By modeling a multicomponent gas, a new source of indirect combustion noise is identified, which is named compositional indirect noise. The advection of mixture inhomogeneities exiting the gas-turbine combustion chamber through subsonic and supersonic nozzles is shown to be an acoustic dipole source of sound. The level of mixture inhomogeneity is described by a difference in composition with the mixture fraction. An n-dodecane mixture, which is a kerosene fuel relevant to aeronautics, is used to evaluate the level of compositional noise. By relaxing the compact-nozzle assumption, the indirect noise is numerically calculated for Helmholtz numbers up to 2 in nozzles with linear velocity profile. The compact-nozzle limit is discussed. Only in this limit, it is possible to derive analytical transfer functions for (i) the noise emitted by the nozzle and (ii) the acoustics traveling back to the combustion chamber generated by accelerated compositional inhomogeneities. The former contributes to noise pollution, whereas the latter has the potential to induce thermoacoustic oscillations. It is shown that the compositional indirect noise can be at least as large as the direct noise and entropy noise in choked nozzles and lean mixtures. As the frequency with which the compositional inhomogeneities enter the nozzle increases, or as the nozzle spatial length increases, the level of compositional noise decreases, with a similar, but not equal, trend to the entropy noise. The noisiest configuration is found to be a compact supersonic nozzle.

Introduction

In order to reduce NOx emissions from aeronautical gas turbines, the objective is to burn in a lean regime. Lean flames, especially premixed and stratified flames, burn very unsteadily, because they are sensitive to the turbulent environment of the combustion chamber. Such an unsteady combustion environment is the cause of two unwanted phenomena in aeroengines: (i) combustion noise and (ii) thermoacoustic instabilities (also known as combustion instabilities).

On the one hand, after fan and jet noise, combustion noise is the dominant cause of noise pollution generated by the whole turbojet. Although methods to mitigate fan and jet noise have been in place for a decade—such as ultra-high bypass ratio turbofan engines, acoustic liners, and fan blade geometric design—combustion noise is bound to increase with the implementation of low-emission combustors, high power-density engine cores, or compact burner designs [13].

On the other hand, thermoacoustic instabilities occur when the heat released by the flame is sufficiently in phase with the acoustic waves [4]. Therefore, the reflection or generation of acoustics at the nozzle downstream of the combustor has the potential to enhance the flame unsteadiness, which, in turn, generates larger acoustic oscillations. Thermoacoustic oscillations can cause structural damage and cracking resulting in the reduction of the combustor lifetime by a factor of two or more [2,5,6].

Both combustion noise and thermoacoustic instabilities can be caused by direct and indirect mechanisms.

  • (1)

    The heat released by the unsteady flame is a powerful monopole source of sound, and the acoustics emitted by it propagate in the combustion chamber through the turbine and are distorted by mean-flow gradients. The sound that is transmitted through the downstream-engine component causes noise pollution [713], whereas the sound that is reflected at the nozzle can create a thermoacoustic feedback [5,6,14]. The authors in Refs. [2] and [3] discussed this aspect in more detail with additional references.

  • (2)

    Indirect mechanisms arise from the interaction between nonacoustic perturbations exiting the combustion chamber and the nozzle downstream of the combustor. The two well-known mechanisms for indirect combustion noise and instability consist of entropy perturbations and vorticity accelerated through the nozzle [1519]. From a thermoacoustic point of view, the acoustics generated at the nozzle and traveling back to the combustion chamber can become the key feedback mechanism for a very low frequency combustion instability, called “rumble.” This mechanism has been demonstrated by numerous studies, among others [2022]. Isolating the contribution of indirect mechanisms to combustion noise and instability is an active research area [23]. In order to calculate the transfer functions for the acoustic transmission and reflection, different theoretical approaches were employed [15,2429], and experimental studies were carried out [30,31]. Dissipation and differential convection of entropy perturbations were analyzed by direct numerical simulation [32] and large-eddy simulation and experiments [33]. These studies showed that indirect mechanisms require consideration in the analysis of combustion noise and thermoacoustic instability.

As for the far-field emissions, inhomogeneities and mean-flow distortions that are generated in the engine core can modulate the acoustic-source distribution in a jet-exhaust [3,34], thereby modulating the noise level.

Common to all of the investigations of indirect mechanisms is the assumption that the gas mixture has a homogeneous composition. However, mixture inhomogeneities can arise from incomplete mixing, air dilution, and variations in the combustor exhaust gas compositions. It was first shown theoretically for subsonic compact nozzle flows [3] that inhomogeneities in mixture composition represent an additional source for indirect noise, which was then extended to supersonic nozzle flows with/without shock waves and examined through detailed studies [35]. (A compact nozzle is a nozzle with Helmholtz number equal to zero.) The main objective of this paper is to extend this analysis by quantifying the importance of this indirect mechanism in subsonic and supersonic nozzles relaxing the compact-nozzle assumption.

The paper is structured as follows. The differential equations for multicomponent gas mixtures are derived, and compositional inhomogeneities are expressed as a function of the mixture fraction. By inspection of the equations, the source of sound due to accelerated compositional inhomogeneities is identified. The equations are cast in compact-nozzle invariants (mass flow rate, total enthalpy, entropy, and mixture fraction) and solved numerically for a range of Helmholtz numbers. A parametric study with respect to the Helmholtz number and nozzle Mach numbers shows the relative contributions between compositional/entropy/direct mechanisms to the sound generated by a nozzle with linear-velocity profile.

Governing Equations

The multicomponent gas problem is governed by the equations of continuity, momentum, energy, and species transport, respectively [3638]
DρDt+ρ·u=0
(1)
ρDuDt+p=0
(2)
ρDhDt=DpDt
(3)
DYiDt=0
(4)
where ρ is the density; u is the velocity; p is the pressure; h is the enthalpy; the subscript i denotes the i-th mass fraction Yi; Ns is the number of species. The material derivative is D/Dt=/t+u·(·), where is the nabla operator. The entropy s is defined through Gibbs' relation for a multicomponent gas
Tds=dhdpρiNsμiWidYi
(5)
where μi is the chemical potential, which will be shown to be crucial in this analysis, and Wi is the molar mass. It is assumed that the flow is advection-dominated (no diffusion effects); it is not chemically reacting; the gas is ideal and calorically perfect; the heat-capacity ratio, γ, is constant; the flow is quasi one-dimensional and isentropic. The frozen-flow assumption is valid if the Damköhler number is small, i.e., Da=τc/τchem1 [35], where τc is the advective time scale, and τchem is the chemical time scale. This condition can be reformulated as He/(Mfτchem)1, where He=fL/c¯a is the Helmholtz number, M is the Mach number, f is the perturbation frequency, L is the nozzle length, and c¯a is the mean-flow speed of sound at the inlet. Under all the above assumptions, Eqs. (1)(4) simplify to
DρDt+ρux=0
(6)
ρDuDt+px=0
(7)
DDt{log(p1/γρ)}1cpDsDtΨDZDt=0
(8)
DsDt=0
(9)
DZDt=0
(10)

where the one-dimensional (1D) material derivative is redefined as D/Dt=/t+u/x. Thermodynamic and transport variables are functions of the mixture fraction Z, i.e., Yi=Yi(Z),cp=cp(Yi(Z))=cp(Z), etc. The calculation of the chemical potential function, Ψ, is briefly explained in the Calculation of the Chemical Potential Function section. The flame structure is obtained from the steady-state solution of conservation equations for continuity, species, and energy, which are solved using the cantera software package [39].

Linearization.

The variables are split as Φ=Φ¯+Φ, where Φ¯ is the mean-flow component, and |Φ|1 is the unsteady fluctuation. By grouping the steady terms, the equations for the mean flow read
(ρ¯u¯)x=0
(11)
(p¯+ρ¯u¯2)x=0
(12)
p¯1γρ¯=const
(13)
s¯x=0
(14)
Z¯x=0
(15)
By grouping the unsteady terms, the linearized equations, which govern the unsteady dynamics of the fluctuations, read
D¯ρDt+uρ¯x+ρ¯ux+ρu¯x=0
(16)
ρ¯D¯uDt+ρu¯u¯x+ρ¯uu¯x+px=0
(17)
D¯Dt{pγp¯sc¯pρρ¯Ψ¯Z}+u¯ZΨ¯x=0
(18)
D¯Dt{sc¯p}=0
(19)
D¯ZDt=0
(20)
where the linearized material derivative is D¯/Dt=/t+u¯/x. Note that c¯p is a constant because it depends only on Z¯, which is a constant. By integrating Eq. (18) along a characteristic line t=xds/u¯(s), the density ρ can be eliminated, which yields
t{pγp¯}+u¯x{pγp¯}+u¯x{uu¯}=0
(21)
t{uu¯}+u¯x{uu¯}+u¯M¯2x{pγp¯}=[2uu¯(γ1)pγp¯]u¯x+(sc¯p+Ψ¯Z)u¯x
(22)

This set of equations tends to the single-component gas equations of Marble and Candel [15] as either Z=0 or Ψ¯=0. The term [2(u/u¯)(γ1)(p/γp¯)]u¯/x in Eq. (22) represents refraction and reflection of the acoustics due to the mean-flow gradient. As already pointed out by Marble and Candel [15], the unsteady interaction between the entropy disturbance s and the mean-flow gradient is a dipole source term. New to this analysis is the identification of the term Ψ¯Zu¯/x as a second source of indirect noise, again through the action of an acoustic dipole. Physically, not only do density variations create noise through entropy mechanisms, but also differences in species generate noise through the chemical potential, when mean-flow gradients are present. The compositional dipole source of sound may augment or reduce the effect of the entropy source (Eq. (22)). The purpose of this paper is to quantify the relative importance between noise induced by entropy perturbations and compositional inhomogeneities in an n-dodecane–air mixture.

Invariants.

In a nozzle flow under the assumptions made in the Governing Equations section, it is convenient to work with the normalized mass flow rate, Im˙C=(m˙/m˙¯); total enthalpy, IhT=(hT/h¯T); entropy, Is=(s/c¯p); and mixture fraction, Iz=Z. (The total enthalpy is the sum of the enthalpy and the specific kinetic energy, i.e., hT=h+1/2u2.) These are called “Invariants.” The noncompact nozzle invariants tend to the compact-nozzle invariants as the mean-flow gradients tend to zero. Therefore, in a general noncompact nozzle, the invariants are functions of space (and time) due to the mean-flow gradients. Nonetheless, the entropy and mixture fraction invariants are the same as those in the compact nozzle, Is=IsC and Iz=IzC, where IC denotes an invariant for the compact-nozzle, because their dynamics are not affected by mean-flow gradients (see Eqs. (19) and (20)). A nondimensionalization similar to that proposed by Duran and Moreau [28] is employed here: x=ηL,t=τ/f,u¯=ũc¯a, and c¯=c̃c¯a. The chain rule provides the transformation for the derivatives, /tf/τ and /x1/L/η. By assuming that the flow is forced harmonically, the state vector can be decomposed as q(τ,η)=q̂(η)exp(2πift)=q̂(η)exp(2πiτ). The system of differential equations (21) and (22) is linear with spatially dependent coefficients; therefore, it can be cast in matrix form as
2πiHeq̂=E(η)q̂η
(23)
where q=[Im˙C,IhTC,IsC,IzC]T and
E(η)=ũ[11+(γ1)2M¯2(γ1)M¯21(γ1)M¯2Ψ¯(γ1)M¯2γ11+γ12M¯21γ11+γ12M¯2γ11+γ12M¯2Ψ¯00100001]
(24)

Numerical integration of Eq. (23) enables the calculation of the nozzle transfer functions.

Characteristic Decomposition.

By employing a characteristic decomposition of the invariants, four independently evolving solutions at each side of the nozzle are identified, as shown in Fig. 1(a). They correspond to the downstream and upstream propagating acoustic waves, the advected entropy perturbation, and the compositional inhomogeneity, respectively, π±=(1/2)((p/γp¯)±(u/c¯)),σ=(s/c¯p),ξ=Z. The numerical procedure and boundary conditions used to solve Eq. (23) are briefly explained in the Results section.

Calculation of the Chemical Potential Function.

By definition, the chemical potential of the substance i is the partial derivative of the Gibbs free energy (or, simply, Gibbs energy), G, with respect to the number of moles of the same substance, i.e., μi=(G/ni)T,p,nj=i. Hence, μi=μi(Yi(Z))=μi(Z). The chemical potential function, Ψ, is calculated by finite difference from flamelet calculations
Ψ=1cpTi(μiWiΔhi°)dYidZ=1cpTgZ
(25)

where Δhi° is the formation enthalpy of the i-th species, and g is the specific Gibbs energy of the mixture. The analysis is applied to a combustion configuration to quantify the relative contribution of compositional noise to the overall combustion noise. For this, we consider an idealized configuration in which the combustor exhaust-gas composition enters the nozzle. This exhaust-gas composition is represented from the solution of a series of one-dimensional strained diffusion flames [40] that include the equilibrium composition, typically observed at low-power cruise conditions, and highly strained combustion conditions representative of high-load operation. The flame solutions are generated by considering n-dodecane (C12H26), a kerosene surrogate, as fuel and air in the oxidizer stream. The reaction chemistry is described by a 24-species mechanism [41], which provides an accurate flame representation at these conditions.

The degree of straining, i.e., the deviation from equilibrium, is characterized by the scalar dissipation rate, χ=2α|Z|2, where α is the diffusivity of the mixture fraction, and χ is evaluated at stoichiometric condition, corresponding to a value of Zst=0.063. This study considers an operating condition at χst=1 s−1 (quasi unstrained condition near equilibrium), which is expected to be close to the condition exiting a gas-turbine combustion chamber. This operating condition is depicted by a cross in the S-shaped curve stable branch in Fig. 2, which is obtained by solving the steady flamelet equations [40] with boundary conditions Tfuel=Tox=295 K at p = 1 bar. The chemical potential function and its gradient with respect to the Mach number are shown in Fig. 3. The structure of the flame together with the chemical potential function and specific Gibbs energy are shown in Fig. 4.

Compact Nozzles: Zero Helmholtz Number

Before studying the effect of the Helmholtz number, the compact nozzle limit is discussed [35]. In the compact-nozzle limit He = 0, Eq. (23) reduces to a set of jump conditions across the nozzle, which is viewed as a discontinuity
Im˙Cab=0,IhTCab=0,IsCab=0,IZCab=0
(26)
where Φab=Φ(b)Φ(a), and the indices a and b denote the conditions at the inlet and outlet, respectively (Fig. 1). In a choked nozzle, the variables are constrained by the conditions that the mass flow rate attains a maximum, which yields the additional condition [35]
s2c¯p+12Ψ¯ZuM¯c¯+γ12γpp¯=0
(27)

The transfer functions for the incoming/outgoing acoustic waves generated by acoustic, entropy, and compositional perturbations that enter subsonic/supersonic compact nozzles are presented in Ref. [35] or can be calculated in the proposed framework by setting He = 0.

Results

The in-house solver for integration of Eq. (23) relies on an adaptive fourth-order Runge–Kutta algorithm. It imposes the relevant boundary conditions by iteratively solving the system via an optimization procedure. Particularly delicate is the treatment of the choked location, which creates a singularity in the equations. To numerically treat this singularity, transonic problems are split into two subdomains: one extending from the nozzle inlet (a state) to a small distance before the choked throat ηthε, and a second extending from that same distance ε downstream of the throat to the nozzle exit (b state). In the subsonic domain, the downstream-propagating perturbation amplitudes (π+,ξ,σ) are imposed at the inlet state, and Eq. (27) is imposed near the throat. In the supersonic domain, the perturbation values ηε from the downstream edge of the subsonic solution are imposed at η+ε. Throughout this study, the value of ε is prescribed as 105, and solutions were observed to be insensitive to further reductions in ε or changes in the solver tolerances. The code has been verified both in the limit of zero imposed compositional fluctuations ξ = 0 as well as for compositional fluctuations in the limit He = 0 and found to agree favorably with earlier studies (Fig. 6). We consider three nozzle flows based on the linear velocity profile [15]. Figure 5(a) shows the area variation and Fig. 5(b) shows the corresponding mean-flow Mach number. We study (i) a subsonic nozzle (blue-solid line), (ii) a transonic nozzle (red-dashed line), and (iii) a supersonic nozzle (black-dotted line). The latter two are those studied in Ref. [28], against which we compare our solutions.

The modulus of the transfer function between an entropy input and (i) the outgoing acoustic wave, |πb+/σ| is shown in Fig. 6(a), and (ii) the acoustic wave at the outlet, |πb/σ|, is shown in Fig. 6(b). The reference solution is taken from Figs. 5 and 6 in Ref. [28]. The subsonic solution in blue-solid line was not discussed in Ref. [28], hence, no comparison can be made. Both transonic and supersonic cases, for He = 0, tend to the compact-nozzle solutions [15,35], which provide generally over-estimating transfer functions (horizontal lines in Fig. 6). The solutions obtained with the in-house solver compare satisfactorily with those of Ref. [28]. A marginal discrepancy is observed in the transonic nozzle, which can be due to the numerical methods with the treatment of the stiffness at the throat. Note that the subsonic nozzle solution |πb/σ| is zero (Fig. 6(b)), because it is an input of the problem (set to zero here), and not an output, as opposed to the choked-flow nozzle. As already discussed in the literature [28], the effect of the nozzle noncompactness on the entropy noise production is non-negligible and has a monotonic decaying trend with increasing the Helmholtz number, He. The higher the Helmholtz number, the smaller the effect of entropy fluctuations in the sound generation.

Effect of the Helmholtz Number on the Transfer Functions for Compositional Inhomogeneities.

The flamelet solutions in Fig. 4(a) can be interpreted as an idealized representation of the gas composition exiting the combustor. The combustor operates at a lean global equivalence ratio ϕ=0.3 corresponding to a mean mixture fraction Z¯0.02, with Z¯=ϕZst/[Zst(ϕ1)+1] [40], and the corresponding thermochemical state is then taken from the flame solution of Fig. 4. In addition to temperature fluctuations, which are known to generate entropy noise, turbulence, and other unsteady effects give rise to fluctuations in Z, having the potential to produce compositional noise downstream. To assess the compositional noise that is generated, the combustor exhaust composition is isentropically compressed/expanded through an ideal nozzle with the mean mixture composition assumed to be frozen at Z¯0.02.

The effect of the nozzle Helmholtz number is investigated for the three nozzle configurations in the range 0He2. In practical terms, this range physically corresponds to perturbation frequencies of 0f6000Hz for a nozzle of length L0.31 m and an adiabatic flame temperature of 2100 K.

Figure 7 shows the sound generated by acoustic forcing. Physically, this noise is caused by the passage of acoustic waves generated by the flame in the combustion chamber, which is a monopole source of sound [2,3]. Unlike the transfer function for entropy perturbations, the reflected acoustic wave transfer function overshoots for He up to 0.7 in the supersonic nozzle (Fig. 7(b)), i.e., it is not a monotonic function of He. The phases have a fairly linear trend.

Figure 8 shows the modulus (top row) and phase (bottom row) of the transfer functions between a compositional input and the sound generated. First, we note that the Helmholtz number has a significant effect on the transfer functions. The modulus tends to decrease monotonically as the Helmholtz number increases. Second, the compact nozzle solutions, which are strictly valid for He = 0, over-estimate the noise level, in particular in the supersonic case (black-dotted line). Third, in agreement with the compact-nozzle solutions, the level of noise generated by compositional inhomogeneities tend to increase by almost an order of magnitude as the flow becomes supersonic. However, even in the subsonic and transonic cases (blue-solid and red-dashed lines, respectively) the noise generated by compositional inhomogeneities is non-negligible and comparable to, if not greater than, the noise generated by entropy perturbations (Fig. 6) and direct noise (Fig. 7). The phase is likewise markedly influenced by the Helmholtz number (Figs. 8(b) and 8(d)). It is fairly linear for He1.2 in the supersonic case and throughout the entire range of He for the subsonic case. This physically signifies that the finite nozzle spatial extent tends to delay the wave response by a nearly constant time delay. The phase becomes nonlinear for higher Helmholtz number in the supersonic case.

In general, we observe that the behavior of the transfer functions due to compositional inhomogeneities (Fig. 8) is qualitatively comparable to the response to entropy perturbations (Fig. 6). The physical reason can be explained by inspection of the governing Eq. (22). Indeed, the compositional-inhomogeneity term appears beside the entropy term.

The spatial dependencies of the transfer functions for the supersonic nozzle, parameterized with the Helmholtz number, are shown in Fig. 9. Again, the modulus of the transfer function decreases, while the phase delay increases, as the Helmholtz number increases. This holds true also at each spatial location. The compact nozzle assumption over-estimates the transfer functions, except for a small range close to η = 0, where, in fact, the compact assumption holds. The five lines collapse onto each other in Fig. 9(b), because the compositional perturbation is imposed, thus it does not vary in space because it is an invariant.

A Quantitative Estimate of the Compositional Noise Level in a Gas Turbine.

We estimate the ratio between composition noise and entropy noise by multiplying the corresponding transfer functions by the factor ξa/σa=Z/(T/T¯) [35]. This factor is estimated by considering that the mixture composition at the combustor exit reaches equilibrium with a mean temperature of T¯a=2100 K, corresponding to an equivalence ratio of ϕ=0.3 and mean mixture fraction of Z¯=0.02 at conditions shown in Fig. 4(a). The mixture-fraction distribution at the combustor exit is represented by a beta-distribution, β(Z). The fluctuation magnitude is estimated on the safe side as Z=ζZ¯(1Z¯) [42]. By considering a combustor in which the mixing is nearly completed with ζ=103, the temperature fluctuation can be evaluated from
T=01[T¯(z)T¯a]2β(z)dz
(28)

Substituting these values, we obtain a fluctuation ratio of ξa/σa102, which, multiplied by the ratio in Fig. 10, indicates that the noise ratio at subsonic condition is of order 101, but it is of order 1 in the supersonic case.

Comparison With Experimental Data.

The computationally cheap predictions offered by the compact-nozzle transfer functions were compared with experimental data in Rolland et al. [43]. The reader may refer to their paper for more details. In their experiment, they injected air-helium perturbations, which are compositional inhomogeneities, into an air flow in a choked duct. Then, they measured the pressure signal and identified the backward-traveling acoustic wave, πa, generated by the compositional inhomogeneity being accelerated through the choked nozzle. The time pressure signal was acquired with microphones. By decomposing the pressure signal as direct and compositional noise contributions, they succeeded at accurately reconstructing the measured pressure signal. The contribution of the compositional-inhomogeneity induced noise was identified in their Figs. 13 and 14. In summary, it was experimentally shown that compositional inhomogeneities generate sound, which, in compact nozzles, can be predicted by the algebraic transfer functions of Ref. [35] or setting He = 0 in this paper.

Conclusions

By modeling inhomogeneities in a multicomponent gas exiting a combustion chamber and entering a nozzle, it is shown that the acceleration of compositional inhomogeneities generates sound. This source of sound is shown to act like an acoustic dipole, as accelerated entropy perturbations do. Key to the analysis is the calculation of the chemical potential function, which depends on the chemical potential, the mean-flow Mach number, and the gas composition. In this paper, an n-dodecane fuel with a 24-species mechanism is considered, because it is relevant to kerosene fuels of aero-engines.

By relaxing the assumption on the nozzle compactness with respect to the wavelength of the disturbances, three nozzle configurations are analyzed: a subsonic nozzle, a transonic nozzle, and supersonic nozzle without shock waves. Unlike the compact nozzle limit, the equations do not have an algebraic solution, thus they are numerically integrated. It is found that

  • (1)

    the sound generated by a compositional inhomogeneity has a generally monotonic trend with increasing the Helmholtz number. This physically occurs when the frequency at which the compositional inhomogeneities enter the nozzle is high, or when the nozzle has a non-negligible spatial extent (or a combination of both).

  • (2)

    the indirect noise produced is lower when the nozzle is noncompact. In other words, the compact-nozzle theory over-estimates the level of indirect noise generated by compositional inhomogeneities.

  • (3)

    in supersonic flows, the noise generated by compositional inhomogeneities may be comparable to, or even exceed, the direct noise and indirect noise created by entropy perturbations.

This suggests that compositional-inhomogeneity induced sound requires consideration with the implementation of compact burners, high power-density engine cores, and advanced low-emission combustors, both for noise and thermoacoustic instability reduction.

Funding Data

  • Ames Research Center (Grant No. NNX15AV04A).

  • Royal Academy of Engineering (Research Fellowship).

Nomenclature

     
  • c =

    speed of sound

  •  
  • cp =

    specific heat at constant pressure of the mixture, cp=iNscp,iYi

  •  
  • cp,i =

    specific heat at constant pressure of species i

  •  
  • D¯/Dt =

    linearized material derivative, D¯/Dt=/t+u¯/x

  •  
  • f =

    forcing frequency

  •  
  • g =

    specific Gibbs free energy

  •  
  • h =

    enthalpy

  •  
  • hT =

    total enthalpy

  •  
  • He =

    Helmholtz number, He=Lf/c¯a

  •  
  • IhT =

    total-enthalpy invariant

  •  
  • Im˙ =

    mass-flow rate invariant

  •  
  • Is =

    entropy invariant

  •  
  • Iz =

    mixture-fraction invariant

  •  
  • L =

    nozzle characteristic length

  •  
  • M =

    Mach number

  •  
  • ni =

    moles of species i

  •  
  • Ns =

    number of species

  •  
  • p =

    pressure

  •  
  • q =

    vector of compact-nozzle invariants, q=[Im˙C,IhTC,IsC,IzC]T

  •  
  • R =

    gas constant, R=RiNsYi/Wi

  •  
  • s =

    entropy

  •  
  • u =

    axial velocity

  •  
  • u =

    velocity

  •  
  • ũ =

    nondimensional velocity, ũ=u¯/c¯a

  •  
  • Wi =

    molar mass of species i

  •  
  • Yi =

    mass fraction of species i

  •  
  • Z =

    mixture fraction

Greek Symbols

    Greek Symbols
     
  • γ =

    heat capacity ratio, γ=cp/cv

  •  
  • Δhi° =

    formation enthalpy of species i

  •  
  • η =

    nondimensional axial coordinate, η=x/L

  •  
  • μi =

    chemical potential of species i

  •  
  • ρ =

    density

  •  
  • τ =

    nondimensional time, τ=tf

  •  
  • ϕ =

    global equivalence ratio

  •  
  • Ψ =

    chemical potential function, Ψ=(1/cpT)iNs((μi/Wi)Δhi°)(dYi/dZ)

Subscripts

    Subscripts
     
  • a =

    nozzle inlet

  •  
  • b =

    nozzle outlet

  •  
  • i =

    ith species

Superscripts

    Superscripts
     
  • C =

    compact nozzle

Mathematical

    Mathematical
     
  • ¯ =

    mean-flow quantity

  •  
  • R =

    universal gas constant

  •  
  • ̃ =

    nondimensional mean-flow quantity

  •  
  • =

    perturbed quantity

References

1.
Chang
,
C. T.
,
Lee
,
C.-M.
,
Herbon
,
J. T.
, and
Kramer
,
S. K.
,
2013
, “
NASA Environmentally Responsible Aviation Project Develops Next-Generation Low-Emissions Combustor Technologies (Phase I)
,”
J. Aeronaut. Aerosp. Eng.
,
2
(
4
), p.
116
.
2.
Dowling
,
A. P.
, and
Mahmoudi
,
Y.
,
2015
, “
Combustion Noise
,”
Proc. Comb. Inst.
,
35
(
1
), pp.
65
100
.
3.
Ihme
,
M.
,
2017
, “
Combustion and Engine-Core Noise
,”
Annu. Rev. Fluid Mech.
,
49
(
1
), pp.
277
310
.
4.
Rayleigh
,
L.
,
1878
, “
The Explanation of Certain Acoustical Phenomena
,”
Nature
,
18
(
455
), pp.
319
321
.
5.
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.
6.
Culick
,
F. E. C.
,
2006
, “
Unsteady Motions in Combustion Chambers for Propulsion Systems
,” RTO AGARDograph, North Atlantic Treaty Organization, Neuilly-Sur-Seine, France, Report No.
AG-AVT-039
.http://www.dtic.mil/docs/citations/ADA466461
7.
Strahle
,
W. C.
,
1978
, “
Combustion Noise
,”
Prog. Energy Combust. Sci.
,
4
(
3
), pp.
157
176
.
8.
Hurle
,
I. R.
,
Price
,
R. B.
,
Sugden
,
T. M.
, and
Thomas
,
A.
,
1968
, “
Sound Emission From Open Turbulent Premixed Flames
,”
Proc. R. Soc. London A
,
303
(
1475
), pp.
409
427
.
9.
Singh
,
K. K.
,
Zhang
,
C.
,
Gore
,
J. P.
,
Mongeau
,
L.
, and
Frankel
,
S. H.
,
2005
, “
An Experimental Study of Partially Premixed Flame Sound
,”
Proc. Comb. Inst.
,
30
(
2
), pp.
1707
1715
.
10.
Rajaram
,
R.
, and
Lieuwen
,
T.
,
2003
, “
Parametric Studies of Acoustic Radiation From Premixed Flames
,”
Combust. Sci. Technnol.
,
175
(
12
), pp.
2269
2298
.
11.
Candel
,
S.
,
Durox
,
D.
,
Ducruix
,
S.
,
Birbaud
,
A.-L.
,
Noiray
,
N.
, and
Schuller
,
T.
,
2009
, “
Flame Dynamics and Combustion Noise: Progress and Challenges
,”
Int. J. Aeroacoust.
,
8
(
1–2
), pp.
1
56
.
12.
Zhao
,
W.
, and
Frankel
,
S. H.
,
2001
, “
Numerical Simulations of Sound Radiated From an Axisymmetric Premixed Reacting Jet
,”
Phys. Fluids
,
13
(
9
), pp.
2671
2681
.
13.
Ihme
,
M.
,
Pitsch
,
H.
, and
Bodony
,
D.
,
2009
, “
Radiation of Noise in Turbulent Non-Premixed Flames
,”
Proc. Comb. Inst.
,
32
(
1
), pp.
1545
1553
.
14.
Dowling
,
A. P.
,
1997
, “
Nonlinear Self-Excited Oscillations of a Ducted Flame
,”
J. Fluid Mech.
,
346
, pp.
271
290
.
15.
Marble
,
F. E.
, and
Candel
,
S. M.
,
1977
, “
Acoustic Disturbance From Gas Non-Uniformities Convected Through a Nozzle
,”
J. Sound Vib.
,
55
(
2
), pp.
225
243
.
16.
Morfey
,
C. L.
,
1973
, “
Amplification of Aerodynamic Noise by Convective Flow Inhomogeneities
,”
J. Sound Vib.
,
31
(
4
), pp.
391
397
.
17.
Cumpsty
,
N. A.
,
1979
, “
Jet Engine Combustion Noise: Pressure, Entropy and Vorticity Perturbations Produced by Unsteady Combustion or Heat Addition
,”
J. Sound Vib.
,
66
(
4
), pp.
527
544
.
18.
Ffowcs Williams
,
J. E.
, and
Howe
,
M. S.
,
1975
, “
The Generation of Sound by Density Inhomogeneities in Low Mach Number Nozzle Flows
,”
J. Fluid Mech.
,
70
(
3
), pp.
605
622
.
19.
Howe
,
M. S.
,
2010
, “
Indirect Combustion Noise
,”
J. Fluid Mech.
,
659
, pp.
267
288
.
20.
Polifke
,
W.
,
Paschereit
,
C. O.
, and
Döbbeling
,
K.
,
2001
, “
Constructive and Destructive Interference of Acoustic and Entropy Waves in a Premixed Combustor With a Choked Exit
,”
Int. J. Acoust. Vib.
,
6
(3), pp.
135
146
.
21.
Goh
,
C. S.
, and
Morgans
,
A. S.
,
2013
, “
The Influence of Entropy Waves on the Thermoacoustic Stability of a Model Combustor
,”
Combust. Sci. Technol.
,
185
(
2
), pp.
249
268
.
22.
Morgans
,
A. S.
, and
Annaswamy
,
A. M.
,
2008
, “
Adaptive Control of Combustion Instabilities for Combustion Systems With Right-Half Plane Zeros
,”
Combust. Sci. Technol.
,
180
(
9
), pp.
1549
1571
.
23.
De Domenico
,
F.
,
Rolland
,
E. O.
, and
Hochgreb
,
S.
,
2017
, “
Detection of Direct and Indirect Noise Generated by Synthetic Hot Spots in a Duct
,”
J. Sound Vib.
,
394
, pp.
220
236
.
24.
Stow
,
S. R.
,
Dowling
,
A. P.
, and
Hynes
,
T. P.
,
2002
, “
Reflection of Circumferential Modes in a Choked Nozzle
,”
J. Fluid Mech.
,
467
, pp.
215
239
.
25.
Goh
,
C. S.
, and
Morgans
,
A. S.
,
2011
, “
Phase Prediction of the Response of Choked Nozzles to Entropy and Acoustic Disturbances
,”
J. Sound Vib.
,
330
(
21
), pp.
5184
5198
.
26.
Moase
,
W. H.
,
Brear
,
M. J.
, and
Manzie
,
C.
,
2007
, “
The Forced Response of Choked Nozzles and Supersonic Diffusers
,”
J. Fluid Mech.
,
585
, pp.
281
304
.
27.
Giauque
,
A.
,
Huet
,
M.
, and
Clero
,
F.
,
2012
, “
Analytical Analysis of Indirect Combustion Noise in Subcritical Nozzles
,”
ASME J. Eng. Gas Turbines Power
,
134
(
11
), p.
111202
.
28.
Duran
,
I.
, and
Moreau
,
S.
,
2013
, “
Solution of the Quasi-One-Dimensional Linearized Euler Equations Using Flow Invariants and the Magnus Expansion
,”
J. Fluid Mech.
,
723
, pp.
190
231
.
29.
Duran
,
I.
, and
Morgans
,
A. S.
,
2015
, “
On the Reflection and Transmission of Circumferential Waves Through Nozzles
,”
J. Fluid Mech.
,
773
, pp.
137
153
.
30.
Bake
,
F.
,
Richter
,
C.
,
Mühlbauer
,
C.
,
Kings
,
N.
,
Röhle
,
I.
,
Thiele
,
F.
, and
Noll
,
B.
,
2009
, “
The Entropy Wave Generator (EWG): A Reference Case on Entropy Noise
,”
J. Sound Vib.
,
326
(3–5), pp.
574
598
.
31.
Kings
,
N.
, and
Bake
,
F.
,
2010
, “
Indirect Combustion Noise: Noise Generation by Accelerated Vorticity in a Nozzle Flow
,”
Int. J. Spray Combust. Dyn.
,
2
(
3
), pp.
253
266
.
32.
Morgans
,
A. S.
,
Goh
,
C. S.
, and
Dahan
,
J. A.
,
2013
, “
The Dissipation and Shear Dispersion of Entropy Waves in Combustor Thermoacoustics
,”
J. Fluid Mech.
,
733
, p.
R2
.
33.
Giusti
,
A.
,
Worth
,
N.
,
Mastorakos
,
E.
, and
Dowling
,
A.
,
2017
, “
Experimental and Numerical Investigation Into the Propagation of Entropy Waves
,”
AIAA J.
,
55
(
2
), pp.
446
458
.
34.
O'Brien
,
J.
,
Kim
,
J.
, and
Ihme
,
M.
,
2016
, “
Investigation of the Mechanisms of Jet-Engine Core Noise Using Large-Eddy Simulation
,”
AIAA
Paper No. AIAA 2016-0761.
35.
Magri
,
L.
,
O'Brien
,
J.
, and
Ihme
,
M.
,
2016
, “
Compositional Inhomogeneities as a Source of Indirect Combustion Noise
,”
J. Fluid Mech.
,
799
, p.
R4
.
36.
Williams
,
F. A.
,
1985
,
Combustion Theory
,
Perseus Books
,
Reading, MA
.
37.
Poinsot
,
T.
, and
Veynante
,
D.
,
2005
,
Theoretical and Numerical Combustion
, 2nd ed., R. T. Edwards, Philadelphia, PA.
38.
Lieuwen
,
T. C.
,
2012
,
Unsteady Combustor Physics
,
Cambridge University Press
, Cambridge, UK.
39.
Goodwin
,
D. G.
,
Moffat
,
H. K.
, and
Speth
,
R. L.
,
2017
, “
Cantera: An Object-Oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes
,” Version 2.2.1, Zenodo, Geneva, Switzerland.
40.
Peters
,
N.
,
2000
,
Turbulent Combustion
,
Cambridge University Press
,
Cambridge, UK
.
41.
Vie
,
A.
,
Franzelli
,
B.
,
Gao
,
Y.
,
Lu
,
T.
,
Wang
,
H.
, and
Ihme
,
M.
,
2015
, “
Analysis of Segregation and Bifurcation in Turbulent Spray Flames: A 3d Counterflow Configuration
,”
Proc. Comb. Inst.
,
35
(
2
), pp.
1675
1683
.
42.
Dimotakis
,
P. E.
, and
Miller
,
P. L.
,
1990
, “
Some Consequences of the Boundedness of Scalar Fluctuations
,”
Phys. Fluids A
,
2
(
11
), pp.
1919
1920
.
43.
Rolland
,
E. O.
,
De Domenico
,
F.
, and
Hochgreb
,
S.
,
2017
, “
Direct and Indirect Noise Generated by Injected Entropic and Compositional Inhomogeneities
,”
ASME
Paper No. GT2017-64428.