Toward the study of combustion instabilities in aircraft engines

Toward the study of combustion instabilities in aircraft engines

A. Chatelier ^∗

The prediction of combustion instabilities is crucial for aircraft engines manufacturers to ensure the reliability and the life span of their engines. To study these phenomena, numerical simulations are more and more often performed. They offer advantages: a wide range of parameters is available, they are safe and inexpensive compared to experiments.

This work presents a first step in the general study of combustion instabilities. Firstly, several tools for acoustics and combustion are validated. The tests are performed on simple cases which have reference solutions from analytical resolution or measurements, such as one dimensional flames for combustion and rectangular duct for acoustics. The parameters of these cases are close to the more complex ones. Then, the dynamics of a laminar flame are studied. Finally, a laboratory scale configuration is explored. Kinetics for a methane-air mixture are validated as well as the TFLES model for combustion. The AVSP code for acoustics correctly determines the eigenmodes of a simple configuration.

The TFLES model on a pulsed flame has a significant impact on its dynamics, depending on the thickening factor and the model used. An unstable mode is found for reacting RANS computations of a laboratory scale configuration. An unstable mode is also predicted by AVSP computations with an active flame model. Even if further work is needed to develop these tools, the first results indicate that they can quickly yield predictions about combustion instabilities.

I. Introduction

The environmental legislation becoming more and more strict, aircraft engine manufacturers developed new concepts of less polluting combustion chambers with a focus on NOx emissions. However, these new chambers, designed to operate at low equivalence ratio, may experience combustion instabilities that can damage or, even worse, destroy them. Trying to predict numerically these instabilities is thus a crucial challenge for the developers of CFD codes.

The first scientist to study combustion instabilities, i.e. the interaction between acoustics and combustion, was Lord Rayleigh with his singing flame. ¹ He defined what is called nowadays the Rayleigh criterion in combustion. Another major researcher to focus on combustion instabilities was Crocco, ² during the 1950s.

In the 1990s, McManus et al. ³ made a review of the progress on the control of these instabilities. Recently, researchers focused on ways to model flames for acoustic purposes, for instance Giauque ⁴ on the Flame Transfer Function (FTF).

With the currently available computation resources, unsteady numerical simulations of reacting flows, on technical scale configurations can not resolve the entirety of flame characteristic lengths and their interactions with turbulence, due to computational costs. For the conditions usually encountered inside combustion chambers, the heat thickness of the flame front is very small (usually below one millimetre) and the reaction zone even smaller (one tenth of the heat thickness). Standard LES (Large Eddy Simulation) meshes, designed initially to resolve the most energetic turbulent structures, have mostly a characteristic mesh size larger than the heat thickness. Thus they are unable to resolve properly the instantaneous flame front. This is why numerous models, such as TFLES (Thickened Flame for LES) have been developed to take into account the limits of standard LES meshes in combustion. However, these models have been designed to recover integral properties of the flame, for example its consumption speed, but that can be done to the detriment of its dynamics. Unfortunately, this dynamics has a key role in the combustion instabilities, in which the heat

∗

Intern at DEFA (Department of Fundamental and Applied Energetics), ONERA, 29 avenue de la Division Leclerc, 92322,

Chatillon, France and Double-Degree Student at KTH, Stockolm, Sweden and ENSTA ParisTech, Palaiseau, France

release coming from the combustion is coupled with the acoustics of the chamber: a poor resolution of the flame dynamics modifies this coupling and leads to an incorrect prediction of instabilities, if they exist.

The structure of this article is the following: the numerical tools are presented, then the combustion and acoustic models are validated. The interaction between acoustics and combustion is studied on a pulsed laminar flame. Finally, the combustion instabilities are explored on a laboratory scale configuration, using the tools validated previously.

II. Numerical Tools

II.A. CEDRE

II.A.1. Main characteristics of CEDRE

CEDRE is a research code developed at ONERA and designed to solve multi-physics problems for energetics and propulsion, with several solvers. The solver used in this work is CHARME, which solves the Navier- Stokes equations for multispecies reacting flows. Using Favre’s decomposition (for more details, see the appendix A on the filtering of Navier-Stokes equations) and the Einstein’s notations, the filtered or averaged Navier-Stokes equations are recast in RANS and LES models as Eqs (1), (2), (3) and (4):

∂ρ

∂t + ∂

∂x i

(ρ u e i ) = 0, (1)

∂ρf Y k

∂t + ∂ρ u e i Y f k

∂x _i + ∂J k,i

∂x _i − ˙ω _k = − ∂

∂x _i

 ρ^ u

_i Y _k



= − ∂

∂x _i

 J _k,i ^t/sgs 

, (2)

∂ρ u e _i

∂t + ∂

∂x j

(ρ u e _i u e _j ) + ∂p

∂x i

− ∂τ _ij

∂x j

= − ∂

∂x j

 ρ ] u

_i u

_j 

= − ∂

∂x i

 τ _ij ^t/sgs 

, (3)

∂ρ e e t

∂t + ∂

∂x _i (ρ u e i e e t ) − ∂

∂x _j [(τ − pδ ij )u i ] + ∂J e

,i

∂x _i = − ∂

∂x _j h

ρ ] u

_i e

_t + τ _ij

− p

δ ij
u

_i i

(4)

= − ∂

∂x _j



J _e ^t/sgs

+ τ _ij ^t/sgs e u i



. (5)

The terms on the right hand side (RHS) need to be closed, i.e. modelled, in order to solve these equations.

The closure uses classical analogies (e.g. Chaouat ⁵ ). The first term that has to be closed is the RHS of the momentum equation, Eq. (3). In RANS, an analogy is made between the Reynolds tensor τ _ij ^t and the viscous strain tensor τ ij , as shown in Eq. (6) in BVM (Boussinesq Viscosity Model) approach:

τ _ij ^t ≡ −ρ ] u

_i u

_j = µ t

 ∂ u e _i

∂x j

+ ∂ u e _j

∂x i

− 2 3

∂ f u _k

∂x k

δ ij



− 2

3 ρkδ ij . (6)

In the present work, turbulent viscosity µ _t is modelled with the two-equation k − L model:

µ t = ρL √

k. (7)

In LES, the approach is similar and uses a classical Smagorinsky model:

τ _ij ^sgs ≡ −ρ ] u

_i u

_j = µ sgs

 ∂ u e i

∂x j

+ ∂ u e j

∂x i

− 2 3

∂ f u k

∂x k

δ ij



. (8)

The subgrid scale (sgs) viscosity µ sgs is defined in Eq. (9) with C s being a constant of the model (usually 0.1 < C s < 0.2), ∆ the size of the spatial filter and e S the resolved velocity gradient tensor:

µ sgs = ρ(C s ∆) ² p

2 e S : e S. (9)

The closure of the specie equation (Eq. (2)) and total energy equation (Eq. (4)) is made on the assumption that the turbulent flux follows a gradient law as an analogy with the Fick and Fourier laws, as in Eqs (10) and (11). It is valid in both RANS and LES. This model requires the introduction of two non-dimensional numbers, the turbulent Schmidt number Sc _t/sgs,k and the turbulent Prandtl number P r _t/sgs,k :

J _k,i ^t/sgs = −D ^t/sgs _k ∂ f Y _k

∂x i

with D ^t/sgs _k = µ t/sgs /(ρSc t/sgs,k ), (10)

J _e ^t/sgs

= q ^t/sgs _i +

n

X

k=1

h k J _k,i ^t/sgs = −λ ^t/sgs _k ∂T

∂x i

+

n

X

k=1

h k J _k,i ^t/sgs with λ ^t/sgs _k = µ _t/sgs c p /P r _t/sgs,k . (11)

II.A.2. Numerical methods in CHARME

CHARME is based on a cell-centered finite volume methodology for unstructured meshes of general polyhe- dra. The interpolation is based on MUSCL (Monotone Upstream Schemes for Conservation Laws) schemes.

Both explicit and implicit methods of time integration are implemented in CHARME. The method mainly used in this work is the backward Euler implicit method.

II.B. AVSP

II.B.1. Main characteristics of AVSP

The AVSP code, from CERFACS, is designed to solve linear acoustic problems on 2D or 3D configurations.

It solves the Helmholtz equations assuming harmonic waves (Eqs (12), (13), (14) and (15) from Poinsot and Veynante ⁶ for non-reacting flows) where p _ω is the harmonic pressure variation, ~ u _ω the harmonic velocity variation, ρ _ω the harmonic density variation, k = ω/c ₀ the wave number, ω the pulsation and c ₀ the speed of sound:

−iωp ω + ρ 0 c ² ₀ ∇.~ u ω = 0, (12)

−ρ 0 iω~ u _ω + ∇p _ω = 0, (13)

p ω = c ² ₀ ρ ω , (14)

which leads to:

∇ ² p ω + k ² p ω = 0. (15)

The quantities p 1 , ~ u 1 and ρ 1 are defined in Eq. (16):

p ₁ = <(p _ω e ^−iωt ), ~ u ₁ = <(~ u _ω e ^−iωt ), ρ ₁ = <(ρ _ω e ^−iωt ). (16) These quantities are used to define the acoustic impedance Z and the reflection coefficient R in Eq. (17):

Z = 1 ρ ₀ c ₀

p 1

u ₁ , R = Z + 1

Z − 1 . (17)

II.B.2. Numerical methods in AVSP

AVSP deals with triangles in 2D and tetrahedra in 3D. Two options are available: multiple modes, where modes are computed with increasing frequencies or single mode, where the target frequency is specified as an input. In multiple modes option, two boundary conditions are available. Either the acoustic pressure variation p ₁ is null, which corresponds to an open end or the acoustic velocity variation u ₁ is zero, which is equivalent to a wall. In single mode option, more boundary conditions are available for specific purposes, for instance a given impedance can be set as a constant or from experimental data. Also, an active flame model is available for the study of combustion instabilities.

III. Validation of combustion models

This section aims to compare computational and experimental results on one-dimensional laminar pre- mixed flames. It is possible to express the laminar flame speed S _L as Eq. (18). The derivation of this formula in shown in the appendix C. This expression is valid for the fuel, as well as for the oxidizer:

S L = ω ¯˙ k

ρ u (Y kb − Y ku ) . (18)

Values of ρ u , Y kb and Y ku are computed once, knowing the composition of both the burned and unburned gases. ¯˙ ω k is the integral on the domain of the source term ˙ ω k . The finer the grid along the x direction, the better the term ¯˙ ω _k is approximated. The experimental results for this part of the study come from the experiments of Vagelopoulos ⁷ for the methane-air premixed flame. The computations are done using CEDRE. The computational domain is a two-dimensional rectangle, 5 · 10 ⁻² m-wide in the x direction and 1 m-high in the y direction. Each cell is 5 · 10 ⁻⁵ m-wide in the x direction and 1 m-high in the y direction.

The domain is therefore composed of 1000 cells. The initial position of the flame front is at x = 2.5 · 10 ⁻² m with the origin on the left boundary. On the left side of the flame, the domain is initialised with fresh gases while the right side is initialised with burnt gases. The equivalence ratio of the computation is φ = 1.05. The kinetics used in this section is CM2-CH4-2R. ⁸ It involves six species and two reactions, shown in Eq. (19).

Table [1] sums up the composition and state in fresh and burned gases for this equivalence ratio.

( CH ₄ + ³ ₂ O ₂ −→ CO + 2H ₂ O

CO + ¹ ₂ O ₂ ←→ CO ₂ (19)

Quantity Value in fresh gases Value in burned gases

Y CH

[–] 0.05781 0.

Y O

[–] 0.22 0.001336

Y H

O [–] 0. 0.1298

Y CO

[–] 0. 0.1247

Y CO [–] 0. 0.02156

Y _N

[–] 0.72219 0.722604

T [K] 300 2261.62

P [Pa] 101325 101325

Table 1: Specie composition, temperature and pressure in fresh and burned gases

III.A. Methane-air chemical scheme

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Flame speed (m/s)

Equivalence ratio (-) Experimental data

GRI-Mech 3.0 2S_CH4_BFER with PEA correction CM2_CH4_2R

Figure 1: Comparison of the laminar flame speed between experiments and computations.

Fig. 1 shows a comparison between three chemical schemes (GRI-Mech 3.0, ⁹ 2S CH4 BFER with PEA

correction described by Franzelli et al. ¹⁰ and CM2 CH4 2R) and experimental results from Vagelopoulos

and Egolfopoulos ⁷ for an equivalence ratio between 0.7 and 1.4. GRI-Mech 3.0 ⁹ is the reference detailed chemical scheme for natural gases, including methane. One can notice that the experimental data and the GRI-Mech 3.0 do not give exactly the same flame speed. The discrepancy may have two origins: firstly, GRI-Mech 3.0 is a detailed scheme but does not describe perfectly all the elementary reactions that occur in reality; secondly, the experimental results have uncertainties of measurement. Despite this, the evolution is well reproduced. 2S CH4 BFER and CM2 CH4 2R are two reduced chemistry schemes that are very similar.

CM2 CH4 2R leads to poor results for rich mixture (i.e. φ > 1.1). Franzelli et al. proposed a PEA (Pre Exponential factor Adjustment) correction to fit 2S CH4 BFER with GRI-Mech 3.0 for rich mixtures. This is confirmed by the results shown in Fig. 1.

The reduced schemes reproduce well the flame speed for lean mixture (φ < 1). The scheme that is used in this work is CM2 CH4 2R, dedicated to lean combustion.

III.B. TFLES model

III.B.1. Presentation of the model

TFLES stands for Thickened Flame for LES. For common LES simulations, the size of the flame front is usually smaller than the smallest cell size. It means that either the mesh can be refined or the flame front can be thickened. The first option is most of the time not realistic due to computational costs. The second option is the principle of TFLES. The laminar speed S _L and thickness δ _L of a flame can be expressed as shown in Eq. (20):

S L ∝ pD _th ^u A, δ L ∝ D ^u _th S L

. (20)

Assuming that the heat diffusion coefficient in Eq. (20) can be considered as D th regardless the position in space and that Le k = 1, so that the diffusion coefficients can be written as a unique coefficient D, the previous equation becomes Eq. (21):

S _L ∝ √

DA, δ _L ∝ D S L

. (21)

In order to thicken the flame front δ L , the coefficient diffusion D is multiplied by a factor F . Meanwhile, the laminar flame speed has to remain constant, so the preexponential factor A is divided by F , as in Eq. (22):

S ˘ L ∝ r

F D A F = √

DA ⇒ ˘ S L = S L , ˘ δ L ∝ F D

S _L ⇒ ˘ δ L = F δ L . (22) The next step in the use of the TLFES model is the application of the thickening factor. One has two possibilities: either the TFLES model is applied everywhere in the domain, or the model is applied specifically in the region where the reaction occurs (i.e. the flame zone). The first case is the simplest one and is a reasonable first step when studying the effect of the TFLES model on a flame. It will be called here standard TFLES. The second case requires a criterion that determines where to apply the model. This one will be called dynamic TFLES. The idea behind dynamic TFLES is that the thickening factor is computed in each cell of the domain. In order to apply the model only in the flame, one requires a flame sensor. Eq. (23) is an example of a pseudo reaction rate that has its maximum in the flame. Subscripts F and O stand for Fuel and Oxidizer. Γ is a constant and is usually set to 0.5. T a is the activation temperature for the considered reaction. Eq. (24) is the expression of the sensor α. Ω 0 is the pseudo-reaction rate computed in the laminar 1D flame. C α is a parameter that determines the thickness of the thickened zone. Finally, Eq. (25) gives the expression of the thickening factor as a function of the maximum thickening factor F max and the sensor α:

Ω = Y F Y O exp



−Γ T _a T



, (23)

α = tanh

 C _α Ω

Ω 0



, (24)

F (α) = 1 + (F max − 1)α. (25)

III.B.2. Validation of the TFLES model on a 1D flame

In order to confirm that the TFLES model gives a correct flame speed, it is tested on the same case as in

section III.A with a coarser mesh (the cells are four times larger). The results are shown in Fig. 2:

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

0 50 100 150 200 250

Laminar flame speed (m/s)

Iteration

F = 1 F = 2 F = 3 F = 4

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

0 50 100 150 200 250

Flame thickness (m)

Iteration

F = 1 F = 2 F = 3 F = 4

Figure 2: Convergence of the computation of S L and δ L for several thickening factors F

Fig. 2 shows that the convergence of the flame speed depends on the thickening factor. Computations with F = 1 and F = 2 present oscillations which show that F is not large enough to make the model converge on this given mesh. Also, the flame thickness is not correct for these two thickening factors. The results for F = 3 are much closer to what was expected. Finally for F = 4, the flame speed S L

is 40.4 cm/s while the flame thickness δ L

is approximately 1.48 · 10 ⁻³ m. Therefore the flame speed is underestimated (by less than 3%) and the flame thickness is overestimated (by 4%) compared to the results without the TFLES model. This margin of error remains acceptable.

IV. Validation of acoustic models

IV.A. Boundary conditions and acoustics

The study of the interaction between acoustics and combustion implies that the acoustic waves are properly propagated in the computational domain. The boundary conditions must be able to reproduce the correct acoustic impedance of the configuration.

Inlet Outlet

Computational domain length Physical domain length

Physical outlet Physical inlet

Figure 3: Acoustic impedance modelling for an ideal one-dimensional duct

Fig. 3 presents an ideal case for acoustic modelling. Two boundary conditions must be set to reproduce the correct acoustic behaviour: the inlet and the outlet. The computational domain is usually different from the experimental domain in order to decrease the computational cost. If the pressure is fixed at a boundary, then p 1 = 0 which leads to Z = 0 and R = −1. If the velocity is fixed, then u 1 = 0 so that Z = ∞ and R = 1. These two conditions will be called standard boundary conditions. If the physical boundaries can be considered very far from the computational ones, non-reflecting boundary conditions can be used. This means that acoustic waves travel without reflection, so that R = 0 or R = ∞ (and Z = −1 or Z = 1).

The previous conditions are ideal. For some configurations, it is not possible to use them because the

actual boundary does not fit any of the previous cases. In order to set properly the boundary conditions,

the acoustic impedance must be measured during experiments.

IV.B. Behaviour of CEDRE boundary conditions Pulsed pressure at the outlet boundary

In this case, the pressure of the outlet boundary of a one dimensional domain is pulsed with a single 4 Pa-amplitude pulse compared to the initial value (101325 Pa).

Inlet Outlet

Symmetry

Symmetry Probe position 2 cm

Figure 4: Computational domain for the test of boundary conditions

Time (s)

Pressure (Pa)

0.0002 0.0004 0.0006 0.0008 0.001

101322 101324 101326 101328

Figure 5: Pressure over time at a given position for the standard boundary condition at 20% of the total length of the domain.

Time (s)

Pressure (Pa)

0.0002 0.0004 0.0006 0.0008 0.001

101324 101325 101326 101327 101328 101329

Figure 6: Pressure over time at a given position for the non-reflecting boundary condition at 20%

of the total length of the domain.

Fig. 5 shows the behaviour of the standard boundary condition on the pressure. The position is roughly at 20% of the total length of the domain. The first peak in pressure corresponds to the normal propagation from the right of the pressure wave. However the second peak short after shows that the pressure peak was reflected entirely by the boundary condition. The last two pressure drops come from the reflection of the pulse on the outlet boundary. They are inverted because the pressure is fixed at the outlet boundary at 101325 Pa. Fig. 6 shows what happens with non-reflecting boundary conditions. The position is the same as in Fig. 5. In this case, the peak reaches the limit but is not reflected and leaves the domain. This validates the behaviour of the non-reflecting boundary condition.

IV.C. AVSP validation

In order to validate the results of AVSP, the code is tested on simple configurations that can be solved analytically. The configuration considered here is a 2D rectangular duct closed on the left and opened on the right and enclosed by walls above and below, as shown in Fig. 7.

The resolution of the Helmholtz equation for harmonic waves gives the eigenfrequencies for this configu- ration in Eq. (26), where m and n are integers:

f m,n = c 0

s

 2m + 1 4L

 2

+  n 2h

 ²

. (26)

For L = 1 m and h = 0.5 m, the results of AVSP and the analytical solution are shown in Table 2.

Closed end, u ₁ = 0 Opened end, p ₁ = 0 Wall, u 1 = 0

Wall, u 1 = 0 L h

Figure 7: Reference configuration for the validation of the AVSP code

Mode (m,n) Analytical solution (Hz) AVSP solution (Hz) Error (in %)

(0,0) 86.8 86.8 0

(1,0) 260.4 260.3 -0.04

(2,0) 434.0 433.3 -0.16

(3,0) 607.6 606.4 -0.20

(0,1) 357.9 357.2 -0.20

(1,1) 434.0 433.6 -0.09

(2,1) 555.8 555.0 -0.14

(3,1) 699.8 694.0 -0.83

(0,2) 699.8 698.4 -0.20

(1,2) 741.6 736.1 -0.74

Table 2: Comparison of eigenfrequencies between analytical and AVSP solutions

The results given by AVSP on this configuration are globally close from the analytical results given by linear acoustics. One can notice that the highest the frequency, the largest the error tends to be. Also, when the eigenfrequency is double (for example the modes (2,0) and (1,1) have the same eigenfrequency), AVSP finds two different frequencies.

Figure 8: Pressure variations in arbitrary units for f = 260.3 Hz (mode (1,0)) and f = 555.0 Hz (mode (2,1))

Fig. 8 shows two examples of results from AVSP on the absolute pressure variation (the scale is arbitrary).

The first one corresponds to the so-called three-quarter wave mode (because f = 3/4 × c 0 /L) and is purely

longitudinal. The second one shows a combination of the second longitudinal and the first transversal modes

(therefore begin the (2,1) mode).

Figure 9: Pressure variations in arbitrary units for f = 433.3 Hz and f = 433.6 Hz (modes (2,0) and (1,1))

Fig. 9 shows what happens when an eigenfrequency is double. The figure on the left should show a purely longitudinal mode (similar to the Fig. 8 on the right without the transversal component). The second one should show a combination of the first longitudinal and first transversal modes.

This result can be explained with the method of resolution of AVSP. The code solves the Helmholtz equation for eigenfrequencies. When an eigenfrequency is double, the subspace associated to this frequency is of dimension two. The code is not able to distinguish between the two physical eigenmodes that exist for this frequency. Therefore the result is a linear combination of the two pressure modes associated with the same frequency.

V. Interaction between acoustics and combustion

V.A. n − τ model in AVSP

Combustion instabilities exist because of the interaction between acoustics and combustion. A first step in their analysis is the study of eigenmodes of the configuration. To do so, the code AVSP from CERFACS has been used.

Firstly, the eigenmodes of a configuration are computed with no combustion. The assumptions in this computation are those of linear acoustics. Secondly, the eigenmodes are computed again with an active flame model. The n − τ model introduced by Crocco ¹¹ is used to perform this. As its name suggests, it requires two parameters, n and τ . n is called the interaction index, and describes the intensity of a change in velocity on the heat release. τ represents the time delay between variations in velocity and heat release.

This model links the linearised heat release term ˙ Ω ¹ _T with the linearised acoustic velocity u 1 (Eq. (16)) at a certain position a, as shown in Eq. (27):

Ω ˙ ¹ _T = S 1 ρ 1 c ² ₁

γ − 1 nu 1 (a, t − τ ). (27)

One has several ways to determine these parameters. The first one is the analysis of experimental data.

The second one is to use values within an order of magnitude. The complete description is available in the technical report of CERFACS written by Ghani. ¹²

The introduction of the n−τ model modifies the eigenfrequencies. Their real part is shifted by the presence of the flame and their imaginary part can become positive, which can lead to a combustion instability.

V.B. Pulsed flame

The computations of this section are derived from the EM2C experiment ¹³ for the original experimental setup and results and from the work of Auzillon ¹⁴ for the input data and computation results. The problem is considered two-dimensional and axisymmetrical. The goal of this section is to study the dynamics of a laminar flame when pulsed by a sinusoidal velocity (Eq. (28)) for several TFLES factors:

v(t) = ¯ v + √

2∆v cos(2πf t). (28)

The computational domain is 400 mm-wide and 600 mm-high and is composed of two physical domains.

The first one represents the burner, with 30mm height and 11mm radius. The second one is the atmosphere

with a 1mm-wide and 30mm-high wall to separate it from the burner. The two domains are connected at the exit of the burner.

CH

/air mixture

Inlet 11 mm

30 mm

Outlet 400 mm

600 mm

Axis of symmetry

(a) Computational domain and boundary conditions. (b) Close up view of the mesh Figure 10: Pulsed flame configuration

Fig. 10a shows the computational domain and the boundary conditions and Fig. 10b shows a part of the mesh. The data used to perform the computations are shown in Tables [1] and [3]. The kinetics used in this section is CM2-CH4-2R (Eq. (19)).

Input data Symbol Value(s)

Equivalence ratio φ 1.05

Diameter of the burner (mm) d 22

Mean injection velocity (m/s) ¯ v 0.97

Variable injection velocity (m/s) ∆v 0.19

Pulsation frequency (Hz) f 62.5

Pulsation period (s) T ⁰ 0.016

Thickening factors for standard TFLES F 2.5 4 5.7

Mesh sizes (number of cells) for each F – ≈ 240k ≈ 180k ≈ 150k Characteristic length of the smallest cells l c 1.25 · 10 ⁻⁴ 2 · 10 ⁻⁴ 2.85 · 10 ⁻⁴

for each F (in the flame region) (m)

Table 3: Data for the pulsed flame.

F = 2.5 F = 4 F = 5.7 DNS data

X (m)

Y ( m )

0 0.005 0.01 0.03

0.04 0.05 0.06

X (m)

Y ( m )

0 0.005 0.01 0.03

0.04 0.05 0.06

X (m)

Y ( m )

0 0.005 0.01 0.03

0.04 0.05 0.06

X (m)

Y ( m )

0 0.005 0.01 0.03

0.04 0.05 0.06

t = t i t = t _i + T ₀ /4 t = t _i + T ₀ /2 t = t _i + 3T ₀ /4 Figure 11: Flame front (c = 0.5) for three thickening factors over a cycle of velocity excitation

Fig. 11 shows the comparison between the DNS data from Auzillon ¹⁴ and the standard TFLES model for three thickening factors. The thickening factor has a crucial impact on the flame front, as it was expected.

The curvature of the flame front is smoothed when F increases. Over a cycle of excitation, the thickening factor also has an effect on the phase shift between the thickened flame and the DNS results: the flame front is higher along the y axis when F increases for the first two instants when the speed increases while it is lower or at the same level when the speed decreases.

VI. Application to a laboratory scale configuration: PRECCINSTA

VI.A. The PRECCINSTA configuration

The PRECCINSTA configuration, shown in Fig. 12a, is an experiment designed by DLR in Stuttgart with a Turbomeca helicopter injector (presented for instance by Lartigue and Meier ¹⁵ ). It is composed of a plenum (where the air is injected), an injector (where the fuel is injected) and a chamber (where the combustion occurs). The plenum is used as a tranquillisation chamber. The injector is a swirler and was initially designed by Turbomeca for helicopter combustors. This injector was then modified in order to inject methane. It was chosen because it was likely to create combustion instabilities in certain regimes.

In this paper, the studied case is simplified. The fuel and the air are considered premixed and injected

in the plenum.

(a) The PRECCINSTA configuration. (b) A 2D view of the computational domain for the PRECCINSTA configuration.

Figure 12

The computational domain used in this section is shown in Figure 12b. The large cone is the atmosphere.

Input data Symbol Value(s)

Equivalence ratio φ 0.75

Diameter of the inlet (mm) d ≈ 25

Flow rate (kg/m ² /s) ρu 26.009

Mesh size for RANS (number of cells) – 970,000 Characteristic length of the smallest cells (m) l _c ≈ 1 · 10 ⁻³

Turbulence model for RANS – k − l

Turbulence rate in RANS T u 0.036

Characteristic length of turbulence (m) in RANS l 0.02 Table 4: Data for the PRECCINSTA case

Quantity Value in fresh gases Value in burned gases

Y CH

[–] 0.041984 0.

Y O

[–] 0.22331 0.055968

Y H

O [–] 0. 0.094294

Y CO

[–] 0. 0.114791

Y CO [–] 0. 0.000244

Y N

[–] 0.734706 0.734703

T [K] 300 1931.938

P [Pa] 101325 101325

Table 5: Specie composition, temperature and pressure in fresh and burned gases

VI.B. RANS computations

The data used to initialize the RANS computations are shown in Table 4.

Figure 13: Axial velocity field and streamlines at z = 0 from RANS computations.

Fig. 13 shows the axial velocity field at z = 0 for RANS computations. The effect of the swirler on the velocity is seen. A large recirculation zone appears downstream of the injector, and two smaller are located at the bottom of the chamber.

VI.C. RANS computations with combustion

The data used to perform the RANS computations with combustion are shown in Tables 4 and 5. The kinetics used are the same as previously. The initial state used is the result of the RANS computations without combustion.

Figure 14: Temperature field at z = 0 from RANS computations with combustion before the instability appears

Fig. 14 shows the temperature field given by the RANS computations with combustion. The flame was

initialised just after the exit of the injector, by changing the composition from fresh gases to burnt gases.

Figure 15: Pressure field at z = 0 for two instants, showing the coupling

Fig. 15 shows what happens during the computations: an acoustic coupling appears between the plenum and the chamber.

94000 96000 98000 100000 102000 104000 106000 108000 110000 112000

0.2 0.201 0.202 0.203 0.204 0.205 0.206 0.207 0.208 0.209 0.21

Pressure (Pa)

Time (s)

Plenum pressure Chamber pressure

0 200000 400000 600000 800000 1e+06 1.2e+06 1.4e+06 1.6e+06 1.8e+06

0 200 400 600 800 1000

Amplitude (a.u.)

Frequency (Hz)

FFT result

Figure 16: Pressure in the chamber and the plenum over time and Fast Fourier Transform of one signal

Fig. 16 shows the pressure in the plenum and the chamber over time and the Fast Fourier Transform of one signal. One can note that they are almost in opposition of phase. When the pressure reaches the maximum in the plenum, it is close to the minimum in the chamber.

VI.D. AVSP computations

The geometry of the 3D PRECCINSTA configuration in the plane z = 0 has been simplified as a 2D domain in order to compute the eigenmodes in AVSP. The modes that are found are therefore only the ones in this plane. This approach is consistent with the mode observed in the RANS computations with combustion.

Figure 17: Absolute pressure variations for the first mode f = 389.5 Hz without active flame

The first mode observed in AVSP without active flame is shown in Fig. 17 and is mainly longitudinal.

The active flame model is then enabled. It is initialized as a compact flame (a field of n and τ ), 0.01 m wide along x and as large as the domain along y. It is located directly after the injector. The resolution of the eigenproblem with an active flame is made on a single point, i.e. a target frequency is set before the computations are run. In this section, the computations are focused on the first mode (initially f ₀ = 389.5 Hz). The period associated to this frequency is T ₀ = 2.57 · 10 ⁻³ s.

The n parameter is set at 10 ⁸ J/m. This value is chosen based on the typical order of magnitude of n. ¹² Two values of τ have been chosen: τ ₁ = 0.001 s and τ ₂ = 0.002 s. The first value is below T ₀ /2 while the second one is between T 0 /2 and T 0 . According to Poinsot and Veynante, ⁶ one might expect that computations with τ 1 converge on a stable mode, while the ones with τ 2 on an unstable mode. An effect that has to be studied is the choice of the target frequency (f t ). Two values have been studied: f t,1 = 200 Hz and f t,2 = 400 Hz.

Case τ (s) f t (Hz) Resulting eigenfrequency (Hz) 1 τ 1 f t,1 423.8 − i 4.28 · 10 ⁻³ 2 τ 1 f t,2 423.8 − i 2.80 · 10 ⁻³ 3 τ 2 f t,1 423.8 − i 2.80 · 10 ⁻³ 4 τ 2 f t,2 423.8 + i 3.94 · 10 ⁻³

Table 6: Results of the AVSP computations with active flame

Table 6 presents the results from the AVSP computations with active flame. One can notice that in all cases, the real part of the eigenfrequency is identical. However the imaginary part differs. In cases 1 and 3, the target frequency is set far from the original frequency f 0 . The algorithm in these cases correctly predicts the real part, but the imaginary part is not converged with the default parameters of the algorithm. When setting the target frequency much closer to f 0 , the imaginary part is different. In case 1, it is negative which denotes a stable mode while in case 4 it is positive thus unstable. The FFT of the signals of Fig. 16 gives a maximum amplitude around 400 Hz. However the signals have a length of 0.01 seconds, which is not sufficient to have a good frequency resolution, given that the phenomenon occurs at arout 400 Hz. Further computations must be done to improve the accuracy of the results.

VII. Conclusion

The present work shows that several tools are available to predict combustion instabilities. The use of an acoustic code on a simple geometry can quickly highlight the unstable behaviour of a configuration.

Meanwhile, this unstable mode can be found with computations on the complete configuration. However, one must be careful with the model used for combustion. For instance, the TFLES model has a crucial impact on the dynamics of the flame.

Appendix

A. Filtering of Navier-Stokes equations

The Navier-Stokes equations are shown in Eqs (29), (30), (31) and (32):

∂ρ

∂t + ∇ · (ρV ) = 0, (29)

ρ DY k

Dt = ∂ρY k

∂t + ∇ · (ρY k V ) = −∇ · J k + ˙ ω k , (30)

ρ DV

Dt = ∂ρV

∂t + ∇ · (ρV ⊗ V ) = ∇ · (τ − pI) + S m , (31)

ρ De t

Dt = ∂ρe t

∂t + ∇ · (ρe t V ) = −∇ · J e

+ ∇ · [(τ − pI)V ] + S e

. (32)

In Eq. (30), ˙ ω k is the production (if positive) or consumption (if negative) rate of specie k. J k is the

diffusion flux of specie k and can be written as J k = ρV k Y k with V k being the diffusion velocity of specie

k. P n

k=1 J k = 0 and P n

k=1 ω ˙ k = 0 have to be verified in order to ensure the global mass conservation. J k

is based on the phenomenologic diffusion law of Fick, which is a common approximation, with a diffusion coefficient D _k , as shown in Eq. (33):

J _k = −ρD _k ∇Y _k . (33)

In Eq. (31), τ is the viscous strain tensor and S m is the source term that can be expressed as S m = ρ P n

k=1 Y k f _k + . . . , where f _k are the volume forces. In Eq. (32), J e

= q + P n

k=1 h k J k is the energy flux , where h k is the enthalpy of each specie and q is the heat flux, S e

= S m · (V + V k ) + S rad + . . . is the source term, where S rad is the source term from radiation.

It is assumed that any quantity ϕ can be decomposed as the sum of a main part ¯ ϕ and a fluctuation ϕ ⁰ . In RANS computations, ϕ is the mean part of the quantity, while it is the resolved part of the quantity in LES.

In RANS, Eq. (34) are verified, but not in LES, where new correlations appear. The Favre decomposition (or filtering) is an extension of the first decomposition, where ϕ = ˜ ϕ + ϕ ⁰⁰ , with ˜ ϕ and ϕ ⁰⁰ defined in Eq. (35):

ϕ = ϕ, ϕ ⁰ = 0, (34)

ϕ = e ρϕ

ρ and ρϕ ⁰⁰ = 0, (35)

For example: ρu _i u _j = ρ u e _i u e _j + ρ ] u

_i u

_j . (36) It is designed to take into account the variability of the density in compressible flows. Using this filtering, there will be no correlated part in the equation for conversation of mass.

B. Classical quantities in combustion

The Prandtl number P r k , which compares the kinematic viscosity with the heat conductivity, is defined for each specie k in Eq. (37):

P r k = µ k c pk

λ k

= ν D th

. (37)

The Schmidt number Sc k , which compares the kinematic viscosity with the specie diffusion, is defined for each specie k in Eq. (38):

Sc k = µ k

ρ k D k

= ν D k

. (38)

The equivalence ratio φ is defined for a generic reaction between a fuel F and an oxidant O, resulting in a product P, as shown in Eq. (39):

F + sO → (1 + s)P. (39)

The parameter s depends on the reaction and is called the mass stoichiometric coefficient. The general definition of φ is given in Eq. (40). For a premixed laminar flame, the fuel and the oxidant are mixed and injected at the same mass rate ˙ m. Therefore the equivalence ratio has a simplified expression in Eq. (41):

φ = s m ˙ F

˙

m _O , (40)

φ = s Y F m ˙ Y _O m ˙ = s Y F

Y _O =  Y F

Y _O

 /  Y F

Y _O



st

. (41)

The subscript st denotes the stoichiometry. The progress variable c is introduced to be able to locate a position in the flame in an unambiguous way. For instance, c for premixed methane-air flame can be defined as in Eq. (42):

c = Y CO + Y CO2

(Y CO + Y CO2 ) _eq(φ) . (42)

The composition of the mixture when the reaction reaches equilibrium (subscript eq) depends on the equiv-

alence ratio φ defined in Eq. (41). The definition of c is based on the necessity to have an increasing and

monotonous variable in the flame to describe its structure.

C. Derivation of the laminar flame speed for a simplified 1D-flame

For a simplified 1D-flame (incompressible, stationary and using the Fick law of diffusion), Eq. (30) becomes Eq. (43). Under these assumptions, the flow rate ˙ m is constant and can be expressed as ρu or ρ u S L , where u is the subscript for unburned gases (while b corresponds to burned gases):

ρu dY k

dx = d dx

 ρD k

Y k

dx



+ ˙ ω k . (43)

Eq. (43) can be recast as Eq. (44) and integrated in the x direction as in Eq. (45):

ρ u S L

dY k

dx = − dJ k

dx + ˙ ω k , (44)

ρ _u S _L Z +∞

−∞

dY _k = − Z +∞

−∞

dJ _k + Z +∞

−∞

˙

ω _k dx. (45)

The diffusion flow rate J k is zero far from the flame (since it is a premixed flame). Therefore the first term of the right hand side is equal to zero. By writing the second term as ¯˙ ω k , Eq. (18) is obtained.

Acknowledgements

The author would like to thank Dr. Nicolas Bertier and Aymeric Boucher for their patience, support and invaluable scientific input, the DEFA department and ONERA for their welcome and Dr. David Eller for his supervision.

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