domain: spi phys

text:
Alexandre Fougnie
email: alexandre.fougnie@centralesupelec.fr

Thomas Schmitt
email: thomas.schmitt@centralesupelec.fr

ANALYSIS AND LOW ORDER MODELING OF THE ACOUSTIC DAMPING FOR HIGH-FREQUENCY COMBUSTION INSTABILITIES PREDICTION IN LIQUID ROCKET ENGINES

Keywords: liquid rocket propulsion, thermo-acoustic instabilities, high-frequency instabilities, low order modeling, acoustic damping

The phenomena liable for acoustic damping in thermo-acoustic instabilities in liquid rocket engines are investigated. Modelings to account for the nozzle impedance, in compact hypothesis, and for the visco-thermal losses at walls are derived according to the Galerkin framework, where pressure variations are decomposed over the acoustic eigenmodes. Models are implemented in a low order tool and compared successfully to analytic and numerical predictions. The turbulent/acoustic interaction, responsible for acoustic absorption at the jets, is reviewed, and an in-progress strategy is presented.

INTRODUCTION

High frequency combustion instabilities represent one of the main challenging phenomena to predict when designing a new liquid rocket engine, and this since the beginning of the space era [START_REF] Crocco | Theory of combustion instability in liquid propellant rocket motors[END_REF] [START_REF] Harrje | Liquid Propellant Rocket Combustion Instability[END_REF]. It arises from the coupling between the flames' unsteady heat release, the combustion chamber's acoustics, and the injection dynamics, leading to tremendous pressure oscillations in some configurations. It involves mainly transverse acoustic waves, submitting the flames to transverse oscillating velocity solicitations and the injection units to oscillating mass flow rate. Thus, the prediction of these instabilities is crucial for the safety and nominal running of the engine. However, the global comprehension of the coupling involved still requires substantial analysis of the mechanisms at stake, and consequently the qualification of an engine demands long and expensive real-size tests [START_REF] Oefelein | Comprehensive review of liquid-propellant combustion instabilities in F-1 engines[END_REF] on dedicated test benches. Progress in numerical simulation permits a more refined physics analysis but can still not faithfully reproduce all the relevant couplings and mechanisms.

Low order methodologies have been actively developed throughout recent years to evaluate the level of instability without requiring massive tests or computations. The numerical cost reduction is achieved by simplifying the physics modeled. In collaboration with CNES and ArianeGroup, EM2C laboratory is developing the StaHF (Stability High Frequency) [START_REF] Éry | M écanismes d'instabilit és de combustion haute-fr équence et application aux moteurs-fus ées[END_REF] code based on a modal expansion of the pressure fluctuations over the acoustics eigenmodes [START_REF] Culick | Unsteady Motions in Combustion Chambers for Propulsion Systems[END_REF]. This approach requires models to translate the interactions and behaviors responsible for the evolution of the instabilities. In particular, it is crucial to account for the combustion response to acoustic perturbations and the damping of acoustic waves. From previous work, [START_REF] Crocco | Theory of combustion instability in liquid propellant rocket motors[END_REF] [START_REF] Éry | M écanismes d'instabilit és de combustion haute-fr équence et application aux moteurs-fus ées[END_REF], combustion models have been developed and implemented on low order codes. However, only a little scattered information is available about the acoustic losses within the engine, and no substantial numerical/experimental studies specifically applied to rocket engines with corresponding physical analysis and modeling are available.

Consequently, three mechanisms of acoustic damping are reviewed; nozzle losses, viscous and thermal losses at walls, and turbulence/acoustic interaction. Specific attention about each phenomenon is given in the corresponding section. First, §2 presents some generalities about thermo-acoustic instabilities. Then, §3 outlines the methodology chosen for the development of the StaHF low order tool. The mechanisms of damping are analyzed and modeled in §4, §5 and §6. Finally, §7 draws a conclusion and some perspective for future work.

THERMO-ACOUSTIC INSTABILITIES AND DAMPING GENERALITIES

In the most straightforward linear formulation, the study of the thermo-acoustic stability of a system amounts to the study of the evolution of pressure oscillations p ′ of pulsation ω in the form of a harmonic oscillator p ′ (x, t) = p 0 (x)e -σpt cos(Ωt), Eq.1

with Ω = ω 2 -σ 2 p . The stability is thus linked to the sign of the coefficient σ p . If σ p > 0, p ′ decreases, and the system is deemed stable, the oscillations are damped. Otherwise, p ′ increases and the system is unstable. The challenge is to determine the sign of σ p from the physical phenomena involved. One can decompose the coefficient as [START_REF] Culick | Unsteady Motions in Combustion Chambers for Propulsion Systems[END_REF]

σ p = i σ phenomena,i
Eq.2 with a coefficient of damping, or growth depending on the sign, must be evaluated for each contribution and for a specific acoustic frequency. Classifying the contributions as either producing or destroying acoustic energy, the growth of the instabilities depends on the competition between these two groups, which varies with the acoustic frequencies of interest. In an energetic form, the problem is traditionally written as [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF]

∂E a ∂t + V ∇.(p ′ u ′ )dV = R a + D, Eq.3 with E a = V 1 2 p ′2 ρc + 1 2 ρ|u ′ | 2 dV Eq.4
the acoustic energy, ρ the mean density, c the mean sound speed and u ′ the acoustic velocity oscillations, V ∇.p ′ u ′ dV = S p ′ u ′ .ndS is the acoustic flux at the boundaries,

R a = V γ -1 γ p p ′ q′ dV Eq.5
the Rayleigh term being the flame response, representative of the coupling between the acoustic perturbation and the combustion, with γ the specific heat ratio, p the mean pressure and q′ the unsteady heat release. D is the dissipated acoustic power. It comes

σ p = D -S p ′ u ′ .ndS + R a 2E a .
Eq.6

The previous formulation remains restrictive because the flame does not always act as an excitation source according to its response time [START_REF] Mcmanus | A review of active control of combustion instabilities[END_REF], and other phenomena can be responsible for the production of acoustic energy.

Another point to mention is the link with acoustic frequencies. The determination of σ and Ω is equivalent to accessing the modifications of the real and imaginary frequencies of the acoustic modes induced by these phenomena, as explained in [START_REF] Culick | Unsteady Motions in Combustion Chambers for Propulsion Systems[END_REF]. The frequency of the m th mode can be decomposed as

f m = f Re m + δf Re m + jf Im m , Eq.7
with f Re m the modal frequency of the normal mode, without source term or boundary conditions (BC), δf Re m the deviation from the initial frequency due to combustion, BC, and other phenomena, and f Im m the imaginary part, initially null but rising due to source terms and BC. So

p ′ (x, t) = p 0 (x)R e e -jωmt = p 0 (x)e 2πf Im m t cos(2π(f Re m + δf Re m )t).
Eq.8

It highlights the link between the imaginary modal frequency and the growth/damping rate, and the modification of the real frequency by the various phenomena present. From a design stage perspective, it is also essential to access the correct values of mode frequencies at which instabilities are likely to occur.

One of the main difficulties in treating instabilities is the numerous phenomena and couplings present within the engine. Also, the transposition of studies on academic experimental benches to real size engines are not straightforwards, frequencies are much higher, the combustion more powerful, and where test benches have from one to a few dozen injectors, real size engines can be teamed to hundreds of injection units. Let us come back to the physical behavior within the engine. Three large families of fluctuation modes evolve in a gas [START_REF] Chu | Non-linear interactions in a viscous heat-conducting compressible gas[END_REF], see Fig. 1 and interact with each other, mainly in specific areas; the acoustic modes with disturbances propagating at the speed of sound, the vorticity modes, as the vortex from turbulence near the injection, and the entropy modes that are hot spot generated mainly in the flames region. As soon as an average flow is considered, the last two are present, convecting the perturbations. Most studies only consider the evolution of the acoustic modes, but some coupling models exist, such as [START_REF] Marble | Acoustic disturbance from gas non-uniformities convected through a nozzle[END_REF] where the hot spots convected towards the nozzle exit are transformed into acoustic perturbations injected back into the domain. One of the great difficulties of the low order appears to model as well as possible the involved mechanisms, thus including as many as possible relevant couplings that need to be carefully identified while maintaining a clear and numericallylight framework. The main regions responsible for coupling and acoustic generation/destruction are; 1) the flames and injection area, 2) the nozzle with high Mach flow, 3) the walls, and 4) the injection units. Fig. 1 represents in a non-exhaustive and simplified way the main phenomena of interest. However, most of them are out of the scope of the present study. Flame and excitation models have already been developed [START_REF] Crocco | Theory of combustion instability in liquid propellant rocket motors[END_REF] [4] [START_REF] Nez | Experimental and numerical characterizations of acoustic damping rates in a coupled-cavity configuration[END_REF], but few analyses and models concerning the acoustic damping allow an adequate prediction of the level of instabilities. Mainly, the following mechanisms are reviewed [START_REF] Webster | Analysis of pressure dynamics, forced excitation and damping in a high pressure LOx/H2 combustor[END_REF] 1. the losses at the nozzle region; 2. the visco-thermal losses at walls; 3. the losses from the acoustic/turbulence interaction at the jets location.

Before looking in detail at these different points, the low order framework used to predict the pressure variations is presented, guiding our modeling efforts.

LOW ORDER FRAMEWORK

The low order approach

The low order approach simplifies the conservation equations, applied in general on mesh coarser than for more classical numerical computations as Large Eddy Simulations (LES), in only keeping in the calculation the physical variables and the phenomena of interest regarding the case study. Consequently, the numerical resources needed to perform the analysis are reduced as well as the computation time, allowing to quickly estimate the stability of the configuration of interest for varying geometrical parameters and operative conditions. Without this tool, the LES computations would require disproportionate numerical resources or numerous, long and costly real-size tests on dedicated engines. With a simplified physical representation, the low order approach enables investigating various cases for a reasonable financial and numerical cost.

Several methods have been developed so far for the study of combustion instabilities; as the use of Riemann invariant [START_REF] Li | Open Source Combustion Instability Low Order Simulator ( OSCILOS-Long ) Technical report Open Source Combustion Instability Low Order Simulator[END_REF] through acoustic networks, but often limited to longitudinal waves; the Galerkin expansion, by expanding the acoustic perturbations over the acoustic modes [START_REF] Culick | Unsteady Motions in Combustion Chambers for Propulsion Systems[END_REF] recently extended to the state-space approach [START_REF] Laurent | A novel modal expansion method for low-order modeling of thermoacoustic instabilities in complex geometries[END_REF]; the coupled Riemann-Galerkin hybrid approach; or the resolution of the Helmholtz equation using finite elements as in [START_REF] Nicoud | Acoustic Modes in Combustors with Complex Impedances and Multidimensional Active Flames[END_REF].

Galerkin expansion

The code developed the EM2C laboratory, StaHF [START_REF] Éry | M écanismes d'instabilit és de combustion haute-fr équence et application aux moteurs-fus ées[END_REF], is based on the Galerkin expansion, introduced first by [START_REF] Zinn | Application of the Galerkin Method in the Solution of Non-linear Axial Combustion Instability Problems in Liquid Rockets[END_REF] and extensively developed by [START_REF] Culick | Unsteady Motions in Combustion Chambers for Propulsion Systems[END_REF]. This method consists in decomposing the acoustic pressure oscillations on the base formed by the acoustic normal eigenmodes of the engine. Temporal and spatial contributions are then isolated. Also it allows to work on complex geometries with longitudinal, transverse and coupled modes. Thus, the pressure oscillations are written as

p ′ (x, t) = ∞ n=1 η n (t)Ψ n (x),
Eq.9

where Ψ n are the acoustic eigenmodes, and η n are the temporal evolution of each mode. The particularity of this expansion is to consider the modal base as orthogonal. This condition is met when the modes are computed without source terms and with homogeneous boundary conditions (BC), so that of Neumann where ∇Ψ n (x s ).n = 0 on the boundary, with x s located on the boundary, or Dirichlet where Ψ n (x s ) = 0 on the boundary, or mixed type, with Neumann BC on some parts and Dirichlet BC on others. For simple configurations, the base (Ψ n ) n≥1 can be determined analytically, but for more complex geometries, the use of a Helmholtz solver is necessary, as AVSP from Cerfacs [START_REF] Nicoud | Acoustic Modes in Combustors with Complex Impedances and Multidimensional Active Flames[END_REF]. As a consequence of the orthogonality, the norm is

V Ψ n Ψ m dV = Λ n δ nm with Λ n = V Ψ 2
n dV, Eq.10 which represents the main aspect of the modal expansion. To establish the differential equation that translate the behavior of (η n ) n≥1 , and so of p ′ , a first order linearization is applied to the conservation equations of mass Eq.11, momentum Eq.12, energy Eq.13 and the perfect gas equation of state Eq.14 that are ∂ρ ∂t + ∇.ρu = 0, Eq.11 ρ ∂u ∂t + u.∇u = -∇p, Eq.12 ρT ∂s ∂t + u.∇s = q, Eq.13 p = ρ γ e s/cv , Eq.14 with ρ the density, u = (u 1 , u 2 , u 3 ) the velocity vector, p the pressure, T the temperature, s the entropy of the flow, q the heat release per unit volume, γ the specific heat ratio and c v the specific heat of the mixture at constant volume. Each quantity F(x, t) is decomposed into a mean component F(x) and a fluctuating component F ′ (x, t). Then, using a zero Mach hypothesis that state that ū is of first order, the pressure differential equation Eq.15 is obtained:

∂ 2 p ′ (x, t) ∂t 2 -γ p∇.( 1 ρ ∇p ′ (x, t)) = (γ -1) ∂ q′ ∂t Eq.15
Then, the expansion Eq.9 is used, with the modes verifying the Helmholtz equation Eq.16 with homogeneous BC, with ω m = 2πf m the mode pulsation and f m its frequency;

γ p∇. 1 ρ ∇Ψ m + ω 2 m Ψ m = 0. Eq.16
By considering a finite number of mode N , Eq.15 * Ψ m and Eq.16 * p ′ gives

∂ 2 p ′ ∂t 2 Ψ m + ω 2 m Ψ m p ′ + γ p∇. 1 ρ ∇Ψ m p ′ -γ p∇. 1 ρ ∇p ′ Ψ m = (γ -1) ∂ q′ ∂t Ψ m .
Eq.17

Then an integration of Eq.17 over the volume V of the domain and the use of the projection Eq.9 give:

N n=1 ηn (t) V Ψ n Ψ m dV + ω 2 m N n=1 η n (t) V Ψ n Ψ m dV + V γ p p ′ ∇. 1 ρ ∇Ψ m -Ψ m ∇. 1 ρ ∇p ′ dV = (γ -1) V ∂ q′ ∂t Ψ m dV,
Eq.18 and using the orthogonality property and the Green identity, Eq.19 is obtained for each mode m;

ηm (t) + ω 2 m η m (t) + 1 Λ m S c2 p ′ ∇Ψ m .n -c2 Ψ m ∇p ′ .ndS = γ -1 Λ m V ∂ q′ ∂t Ψ m dV.
Eq.19

At this point, it can be noted that the surface integration term corresponds to the BC and in the energetic formulation Eq.15, to the term ∇.(p ′ u ′ ). It will be further discussed and simplified, noted hereafter as S m BC . Thus,

ηm (t) + ω 2 m η m (t) = γ -1 Λ m V ∂ q′ ∂t Ψ m dV + S m BC .
Eq.20

Modal amplitude differential equation

On Eq.20, the first term of the right-hand side corresponds to a source term translating the effect of the combustion under an acoustic solicitation. It requires a formulation of q′ , the more common being the sensitive time lag model [START_REF] Crocco | Theory of combustion instability in liquid propellant rocket motors[END_REF], that supposes the heat release as proportional to the pressure variation, through the interaction index n, but delayed of a time lag τ ;

q′ (x, t) = n q ū u ′ (x, t -τ ).
Eq.21

Other models have been developed later on [START_REF] Éry | M écanismes d'instabilit és de combustion haute-fr équence et application aux moteurs-fus ées[END_REF]. More generally, the concept of the Galerkin approach is to develop, as source term, models to account for the combustion, the acoustic damping, some experimental excitation devices, etc, so that Eq.20 writes as

ηm (t) + ω 2 m η m (t) = S m comb + S m damp + S m exc .
Eq.22 The source terms can be developed based on experimental or numerical considerations through transfer functions for instance. Or, as it will be seen later on, by deriving the present framework including extra terms in the conservation equations, imposing specific conditions for the boundary term in Eq.19 or via energetic considerations (cf §3.4).

Notes

A few remarks;

1) The mean flow has not been assumed to be zero, but of Mach order 1, so there is no ū term explicitly in the presented framework. However, the influence of the mean flow exiting the engine can be accounted through an impedance BC, see §4. The consideration of the high Mach number in the framework is not treated here and will be added later, M = ū/c being of the order of 0.3 in rocket engines.

2) No further discussion will be made on the flame influence and modeling.

3) A parallel can be made with the remarks and formulations of §2. As a general and simplified formulation, the source terms, reduced to a driving and damping component can be written as

S m driv = -2σ m driv ηm (t) + ∆ω m driv η m (t) S m damp = -2σ m damp ηm (t) + ∆ω m damp η m (t),
Eq.23 so that

ηm (t) + 2(σ m damp + σ m driv ) ηm (t) + (ω 2 m -∆ω m damp -∆ω m driv )η m (t) = 0
Eq.24

The phenomenon modeled hereafter can usually be seen as responsible of growth/damping and of frequency shift. In some cases, it is easier to develop a model for the energetic term of Eq.15, then the growth/damping coefficients are found using

σ m damp = D 2E a and σ m driv = R a 2E a .
Eq.25

NOZZLE ACOUSTIC LOSSES

This section focuses on applying to our framework an impedance to account for the acoustic losses at the domain boundaries, corresponding to the ∇.p ′ u ′ term of Eq.15 and S m BC term of Eq.20. In a rocket engine application, this can be used to model the influence of the nozzle, which is a high Mach region, and therefore where the calculation of modes becomes uncertain because of low Mach hypothesis in Helmholtz solvers.

Acoustic impedance of the nozzle

The acoustic impedance is defined as

Ẑ(x s , ω) = p(x s ) pcû(x s ).n , Eq.26
with the n the normal vector at the outlet and x s the position along the outlet plane. The symbol . denotes a complex quantity. The impedance is linked to a reflection coefficient R, ratio between the reflected acoustic pressure amplitude and the incident acoustic wave as

R = Ẑ -1 Ẑ + 1 . Eq.27
Compact nozzle For a compact nozzle, meaning the acoustic wavelengths are large compared to the nozzle dimensions, the impedance is a real number and independent of the acoustic frequency. Z is a function of the Mach number M 1 at the nozzle inlet, proposed by [START_REF] Marble | Acoustic disturbance from gas non-uniformities convected through a nozzle[END_REF], for a supersonic nozzle (M t = 1 at the throat). The reflection coefficient is

R sup = 1 -1 2 (γ -1)M 1 1 + 1 2 (γ -1)M 1 , Eq.28 giving, Z = 2 (γ -1)M 1 .
Eq.29

For a subsonic nozzle, where the flow does not become supersonic,

R sub = M 2 -M 1 1 -M 1 1 + M 1 M 1 + M 2 1 -1 2 (γ -1)M 1 M 2 1 + 1 2 (γ -1)M 1 M 2 ,
Eq.30 with M 2 the Mach number at the nozzle outlet. Fig. 2 shows the dependance of Z with M 1 , for both subsonic and supersonic nozzles. For low Mach numbers M 1 < 0.3, the impedance is large Z > 20 (γ = 1.4) for the supersonic configuration and small Z < 0.1 for the subsonic one. Thus, in the compact hypothesis, for a chocked nozzle (supersonic), the nozzle behave acoustically as a solid wall, validating the mode computation using a Neumann BC u ′ (x s ).n. On the contrary, a subsonic nozzle would require to compute the normal modes using a Dirichlet condition p ′ (x s ) = 0. Non-compact nozzle The case of a non-compact nozzle is more complex to treat because it is necessary to account for the geometry of the nozzle and that the impedance becomes a complex value and depend on the acoustic frequency. First, [START_REF] Crocco | Behavior of supercritical nozzles under three-dimensional oscillatory conditions[END_REF] developed a 3D admittance nozzle theory for constant longitudinal amplitude oscillations, extended by [START_REF] Bell | The prediction of threedimensional liquid-propellant rocket nozzle admittances[END_REF] to account for amplitude variations. Then, [START_REF] Marble | Acoustic disturbance from gas non-uniformities convected through a nozzle[END_REF] proposed an acoustic and entropic impedance using a linearly varying velocity in the nozzle. Later, [START_REF] Nicoud | About the zero Mach number assumption in the calculation of thermoacoustic instabilities[END_REF] and [START_REF] Duran | Solution of the quasione-dimensional linearized Euler equations using flow invariants and the Magnus expansion[END_REF] conceived a robust numerical methodology based on the Magnus expansion, then extended by [START_REF] Duran | On the reflection and transmission of circumferential waves through nozzles[END_REF] to be applied to transverse waves. Fig. 3 shows the real and imaginary part of the nozzle reflection coefficient with respect to the acoustic frequency for longitudinal perturbations, for an arbitrary combustor with nozzle geometry. Similar profiles are found in the given references. It shows that R evolves from a high value R e (R) ≈ 1, corresponding to the value found using the compact nozzle hypothesis, to values close to R e (R) ≈ 0. Also I m (R) undergoes important evolution. In other references, for specific nozzle geometry and velocity profile, the reflection coefficient becomes negative in some frequency ranges. Thus the nozzle acts acoustically closer to a chocked nozzle (as u ′ (x s ).n = 0) for certain frequencies and for other frequencies, closer to an open exit p ′ (x s ) = 0. The longitudinal waves are highly damped and the BC influences the modes' shape.

γ = 1.2 γ = 1.7 γ = 1.4 0 0.1 0.2 0.5 0 0.1 0.2 0.3 0.4 0.4 0.3 M 2 = 0.05 M 2 = 0.1 M 2 = 0.2 supersonic subsonic M 1 M 1 10 1 10 -1
It underlines one of the main limitations of the Galerkin approach; limited to modes where u ′ (x s ).n = 0 or p ′ (x s ) = 0 conditions have been imposed at the exit when computing the mode to ensure the orthogonality property. However, the impedance BC would require modes computed using both conditions for some applications, depending on the frequency. A solution has been recently proposed to overcome this issue using a state-space approach and a dual base decomposition [START_REF] Laurent | A novel modal expansion method for low-order modeling of thermoacoustic instabilities in complex geometries[END_REF].

In the case of a rocket engine, it has been shown that the nozzle losses are high and preponderant for the longitudinal modes, but of secondary importance for transverse and coupled modes [START_REF] Öglmeier | Validation of Transverse Instability Damping Computations for Rocket Engines[END_REF], which are the principal modes of interest in highfrequency instabilities, and that the compact nozzle model of [START_REF] Marble | Acoustic disturbance from gas non-uniformities convected through a nozzle[END_REF] can be applied to transverse modes [START_REF] Stow | Reflection of circumferential modes in a choked nozzle[END_REF]. The damping is then to be dominated by the phenomena near the plate injection as the turbulence/acoustic interaction (see §6). The hypothesis of a compact nozzle is thus retained and the influence of the modal shape modification is neglected for the proposed modeling.

Low order Z model

The hereafter model is restricted to Ẑ as a real number, following the compact hypothesis, but is also valid for a complex impedance, not shown here for clarity. The modeling is based on the approach suggested in [START_REF] Culick | Unsteady Motions in Combustion Chambers for Propulsion Systems[END_REF] [23] [START_REF] Bonciolini | Low order modelling of thermoacoustic instabilities and intermittency: Flame response delay and nonlinearity[END_REF].

|Z| > 1 For |Z| > 1, the modal shape is close to the shape computed using a ∇Ψ m .n = 0 condition on the boundary of interest. The base used in the low order framework is thus verifying ∇Ψ m .n(x s ) = 0 and the modes are still orthogonal. To account for the impedance, the condition ∇p ′ .n = -f (Z) has to be imposed as BC. The corresponding term of Eq.20 is, supposing c uniform on the boundary surface,

S m BC = - c2 Λ m S f (Z)Ψ m dS.
Eq.31

The relation

∇p ′ .n = -jω ρu ′ .n Eq.32
is obtained from Eq.12, giving with the impedance formulation, f (Z) = jω cZ p ′ . Eq.33

Thus the source term becomes

S m BC = - c Λ m Z S jωp ′ Ψ m dS = - c Λ m Z N n=1 ηn S Ψ n Ψ m dS.
Eq.34

|Z| < 1 For |Z| < 1, the modal shape is close to the shape computed using a Ψ m = 0 BC. However p ′ is not assumed null on the boundary and the new relation p ′ (x s ) = -g(Z) is imposed. Similarly, the source term is

S m BC = c2 Λ m S g(Z)∇Ψ m .ndS, Eq.35
and the g relation is now

g(Z) = cZ jω ∇p ′ .n. Eq.36
Consequently, it comes, using t η n dt = -1

ω 2 n ηn , S m BC = - c3 Z Λ m S 1 jω ∇p ′ .n∇Ψ m .ndS = c3 Z Λ m N n=1 ηn ω 2 n S
∇Ψ n .n∇Ψ m .ndS.

Eq.37

Z Im(f) x10 2 x10 2 0 0 0 0 -1 -1 1 1 -2 -2 2

1D verification

The previously obtained model is applied to the one dimensional geometry of [START_REF] Nicoud | Acoustic Modes in Combustors with Complex Impedances and Multidimensional Active Flames[END_REF], of length L = 0.4m and c = 450m s -1 , where the impedance is applied on one side of the domain.

The imaginary frequencies predicted by the analytic analysis, the AVSP solver [START_REF] Nicoud | Acoustic Modes in Combustors with Complex Impedances and Multidimensional Active Flames[END_REF] and the low order model are compared Fig. 4. The imaginary frequencies arise from the impedance, well represented by the model Eq.34 for instance, reduce to a 1 mode equation

ηm + c Λ m Z S Ψ 2 m dS 2σ m BC =-4πf Im m ηm + ω 2 m η m = 0.
Eq.38

For positive Z > 0, f Im m is negative corresponding to a positive σ m BC , meaning acoustic energy exits through the nozzle, pressure oscillations are damped. Also, f Im m is independent of the mode number and the model indicates that Z real does not produce a frequency shift, other than the one present in the pseudo-pulsation, results in agreements with [START_REF] Nicoud | Acoustic Modes in Combustors with Complex Impedances and Multidimensional Active Flames[END_REF]. However, the model lacks of prediction near the value |Z| = 1 for the interval 2 > |Z| > 0.5. But this interval is satisfactory for transverse acoustic wave in liquid rocket engines [START_REF] Öglmeier | Validation of Transverse Instability Damping Computations for Rocket Engines[END_REF].

3D comparison

The modeled source term is applied on a cylindrical geometry, representative of a rocket engine, and corresponding to the BKD test bench dimensions [START_REF] Schulze | Linear stability assessment of a cryogenic rocket engine[END_REF], for a sound speed of c = 1300m s -1 . The nozzle is truncated because it is a high Mach region, and a real positive impedance is applied at the location of the nozzle entrance. Imaginary frequencies are computed with the solver AVSP applying the impedance BC and converted into a damping coefficient σ m BC for the first longitudinal 1L and transverse 1T modes. Then, the 1L and 1T modes have been computed using a homogeneous Neumann BC for the nozzle (for the case |Z| > 1 is applied, designated as Neumann-like mode) and for a homogeneous Dirichlet BC for the nozzle (for the case |Z| < 1 is applied, designated as Dirichlet-like mode). The low order impedance model implemented in StaHF is applied for a wide range of Z values and the resulting damping is compared to the numerical (AVSP) solution Fig. 5. The values of impedance presented here do not correspond to a specific engine loading point, but were set to cover a wide range of values. There is a satisfactory agreements between the low order predictions and the Helmholtz solver results.

Modal coupling at the boundary

The present model benefits from the summation over each modal amplitudes seeing in Eq.34 and Eq.37. It signifies that the impedance influence over a specific mode is dependent on the temporal evolution of the other modes, and the damping coefficient σ m BC is time-dependent, having a more sophisticated form than in Eq.38. According to this formulation, there is a modal coupling at the boundary. Such modeling allows reproducing physical wave behavior, such as the propagation of a traveling wave in an impedance tube, with reflection on one of the tube sides, having a given impedance value. This example is not represented here to remain succinct, but without this modal coupling, the reflection would not happen, and the wave would be constantly damped through the propagation within the tube since σ m BC is constant for each mode (as in Eq.38). The modal coupling also explains that the BC can be a source of modal non-orthogonality because of interdependence [START_REF] Nicoud | Acoustic Modes in Combustors with Complex Impedances and Multidimensional Active Flames[END_REF].

VISCOUS AND THERMAL DISSIPATION AT WALLS

Generalities

The viscous losses arise from a no-slip condition at the wall interface. The acoustic velocity takes a certain profile in the acoustic viscous boundary layer δ ν expressed as

u ′ (x, z, t) = u x (x) e -y δν cos ωt - y δ ν -cos(ωt) ,
Eq.39 with

δ ν = 2ν ω ,
Eq.40 ν being the kinematic viscosity, according to the formalism shown in Fig. 6. The velocity gradient between the bulk region and the wall leads to a shear stress, and consequently dissipation by friction. Thermal losses are due to the isothermal wall condition, with acoustic oscillations considered as adiabatic 1 . This adiabatic condition is verified when λ a , the acoustic wavelength, satisfies λ a > 2πD th /c, which is always true in a rocket engine, where D th is the thermal diffusivity. As for the acoustic velocity, there is a temperature gradient between the wall surface and the acoustic temperature oscillations outside the boundary layer. Thus an oscillating heat flux appears, leading to thermal losses. The acoustic thermal boundary layer δ th is defined as 1 Adiabatic acoustic is the classical characteristic of the acoustics. It means that the rate of acoustic change in the medium is fast enough so that no heat exchange happens between within the medium. For very high frequencies, the acoustic oscillations are no longer adiabatic but isothermal. [26] first predicted the sound attenuation in a tube of standing wave without mean flow with a simple formulation. A wider modeling, applicable for any geometries has been proposed by [START_REF] Searby | Prediction of the efficiency of acoustic damping cavities[END_REF] based on the formalism of Eq.6. The viscous σ ν and thermal σ th damping coefficients are expressed as

σ ν = S de dt ν dS 2E a
and σ th = S de dt th dS 2E a , Eq.42

V being the volume of the domain and S the surface of hard walls. The energy derivatives are

de dt ν = 1 2 ρu 2 x ων 2
Eq.43 and

de dt th = 1 2 (γ -1) p ′2 ρc 2 ωD th 2 .
Eq.44

This formulation is adaptable for any geometry and can be implemented in low order codes, but it does not account for the effect of the mean flow and turbulence near the walls. [START_REF] Howe | The damping of sound by wall turbulent shear layer[END_REF] extended the losses formulation to the dissipation over an infinite plane, considering the impact of the wall turbulent shear flow for low Mach number. The complete formulation is not presented here, but the result of the influence of the near-wall turbulence in represented Fig. 7. Π is the damping per area unit taking into account the turbulence, Π 0 is the no-flow approximation, and v + is the friction velocity. It can be seen that for large values of ων/v 2 + , meaning when the turbulence is weak and/or the acoustic frequency is large enough, the influence of the turbulence on the sound dissipation is negligible. It corresponds to a configuration wherein Fig. 6 the viscous sublayer is large compared to the acoustic boundary layer δ turb ≫ δ ν . However, for high turbulent configurations, when the boundary layers are similar to δ turb ≈ δ ν , omitting the turbulent shear flow impact can lead to underestimation of the damping of several factors. Figure 7: damping rate per surface area with the influence of wall turbulence Π scaled by the damping rate without considering turbulence Π 0 , with respect to ων/v 2 + measuring the acoustic/turbulence intensity, from [START_REF] Howe | The damping of sound by wall turbulent shear layer[END_REF].

Low order formulation

A derivation of a model for viscous and thermal effects is proposed, using the Galerkin expansion. First, the viscous stress tensor τ is added on the conservation equation Eq.12, and the derivation of the acoustic equation, with the modal projection gives

ηm + ω 2 m η m - 1 Λ m S c2 Ψ m ∇p ′ .ndS = - 1 Λ m V γ p∇. 1 ρ ∇.τ ′ Ψ m dV.
Eq.45

The boundary term is simplified using ∇Ψ m .n = 0 but the second part of the term is kept to apply the viscous BC condition at the wall ∇p ′ .n = (∇.τ ′ ).n. After some calculations and expressing the tangential acoustic velocity at the wall, similarly as [START_REF] Culick | Unsteady Motions in Combustion Chambers for Propulsion Systems[END_REF], it comes

ηm + ω 2 m η m = - c2 Λ m ω m ν 2 1 ω 2 m S (∇ t Ψ m ) 2 dS damping term 2σ m ν ηm ,
Eq.46 where ∇ t Ψ m it the tangential gradient of the m mode at the wall. Similarly, by keeping the thermal term ∇.q f = -∇. (λ∇T ′ ) in Eq.12, it comes

ηm + ω 2 m η m = - 1 Λ m V (γ -1)
∂∇.q f ∂t Ψ m dV. Eq.47

Then using properties on the acoustic temperature gradient explained in [START_REF] Mbailassem | Sound absorption prediction of linear damped acoustic resonators using a lightweight hybrid model[END_REF], the source term due to thermal losses at wall writes

ηm + ω 2 m η m = - 1 Λ m (γ -1) D th ω m 2 S Ψ 2 m dS damping term 2σ m th ηm
Eq.48

Application The models Eq.46 and Eq.48 are applied analytically and numerically on tube and box geometries, and compared to prediction of [START_REF] Searby | Prediction of the efficiency of acoustic damping cavities[END_REF]. The comparison shows satisfactory results with relative errors of less than 10 -2 % for the computed damping rate. In particular, for all the presented modeling (StaHF, [START_REF] Kirchhoff | Ueber den Einfluss der W ämeleitung in Einem Gase ouf die Schallbewegung[END_REF] and [START_REF] Searby | Prediction of the efficiency of acoustic damping cavities[END_REF]), the damping coefficient writes, for a tube of radius and length L

σ m = ω 2 (γ - ω D (1+ 2R
).

WORK TURBU-LENCE/ACOUSTIC INTERACTION

The last phenomenon identified as responsible for the damping of acoustic waves, except the coupling with the combustion, is the interaction between the transverse acoustic perturbations and the turbulence near the injection plate, due to the jets. When the turbulence flow at the injection units' outlet is submitted to transverse acoustic velocity fluctuations, part of the acoustic energy is absorbed and redirected to turbulent energy [START_REF] Reynolds | The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments[END_REF]. Depending on the characteristics of the turbulence and acoustics, sound waves impose stretching to vorticies, transferring energy [START_REF] Howe | On the Absorption of Sound by Turbulence and Other Hydrodynamic Flows[END_REF]. Besides absorption, scattering and refraction of the incoming waves take place. Part of the incident wave is redistributed towards other acoustic frequencies and directions of propagation [START_REF] Lighthill | On the energy scattered from the interaction of turbulence with sound or shock waves[END_REF] [START_REF] Noir | Absorption of sound by homogeneous turbulence[END_REF], as represented in Fig. 8. The scattering and reflection phenomena participate in the modal coupling between the longitudinal and transverse acoustics modes, following §2, but are not of concern in the present study. Most studies do not consider the combustion process in the study of this interaction. It simplifies the analysis to characterize the acoustic environment of the system by accounting for the turbulent flow. The combustion has to be added a-posteriori after mastering the turbulence/acoustic interaction in a coldflow environment. Moreover, the referred literature focuses mainly on the absorption of sound generated from the turbulence to evaluate the noise emitted within internal engines.

Term responsible for absorption

Following the approach of [START_REF] Reynolds | The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments[END_REF], a specific procedure is applied to the conservation equations of Navier-Stokes. First, each quantity is decomposed into a mean, an acoustic and a turbulent contribution, as suggested in [START_REF] Chu | Non-linear interactions in a viscous heat-conducting compressible gas[END_REF]. The signal F(x, t) is expressed as

F(x, t) = F(x) + F ′ (x, t) + F(x, t),
Eq.50 F(x) being the mean component, F ′ (x, t) the turbulent component and F(x, t) the acoustic component. The time average operator . defines the mean component and the phase average ⟨.⟩ the acoustic one,

F ′ = F = 0 and ⟨F⟩ = F + F.
Eq.51

The velocity field is decomposed into the three contributions; u = ū + u ′ + ũ. The mean and turbulent velocity fields are supposed divergence free, ∂ ūi /∂x i = ∂u ′ i /∂x i = 0 (with a summation index notation), and the acoustic fields irrotational, ∇×ũ = 0. The pressure and density fluctuations are supposed to be mainly from acoustic origin, thus p = p+ p, and ρ = ρ + ρ, using the approximation ρ(1 + ρ/ρ) ≈ ρ. The momentum conservation Eq.12 writes, keeping the viscous stress term,

∂u i ∂t + u j ∂u i ∂x j + 1 ρ ∂p ∂x i = 1 ρ ∂τ ij ∂x j , Eq.52
and applying time and phase average operators, the equation for the temporal evolution of u ′ i and ũi are isolated, giving

∂u ′ i ∂t = -ūj ∂u ′ i ∂x j -u ′ j ∂ ūi ∂x j -u ′ j ∂ ũi ∂x j -ũj ∂u ′ i ∂x j + ∂ ∂x j ⟨u ′ i u ′ j ⟩ -u ′ i u ′ j + 1 ρ ∂τ ′ ij ∂x j
Eq.53 and

∂ ũi ∂t = -ūj ∂ ũi ∂x j -ũj ∂ ūi ∂x j - ∂ ∂x j ⟨u ′ i u ′ j ⟩ -u ′ i u ′ j + ∂ ∂x j ũi ũj -ũi ũj -ũi ∂ ũj ∂x j + ũi ∂ ũj ∂x j - 1 ρ ∂ p ∂x i + 1 ρ ∂ τij ∂x j ,
Eq.54

Then, by multiplying Eq.53 by u ′ i , and Eq.54 by ũi , phase averaging and time averaging, it comes the conservation equations for the kinetic energy u ′ i u ′ i and for the velocity part of the acoustic energy ũi ũi ;

1 2 ∂u ′ i u ′ i ∂t = - 1 2 ūj ∂u ′ i u ′ i ∂x j -u ′ i u ′ j ∂ ūi ∂x j - 1 2
∂u ′ i u ′ i u ′ j ∂x j -⟨u ′ i u ′ j ⟩ ∂ ũi ∂x j term of interest - 1 2 ũj ∂⟨u ′ i u ′ i ⟩ ∂x j + u ′ i ρ ∂τ ′ ij ∂x j
Eq.55 and

1 2 ∂ ũi ũi ∂t = - 1 2 ūj ∂ ũi ũi ∂x j -ũi ũj ∂ ūi ∂x j + 1 2 ũi ũi ∂ ũj ∂x j - ∂ ũi ⟨u ′ i u ′ j ⟩ ∂x j + ⟨u ′ i u ′ j ⟩ ∂ ũi ∂x j term interest - 1 2 ∂ ũi j - ũi ∂ i + ρ τij j
Eq.56 ⟨u i u ′ ũi j is identified as "term of in both Eq.55 and Eq.56. It is the only similar term in both equations, with asign for the turbulent energy evolution and a + sign for the acoustic energy, meaning it represents a transfer of energy from the acoustic to the turbulence (or the other way depending on the sign), due to the acoustic/turbulence interaction. Consequently, studies have proposed formulations for this term, either for a general application [START_REF] Noir | Absorption of sound by homogeneous turbulence[END_REF] or especially to predict the enhancement of the acoustic absorption near the walls [28] [34] retrieving the results of Fig. 7.

Model from literature

Some analyses have derived an analytic estimate of the absorption of the sound generated aerodynamically by a jet. At an arbitrary point in the jet, the sound is emitted due to vortices, and propagates through the jet's turbulence, which absorbs part of it. The objective was to provide the Lighthill theory, which predicts the generation of noise by turbulence, with the contribution of absorption. Considering isotropic homogeneous turbulence, [START_REF] Crow | Visco-Elastic Character of Fine-Grained Isotropic Turbulence[END_REF] used the viscoelastic property of the turbulence. Under the effect of an acoustic disturbance, the turbulence is stretched and then relaxes to return to isotropy. Energy is given to the turbulence during strain and dissipated by viscous effect. Thus, the viscoelastic property implies that the turbulence, and thus the Reynolds stress, has a memory of its past state through a relaxation time. Based on the local increase in sound velocity due to deformation of the turbulence, [START_REF] Crow | Visco-Elastic Character of Fine-Grained Isotropic Turbulence[END_REF] predicted a formulation for the damping coefficient;

σ turb ≃ ω 5 M 2 rms f t 1 + f 2 t ,
Eq.57

with ω the acoustic pulsation, f t a reduced frequency such that f t = Ω t /ω, with Ω t related to the return to isotropy time, and M rms = u 2 i /c the root mean square (RMS) Mach number. Considering also a viscoelastic turbulence and manipulating Eq.55 to evaluate the term ⟨u ′ i u ′ j ⟩∂ ũi /∂x j , [START_REF] Noir | Absorption of sound by homogeneous turbulence[END_REF] obtained the following estimation; σ turb ≃ 2.3 ϵ c2 , Eq.58 equivalent to Eq.57 at high frequencies (ω ≫ Ω t ), using the hypothesis that Ω t ∼ ϵ/u 2 i , and ϵ being the turbulent dissipation rate. With a different approach, [START_REF] Howe | On the Absorption of Sound by Turbulence and Other Hydrodynamic Flows[END_REF] used the Lighthill equation, splitting the velocity field into an initial turbulent component and a total contribution caused by the wave. The estimation made leads to, at high frequency and as suggested by previous studies,

σ turb ∼ ϵ c2 .
Eq.59

In this case, the absorption will not depend on the acoustic frequency but the characteristics of the turbulence. An estimation at low frequency is also made, but it is necessary to determine what low and high frequency mean regarding the combustion instability application. Also, the objective is to estimate this absorption coefficient for an imposed acoustic solicitation, both in frequency and amplitude. Therefore, an approach is needed to vary the acoustics as a function of the resonant modes (in connection with the Galerkin method), change the turbulence as a function of the jet's position, and test the hypothesis of isotropic homogeneous turbulence.

Ongoing study

The turbulence component of the velocity is generated using a spectral method of [START_REF] Kraichnan | Diffusion by a Random Velocity Field[END_REF] for homogeneous isotropic turbulence, and the non-isotropic character within the jet can be achieved using the method of [START_REF] Smirnov | Random Flow Generation Technique for Large Eddy Simulations and Particle-Dynamics Modeling[END_REF]. The generation procedure demands knowing the specificity of the turbulence, including turbulent kinetic energy k, dissipation rate ϵ, and the (u ′2 compared to v ′2 and w ′2 ) RMS values for the anisotropy. The present work focuses on the quality of the turbulent velocity field generation and the phase averaging of the velocity signal analytically.

Remarks

The same process of averaging can be applied to obtain the mean flow evolution, as in §6.1. Without further details, the term ũi ũj ∂ ūi ∂xj appears as a source term in the mean flow equation, and is also in Eq.56, with the opposite sign, translating an exchange of energy between the non-uniform mean flow and the acoustics. An equivalent term is found to link the mean flow with the turbulent level; u ′ i u ′ j ∂ ūi ∂xj corresponding to the production of vorticity.

SUMMARY AND CONCLUSION

The three phenomena mainly responsible for acoustic damping in high-frequency combustion instabilities are the nozzle losses, the thermo-viscous losses at walls, and the turbulence/acoustic interaction at the jets. A fourth phenomenon not addressed here is the absorption of acoustics by the injection plate and will be tackled in the future of this work. For a compact nozzle applicable to transverse waves, an impedance model has been pro-posed and allows to find analytic and numerical predictions. The visco-thermal losses are also modeled and compared to the literature prediction but must be reinforced by accounting for the near-wall turbulence. Finally, a strategy is under development for the analysis of jet absorption. Moreover, it will be necessary to also evaluate the losses due to scattering and of the dominant acoustic mode towards Applications parisons be made LES progress The presented order be by using conservation equations on velocity's contributions (mean, turbulence, acoustic, and entropy). It allows having an estimate of the amount of vorticity and entropy produced at the level of the jets and flames, absorbing part of the acoustics, being convected by the mean flow through the combustion chamber, and interacting back with the acoustics when reaching the nozzle.

  + ∫ S p'u'.n dS = Ra + D wall + D turb

Figure 1 :

 1 Figure 1: Non-exhaustive schematic representation of some of the mechanisms involved in high frequency combustion instabilities within rocket engines. Mainly, the perturbations source is the flame coupling producing energy via the R a term. The acoustics is damped at the nozzle location ∇.(p ′ u ′ ), near the walls D wall and by interaction with the jets' turbulence D turb .

ZFigure 2 :

 2 Figure2: Representative impedance of a supersonic (left) and subsonic (right) nozzle, with respect to the inlet Mach number, from[START_REF] Marble | Acoustic disturbance from gas non-uniformities convected through a nozzle[END_REF]

Figure 3 :

 3 Figure 3: Real (left) and imaginary (right) part of the reflection coefficient, compute with the NOZZLE tool[START_REF] Nicoud | About the zero Mach number assumption in the calculation of thermoacoustic instabilities[END_REF] of AVSP, for an arbitrary nozzle geometry with a chocked nozzle. f is the frequency, L the nozzle length and c the sound speed.

Figure 4 :

 4 Figure 4: Comparison of the imaginary frequency on a 1D domain, for a real impedance imposed on one side, from analytic, numerical and model.

Figure 5 :

 5 Figure 5: Comparison of the imaginary frequency predicted from numerical analysis (AVSP) and from the StaHF model, for both 1L and 1T modes in a cylindrical geometry at dimension close to the BKD test bench, where a real impedance is imposed at the exit.

Figure 6 :

 6 Figure 6: Representation of the acoustic viscous and thermal boundary layers, with the near-wall velocity profile, along the viscous sublayer.

Figure 8 :

 8 Figure 8: Transverse acoustic wave through a turbulent jet; some of the acoustics is absorbed by the turbulence or scattered to other directions of propagation and frequencies.

ACKNOWLEDGMENT

This work is part of an ongoing PhD thesis at EM2C laboratory, cofounded by CNES, the French National Space Agency and ArianeGroup. A part of this work was performed using HPC resources from the mesocentre computing center of Ecole Cen-traleSup élec and Ecole Normale Sup érieure Paris-Saclay supported by CNRS and R égion Ile-de-France.