode. In case (c), mean flame asymmetries that represent strong flame–flame interactions were considered and were shown to have a significant effect on growth/decay, depending on the nature of the asymmetry and mode being considered.

To summarize current knowledge, it is understood that the $FT$ route can be intrinsically destabilizing. However, what is not clear is whether these $RI>0$ values are sufficiently strong to overcome the RI values associated with the other indirect pathways that can have $RI<0$ and $RI>0$ values. This motivates the study presented in this paper, wherein we consider the response of the flame to the induced high-frequency axial velocity oscillations ($FTL,FL$ in Fig. 1). As mentioned before, these induced axial velocity fluctuations have the same spatial distribution as the mode that induces them. While it was shown in prior work [20] that nonaxisymmetric disturbances in axial motions lead to no global flame response for compact flames, in this study the axial velocity oscillations are at the high-frequency corresponding to the transverse mode and thus the Rayleigh Index (see Eq. (2), not just the spatially integrated heat release oscillations) due to these induced motions must be calculated to determine the eventual growth rate due to both motions. The new main parameters in this comparison are the transfer function combination ($FTLFL$), the azimuthal-radial transverse mode in question ($l,ma$) and the axial phase speed of the induced disturbances (due to excited vortical flow disturbances). For the purpose of this analysis, we shall focus only on axisymmetric mean flames but shall consider both combustor axis centered flames and axis offset flames in a cylindrical can-combustor geometry.

## Mathematical Formulation

In this section, we present the modeling framework for the space–time dynamics of a thin premixed flame front followed by the acoustic pressure and velocity models for a cylindrical can combustor. The presented formulation is a reiteration of prior efforts by the authors and is only presented briefly here [2729].

### Flame and Heat Release Dynamics.

For the purpose of the analysis, we assume a thin premixed flame with a single valued flame position as shown in the flame schematic in Fig. 2. The space–time dynamics of the flame position is governed by
$∂ξ∂t+ur∂ξ∂r+uθr∂ξ∂θ+sL1+(∂ξ∂r)2+(1r∂ξ∂θ)2=uz$
(4)
Fig. 2
Fig. 2
Close modal
This equation is derived from the level-set equation under certain assumptions detailed in prior work [2729]. The flame base is assumed to be fixed to the center-body rim
$ξ(r=R,θ,t)=0$
(5)
We linearize the flame position and velocity by decomposing them into a time-invariant mean (subscript 0) and a first-order perturbation (subscript 1) as
$ui(r,θ,t)=ui,0(r)+ui,1(r,θ,t)ξ(r,θ,t)=ξ0(r)+ξ1(r,θ,t)$
(6)
Here, we have assumed the mean quantities to be axisymmetric for this analysis. Substituting the decomposition in Eq. (4), the governing equations for the mean and perturbed flame position are
$ur,0dξ0dr+sL1+(dξ0dr)2=uz,0$
(7)
$∂ξ1∂t+(ur,0+sLξ0,r1+ξ0,r2)∂ξ1∂r+uθ,0r∂ξ1∂θ=uz,1−ur,1dξ0dr$
(8)

Note that one of the assumptions used here is that the flame speed is held constant (no flame-stretch effects). Using these equations with an assumed mean flow-field and the perturbation velocity field from the natural acoustic mode or induced axial fluctuations, the mean flame shape and flame position dynamics are determined. Next, we focus on the velocity and pressure field due to the natural transverse acoustic mode.

### Natural Transverse Acoustic Modes.

For the analysis in this paper, a cylindrical combustor with radius $RC$ is used. The polar coordinate system used for the cylindrical combustor's axis and for the nozzle injector is shown in Fig. 3.

Fig. 3
Fig. 3
Close modal
The purely transverse acoustic pressure and velocity fields for radial-azimuthal mode ($l,ma$) in the combustor coordinate system (subscript c) are given by [13]
$p̂c,1(rc,θc)ρ0c02=Jma(αl,marc)cos(maθc)$
(9)
$ûrc,1(rc,θc)c0=i2[Jma+1(αl,marc)−Jma−1(αl,marc)]cos(maθc)ûθc,1(rc,θc)c0=imaαl,marcJma(αl,marc)sin(maθc)$
(10)
Here, $Jma$ is the Bessel function of the first kind, of order $ma$. The overhat notation is used to denote frequency domain equivalents of the time-domain quantities. The coordinates have been normalized by the combustor radius, $RC$ and the velocities have been normalized by $c0$. The nondimensional Eigen-frequency is given by
$ω0RCc0=αl,ma$
(11)
where, $αl,ma$ are determined as roots of
$J′ma(αl,ma)=0$
(12)
The combustor-centered radial and azimuthal velocities in Eq. (9) are transformed to the flame-centered coordinate system using the transformation
$rc(r,θ)θc(r,θ)→{rc2=RCB2+r2+2RCBr cos(θ−θCB) tan θc=RCB sin θCB+r sin θRCB cos θCB+r cos θ$
(13)
For no offset (i.e., $RCB=0$), we retrieve: $rc≡r, θc≡θ$. The velocity field in the cylinder's coordinate system is transformed to the flame's coordinate system using
$ûr,1(r,θ)={ûrc,1(rc(r,θ),θc(r,θ))cos(θ−θc(r,θ))+ûθc,1(rc(r,θ),θc(r,θ))sin(θ−θc(r,θ))ûθ,1(r,θ)={−ûrc,1(rc(r,θ),θc(r,θ))sin(θ−θc(r,θ))+ûθc,1(rc(r,θ),θc(r,θ))cos(θ−θc(r,θ))$
(14)

For a given acoustic azimuthal mode $ma$, when there is no flame offset, the only azimuthal mode in the flow-field is also $ma$. However, for an offset flame, the flow-field at the flame consists of an infinite range of helical modes due to the nature of the transformation in Eqs. (13) and (14). Note that for the axisymmetric flame, the azimuthal velocity component $ûθ,1$ does not affect the flame response. However, note that the azimuthal velocity component in the acoustic field $ûθc,1$ does affect the flame response through the radial velocity component $ûr,1$. In prior work, the nozzle offset has been shown to have a strong effect on the growth/decay rate of the modes [28]. Thus, in this paper we consider this geometric parameter as well when comparing the response between transverse and axial fluctuations. More importantly, the nozzle location results in differing induced axial fluctuations for the same transverse mode.

### Induced Axial Velocity Disturbances.

Depending on the nature of the acoustic mode in the combustor and its pressure field, axial velocity oscillations are induced at the nozzle. As mentioned before, the location of the nozzle within a given mode also has impacts on both the amplitude and nature of azimuthal distribution of the induced velocity.

For the purpose of modeling in this paper, we do not consider the geometric and acoustic details of the nozzle injector or any upstream section. Rather, we assume that the induced axial velocity fluctuations can be related to the pressure field at the nozzle–combustor juncture through an impedance ($ρ0c0ẐTL$), following Blimbaum et al. [18]. This impedance parameter is related to the transfer function $FTL$ shown in Fig. 1. Prior work [18] has shown that the induced axial velocity is related to the spatially averaged pressure field at the nozzle but incorporates the same azimuthal distribution as the pressure distribution [19]. Thus, we consider the radially averaged pressure field at the nozzle and the induced axial velocity to be related at each azimuthal location through a specified impedance $ẐTL$; i.e., using Eqs. (9) and (13), the induced axial fluctuations at the nozzle outlet are given by
$ûz,N(θ)c0=∫nozzlerJma(αl,marc(r,θ))cos(maθc(r,θ))drẐTL∫nozzlerdr$
(15)
An additional model is also needed to relate the axial velocity at the nozzle outlet to that at the flame. Here we use a traveling wave model (to denote the convection of excited vortical flow fluctuations), so that the axial velocity at the flame is given by
$ûz,1(θ,z)=ûz,N(θ)eiαl,maz$
(16)

Note that this induced axial velocity model requires further study for its validity and is not a scope of this work. This induced axial velocity model is used in Eq. (8) to determine the role of axial fluctuations on the overall growth/decay rate. This growth/decay rate is represented by the Rayleigh index, discussed next.

### Mode Stability—Rayleigh Index.

Using the velocity disturbances in the domain, we can determine the growth/decay rate for a mode for the direct excitation acoustic field and the induced axial velocity field separately. Note that the overall growth rate is a superposition of these two for linear dynamics and thus we can determine the values for these two fields separately.

In this analysis, we assume a constant flame speed and thus the spatio-temporal unsteady heat release is proportional to the local flame surface area oscillations
$dA=rdrdθ1+(∂ξ∂r)2+(1r∂ξ∂θ)2$
(17)
Using the perturbation expansion in Eq. (6), the mean and fluctuating heat release are then proportional to
$q˙0(r)=1+ξ0,r2(r)q˙1(r,θ,t)=ξ0,r(r)1+ξ0,r2(r)ξ1,r(r,θ)$
(18)
The Rayleigh index in Eq. (2) is then evaluated using Eqs. (9) and (18) as
$RI=∫tdt∫rξ0,rrdr1+ξ0,r2∫θ=02π[Re{p̂1e−iω0t}Re{ξ̂1,r(r,θ)e−iω0t}]dθ$
(19)
The growth rate for the mode as shown by the RHS of Eq. (1) is the Rayleigh index normalized by the acoustic energy of the mode and is defined as
$ℜ=RI2Ea$
(20)
Here, the integrated acoustic energy at the chosen Eigen-frequency is given by
$Ea=∫tdt∫rrdr∫θ=02π[Re{p̂1e−iω0t}Re{p̂1e−iω0t}+12Re{û1e−iω0t}·Re{û1e−iω0t}]dθ$
(21)

Note that both RI and $Ea$ are evaluated over the two-dimensional transverse domain only since we focus on purely transverse acoustic modes. This growth rate can be calculated for different values of the control parameters and can then be used to determine if it is positive or negative for a given mode for a chosen flame/flow configuration. Thus for a given mode, the growth rate due to direct excitation (denoted as $ℜT$) will be compared to that due to the induced axial excitation (denoted as $ℜL$) to determine the significance of the direct excitation mechanism.

## Example Illustration

In this section, we present an example illustration to compare the effect of different control parameters on the relative significance of the different excitation mechanisms. For convenience of notation, the velocity field along coordinate j is decomposed into azimuthal components (mode number $m$) as
$ûj,1(r,θ)=∑m=−∞+∞Γ̂j,m(r)eimθ$
(22)
The following mean-flow and flame configuration is assumed similar to earlier work:
$uz,0=M0ur,0=0uθ,0=Ωrc0}⇒{ξ0(r)=(r−βR)cotψsLc0=M0 sin ψ$
(23)
$βR=RRC βRf=RfRC βf=LfRC$
(24)
Note that only $βR, βRf$ are the free parameters since
$βf=(βRf−βR)cotψ$
(25)

The local flame dynamics solution is calculated and used in the unsteady heat release rate calculation in Eq. (18) following which the Rayleigh index can be calculated. We shall denote the growth rate due to the transverse excitation (direct mechanism) as $ℜT$ and that due to the induced axial fluctuations (indirect mechanism) as $ℜL$. We shall now compare these two quantities for variations in key control parameters.

### Flame Length Effects.

In this example, we consider the effect of flame length (through the flame angle ($ψ$)) on the growth rates. First consider the response to the axisymmetric transverse radial mode $(l,ma)=(2,0)$ and its comparison to that due to the induced axial fluctuations shown in Fig. 4. For the purpose of this comparison, we set: $βRf=3βR=0.15$, $M0=0.1$ for a nozzle centered in the combustor (i.e., $RCB=0.0$). The nozzle impedance is set as $ẐTL=1$, which corresponds to the upstream nozzle traveling wave case; i.e., no reflected downstream acoustic waves exist.

Fig. 4
Fig. 4
Close modal

As seen in the figure, the growth rate for the transverse mode (subscript T) is strictly positive for all cases as shown in prior work. However, note the much larger magnitudes of the oscillatory positive and negative growth rate for the induced axial fluctuations (subscript L). This oscillatory behavior has been alluded to before in the Introduction in the context of stability bands, as the pressure and heat release move in and out of phase. An important takeaway from this figure is that the induced axial contribution is much larger, i.e., while the natural transverse mode may have a positive growth rate, the overall growth rate can be negative in certain bands of the flame angle space. This has strong implications on the importance of the direct excitation mechanism in assessing mode behavior. As the growth rate of the induced axial fluctuations scale as $1/|ẐTL|$ (see Eq. (15)), it is possible that under nozzle antiresonance conditions (i.e., where the induced axial velocity is near zero), that the direct term could dominate; however, this result shows that overall the direct term has a small magnitude relative to typical values for axial excitation.

Next, consider the response to the first transverse mixed mode $(l,ma)=(1,1)$ and its comparison to that due to the induced axial fluctuations shown in Fig. 5. In contrast to the radial mode, the mixed mode has bands of positive and negative growth rate (subscript T) as shown in prior studies by the authors. An important result from this plot is that unlike the radial mode case, the mixed mode shows a stronger dependence of the dynamics on the direct excitation mechanism as evidenced by the total growth rate (dashed curve) and the growth rate of the transverse mode (subscript T) closely following each other. The induced mechanism is nearly zero in this case, because the acoustic pressure has a node at the nozzle centerline. Using the axial velocity model, Eq. (16), this implies that $uz,1$ is positive on one-half of the flame and negative on the other; only the fact that the flame is noncompact causes $ℜL≠0$ in this case. A similar conclusion would hold for all transverse modes where $ma≠0$. This implies that the direct excitation mechanism can be the dominant contributor to the growth rate, albeit with very weak amplification values. Thus, depending on the mode in question, the dominant mechanism changes.

Fig. 5
Fig. 5
Close modal

### Nozzle Location Effects.

In this example, we consider the effect of the nozzle position ($RCB, θCB$ in Fig. 3) on the growth rate comparisons. First, consider the symmetric radial mode, (2, 0) where only $RCB$ has an effect (no effect of $θCB$). The comparison between the growth rates is as shown in Fig. 6. For this case, we choose $ψ=40 deg$ and we start with $RCB≥2$ so that the flame surface does not pass through the origin—this is practically seen in combustors where the offset nozzle locations are chosen such that there is enough room for the center flame. Note that as the nozzle is offset radially, the growth rate from the induced axial fluctuations varies greatly. Moreover, the growth rate changes sign at a certain radial offset location while the growth rate from the induced axial excitation remains negative. However, the overall growth rate hovers about the induced axial response (subscript L) but is clearly affected in a non-negligible way by the transverse mode (see dashed curve) indicating that for certain flame angles, for nozzles that are radially offset from the combustor axis, the direct excitation mechanism is important for the first radial mode. Although not shown here, similar qualitative behavior is seen for the case of radial offset effects on the growth rate comparison for the 1-T mode.

Fig. 6
Fig. 6
Close modal

Next, we focus on azimuthal offset effects for the 1-T mode. For this comparison, we consider the radial offset fixed at $RCB=0.3$ as the nozzle is azimuthally moved around in the combustor. First consider the flame angle case of $ψ=π/4$ shown in Fig. 7(a). Note that as the nozzle is moved azimuthally around the combustor, the growth rate due to the induced axial fluctuations is insensitive to this movement. Also note that around $θCB=90 deg, 270 deg$, the growth rate is nearly 0 since this corresponds to the pressure nodes for the 1-T node. In contract, the direct excitation is very sensitive to azimuthal movement, as evidenced by the sharp change of the growth rate from negative to positive around $θCB=90 deg, 270 deg$. Moreover, it is clearly seen that the overall growth rate is mostly controlled by the transverse mode. This implies that outer nozzles are strongly sensitive to the direct excitation mechanism and it is possible for the growth rate to be strongly controlled by this mechanism.

Fig. 7
Fig. 7
Close modal

Next, consider the flame angle case of $ψ=π/3$ shown in Fig. 7(b). Here, the sensitivity of the direct excitation growth rate (subscript T) is lesser than that due to the induced axial mechanism. This results in the overall growth rate (dashed curve) being determined largely by the induced axial motions (subscript L) resulting in a behavior contrary to that seen in the previous case. Thus, the azimuthal offset has differing effects on the relative response, depending on the flame length.

Thus, for nozzles centered in the combustor, the induced axial excitation dominates the growth rate for the radial (2, 0) mode while the transverse excitation dominates the growth rate for the 1-T mode. For nozzles that are offset from the combustor axis, the transverse excitation mechanism was seen to be dominant for both the radial (2, 0) mode as well as the mixed 1-T mode. This has important implications on modeling the combustion dynamics of multinozzle can combustion systems under transverse excitation.

## Conclusions

High-frequency transverse combustion instabilities have received increased attention in recent literature. In this paper, we focused on the velocity coupling pathway and discussed several routes for transverse acoustic excitation to cause flame response. In prior research, the authors focused on the direct acoustic excitation mechanism and using a level-set model for the premixed flame, computed the growth rate for the radial and 1-T modes for different control parameters. However, the authors have separately shown that transverse acoustic motions in the combustor induce axial motions at the nozzle–combustor juncture. In this paper, we compared the growth rates from the direct excitation mechanism and this induced axial response mechanism in order to determine the relevance of the different mechanisms on the overall growth rate for a given transverse mode. For nozzles centered in the combustor, the radial mode response was seen to be insignificant when compared to the induced axial response. In contrast, the opposite was seen for the 1-T mode. When the nozzles were offset from the combustor axis, it was shown that depending on the radial offset, azimuthal offset, and flame angle, either mechanism could be dominant or both mechanisms could be collectively important. This implies that depending on flame properties and the nozzle in question (in the multinozzle can), modeling the growth rate may require the inclusion of the induced axial mechanism along with the direct excitation mechanism. While the results are based on a first principles-based assumption for the induced axial motions, further detailed studies are required on the functional forms and nature of these induced axial motions in order to make accurate comparisons of the growth rate due to these two modes. This requires a combination of detailed experiment measurements and finite element simulations in order to better understand the multidimensional nature of acoustic modes and the induced vortical flow fluctuations at the different nozzle injectors.

## Funding Data

• This work has been supported by the U.S. Department of Energy (DOE) through the University Turbine Systems Research (UTSR) program under Contract No. DE-FE0031285 monitored by Mark Freeman. The numerical computations over the parametric space were performed on the Comet (SDSC) and Bridges (PSC) clusters through XSEDE charge Grant No. TG-CTS160017 (Funder ID: 10.13039/100000015) under PI Dr. Vishal Acharya.

## Nomenclature

• $c0$ =

speed of sound

•
• $Ea$ =

volume integrated acoustic energy of a mode

•
• $l$ =

•
• $Lf$ =

flame height

•
• $ma$ =

acoustic azimuthal mode number

•
• $M0$ =

Mach number, $=u0/c0$

•
• $p$ =

acoustic pressure

•
• $q˙$ =

unsteady heat release rate per unit volume

•
• $R$ =

radius of center-body for flame stabilization

•
• $ℜ$ =

nondimensional growth rate of mode, $=RI/2Ea$

•
• $RC$ =

•
• $RCB$ =

•
• $Rf$ =

•
• $RI$ =

Rayleigh Index

•
• $sL$ =

laminar flame speed

•
• $St$ =

Strouhal number, $=ω0Lf/u0$

•
• $ui$ =

velocity along coordinate direction i

•
• $u0$ =

characteristic velocity

•
• $V$ =

combustor control volume

•
• $ẐTL$ =

normalized acoustic impedance of nozzle

•
• $αl,ma$ =

nondimensional frequency for mode $(l,ma)$

•
• $βf$ =

normalized flame height, $=Lf/RC$

•
• $βR$ =

normalized center-body radius, $=R/RC$

•
• $βRf$ =

normalized radial flame extent, $=Rf/RC$

•
• $Γi,m$ =

azimuthal mode amplitude for i-direction velocity

•
• $ξ$ =

local flame position

•
• $θCB$ =

azimuthal offset of flame center-body

•
• $ρ0$ =

density

•
• $ω0$ =

angular frequency of acoustic mode

•
• $Ω$ =

azimuthal flow frequency for swirling flow

•
• $()̂$ =

Fourier transformed variable

•
• $()c$ =

quantity in combustor coordinate system

•
• $()r$ =

•
• $(),r$ =

•
• $()z$ =

axial component

•
• $(),z$ =

axial derivative

•
• $()θ$ =

azimuthal component

•
• $(),θ$ =

azimuthal derivative

•
• $()0$ =

time-averaged component

•
• $()1$ =

## References

1.
Lieuwen
,
T. C.
, and
Yang
,
V.
,
2005
,
Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling
,
American Institute of Aeronautics and Astronautics
,
Reston, VA
.
2.
Mongia
,
H. C.
,
Held
,
T. J.
,
Hsiao
,
G. C.
, and
Pandalai
,
R. P.
,
2005
, “
Incorporation of Combustion Instability Issues Into Design Process: GE Aeroderivative and Aero Engines Experience
,”
Prog. Astronaut. Aeronaut.
,
210
(
43
), Chapter 3.https://arc.aiaa.org/doi/abs/10.2514/5.9781600866807.0043.0063
3.
Cohen
,
J.
,
Hagen
,
G.
,
Banaszuk
,
A.
,
Becz
,
S.
, and
Mehta
,
P.
,
2011
, “
Attenuation of Gas Turbine Combustor Pressure Oscillations Using Symmetry Breaking
,”
AIAA Paper No. 2011-60
.10.2514/6.2011-60
4.
Smith
,
K. O.
, and
Blust
,
J.
,
2005
, “
Combustion Instabilities in Industrial Gas Turbines: Solar Turbines' Experience
,”
Combustion Instabilities in Gas Turbine Engines—Operational Experience, Fundamental Mechanisms, and Modeling
,
AIAA
,
Reston, VA
, pp.
29
42
.https://arc.aiaa.org/doi/10.2514/5.9781600866807.0029.0041
5.
Krebs
,
W.
,
Bethke
,
S.
,
Lepers
,
J.
,
Flohr
,
P.
,
,
B.
,
Johnson
,
C.
, and
Sattinger
,
S.
,
2005
, “
Thermoacoustic Design Tools and Passive Control: Siemens Power Generation Approaches
,”
Combustion Instabilities in Gas Turbine Engines Operational Experience, Fundamental Mechanisms and Modeling
,
AIAA
,
Reston, VA
.https://arc.aiaa.org/doi/10.2514/5.9781600866807.0089.0112
6.
Zellhuber
,
M.
,
Meraner
,
C.
,
Kulkarni
,
R.
,
Polifke
,
W.
, and
Schuermans
,
B.
,
2013
, “
Large Eddy Simulation of Flame Response to Transverse Acoustic Excitation in a Model Reheat Combustor
,”
ASME J. Eng. Gas Turbines Power
,
135
(
9
), p.
091508
.10.1115/1.4024940
7.
Sewell
,
J. B.
, and
Sobieski
,
P. A.
,
2005
, “
Monitoring of Combustion Instabilities: Calpine's Experience
,”
Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling
, Vol.
210
, AIAA, Reston, VA, pp.
147
162
.https://arc.aiaa.org/doi/10.2514/5.9781600866807.0147.0162
8.
Ebrahimi
,
H.
,
2006
, “
Overview of Gas Turbine Augmentor Design, Operation, and Combustion Oscillation
,”
AIAA Paper No. 2006-4916
.10.2514/6.2006-4916
9.
Rogers
,
D. E.
, and
Marble
,
F. E.
,
1956
, “
A Mechanism for High-Frequency Oscillation in Ramjet Combustors and Afterburners
,”
J. Jet Propul.
,
26
(
6
), pp.
456
462
.10.2514/8.7049
10.
Elias
,
I.
,
1959
, “
Acoustical Resonances Produced by Combustion of a Fuel‐Air Mixture in a Rectangular Duct
,”
J. Acoust. Soc. Am.
,
31
(
3
), pp.
296
304
.10.1121/1.1907715
11.
,
W.
, and
Noreen
,
A.
,
1955
, “
High-Frequency Oscillations of a Flame Held by a Bluff Body
,”
ASME Trans.
,
77
(
6
), pp.
855
891
.
12.
King
,
C. R.
,
1957
, “
Experimental Investigation of Effects of Combustion-Chamber Length and Inlet Total Temperature, Total Pressure, and Velocity on Afterburner Performance
13.
Lieuwen
,
T. C.
,
2012
,
,
Cambridge University Press
,
Cambridge, UK
.
14.
Rayleigh
,
L.
,
1878
, “
The Explanation of Certain Acoustical Phenomena
,”
Roy. Inst. Proc.
,
8
, pp.
536
542
15.
Huang
,
Y.
, and
Yang
,
V.
,
2009
, “
Dynamics and Stability of Lean-Premixed Swirl-Stabilized Combustion
,”
Prog. Energy Combust. Sci.
,
35
(
4
), pp.
293
364
.10.1016/j.pecs.2009.01.002
16.
Candel
,
S.
,
Durox
,
D.
,
Schuller
,
T.
,
Bourgouin
,
J.-F.
, and
Moeck
,
J. P.
,
2014
, “
Dynamics of Swirling Flames
,”
Annu. Rev. Fluid Mech.
,
46
(
1
), pp.
147
173
.10.1146/annurev-fluid-010313-141300
17.
Venkataraman
,
K.
,
Preston
,
L.
,
Simons
,
D.
,
Lee
,
B.
,
Lee
,
J.
, and
Santavicca
,
D.
,
1999
, “
Mechanism of Combustion Instability in a Lean Premixed Dump Combustor
,”
J. Propul. Power
,
15
(
6
), pp.
909
918
.10.2514/2.5515
18.
Blimbaum
,
J.
,
Zanchetta
,
M.
,
Akin
,
T.
,
Acharya
,
V.
,
O'Connor
,
J.
,
Noble
,
D.
, and
Lieuwen
,
T.
,
2012
, “
Transverse to Longitudinal Acoustic Coupling Processes in Annular Combustion Chambers
,”
Int. J. Spray Combust. Dyn.
,
4
(
4
), pp.
275
297
.10.1260/1756-8277.4.4.275
19.
O'Connor
,
J.
,
Acharya
,
V.
, and
Lieuwen
,
T.
,
2015
, “
Transverse Combustion Instabilities: Acoustic, Fluid Mechanic, and Flame Processes
,”
Prog. Energy Combust. Sci.
,
49
, pp.
1
39
.10.1016/j.pecs.2015.01.001
20.
Acharya
,
V.
,
Shin
,
D.-H.
, and
Lieuwen
,
T.
,
2012
, “
Swirl Effects on Harmonically Excited, Premixed Flame Kinematics
,”
Combust. Flame
,
159
(
3
), pp.
1139
1150
.10.1016/j.combustflame.2011.09.015
21.
Schwing
,
J.
,
Sattelmayer
,
T.
, and
Noiray
,
N.
,
2011
, “
Interaction of Vortex Shedding and Transverse High-Frequency Pressure Oscillations in a Tubular Combustion Chamber
,”
ASME Paper No. GT2011-45246
.10.1115/GT2011-45246
22.
Schwing
,
J.
,
Grimm
,
F.
, and
Sattelmayer
,
T.
,
2012
, “
A Model for the Thermo-Acoustic Feedback of Transverse Acoustic Modes and Periodic Oscillations in Flame Position in Cylindrical Flame Tubes
,”
ASME Paper No. GT2012-68775
.10.1115/GT2012-68775
23.
Schwing
,
J.
, and
Sattelmayer
,
T.
,
2013
, “
High-Frequency Instabilities in Cylindrical Flame Tubes: Feedback Mechanism and Damping
,”
ASME Paper No. GT2013-94064
.10.1115/GT2013-94064
24.
Hummel
,
T.
,
Schulze
,
M.
,
Schuermans
,
B.
, and
Sattelmayer
,
T.
,
2015
, “
Reduced Order Modeling of Transversal and Non-Compact Combustion Dynamics
,”
Proceedings of the 22nd International Congress on Sound and Vibration
,
Florence, Italy
, July
12
16
.https://www.researchgate.net/publication/293803298_REDUCED-ORDER_MODELING_OF_TRANSVERSAL_AND_NON-COMPACT_COMBUSTION_DYNAMICS
25.
Hummel
,
T.
,
Berger
,
F.
,
Hertweck
,
M.
,
Schuermans
,
B.
, and
Sattelmayer
,
T.
,
2017
, “
High-Frequency Thermoacoustic Modulation Mechanisms in Swirl-Stabilized Gas Turbine Combustors—Part II: Modeling and Analysis
,”
ASME J. Eng. Gas Turbines Power
,
139
(
7
), p.
071502
.10.1115/1.4035592
26.
Méry
,
Y.
,
2017
, “
Impact of Heat Release Global Fluctuations and Flame Motion on Transverse Acoustic Wave Stability
,”
Proc. Combust. Inst.
,
36
(
3
), pp.
3889
3898
.10.1016/j.proci.2016.08.009
27.
Acharya
,
V. S.
, and
Lieuwen
,
T. C.
,
2018
, “
Modeling Premixed Flame Response to Transverse Acoustic Modes
,”
AIAA Paper No. 2018-1182
.10.2514/6.2018-1182
28.
Acharya
,
V.
, and
Lieuwen
,
T. C.
,
2018
, “
Effects of Transverse Nozzle Location on High-Frequency Transverse Combustion Instabilities in Can Combustors
,”
Spring Technical Meeting of the Eastern States Section of the Combustion Institute
,
Combustion Institute
,
State College, PA
, Mar.
4
7
.
29.
Acharya
,
V. S.
, and
Lieuwen
,
T. C.
,
2019
, “
Premixed Flame Response to High-Frequency Transverse Acoustic Modes: Mean Flame Asymmetry Effects
,”
AIAA Paper No. 2019-0671
.10.2514/6.2019-0671