e ect of the fluid structure interaction on the aeroacoustic instabilities...

Effect of the Fluid Structure Interaction on the

Aeroacoustic Instabilities of Solid Rocket Motors

RICHARD J.∗CERFACS, 42 avenue Gaspard Coriolis, 31057 Toulouse, FRANCE

NICOUD F.†Universite Montpellier 2, 2 place Eugene Bataillon, 34095 Montpellier, FRANCE

Nowadays the industrial optimization needs the treatment of coupled phenomena suchas aero-thermal or aero-elasticity. The following paper presents the construction and vali-dation of a coupled software assembly, through the realization of simple test cases with aknown analytical solution, with the aim to subsequently use it as a solver for an industrialconfiguration coming from the solid propulsion.

I. Introduction

Large solid propellant rocket motors suffer from aero-acoustic instabilities arising from a coupling betweenthe burnt gas flow and the acoustic eigenmodes of the combustion chamber.3 During the firing, theseinstabilities generate pressure oscillations leading to thrust oscillations that could endanger the launcherintegrity. Given the size and cost of any single firing test or launch, being able to predict and avoid theseinstabilities at the design level is of the first industrial importance. The full scale calculations of the Ariane5 Solid Rocket Motors (P230) have allowed to highlight the impact of fluid-structure interaction on someelements of the structure such as the frontal thermal inhibitors. Inhibitors are designed with elastomerswhich burn at a lower speed than propellant, creating diaphragms with small rigidity across the flow. Thelarge deformation and the flapping of these thermal inhibitors have already been calculated in a fully coupledway in the region of the second segment28 . It was shown that they have a strong influence on the amplitudeof the thrust oscillations4,24 . However, interaction between the structural response of some parts of thelauncher, like the head end or the propergol, with the instabilities have not been investigated. The objectiveof this paper is to evaluate and validate the articulation of the numerical tools needed for these investigations,through the realization of simple test cases for which analytical solution can be obtained. This will lead toa first application on a subscaled version of the Ariane 5 SRM.

II. Presentation of the numerical software

Classically, the fluid-structure interaction problem consists in solving simultaneously both the fluid (1)and the structural (2) equations where some variables of one work as a source term for the other.

d

dt(AW ) + F c(W,~x, ~x) = R(W,~x) (1)

Md2

dt(~U) +D

d

dt(~U) +K~U = fext(W (~x, t), ~x) (2)

In this semidiscrete formulation already presented by Lesoinne,16 dots stand for time-derivatives, ~x is thedisplacement or position vector of the moving fluid grid points, W is the fluid state vector, A results fromthe discretization of the fluid equations, F c = F − xW is the vector of Arbitrary Lagrangian Eulerian (ALE)

∗Ph.D. Student, CERFACS, FRANCE, AIAA Member†Prof, UMR CNRS 5149, University Montpellier 2, FRANCE

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American Institute of Aeronautics and Astronautics

convective fluxes, F is the vector of convective fluxes, R the vector of diffusive fluxes, ~U the structuraldisplacement vector, and finally M, D and K are respectively the mass, damping and stiffness matrices ofthe structural system. In order to solve the problem of fluid-structure interactions, we have chosen to usea Conventional Serial Staggered (CSS) method with subcycling, which has already been well studied andadvocated7,21,22 . Softwares we used are shortly presented in this section, then the CSS method is reminded.

A. AVBP

Two different modeling tools are used for this work. First of all, aero-acoustics calculations are performedwith the Large Eddy Simulation (LES) solver AVBP26 developed at CERFACS and IFP Energies Nou-velles, which has already been validated for aero-acoustic applications25 . AVBP is designed to solve theNavier-Stokes equations for three-dimensional compressible flows over unstructured meshes. The numericaldiscretization is based for the presented simulations, on a Lax-Wendroff scheme which is 2nd order in bothspace and time. The sub-grid scale terms are modeled with the Smagorinsky or the WALE model.20

B. MARC

The structural deformations are computed with the structural analysis software MARC, using a finite ele-ment method (FEM) and developed by MSC-SOFTWARE. It is well suited for the treatment of non-linearmaterials, commonly encountered in solid propulsion. It allows static, dynamic and modal computations.29

The numerical scheme used in our computations is the trapezoidal Newmark method12 which has alreadybeen advocated for coupling applications11.22 It consists in writing the structure’s equilibrium at the timetn+1, knowing its state at tn, and the external force f at tn+1 :(

4∆t2s

M +2

∆tsD +K

)un+1 = fn+1 +

(4

∆t2sM +

2∆ts

D

)un +

(4

∆tsM +D

)un +Mun (3)

C. PALM

The PALM coupler developed at CERFACS is used to synchronize these two solvers.2 It also providesdata transmission between the different tools at the fluid-structure interface without large intrusion in thedifferent solvers. As a first step, the interpolation between the non conformal meshes is achieved with a firstorder method.

D. Coupling architecture

In most cases the time step ∆tf for the fluid is smaller than the structural one ∆ts. Then we consider thatthe coupling time step ∆tc is equal to ∆ts. The following steps are done for each coupling step of the CSSmethod in order to go from tn to tn+1 = tn + ∆ts :

• 1: Predict the structural un+1p displacement at time tn+1. A common predictor is :

un+1p = un + α0∆tsun + α1∆ts(un − un−1) (4)

It is worth noticing that the choice of α0 = 1 and α1 = 0 give a rise to a first order predictor, whileα0 = 1 and α1 = 0.5 defines a second order one.

• 2: Advance the fluid system to tn + ∆ts while updating the position of the fluid grid in order tomatch the position un+1p at the end of the coupling steps. Because ∆tf < ∆ts this step is achieve bysubcycling the fluid solver.

• 3: Transfer the fluid pressure Pn+1S to the structure. Note that Pn+1

S does not have to be the fluidpressure at the interface at time tn+1. Many choices for Pn+1

S can be found in the literature and wekept three of them which are explained in details later.

• 4: Finally integrate the structure to tn+1.

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In order to ensure the Geometric Conservation Law (GCL)17 the velocity of the fluid/structure interface iskept constant during the subcycling and is defined as un+1p−unp

∆ts. If no thermal inhibitors are considered,

the assumption of small deformations is well justified in the context of solid propulsion described in theintroduction. This approximation allows to model the wall motion in the flowsolver by a transpiration/suctionboundary condition, which reproduces correctly the coupled solid surfaces motion speed8,9, 21 . This approachwas preferred as a first step because it avoids considering mesh deformation, thus the overall chain complexity.

III. 0D test configuration

A. Device description

The coupled system is composed of a fluid portion in an adiabatic room, closed at its left-hand side by afixed wall and at its right-hand side by a mobile piston. It has a section S0 and a length L0 when the pistonis at x = 0. The piston has a mass m and is attached to an external fixed point with a spring of stiffness k.Displacement, velocity and acceleration of the piston at time t are respectively x(t), x(t), and x(t).

P0 K

X X=0

L0

Figure 1. 0D coupling system

The following assumptions are first made for thesystem :

• The gas is perfect with an adiabatic coeffi-cient γ

• The gravity effect is neglected

• The spring mass is neglected

• The spring is considered with no dissipation

B. 0D Modelling

As presented by Lefrancois and Boufflet,15 applying the Newton’s second law to the piston and consideringthat its position at rest corresponds to x(t) = ˙x(t) = ¨x(t) = 0 and P = P0 leads to:

mx(t) = −k x(t) + S0 (P (x(t), t)− P0) (5)

All fluid variables are supposed to be uniform in the chamber (0D assumption), which means that P (x(t), t) =P (t). Due to adiabatic walls, there is no thermal flux between the gas and the chamber. Then assuming thecompression is done in reversible way allows to apply Laplace’s law, it reads:

P (x(t), t) = P (t) =P0 L

γ0

(L0 + x(t))γ(6)

By injecting (6) in (5) and linearizing for small displacements around x = 0, we get :

mx(t) = −k x(t) + S0−γ P0

L0x(t) (7)

Solving (7) gives us both the temporal evolution of the piston position and the chamber’s pressure, as :

x(t) = x0 cos

(√k + γP0S0/L0

mt

)(8)

P (t) = P0 +γP0

L0x0 cos

(√k + γP0S0/L0

mt+ π

)= P0 + P1 cos

(√k + γP0S0/L0

mt+ π

)(9)

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C. Energy consideration

Looking at equations (8) and (9)we see that pressure and displacement vibrate at the same circular frequency

ω =√

k+γP0S0/L0m , but with a phase difference φ = π.

Following Piperno and Fahrat22 it is possible to calculate the numerical amount of energy imbalance createdat the fluid structure interface ∆EN after N oscillations for a general case including structural dissipationand for different choices of Pn+1

S . According to their results ∆EN ∼ Nπ(δEF + δES), where δEF can beinterpreted as an energy transferred from the fluid to the structure as viewed from the fluid, and δES is thesame energy but as viewed from the solid. In our particular case with subcycling we have :

δEF = ke

[(α0 − 1)h+

(14− 7α0

12+

3α1

2

)h3

]+O(h4) (10)

Where ke = x0 P1 cos(φ), and h = ω∆tc. δES is depending on the choice of the pressure transmitted to thestructure Pn+1

S .For the first choice, Pn+1

S = Pn+1:δES = O(h4) (11)

For the second choice, Pn+1S = 1

∆tc

∫ tn+1

tnP (t)dt:

δES = ke

[−h

2+h3

8

]+O(h4) (12)

For the third choice, Pn+1S = 2

∆tc

∫ tn+1

tnP (t)dt− PnS :

δES = O(h4) (13)

Using a first order predictor (which means α0 = 1 and α1 = 0) and noticing that in this case ke < 0, thefirst and third choice of Pn+1

S produce energy at third order, which is not an issue with regard to the secondorder accuracy of the fluid and structural solvers. On the other hand, the second choice of Pn+1

S will createenergy at the first order. In this case, the coupling of the fluid and structure solvers induces a large error,which must be seen in our simulations as a linear growth of the oscillations’s amplitudes

Figure 2. Fourier’s Transform of the displacement Figure 3. Displacement’s amplitude

D. Numerical Results

The structure is modeled in two dimensions for this test case, the spring being represented in MARC by abloc of material of Young’s Modulus EM . It has a section SM0 = S0 and an unstretched length of LM0.Assuming that the Poisson coefficient is set to zero, and that the displacements are small, this is equivalentto a spring of stiffness k = EMS0

LM0. Considering the 0D assumption, AVBP is replaced in the coupled software

assembly by a simple 0D program which uses the information from MARC to compute Pn+1S according to

the Laplace’s law. Figure(2) shows that the three choices of Pn+1S give a good agreement with theoretical

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results, the piston indeed is vibrating at the predicted frequency. Figure(3), underlines the fact that the firstand third choices of Pn+1

S seem to well conserve the energy of the system, whereas the second choice leadsto a linear growth of the amplitude, thus creating energy. These results are in complete agreement with thetheoretical considerations of section C.

IV. Acoustic test configuration

This one dimensional test case is designed in order to account for the acoustics in a fluid-solid coupledsystem.

A. Device description

The set-up is composed of an adiabatic chamber filled with gas, closed at the left-hand side by a fixed walland at the right-hand side by a deformable block of rubber which is fixed on its right side, see figure (4).The fluid chamber has a section S0 and length l0 when the block of rubber is at rest at x = l0. The rubber ofdensity ρS , has a Young modulus E and a Poisson ratio ν, and is fixed at x = L0. Displacement (or position,depending the contest), velocity, and acceleration of a point ~U of the solid at time t are respectively ~U(t),~U(t), and ~U(t).Starting from rest, the fluid part will be disturbed with a pressure Dirac impulsion. Aftersome iterations, coupled eigenfrequencies appear while looking at the pressure signal in the chamber.

Solid Fluid

Interface reflection/transmission

x

A+S

A-S

A+F

A-F

reflection reflection

0 l0 L0

Figure 4. Acoustic coupling system

The following assumptions are first made for the system :

• The gas is perfect with an adiabatic coefficient γ

• There is no external volume forces nor on the solid, neither on the fluid

• Both fluid and solid are considered with no dissipation

• The deformations of the solid are small, which allows to approximate the interface location by l0

B. Acoustic Modeling

1. Speed of sound in a solid

We shortly recall how to find the speed of sound Cs of a compression wave in an isotropic solid. The generalmethod for every wave, in a general solid can be found in literature.1 The Newton’s second law applied toa point ~U of the solid leads to :

div(¯σ(~U(t), t)) + ~fextvol (~U(t), t) = ρS(~U(t), t) ~U(t) (14)

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where ¯σ is the stress tensor and ~fextvol is the sum of the external body forces (which is null according to theassumptions done) . Introducing the fourth order rigidity tensor ¯C, which links the displacement gradient tothe stress tensor, leads to the system (15) which can be recast as (16), where the subscripts [i, j, k, l] ∈ (1, 2, 3)are relative to components in cartesian coordinates:

∂σij∂xj

= ρS∂2Ui∂t2

σij = Cijkl∂Ul∂xk

(15) Cijkl∂2Ul∂xk∂xj

= ρS∂2Ui∂t2

(16)

Let us consider the expression of a plane wave propagating in the ~n direction (17) as presented figure(5):

Ui = U0i f

t−∑j

nj ∗Mj

CS

(17)

n U

Φ2

Φ1 Φ0

€

M ∈Φ⇔OM ⋅ n = cte

Figure 5. Plane wave Φ propagating in the ~n direction

Associating (17) and (16) leads to the Christoffel’s equation (18) .

ρSC2SU

0i = Cijkl + njnkU

0l = ΓilU0

l (18)

We can notice that U0i = U0

l ∗ δil, which allows to rewrite(18) as :

(Γil − ρSC2s δil)U

0l = 0 (19)

This result means that ~U0 is an eigen-vector of Γil and ρS ∗C2s is the corresponding eigen-value. In our case

of compression plane wave in an isotropic material we have the following expression:

¯Γ =

(1−ν)E

(1+ν)(1−2ν) 0 0

0 E2(1+ν) 0

0 0 E2(1+ν)

CS =

√(1− ν)E

ρS(1 + ν)(1− 2ν)(20)

2. Analytical solution

Considering an eigen circular frequency ωc of the coupled system, we have the following expression for thespeed of propagation in the Fluid (CF ) and solid (CS) respectively:

CF =√γRT and kF = ωc

CGCS =

√(1−ν)E

ρS(1+ν)(1−2ν) and kS = ωc

CS

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P being the oscillating pressure and U the oscillating velocity, the wave propagation for both fluid and solid,can be written as :

PF,S = A+F,Se

jkF,Sx +A−F,Se−jkF,Sx (21)

UF,S =1

ρF,SCF,S(A+

F,SejkF,Sx −A−F,Se

−jkF,Sx) (22)

where KF,S = ω/C is the wave number.Using (21) and (22) along with the appropriate boundary and jump conditions

• UF (0) = US(0) = 0 (wall no-slip condition for the fluid and embedding for the solid)

• UF (l0) = US(l0) (velocity continuity at the fluid structure interface located at l0)

• PF (l0) = PS(l0) (Stress continuity at the fluid structure interface located at l0)

one can find that ωc has to be the solution of the following dispersion relation (23) which can be solvednumerically. This relation allows to find the coupled eigenmodes of the system which, depending from thesub-systems properties, may differ from the eigen-frequency of the fluid and solid taken alone.

tan( ωc

CS(l0 − L0))

tan( ωc

CF(l0))

=ρSCSρFCF

(23)

C. Numerical Results

Both fluid and structure are modeled in two dimensions for this test case. Since only longitudinal 1D solutionare sought for, symmetric conditions have been used for the fixed bottom and top walls. The solid is modelledin MARC by a bloc of material of Young’s Modulus E = 4.274 105 Pa and Poisson’s coefficient ν = 0. It hasa section SM0 = S0 and an unstretched length LM0 . The calculation with the first and second choice of Pn+1

S

have been achieved using coupling time step ten times bigger than the fluid one (ie ∆tc = ∆ts = 10∆tf ),and frequencies results are represented in figure(6). Unfortunately the third choice for Pn+1

S appeared to bequickly numerically unstable. We don’t have any valuable explanation for this behavior at the moment.

Figure 6. Fourier’s Transform of the pressure at the middle of the chamber for the first choice of P n+1S on the

left hand side and for for the second choice on the right hand side

It’s worth noticing on Fig.(6) that each choice of Pn+1S presents a good capacity to retrieve the coupling

eigen-frequency of the system. Nevertheless, looking at the spectrum of the second choice of Pn+1S (ie

Pn+1S = 1

∆tc

∫ tn+1

tnP (t)dt), one can notice that some modes are damped while others seem to be fed in

comparison with the spectrum obtained with Pn+1S = Pn+1. These results stem from the same energy

consideration as in the 0D part. Writing pressure and displacement at the fluid structure interface for one of

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the coupled eigen-modes leads to Eq.(24) which shows that the pressure and the structure are vibrating atthe same circular frequency ωc but with a phase depending on kG and l0. It can be checked that the phaseφ is π if 1

2 sin( 4πfcl0CG

) > 0 where fc is an eigenfrequency of the coupled system, and φ = 0 in other cases.Using the same scalar ke and h as in the 0D part, it comes out that the second choice of Pn+1

S will feed inenergy some modes and damp the others. This can be visualized in Fig.(7).

UG(l0, t) =−2A+

G

ρGCGωcsin(kGl0)e−jωct

PG(l0, t) = 2A+G cos(kGl0)e−jωct

(24)

Hz

Fed mod

es

Dam

ped mod

es

1st coupled acous2c mode

2nd coupled acous2c mode

3rd coupled acous2c mode

4th coupled acous2c mode

Figure 7. Fed and Damped modes

The physical and numerical behavior of the coupling chain with transpiration boundary condition hasbeen advocated thanks to this acoustic test case. We can now go for an industrial application , which is theSolid Rocket Motor of Ariane 5.

V. Application to the P230 configuration with fluid-structure interaction

A. Flow configuration

The Ariane 5’s Solid Rocket Motor (SRM), named P230, is one the most impressive ever constructed inEurope. It is about 37 meters high for a radius of 1.5 meters, with an initial mass of about 280 tons. Thetwo SRM cover 90% of the thrust at takeoff, burning out during 130 s and consuming approximately 2 tonsof propergol each second. They are finally ejected when the rocket has reached an altitude of approximately70 km and a speed of 2000 m/s.Their corresponding thrust is unfortunately not stationnary during the firing, and some oscillations mayoccur. This might jeopardize the integrity of the payload due to vibrations. The mechanism has been inves-tigated extensively over the last decades5,6, 13,14,18,27 and linear theory has been derived.10 This mechanismis a coupling between the acoustics modes and the hydrodynamic perturbation, and is represented on Fig.(8).An unstable shear layer in the mean flow produces vortices which are convected until they impact the headof the nozzle. The acoustic wave generated by this impact can move back upstream since the flow is stillsubsonic. It perturbs the unstable shear layer, intensifying the generation of vortices. Such an aero-acousticmechanism can lead to high amplitude fluctuations when the underlying frequency is close to the frequencyof an acoustic mode of the whole geometry.

Experiments show that mainly the first and second acoustic modes are excited during three outburst, asrepresented on Fig.(9), the first being the stronger one. Looking at Fig.(10) which displays the two modesshapes, one can deduce the impact of the acoustic on the P230. The pressure oscillations steming from thefirst acoustic mode will lead to a non zero sum of forces on the structure which may induce a rigid bodymotion of the rocket. By contrast the second acoustic mode shape gives a nearly zero sum of forces and hasthe potential to induce oscillations of the full length of the P230.

B. Computations global description

The objective is to quantify the impact of fluid-structure interactions on the amplitude of these pressureoscillations. Computations are performed at a combustion time of 86 seconds which corresponds to thesecond out of the three outbursts. The chamber geometry at this particular time is obtained thanks to asurface burnback computations and is furnished by Snecma Propulsion Solide. The average deformation of

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1 2 3

4 5

Vortex genera0on (1)

Vortex convec0on (2)

Vortex impact (3)

Acous0c exita0on (3)

Acous0c feedback (3)

Figure 8. Instabilities mechanism

Figure 9. Pressure spectrum in Ariane 5 SRM (Courtesy of CNES)

the thermal inhibitors is pre-calculated by performing a few successive fluid and solid computations untilconvergence is reached. This approach is justified when a stationary solution is sought for but cannot replacethe coupling of the two solvers when dealing with the dynamics of the rocket. The global rigidity of thestructural elements, excepting the thermal inhibitors, allows the assumption of small deformations. Thisproperty enables the use of the transpiration boundary condition as aforementioned. The computationaldomain including the fluid and the solid domain is displayed on Fig.(11). The structure is fixed on hisleft-hand side, in the same way than during flight or experiment. Different materials as propellant, steel andthermal inhibitors are considered in the computation each with its own mechanical properties.

Calculations are done on axisymetric configurations for the fluid and the structure. In order to reducethe CPU/memory effort in this feasibility study, it has been chosen to reduce the geometry of the P230 by afactor 1/15, thus considering a system which size is comparable to the subscaled setup used for past studiescarried out by ONERA.23 The material characteristics of the structure were adjusted to reproduce the samecrossover of eigenmodes between the fluid and solid domains as in scale 1.

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ΔFL ΔFR ΔF = ΔFL+ ΔFR

1st Acous.c Mode +ΔP

-‐ΔP

ΔF ≈ 0 ΔFL

ΔFR

2nd Acous.c Mode

+ΔP +ΔP

Figure 10. Impact of pressure oscillation on thrust oscillation

C. Calculation without Fluid Structure Interaction (FSI)

The objectives of the calculation are double. First of all it has to demonstrate the ability of the fluid solverto reproduce the classical vortices generation6 and the self excitation of the first and second acoustic modes.Secondly, it produces a reference case in order to evaluate the impact of fluid-structure interaction.Results from purely acoustic calculations are presented on Fig.(12) and Fig.(13). The AVSP Helmholtzsolver developped at CERFACS was used for this purpose.19 The configuration is treated as a closed cavity,the nozzle being cut off where the flow becomes supersonic. Velocity fluctuations being zeros at this pointallows to treat it as a wall condition with zero wall normal velocity. One can notice that the frequency isindeed the one from the full scale configuration multiplied by 15. A snapshot of the flow obtained by solvingthe full Navier-Stokes equations is displayed on Fig.(14). It shows the presence of vortices generated near thepropergol and at the extremity of the thermal inhibitor during the calculation. A Power Spectral Density(PSD) of the pressure signal at the head end of the configuration is displayed in Fig.(15). One can noticethat the acoustic modes are actually excited. Following the procedure which is used in the industry for thesignal analysis of pressure oscillations, the levels of acoustic modes are reproduces by an integration of thePSD on a certain frequency range which goes from 225 to 375 Hz for the first acoustic mode and from 525to 720 Hz for the second one. Results are presented in Fig.(19). The order of the corresponding pressureoscillations being 2000 Pa, one can assess the typical value of the acoustic velocity as Uac Pac/ρ0c0. Withρ0 = 4kgm−3 and c0 = 1500ms−1, this leads to Uac = 0.3ms−1.

D. Calculation with fluid structure interaction

The structural response of the P230 is now accounted for (see Fig.16). Looking at the shape of the acousticmodes, it appears that they may easily interact with the structure through these front and aft-end. Fromthe fluid to the solid, the procedure used consists in transferring the pressure Pn+1

S calculated in the rocketchamber to the whole structure so we get the full response of the latter during the calculation. From thesolid to the fluid, only the structural responses of the front and aft-end are transmitted. This is well justifiedsince the radial displacements are negligible in comparison with longitudinal displacements. From front andaft-end (Fig.(17) and Fig.(18)), it is worth noticing that velocities obtained are about 10 % of the acousticvelocity magnitude obtained in the calculation without FSI. This actually leads to the 40% amplification ofthe pressure oscillations magnitude which are presented on Fig.(19) in comparison with the uncoupled case.

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Figure 11. P230 configuration

Figure 12. First acoustic mode at 333 Hz

Figure 13. Second acoustic mode at 664 Hz

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Figure 14. Vorticity generated near the propergol and at the thermal inhibitor

Figure 15. PSD of the pressure signal taken at the front-end extremity of the configuration

Front-‐end

(small displacements)

Thermal inhibitors (large displacements)

A8-‐end

(small displacements)

Structural response (Solid Fluid)

Transmi;ed pressure (Fluid Solid)

Figure 16. Fluid Structure interaction on the P230 configuration

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Figure 17. Velocity of the front-end Figure 18. Velocity of the aft-end

0

500

1000

1500

2000

2500

3000

3500

4000

4500

First Acous1c Mode Second Acous1c Mode

Pa

With

out fl

uid-‐structure interac1on

With

fluid-‐structure interac1on

Figure 19. Integral of PSD of the pressure signal taken at the forward extremity of the configuration

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VI. Conclusion

A computational chain was developed that couples a fluid and a structure solver. Special care was takento design the interface condition and the retained methodology showed good conservativity property androbustness. The results obtained when considering two academical configurations were in full agreementwith the theoretical developments. At last, the computational chain was applied to a subscaled version ofthe P230 solid rocket motor. Results show that the amplitude of the self-sustained pressure is increasedsignificantly when accounting for the fluid-structure interaction.

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e ect of the fluid structure interaction on the aeroacoustic instabilities...

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