Analysis of combustion noise sources using Doak’s momentum potential theory

Ra ff aele D’Aniello 1

German Aerospace Center (DLR) Department of Engine Acoustics, Müller-Breslau-Straße 8, Berlin - Germany

Karsten Knobloch 2

German Aerospace Center (DLR) Department of Engine Acoustics, Müller-Breslau-Straße 8, Berlin - Germany

Carolin Kissner 3

German Aerospace Center (DLR) Department of Engine Acoustics, Müller-Breslau-Straße 8, Berlin - Germany

ABSTRACT Noise emissions of modern lean combustors are related to di ff erent sources. Direct combustion noise is generated by heat release fluctuations, while indirect noise sources are caused by the acceleration of entropy and vorticity inhomogeneities through the nozzle-guide-vane at the combustor exit. The latter noise source is characterised by the coupling of fluctuations. When convected in non-uniform mean flow, this coupling can e.g. cause vortical or entropic fluctuations to be partly transferred into acoustics. Due to the complexity of the sources, a clear and quantitative separation of the di ff erent phenomena in terms of primitive variables presents a significant challenge. This study therefore proposes an alternative framework for the description of combustion noise generation based on Doak’s Momentum Potential Theory (MPT). The MPT defines a Generalised Acoustic Field (GAF) and describes the sound production in terms of mean energy fluxes carried by the respective acoustic, thermal and turbulent fluctuating momentum components. To confirm the ability to identify the di ff erent combustion noise sources, the method was applied to Large-Eddy Simulation data of a non-reacting swirl-combustor simulator. The coherent character and spectral behaviour of the GAF were investigated using a Spectral Proper Orthogonal Decomposition analysis and the GAF mean fluxes correlated to the source distributions.

1. INTRODUCTION

Noise emissions associated with combustion processes can significantly contribute to the overall engine-core noise of aeronautical engines and gas turbines [1]. They typically feature a broadband, low-frequency behaviour that can a ff ect the dynamics of the flow in the combustion chamber as well as of the jet-exhaust gas [2]. Nevertheless, they are of particular interest for modern combustor devices, which are characterised by high-temperature, unsteady combustion processes [3]. In fact, the heat release fluctuations resulting from the dynamics of lean premixed flames cause sound production within the combustion chamber, which commonly referred to as direct combustion noise. Another noise source, called indirect combustion noise, is related to the acceleration of flow inhomogeneities-

1 ra ff aele.daniello@dlr.de

2 karsten.knobloch@dlr.de

3 carolin.kissner@dlr.de

such like entropy, vorticity or mixture inhomogeneities through the nozzle-guide-vane (NGV). Even though highly dependent on geometry and operating conditions, the indirect combustion noise can dominate combustion noise [4] and is thought to play an important role in the feedback mechanisms which cause combustion instabilities [2]. A proper understanding of the combustion noise production thus represents an important milestone in realising safer, more e ffi cient and consequently greener combustor devices. The transfer mechanism between vortical, entropic and acoustic fluctuations, which is typical for the production of indirect combustion noise, can be explained by mode coupling [3]. In general, if the fluctuations propagate in a uniform mean flow, three uncoupled types of disturbances (i.e. acoustic, vortical and entropic modes) can propagate with the flow without any mutual interaction. Conversely, if the mean flow is non-uniform as is typical in a combustion chamber, the disturbances couple with each other. For instance, entropy disturbances can then be transferred into acoustic perturbations [5]. Chu et al. [6] showed, however, that the mutual influence between the modes generally results in a complex interdependence of the primitive variables due to the non-linearity of the model flow. This, in turn, makes the distinction between the di ff erent noise sources as well as the quantification of the respective sound production challenging. Conventional combustion noise models tend to focus on the propagation of modes and to make strong simplifying assumptions for the sources characterisation. As a consequence, these models, which mostly rely on the works of Marble and Candel [7] and Cumpsty and Marble [8], define unambiguous transfer functions for acoustic fluctuations, although several extensions exist. The main assumptions include an isentropic propagation of the disturbances, a quasi-one-dimensional framework and a compactness of the nozzle. Moreover, the sources, typically treated as a known input for the system, are modelled as periodic oscillations. Several studies question these assumptions. For instance, De Domenico et al. [9] showed that a non-isentropic extension is necessary to capture the indirect combustion noise components and match the observed experimental behaviour. Weilenmann et at. [10] compared di ff erent one-dimensional models for the propagation of entropy waves with experimental and simulation data showing that the waves should be modelled by using at least a two-dimensional approximation. Semlitsch et al. [11] applied mode decomposition techniques to analyse the generation of entropy and vorticity waves for CFD simulation data of a realistic gas turbine combustor configuration. Their findings point out the unsteady and three-dimensional character of the coherent structures featured by the fluctuations, which question idealised wave assumptions. A complex and at the same time reliable description of the acoustic sources is however crucial to match the acoustic emissions at the NGV entrance. Therefore, this study proposes an alternative framework for the description of combustion noise generation based on the Momentum Potential Theory of Energy Fluxes carried by Momentum Fluctuations (MPT) [12]. The MPT was developed by P.E. Doak and will be applied for the first time -to the best of the authors’ knowledge- in the context of combustion dynamics in this work. The MPT has two main benefits: firstly, it identifies the Total Fluctuating Enthalpy (TFE) as a general acoustic variable, the so-called Generalised Acoustic Field (GAF) that can be used to quantify the acoustic properties of a non-isentropic system without using any strong assumption. Secondly, the mean fluxes of the GAF are connected with three clearly separated turbulent, acoustic and thermal sources, which only depend on flow fluctuating quantities. Jordan et al [13] used these properties of the MPT to analyse the acoustic radiation due to a solenoidal wave-packet perturbation. They show how the theory can be used to identify propagating components of the packet and the respective source behaviour. Unnikrishnan and Gaitonde [14] were the first to apply the MPT to Large-Eddy simulation data of a realistic case by examining the dynamics of a supersonic cold jet. This as well as successive studies on turbulent jets [15, 16] highlighted the potential of the MPT not only for distinguishing di ff erent modes, but also for understanding the physical mechanism responsible for the generation and interaction of such modes. Furthermore, Prasad et al. [17] showed how the MPT can be used in the form of the Doak’s analogy [18] in order to correlate the evolution of the GAF with the space-time behaviour of the sources.

The main contribution of the research presented here is the extension of the applicability domain of the MPT to combustion acoustics problems. The flexibility of using the MPT in terms of mean fluxes or in form of the Doak’s analogy as well as the clear separation of the turbulent, acoustic and thermal sources makes the MPT a suitable candidate for describing and analysing unsteady flows of combustion chambers. To confirm its ability to distinguish between the di ff erent combustion noise sources, the mean fluxes formulation of the MPT is presented and applied for the analysis of Large- Eddy Simulation data of a non-reacting swirl combustor simulator. Specifically, the influence of entropy and vorticity disturbances, respectively due to cooling air entering the chamber and swirler activity, are investigated. The physical mechanisms responsible for the GAF production and the distribution of the di ff erent sources are analysed with the help of the MPT formulation. Moreover, the spectral behaviour of the GAF is investigated using a Spectral Proper Orthogonal Decomposition (SPOD) analysis. The paper is organised as follows. An overview of the MPT general principles as well as the definition of GAF are reported in § 2. The model for the description of combustion noise sources based on the MPT formulation is presented subsequently in § 3. The simulation setup and the results from the application of the model to the simulation data are then analysed in § 4. Finally, the main findings are summarised in § 5.

2. OVERVIEW OF DOAK’S MOMENTUM POTENTIAL THEORY

Doak’s Momentum Potential Theory (MPT) [12] is closely related to the works of Myers [19] and Jenvey [20], which focus on the exchange mechanisms between acoustic and non-acoustics fields for turbulent and homentropic flows. The peculiarity of the MPT is the explicit emphasis on the concept of local fluctuating dynamical equilibrium [12], that is the notion of an equilibrium between flow fluctuating quantities only. The theory assumes no strong assumptions but the time-stationarity of the flow. The primary variable in the MPT formulation is the momentum density ρ u i , where ρ and u i represent local density and velocity of the fluid respectively. The starting point for deriving the MPT is the Helmholtz decomposition of the vector ρ u i in its solenoidal and irrotational parts:

ρ u i = B i − ∂ψ

∂ x i , ∂ B i

∂ x i = 0 . (1)

In Equation 1, B i represents the solenoidal, divergence-free part of ρ u i , while ∂ψ/∂ x i is the irrotational potential obtained according to the Helmholtz theorem. For time-stationary flows, the density can be split into its mean and fluctuating parts:

ρ ( t , x i ) = ρ ( x i ) + ρ ′ ( t , x i ) . (2)

Using Equations 1 and 2, the conservation of mass can be expressed as:

∂ρ ′

∂ t − ∂ 2 ψ ′

∂ x 2 i = 0 , ∂ 2 ψ

∂ x 2 i = 0 . (3)

As shown in Equation 3, the time-stationarity of the density field implies that the continuity equation can be written as a Poisson di ff erential equation for the potenti al ψ ′ , which only depends on ρ ′ . It is worth noting that according to Equation 1 and 3, the mean value ψ is irrotational as well as solenoidal at the same time and can be assumed to be zero - without loss of generality. Thus, the momentum mean value is given by B i , i.e.:

ρ u i = B i = B i − B ′ i . (4)

On basis of constitutive thermodynamic relationships [12], the quantity ψ ′ can be further decomposed as the sum of an acoustic and a thermal component. This means that Equation 3 can be rewritten as:

∂ ( ψ ′ = ψ ′ A + ψ ′ T ) ∂ x i = ∂ρ ′

∂ t ,

(5)

∂ 2 ψ ′ A ∂ x 2 i = 1

∂ t , ∂ 2 ψ ′ T ∂ x 2 i ∂ρ

c 2 ∂ p ′

∂ S ′

!

∂ t ,

∂ S

p

where p ′ and S ′ represent fluctuating pressure and entropy respectively. The potentials ψ ′ A and ψ ′ T (as well as the total potential ψ ′ ) are defined as the solution of at least two Poisson equations, which are dependent on fluctuating density, entropy and pressure. The momentum density can then be rewritten as:

ρ u i = B i + B ′ i − ∂ψ ′ A ∂ x i − ∂ψ ′ T ∂ x i . (6)

In Equation 6, the primary variable is expressed as a s uperposition of four clearly separated and uniquely defined components: the mean component B i , the turbulent component B ′ i , the acoustic component ∂ψ A /∂ x i and the thermal component ∂ψ T /∂ x i . Equation 6 represents the core of the MPT and introduces, in the context of Doak’s formulation, an unambiguous definition for the words "acoustic", "thermal" and "turbulent" [12]. The definitions of the fluctuating momentum components are useful in providing a relation for the mean energy flux that only depends on fluctuating quantities. First, the energy transport equation can be expressed in terms of the total enthalpy H = h + 1

2 u 2 i , defined as the sum of the specific enthalpy h and of the kinetic energy per unit mass 1

2 u 2 i . Using Equation 6, it can be sho wn th at, similar to the momentum, a superposition principle exists for the total enthalpy mean flux H ρ u i too. Then, a relationship for the production of this quantity can be obtained averaging the energy transport equation, that reads:

H ρ u i z }| {

H B i + H ′ B ′ i − H ′ ∂ψ ′ A ∂ x i − H ′ ∂ψ ′ T ∂ x i | {z }

  S i j u j + λ ∂ T

 + Q T , (7)

∂ ∂ x i

! = ∂

∂ x i

∂ x i

H ′ ( ρ u i ) ′

In Equation 7, the H ρ u i production is balanced by the sources on right-hand side, with the quantity S i j , T and Q T representing the viscous stress tensor, the static temperature and the rate of heat addition per unit volume respectively. As previously noted, one of the main objective of the MPT is, however, the definition of a local fluctuating dynamical equilibrium. This is achieved by rearranging Equation 7 so that only fluctuating quantities are involved in the balance:

! ′ + ( ρ T ) ′ ∂ S ′

ρ ∂ S i j

∂ ∂ x i [ H ′ ( ρ u i ) ′ ] = − ( ρ u i ) ′ ( Ω × u ) ′ i + ( ρ u i ) ′ T ∂ S

∂ x i + 1

∂ t (8)

∂ x j

Equation 8 represents a balance of the energy mean flux due to the momentum fluctuations that only depends on kinematic and internal thermodynamic variables. The sources of the energy mean flux H ′ ( ρ u i ) ′ on the right side are three outflowing components (production rates). The first is related to the fluctuating Coriolis acceleration ( Ω × u ) ′ i due to the presence of the vorticity Ω . The second source is a dissipative deceleration per unit mass due to entropy gradients and viscous e ff ects. The third source is again dissipative and directly connected to thermal di ff usion e ff ects due to the fluctuations of entropy. Moreover, Equation 8 is an important starting point for applying the MPT to acoustic problems. In fact, Doak interprets the mean flux of the Total Fluctuating Enthalpy (TFE) H ′ as a

generalised acoustic intensity or as a measure of the acoustic radiation to the far-field. To demonstrate this, Equation 8 can be rewritten as follows:

∇· J = q J , (9)

where J represents H ′ ( ρ u i ) ′ and q J the sources on the right-hand side of Equation 8. By definition, the TFE total power radiated from any fluid volume V can be expressed as the integral over V of the sources q J , that is:

W V TFE = Z

V q J . (10)

Although, under the assumption that the radiation condition holds true in the far-field, the TFE mean flux reduces to:

H ′ ( ρ u i ) ′ ≈ H ′ ( −∇ ψ A ) ′ → p ′ u ′ i + O ( p ′ 2 u ′ i , ... ) . (11)

This means that the energy mean flux due to the momentum fluctuations can be interpreted as a generalised acoustic intensity and the TFE as a Generalised Acoustic Field (GAF). In fact, choosing a domain far away from the sources and according to the Gauss theorem, the quantity W TFE approximates the radiated acoustic power and q j represents the acoustic source. That is:

W Tot TFE = Z

V far q j ≈ W Acou . (12)

The TFE can be further decomposed into its mean, turbulent, acoustic and thermal components:

H = H + H ′ = H + H ′ B + H ′ A + H ′ T , (13)

where the fluctuating components H ′ B , H ′ A and H ′ T can be found following Jenvey’s approach [20]:

H ′ B = c 2 M i B ′ j ρ , (14a)

∂ψ ′ A ∂ x i

H ′ A = (1 − M i 2 ) p ′

! , (14b)

ρ − cM i

ρ

H ′ T = c 2

γ − 1 [1 + ( γ − 1) M i 2 ] S ′

c p = H S ′

c p . (14c)

Thus, a decomposition in clearly defined turbulent, acoustic and thermal components is achieved for the TFE as well. In particular, the acoustic component H ′ A can be used to further approximate the GAF radiation to the far-field.

3. MPT FOR ANALYSIS OF COMBUSTION NOISE SOURCES

Doak provides several reformulations of Equation 8. Among these, one is of particular interest with regards to the application of the MPT for the analysis of combustion noise sources:

S Acou z }| { ∂ψ ′ A ∂ x i

H ′ ( ρ u i ) ′ z }| { H ′ B ( ρ u i ) ′ + H ′ A ( ρ u i ) ′ + H ′ T ( ρ u i ) ′  = − B ′ i α ′ i |{z} S Turb

∂ψ ′ T ∂ x i

! α ′ i + ( ρ T ) ′ ∂ S ′

∂ ∂ x i

! α ′ i +



. (15)

∂ t | {z } S Therm

The balance expressed in Equation 15 relates the TFE mean flux H ′ ( ρ u i ) ′ to the behaviour of the three sources on the right-hand side S Turb , S Acou and S Therm . In the source terms, the new factor α ′ i appears. This variable is defined as follows:

α ′ 3 , i z }| { 1

α ′ 1 , i z }| { ( Ω × u ) ′ i − T ∂ S

! ′

! ′ .

ρ ∂ S i j

α ′ i =

∂ x i

∂ x j

| {z } α ′ 2 , i

The three terms α ′ 1 , i , α ′ 2 , i and α ′ 3 , i can be interpreted as the main physical mechanisms responsible for the production of TFE mean fluxes. The term α ′ 3 , 1 depends on the viscous stresses and is therefore of minor interest for combustion acoustics problems. Thus, the factor α ′ i is approximated by neglecting α ′ 3 , i as: α ′ i ≈ α ′ 1 , i + α ′ 2 , i ,

! ′ . (16)

α ′ 1 , i = ( Ω × u ) ′ i , α ′ 2 , i = − T ∂ S

∂ x i

The first mechanism is related to the fluctuating Coriolis acceleration α ′ 1 , i , obtained as the cross product of vorticity Ω i and velocity. This term becomes relevant in flow region where turbulence or strong accelerations play an important role. The swirling process or the flow acceleration through the NGV are typical examples of such regions. The second mechanism described by α ′ 2 , i is conditioned by entropy gradients. This can become relevant downstream of a premixed flame or in the film cooling region. Both α ′ 1 , i and α ′ 2 , i interact separately with acoustic, thermal and turbulent momentum fluctuations components. These can be seen as "the carriers" of TFE mean fluxes and determine how the generation mechanisms interact with di ff erent acoustic, thermal and turbulent modes. In a sense, the MPT therefore achieves a new interpretation of the generation of indirect combustion noise that can be summarised as follows:

– The processes which drive the coupling of modes are described by the factors α ′ i . These directly relate to the fluctuating Coriolis acceleration and the entropy gradient fluctuations.

– The terms S Turb and the α ′ i -dependent part of S Therm define the interaction of turbulent and thermal fluctuations with the production mechanisms included in α ′ i and can potentially be related to the production of vorticity and entropy noise. The additional term S Acou is a source due to the interaction between the acoustic momentum potential and α ′ i and is expected to be small for realistic combustion chamber flows.

Moreover, the last term of the thermal source, namely ( ρ T ) ′ ∂ S ′

∂ t , does not depend on α ′ i . However, one may argue that, because of the strong relation between entropy, temperature and heat release fluctuations, this term represents a measure for the direct combustion noise source. The application of the MPT for the analysis of combustion noise could therefore provide a clearer physical understanding of all combustion noise sources.

4. APPLICATION TO LES DATA OF A NON-REACTING COMBUSTOR SIMULATOR

To demonstrate the suitability for combustion noise analysis, the MPT formulation is applied to Large- Eddy Simulation (LES) data of a non-reacting combustor simulator. The simulator geometry is a simplified version of the FACTOR configuration, a full annular combustor configuration developed to investigate the interaction between combustor and NGV [21]. For this early stage of the study, the NGV is neglected and the simulation domain is reduced to a periodic 18 ◦ sector of the full geometry, containing a single swirler as shown in Figure 1. The picture highlights the boundary locations and the corresponding boundary conditions are reported in Table 1. Cooling air enters the combustion

Figure 1: Geometry of simulation domain with axial (red) and cross (green) analysis planes

Table 1: Boundary conditions

Variable Value

Main inlet mass flow 3.1 kg / s

Main inlet total pressure 147 kPa

Main inlet total temperature 512 K

Inner coolant mass flow 0.66 kg / s

Outer coolant mass flow 0.94 kg / s

Coolant temperature 298 K

Outlet static pressure 141 kPa

chamber from outer and inner cooling source regions. Since no combustion occurs, the film cooling represents, together with the activity of the swirler, the main disturbance for the flow in the combustion chamber. The simulation data are sampled with a frequency of 10 kHz on the analysis planes shown in Figure 1. Approximately 1400 time steps after convergence of the mean flow are considered for post-processing. For the sake of brevity, only the results on the plane y 0 , as a measure of the overall development of the fluctuations, are reported in the following. At first, the GAF components are computed using the relations in Equation 14. For this purpose, the fluctuating momentum components B ′ i , ∂ψ ′ A /∂ x i and ∂ψ ′ T /∂ x i are needed. They are dependent on the flow fluctuations and can be obtained solving at least two of the Poisson equations expressed in Equation 5. Since density and pressure fluctuations are directly given by the simulation, the system is solved for the total potential ψ ′ and its acoustic component ψ ′ A . Using the definitions given in § 2, the turbulent and thermal components B ′ i and ψ ′ T can then be determined. To solve the di ff erential equations for the potentials, a solver based on Finite Element Methods (FEM) is used. All fixed walls are set as zero-value Dirichlet boundary. Close to the boundaries in the source regions, acoustic e ff ects and density fluctuations are assumed to be very small, thus justifying the use of fixed value of zero at the respective boundary conditions - identical to the treatment at the wall boundary conditions. For the analysis plane y 0 , the outlet boundary condition is set using the results obtained on the x 7 -station, where the Poisson equations are solved in cylindrical coordinates, with periodicity on the sides given by the simulation setup. Mean values for Mach number, speed of sound as well as temperature and entropy fields are obtained from the CFD data. Once momentum fluctuations and GAF components are computed, a Spectral Proper Orthogonal Decomposition (SPOD) [22] analysis is performed for both quantities to extract coherent structures and to examine the spectral behaviour. Analogously, pressure and entropy fluctuations are analysed. The blocks for the SPOD contain 400 time steps each and feature an overlap of 60%. With these settings, seven modes are obtained for each frequency. The respective results are reported in § 4.2. The results of the MPT application, described in § 3, are presented in § 4.3.

4.1. CFD Simulation Setup The in-house CFD-solver TRACE [23] of the Institute of Propulsion Technology at DLR is used to perform the simulation. The unstructured, hex-dominant computational mesh contains about 13 million cells. Convective fluxes are discretised using Roe‘s flux di ff erence splitting and a 2nd order accurate MUSCL reconstruction with the slope limiter of Venkatakrishnan [24]. For the discretisation of the viscous fluxes derivatives, central di ff erences are chosen, whereas an explicit Runge-Kutta scheme of 3rd order with a constant time step of ∆ t = 5e-8 s is used for time integration. The sub- grid scale turbulence model is set as the WALE model [25] and the gas is considered as calorically perfect, with Sutherland’s law used for molecular viscosity computation. The e ff ect of the coolant is considered by using thin volume sources that are modelled with the Adiabatic Homogeneous Model [26].

4.2. SPOD analysis The SPOD decomposes the fluctuations in orthogonal, space-time developing structures - referred to as "modes"- which evolve with respect to a determined frequency. A SPOD output consists of two parts: the modal energies and the mode amplitudes. The modal energies are obtained as the eigenvalues of an eigenvalue problem solved for each of the analysed frequencies. The modes are ordered from the most dominant (so-called "leading mode") to the less energetic ones depending on the choice of the energy norm, which is set here as the data variance. The eigenvectors extracted from the same problem represent the modal amplitudes and express a measure for the distribution of modes in space and time. The first two quantities analysed with the SPOD are α ′ 1 , x and α ′ 2 , x on the plane y 0 , where the index x represents the direction of mean flow propagation. The obtained modal energies are normalised with respect to the overall SPOD energy (sum of all eigenvalues) and shown in Figure 2. Each line corresponds to a mode evolving in the Fourier space, where the black line indicates the leading mode and the line shading indicate subsequent order modes. The spectra lack distinct features over most of the frequency range with the exception of two peaks, which are clearly recognisable. The peak f 0 , at approximately 575 Hz , is present for both α ′ 1 , x and α ′ 2 , x , whereas the second peak f 2 , at about 1775 Hz , is only observable in the α ′ 1 , x -spectrum. A possible explanation of how these parameters influence the mode coupling is given in Figure 3, where the spectral behaviour of the modal energy for the TFE component H ′ A is compared with the output obtained for the pressure fluctuations. For the pressure, three peaks can be clearly seen in the leading mode: the f 0 and f 2 peaks

Normalised SPOD energy [-]

10 − 2

10 − 2

10 − 3

10 − 3

10 − 4

10 − 4

f 0 f 2 10 − 5

f 0 10 − 5

Frequency [Hz]

Frequency [Hz]

(a) α ′ 1 , x

(b) α ′ 2 , x

Figure 2: Normalised SPOD energy on plane y 0 for α ′ 1 , x and α ′ 2 , x

10 − 1

10 − 1

Normalised SPOD energy [-]

10 − 2

10 − 2

10 − 3

10 − 3

10 − 4

10 − 4

10 − 5

10 − 5

f 0 f 1 f 2 10 − 6

f 0 f 1 f 2 10 − 6

Frequency [Hz]

Frequency [Hz]

(a) p ′

(b) H ′ A

Figure 3: Normalised SPOD energy on plane y 0 for p ′ and H ′ A

(a) f 0 ≈ 575 Hz

(b) f 2 ≈ 1775 Hz

*. a mode amplitude [-] | -0.01 0 0.01

Figure 4: SPOD amplitudes for H ′ A on plane y 0 at f 0 and f 2

correspond to the peaks in the α ′ 1 , x spectrum and a third peak at the frequency f 1 close to 1375 Hz . Interestingly, the f 1 -peak is damped in the H ′ A leading mode, which otherwise develops similarly to the pressure fluctuations over the entire frequency range. The damping of f 1 highlights the role of the mechanisms described by α ′ i that seem to work amplifying or rather selecting some spectral characteristics of the GAF, represented in this case by the peaks on f 0 and f 2 . Interestingly, it could be shown that the modal amplitudes relative to f 0 and f 2 , here shown in Figure 4 for H ′ A , remain almost equal when compared to the same output obtained for the pressure. This could mean that the MPT decomposition only reallocates the spectral energy of the fluctuations, while their coherence remains unchanged.

4.3. Combustion noise sources on basis of the MPT Following the arguments of § 2, the total TFE radiation power can be computed integrating the sources on the right-hand side of Equation 15. The results of these integrals over the section y 0 are shown in Figure 5. As expected, the main contribution is represented by the turbulent source. The α ′ i -dependent components of the thermal source work as sinks, whereas a small acoustic source can be observed. All other co ntributi ons are negligi ble. Figure 6 shows the distributions of the three main contributions S ( α 1 ) Turb = B ′ i α ′ 1 , i , S ( α 2 ) Turb = B ′ i α ′ 2 , i and S ( α 2 ) Therm = ∂ψ ′ T /∂ x i α ′ 2 , i for the section y 0 . These describe all most import mechanisms of TFE generation. In fact, S ( α 2 ) Therm and S ( α 2 ) Therm , dependent on α ′ 2 , i , are connected with the e ff ects of the cooling film in source regions, represented by the mixing process and the shear flow generating at chamber entrance. The source S ( α 1 ) Turb , dependent on α ′ 1 , i , considers the swirling process and hydrodynamic e ff ects in general, e.g. the acceleration of the flow

hh, mode amplitude [-] -0.01 0 0.01

1 , 500

m 3 ]

Source magnitudes [ W

1 , 000

500

0

− B ′ i α ′ 1 , i − B ′ i α ′ 2 , i ∂ψ ′ A ∂ x i α ′ 1 , i

∂ψ ′ A ∂ x i α ′ 2 , i

∂ψ ′ T ∂ x i α ′ 1 , i

∂ψ ′ T ∂ x i α ′ 2 , i ( ρ T ) ′ ∂ S ′

∂ t

gle) Su, (W/m) 7 = -500000 0 500000

Figure 5: Source contributions to TFE radiation for plane y 0

(a) S ( α 1 ) Turb = − B ′ i α ′ 1 , i

(b) S ( α 2 ) Turb = − B ′ i α ′ 2 , i

∂ψ ′ T ∂ x i α ′ 2 , i

(c) S ( α 2 ) Therm =

Figure 6: Distribution of principal sources of TFE radiation on plane y 0

> —

in the convergent part of the chamber. Finally, it is interesting to examine how the s ources dis tribution influences the GAF intensity. To this end, Figure 7 shows the x -component of the H ′ A ( ρ u i ) ′ on plane y 0 , as a measure of the acoustic fluxes generated in and exiting from the combustor simulator. As expected, the acoustic production is small (between ± 1% of the TFE absolute maximum) and mostly results from hydrodynamic e ff ects related to the swirler activity and flow acceleration. Only a small part of the acoustic intensity is generated in the cooling regions or dissipated in the shear layer.

5. CONCLUSIONS

This study proposed a new approach for the analysis of combustion noise sources based on Doak’s MPT. With the help of the MPT formulation, a possible physical description as well as a clear separation of all direct and indirect combustion noise sources could be achieved. Furthermore, a GAF was identified in the TFE, whose mean fluxes were balanced by the previously identified sources. The production of TFE mean fluxes was found to be related to two main mechanisms, described by the factor α ′ i : the fluctuating Coriolis acceleration ( Ω × u ) ′ i and the term  − T ∂ S

∂ x i  ′ . In the interaction

ier) Therm 0 (W/m? 50000

Figure 7: x -component of GAF intensity H ′ A ( ρ u i ) ′ on plane y 0

between α ′ i and the turbulent, acoustic and thermal fluctuating momentum components obtained by a Helmholtz decomposition of the momentum fluctuations, a possible description of the mode coupling was identified. To test the ability of describing combustion noise generation, the MPT was applied to Large-Eddy Simulation data of a non-reacting combustor simulator, where the cooling film and swirler activity represented the main entropy and vorticity disturbances respectively. The spectral behaviour and the coherence of the GAF and pressure fluctuations were investigated by means of an SPOD analysis. The factor α ′ i seemed to only redistribute the spectral energy of the GAF, leaving their coherence unchanged. The analysis of the principal source contributions were able to describe and separate all relevant phenomena responsible for the TFE production. Finally, the hydrodynamic e ff ects were found to be the first source of GAF radiation. All of these findings highlight the potential of the MPT as a really promising formulation to distinguish between the sources and to describe the generation of combustion noise in realistic combustor configurations.

ACKNOWLEDGEMENTS

The work presented in this paper is carried out in the project MUTE (Methoden und Technologien zur Vorhersage und Minderung von Triebwerkslärm, grant number 20T1915D), which is supported within the framework of the German Luftfahrtforschungsprogramm VI by the German Federal Ministry for Economic A ff airs and Energy (BMWK) on the basis of a resolution of the German Bundestag. The authors acknowledge the continuous support of Rolls-Royce Deutschland Ltd. & Co KG discussing the progress of the work. Moreover, they wish to gratefully acknowledge FACTOR Consortium for the kind permission of publishing the content herein. FACTOR was a Collaborative Project co-funded by the European Commission within the 7th Framework Programme (2010-2017) under the Grant Agreement no. 265985. Finally, the authors would like to express their gratitude to Simon Gövert and the department of Combustion Chamber Simulation at the DLR for their help in setting up and performing the CFD simulations.

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