Tharikaa Ramesh Kumar and Dr. Joseph MajdalaniAuburn University, Department of Aerospace Engineering

Photo Acknowledgements:SLS Rocket: <http://www.nasa.gov/sites/default/files/thumbnails/image/sls_wireframe_1_dac3.jpg>Background: <http://i.imgur.com/8xXfKkj.jpg>

►COMBUSTION ACOUSTIC INTERACTIONThese interactions cause enhanced heattransfer rates and resonance resulting in structuralvibrations, increased noise levels to seriousmechanical failure.

0 100 200 300 400 500 600

0.1

0.2

0.3

0.4

0.5

0

Uns

tead

y A

xial

Wav

e (m

/s)

fr (Hz)

FFT Present Acoustic

168.4 337.8 507.4170.1 340.3 510.4

► Continuity, Momentum and Energy Equations

► Nondimensionalization

► Perturbation

► The Biglobal Stability Approach

► Majdalani Mean Flowfield ( Compressible, Closed-form, Cylindrical )

► Eigenvalue Problem

( ) 0tρ ρ∂+∇⋅ =

∂u ( ) 4

3D ( )D

Pt

ρ µ µ= −∇ − ∇× ∇× + ∇ ∇⋅u u u

2D DD Dp

T Pc Tt t

ρ κ= +Φ + ∇

2; ; ; ; ; ;

; ; ; ;

w

w w w w

xz r rz z

w w w w w w

cz r P Tz r t t a p Ta a a a c T

U u uU U uu U u V u W u v wU U U c c c

θ θθ

θ ρθ ρρ ρ

′ ′= = = = ∇ = ∇ = = =

= = = = = = = = =

1 1 1 1

1 1 1 1 1

= ; =

= ; = ; = ; =z w r w

w

u u U u M U u u v V v M V v

u w W w M W w p P p T T Tθ ρ ρ ρ

= + = + = + = +

= + = + + + +

( ) ( )1 , exp( )expi rg g z r t i m tω θ ω= −

( ) { }( )2 2 2 2 21 1 1w2 4 3sin 1 1 cos( ) 2x r M x rψ π π π = − + +

ij i ij iA f B fω=

, , , , ,

, , , , ,

, , , , ,

, , , , ,

, , , , ,

continuity-momentum

= -momentu

m-momen

tu

z r

c u c v c w c c T

z u z v z w z z T

r u r v r w r r Tij

u v w T

e u e v e w e e T

U U U T

A A A A AA A A A A zA A A A AA rA A A A AA A A A A

θ

ρ

ρ

ρ

θ θ θ θ ρ θ

ρ

ρ

θ

↓ ↓ ↓ ↓ ↓

→ → →

→

menergy→

ˆ = + + + u U u u u= + u U uBASE UNSTEADY

(TRADITIONAL)

One swoop strategy!(lesser work, time and cost)

Cylindrical Compressible Navier-Stokes Solver - a more reliable predicting scheme

Predicted Vorticoacoustic and hydrodynamic modes without the need todecompose the equations using the same eigensolver with the accurateprediction of frequency shifts due to mean flow effects.

Capturing the fully coupledresponse between thehydrodynamic and acousticwaves is more reliable than thetraditional way of prediction.

ONERA’s Cold flow facility

When wall injection is turned on,►When mean flow speed is

increased, a leftward shift infrequencies is observed.

►An increase in Reynolds number isaccompanied by a slightamplification in temporal growthrate. We also observed thatviscosity is a stabilizing factor as itleads to damping in thevorticoacoustic waves.

►Considering the aspect ratio, it isevident that the frequency of thevorticoacoustic modes lowerproportionately, but their stabilityremains unaffected.

►Conversely, hydrodynamic modesbecome less stable in moreelongated chamber.

►PREDICTING COMBUSTION INSTABILITIESDuring the developmental stages of Titan IV, Ariane4 and 5, Ares and Reusable SRM, strongvorticoacoustic waves have been reported. So, thestability schemes are targeted towards predictingthe various frequency modes in cold flows.